The Priority of Propositions. A Pragmatist Philosophy of Logic 3031252284, 9783031252280

Table of contents :
Preface
Reference
Acknowledgements
About this Book
Contents
About the Author
Abbreviations
Part I: The Pragmatist Basis
Chapter 1: Pragmatism and Metaphysics: The General Background
1.1 Metaphysics
1.2 The Conceptual Articulation of Reality
1.3 Assertion
1.4 Propositions and the Formality of Logic
1.5 Arguments, Inferences, and Argumentations
References
Chapter 2: Groundbreaking Principles
2.1 Five Principles
2.2 Two Models of Propositional Individuation
2.3 Propositional Identification
2.4 Logical Propositions
2.4.1 Three Alternative Approaches
2.5 Logic as a Science
References
Chapter 3: Semantic and Pragmatic Hints in Frege’s Logical Theory
3.1 Frege’s Projects
3.2 The Representation of Abstract Reality
3.3 The Analysis of Discourse
3.4 Two-factor Semantics and the Meaning of Identity
3.5 Special Notions
3.5.1 The Judgement Stroke
3.5.2 The Predicables ‘Is True’ and ‘Is a Fact’
3.5.3 Implicatures and Presuppositions
References
Part II: Logical Constants
Chapter 4: Implying, Precluding, and Quantifying Over: Frege’s Logical Expressivism
4.1 Logical Expressivism
4.2 The Conditional and Negation
4.3 Negation, Incompatibility, Falsehood
4.4 Expressions of Quantity and Relations Between Concepts
References
Chapter 5: Lessons from Inferentialism and Invariantism
5.1 What Is the Issue with Logical Constants?
5.2 Analytically Valid Arguments
5.3 Inferentialist Approaches
5.4 The Erlangen Programme
5.5 Invariant Terms of Logic
5.6 A Pragmatist Excursus
References
Chapter 6: The Inference-Marker View of Logical Notions: What a Pragmatist Proposal Looks Like
6.1 The Proposal
6.2 Some Consequences of (IMV)
6.2.1 Concepts and Propositions
6.2.2 Monadic and Binary Operators
6.3 Inferential Significance
6.4 Genuine Logical Notions
References
Part III: Further Applications of Propositional Priority
Chapter 7: Grue, Tonk, and Russell’s Paradox: What Follows from the Principle of Propositional Priority?
7.1 Paradoxes
7.2 Goodman’s ‘Grue’
7.3 Prior’s ‘Tonk’
7.4 Russell’s Paradox
7.5 Taking Stock
References
Chapter 8: Visual Arguments: What Is at Issue in the Multimodality Debate?
8.1 Multiple Modes
8.2 Non-linguistic Aspects of Linguistic Communication
8.3 Sentences, Pictures, and Relational Linguistic Pragmatism
8.4 Affordances
8.5 Ineffability and Conceptual Articulation
8.6 Visual Thinking in Mathematics
8.7 Some Conclusions
References
Chapter 9: Truth and Satisfaction: Frege Versus Tarski
9.1 The Scope of Tarski’s Proposal
9.2 Physicalism and the Unity of Science
9.3 Correspondence and Deflationism
9.4 Satisfaction
9.5 Frege on Truth and Judgeable Contents
References
Chapter 10: Truth Ascriptions as Prosentences: Further Lessons of the Principle of Propositional Priority
10.1 Why Truth Is So Elusive
10.2 The Pragmatist Strategy: Truth Ascriptions and the Fregean Principle of Context
10.3 Proforms
10.4 Pragmatism, Expressivism, and the Priority of the Proposition
10.5 The Prosentential Approach to Truth
10.6 Truth and Assertion
References
Index

Citation preview

Synthese Library 470 Studies in Epistemology, Logic, Methodology, and Philosophy of Science

María José Frápolli

The Priority of Propositions. A Pragmatist Philosophy of Logic

Synthese Library Studies in Epistemology, Logic, Methodology, and Philosophy of Science Volume 470

Editor-in-Chief Otávio Bueno, Department of Philosophy, University of Miami, Coral Gables, USA Editorial Board Members Berit Brogaard, University of Miami, Coral Gables, USA Steven French, University of Leeds, Leeds, UK Catarina Dutilh Novaes, VU Amsterdam, Amsterdam, The Netherlands Darrell P. Rowbottom, Department of Philosophy, Lingnan University, Tuen Mun, Hong Kong Emma Ruttkamp, Department of Philosophy, University of South Africa Pretoria, South Africa Kristie Miller, Department of Philosophy, Centre for Time, University of Sydney, Sydney, Australia

The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology, all broadly understood. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines. Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. In addition to monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.

María José Frápolli

The Priority of Propositions. A Pragmatist Philosophy of Logic

María José Frápolli Department of Philosophy I University of Granada Campus Cartuja, Granada, Spain

ISSN 0166-6991     ISSN 2542-8292 (electronic) Synthese Library ISBN 978-3-031-25228-0    ISBN 978-3-031-25229-7 (eBook) https://doi.org/10.1007/978-3-031-25229-7 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Fran, Victoria, and Joan

Preface

What I offer in the following pages is a narrative, one which vindicates the centrality of logic to philosophy and its study. A narrative aims to provide clues for looking at things with fresh eyes. I aspire to give new life to some older intuitions and apply them to contemporary discussions. I also seek to engage the reader in a conceptual adventure that should change how they see logic and its philosophy. Sometimes, even the smallest change of perspective has the effect of placing worn-out problems under a more favourable light and, if we are lucky, opening unexpected ways of dealing with recalcitrant difficulties. The narrative I propose integrates insights from sources that are sometimes very distant from each other, but I offer it as a coherent general argument for understanding logical issues; for there is indeed a reason for the coherent integration of distant intuitions. As a pragmatist, I have no choice but to think that we all share the most basic insights about the functioning of the concepts that we put to work in discursive actions. And this is so because we share the same kind of brain and the world around us, and we have undergone similar adaptive situations that have resulted in the networking of analogous conceptual systems. Philosophers and logicians often disagree about the theoretical system by which they implement those shared basic insights, but this is a different story that does not neutralise the communal background. The Priority of Propositions is not a historical essay; it does not seek to offer a non-committed description of authors and schools. This book is a philosophical essay, intended to be a contribution ‘to that peculiar genre of creative non-fiction’ to which, according to Brandom (1994, p. xi), philosophical works belong. In it, I pour the insights I have gained after decades devoted to studying, thinking, and debating the topics included in it. I am convinced—and am hoping to convince the reader— that what is taught in standard logic courses, and what we publish in journals that include ‘logic’ in their titles, is at most only tenuously connected with the project that Frege initiated in his Begriffsschrift. And nevertheless, it is Frege’s project and what follows from it that explains the relevance of logic to philosophy and science.

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My reading of Frege, in particular, and many of the analyses I offer of central topics in the philosophy of logic are largely discordant with the received view. It is this discordance that encouraged me to write this book, though. Had I had a standard approach to authors and themes, I would not have found the motivation for the long and solitary hours that the writing of a book demands. I have assumed the task of writing it down because the narrative I offer sheds some light on and corrects the distorted perspective derived from an excessive interest in formalism, or so I hope. I owe my position on the philosophy of logic to some of the greatest philosophical minds of the last hundred years or so: Gottlob Frege, Frank Ramsey, Ludwig Wittgenstein, C. J. F. Williams, and Robert Brandom, to name a few. These are my philosophical heroes, but there are many others whose influence will be apparent as the book progresses. Susan Haack deserves a special mention. At the beginning of my career, she planted in me the seed of pragmatism in the philosophy of logic, a pragmatism respectful of the classical tradition in logic. During the many occasions when we have discussed philosophy, she has been an honest mentor, always putting forward the most challenging and insightful questions and never allowing herself to soften a truth. She also conducted her mentoring with immense generosity and personal warmth. These are all philosophical debts that I am proud to acknowledge. As for my comprehensive view, I claim full responsibility. This book is dedicated to my students, to many generations of them, who have encouraged me to be coherent. For a philosopher, no scenario is more terrifying than a classroom full of smart, interested students; none is more challenging and rewarding, either. I have had the good fortune of finding myself in this situation every single academic year of my career. From my students, I have learned that the story that we tell when we teach logic does not make much sense, even if the technicalities of any logic course are profitable tools as long as their purpose and scope are clearly delimited. A successful logic course should provide students with the powerful method of conceptual analysis. It should explain why what Wittgenstein once called ‘surface grammar’ is not a reliable source of logical knowledge, and that the elements of arguments and inferences are propositional contents. It should promote logic among students by showing them that there is no method for doing philosophy other than sound argumentation, and that this involves, as an essential condition, substantial knowledge of how language works. To conclude, a word to my fellow women philosophers, present and future. Academic life is merciless to women. We are too often ‘invited’ to make unacceptable choices, urged to put up with intolerable treatment, forced to acquiesce to condescending attitudes, advised to suppress reasonable reactions, and, even when ‘well-behaved’, we are made invisible, our work is ignored, our voices silenced. For all this, my female colleagues deserve my acknowledgement and deepest respect. To my female students, I wish them strength and courage. A professional life devoted to the growth of knowledge and its transmission to future generations is

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enormously fulfilling. Only each woman can decide whether the heroism and pain involved are really worth it. Granada, Spain

María José Frápolli

Reference Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press.

Acknowledgements

I want to express my deepest gratitude to a large group of persons and institutions: To Otávio Bueno, who kindly invited me to submit the project of this book to Synthese Library and encouraged me to take up again a long-cherished project, often initiated and systematically postponed. To an anonymous referee, whose comments compelled me to be extremely accurate and clear in my assertions. To my students, former and present, some of whom are now my colleagues, and to some of my colleagues who never were my students. Juan Acero, Alba Moreno, Llanos Navarro, Eduardo Pérez, Esther Romero, Manuel de Pinedo, and Neftalí Villanueva. To the participants of the Colombian reading group organised by Ángela Bejarano and José Andrés Forero, who were generous enough to dedicate some sessions to the discussion of a draft of this book: Ángela Bejarano, Tomás Barrero, José Andrés Forero-Mora, Miguel Ángel Pérez, and Kurt Wischin. To the organisers and participants of the postgraduate reading group of the Interuniversity PhD Programme on Logic and Philosophy of Science, who eagerly allowed me to explain my take on the philosophy of logic, and who aptly spotted unclear claims and half-baked thoughts: Violeta Conde, José Alejandro Fernández Cuesta, Pedro Antonio García Jorge, José Javier González López, Francisco Javier Gutiérrez Cózar, Tomás Hernández Mora, Rodrigo López Orellana, Eva Martino, José Gerardo Moya, Elberto Plazas, Manuel de la Cruz Recio, Jean Paul Rossi, Roberto Sánchez, Manuel J. Sanchís, Julián Valdés, and Juan Antonio Sánchez Guzmán. The Revista de la Sociedad de Lógica, Metodología y Filosofía de la Ciencia en España devoted part of its issue n° 66 (January 2022) to this book (https://www. solofici.org/revistas/). The issue includes comments by several colleagues and my answers to them. These colleagues were Juan José Acero, María José Alcaraz, Tomás Barrero, Ángela Bejarano, Cristina Corredor, José Andrés Forero-Mora, Concha Martínez Vidal, Eduardo Pérez Navarro, and Kurt Wischin. To all of them, I am deeply grateful. My gratitude is further extended to the president of xi

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Acknowledgements

the Sociedad, Cristina Corredor, and to the editor of the journal, David Pérez Chico, for their generosity and support. Other colleagues have enriched my views and prevented me from leaving mistakes and inaccuracies in specific chapters and sections of the book. I acknowledge their advice and help in the corresponding places. The research that has led to this book has been possible thanks to the financial support of the European Commission’s programme Horizon 2020, through a Marie-­ Skłodowska-­ Curie action (EMEHOC 653056), the Spanish Ministerio de Ciencia, Tecnología, Conocimiento e Innovación PID2019-109764RB-100, and the Consejería de Conocimiento, Investigación y Universidad of the Junta de Andalucía (P18-FR-2907 and B-HUM-459-UGR18).

About this Book

Philosophy of logic, in the sense I understand it, is a branch of the philosophy of language. It is part of the more general task of understanding the role of concepts as they are used by speakers to perform and express certain discursive actions. Concepts such as argument, existence, identity, inference, generality, logical constant, logical form, logical truth, proposition, and truth cannot be approached without deep knowledge of how language works and the plurality of ways in which it works. In particular, they cannot be understood independently of the practices— actual and virtual—of rational, discursive creatures. This is the general background of this book. The whole book is a development of a principle that Frege incorporates in his works although he is not its first proponent, the Principle of Propositional Priority: that propositions take priority over concepts, both in a logical and chronological sense. I have two reasons to make extensive use of ideas that I find in Frege, some of them developed, some in germ, and some merely suggested. First, I want to acknowledge Frege’s essential influence in my position concerning the philosophy of logic and the philosophy of language; secondly, I want to reassure the reader that my views are not revisionist but are instead deeply rooted in the works that we all recognise as giving rise to contemporary logic. I’m aware that the Frege that people learn and teach today lacks the semantic and pragmatic depth I infer from him. While Frege sometimes expresses himself in ways that may appear to contradict my claims on those parts of his work that interest me most, my reading is nevertheless faithful to the letter and the spirit of his philosophical project, as the many passages of his works that I quote in this book make abundantly clear. This does not preclude the fact that, at some points, this book goes far beyond his assumptions and claims. Whenever this happens, I will make it explicit. The formalist attitude towards logic that Frege rejected, which has ironically turned out to be the received view in logic and in Fregean studies, ended up, on the one hand, undermining the very ground upon which his work rested, due partly to the success of Hilbert’s programme and partly to Russell’s interpretation of Frege’s xiii

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About this Book

writings. A misguided understanding of Frege’s rejection of psychologism—his other philosophical adversary together with formalism—prevented, on the other hand, philosophers of logic and mathematics from appreciating the many pragmatist hints in his approach to logic and philosophy of language, as if the objective foundations he intended for these disciplines demanded the removal of agents and their practices from the overall picture. Frege’s semantic finesse foresaw many of the subtleties that were developed at a later stage by various philosophical schools with a pragmatic orientation; it is not by chance that he holds the honorific title of father of the philosophy of language. For this reason, I have chosen to present my position on the philosophy of logic as inspired by Frege’s work, although I will complement it with the many insightful resources that have proved their explanatory utility in the long period between Frege’s work and the present time. Three major topics, essentially interconnected, constitute the core of the philosophy of logic as a discipline: logical consequence, logical form, and logical constants. Any one of these inevitably ends up featuring in the discussion of the other two. In this book, the meaning of the expressions that represent logical constants takes the central stage. Logical form will not be discussed separately, although the different reasonable senses that are given to this expression in different contexts will be highlighted when the occasion arises. Finally, logical consequence raises the issues of the bearers of logical properties and the kinds of items that are connected by logical relations and, with them, the issue of the arguments of the predicative notions that we linguistically represent as logical constants. In this indirect way, the topic of logical consequence will be present from the beginning to the end, in every chapter and every argument. Some allegedly inextricable difficulties in logical theory, among them the definition of logical constants and the role of logical and semantic paradoxes, allow alternative explanations in the pragmatist approach that this book incarnates. The scope and consequences of metaphysical and epistemological arguments, criticisms, and obstacles for the purpose of a reasonable understanding of the practice of drawing inferences, and the theory that explains it, will fall into place once the background of these discussions, their assumptions, and some of the insights that originated the modern approach to the study of logic and language are spelt out. The attitude to the philosophy of logic that I find in Frege’s writings, which combines rationalist aspects with traits of what has been called ‘analytic pragmatism’, and which this book aims to promote, will help logic to recover its place in the network of activities that agents perform with words. The difficult co-existence between logical theory and other philosophical and scientific studies has been an unfortunate side effect of its excessive mathematisation. Seeing judgeable contents as products of human actions and acknowledging their inferential relations to be logic’s sole concern will help to steer the course. The ten chapters of this book develop a speakers-friendly view of the philosophy of logic. Touching upon topics that are essentially connected, the chapters are intended to be as self-contained as possible. Repetitions are a collateral result of this self-containment, which sometimes makes it unavoidable to summarise in one

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chapter what has already been thematised by another. In any case, a certain amount of repetition is always beneficial, like the soft rain that gently waters the meadows. The book is divided into three parts. In the three chapters of Part I (“The Pragmatist Basis”), I lay down the general background (Chap. 1), identify the principles that frame the discussions in the book (Chap. 2) and pinpoint them in Frege’s work (Chap. 3). The three chapters of Part II (“Logical Constants”) are dedicated to the analysis of logical constants. In Chap. 4, I discuss Frege’s insights about logical concepts. Chapter 5 analyses the weaknesses of inferentialism and invariantism as theories of the meaning of logical terms. In Chap. 6, I put forward my own proposal for logical constants, which I call the ‘Inference-Marker View’. Finally, Part III (“Further Applications of Propositional Priority”) includes a sample of the effects of adopting the theoretical perspective of this book to deal with recalcitrant difficulties essentially related to logical theory: paradoxes (Chap. 7), the nature of arguments (Chap. 8), and the definition of truth (Chaps. 9 and 10).

Contents

Part I The Pragmatist Basis 1

 Pragmatism and Metaphysics: The General Background ������������������    3 1.1 Metaphysics��������������������������������������������������������������������������������������    3 1.2 The Conceptual Articulation of Reality��������������������������������������������    8 1.3 Assertion ������������������������������������������������������������������������������������������   11 1.4 Propositions and the Formality of Logic������������������������������������������   16 1.5 Arguments, Inferences, and Argumentations������������������������������������   23 References��������������������������������������������������������������������������������������������������   26

2

Groundbreaking Principles��������������������������������������������������������������������   29 2.1 Five Principles����������������������������������������������������������������������������������   29 2.2 Two Models of Propositional Individuation ������������������������������������   34 2.3 Propositional Identification��������������������������������������������������������������   40 2.4 Logical Propositions ������������������������������������������������������������������������   42 2.4.1 Three Alternative Approaches����������������������������������������������   44 2.5 Logic as a Science����������������������������������������������������������������������������   46 References��������������������������������������������������������������������������������������������������   49

3

 Semantic and Pragmatic Hints in Frege’s Logical Theory������������������   53 3.1 Frege’s Projects��������������������������������������������������������������������������������   53 3.2 The Representation of Abstract Reality��������������������������������������������   56 3.3 The Analysis of Discourse����������������������������������������������������������������   58 3.4 Two-factor Semantics and the Meaning of Identity��������������������������   62 3.5 Special Notions ��������������������������������������������������������������������������������   65 3.5.1 The Judgement Stroke����������������������������������������������������������   66 3.5.2 The Predicables ‘Is True’ and ‘Is a Fact’������������������������������   68 3.5.3 Implicatures and Presuppositions ����������������������������������������   72 References��������������������������������������������������������������������������������������������������   74

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Part II Logical Constants 4

 Implying, Precluding, and Quantifying Over: Frege’s Logical Expressivism ��������������������������������������������������������������������������������������������   79 4.1 Logical Expressivism������������������������������������������������������������������������   79 4.2 The Conditional and Negation����������������������������������������������������������   82 4.3 Negation, Incompatibility, Falsehood ����������������������������������������������   87 4.4 Expressions of Quantity and Relations Between Concepts��������������   94 References��������������������������������������������������������������������������������������������������  100

5

 Lessons from Inferentialism and Invariantism ������������������������������������  103 5.1 What Is the Issue with Logical Constants? ��������������������������������������  103 5.2 Analytically Valid Arguments ����������������������������������������������������������  108 5.3 Inferentialist Approaches������������������������������������������������������������������  110 5.4 The Erlangen Programme ����������������������������������������������������������������  114 5.5 Invariant Terms of Logic������������������������������������������������������������������  117 5.6 A Pragmatist Excursus����������������������������������������������������������������������  119 References��������������������������������������������������������������������������������������������������  122

6

 The Inference-Marker View of Logical Notions: What a Pragmatist Proposal Looks Like��������������������������������������������������������������������������������  125 6.1 The Proposal ������������������������������������������������������������������������������������  125 6.2 Some Consequences of (IMV)����������������������������������������������������������  130 6.2.1 Concepts and Propositions����������������������������������������������������  133 6.2.2 Monadic and Binary Operators��������������������������������������������  136 6.3 Inferential Significance ��������������������������������������������������������������������  139 6.4 Genuine Logical Notions������������������������������������������������������������������  141 References��������������������������������������������������������������������������������������������������  146

Part III Further Applications of Propositional Priority 7

Grue, Tonk, and Russell’s Paradox: What Follows from the Principle of Propositional Priority?����������������������������������������  151 7.1 Paradoxes������������������������������������������������������������������������������������������  151 7.2 Goodman’s ‘Grue’����������������������������������������������������������������������������  153 7.3 Prior’s ‘Tonk’������������������������������������������������������������������������������������  158 7.4 Russell’s Paradox������������������������������������������������������������������������������  162 7.5 Taking Stock ������������������������������������������������������������������������������������  172 References��������������������������������������������������������������������������������������������������  173

8

Visual Arguments: What Is at Issue in the Multimodality Debate? ����������������������������������������������������������������  175 8.1 Multiple Modes��������������������������������������������������������������������������������  175 8.2 Non-linguistic Aspects of Linguistic Communication����������������������  180 8.3 Sentences, Pictures, and Relational Linguistic Pragmatism ������������  185 8.4 Affordances��������������������������������������������������������������������������������������  188 8.5 Ineffability and Conceptual Articulation������������������������������������������  190

Contents

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8.6 Visual Thinking in Mathematics ������������������������������������������������������  193 8.7 Some Conclusions����������������������������������������������������������������������������  195 References��������������������������������������������������������������������������������������������������  196 9

 Truth and Satisfaction: Frege Versus Tarski ����������������������������������������  199 9.1 The Scope of Tarski’s Proposal��������������������������������������������������������  199 9.2 Physicalism and the Unity of Science����������������������������������������������  202 9.3 Correspondence and Deflationism����������������������������������������������������  205 9.4 Satisfaction����������������������������������������������������������������������������������������  207 9.5 Frege on Truth and Judgeable Contents��������������������������������������������  212 References��������������������������������������������������������������������������������������������������  218

10 Truth  Ascriptions as Prosentences: Further Lessons of the Principle of Propositional Priority����������������������������������������������  221 10.1 Why Truth Is So Elusive ����������������������������������������������������������������  222 10.2 The Pragmatist Strategy: Truth Ascriptions and the Fregean Principle of Context����������������������������������������������  224 10.3 Proforms������������������������������������������������������������������������������������������  228 10.4 Pragmatism, Expressivism, and the Priority of the Proposition����������������������������������������������������������������������������  234 10.5 The Prosentential Approach to Truth����������������������������������������������  236 10.6 Truth and Assertion������������������������������������������������������������������������  243 References��������������������������������������������������������������������������������������������������  247 Index������������������������������������������������������������������������������������������������������������������  251

About the Author

María  J.  Frápolli  is a Professor of Logic and Philosophy of Science at the Department of Philosophy I, University of Granada (Spain). From 2006 to 2012, she held the presidency of the Society of Logic, Methodology and Philosophy of Science in Spain. Currently, she chairs the Spanish Society for Women in Philosophy (Analytic branch). From 2015 to 2017, Prof. Frápolli held a Marie-SkłodowskaCurie grant at the Department of Philosophy, University College London and, from 2017 to 2020, was Honorary Professor in the same department. She has worked on the philosophy of language, logic, and mathematics, always from a pragmatist and naturalist standpoint. Some of her books are the following. As author: The Nature of Truth. An Updated Approach to the Meaning of Truth Ascriptions, Springer (2013). As editor: Expressivisms, Knowledge and Truth, Cambridge University Press, (2019); Saying, Meaning, and Referring: Essays on François Recanati’s Philosophy of Language, Houndmills, Basingstoke, Hampshire (UK), Palgrave Studies in Pragmatics, Language and Cognition (2007); and F.  P. Ramsey. Critical Reassessments. London (UK), Continuum Studies in British Philosophy, (2005).

xxi

Abbreviations

[DI] (Descriptive ineffability): Speakers are never fully satisfied when they paraphrase expressive content using descriptive, i.e. nonexpressive, terms. (FPE) (First Principle of Effability): Each proposition or thought can be expressed (=conveyed) by some utterance of some sentence in any language. (HP) (Hume’s Principle): The number that belongs to the concept F is the same as the number that belongs to the concept G if, and only if, a 1-1 correlation can be established between the items that fall under F and the items that fall under G. (HP*) (Hume’s Principle)*: The extension of the concept ‘equal to the concept F’ is identical to the extension of the concept equal to the concept G is true if, and only if, the same number belongs to the concept F as to the concept G. (IMV) (The Inference-Marker View): Logical constants are binary higher-level predicables that have 0-adic predicables as their arguments. The terms that represent them linguistically neither name nor describe. Their function is rather to show ­inferential connections between their arguments. (IMV)weak (The Inference-Marker View weak): Logical constants are binary higherlevel predicables whose arguments are n-adic propositional functions (n ≥ 0). They don’t name any kind of entity, nor do they describe any aspects of the world, but are natural language devices for making explicit inferential relations between concepts and propositional contents. (Inf. Val.)ling (Inferential Validitylinguistic): The validity of inferences and the meaning of some distinguished terms occurring in their expression are essentially related. (Inf. Val.)con (Inferential Validityconceptual): At least some of the concepts involved in inferences are essentially related to their validity. (MIR) (Meanings Involve Referents): Any characterisation of the meaning of a sentence S that contains a referential occurrence of a singular term, a, must make use of a or some co-­referring term. (NT) (The Norm of Truth for Assertion): Only assert what is true. (NB) (The Norm of Belief for Assertion): Only assert what you believe. (NK) (The Norm of Knowledge for Assertion): Only assert what you know. (OI) (The Organic Intuition): To be a proposition is to possess propositional properties and stand in propositional relations. xxiii

xxiv

Abbreviations

(PA) (The Principle of Assertion): Assertion is the minimal act required to produce outputs with logical properties. (PComp) (The Principle of Compositionality): The meaning of a complex expression is completely determined by its ingredients and how they are combined. (PCont) (The Principle of Context): Only in the context of a sentence does a word have meaning. (PGS) (The Principle of Grammar Superseding): Grammatical analysis is not a source of logical knowledge. (PH) (The Principle of Homogeneity): All steps in arguments, understood as products, belong to the same logico-semantic category. (PI) (The Principle of Individuation): Two propositions are one and the same if and only if they follow from the same set of propositions, and the same set of propositions follow from them. (PII) (The Principle of Inferential Individuation): Two sentences (uttered by an agent in context) express one and the same proposition if and only if their contents follow from the same set of propositions, and the same set of propositions follows from them. (PPP) (The Principle of Propositional Priority): The primary bearers of logical properties are propositions and sets of propositions. (RT) (The Recarving Thesis): f (α) = f (β) ↔ α~β (SD) (Syntactic Decisiveness): If an expression exhibits the characteristic syntactic features of a singular term, then that fact decisively determines that the expression in question has the semantic function of a singular term (reference). (SS) (Surface Syntax): The two sides of an abstraction principle have the syntactic and semantic forms that they appear to have.

Part I

The Pragmatist Basis

Chapter 1

Pragmatism and Metaphysics: The General Background

Abstract  In this introductory chapter, I set out the background against which the theses and arguments of the rest of this book must be understood. I explain my (minimal) version of pragmatism, and how I use the central concepts that this book is about. My inspiration has been some claims defended or suggested by Frege, and some further developments of Fregean inferentialism promoted by Robert Brandom. I discuss my take on assertion as an activity for whose outcome the agent is accountable. I also offer a complete characterization of the kind of item that propositions are, giving their individuation and identification criteria. In this task, I follow some Fregean insights related to Hume’s Principle and the Julius Caesar problem for the definition of number. Frege is also the source of the approach to the existence of abstract objects that I endorse. Existence is instantiation: a second-level concept. It is legitimate to discuss what is meant by existential claims in science. Nevertheless, the claim that numbers exist, or that propositions exist, is not something that is up for discussion; we simply cannot doubt it, since they are presupposed in our mathematical and assertive practices, respectively. Our very nature as producers and consumers of reasons rests on the assumption that judgeable contents, i.e., propositions, are exchanged in those discursive practices that define human rationality. Finally, in the last section, I explain what I will understand by the notions of argument, argumentation, and inference in the chapters ahead, and discuss material inferences that I contend, following Danielle Macbeth, were accepted by Frege as genuine logical inferences. Keywords  Assertion [assertible] · Claimable · Compositional [compositionalist · Compositionality] · Existence · Hume’s principle · Judgeable content · Julius Caesar problem · Pragmatism [pragmatist] · Proposition · Truth

1.1 Metaphysics All philosophical books deal with metaphysics. Philosophers are not physicists; we do not deal with ordinary objects and their first-order properties, or with the objects and properties discovered or postulated by the natural sciences. Abstract entities, © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_1

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1  Pragmatism and Metaphysics: The General Background

concepts and propositions, are the objects of our interest and, among these, we specialise in higher-level concepts, i.e., in concepts that we apply to other concepts or compounds of concepts, and in concepts belonging to what Wittgenstein understood as logical grammar, i.e., concepts that express categories. One might think that this general claim does not cover ontology, but it does. When philosophers ask about the structure of reality and its ingredients, they ask for those kinds of objects, for those categories, that should be listed in a complete description of reality. They are interested in knowing whether properties, or numbers, or events exist,1 i.e., whether those concepts are instantiated. Discussing existence is thus to discuss a property of concepts. Even philosophers who implement experimental methods in philosophy do so in order to tap into speakers’ intuitions2 regarding the application of certain concepts, seeking to determine under which circumstances they would use one concept rather than a close alternative, and to what extent certain applications of concepts are perceived as correct by competent users of language. Experimental philosophy aims, thus, at refining the ‘contours of our conceptual competence’ (Aguiar et al., 2014, p. 191), by appealing to speakers’ insights in those particular contexts that are relevant to various philosophical enterprises.3 C. J. F. Williams used to say that the philosophy of logic was the rightful heir to Aristotle’s metaphysics in the twentieth century. And I believe this to be true. We  Typical ontological and/or metaphysical questions are: ‘Aside from concrete objects, are there also abstract objects like numbers and properties? Does every event have a cause? What is the nature of possibility and necessity?’ (Manley, 2009, p. 1). They also enquire about the relations between parts and the wholes to which they belong and the relations between the parts that belong to a single whole, as well as about the nature of the ontological commitments of our discourse, etc. For a recent survey, see Chalmers et al. 2009. 2  I will refer abundantly to speakers’ intuitions in what follows. Williamson (2007) and Cappelen (2012) put on the table the debate about the role of intuitions in philosophy. I do not have any precise account of what intuitions are, and I cannot have one since this is a pre-theoretical notion. Competent speakers are usually sure about clear cases of the application of a concept in a particular circumstance and are able to verbalise what would follow from the application of some concepts and what would be precluded by their application. For instance, the application of the concept water to some stuff licenses the application of ‘liquid’ under certain circumstances concerning temperature, pressure, etc. This makes room for borderline cases in which it is difficult to decide one way or another. Even if viruses are a debatable case, we still would not hesitate to deem a dog alive and a rock inert. Contextual factors affect our willingness to apply some concepts to particular situations; see for instance the experiments of epistemic contextualism concerning when to say that an agent knows something (DeRose, 1999). And we have some strong feelings about whether or not to withdraw or not the application of proper names to individuals who, after all, do not bears the properties that we formerly attributed to them (Kripke, 1980). I refer to these phenomena when I appeal to speakers’ intuitions, i.e., to those inclinations towards linguistic practices that we learn when we learn to speak. Nevertheless, I do not believe that intuitions are the evidence for philosophical theories or that they are final. They are, like conjectures in mathematics, contents or claims that seem to us to be true. But some of them are false, and the truth of those that are tre has to be justified independently. 3  In Aguiar et al. (2014), the three authors advocate for shaping experimental ethics along the lines of experimental economics, as a way to avoid some criticisms directed against experimental philosophy by Antti Kauppinen (2007). What these authors say about experimental ethics, its aims and difficulties is easily exportable to the general enterprise of experimental philosophy. 1

1.1 Metaphysics

5

philosophers of logic work on generality, on inference, on existence, on identity, on truth and, as Aristotle did, we keep an eye on the meanings of terms, i.e., on concepts, and what we do with them. Blackburn saw himself as doing ‘conceptual engineering’ (Blackburn, 1999, p. 2): a suggestive description of our work as philosophers that has enjoyed continuity in Herman Cappelen’s project (Burgess et  al., 2020; Cappelen, 2018, 2020; Chalmers, 2020). Carnap seems to have been the §origin of this metaphor (Isaac, 2020, p. 3, n. 3). We owe gratitude to Richard Creath, in his edition of the Quine—Carnap correspondence (Creath, 1990), for drawing our attention to this enlightening analogy in Carnap’s work (Chalmers op. cit., p. 4). I like some of the images this metaphor evokes. It elicits the feeling that we philosophers are workers like many others, with the only difference being that the stuff that defines our work is imperceptible. Our work on concepts sometimes results in the modification of the original stuff and the construction of new ways of dealing with old and new realities. We philosophers are not special and, if this evocation is intended, I agree with it. I am also very sympathetic to the work of some women philosophers—Haslanger, Kukkla, Manne and Saul, among others—who analyse race and gender concepts and expose their mostly unfair contents, which rest on and prompt damaging inferences about the groups concerned (Haslanger, 2011; Kukkla, 2014; Manne, 2020; Saul, 2012, 2019). Besides, as Chalmers notes, the work of philosophers of language has always involved the analysis of concepts, which sometimes includes meliorative proposals. Indeed, the different notions of meaning provided by different philosophers and theories reveal this philosophical activity (Chalmers, 2020, p. 3). Nevertheless, the background that supports the discussions of this book markedly differs from some assumptions that seem to inspire Cappelen’s project and which he summarises in his so-called ‘Master Argument’. Part of my discomfort with this project comes from the kind of meta-semantics approach that it seems to adopt, in which the focus is on isolated terms within compositional structures (Cappelen, 2020, part II, Chalmers op. cit.). If conceptual engineering is actually about concepts, as opposed to terms, then compositionality does not play a role and the focus should be instead on the material inferences in which those concepts intervene. I also find quite disturbing the idea that some widely used non-empirical concepts that are central to our discursive life can be utterly defective, as some philosophers seem to believe about truth. The fact that Tarski and his followers— and, in general, those working in the formalist tradition in semantics—harbour this feeling about natural language terms is understandable. The idea that this feeling can be imported into the realm of conceptual engineering, as Cappelen, Plunkett and Chalmers do in connection with Sharp’s book on truth (Cappelen & Plunkett, 2020, p. 6, Chalmers op. cit., Sharp, 2013), is less defensible. Nevertheless, I agree that our ordinary concepts derive from practices, assumptions, and ideologies that evolve and have to be revised. Moreover, some philosophical and scientific notions are inadequate. Substituting inadequate concepts for adequate ones is (part of) what progress means in science and in philosophy. I see the roots of my work as essentially Fregean. As I will explain in the next chapter, I endorse the Principle of Propositional Priority. Propositions, and not

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1  Pragmatism and Metaphysics: The General Background

concepts, are the minimal units of discursive life and logical practices; they are what is put forward when we say something, in a technical sense of ‘saying’ that contrasts with the mere production of sounds—even if these sounds are organised into well-­ formed sentences of some recognisable language. Propositions are essentially assertables, they are the products of actual assertions and the possible contents of possible assertoric acts. This is Frege’s view, although the Fregean style exchanges ‘assertable’ for ‘judgeable’. Ramsey’s version would use beliefs instead of assertions and judgements. Propositional references are believables for Ramsey, i.e. the actual or virtual contents of our beliefs (see, for instance, Ramsey, 1927/1991). In plain nonphilosophical English, then, propositions are what we believe, assert, and judge; they are what speakers mean by declarative sentences in context and the contents of the that-clauses attached to some verbs. This point will be taken up in the next section. I am a pragmatist. I consider the actual practices of rational agents to be the starting point for philosophical analysis. Those practices are our destination too since, even if concepts and intuitions are refined through philosophical work, philosophy has to test its products against the intuitions of competent, rational agents. What philosophers of language have done with the ordinary notion of ‘saying’ and the locution ‘what is said’ provides some examples of this procedure. We all know what it is to say something. But philosophers of language have made the notion precise and distinguished it from cases of “saying” that are less central to their theoretical tasks. Something similar can be said of ‘meaning’. Ordinary speakers have a pre-­ theoretical notion, but the job of the philosophy of language has been to adapt, extend and cleanse it, identifying more specific kinds of meaning, so as to offer more sophisticated analyses of what we do with words. These analyses have to serve common purposes too, and competent speakers have something to say about their adequacy, which sometimes also implies changing some everyday perceptions. There must be what Rawls and, after him, philosophers of economics have called a ‘reflective equilibrium’ (Rawls, 1971). Think of Kripke’s 1970 Princeton Lectures (Kripke, 1980) and his introduction to the notion of rigid designation. In this task of dealing with concepts, philosophy is not different from other scientific enterprises, where theorists introduce new terms and modify some old ones with the ultimate purpose of understanding and explaining phenomena that, directly or indirectly, affect human lives. In fact, at the limits of the scientific enterprise, philosophers and scientists work hand in hand. The classic example is the notion of simultaneity that radically changed with Einstein’s work (see, for instance, Jammer, 2006). But there are hundreds of contemporary examples in which the work of philosophers and scientists is indistinguishable. Philosophy of biology offers some of them, with its analyses of concepts such as life, autonomy, function, and even death, which produce new material even for ethical and political debates. This is the aspect of our work with concepts that I share, at least in the letter, with Cappelen’s project. Being a pragmatist also means not bothering to engage in artificial discussions with no real interests, either for ordinary people or for the development of science. This is a consequence of Peirce’s pragmatist maxim (see e.g. Peirce, 1868, and 1878, p. 294, both in Peirce, 1932), which establishes that the meanings of terms and the information that concepts somehow codify are seen in the consequences of

1.1 Metaphysics

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their application. Haack explains that Peirce’s maxim has, besides a ‘constructive’ interpretation, a ‘critical’ one (Haack, 2019, p. 23). It is the critical interpretation that interests us here since it advises against engaging in disputes without any practical impact. Catholics and Protestants discussing transubstantiation was one of the debates that Peirce had in mind; philosophical disputes involving the notions of truth and reality were others. Pairing propositions with assertion, belief and judgement, as I did above, prompts one of those questions-without-any-practical-impact that are sometimes so dear to philosophers: the question of whether propositions pre-exist their assertion or are instead created in the act of assertion. The terms ‘pre-exist’ and ‘create’ already place this debate on the wrong path, since they produce the false impression that language and what we do with it are separate realities. This is a central assumption of the paradigm that has dominated philosophy for centuries, and is common to almost all philosophical schools, from realism and rationalism to positivism and empiricism. This comes with the complementary metaphysical assumption that reality and its linguistical conceptualization are independent of each other. For the kind of pragmatism that I defend, language and thought are two aspects of a single reality. This monistic claim admits of alternative interpretations, all of which I endorse. The first one is a sort of evolutionary hypothesis: that language and thought have an indistinguishable developmental history. Brains, their functions and communicative practices have evolved together in a continuous interplay. We could not think the way we think if we did not have the language we have. We could not have developed the articulated communicative systems that define us as humans—as ‘sapients’ in Brandom’s terminology—independently of the brain we have and the possibilities it opens up in terms of conceptual development. A second interpretation, which I will take up in the next section, is that what we think, believe, know, assert and claim all share their contents. If anything makes sense as the content of a state of thinking, etc., it necessarily makes sense as the content of any one of those mental states and intentional acts. Pragmatism is essentially anti-dualist. It rests on a—sometimes implicit— assumption that reality is one and that we are part of it. The methodological benefit of making distinctions cannot be denied. Nevertheless, we should not forget that these distinctions belong to the mind that judges, evaluates and explains, and not to reality (whatever this is). An example will clarify the point. Communicative practices are a continuous phenomenon that theorists divide into syntactic, semantic and pragmatic aspects. This tripartite distinction has proved its utility, although the divide between semantics and pragmatics is now more of a theoretical issue in itself than an aid for analysis. But it would be misleading to think that these three aspects, from which we approach the study of language, represent actual layers that constitute a phenomenon that is ‘out there’ in communication. Truth Conditional Pragmatics, for instance, explains that speakers do not have direct access to the semantic layer of communication, i.e. to the minimal proposition (Recanati, 2003). And it is hard to believe that speakers have syntactic intuitions separated from their semantic intuitions. The debate around Chomsky’s example (Chomsky, 1957), “Colorless green ideas sleep furiously”, showed that purely syntactic correctness

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1  Pragmatism and Metaphysics: The General Background

isolated from semantic value and pragmatic function is more a technical product of a particular linguistic proposal than something that speakers actually understand and identify.

1.2 The Conceptual Articulation of Reality Propositions are assertables, claimables, believables, and judgeables. In Brandom’s words: What is asserted in an act of asserting, what is assertible, is a propositional content. Assertible contents, assertibles, are also believables and judgeables; states of beliefs and acts of judgements can accordingly be expressed as assertions. (Brandom, 2000, p. 189)

The issue of which one of these acts takes priority over the rest has shaped philosophy from Descartes until the present day. Modern philosophy ‘took for granted a mentalistic order of explanation’ (Brandom, op. cit., p. 5). In the twentieth century, this order reversed, and philosophy underwent a linguistic turn whose peak Brandom locates in Dummett’s understanding of judgement as ‘the interiorization of the external act of assertion’ (loc. cit.). Both traditions, modern philosophy and the philosophy derived from the linguistic turn, embrace the mutual independence of language and thought. For modern philosophers, concepts and contents pre-exist their expression, and language is only their perceptible wrapping. This seems to have been Frege’s approach, who stressed the connection between internal acts of judgement and external acts of assertion, but always gave priority to the thought over its expression (see, for instance, Frege, 1918–1919, p.  354). The earlier Wittgenstein held a similar view (Tractatus 3.1). The philosophy of the early linguistic turn, by contrast, kept faithful to the traditional empiricist stance of prioritising what is observable over its unobservable counterparts, and to the promotion of the mind as an empty screen that only reflects and processes what senses send to it. Both narratives are defied by pragmatism, an essentially non-dualist approach that privileges continuity and gradual explanations over the binary kind of thinking so dear to the analytic tradition. In the pragmatist tradition, language and thought are inseparable; they are ‘two sides of one coin’ (Brandom, 2000, p. 6). Besides the evolutionary hint given in the previous section and the thesis with which this section begins, Brandom adds a further interpretation of the unbreakable link between language and thought. Faithful to his pragmatist approach, Brandom stresses that the activities of believing and asserting can only be explained with reference to each other, i.e. they cannot ‘be made sense of independently’ of each other (Brandom loc. cit.). To all this, he adds a further linguistic remark, that the ‘[l]inguistic expressions whose freestanding utterances have the default significance of assertions are (declarative) sentences’ (Brandom op.  cit., p.  189). Propositions are what is asserted in those acts in which we use ‘freestanding’ utterances of declarative sentences. The standard characterization of propositions as truth-bearers only gets us a grammatical step further.

1.2 The Conceptual Articulation of Reality

9

The radical Cartesian divide between language/thought and reality is a common presupposition of modern and contemporary philosophy that pragmatism also rejects. This divide shapes the particular epistemological character that modern philosophy presents, and explains the undue weight given to philosophical scepticism. In the philosophy of the linguistic turn in its dominant version, the divide assumes a semantic guise and is responsible for the difficulties that philosophers have encountered in providing an acceptable definition of the correspondence relation that truth allegedly constitutes between language and the world. That propositions are assertables, claimables, and believables, together with the essential connection between those acts and the discourse of truth, is the first source of plausibility for some theories of truth, particularly those in the correspondentist family. The so-­ called ‘identity theory of truth’ (Candlish, 1989), in its turn, is an attempt to mend the dysfunctions of the correspondentist intuition while preserving the interpretation of propositions as assertables, believables, and judgeables. Candlish attributes the first formulations of this theory to Bradley, but also detects it in Moore and Russell (see also Baldwin, 1991). It has been more recently elaborated by Dodd and Hornsby (Dodd, 1996; Dodd & Hornsby, 1992; Hornsby, 1997), who also include Frege and MacDowell among its early practitioners. The identity theory of truth is a conspicuous example of how philosophers upgrade to the category of a metaphysical statement something that is nothing more than a grammatical remark. As we will see in Chap. 10, the correspondence theory of truth is another outstanding example of a grammatical generalisation presented as a substantive proposal. In the kind of pragmatism that inspires my work, by contrast, our minds, our use of language, and the world that we have access to form a continuum that can only be dissected artificially. We are (part of) the world and our mental and linguistic life cannot be severed from the natural and social environment in which our lives as rational creatures develop (see Frápolli, 2022). Various philosophical areas and schools have offered elaborations of the pragmatist take on reality as a unified continuum to which we belong. In the philosophy of science of the twentieth century, Norwood Russell Hanson put forward the slogan that observation is theory-laden, which inspired his seminal book of 1958 (Hanson, 1958). The many different insights that led Kuhn to propose his, admittedly vague, notion of a paradigm (Kuhn, 1962) are a further example. Both Hanson and Kuhn argued that there is nothing like a neutral approach to reality or a direct perception of things as they really are. They shattered the positivist myth of the naked eye as the origin of knowledge and science. ‘Seeing’, Hanson claims, ‘is a “theory-laden” understanding’ (Hanson op. cit., p. 19). Kuhn, in a similar vein, explained that the concepts and knowledge with which we approach reality, and which shape the paradigm we are in, determine what we see and allow the identification of those ‘anomalies’ that defy our expectations (Kuhn op. cit., p. 57). Perception is only possible from some conceptual scaffolding: Surveying the rich experimental literature from which these examples are drawn makes one suspect that something like a paradigm is prerequisite to perception itself. What a man see depends both upon what he looks at and also upon what his previous visual-conceptual

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1  Pragmatism and Metaphysics: The General Background experience has taught him to see. In the absence of such training there can only be, in William James’s phrase, “a bloomin’ buzzin’ confusion”. (Kuhn op. cit., p. 113)

So, there is no such thing as an uncontaminated description of the world in itself— although this idea is nothing new, as Kant could attest. From a different perspective, the same intuition of unicity is patent in Brandom’s expressivism. Its most publicised feature relates to the meaning of normative concepts, whose role is making explicit normative statuses. But there is a complementary interpretation of the expressivist picture that accounts for the common role of first-level concepts in assertion. In Brandom’s view, it is through language that reality gets conceptualised, and only conceptualised reality affects our rational life. Practitioners of assertive actions not only make explicit under the form of conceptual articulation their dealings with the world—both physical and social—that surrounds them; they also process them in a way that enables us to enter into the discursive practices that make us sapients. Conceptualised reality, the reality with which we are mostly concerned in our specific human activities, is a reality shaped by our conceptual system. In Brandom’s words: First, we might think of the process of expression in the more complex and interesting cases as a matter not of transforming what is inner into what is outer but of making explicit what is implicit. This can be understood in a pragmatist sense of turning something we can initially only do into something we can say: codifying some sort of knowing how in the form of a knowing that. Second, as is suggested by the characterization of a pragmatist form of expressivism, in the cases of most interest in the present context, the notion of explicitness will be a conceptual one. The process of explicitation is to be the process of applying concepts: conceptualizing some subject matter. (Brandom op. cit., p. 8)

Assertion is, in Brandom’s universe, the basic discursive activity; contra Wittgenstein, assertion is language’s downtown. An inferential approach to propositional individuation like Brandom’s cannot proceed otherwise. As we will see in the next chapter, Frege takes a similar path. Thus, the Cartesian divide closes up in assertion, since assertion is where language meets the world. Some contemporary representatives of the ‘political turn’4 in analytic philosophy have gone a long way to highlight the fact that only with the appropriate repertoire of concepts can some social phenomena be perceived and understood. This is the phenomenon that Miranda Fricker has analysed under the label of ‘hermeneutical injustice’ (Fricker, 2007, chapter 7, Medina, 2012, Medina, 2017). Before the introduction of notions such as abuse, obstetric violence, and bullying, not even the victims were aware of their status as victims. Concepts without intuition are empty, intuition without concepts is blind.

 The label ‘political turn’ applied to contemporary analytical philosophy is due to Manuel de Pinedo. The group of analysis and language of the Department of Philosophy I of the University of Granada uses it now to cover the kind of approach to politically laden terms and expressions, slurs, dogwhistles, terms linked to gender and race, etc., that philosophers such as Fricker (2007), Haslanger (2012), Medina (2012), and (Saul, 2012, 2019) have recently brought about (see Bordonaba et al., 2022). 4

1.3 Assertion

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The case against the Cartesian divide has been made in enough detail by the philosophers I have mentioned in this section and by many others, from Kant to the present day. To an unprejudiced mind, that the res cogitans and the res extensa owe their intelligibility to each other should be quite uncontroversial. But from the paradigm defined by the Cartesian divide, the pragmatist narrative seems to be too close to an idealist position in which, without renouncing the gap, everything ends up falling on one side of the divide, the mind or the subject in this case. Pragmatism thus seems to bridge the gap at the price of losing the world. This, however, is a stultified criticism that pragmatism should simply ignore (but see Frápolli, 2022). It happens only too often that criticisms originating from a particular paradigm are expanded to cover developments in a competing one in which they do not make any sense. And it is not uncommon for proponents of alternative views to feel the need to answer to objections that, without affecting their own positions, have been with us for so long that we uncritically take them to touch some vital nerve. What is needed is not a secure passage from the world to our knowledge of it, or from reality to our conceptual system. What is needed, independently of the semantic or epistemic paradigm that we find ourselves in, is an explanation of the essential distinction between what is subjective and what is objective, between those aspects of our rational life that depend on our will and those that resist it. Pragmatism has the mechanisms to mark this distinction without opening a breach that all attempts to account for knowledge and truth must subsequently struggle to bridge.

1.3 Assertion Assertion is the site where language and the world come together. In contemporary philosophy of language, assertion is a topic in itself. Different authors have put forward their own accounts of assertion, sometimes as a speech act, and sometimes as an action that transcends what we do with words (see, for instance, Schiffer, 1972, p. 126, MacFarlane, 2011, p. 82). My interest focuses on discursive actions, actions in which the linguistic application of concepts is involved, but, as I discuss in Chap. 8, I acknowledge the performance of actions with conceptual content through the use of alternative, non-linguistic vehicles. MacFarlane (op. cit., p. 81) classifies the different proposals about assertion that are currently alive in the philosophical arena into four groups that we can label as follows: (1) the expressive account: asserting is expressing beliefs; (2) the game-theoretical account: asserting is to make a move in a language game subject to constitutive rules; (3) the informational account: asserting is to add information to the shared background of a conversational setting; (4) the normative account: asserting is to undertake a specific commitment. Taking sides among the vast list of particular theoretical developments in competition is neither necessary nor useful. The pragmatist attitude of paying attention to the intuitions of competent speakers advises us to take these proposals seriously since all of them highlight aspects of assertion that deserve scrutiny. In fact, all of

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these approaches are actually correct. In asserting, we do all these things, even though different particular instances might give more weight to one of them over the others. Probably, as it often happens, particular developments make specific assumptions that preclude other particular developments. But all the intuitions that they develop are present in all (or most) acts of assertion. There has been some debate about whether, in every act of assertion, speakers express their beliefs, since it might be the case that a speaker asserts a content just for the sake of the conversation, without actually committing herself to its truth. This possibility does not constitute a serious objection to the expressive and normative approaches. As I will explain in Chap. 10, those cases can be explained as including an implicit circumstance-shifting operator (Lewis, 1980): an operator that directs the audience to the world in which the content can be asserted without reservations. For the sake of the conversation, we make as if we were fully committed to the asserted content. This kind of pretence is ubiquitous in language (see, for instance, Recanati, 2000). A different debate relates to the norms that govern the move that assertion makes in a language game. The three main candidates for the norm of assertion are truth (NT), belief (NB), and knowledge (NK): (NT)  Only assert what is true. (NB)  Only assert what you believe. (NK)  Only assert what you know. Williamson instigated this debate not only in the philosophy of language but also in epistemology (Williamson, 1996, 2000) with his defences of the priority of knowledge and knowledge as the constitutive norm of assertion (also see Goldberg, 2015). Nevertheless, as I have discussed elsewhere (Frápolli, 2019), these three norms are equivalent in practice. From the first-person perspective, a speaker cannot distinguish between what she knows and what she is merely justified in believing, as Gettier cases show. And the same can be said of truth. If (NT) is interpreted reasonably, it is indistinguishable from (NB) and (NK). One of its unreasonable-and-­yetnot-uncommon interpretations, i.e. the one that advices us only to assert what is ‘objectively’ true, condemns us to silence. The expressive and normative approaches have also been contested on the basis of their failure to distinguish between what one explicitly asserts, and what asserting it assumes or implies, thus blurring the borders between what is said, what is pragmatically implicated, and what is presupposed (see, for instance, MacFarlane, 2011, pp. 81–2). Here the borders are unclear, and the philosophy of language offers its help by articulating the different kinds of meanings that might be at play in an utterance and their relationship with the utterance’s content. As a rule, I will assume that one is primarily committed to what is—literally or metaphorically—said, and secondarily committed to what directly follows from it, in a way that is always contextually modulated. There is no a priori limit to the consequences that follow from what a speaker says to which she commits, but some underdetermination at this point is what one should expect from the pragmatist methodology, which often

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favours gradual transitions over rigid demarcations, and favours characterizing and explaining over classifying and delimiting. Frege explicitly stresses an expressive approach to assertion—in ‘Logic’ (Frege, 1879–1891, p. 2) and in ‘Thought’ (Frege, 1918–1919, p. 356)—which is congenial to the priority that he confers to thought over language. Brandom’s view, on the other hand, includes intuitions that belong to the four kinds of approaches found in MacFarlane’s classification. My approach is thoroughly Brandomian. Brandom sometimes calls his overall view ‘analytic pragmatism’, for good reasons (see, for instance, Brandom, 2008, pp. 31, 32, and 2014, Lecture Three). The label is appropriate for covering the work and methodology of most representatives of both traditions: analytic philosophy and pragmatism. Pragmatics, as a specific approach to meaning and content, combines analysis with the contribution of context that includes speakers’ intentions. Carnap, Ryle, Putnam, Quine, and Haack are analytic pragmatists, although Haack explicitly rejects this categorisation of her work (Haack, 2020, p. 96). All of these are philosophers with a taste for precision and a sensitivity to understanding the role of speakers in communication. I also count Frege among them, in particular for his many far-reaching insights into the functioning of natural language, as will be seen in Chaps. 3 and 4. Crude empiricism is neither an essential characteristic of the linguistic turn nor of the analytic tradition that derived from it. Respect for science, by contrast, is, as is the attitude of not presenting the work of philosophers as competing with the work of scientists. Brandom’s account of assertion combines the analytic stance of paying attention to the stimuli that explain the correctness of specific linguistic reactions, together with a pragmatist sensitivity to the consequences of our acts. It keeps tabs on ‘upstream’ as much as ‘downstream’ aspects (MacFarlane, 2011, p.  91; Shapiro, 2020, pp.  86ff.). ‘Stimuli’ and ‘reactions’, however, sound more Quinean than Brandomian. Brandom speaks instead of the circumstances of application of particular terms and the proprieties of uses of words and sentences (see e.g. Brandom, 2000, p. 185). The exclusive focus on stimuli and proprieties defines the specific analytic version of the intuition that the meaning of expressions answers to the world. The exclusive focus on consequences shapes the pragmatist implementation of the intuition that meaning is related to practical applications. Brandom argues that both are necessary for understanding the pragmatic aspects of assertion, which are explained in terms of normative statuses, and its semantic aspects, which individuate propositional contents inferentially, i.e. by the correctness of material inferences. The structure of assertions is complex. The inclusion of the circumstances and consequences of the use of declarative sentences shows the two essential kinds of appraisal to which assertions are subject (Brandom 2000, p. 187). An act of assertion can fail at both ends, either because the conditions for the application of its linguistic aspects do not hold, or because the consequences of the acts of uttering are not complied with. In the normative terms in which Brandom casts his approach, for an agent to be in a position to perform an act of assertion, she has to be in possession of the appropriate entitlements. The normative term ‘entitlement’ plays the role of justifications and reasons in alternative approaches. A speaker who performs

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a successful assertion must have some support for the proposition that she is putting forward. The agent of an assertive action also commits herself to the consequences that follow from her asserting, which include the indirect assertion of the contents that follow from the asserted content. Asserting is part of a sophisticated social structure. For a linguistic practice to count as an act of assertion, it has to be performed in the context of a specific kind of language game, which Brandom, following Sellars, understands as the ‘game of giving and asking for reasons’ (Brandom, 1994, p. 189). To assert is to make a move in this game, which is performed by the utterance of a declarative sentence, whose content one is entitled to and whose consequences one endorses. If he had to choose a norm for assertion, Brandom would choose belief (Brandom, 1994, p.  153), although, as I have said before, the three norms are equivalent, particularly within a view in which truth is expressive, as is the case with Brandom’s view. Propositions are thus those entities whose production is essentially in need of reasons, and which can serve as reasons for other contents. They are those entities that can occur as the premises and conclusions of inferences and are individuated by those correct material inferences of which they are part. The pragmatist slogan that meaning is use acquires in Brandom’s approach a very specific form that connects the pragmatic aspects of assertings with the semantic aspects of the inferential individuation of propositions: The link between pragmatic significance and the inferential content is supplied by the fact that asserting a sentence is implicitly undertaking a commitment to the correctness of the material inference from its circumstances to its consequences of application. (Brandom op. cit., p. 63)

Assertion, considered as a game, and its content, individuated as the ‘undertaking of a commitment’, highlight the social character of Brandom’s view. Entitlements and commitments are statuses of the agents that take part in the game; they play the part of the pragmatic force. But not everything in assertion depends on force and the agent’s attitude. Pragmatism is not a kind of internalism—in the first place, because assertion is a social practice that is subject to the assessment of other practitioners; and second, because the successful performance of assertion affects the normative statuses of those involved in the game. Understanding the content of an assertion means understanding the commitments one undertakes by accepting it, and the commitments that its acceptance precludes: To play such a game [the game of giving and asking for reasons] is to keep score on what various interlocutors are committed and entitled to. (Brandom op. cit., p. 165)

In the third place, the individuation of propositions transcends the speaker’s attitudes. The ‘finer-grained normative vocabulary of commitment and entitlement, and hence of incompatibility’ (Brandom op. cit., p. 198) provides the conceptual structure to explain the independence of propositions from the attitudes of particular agents, and the objectivity of communication.

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To return to the issue with which I have concluded the previous section, what is needed to understand communication is not a way to bridge the alleged gap between what is internal and what is external to the subject, but rather to explain the difference between what is subjective and what is objective. Objectivity involves the perspective of others. Brandom criticises assertability theories that confine the correction of assertion to the speaker’s attitudes. Assertability theories forget the objective aspects that only an external perspective provides. This external perspective is often explained in terms of truth conditions. The standard explanation of assertion in terms of truth conditions usually has very little explanatory power. It goes no further than the grammatical remark that, in assertion, speakers put forward something as true. Nevertheless, truth-conditions talk touches on an aspect of linguistic communication that is missing in assertability proposals. Assertability conditions (subjective proprieties) are not truth conditions (objective proprieties). A distinction between a first-person perspective that accounts for the agent’s entitlements, and the third-person perspective that confers objectivity on the contents is essential, and Brandom’s proposal includes the mechanisms to account for it. Identity of assertability conditions is not identity of content, something that assertability proposals cannot explain. Consider the following examples, (1) and (2), (1) The book is red, (2) At this moment and from my present position with this light etc., I can properly assert that the book is red. Their assertability conditions are identical, and nevertheless, they convey different information. They have, in the standard terminology, different truth conditions. I might be in a position as to be entitled to assert (2), even if (1) is false. My entitlements to assert (2) might not be enough for me to assert (1), and (2), but not (1), is incompatible with the truth of (3), (3) I’m blind. Moreover, (1), but not (2), entails (4), (4) The book is not brown. Inferential semantics, i.e. an approach to the contents of assertions in which they are individuated by their inferential connections, and a normative pragmatist approach to assertion that places it in a socio-normative structure, can account for the objective aspects of communication. We do not need to bridge any gap or take a God’seye perspective to reach the world. As sapients, we are just where we should be. In any case, there is no other place.

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1.4 Propositions and the Formality of Logic Propositions—what we say, assume and conjecture—determine the topics of discourses, debates and theories. They constitute their material content. A presupposition of the narrative that I am proposing in this book is that only when these topic-determining items are around, does the job of logic begin. This presupposition follows what I call the Principle of Propositional Priority, which will be discussed in Chap. 2. Logic takes care of some links between propositions, but neither determines what these propositions are about nor establishes new links among them. It is in this sense that logic is ‘topic-neutral’, an expression that (Ryle, 1954) put on the table, even if only to reject it as an insufficient delimitation for logical words. Logical notions mark connections without affecting the items thus connected. In Tractatus 5.4611, Wittgenstein gives voice to this intuition by declaring that ‘[l]ogical operations are signs of punctuation’ (see also Došen, 1989). Even if the identification of logical terms with punctuation signs is more than debatable, the intuition is clear: logical terms mark the boundaries of propositions, their role being transversal to any discursive enterprise. The transversality of logic is, thus, a reasonable interpretation of its formality. Logic can be said to be formal because it is not essentially linked to any particular kind of content. This feature of logic, which is broadly recognized, should make those who consider logic to be part of mathematics or to represent mathematical ways of reasoning think twice. As Frege said in his Begriffsschrift, the analogy of his conceptual writing with arithmetic is seen in the use of letters (Frege, 1879, p. 6), i.e. the analogy relates to external aspects of his proposal and does not expose any essential similarity. Frege’s own explanation of the analogy does not, however, do justice to the connections of his conceptual writing with the language of arithmetic, though, whose major likeness is the use of the categories of function and argument applied to the analysis of judgeable contents. Propositions are missing in post-Fregean logic. Only the links between them, i.e. logical constants, remain. The elimination of propositions proceeds in two steps. First, there is a selection of those terms that represent logical notions in arguments; then, in a second step, the rest of the terms are removed and substituted by variables of the appropriate category. The result is contentless skeletons. This procedure explains why the philosophy of logic is practically reduced to the analysis and definition of (formal) logical constants, which determine the central issues of validity and logical form. In Chaps. 4, 5, and 6 the meanings of logical terms will be discussed. Yet, in the practice of producing and assessing arguments, in science and everyday life, propositions, i.e. the steps or elements of arguments, play an essential part. We are interested in knowing whether such-and-such a propositional content follows from some other(s), whether the transition from such-and-such information to some other piece of information is safe, whether we are justified in assuming that it is the case that so-and-so once such-and-such has been accepted, etc. The process of erasing propositions and leaving behind their empty imprints has been deeply damaging to the whole enterprise of logical theory. Logic has turned into a soulless

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theory, such that we do not even know its subject. Turning propositions into variables highlights the instrumental role of logical constants but obscures the fact that inferential connections depend on the ‘inner nature of propositions’, as Frege remarked (Frege, 1879, p. 5). The elimination of propositions affects logicians’ legitimate interest in the meaning of logical constants and jeopardises the success of the enterprise of defining them since, unless we understand the terms linked by logical relations, the nature of the relations themselves becomes mysterious. This issue will be taken up in Chap. 6. Understanding logic as a part of mathematics, or as essentially related to it, has not helped either. Mathematics deals with numbers, their kinds and relations, whereas logic deals with arguments, i.e. relations among propositions. Frege, who gave the first correct definition of natural number in the history of mathematics, also offered an inferential approach to judgeable contents and was able to distinguish between them as semantic entities (senses or thoughts), their truth values, and the pragmatic force with which they can be put to work. He did not confuse his semantic (logical) project with his foundational project in arithmetic, although his foundational project pushed him to refine the conceptual instruments of analysis, and those refinements gave rise to the philosophy of language as the discipline we know now. Frege’s work begins with the notions of assertion and judgeable content (Frege, 1879, §2 and §3). This point of departure is unequivocally pragmatist, in several senses. Firstly, it places a human action, an assertion, at the forefront of his theoretical approach and characterises judgeable contents by their nature as assertables. Second, it gives priority to complete propositions over their ‘parts’—concepts— since pragmatic force can only be attached to complete propositions (Brandom, 1994, p. 82). Propositions are linguistically expressed by means of sentences, and this also gives priority to sentences over terms. Frege’s logic is not a logic of terms. Brandom follows this Fregean tradition of the priority of propositions, which he ultimately traces to Kant’s philosophy (Brandom op. cit., pp. 80–83). The priority of the propositional is a consequence of the pragmatist approach to meaning that Brandom summarises in the slogan ‘Semantics Must Answer to Pragmatics’ (Brandom op. cit., pp. 83ff.), that Frege implements in some parts of his philosophy—as it will be argued in Chaps. 2 and 3—and that I also assume. I will deal with the nature of propositions in Chap. 2 but, at this point, a brief overview will be useful. The definition of any abstract entity involves two kinds of differentiated criteria, individuation criteria and identification criteria. Individuation criteria tell us when two superficially distinct items are in fact one and the same. Identification criteria help us to pinpoint the appropriate kind of item. Propositions, as abstract entities, are no exception. It is common for logicians and philosophers of logic to doubt, at least methodologically, the existence of propositions. Philosophers of mathematics take a similar attitude towards numbers. Some caution related to new abstract entities is reasonable in unknown theoretical realms. An example is Cantor’s transfinite arithmetic. If the development of one’s research leads to a point at which some novel entities must be postulated, it is advisable to be sure that these entities exist at all. The difficulty, at this point, is to figure out what ‘exist’ means. Surely, the existence of abstract

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entities cannot be equated with any kind of spatiotemporal placement. Philosophers have struggled to answer existential questions that have led to a broad variety of proposals, from the decidedly Platonist answers that understand abstract objects along the lines of physical objects to the radical nominalist views that, at most, assume their existence as useful fiction (Balaguer, 2008), or as an indispensable hypothesis (Field, 1980). The answer, I contend, is much simpler, and it can even be found in the practice and theory of allegedly realist philosophers and mathematicians. From a pragmatist perspective, the distinction between realism and antirealism does not carry any actual weight since it is a distinction without a difference. Cantor saw that his transfinite numbers were genuine numbers after checking that they possessed precise relations with other abstract entities that were already accepted and that, from them, a specific kind of arithmetic could be developed (see Frápolli, 2015, pp. 335–6). This was an example of a practical solution. Around the same time, Frege supplied the theory that supports this kind of solution. For Frege, existence is instantiation, a second-level property that applies to concepts if, and only if, their extension is not empty. I believe that the Fregean explanation is correct and that it provides us with everything that we need to understand existence. Frege’s deepest insights, which prompted the right perspective on logic, were the identification of higher-level concepts, and a way to represent their scope. In the same sense in which philosophy of language offers everything we need to understand truth, as I will explain in Chap. 10, Fregean logic offers everything we need to understand existence. As Coffa has argued, Frege, who belongs to the ‘semantic tradition’ in logic, used the word ‘logic’ as we use ‘semantics’ (Coffa, 1991, p. 64). I not only agree but would add that sometimes it is our pragmatics that is the right correspondence for some central Fregean notions, although from my perspective there is no clear-cut divide between the two disciplines. It is for this reason that Frege opens his Begriffsschrift with the judgement stroke, and highlights, in his Grundgesetze, the distinction between sense and meaning as being of the utmost importance for his logic. Truth and existence, and all other notions with any logical significance, are higher-level and their import is entirely intra-systemic. Neither is existence a property of the objects of the universe nor does truth reflect ‘how things are’ out there. In contrast with ‘truth’, ‘[t]he explanation of the meaning of “exist” and “be” is not even a matter of semantics: it is a matter of syntax’ (Williams, 1981, p. x). Admittedly, some instances of existential terms seem to play a first-level role, and the verb ‘to be’, even in locutions such as ‘there is’ and ‘there are’, sometimes has a locative function that makes it natural to interpret apparently existential sentences as placing objects in space. These uses have explanations that do not defy the instantiation interpretation given by Frege (Williams, 1981, p. 15). Embedded contexts also pose a challenge to the interpretation of existence as higher-level. Sentences such as (5) and (6) make perfect sense and, in them, existence seems to be first-level, (5) Joe Biden does not know that María José exists. (6) María José might not have existed.

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Again, (5) and (6) can be reformulated as to show that existence is not predicated of a particular individual, but its role linked to predicables of the form ‘a person with such-and-such characteristics’, ‘somebody called “María José”’, etc. Williams gives detailed explanations of embedded uses of existential clauses and shows that their challenge is merely apparent (Williams, 1981, chapter IV). In any case, there would be no theoretical difficulty in accepting a first-level reading for some uses of allegedly existential terms, and thus assuming first and second-level existential concepts. This acceptance would show that there is more than one concept involved in uses of sentences with terms that we identify as existential. This is not my position, though. I believe that existence, truth, good, and a substantial amount of those notions privileged by philosophers are higher-level and, as such, cannot be identified with any set of first-level concepts. Sometimes, their higher-level nature has led to them being qualified as undefinable, as happened with the Fregean explanation of truth and the Moorean account of good. Indeed, deeming them undefinable is closer to the truth than giving their definition in terms of first-order properties. I, like everybody else once we consider the generalised acceptance of Frege’s view of quantifiers, follow Frege in his characterization of existence. I wonder how it is possible for this acceptance to coexist with sophisticated discussions between realists and anti-realists in the philosophy of mathematics, and with widespread suspicion about the existence of abstract entities. If existence is instantiation, we cannot doubt that numbers exist. The only consistent way to doubt their existence would be to cease to use numerical expressions altogether. Otherwise, any time that we put forward as true a proposition that includes numerical information, we presuppose the existence of numbers. And this is so because (8) follows directly from (7), with no need for any further qualification, (7) 2 is a number, (8) Something is a number. The inference from (7) to (8) is all that is involved in the attribution of existence (Gibbard, 2012, p. 113). This statement can be further qualified to cover what is at issue in indirect existential proofs. But in those cases, what we have is the modal general assumption that not all instances of a particular existential claim can be false, even if we do not know which one is true. Doubting the existence of numbers clashes with standard scientific practices, and with a significant part of our linguistic actions. Doubting the existence of propositions is even more unjustified. In these very pages, I am using some English sentences to say something, to put forward some arguments, to make some points—but these sentences are not what I’m saying. I could have said exactly the same things, put forward exactly the same arguments,

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and make exactly the same points in Spanish, i.e. using completely different sentences belonging to a completely different language. In our linguistic actions we utter sentences, but say things that cannot be identified with them. What I am saying and what you are understanding, accepting, rejecting, or doubting are propositions, which we can refer to using different terms, such as ‘information’, ‘content’, or ‘what is said’. The customary focus on sentences, proper to the early years of the analytic school, does not resist the slightest superficial analysis. Even so, it is still the standard view among logicians and philosophers of logic and is also common within some special areas of philosophical discussion, like debates about the nature of truth. I will discuss this point in Chaps. 9 and 10. Fortunately, Frege did not confuse what is said with the linguistic expressions used to say it, and clearly distinguished between sentences (Sätze) and propositions,5 which he called ‘judgeable contents’ (beurtheilbarer Inhalten) and ‘thoughts’ (Gedanke), and which he characterised as the senses of declarative sentences and propositional questions. Numbers and propositions are thus abstract entities whose existence we directly assume in our discursive practices. My characterisation of propositions follows some techniques that Frege used in his characterization of numbers. It is a complete characterisation since it provides us with everything we need in order to understand their nature and to explain the role they play in scientific and everyday discussions. Let us first recall the basic lines on which Frege built up his definition of number. In (Frege 1884, §62 and §63), Frege begins by mentioning what is now known as ‘Hume’s Principle’, (HP): (HP) The number that belongs to the concept F is the same as the number that belongs to the concept G if and only if a 1–1 correlation can be established between the items that fall under F and the items that fall under G.

Frege assumes Leibniz’s definition of identity (Frege 1884, §65) but still considers (HP) unsatisfactory because it gives us no clue about the kind of objects to which the definition applies. As a solution, he proposes to identify numbers with the extensions of concepts, which leads to (HP*), (HP*) The extension of the concept ‘equal to the concept F’ is identical to the extension of the concept ‘equal to the concept G’ is true if, and only if, ‘the same number belongs to the concept F as to the concept G’. (Frege 1884, §69)

 The English term ‘proposition’ suffers the same ambiguity as the German ‘Satz’ and the Latin ‘propositio’. All of these can refer to expressions, and thus be synonymous with ‘sentence’, or else refer to the contents of sentences, to thoughts or judgeable contents in Fregean terminology, or to premises and hypotheses in the terminology of medieval logic (see Kneale and Kneale, op. cit., p. 361). Many translations of Frege’s writings into English render ‘Satz’ as ‘proposition’, and this circumstance makes it difficult to identify Frege’s views. Nevertheless, Frege was crystal-clear about the distinction between thought (Gedanke) and conceptual content (begrifflicher Inhalt), on the one hand, and its linguistic expression, the sentence (Satz), on the other. In what follows, unless I explicitly state otherwise, I will use ‘proposition’ to refer to the contents of assertions, i.e. to Fregean thoughts. 5

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In his mature work (Grundgesetze, 1893/2013, §66 and §67), Frege returned to this identification problem, but this time numbers were identified with courses of values, which gave rise to Russell’s paradox (but see Chap. 7). In his rejection of his former view, he replicates his doubts about the identification problem. The definition, he admitted, cannot be used to determine whether, for instance, Julius Caesar is a number. Among philosophers of logic, this problem is now known as the ‘Julius Caesar problem’ (MacBride, 2003, p.  104), or simply as the ‘Caesar problem’ (Rosen, 2001, p. 229). Frege did not concede too much importance to it (MacBride, loc. cit.; Rosen, loc. cit.), but contemporary philosophers of logic do. The exact nature of the problem exposed by the Julius Caesar example is not completely clear (Greimann, 2003; Salmon, 2018), nor is the logical status of (HP) and (HP*) (Hale & Wright, 2001). But neither of these issues affects my argument here. But it is interesting to note that, as often happens with classical debates in the philosophy of logic and mathematics, this issue is not confined to those disciplines or to the application of the notion of number. The Julius Caesar problem touches upon the general issue of how to delimitate a concept’s domain. In fact, Frege’s concern when he mentioned the Julius Caesar example was to avoid vagueness. The concern, justified as it was for the foundation of mathematics, does not extend to other areas of analysis in which vague concepts are perfectly acceptable. This is an issue in which the disparity in methodologies and presuppositions between the analytic and the pragmatist approaches is exposed in all its sharpness. From the pragmatist perspective that I have depicted in previous sections, concepts cannot be dissociated from their application conditions. And application conditions involve discriminating between those situations in which the assertion of a content that includes the concept is appropriate and those situations in which it is not. A concept’s domain is something that we learn when we learn how to use it. A concept cannot be said to be mastered and the meaning of a word cannot be said to be known unless the central cases that fall under the concept, as well as those cases that clearly lie beyond it, are clearly identified. Understanding the concept of number essentially involves knowing that Julius Caesar does not belong to its extension. Admittedly, in some specific situations, a concept’s domain is a debatable issue. This happens, for instance, in science when new concepts are introduced. In everyday life, the boundaries of ordinary concepts may come under scrutiny if some inferences in which they occur are no longer supported, such as when we reject the offensive connections that some concepts may have had in the past (see Brandom, 2000, pp. 70ff., Plunkett, 2015). Introductions of new concepts and variations of old ones are not blind, though. They are guided by theoretical and practical goals. And, in a pragmatist setting, common sense must always be present. Frege shared this feeling. In Frege’s writings, in the way in which he approaches the foundational questions, the reasonable concept-user outweighs the mathematician. In relation to the Julius Caesar problem, this seems to be the case. Rosen, for instance, says: In the Foundations of Arithmetic Frege makes a stronger claim for what has come to be called ‘The Caesar Problem’. Frege apparently holds that any philosophical account that

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1  Pragmatism and Metaphysics: The General Background fails to vindicate common sense on this point is automatically, for that very reason, unacceptable. But why should this be? (Rosen, loc. cit.)

Because he was a pragmatist. This one would be a sensible answer to Rosen’s question. This answer is, spite all appearances, not a radical one. It simply means that Frege wanted theoretical developments in the foundations of arithmetic to be connected to how mathematicians actually use the concepts involved, without adding to the difficulties that the field already faces any new artificial issues that, in Rosen’s words, would be ‘almost completely epiphenomenal’ (Rosen, loc. cit.). Let us now move on to the analysis of propositions. A similar pragmatist attitude and similar steps and presuppositions that Frege displayed in the definition of numbers can be put to work in the task of characterizing propositions. Dealing with propositions by analogy with numbers is not a new strategy (Soames, 2020). I want to go a step forward, though. I contend that propositions can be completely characterized using the two kinds of criteria mentioned above—individuation and identification—and that Frege had the tools to do it. Fregean propositions are unstructured entities with inferential individuation criteria, as I will argue in Chap. 2. In his Begriffsschrift (Frege, 1879, Prologue and §3), Frege suggests something like the following principle of individuation, (PI): (PI) Two propositions are one and the same if and only if they follow from the same set of propositions and the same set of propositions follow from them.

In Chap. 2, I offer a slightly different version, (PII), in which the Fregean formulation is enriched by mentioning the agents and sentences by means of which propositions are expressed. Both versions are equivalent.6 (PI) plays for propositions the role that (HP) and (HP*) play for numbers. The Julius Caesar problem for propositions is the problem of identifying the kind of item that propositions are. The complete answer will be given in Chap. 2, but at this point it will be enough to say that the identification criterion that I will provide proceeds top-down. It identifies propositions by their properties and relations instead of by their virtual components. I take this to be a Fregean strategy too. As said above in section two, propositions are assertable, claimable items. They are the items that we believe and know, that can be the premises and consequences of inferences, and ‘something for which the question of truth can arise at all’ (Frege, 1918–1919, p.  353). This is the identification criterion for propositions, which guides what I have called the ‘organic intuition’, (OI), (Frápolli, 2019, p. 89): (OI) To be a proposition is to possess propositional properties and stand in propositional relations.

 In Begriffsschrift §3, Frege only stresses one side of this equivalence: two judgeable contents are one and the same if, from them, the same consequences follow. But Frege’s characterisation is equivalent to (PI) and (PII), and these two principles make the import of Frege’s approach explicit. In the Prologue, he includes the propositions from which a proposition follows to characterise safe ways of establishing the truth of propositions. 6

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Now, (PI) and (OI) give a complete characterization of propositions by giving their individuation and identification conditions. Propositions are claimable items capable of being elements of inferences whose individuation criterion rests on the identity of their inferential connections.

1.5 Arguments, Inferences, and Argumentations If anything, logic deals with inferences and arguments. Argumentation theory has rescued the classical debate about what arguments are and their differences, if any, with related items such as argumentations and inferences. I will use ‘argument’ and ‘inference’ as synonyms. The term ‘argument’ (and also ‘inference’) admits of different definitions. As it has been my strategy so far, I will offer a minimal characterisation of this notion suitable for understanding the use I make of it, but I will not discuss the various proposals and their relative advantages over their competitors (for a recent survey, see Goodman, 2018). All of these surely touch upon aspects that deserve to be acknowledged. Considering the literature on the topic, and how terms such as ‘argument’, ‘argumentation’, ‘inference’, ‘reasoning’ and the like are used, a distinction between dynamic and static approaches can be detected. When the analysis of arguments is approached from a dynamic perspective, i.e., as some kind of movement from premises to conclusion, the more frequently used terms are ‘argumentation’ and ‘reasoning’ (see, for instance, Tseronis & Forceville, 2017), while ‘argument’ is usually reserved for the static perspective that focuses on the relations between propositions. And sometimes the distinction is marked by the use of specific pairs, such as ‘arguments-as-activity’ and ‘arguments-as-objects’ (Goodman, 2018, p.  3), or ‘arguments-as-acts’ and ‘arguments-as-products’. Because this is a book on the philosophy of logic, I will be interested in the static sense, i.e. in arguments-as-products. But, as its general background is pragmatist, we cannot forget that these are products of human actions, be they actual or virtual. Two general kinds of dynamic proposals are worth distinguishing: the kind of argument(ation) that proceeds backwards, and the kind that proceeds forwards. In the first case, the stress is placed on the conclusion for which reasons are provided (Groarke, 2015; Hitchcock, 2007). This kind of proposal is perfectly congenial to Brandom’s view of assertion, as well as to Walton’s view of claims: ‘A claim is an upholding of some particular proposition that is potentially open to questioning’ (Walton, 1990, p. 409). In more Brandomian terminology, a speaker who makes an assertion, and who thereby agrees to play the game of giving and asking for reasons, commits herself to provide the reasons she has for asserting the content if challenged. Assertions are thus potential arguments with implicit premises. Forward argument(ation)s are more common in the realm of science, in which some hypotheses can be put forward in order to see what follows from them, or how far we can go once they have been assumed. Arguments as products are a methodological abstraction. Logicians, philosophers of language and philosophers of logic consider them to check on the relative

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relations between the propositions involved, understood as semantic items that are unrelated to real practices. This methodological abstraction is unavoidable and perfectly respectable. Without it, science would collapse. In the static sense, arguments are sets of propositions, not sentences. This point should not need much discussion, notwithstanding the received view in logic. That propositions are the bearers of logical relations is a basic presupposition of this book, a presupposition that I consider completely safe. The alternative option, i.e. understanding arguments as sets of sentences, is, I contend, hardly defensible if one has some familiarity with recent, and not so recent, discussions in the philosophy of language, and some sensitivity to what speakers actually do. Goodman offers a beautiful example in support of propositions against sentences as the elements of arguments: I can surely consider Anselm’s Ontological Argument. When I do so, however, I am not considering any set or collection of sentence tokens uttered or inscribed by Anselm—I am not in contact with them. Nor am I considering any set or collection of sentence types that those tokens were instances of—I am ignorant of Latin. I thus think it is eminently reasonable to regard arguments in the logical sense as sets of collections of propositions, the abstract semantic contents of the types of sentences once tokened by Anselm to express his Ontological Argument. (Goodman op. cit., n. 5)

Frege’s arguments can be assessed without knowing any German, and Tarski’s arguments without knowing any Polish. An invalid argument in Spanish continues to be invalid in any correct translation into any other language, among other things because the preservation of the logical properties is a basic test for a translation’s correctness. Even if arguments as products are sets of propositions, not every set of propositions is an argument. As Goodman notes, a mere list does not qualify as such (Goodman op. cit.); the intention with which a set of propositions is put forward cannot be ignored. In a genuine argument, the propositions that act as premises are intended to give support to the proposition that occurs as the conclusion. The support given by the premises to the conclusion admits of different degrees of strength, logical consequence being the upper limit that defines monotonic reasoning. Non-­ monotonic reasoning also gives rise to arguments as products in which the support given by the premises to the conclusion is contextual. Within the scientific fiction that arguments as products represent, the speaker’s intention is a conditional idealization. There is a presumption that the premises support the conclusion, even if no actual agent is involved. As happens with the antecedents of conditionals, premises are conditionally, provisionally, or pretendedly asserted, as Frege saw. The assertion of the premises initiates a movement from them to the conclusion, a movement of which the argument-as-product is the static outcome. In the case of monotonic reasoning, the support given by the premises to the conclusion can be stated in conditional normative terms: whoever is entitled to assert the premises, is likewise entitled to assert the conclusion, and whoever is committed to the premises is thereby committed to the conclusion. The entitlements required and the commitments acquired in the assertion of the conclusion are

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included in those involved in the assertion of the premises. The assertion of the conclusion does not demand additional normative statuses. Before concluding the section and the chapter, it might be useful to insist that the kind of arguments that I have in mind are what Brandom, following Sellars, calls ‘material arguments’. Material arguments are also what is at issue in argumentation theory. Formal arguments, i.e., logically valid forms cast in a formal language, count as arguments only in a derivative way. The activity of drawing inferences essentially involves conceptual contents. Sentences and formulae are the vehicles of conceptual, propositional information, but, as linguistic items, fall short of being the primary bearers of logical properties. At this point too, I take my approach to be completely Fregean. Arguments represented in a logical language, such as that of the Begriffsschrift, include only the addition of a higher degree of perspicuity. As Macbeth notes, Frege’s discomfort with the arguments used in Euclid’s Elements was not that they were invalid or enthymematic, but rather that the rules of inference used in them were not explicit from the beginning (Macbeth, 2005, positions 221–242). These non-explicit rules were not laws of logic, but ‘materially valid rules’ that governed the use of the geometrical terms in the axioms. In other words, inferences in geometry rested on the meaning of extra-logical terms that made explicit the inferential import of the terms involved. They were analytically but not logically valid. Presumably, the wide variety of concepts involved in material inferences explains the wide variety of rules of inferences found in ordinary and scientific arguments. In fact, Frege’s limitation of the accepted rules of inference to just one, the modus ponens, was driven by his purpose of making explicit the list of all permissible transitions. If so, then modus ponens unifies them all since the conditional makes explicit what was implicit in meaning transitions. Nevertheless, this does not mean that materially valid rules are equivalent to modus ponens. Modus ponens is a formal rule that expresses the formal claim that if the antecedent is asserted, then the consequent cannot be denied, disregarding the contents of antecedent and consequent. Because Frege assumes that his Begriffsschrift neither establishes new truths (Frege, 1879, p. 13) nor new inferential connections, which in any case depend on the inner structure of propositions, the limitation of inference rules to modus ponens is all that is needed to expose inferential chains of thoughts for their assessment. The combination of (PI), the claim that inferences rest on the inner structure of propositions, the role of conditional to express the steps in a derivation, and the acceptance of materially valid inferences make a strong case in favour of Frege’s semantic inferentialism and the expressive role of conditionals. And now a word of conclusion. In these pages, I have explained the extent of my pragmatism and also how I understand the essential notions that will be discussed in the rest of the book. Two ideas should be kept in mind, the first of these being that we do not need to engage in the discussion of artificial and scholastic problems or accept theoretical options that turn their backs on the basic intuitions of competent speakers. The second one is that this pragmatist path leads straight back to Frege’s work.

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References Aguiar, F., Gaitán, A., & Rodríguez-López, B. (2014). Robust intuitions, experimental ethics, and experimental economics: Bringing reflective equilibrium into the lab. In C. Lütge, H. Rush, & M. Uhl (Eds.), Experimental ethics. Toward an empirical moral philosophy (pp. 191–208). Palgrave Macmillan. Balaguer, M. (2008). Fictionalism in the philosophy of mathematics. Stanford Encyclopedia of Philosphy. Baldwin, T. (1991). The identity theory of truth. Mind, 100, 35–52. Blackburn, S. (1999). Think: A compelling introduction to philosophy. Oxford University Press. Bordonaba, D., Fernández Castro, V., & Torices Vidal, J. R. (2022). The political turn in analytic philosophy. Reflections on Social Injustice and Oppression. De Gruyter. Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press. Brandom, R. (2000). Articulating reasons. An Introduction to Inferentialism. Harvard University Press. Brandom, R. (2008). Between saying and doing: Towards and analytic pragmatism. Oxford University Press. Brandom, R. (2014). Analytic pragmatism, expressivism and modality. The 2014 Nordic Pragmatism Lectures, Nordic Pragmatism Network (www.nordprag.org). Burgess, A., Cappelen, H., & Plunkett, D. (2020). Conceptual engineering and conceptual ethics. Oxford University Press. Candlish, S. (1989). The truth about F. H. Bradley. Mind, 98, 331–348. Cappelen, H. (2012). Philosophy without intuitions. Oxford University Press. Cappelen, H. (2018). Fixing language: An essay on conceptual engineering. Oxford University Press. Cappelen, H. (2020). Conceptual engineering: The master argument. In A. Burgess, H. Cappelen, & D. Plunkett (Eds.), (pp. 132–152). Cappelen, H., & Plunkett, D. (2020). A guided tour of conceptual engineering and conceptual ethics. In A. Burgess, H. Cappelen, & D. Plunkett (Eds.), (pp. 1–34). Chalmers, D. (2020). What is conceptual engineering and what should it be? Inquiry: An Interdisciplinary Journal of Philosophy., 1–18. Chalmers, D., Manley, D., & Wasserman, R. (Eds.). (2009). Metametaphysics. New essays on the foundations of ontology. Clarendon Press. Chomsky, N. (1957). Syntactic structures. Walter de Gruyter. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. Creath, R. (1990). Dear Carnap, dear Van. The quine – Carnap correspondence and related work. University of California Press. DeRose, K. (1999). Contextualism: An explanation and defense. In J. Greco & E. Sosa (Eds.), The Blackwell guide to epistemology (pp. 187–205). Blackwell Publishers. Dodd, J. (1996). Resurrecting the identity theory of truth: A reply to Candlish. Bradley Studies, 2, 42–50. Dodd, J., & Hornsby, J. (1992). The identity theory of truth: Reply to Baldwin. Mind, 101, 319–322. Došen, K. (1989). Logical constants as punctuation Marks. Notre Dame Journal of Formal Logic, 30(3), 362–381. Field, H. (1980). Science without numbers. Princeton University Press. Frápolli, M. J. (2015). Non-representational mathematical realism. Theoria, 30(3), 331–348. Frápolli, M. J. (2019). Propositions first. Bitting Geach’s bullet. In M. J. Frápolli (Ed.), Expressivisms, knowledge and truth (Royal Institute of philosophy supplement 86) (pp. 87–110). Cambridge University Press. Frápolli, M. J. (2022). Tracking the world down. How inferentialism accounts for objective truth. Philosophical Topics, vol. 50, n 1, pp. 83–107.

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Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jean van Heijenoort (1967). From Frege to Gödel. A source book in mathematical logic, 1879–1931, (pp. 1–82). Harvard University Press. Frege, G. (1879–1891). Logic. In G. Frege (1979). Posthumous writings (pp. 1–8). Basil Blackwell. Frege, G. (1884/1953). The Foundations of Arithmetic. A logic-mathematical enquiry into the concept of number. Translated by J. L. Austin. Second Revised Edition. New York, Harper Torchbooks/The Science Library, Harper & Brothers. Frege, G. (1893/2013). Gottlob Frege: Basic Laws of arithmetic. Edited by Philip A. Ebert and Marcus Rossberg. Oxford University Press. Frege, G. (1918–1919). Thoughts. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 351–372). Basil Blackwell. Fricker, M. (2007). Epistemic injustice. Power and the ethics of knowing. Oxford University Press. Gibbard, A. (2012). Meaning and normativity. Oxford University Press. Goldberg, S. (2015). Recent work on assertion. American Philosophical Quarterly, 52(5), 365–380. Goodman, J. (2018). On defining ‘Argument’. Argumentation, 32, 589–602. Greimann, D. (2003). What is Frege’s Julius Caesar problem? Dialectica, 57(3), 261–278. Groarke, L. (2015). Going multimodal: What is a mode of arguing and why does it matter? Argumentation, 29, 133–155. Haack, S. (2019), “Reconceptualizing rationality: The growth of meaning and the limits of formalism”. Lógica, Argumentación y Pensamiento Crítico. Alcances, relaciones y aplicaciones. Edited by Gabriela Hernández Deciderio, Roberto Casales García and José Martín Castor Manzano. Ediciones del Lirio, (pp. 17–42). Haack, S. (2020). Not one of the boys: Memoir of an academic misfit. Cosmos + taxis. Studies in Emergent Order and Organization, 8(6 + 7), 92–106. Hale, B., & Wright, C. (2001). The Reason’s proper study. Essays towards a Neo-Fregean Philosophy of Mathematics. Oxford University Press. Hanson, N. R. (1958). Patterns of discovery: An inquiry into the conceptual Foundation of Science. Cambridge University Press. Haslanger, S. (2011). Ideology, generics, and common ground. In C. Witt (Ed.), Feminist metaphysics (pp. 179–207). Springer. Haslanger, S. (2012). Resisting reality: Social construction and social critique. Oxford University Press. Hitchcock, D. (2007). Informal logic and the concept of argument. In D. Jacquette (Ed.), Philosohy of logic (pp. 101–129). Elsevier. Hornsby, J. (1997). Truth: The identity theory. Proceedings of the Aristotelian society XCVII, 1–24. Also, in M. Lynch (Ed.) (2001), 663–81. Isaac, M.  G. (2020). How to conceptually engineer conceptual engineering? Inquiry: An Interdisciplinary Journal of Philosophy, 1–24. Jammer, M. (2006). Concepts of simultaneity: From antiquity to Einstein and beyond. John Hopkins. Kauppinen, A. (2007). The rise and fall of experimental philosophy. Philosophical Explorations, 10, 95–108. Kripke, S. (1980). Naming and necessity. Harvard University Press. Kuhn, T. (1962). The structure of scientific revolutions. International Encyclopedia of Unified Science. The University of Chicago Press. Kukkla, R. (2014). Performative force, convention, and discursive injustice. Hypatia, 29(2), 293–330. Lewis, D. (1980). Index, context, and content. In S. Kanger & S. Ōhman (Eds.), Philosophy and grammar. Synthese library (studies in epistemology, logic, methodology, and philosophy of science) (Vol. 143). Springer. Macbeth, D. (2005). Frege’s logic. Harvard University Press. (kindle edition). MacBride, F. (2003). Speaking with shadows: A study of neo-Logicism. British Journal for the Philosophy of Science, 54(1), 104–163.

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MacFarlane, J. (2011). What is assertion? In J. Brown & H. Cappelen (Eds.), Assertion (pp. 79–96). Oxford University Press. Manley, D. (2009). Introduction: A guided tour of Metametaphysics. In D. Chalmers, D. Manley, & R. Wasserman (Eds.), (pp. 1–37). Manne, K. (2020). Entitled. How Male Privilege Hurts Women. Penguin and Random House. Medina, J. (2012). Hermeneutical injustice and polyphonic Contextualism: Social silences and shared hermeneutical responsibilities. Social Epistemology, 26(2), 201–220. Medina, J. (2017). Varieties of hermeneutical injustice. In J. Kidd, J. Medina, & G. Pohlhaus Jr. (Eds.), The Routledge handbook of epistemic injustice. Routledge. Peirce, C. S. (1868). Some consequences of four incapacities. Journal of Speculative Philosophy, 2, 140–157. Peirce, C. S. (1878). How to make our ideas clear. Popular Science Monthly, 12, 286–302. Peirce, C. S. (1932). In C. Hartshorne & P. Weiss (Eds.), Collected papers of Charles Sanders Peirce. (Vol II. Elements of Logic). Harvard University Press. Plunkett, D. (2015). Which concepts should we use?: Metalinguistic negotiations and the methodology of philosophy. Inquiry, 58(7–9), 828–874. Ramsey, F. P. (1927/1991). The nature of truth. In N. Rescher & U. Majer (Eds.), On truth: Original manuscript materials (1927–1929) from the Ramsey collection at the University of Pittsburgh (pp. 6–24). Kluwer Academic Publishers. Rawls, J. (1971). A theory of justice. Harvard University Press. Recanati, F. (2000). Oratio obliqua, oratio recta. The MIT Press. Recanati, F. (2003). Literal meaning. Cambridge University Press. Rosen, G. (2001). Platonism, semiplatonism and the Caesar problem. In R.  Hale & C.  Wright (Eds.), (pp. 229–244). Ryle, G. (1954). Dilemmas. Cambridge University Press. Salmon, N. (2018). Julius Caesar and the numbers. Philosophical Studies, 175, 1631–1660. Saul, J. (2012). Skepticism and Implicit Bias. Disputatio, 5(37), 243–263. Saul, J. (2019). (How) should we tell implicit bias stories? Disputatio, 10(50), 217–244. Schiffer, S. (1972). Meaning. Oxford University Press. Shapiro, L. (2020). Commitments of assertion. In S. Goldberg (Ed.), Oxford handbook of assertion (pp. 75–98). Oxford University Press. Sharp, K. (2013). Replacing truth. Oxford University Press. Soames, S. (2020). What we know about numbers and propositions and how we know it. Organon F, 27(3), 282–302. Tseronis, A., & Forceville, C. (2017). Argumentation and rhetoric in visual and multimodal communication. In A. Tseronis & C. Forceville (Eds.), Multimodal Argumentation and Rhetoric in Media Genres (pp. 1–24). John Benjamins Publishing Company. Walton, D. (1990). What is reasoning? What is argument? Journal of Philosophy, 87, 399–419. Williams, C. J. F. (1981). What is existence? Clarendon Press. Williamson, T. (1996). Knowing and asserting. Philosophical Review, 105, 489–523. Williamson, T. (2000). Knowledge and its limits. Oxford University Press. Williamson, T. (2007). The philosophy of philosophy. Blackwell Publishing.

Chapter 2

Groundbreaking Principles

Abstract  Several principles shape the pragmatist take on logic that I will defend in this book. The essential four are the Principle of Assertion (PA), the Principle of Propositional Priority (PPP), the Principle of Grammar Superseding (PGS), and the Principle of Inferential Individuation (PII). (PA) says that the bearers of logical properties are products of human discursive actions. (PPP) takes up the thesis that the primary bearers of logical properties are complete propositions. (PGS) advises not to look at the linguistic surface in search of logical insights. And finally, (PII) says that only inferential consequences, upstream and downstream, have any effect on propositional identity. In this chapter, I will explain these principles and their consequences and show that all of them can be traced back to Frege’s work. In addition, I will comment on the Fregean Principle of Context (PCont), a linguistic version of (PPP), and the semantic Principle of Compositionality (PComp), which Frege never stated. As a criterion of propositional identification, I propose the Organic Intuition (OI): ‘To be a proposition is to possess propositional properties.’ I then explain why (OI) offers a safe non-circular guide to detecting the presence of propositions, and finally compare genuine propositions with logical propositions. In the last section, I discuss the kind of discipline that logic is. Keywords  Assertion · Bearer · Compositional · Context · Grammar · Inferential [inferentialist · Inferentialism] · Logic · Organic intuition · Pragmatics · Propositional priority

2.1 Five Principles The pragmatist perspective that I will set out in the following pages makes meanings depend on human purposes and actions.1 Understanding the aim of logic involves, among other things, understanding the use that speakers make of logical 1  This claim makes my perspective ‘assimilationist’ in Brandom’s sense (Brandom, 2000, pp. 2ff.), i.e. it understands linguistic actions as continuous with a broader class of human (and maybe also non-human) activities. I do not have a clear position in this debate, though. In any case, inferential

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notions in the context of their rational practices. Rationality is thus a presupposition of my analysis, as it is of any pragmatist approach to meaning and logic. Agents’ rationality has to be tested by their willingness to honestly engage in cooperative enterprises. Speakers show the inferential connections between the propositional contents they entertain using logical terms, as I will discuss in Chap. 6. These inferential connections may rest on the inner nature of the propositions involved (as Frege says in his 1879, p. 5), or else on the general background of beliefs that speakers assume. In the first case, the connections are conceptual and necessary, in the second case they are contingent and possibly epistemic. But in both cases, the meaning of terms used to publicly present them, i.e. logical terms, is stable and determined. There are different characterisations of what it is to be a logical notion, as I will explain later in Chaps. 5 and 6, but all of them, from the Begriffsschrift on, share the intuition that logical notions do not contribute new concepts to the contents of assertions. It is also generally believed that logical notions help to build up complex propositional contents out of simpler ones. Frege explicitly assumes these theses regarding logical constants in the Begriffsschrift when he claims that logical constants help represent chains of inferences (Frege, 1879, p. 5), as well as in ‘On sense and meaning’ when he affirms that some conjunctions do not have a sense (see Frege, 1892, p. 172, n. 16), and in the whole of his paper ‘Compound Thoughts’ (Frege, 1923–1926).2 Throughout his life, he kept faithful to this view of logical constants, which I call ‘expressivist‘. The same expressivist approach is defended by Wittgenstein in the Tractatus (4.0312 and 6.1264). Frege and Wittgenstein display similar intuitions concerning the meaning and role of logical terms3; intuitions that are also evident in the works of most of the logicians who have shaped the discipline in the twentieth century, even if sometimes the specific explanations that they offer and the particular developments of the different logical theories that they put forward differ widely. In general, the role of logical terms makes obvious that logic can only do its job once propositional contents are available. Propositional contents are the logician’s raw material. It is in this sense that contemporary logic is a logic of propositions, and not a logic of terms like Aristotelian and Medieval

activities, as understood in this book, are activities that essentially use concepts and are thus ‘discursive’ or ‘linguistic’ in Brandom’s sense (Brandom loc. cit.). 2  This is not the standard interpretation of the meaning of logical constants in Frege’s work. The received view is that they are functions with a sense and a meaning, like any other functions. In this sense, the received view attributes to Frege a semantically homogeneous theory of meaning in which all terms are either names or functions, each with a sense and a meaning. Frege gives reasons to support this view, which should be understood in the context in which they were given. As a theoretical proposal for covering certain theoretical domains, it is not objectionable. Nevertheless, there are enough hints in Frege’s writings to argue that he was aware that logical terms do not work as ordinary predicates do, as I explain in Chap. 3. 3  The received view of the relations between Wittgenstein and Frege would reject this claim (see, for instance Hacker, 2001). I nevertheless consider Frege and Wittgenstein to be more congenial to each other than is traditionally (see, for instance, Wischin, 2019; Forero-Mora & Frápolli, 2021), and that the received interpretation of Frege’s writings (Baker & Hacker, 1984) does not do justice to the nuances about meaning that Frege addresses in his work.

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logics were, and also were the algebraic approaches developed at the end of the nineteenth century. Propositional Priority (PPP) and Grammar Superseding (PSG) define the rise of modern logic over the previous syllogistic model. The following is a possible formulation of (PPP): (PPP) The primary bearers of logical properties are propositions and sets of propositions.

Propositions are the products of assertive acts and the contents of judgements, in Frege’s sense. Thus, (PPP) is closely related to what we might call the ‘Principle of Assertion’ (PA): (PA) Assertion is the minimal act required to produce outputs with logical properties.

The Begriffsschrift begins with the introduction of the judgement stroke, ‘∣—’ (Frege, 1879, §2), an implicit acknowledgement of (PA). (PPP) and (PA) imply that logical activity requires discursive activity, i.e. that logical terms cannot emerge and do their job unless speakers are already proficient in the performance of discursive practices. Both (PPP) and (PA) are characteristically pragmatist principles: (PA) because it shows that in the beginning there were the practices, and (PPP) because it shows that complete propositions are more basic than concepts. The ‘priority of the propositional’4 is a consequence of the essentially pragmatist approach to meaning that Brandom summarises in the slogan ‘Semantics Must Answer to Pragmatics’ (Brandom op. cit., pp. 83ff.), and which I completely endorse. Propositions constitute steps or elements in inferences and are the contents of some linguistic acts. Nothing that cannot occur as a premise or a conclusion in an argument is a proposition; nothing that cannot be put forward as true in an assertive act is a proposition. These two claims represent (part of) this notion’s identification criteria. Their individuation criterion, on the other hand, is inferential. This is what the Principle of Inferential Individuation, (PII), states: (PII) Two sentences (uttered by an agent in context) express one and the same proposition if and only if their contents follow from the same set of propositions, and the same set of propositions follows from them.

Evidence of (PII) is seen in the inferential individuation that Frege establishes for judgeable contents in the Begriffsschrift, §3, when he provides the examples ‘The Greek defeated the Persian at Platea’ and ‘The Persian were defeated by the Greeks at Platea’ in order to insist that what they share is the only concern of his logical project. It is not uncommon among Frege scholars to acknowledge this inferential tendency in his early writings. Even Hacker, who generally considers Frege’s semantics to be clearly representationalist and compositional, makes room for a different view related to Frege’s first works (Hacker, 2001, p. 198, Wischin op. cit.). Brandom, on the other hand, has championed the revisionary interpretation of Frege which sees the ‘sage of Jena’ (Brandom, 2000, p.  45) as an inferentialist.

 This is Brandom’s version of (PPP) (Brandom, 1994, p. 79ff.), a principle that he attributes to Kant, in the first place, and then to Frege and Wittgenstein. 4

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Nevertheless, not even Brandom is completely clear about the temporal scope of Frege’s inferentialism, sometimes seeming to accept Dummett’s interpretation of the representationalist turn that Frege allegedly undertook around 1890 (Brandom, 2000, p. 51, Wischin op. cit.). As I read Frege, his support of (PPP) and (PII) was constant until the end of his published output, even though, from 1881 on, he took the risky semantic move of assimilating sentences to proper names (Brandom, 1994, p. 81), a move that makes it difficult to see the semantic specificities that derive from the complexity of sentences and allow us to explain inferential connections. I will come back to this issue in Chap. 7. (PII) is the principle that supports the whole argument of (Frege, 1923–1926) and, with respect to (PPP), he explicitly recognises as later in his life as in 1919, that this was one of the few insights that inspired his philosophical thought from the beginning to end: What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the word ‘true’, and then immediately go on to introduce a thought as that to which the question ‘Is it true?’ is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgment; I come by the parts of a thought by analyzing the thought. This marks off my concept-script from the similar inventions of Leibniz and his successors, despite what the name suggests; perhaps it was not a very happy choice on my part. (Frege, 1919, p. 252)

As this text shows, Frege considered (PPP) to be the principle that outlined his take on logic, and which sets it apart from the views of his contemporaries and predecessors. Brandom traces the first complete realisation of this principle back to Kant’s theory of judgement, and according to him the principle subsequently disappeared from the scene until Frege’s work (Brandom op. cit., pp. 79–80). Nevertheless, this is not completely accurate, since a version of (PPP) can also be seen in Bolzano’s Wissenschaftlehre (Bolzano, 1837) (see Dummett, 1991, p. vii, Kneale & Kneale, 1962, pp. 358ff, Sundholm, 2009, p. 269). The option for propositions has as its negative dual the rejection of sentences and, in general, linguistic expressions as the centre of logical analysis. This is what the (PGS) states. A possible formulation of this is: (PGS) Grammatical analysis is not a source of logical knowledge.

(PGS) makes explicit that the syntax of natural languages is not a reliable guide to the logical analysis of propositions. Grammar is not a guide to thoughts in the Fregean sense. Frege stuck to (PGS) throughout his entire life, from the Begriffsschrift, §3, to his Logical Investigations (see, for instance, Frege, 1918–1919b, p. 381). (PGS) sanctions a sharp divide between the level of abstract entities (concepts and propositions) and the perceptible level of their linguistic expressions. According to (PGS), logic is not concerned with languages, neither natural nor artificial, but rather with what we say by employing them in specific linguistic actions, i.e. with propositional contents, and their inferential connections. The relevance of some artificial languages, so-called ‘languages of logic’, rests in the possibility that they offer of representing in the syntactical surface logical

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statuses and logical relations. Neither the language that Frege introduced in the Begriffsschrift nor the first-order languages that derive from it, prescribe the higher-­ level nature of existence and generality, for instance. These are higher-level concepts because of the function they perform in our conceptual system. But logical languages make it easier to visualise their logical status, which is usually unclear in natural languages. The diverse logical calculi we are familiar with are scientific models that represent inferential connections between propositions in certain areas of discourse. Logic, in the sense in which I use the term in this book, is not a science but rather the inferential apparatus of our conceptual or linguistic system. The two contrasting answers to the question of whether logic is a science prompted the bifurcation of the paths the discipline took in the past century: one of them recognisable in Hilbert’s and Tarski’s work, and the other explicitly put forward by Frege. Both of these have given rise to disciplines that share the name ‘logic’. In Hilbert’s and Tarski’s tradition, logic represents a formal project. Frege’s, by contrast, belongs to what Coffa has called ‘the semantic tradition’ (Coffa, 1991, p. 64). This is the tradition I aim to vindicate. All these principles display deep connections with each other. Propositions are essentially assertable items and the bearers of logical relations. The individuation of propositions is inferentially determined, and sentences do not reflect the logical properties of their contents. Frege endorses these four principles and adds a further one, the Principle of Context (PCont), which parallels (PPP) at the linguistic level: (PCont) Only in the context of a sentence does a word have meaning.

Thus, according to (PCont), complete sentences take priority over words at the semantic level. Speakers express propositions through the utterance of complete declarative sentences,5 the only kind of item to which pragmatic force can be attached (Brandom, 1994, p. 82). Linguistic expressions shorter than sentences are not apt to make a move in the game that defines what it is to be a reason for something and, thus, what it is to be rational; Brandom calls it the ‘game of giving and asking for reasons’ (Brandom op. cit., p. 189). (PCont) first occurs in the Grundlagen (1884), although it is applied in the Begriffsschrift, §9, in the discussion of functions and arguments. Afterwards, it is stated, suggested or put to work in works that span Frege’s entire career, and is detectable in his logicist masterpiece, the Grundgesetze (Frege, 1893), as I will argue in Chap. 7. These five principles define Frege’s approach to logic (=semantics) and shape the pragmatist picture that I will develop in this book.

 Although, as Frege knew, neither all nor only sentences express propositions. Some subordinate sentences do not (Frege, 1892) and some questions and adverbs do (Frege, 1918–1919a). 5

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2.2 Two Models of Propositional Individuation Logic deals with propositions in two different senses that will be disclosed as the book proceeds: in the sense in which Frege introduces them in the Preface to the Begriffsschrift, i.e. as the steps in inferences that his conceptual writing purports to represent conspicuously, and also as the material for conceptual analysis. Fregean logic is also an instrument for determining the inferential potential of concepts, abstracted from propositions, and for identifying their logical category. We must thank Frege for the essential distinction between first-level and higher-level concepts, which opened the door to a logically correct characterisation of existence, generalisation, logical constants,6 and alethic and epistemic modalities. The kind of item that propositions are has been explained in the past century via two different models, the building-block model and the organic model (Frápolli & Villanueva, 2016). These two models correspond to the two modes of explanation— analytic and synthetic—that Belnap detects ‘throughout the whole texture of philosophy’ (Belnap, 1962, p. 130). The building-block model is guided by the Principle of Compositionality (PComp) and the organic model by the Principle of Propositional Priority (PPP), one of whose linguistic implementations is the Fregean (PCont). Here we find a possible formulation of compositionality: (PComp) The meaning of a complex expression is completely determined by its ingredients and how they are combined.7

(PComp) is an essential principle of semantic theory that accounts for the learnability, productivity, and systematicity of language. Frege explicitly states (PCont) in the Preface to (Frege, 1884) and also in §62, and, as Linnebø convincingly argues (Linnebo, 2019), it is also present in the Grundgesetze (Frege, 1893/2013, §§29–31). In contrast, there is no clear statement of (PComp) in Frege’s work (see Pelletier, 2001). Wittgenstein includes an exact replica of the Fregean (PCont) in the Tractatus 3.3, ‘Only the proposition has sense; only in the context of a proposition has a name meaning.’8 Both Frege and Wittgenstein sometimes express themselves as if they supported something close to (PComp), and so they have both been customary understood in this way. Nevertheless, their claims in this sense are unclear and compatible with alternative interpretations.

 Logical constants are first-order relations in Fregean theory since they are relations between propositions, which are saturated entities, and hence objects. This is a peculiarity of Frege’s semantics which does not affect the fact that they are relations between entities in which concepts can be identified. 7  See, for instance, (Pelletier, 1994). For an up-to-date discussion of different versions of the principle of compositionality and their scope, see (Frápolli & Villanueva, 2018). 8  By ‘proposition’ Wittgenstein means a sentence with its sense. Its content is, as in Frege’s view, a thought: ‘In the proposition the thought is expressed perceptibly through the senses’ (Tractatus 3.1). The mere linguistic object is the propositional sign (Tractatus 3.12), and ‘the applied, thought, propositional sign is the thought’ (Tractatus 3.5). 6

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In ‘Compound Thoughts’, Frege claims: It is astonishing what language can do. With a few syllables it can express an incalculable number of thoughts, so that even a thought grasped by a human being for the first time can be put into a form of words which will be understood by someone to whom the thought is entirely new (Frege, 1923–1926/1984, p. 390).

Wittgenstein’s version occurs in the Tractatus 3.2: ‘In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign.’ According to the building-block model, propositions are built up out of some pre-existing constituents. This feature defines what Brandom dubs ‘conceptual platonism’ (Brandom, 2000, p. 4). Depending on different authors and schools, these constituents are a combination of concepts, or else of concepts and objects. Russell included objects as ingredients of singular propositions (Russell, 1910, pp. 178ff., Russell, 1919), which hence are called ‘Russellian propositions’ (Pelham & Urquhart, 1995). In the building-block model, propositions are structured entities whose individuation criteria take structure into account. Two propositions are one and the same if, and only if, they contain the same ingredients organised in the same way. The two propositions Desdemona loves Cassio and Cassio loves Desdemona share their ingredients but differ in how they are combined. They are thus two different propositions. The propositions Desdemona loves Cassio and Cassio is loved by Desdemona also share their ingredients and, although they are equivalent, their differences in structure make them different propositions. Note that, in their standard formulations, (PComp) and (PCont) primarily apply to expressions, not to their contents. Expressions are linked to particular languages; their contents, in contrast, are abstract entities that are expressible in different linguistic systems. A presupposition of the building-block model of propositional individuation is that propositions are sentence-like entities that reflect the structure of the sentences by means of which they are expressed. Even Frege, who never relied on a standard formulation of (PComp),9 sometimes expressed himself as if he believed in a structural similarity between sentences and thoughts (see, for instance, Frege, 1918–919b, p. 378). This presupposition is nevertheless far from straightforward and there are solid arguments for Fregean propositions being unstructured (Pérez-Navarro, 2020). Furthermore, given the remarkable variety of natural languages and the fact that all of them are inter-translatable, and assuming a general effability principle (Carston, 2002, p. 33), this presupposition would require that all

 (PComp) has been called ‘Frege’s Principle’, and it is customary to read it into Frege’s substitutional argument for distinguishing sense from meaning in 1892. Nevertheless, as we have mentioned, there is no explicit formulation of this principle in Frege’s work (Pelletier, 2001), and it has a problematic coexistence with the Fregean Principle of Context. The compatibility of these principles has been profusely debated (see, for instance, Dummett, 1995; Hale, 1994). My position is that, as they are stated, they do not interfere with each other (Lewis, 1980; Linnebo, 2008, 2019, p. 104). 9

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languages somehow shared a similar deep structure.10 There is no suggestion that Frege ever adopted such a strong hypothesis. The linguistic level, the level at which (PComp) and (PCont) apply, has to be distinguished from the content level, the level of what is said or expressed by the use of language. Even if (PComp) is generally accepted as a basic semantic principle,11 at the content level the principle does not necessarily hold. (Lewis, 1980) is an example of an approach to meaning that assumes (PComp) as an essential aspect of any semantics for English (Lewis op.  cit., p.  82), and at the same time rejects ­compositionality for the propositional content of sentences (Lewis op. cit., p. 95). Semantic compositionality prompts a level of semantic content that can be said to share with sentences some structural similarities. If words in sentences have senses, then their combination gives rise to a complex abstract entity that mirrors the sentential form. This would be a sense of the Fregean notion of sense in which senses would be structured entities. I will come back to this point later in the chapter. Nevertheless, these structured senses would fall short of being judgeable contents.12 The alternative to the building-block model is the organic model (Frápolli and Villanueva op. cit., Frápolli, 2019). In the organic model, propositions are individuated by their properties and relations, without taking their virtual ingredients into separate consideration. According to the organic model, propositions are non-­ structured entities. The unstructured nature of propositions does not preclude that some structure or other could be imposed on them by means of analysis. Being non-structured does not mean being non-articulable or non-analysable. The Wittgensteinian image of the contrast between propositions as complex and names as simple is a powerful one, which Frege partially squandered by understanding sentences as names in his mature period. Still, Fregean thoughts are essentially analysable by their inferential properties, although their analysability does not imply being made out of parts in any literal sense. As Bronzo argues, ‘the assumption that essential articulatedness implies unique articulation’, as he puts it, only stands ‘if we conceive of the relation between a thought and its parts in accordance with an atomistic model of the part-whole relation, but can be resisted if we adopt an alternative, organic model’ (Bronzo, 2017, p. 25). And Bronzo interprets Frege’s thoughts along the lines I am developing here. In one of his Logical Investigations, Frege issued a comment that put the debate about the nature of propositions (which he called ‘thoughts’ at this time) in the right light: If one thought contradicts another, then from a sentence whose sense is the one it is easy to construct a sentence expressing the other. Consequently the thought that contradicts another  Chomsky’s Innateness Hypothesis makes such an assumption. The Innateness Hypothesis postulates a specific language organ in the human brain in order to explain how children are all able to develop a mastery of their languages, regardless of the complexity of the task and the ‘poverty of stimuli’ (see Chomsky, 1988, pp. 46–7; Pinker, 1994, pp. 237–8). 11  With some illustrious exceptions. (Hintikka & Sandu, 2007) is one of them. 12  This is one of the central tenets of Truth Conditional Pragmatics, the view defended by Recanati (2002, 2003) and Carston (2002). 10

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thought appears as made up of that thought and negation […]. But the words ‘made up of’, ‘consist of’, ‘component’, ‘part’ may lead to our looking at it the wrong way. If we choose to speak of parts in this connection, all the same these parts are not mutually independent in the way that we are elsewhere used to find when we have parts of a whole. The thought does not, by its make-up, stand in any need of completion; it is self-sufficient. (Frege, 1918–1919b, p. 386)

A thought is a self-sufficient entity.13 It is not made out of parts, and it does not stand in need of completion. It is the primary result of an act of assertion, which is the expression of an act of judging.14 The judgement sign, ‘∣—’, indicates that what comes after it is the content of a judgement. Without the vertical stroke, the judgement becomes a ‘mere combination of ideas’. In assertion, ideas which were independent, and which could be represented by predicables and common nouns, become an organic unity. The contents of judgements are propositions, or thoughts, and they are ‘what alone matters to’ Frege in his Begriffsschrift (Preface, p.  6). Expressed thoughts acquire the structure of the sentences by means of which we express them. But that those structures do not belong to the thoughts themselves is seen in the fact that the same thought can be expressed using sentences with different forms, different words, and even belonging to different languages. Grammatical issues, such as the choice of the active or passive voice (and thus the choice of which concepts appear related to the grammatical subject, and which to the grammatical predicate) or, in analysis, which parts of a content should be taken as fixed and which as variable, are alien to logical aims and are usually motivated by pragmatic reasons (Frege, 1879, Preface, §3 and §9). The text just quoted points to something even more basic that connects logic with common sense: that having parts is not something that can be said of abstract entities. Abstract entities admit of analysis, but they can neither be broken down into nor built up out of pieces. They are not physical objects. Frege’s Logical Investigations (Frege, 1918–1919a, b, 1923–1926) are rife with arguments and hints about the unstructured nature of propositions. It is an acceptable, and sometimes very useful, fiction to talk of abstract entities as if they were physical objects. But this methodological fiction should not make us lose sight of the fact that attributing parts to non-physical entities is a category mistake. A similar category mistake occurs when we ask about the realm in which abstract entities (numbers, propositions) stand, as realists (approvingly) and anti-realists (disapprovingly) usually do. Numbers and propositions are not anywhere; they have no spatio-temporal location. The ‘domain of what is objective’ (Frege, 1893, pp. 15–6) or the ‘third realm’ (Frege, 1918–1919b, p. 363) that Frege reserves for  This claim does not imply that thoughts stand anywhere. The sometimes too quick metaphysical consequences drawn from some semantic and logical claims result from the practice of understanding metaphorical discourse as if it were literal. 14  The connection between propositions and assertion is not contingent but a matter of logical grammar. There are propositions because humans engage in acts of assertion, and acts of assertion are essentially acts in which a proposition is put forward. Nevertheless, this by no means implies that all possible propositions must be asserted. What it means is that propositions are assertables, claimables, thinkables, as I have explained in Chap. 1. 13

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them are not physical places, but rather the acknowledgement of their objectivity. This also hints at the specific status that sets them apart from physical objects and from private psychological impressions that can be modified at will. The confusion between what is objective in the conceptual sphere with those external objects that are represented by mental representations is the ‘proton pseudon’ (the first lie), Bolzano contended, of idealist philosophy (Coffa, 1991, p.  31). This is also the proton pseudon of all empiricisms. The domain of what is objective is neither reducible to mental representations nor to physical configurations. Abstract entities are similar to psychological entities in that they cannot be perceived, and similar to physical objects in that they do not need a bearer, as Frege explains again and again (see, for instance, Frege, loc. cit.). Therefore, logic and semantics, whose subjects are concepts and propositions, are neither psychology nor physics. Numbers and propositions are objective; their properties cannot be invented or forged. Two plus two is four, and from that Joan is a computing science student it follows that Joan is a student, and this does not depend on any individual will. Sentences, by contrast, do have parts: they have a detectable structure governed by the rules of grammar15 and, according to the best grammatical theories we know of, they are (mostly) governed by the Principle of Compositionality. It is of the utmost importance for the philosophy of logic and the philosophy of language to draw a sharp distinction between linguistic expressions (sentences, names, predicates, etc.) and the abstract entities that speakers entertain by using them. Unfortunately, the use of ‘proposition’ to refer to both levels is a continuous source of confusion. We philosophers are not linguists; we are not concerned with specific linguistic systems. As Frege explicitly stated in the Preface of his Begriffsschrift, the only concern of logicians is (or should be) the contents of assertive acts, actual or virtual, and their relations, i.e. propositions and inferences, all of them abstract entities. Frege introduces the notion of thought precisely to mark the distinction between what is said, known, believed and discussed, and the linguistic clothing that makes it perceptible. As (PGS) states, not all operations on linguistic expressions have an effect on their contents; some linguistic operations are logically idle: A sentence can be transformed by changing the verb from active to passive and at the same time making the accusative into the subject. In the same way we may change the dative into the nominative and at the same time replace ‘give’ by ‘receive’. Naturally such transformations are not trivial in every respect; but they do not touch the thought, they do not touch

 A different debate is whether sentences are abstract entities. The sentences that we use in our actual linguistic exchanges are not; they are either traces in a surface, or sound waves. We call them ‘sentence-tokens’. From tokens, types can be abstracted. Two utterances of ‘Joan is a computing science student’ produce two different tokens that belong to the same sentence-type. Sentence-­ types are entities abstracted from their tokens. Given the (at least basic) compositional nature of the grammar of natural languages and the identity criteria for sentences, we are justified in applying the discourse of parts and wholes to them. There are nevertheless other abstract entities that are articulated in a specific sense, but whose articulation cannot be explained by resorting to any particular abstraction process. This is the case, for instance, for scientific theories and musical works, both of which are undoubtedly articulated abstract entities. The question at this point is whether they are made of parts, in any non-metaphorical sense. My contention is that they are not. 15

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what is true or false. If the inadmissibility of such transformations were recognized as a principle, then any profound logical investigation would be hindered. It is just as important to ignore distinctions that do not touch the heart of the matter, as to make distinctions which concerns essentials. But what is essential depends on one’s purpose. To a mind concerned by the beauties of language, what is trivial to the logician may seem to be just what is important. (Frege, 1918–1919a, p. 358, italics added)

Frege suggests here a test to distinguish the two models of propositional individuation. Consider again the two examples mentioned above, Desdemona loves Cassio and Cassio is loved by Desdemona. These sentences are different; they have different subjects and different predicates, the first is in active mode, the second in passive mode. According to the building-block model, these two sentences express different propositions, even if they are logically equivalent. In the organic model, in contrast, these two examples are expressions of one and the same proposition; these transformations ‘do not touch the thought’. In the organic model, there cannot be logically equivalent sentences that express different thoughts. This is the analytic equivalence test that discriminates between both models (Frápolli & Villanueva, 2016). The nature of thoughts is not homogenously characterised in Frege’s writings, as I have suggested above. The notion explained in the text just quoted corresponds to what he calls a ‘judgeable content’ in the Begriffsschrift and to what we call here ‘propositional content’ and ‘proposition’. But Frege also points to a different layer of sense that is also covered by the term ‘thought’. In (Frege, 1918–1919b, p. 373), thoughts are the senses of declarative sentences and of propositional questions. This way of characterising thoughts supports the thesis of Fregean thoughts as structured entities, the structure determined by the senses of the words that occur in the sentences and questions concerned. Several authors have acknowledged the tension between these two alternative characterisations of Fregean senses (Penco, 2003; Bronzo, 2017). Penco, in particular, insists that, although Frege was aware of this tension, he did not have the tools to solve it satisfactorily. Frege’s awareness supports my reading of Fregean semantics as being essentially more nuanced than what follows from the received view. The distinction between compositional meaning and an essentially richer content is broadly accepted among contemporary linguists and philosophers of language. This is what Kaplan sometimes16 points to with his distinction between character and content (Kaplan, 1977); it is the distinction thematised by (Recanati, 2003) under the labels ‘literal meaning’ and ‘what is said’,17  Kaplan’s content, one of the two types of meaning that he identifies in indexicals (Kaplan 1979, p. 500), is standardly seen as a contextual ‘enrichment’ of the character of a term, being the character the other kind of meaning that indexicals possess. In this sense of ‘content’, the contents attributed to sentences would be compositional and structured. But Kaplan’s use of ‘content’ is ambiguous: when applied to complete sentences, Kaplanian contents become assertoric contents. This happens, for instance, when he speaks of contents as the bearers of modal and temporal properties (Kaplan, op.  cit., p.  501). In this latter sense, contents are neither linguistic entities nor enrichments of them. 17  Recanati’s what is said, individuated by the Availability Principle, is essentially richer than what follows from the sentence’s linguistic meaning plus saturation (see Recanati, 2003, pp. 20ff.). 16

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and the contrast that (Rabern, 2017) discusses using the terms ‘semantic value’ and ‘content’. Frege was able to pinpoint this distinction, as it can be seen in the following text: With the word ‘true’ the matter is quite different. If I attach this to the words ‘that sea-water is salty’ as a predicate, I likewise form a sentence that expresses a thought. The thought expressed in these words coincides with the sense of the sentence ‘that sea-water is salt’. So the sense of the word ‘true’ is such that it does not make any essential contribution to the thought. If I assert ‘it is true that sea-water is salt’, I assert the same thing as if I assert ‘sea-­ water is salt’. This enables us to recognize that the assertion is not to be found in the word ‘true’, but in the assertoric force with which the sentence is uttered. This may lead us to think that the word ‘true’ has no sense at all. But in that case a sentence in which ‘true’ occurred as a predicate would have no sense either. All one can say is: the word ‘true’ has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate. (1979, pp. 251–3)

As this passage makes explicit, some words (‘true’ in this case) can be meaningful, i.e. possess a semantic value, without adding any information to what is said. The acknowledgement of this subtlety of the notion of meaning did not lead Frege, nevertheless, to the identification of two clearly differentiated notions in his system.

2.3 Propositional Identification So far, I have discussed propositional individuation, i.e. how to determine whether two contents are one and the same. But the task of recognising when we are dealing with propositions at all is a prior one. As mentioned in Chap. 1, I have proposed (Frápolli, 2019, p. 89) that the Organic Intuition, (OI), is the required identification/ recognition criterion: (OI) To be a proposition is to possess propositional properties and stand in propositional relations.

This criterion is not circular as long as the propositional properties and relations are listed independently. And here we are in luck since, even if the (OI) is surely contested by more traditional philosophers who stick to compositionality and structure, the list of propositional properties and relations is widely accepted. I mentioned some of them at the beginning of this section. The first that comes to mind is that propositions are truth-bearers. Frege, for instance, defines propositions (thoughts) as those entities for which truth and falsehood enter into account (Frege, 1918–1919a, p. 353). One can debate whether they are the primary truth-bearers. Unfortunately, since (Tarski, 1935/1956), sentences have been the favoured items to be the primary bearers of truth. I will follow Frege in attributing truth to judgeable contents, i.e. propositions, and also endorse Ramsey’s claim that ‘sentences are not serious candidates’ in this contest (Ramsey, 1927/1991, p. 10). But even if sentences are said to be true or false in a primary or secondary sense, nobody doubts that what we say by means of them—the propositions expressed—are true or false as well.

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Moreover, propositions are the arguments of logical connectives. This claim is part of what is at issue in the Frege-Geach argument against non-descriptivisms: because normative and ethical claims can fall under the scope of negation and conditional, they must express propositions (see Frápolli, 2019). Logical connectives connect and compose truth bearers and are defined by reference to truth values. They are truth functions. We can discuss whether their arguments are actually truth values or the propositions of which these values are values. And Frege, by reinterpreting sentences as names in his mature period, put this debate on the table. But truth values are essentially connected to sentences and their contents, and thus nothing less than a complete truth bearer can play the role of a connective’s argument. I will come back to this point in Chap. 6. Were propositions merely secondary truth-­ bearers, truth functions could still be used to recognise their presence. Propositional-attitude verbs call for a that-clause that introduces a proposition. ‘Know’, ‘believe’ and ‘doubt’ are examples of propositional attitude verbs. In the more standard explanation, they express a relation that holds between an individual and a propositional content. We can discuss whether these verbs relate agents and propositions as objects, but it is beyond question that these verbs point to propositions. Ramsey characterised the whole type by the feature of having ‘propositional reference’ (Ramsey, 1927/1991, p. 7). Thus, according to the mainstream philosophy of language and mind, what agents believe, know, doubt, consider, reject, deny, assume, postulate, etc. are propositions. Verbs that introduce indirect speech—‘say’, ‘declare’, ‘state’, ‘assert’, etc.— constitute a further category of verbs with propositional arguments (Strawson, 1950, p. 57). By saying something, we make explicit our beliefs.18 Frege used assertion to introduce judgeable contents (Frege, 1879, §2) and stated that assertion is the expression of a judgement (Frege, 1918–1919a, p.  356). His distinction between judgeable and non-judgeable contents in the Begriffsschrift established the distinction between propositions—the entities that can be judged and asserted—and other abstract entities that fall short of this task. Propositions are thus what we say, in a specific sense of ‘saying’. Finally, propositions are the entities that perform the roles of premises and conclusions in arguments. As with truth-bearers and logical connectives, the debate about the relative priority of sentences and propositions does not affect the fact that propositions imply and preclude other propositions; that once some have been put forward, some others cannot be denied and still some others cannot be assumed. My take on logic makes propositions the basic entities of logical theory. And truth, assertion, logical connectives, propositional attitude verbs and inferences unequivocally mark the presence of propositions. (OI) is neutral between the two models of propositional individuation, the building-­block and the organic model. Propositions, once pinpointed, can be individuated by their ingredients and organisation or else by their properties and  It is sometimes argued that a speaker can assert something that she does not believe. This is a mistake. Believing is a presupposition of assertion, as Grice’s maxims make explicit. In Chap. 10, I will develop this point through the notion of circumstance-shifting operator. 18

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relations. That is, they can be defined by a constructive movement from their simpler parts, or else ‘from above’, by their properties and the relations they stand in. (OI) thus belongs to the meta-theory of a theory of propositions. Nevertheless, the most natural path, once (OI) has been assumed, is to develop it into an individuation criterion as well. I will follow this path; the path inaugurated by Bolzano and continued by Frege and the pragmatic tradition. Nothing less than a proposition makes a move in a linguistic game. This is also Wittgenstein’s view and, later on, Sellar’s and Brandom’s positions.

2.4 Logical Propositions Logical propositions are not ordinary propositions, and logical truths are not ordinary truths. In the expressions ‘logical proposition’ and ‘logical truth’, ‘proposition’ and ‘truth’ are used in a derivative sense. Ordinary propositions are truth-bearers; they are the products of successful acts of assertion, the items that we know, doubt, reject, intend to establish, etc. They are the premises and conclusions of material arguments, the steps of scientific proofs, the starting points and the ends of deliberations, the contrasting positions in controversies, etc. Truths are propositions we endorse and are prepared to back (see Chap. 10). Logical truths, by contrast, can only be the premises of logical proofs, i.e. of proofs whose job is to show how certain logical principles can be reached from other logical principles. ‘It would be too remarkable’, Wittgenstein said in the Tractatus (6.1263), ‘if one could prove a significant proposition logically from another, and a logical proposition also. It is clear from the beginning that the logical proof of a significant proposition and the proof in logic must be two quite different things.’ Logical propositions represent structures but do not express thoughts. It is not completely clear what kind of item logical propositions and logical truths are. They are sometimes understood as formulae of logical languages, like (1), and sometimes as ordinary sentences with a specific structure, like (2): (1) p v ¬p. (2) Either it’s raining or it’s not. In standard logical theory, logical truths are complex sentences/formulae, i.e. sentences/formulae that include logical terms that connect their sentential parts. But some authors talk of logical truth in connection with sentences in which no logical term occurs. This is the case of Kaplan, who considers (3), (3) I exist.

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to be a truth of logic (Kaplan 1979, p. 540). Kaplan’s deviant usage makes patent the disparity in bearers between logical and ordinary truths, since (3) is a logical truth that is not necessary. Every time that (3) is uttered, it produces a true proposition, but the proposition produced is contingent. The distinction involved is that between the linguistic expression and the content expressed, a difference that Frege wanted to mark by the introduction of the pair of terms ‘sentence’ (Satz) and ‘thought’ (Gedanke, Sinn), and Wittgenstein by the pair ‘propositional sign’ and ‘thought’ (Tractatus 3.5). The standard practice of making sentences the primary truth-bearers, as the mainstream analytic philosophy has done in the past century, dilutes the crucial difference in nature between truths and logical truths. A standard characterisation of logical truths is what Etchemendy has called the ‘substitutional/interpretational’ account (Etchemendy, 1983, p. 236). In the substitutional account, logical truths are skeletons in which the places to be filled by ordinary propositions are marked by propositional variables, while their structure is determined by logical constants. Logically valid arguments are characterised in a similar way. The substitutional/interpretational account associates every ordinary true sentence and every truth-preserving argument—i.e. one which has either false premises or a true conclusion—with a class of sentences or arguments. The associated class is built up from the corresponding true sentence or truth-preserving argument by performing appropriate, systematic and homogeneous substitutions (or interpretations) of their variable expressions. A sentence is said to be a logical truth, and a set of sentences a logically valid argument, if and only if all members of its associated class are true/valid (Etchemendy, loc. cit.). These definitions of logical truth and logical validity occur in virtually every logical handbook (see, for instance, Sagüillo, 2007, pp. 68–69). Bolzano’s Wissenschaftlehre seems to be the origin of this account (Etchemendy loc. cit.), which has its definitive and better-known version in (Tarski, 1936) (see, for instance, Sagüillo, 1997). As the set of logical truths is a subset of the broader set of analytic truths, analytic truths can also be characterised by the same substitutional/interpretational method. Analytic truths are sentences that are true by virtue of the meaning of some of their terms (Hintikka & Sandu, 2007, pp.  14–15). When the terms responsible for their truth are logical terms, the sentences concerned are said to be logically true. Two features of the characterisation of logical truths just given are worth noticing. The first one is that the kind of entity to which we ascribe logical and analytic truth is not the kind of entity to which we ascribe ordinary truth. The second feature is that the characterisation of logical truths and logically valid arguments rests on a prior distinction between logical and non-logical terms. This second feature will be discussed in Chap. 5, but I will make some relevant comments at the end of the next section.

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2.4.1 Three Alternative Approaches Leaving aside Kaplan’s sense, logical truths have traditionally been characterised from two alternative points of view. The first is Russell’s proposal that logical (and mathematical) truths are maximally general. The second follows Wittgenstein’s thesis that logical ‘propositions’ are uninformative. At the time of The Principles of Mathematics, Russell considered mathematics to be part of logic, and thus the characterisation of mathematical propositions affected his view of logical propositions: But for the desire to adhere to usage, we might identify mathematics and logic, and define either as the class of propositions containing only variables and logical constants. (Russell, 1903, p. 11)

Ramsey offered a knockdown counterexample to Russell’s approach. The sentence ‘Any two things differ in at least thirty ways’ is completely general. Still, even if it might well be true, this would be hardly regarded as a logical truth (Ramsey, 1925/1990, p. 167). For Wittgenstein, on the other hand, what distinguishes logical propositions from the rest is their absence of content. Logical propositions are either tautologies (Tractatus 6.1) or contradictions (Tractatus 6.1202), and they do not say anything (Tractatus 6.11). The defining contrast between logical and genuine propositions is stated in (Tractatus 6.113): It is the characteristic mark of logical propositions that one can perceive in the symbol alone that they are true; and this fact contains in itself the whole philosophy of logic. And so also it is one of the most important facts that the truth or falsehood of non-logical propositions can not be recognized from the propositions alone.

Only ordinary propositions have a sense. Wittgensteinian logical propositions are self-neutralising combinations of propositional variables and logical constants. This Wittgensteinian take stresses that logical propositions, because of the configuration of logical constants in them, are either necessarily true or necessarily false. These logical propositions are not genuine propositions, but by-products of logical combinatory processes: The propositions of logic demonstrate the logical properties of propositions, by combining them into propositions which say nothing. This method could be called a zero-method. In a logical proposition, the component propositions are brought into equilibrium with one another, and the state of equilibrium then shows how these propositions must be logically constructed. (Tractatus 6.121)

These neutralised pseudo-propositions are not needed either in science or in everyday life. By saying nothing, logical truths cannot trigger the movement that an inference consists in. ‘Whence it follows that we can get on without logical propositions, for we can recognize in an adequate notation the formal properties of the propositions by mere inspection’ (Tractatus 6.122). Logical propositions only offer information about the linguistic system in which they are represented or, as Wittgenstein would have it, of any possible linguistic system.

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These two approaches, Russell’s and Wittgenstein’s, are attempts to give scientific precision to the correct intuition that logical propositions are contentless, and both attempts were developed along with different accounts of the kind of enterprise that logic is. I will discuss this point in the next section. A third position, less frequently pursued, is the distinctively pragmatist view that logical propositions codify principles of reasoning. The distinction between premises and principles of reasoning is the core of Lewis Carroll’s outstanding paper ‘What the Tortoise Said to Achilles’ (Carroll, 1895), and is also detectable in Wittgenstein’s Tractatus: The significant proposition asserts something, and its proof shows that it is so; in logic every proposition is the form of a proof. Every proposition of logic is a modus ponens presented in signs. (And the modus ponens cannot be expressed by a proposition.) (Tractatus 6.1264)

The combination of (Tractatus 6.234), ‘Mathematics is a method of logic’, and (Tractatus 6.211), In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics others which equally do not belong to mathematics,

points to a similar position. The disparity between genuine and logical propositions makes Wittgenstein claim that it is ‘clear why logic has been called the theory of forms and inference’ (Tractatus 6.1224). Frege’s view of logical terms as marking the links in an inferential chain and Brandom’s view of logical terms as making inferential commitments explicit strongly suggest this view of logical truths. The explicitly stated purpose of the Begriffsschrift shows that Frege considers the role of logic to be that of signalling and testing transitions between propositional contents. These contents are the stuff on which logic performs its task. The Begriffsschrift, the first treatise of contemporary logic, does not contain, in the words of its author, ‘new truths’ (Frege, 1879, p. 13), since the role of logic is methodological. A logical notation like that of the Begriffsschrift is a method of representation, and logic is a method of proof. Brandom, in turn, explains the role of logical terms as being that of making explicit what is implicit in our linguistic inferential actions (Brandom, 1984, p. 115). I will develop Frege’s and Brandom’s insights in Chaps. 5 and 6. The second feature of the substitutional/interpretational procedure that I mentioned above is that any characterisation of logical truths presupposes a distinction between logical and non-logical terms. This is old news. Tarski stated the difficulty at the end of his paper ‘On the Concept of Following Logically’ (Tarski, 1936): In order to realize the importance of the question under consideration from the point of view of certain philosophical conceptions, it is necessary to direct one’s attention to the fact that the division of terms into logical and extra-logical exerts an essential influence on the definition also of such terms as “analytic” or “contradictory”; yet the concept of an analytic sentence –in the intention of some contemporary logicians– is to be a precise formal correlate of the concept of tautology as a sentence which “says nothing about the real world”, a concept which to me personally seems rather murky but which played and still plays a prominent role in the philosophical speculations of L. Wittgenstein and almost the whole

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2  Groundbreaking Principles Vienna Circle. Clearly, further investigations may throw a lot of light on the question which interests us; perhaps one will succeed with the help of some weighty arguments of an objective character in justifying the dividing line traced by tradition between logical and extra-­ logical terms. Personally I would not be surprised however even if the result of these investigations were to be decidedly negative and if hence it would turn out to be necessary to treat such concepts as following logically, analytic sentence or tautology as relative concepts which must be related to a definite but more or less arbitrary division of the terms of a language into logical and extra-logical; the arbitrariness of this division would be in some measure a natural reflection of that instability which can be observed in the usage of the concept of following in everyday speech. (Tarski, 1936, p. 196)

This is an impressive text which shows the depth of Tarski’s understanding of logic. He was not misled by the artificial clearness and precision that a part of analytic philosophy imposed on basic concepts. The divide between logical and non-­ logical notions is context-dependent and gradual in everyday speech, and scientific theories can only make it sharp by tolerating some degree of arbitrariness. Etchemendy and, from a completely different background, Brandom showed that the standard method of fixing some terms while turning the rest into variables, i.e. the ‘substitutional/interpretational’ method, does not in the least illuminate the notions of logical term and logical structure (Etchemendy, 1983, p. 326, Brandom, 2000, p. 55). Some distinction between fixed and variable terms is needed in order to determine the structures that somehow unify sets of valid material arguments, but this distinction is not enough to define logically valid arguments. The method was used by syllogistic logic with a different purpose, and this shows that the method by itself says nothing about the nature of logical validity. To characterise logical validity and logical truth, the identification of what exactly is what makes logical terms logical is needed. And this can only be done by attending to their specific function in language, as I will argue in Chap. 6. Any other way is, as the history of logic shows, doomed to failure.

2.5 Logic as a Science Is logic a science among others? Most philosophers would argue that it is not. This negative answer does not settle the question, though; a positive answer about the place of logic in the system of knowledge is also needed. Jean van Heijenoort famously distinguished between a conception of logic as a calculus and a conception of logic as a language (Van Heijenoort, 1967). Korhonen, more recently, casts this intuitive distinction in terms of a universalist conception of logic, which he attributes to Frege and Russell, and a conception of logic as a theory (Korhonen, 2012). The way in which logicians have understood their task is very varied. Frege put forward an instrumental approach to logic, Russell considered logic to be maximally general. Tarski defended these two views in turn and Wittgenstein interpreted logic as transcendental. In its connection with other sciences, logic can be either the queen or the assistant; the science at the top of the system or else an instrumental aid for assessing inferences. These two alternatives are not neutral, and they carry with

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them significant constraints on the definition of logical constants, as well as on the foundations of the truth of logical theories and the epistemology of logic. Here we have the two contrasting pictures, the logic-as-queen picture and the logic-as-assistant picture. In the logic-as-queen picture, logic is a genuine science whose concern is reality and its most general features. Logical terms are, in turn, denoting expressions that occur in its true sentences. The truths of logic reflect the most abstract descriptions of the real and of the possible, and access to them proceeds either by abstraction or intuition. Logical propositions are, in this picture, genuine contentful propositions, whose difference with the propositions of other sciences is merely one of degree. This seems to have been Aristotle’s view, although the exact interpretation of the scope of Aristotle’s first principles is uncertain, as it is not clear whether Aristotle was establishing metaphysical principles or rules to be used in inferences and argumentations (Kneale and Kneale op. cit., p. 25, pp. 46ff.). In modern times, this view is defended in Russell’s The Principles of Mathematics and in Tarski’s ‘What are Logical Notions?’. Russell’s definition of mathematical propositions as maximally general, as has been mentioned in the previous section, together with his logicist approach to the relations between mathematics and logic (see, for instance, Russell, op. cit. p. 8) speak for his substantive position on the nature of logic. Tarski’s view of logic as a maximally general science derives from his approach to logical notions as invariant under all transformations of the universe onto itself (Tarski, 1986, p. 149), in the spirit of the Erlangen Program and of Felix Klein’s strategy for defining geometrical terms. His definition does not grant logical notions any special status: I shall try to extend [Felix Klein’s] method beyond geometry and apply it also to logic. I am inclined to believe that the same idea could also be extended to other sciences. Nobody so far as I know has yet attempted to do it, but perhaps one can formulate using Klein’s ideas some reasonable suggestions to distinguish among biological, physical, and chemical notions. (Tarski, op. cit., p. 146)

Logic becomes then a science like any other, albeit with the highest degree of abstraction, whose defining notions denote objects, as happens with other knowledge systems that seek to explain the world: I shall not discuss the general question ‘What is logic?’. I take logic to be a science, a system of true sentences, and the sentences contain terms denoting certain notions, logical notions. (Tarski op. cit., p. 145)

The alternative, logic-as-an-assistant picture, takes logic to be a transversal instrument for representing inferences and a step for assessing the validity of any inferences that are explicitly and exhaustively represented. In his Begriffsschrift, Frege acknowledges the instrumental character of his conceptual-script: This ideography, likewise, is a device invented for certain scientific purposes, and one must not condemn it because it is not suited to others. If it answers to these purposes in some degree, one should not mind the fact that there are no new truths in my work. (Frege, 1879, p. 6)

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There are no new truths in the Begriffsschrift, either logical or otherwise, if by ‘truths’ we understand—as Frege does in this context—facts or contents that we endorse (Frege op. cit., p. 13). The unique nature of logic is central to Wittgenstein’ Tractatus, where logic becomes ‘transcendental’ (Tractatus 6.13). It is not my concern here to discuss the many peculiarities of Wittgenstein’s approach to logic in his first masterpiece. For my purposes, it suffices to understand that logic is not an ordinary science, not even the first one. ‘Logic is not a theory but a reflection of the world’, he says in the first part of (Tractatus 6.13). And then, in the next aphorisms: ‘Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-­ propositions’ (Tractatus 6.2), and ‘Mathematical propositions express no thoughts’ (Tractatus 6.21). Logic, philosophy (Tractatus 4.111, Tractatus 4.112), mathematics, and ethics (Tractatus 6.421) are not theories, and thus, they cannot stand ‘beside the natural sciences’ (Tractatus 4.111). It is surely misleading to say that in the Tractatus logic is the assistant, as it is in the Begriffsschrift. But it undoubtedly is not primus inter pares, and it does not consist of a collection of true sentences or propositions, as in (Tarski, 1986). In logical truths, according to Wittgenstein, logical terms do not denote. They do not stand for substantive concepts, but for transitions on propositions, or for operations on truth-­ values. Wittgenstein famously declared that his ‘fundamental thought [was] that the “logical constants” do not represent’ (Tractatus 4.0312). Non-representationalism is a negative semantic thesis that, when applied to the meaning of logical terms, as Wittgenstein does, is known as ‘logical expressivism’. The status of logic as science is not merely a scholars’ concern. The understanding of logical constants and the nature of logical truths are issues that are essentially related to the question of the kind of science, if any, that logic is. These issues are also affected, even if not determined, by some apparently superficial features of the presentations of logical calculi. Gentzen changed the aspect of logical calculi when he introduced his first-order sequent calculus. His innovations had a deeper philosophical motivation and were instrumental to the evolution of logical theory: My starting point was this: The formalization of logical deduction, especially as it has been developed by Frege, Russell, and Hilbert, is rather far removed from the forms of deduction used in practice in mathematical proofs. Considerable formal advantages are achieved in return. In contrast, I intend first to set up a formal system which comes as close as possible to actual reasoning. The calculus was a “calculus of natural deduction.” (Gentzen, 1935/1964, p. 288).

In the context of the present discussion, Gentzen’s main innovation was the elimination of axioms and their substitution by different inference figure schemata which governed the functioning of conjunction, disjunction, negation, conditional, the existential quantifier and the universal quantifier by means of specific introduction and elimination rules, in a way that has since become standard. This move shifted the focus of logic from logical truths to inference rules, i.e. from necessarily true sentences or formulae to transitions between schemata (Kneale, 1957, p. 24; Haack, 1978, p.  20). The focus on logical truths and axioms made it easier to

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assimilate the job of logic to that of geometry, in its many axiomatic systems and presentations. Axiomatic presentations do not determine a particular take on the nature of logic, though. Presentations are just that, presentations, and not all axiomatic systems have the same status (Hintikka & Sandu, 2007, pp.  16–17), but the natural deduction approach highlights the fact that what is characteristic of logic is something else: something that is more connected with transformations of, or transitions from, some formulae into some others than it is with any set of substantive logical truths. I defend the view of logic as an assistant. I agree with Frege, Wittgenstein and Brandom that logic does not establish new truths. And this should make us think twice about the contents of logic as a theoretical discipline. By contrast, the mathematical branch that deals with the properties of formal systems does produce new mathematical knowledge. But this discipline is not what I mean when I speak of logic. ‘Metalogic’ is a standard name for it, and a good one. Logic as an assistant, not being a theory, does not have to deal with the central problem of the epistemology of logic, i.e. the paradox produced by the use of logic to justify itself. This paradox is known as the ‘logocentric predicament’: [L]ogic is epistemically circular in the sense that any attempt to explain or justify logic must presuppose and use some or all of the very logical principles and concepts that it aims to explain or justify. (Hanna, 2006, p. 55)

Logic, in the tradition in which I insert my work, needs no further justification. It is supported by the way we use our concepts; a way that does not depend on our individual will, but which is not immutable either. We learn logic when we learn to talk, and we learn to talk when we learn to live. This is my pragmatist take on the matter, which is perceptible in Wittgenstein and Brandom and not alien to Frege either.

References Baker, G. P., & Hacker, P. M. S. (1984). Frege: Logical excavations. Oxford University Press. Belnap, N. (1962). Tonk, plonk and plink. Analysis, 22(6), 130–134. Bolzano, B. (1837). Wissenschaftler: Versucht einer ausführlichen und grösstentheils neuen Darstellung der logic mit steter Rücksicht auf deren bisherige Bearbeiter (Vol. 1). Herausgegeben von J. E. v. Seidel. Brandom, R. (1984). Reference explained away. The Journal of Philosophy, 81(9), 469–492. Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press. Brandom, R. (2000). Articulating reasons. An Introduction to Inferentialism. Harvard University Press. Bronzo, S. (2017). Frege on multiple analyses and the essential Articulatedness of thought. Journal of the History of Analytical Philosophy, 5(10). Carroll, L. (1895). What the tortoise said to Achilles. Mind, IV(14), 278–280. Carston, R. (2002). Linguistic meaning, communicated meaning and cognitive pragmatics. Mind and Language: Special Issue on Pragmatics and Cognitive Science, 17(1), 127–148. Chomsky, N. (1988). Language and problems of knowledge: The Managua lectures. The MIT Press.

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Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. Dummett, M. (1991). The logical basis of metaphysics. Harvard University Press. Dummett, M. (1995). The context principle: Centre of Frege’s philosophy. In I. Max & W. Stelzner (Eds.), Logik und Mathematik: Frege-Kolloquium Jena 1993. de Gruyter. Etchemendy, J. (1983). The doctrine of logic as form. Linguistics and Philosophy, 6, 319–334. Forero-Mora, J. A., & Frápolli, M. J. (2021). Show me. Tractarian Non-descriptivism. Teorema, XL(2), 63–81. Frápolli, M. J. (2019). Propositions first. Bitting Geach’s bullet. In M. J. Frápolli (Ed.), Expressivisms, knowledge and truth (Royal Institute of Philosophy Supplement 86) (pp. 87–110). Cambridge University Press. Frápolli, M. J., & Villanueva, N. (2016). Pragmatism. Propositional priority and the organic model of propositional individuation. Disputatio, viii(43), 203–217. Frápolli, M. J., & Villanueva, N. (2018). Quantifiers. Hintikka and Frege on quantification concepts. In H. van Ditmarsch & G.  Sandu (Eds.), Jaakko Hintikka on knowledge and game-­ theoretical semantics (Outstanding contributions to logic, Vol. 12). Springer. Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jean van Heijenoort (1967), From Frege to Gödel. A source book in mathematical logic, 1879–1931 (pp. 1–82). Harvard University Press. Frege, G. (1884/1953). The foundations of arithmetic. A logic-mathematical enquiry into the concept of number. (J.  L. Austin, Trans.). Second Revised Ed. Harper Torchbooks/The Science Library, Harper & Brothers. Frege, G. (1892). On Sense and Meaning. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 157–177). Basil Blackwell. Frege, G. (1893/2013). In P. A. Ebert & M. Rossberg (Eds.), Gottlob Frege: Basic laws of arithmetic. Oxford University Press. Frege, G. (1919). Notes for Ludwig Darmstaedter. In G. Frege (1979). Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach (pp. 253–257). Basil Blackwell. Frege, G. (1918–1919a). Thoughts. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Oxford, Basil Blackwell, 351–372. Frege, G. (1918–1919b). Negation. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 373–389). Basil Blackwell. Frege, G. (1923–1926). Compound Thoughts. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 390–406). Basil Blackwell. Gentzen, G. (1935/1964). Investigation into logical deduction. American Philosophical Quarterly, 1(4), 288–3061. Haack, S. (1978). Philosophy of logics. Cambridge University Press. Hacker, P. M. S. (2001). Frege and the early Wittgenstein. In Wittgenstein: Connections and controversies (pp. 191–218). Clarendon. Hale, B. (1994). Dummett’s critique of wrights attempt to resuscitate Frege. Philosophia Mathematica, 2, 122–147. Hanna, R. (2006). Rationality and logic. A Bradford Book. Hintikka, J., & Sandu, G. (2007). What is logic? In D.  Jaquette (Ed.), Philosophy of logic (pp. 13–39). Kaplan, D. (1977). Demonstratives: An essay on the semantics, logic, metaphysics and epistemology of demonstratives and other Indexicals. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes From Kaplan. Oxford University Press. (1989). Kaplan, D. (1979). On the logic of demonstratives. Journal of Philosophical Logic, 8(1), 81–98. Kneale, W. (1957). The province of logic. Mind, 66(262), 258. Kneale, W., & Kneale, M. (1962). The development of logic. Clarendon Press. Korhonen, A. (2012). Logic as a science and logic as a theory: Remarks on Frege, Russell and the logocentric predicament. Logica Universalis, 6, 587–613. Lewis, D. (1980). Index, context, and content. In S. Kanger & S. Öhman (Eds.), Philosophy and grammar (pp. 79–100). D. Reidel Publishing Company.

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Linnebo, Ø. (2008). Compositionality and Frege’s Context Principle. Ms., invited talk given at the CSMN Workshop on Reference (Oslo 2007), and London Logic and Metaphysics Forum (November 2007). Retrieved from http://www.thatmarcusfamily.org/philosophy/Course_ Websites/Readings/Linnebo%20-­%20Compositionality%20and%20Context%20Principle.pdf Linnebo, Ø. (2019). The context principle in Frege’s Grundgesetze. In P. Ebert & M. Rossberg (Eds.), (pp. 90–11). Pelham, J., & Urquhart, A. (1995). Russellian propositions. Studies in Logic and the Foundation of Mathematics, 134, 307–326. Pelletier, F. J. (1994). The principle of semantic compositionality. Topoi, 13, 11–24. Pelletier, F.  J. (2001). Did Frege Believe Frege's Principle? Journal of Logic, Language and Information, 10, 87–114. Penco, C. (2003). Frege: Two theses, two senses. History and Philosophy of Logic, 24(2), 87–109. Pérez-Navarro, E. (2020). Are Frege’s thoughts Fregean propositions? Grazer Philosophische Studien, 97, 223–244. Pinker, S. (1994). The language instinct: How the mind creates language. William Morrow. Rabern, B. (2017). A bridge from semantic value to content. Philosophical Topics, 45(2), 201–226. Ramsey, F. P. (1925). The foundations of mathematics. In D. H. Mellor (Ed.), 1990 (pp. 225–244). Ramsey, F. P. (1927/1991). The nature of truth. In N. Rescher & U. Majer (Eds.), On truth: Original manuscript materials (1927–1929) from the Ramsey collection at the University of Pittsburgh (pp. 6–24). Kluwer Academic Publishers. Recanati, F. (2002). Unarticulated constituents. Linguistics and Philosophy, 25(3), 299–345. Recanati, F. (2003). Literal meaning. Cambridge University Press. Russell, B. (1903). The principles of mathematics. Cambridge University Press. Russell, B. (1910). Philosophical essays. Longmans, Green. Russell, B. (1919). On propositions. What they are and how they mean. Aristotelian Society Supplementary Volume, 2(1), 1–43. Sagüillo, J. M. (1997). Logical consequence revisited. Bulletin of Symbolic Logic, 3(2), 216–241. Sagüillo, J. M. (2007). Validez y Consecuencia Lógica. In Frápolli (Ed.), (pp. 55–81). Strawson, P. (1950/2013). Truth. Virtual Issue, n 1. Proceedings of the Aristotelian Society, pp. 54–74. Sundholm, G. (2009). A century of judgment and inference, 1837–1936: Some strands in the development of logic. In L. Haaparanta (Ed.), (pp. 263–317). Tarski, A. (1935/1956). The definition of truth in formalized languages. In A.  Tarski (1956), (pp. 152–278). Tarski, A. (1936/2002). On the concept of following logically. Translation from the polish and German by Magda Stroiska and David Hitchcock. History and Philosophy of Logic, 23(3), 155–196. Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7, 143–154. Van Heijenoort, J. (1967). Logic as calculus and logic as language. Synthese, 17(3), 324–330. Wischin, K. (2019). Frege’s legacy in the later Wittgenstein and Brandom. Disputatio. Philosophical Research Bulletin, 8(9), 00–00.

Chapter 3

Semantic and Pragmatic Hints in Frege’s Logical Theory

Abstract  Frege is the acclaimed father of twentieth-century logic and at the same time the father of the discipline philosophy of language. Both of these paternities arise from a unique and general project grounded on a profound understanding of the many linguistic and conceptual subtleties that govern the use of language. His logical project, which set him apart from his contemporary fellows and on an underexplored path, participates in the richness of his approach to language and thought, which includes semantic and pragmatic hints that were only recognised much later in the twentieth century. I explain and develop these hints in support of an approach to logic that, far from psychological vagaries, benefits from the analytical depth of acknowledging the role of concepts in human communication. Keywords  Fact · Function · Grice · Identity · Implicature · Judgement stroke · Number · Presupposition · Strawson · Truth [true]

3.1 Frege’s Projects The Grundgeseztze, the Basic Laws of Arithmetic, was Frege’s life’s project. It was the completion of the project that was initiated in the Begriffsschrift (1879) and continued in the Grundlagen der Arithmetik (1884). The overall aim of this life work was to provide a safe foundation for arithmetic and to show that its core notions could be defined on the basis of logical notions and that the principles of inference that had previously been understood as specifically mathematical could be reduced to logical transitions. In pursuit of a foundation for arithmetic, Frege developed a conceptual notation that aided the accurate expression of contents and their analysis in terms of functions and arguments and allowed the steps in inferences to be perspicuously represented. The conceptual notation, his Begriffsschrift, was instrumental to the general project of the foundation of arithmetic and is the origin of the modern conception of logic. The Foundations of Arithmetic and the Basic Laws, in contrast, bore no direct conceptual connection with the auxiliary enterprise of designing an analytical tool for the representation of inferences. They are specific applications of Frege’s logic, which was the topic of the Begriffsschrift. © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_3

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Frege’s logic bears a mere de facto connection to these two tasks of the foundation of arithmetic and the definition of the notion of number. A similar discipline could have been developed along with the project of laying down the foundations of biology, chemistry, or along with the task of analysing ordinary everyday discourse. The instrument of representation that Frege developed in 1879 works for any area of discourse regardless of the kind of content related by its inferential relations. This is a further sense (see Chap. 1, Sect. 1.4) in which logic can be said to be formal: it is a universal representation method, a lingua characteristica in Leibniz’s sense (1880/1, p.  10, Van Heijenoort, 1967, p.  324). As Macbeth emphasises, whereas other sciences—her example is psychology—discover those laws proper to their scope, ‘logic discovers the laws of judgement itself’ (Macbeth, 2005, position 310). Negation, conditionality, the identity of content and generalisation—the specific notions that Frege explicitly characterises in his Begriffsschrift—apply to propositions and concepts of any kind. The project developed in this work mainly consists in the replacement of an unperspicuous representation system with a perspicuous one: a system that reflects in its syntax the semantic relations between propositions, which are the concern of logic. This characteristic of Frege’s conceptual writing has given rise to a further sense in which logic is sometimes said to be formal, albeit this time a mistaken one. This is the sense that became standard in the early works of Hilbert, Tarski and Carnap, according to which logical relations are syntactic relations between uninterpreted formulae. An illustration of this sense of the formality of logic is provided by the following text, taken from Carnap’s Logical Syntax of Language: [T]he development of logic during the past ten years has shown clearly that it can only be studied with any degree of accuracy when it is based, not on judgments (thoughts, or the content of thoughts) but rather on linguistic expressions, of which sentences are the most important, because only for them is it possible to lay down sharply defined rules. (Carnap, 1937, p. 1)

Carnap’s conclusion establishes what we have all learned in the classical handbooks of formal logic, that ‘[i]n this way, logic will become a part of syntax’ (Carnap op. cit., p. 2). Carnap part ways here with Frege’s heritage and contributes to consolidating the style of logical theory that subsequently dominated the twentieth century. This text follows the tradition inaugurated by the algebraic approach to logic defended by Boole, and which connects with Hilbert’s meta-mathematical project, and afterwards with Tarski’s extension to formal semantics. Frege, nevertheless, did not belong to this tradition. His several works defending his Begriffsschrift from any assimilation of it to Boole’s project (see Frege, 1880/1, 1882; Sluga, 1987) are witnesses to the distance that separated Frege’s approach from any mathematisation of logic that intended to go beyond superficial similarities in the use of symbols and analytical notions. The unfavourable reception of the Begriffsschrift among mathematicians, incited in part by Schröder’s unfriendly review (Schröder, 1880; Sluga op. cit., p. 80; but cfr. Vilkko, 1998), is partially explained by their failure to identify in Frege’s work a project of a completely different kind. As Macbeth makes clear, Frege’s conceptual notation was not intended to trace truth conditions: ‘truth

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conditions are easily formulated for any Begriffsschrift sentence’, but it is not Frege’s primarily aim to represent them’ (Macbeth, 2005, position 72). In sharp contrast with the formalists, Frege’s views were inserted in a ‘semantic tradition’ (Coffa, 1991) to which Husserl, and Wittgenstein later on, also belonged: ‘The logical' it would be a serious error to misunderstand what Frege meant by this recurring expression in his early writings. What Frege and Russell called ‘logical', what Husserl called a ‘logical’ investigation, what Meinong called ‘Gegendstandstheorie’ and what Wittgenstein termed a ‘logico-philosophical' observation are close relatives; they should not be confused with what is now called logic, after formalism and set theory have come to dominate the field. Their ‘logic’ was our semantics, a doctrine of content, its nature and structure, and not merely of its ‘formal’ fragments. (Coffa, op. cit., p. 64)

Brandom adds Kant, Carnap, Sellars, and Dummett to the list (Brandom, 1984, passim.). The inclusion of Frege in a semantic tradition is, without any doubt, more accurate than the interpretation of his work as part of a syntacticist school. To make sense of Frege’s assessment of his introduction of the distinction between sense and meaning as an essential step in the development of his logical project, Macbeth explains that the Begriffsschrift does not only codify a theory of quantification, and that some standard criticisms of Frege’s logical projects—and here Macbeth mentions those of Evans and Dummett— rest on a failure to understand the nature of Frege’s enterprise (Macbeth, 2005, position 110). If, by contrast, we follow Coffa in understanding logic as semantics, then Frege’s interpretation of the centrality of the distinction in his work falls perfectly into place. I will go further in the reinterpretation of Frege’s doctrines and highlight some unequivocal pragmatic hints in his writings. Something that we have learned in the philosophy of language of recent decades is that there is no clear-cut divide between semantics and pragmatics. This divide makes sense only as a technical decision within particular theories. Even so, it is mostly arbitrary, since what we do with words cannot be severed from the meaning of those words that we use to do what we do. The meaning of our linguistic actions, and of the systems that we use to perform them, derives from the communicative intentions of speakers and from the aims of the actions they engage in. We have learned to recognise the relevance of the role of agents and contexts in communication, thanks to the work of Wittgenstein, Grice, Austin and Searle, and more recently thanks to the cooperation between the linguists and philosophers who have produced scientific theories of communication such as Relevance Theory (D.  Sperber and D.  Wilson) and Truth-Conditional Pragmatics (F. Recanati and R. Carston). A further step in overcoming the traditional levels of semantics and pragmatics is contemporary inferentialism, as defended by Robert Brandon, which I consider to be a genuine paradigm shift in the analysis of the connections between meaning, actions, and rationality. The recognition of pragmatist aspects in meaning is compatible with a purely semantic analysis of some features of language and logic, and yet it is essential to keep in mind that pure semantics is an abstraction in which we intentionally disregard some aspects of the phenomenon. In the proposal of a conceptual writing for the foundation of arithmetic, Frege focused on semantics and consciously overlooked those facets of meaning that depended on the peculiarities of agents and

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contexts. But he was perfectly aware that everything begins with an agent’s act of expressing a judgement. He also recognised that by using logical notions, speakers give expression, not only to the connections between the results of acts of assertion, but also to the commitments involved in actions of asserting, affirming and rejecting contents. In this chapter, I will show that some notions that philosophers of language now consider to be characteristically pragmatic were also spotted by Frege. But before dealing with these pragmatist hints, let us make a brief detour in order to comment on better-known features of Frege’s approach and their received interpretations.

3.2 The Representation of Abstract Reality Numbers and thoughts are independent of people’s mental lives, without being affected by causal relations as physical entities are. The entities that belong to this intermediate type are not perceptible by the senses but, unlike psychological representations, they do not need an individual bearer either (Frege, 1918–1919a, p. 363). They are at the same time mind-independent and beyond the causal world. Some examples will help to show the differences. My toothache belongs to me, as do my feelings about Brexit or about my children. My children are independent individuals, as are the Moon and the Earth. But the axis of the Earth, the distance from the Earth to the Moon, and the validity of the inference from I gave birth to my children to my children are human beings are abstract and objective entities and relations of the kind that we deal with in our everyday communicative exchanges. If this is all that is at stake when we characterise Frege as a platonist or a realist, then there is no objection. But if by applying these labels it is meant that Frege placed numbers and thoughts somewhere in some special pseudo-spatial realm—even if this is interpreted in a weak manner—then it is false. This would be to mix together objectivity and mind-independence with some kind of metaphorical spatiotemporal location: a confusion which lies at the origin of many metaphysical debates about realism (see Frápolli, 2015) and that cannot be attributed to Frege. In The Foundations of Arithmetic, he explains: I distinguish what I call objective from what is handleable or spatial or actual. The axis of the earth is objective, so is the centre of mass of the solar system, but I should not call them actual in the way the earth itself is so. We often speak of the equator as an imaginary line; but it would be wrong to call it an imaginary line in the dyslogistic sense; it is not a creature of thought, the product of a psychological process, but it is only recognized or apprehended by thought. (Frege, 1884, §26, p. 35)

Besides numbers, concepts, and thoughts, there are also signs, words and sentences, i.e. linguistic expressions through which these entities become somehow perceptible. The relation between language and thought is more complicated than what the metaphor of content and vehicle usually conveys (see Chap. 8). Frege, like the early Wittgenstein, sometimes talked as if he viewed language as an inert container or a lifeless cloth. Frege said that ‘[t]he thought, in itself imperceptible by the senses,

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gets clothed in the perceptible garb of a sentence, and thereby we are enabled to grasp it’ (Frege, 1918–1919a, p. 354), and Wittgenstein claimed that ‘in the proposition the thought is expressed perceptibly through the senses’ (Tractatus 3.1). The later Wittgenstein rejected this view, which seriously misrepresents the nature of language. A more promising picture to have in mind is Brandom’s ‘relational linguistic’ approach (Brandom, 2000, p. 6), according to which believing and asserting are the two sides of the same coin. The slogan that Brandom chooses to characterize his relational linguistic pragmatism is Sellars’s claim that ‘grasping a concept is mastering the use of a word’ (Brandom loc. cit.). Thought, in the sense that characterises us as rational creatures, cannot be understood except in its connection with linguistic practices, and only the application of concepts and the expression of thoughts make some of our practices discursive. Thus, language is the home of philosophers, although this does not turn philosophy—not even philosophy of language—into linguistics. Concepts and conceptual contents lie at the core of the work of philosophers, who are, as Blackburn said, conceptual engineers (Blackburn, 1999, p. 2; but see Chap. 1, Sect. 1.1). The objectivity of concepts and conceptual contents means that their properties and relations do not depend on the will of any particular agent. There is nothing we can do about the fact that two times two is four, nor can we modify the set of inferences that are valid. From The Moon is the only Earth’s satellite, The Earth has only one satellite and There is something that is the Earth’s satellite follow; from Greenhouse gases are producing the global warming, Planet Earth is getting warmer follows. The conceptual relations on which the validity of these inferences rests are objective. We might change the meanings of our words and modify the boundaries of the concepts that we use, but this would require changing the scientific theories that we endorse or, depending of their contents, changing our forms of life. The possibility of conceptual and meaning variance does not conflict with the claim that conceptual relations are objective. This point will not be pursued further. Nevertheless, it is vital to insist on something that, once stated, seems obvious: that the resistance to comprehension that concepts and thoughts exert on rational agents is not the resistance to penetration exerted by physical objects. We have here distinguished between concepts and thoughts, which are abstract entities, and planets and people, which are physical entities. There are also mental entities such as feelings, pains and fears. A higher-level concept such as existence, which Frege characterises in The Foundations of Arithmetic §53, applies to concepts under which abstract, physical and mental objects fall. To grant existence to physical objects and deny it to all others is to fail to understand that existence relates to concepts, not to objects of any kind; and that it does not imply any kind of location. As we saw in Chap. 1, existence is instantiation. Its introduction produces some kind of unburdening of content in which some information is lost, and its elimination yields a case, an instance of a general assertion. Some examples will serve to illustrate this point. The introduction of the existential quantifier in (1), (2), and (3):

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(1) The Moon is smaller than the Earth. (2) 3 is greater that 2. (3) My headache is getting better. gives rise to the existential generalisations, (4), (5), and (6): (4) Something is smaller than the Earth. (5) Something is greater than 2. (6) Something is getting better. We might have used more explicitly existential formulations, such as ‘there is’ or ‘there exists’ instead of ‘something’ but this would not change anything. Examples (4), (5), and (6) are well-formed English sentences, meaningful and true, and all of them include an existential term which makes the sentence an existential generalisation. The quantifier plays the same semantic role in all of them: that of expressing that the extension of the predicable which is, from a logical viewpoint, its argument, is not empty. And nevertheless, in the first case the topic is planets, in the second, numbers, and in the third case, a feeling. Only philosophers convinced by (possibly a poor interpretation of) Quine’s and his many followers’ view of the connection between quantification and ontology have any difficulty in understanding the meaning and truth-conditions of sentences (4)–(6). So, we need not worry about the existence of concepts and propositions, the stuff on which logic operates. They are objective abstract entities with which we deal on a daily basis.

3.3 The Analysis of Discourse Frege’s contribution to philosophy was not restricted to the representation of inferences. He also proposed an instrument for analysing propositions’ inner nature into functions of different levels and their arguments. This distinction was explained in terms of its inferential effects, tested in a substitutional procedure (Frege Begriffsschrift §9, Frege, 1884, §70). In Frege’s writings, the term ‘function’ sometimes refers to the expression, and sometimes to its meaning; and, thus, a certain degree of ambiguity can hardly be avoided. The arguments of functions can be names, i.e. semantically complete notions, or else other functions. In this second case, Frege speaks of ‘second-level’ functions. If the inferential apparatus of the Begriffsschrift is the origin of modern logic, then Frege’s analysis of propositions into functions and their arguments, together with the distinction between content and how it is determined, is the starting points of modern philosophy of language. Frege is commonly praised as being the father of both disciplines, but it is important to keep in mind that this was the same Frege with the same overall position. There are not two Freges, the logician and the philosopher of language, but rather a single

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Frege with a wide-ranging project that involved a methodology of inferential representation and assessment, and a system of conceptual analysis that covered different types of notions: propositions, logical notions, and propositional functions whose linguistic representations induce indirect speech. Frege’s analytical methodology reserves certain terms to name expressions, and other terms to name the entities that those expressions denote. Functions are incomplete, ‘unsaturated’ expressions (Frege, 1891, CP, p.  140), which emerge via the analysis of sentences: Let us assume that the circumstance that hydrogen is lighter than carbon dioxide is expressed in our formula language; we can then replace the sign for hydrogen by the sign for oxygen or that for nitrogen. This changes the meaning in such a way that ‘oxygen’ or ‘nitrogen’ enters into the relations in which ‘hydrogen’ stood before. If we imagine that an expression can thus be altered, it decomposes into a stable component, representing the totality of relations, and the sign, regarded as replaceable by others, that denotes the object standing in these relations. The former component I call a function, the latter its argument. The distinction has nothing to do with the conceptual content; it comes about only because we view the expression in a particular way. According to the conception sketched above, ‘hydrogen’ is the argument and ‘being lighter than carbon dioxide’ the function; but we can also conceive of the same conceptual in such a way that ‘carbon dioxide’ becomes the argument and ‘being heavier than hydrogen’ the function. We then need only regard ‘carbon dioxide’ as replaceable by other ideas, such as ‘hydrochloric acid’ or ‘ammonia’. (Frege, 1879, §9)

Some claims in this text deserve a comment. One of them is that it deals specifically with expressions and not with what they express. Sentences are the starting point from which the agent sets some sub-expressions as fixed and leaves some others as variable. This procedure is an example of (PConxt) and evidence of the (PPP): functions and their arguments are discerned in sentences through the procedure of substituting them with others. There is no (PComp) here according to which the sentences at issue are built up out of previous functions and arguments. Besides, the identification of functions and their arguments is guided by the purposes of analysis and ‘has nothing to do with the conceptual content’. There is more than one way of analysing an expression without affecting its content. The sentence ‘Carbon dioxide is heavier than hydrogen’ expresses a thought, the same thought, regardless of whether we fix the function ‘carbon dioxide is heavier than’ and treat ‘hydrogen’ as its argument, or whether we understand ‘carbon dioxide’ as the argument of the function ‘is heavier than hydrogen’. Analysing a sentence into parts is something that agents do for their own purposes. The intervention of agents is a mark of pragmatism, which in Frege is seen from his first works. In (Frege, 1891, pp. 144ff.), Frege explains the ways in which he has extended the mathematical notion of function. Relational signs such as ‘=‘, ‘’ are now also considered as functions, since they are signs that need some completion if they are to mean anything. These functions can be identified by erasing some singular terms in sentences (formulae) such as ‘2 = 1 + 1′, ‘3 < 4’ or ‘7 > 1’. The range of their arguments is the same as that of the arguments for the traditional functions ‘+’,

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‘−’, ‘.’, etc. Thus open formulae such as ‘2 > x’ are interpreted along the lines of ‘2 + x’. Given some arguments, the values of the function is clear: for the argument ‘3’, the function ‘2 + x’ yields the value ‘5’, and for the argument ‘0’ the value ‘2’. The challenging question was: what could possibly be the values for functions such as ‘2 > x’? Frege gave an original answer: those functions together with appropriate arguments, have truth-values as their values. Thus, for the argument ‘3’, the function ‘2 > x’ yields the value the False, and with ‘0’ as its argument the value the True. If functions are incomplete, unsaturated, expressions, names are those expressions that are saturated. Numerals (e.g. ‘0’, ‘3’, ‘100’) and complex expressions without variables (e. g. ‘3 + 2’, ‘42’) are names. As with the category of function, Frege likewise extends the category of name to cover complete formulae. Thus, ‘3 + 2 = 5’ and ‘4 > 2’ are names for Frege. These innovations, so far restricted to the analysis of arithmetical discourse, were extended to the analysis of natural language sentences and their components. This extension was announced in the Preface of the Begriffsschrift and applied in the Grundlagen, but it is from 1891 that it acquires its final form. Now, not only ‘3’ and ‘3 > 2’ are names, but also ‘Julius Caesar’ and ‘Caesar conquered Gaul’ (Frege, 1891, pp. 146–7), and not only ‘x + y’ and ‘3 > x’ represent functions, but also ‘the capital of’ and ‘conquered Gaul’. The value of the function ‘the capital of’ for the argument ‘the German Empire’ is an object, Berlin. The value of ‘conquered Gaul’ for the argument ‘Caesar’ is another object, albeit of a different kind: the True. Functions that have truth-values as values are called ‘concepts’. As mentioned before, Frege calls unsaturated expressions and their contents ‘functions’. But in the case of concepts he uses a special term, ‘concept-word’ (Frege, 1892a, p. 183), for the expression and reserves ‘concept’ for its content. Frege is aware of the subtleties and ambiguities of his present use of terms such as ‘concept’ and ‘function’. At the end of (Frege, 1892a), where he has been answering Benno Kerry’s criticism about his use of ‘concept’ and ‘object’, Frege says: It may make it easier to come to an understanding if the reader compares my work Function and Concept. For over the question what is that called a function in Analysis, we come up against the same obstacle; and on thorough investigation it will be found that the obstacle is essential, and founded on the nature of our language; that we cannot avoid a certain inappropriateness of linguistic expression; and that there is nothing for it but to realize this and always take it into account. (Frege op. cit., p. 194)

The linguistic puzzles inherent to the discours of abstract entities were acknowledged by Frege, who insisted on the idea that some difficulties in formulating his views derived from the fact that language is, at some points, ill-adapted for the analysis of conceptual contents. This key intuition is the origin of (PGS), which I have introduced in Chap. 1. To deal with these puzzles that language sometimes produces, Frege displays a fair amount of common sense; in (Frege, 1892a), for instance, with respect to functions, and in all his work with respect to the interpretation of truth values, concepts and thoughts. The same cannot be said of some of his commentators, who at some points have created artificial metaphysical problems, which not only are utterly alien to Frege’s thought but—and this is actually the

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relevant aspect—do not in the least help to provide understanding about how language works. Among these artificial problems, the status of the True and the False as objects, the meaning of functions, and the hierarchy of indirect senses are probably the most discussed. If our aim is understanding, we should meet Frege ‘halfway’ and not ‘begrudge a pinch of salt’ (Frege op.  cit., p.  193), although we philosophers have a natural inclination to scholasticism, which is the death of the spirit. At this point, we should pause and recall the main points of Frege’s general proposal. He was interested in the definition of the notion of number, the role played by identity, the meaning of conditionality, negation and generality, and the nature of mathematical modes of inference. Because he was a mathematician, he was interested in providing a unified system with the smallest possible number of undefined notions. Some of his statements thus only serve to maintain the simplicity of the system. His classification of the True and the False as objects has such an intra-­ systemic, superficial explanation, which Frege explicitly acknowledges: We are therefore driven into accepting the truth-value of a sentence as constituting what it means. By the truth-value of a sentence I understand the circumstance that it is true or false. There are no further truth-values. For brevity I call the one the True, the other the False. (Frege, 1892b, p. 163)

He continues: The designation of the truth-values as objects may appear to be an arbitrary fancy or perhaps a mere play upon words, from which no profound consequences could be drawn […]. But so much should already be clear, that in every judgement, no matter how trivial, the step from the level of thoughts to the level of meaning (the objective) has already been taken. (Frege loc. cit.)

In the Grundgesetze, he characterises what he means by ‘object’ along similar lines: The domain of what is admitted as argument must also be extended to objects in general. Objects stand opposed to functions. Accordingly I count as objects everything that is not a function, for example, numbers, truth-values, and the courses-of-values to be introduced below. The names of objects⏤the proper names⏤therefore carry no argument-places; they are saturated, like the object themselves. (Frege, 1893, pp. 35–36)

The explanation Frege gives of the significance he attaches to the two ‘objects’, the True and the False, can be rendered as an instance of the Recarving Thesis, (RT):

( RT ) (The Recarving Thesis ) : f (α ) = f ( β ) ↔ α ~ β .

Applied to this particular case, (RT) says that two sentences are equivalent if, and only if they have the same truth-value. (RT) has been much debated (Linnebø, 2008), and some authors (Field, 1984; Dummett, 1991, chapter 15) have considered it false, in part because it contradicts a generalised syntactic intuition that implies the two following principles, Meanings Involves Referents (MIR) and Surface Syntax (SS):

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3  Semantic and Pragmatic Hints in Frege’s Logical Theory (MIR) (Meanings Involve Referents): Any characterisation of the meaning of a sentence S that contains a referential occurrence of a singular term, a, must make use of a or some co-­ referring term. (SS) (Surface Syntax): The two sides of an abstraction principle have the syntactic and semantic forms that they appear to have. (Linnebø, op. cit., p. 13 ms.)

For logicians and philosophers, the syntactic intuition seems to be very difficult to resist. Neo-logicism, a position that allegedly develops Frege’s view of numbers and arithmetical truths, upholds it in the form of a principle that McBride calls ‘Syntactic Decisiveness’ (SD): (SD) Syntactic Decisiveness. If an expression exhibits the characteristic syntactic features of a singular term, then that fact decisively determines that the expression in question has the semantic function of a singular term (reference). (MacBride, 2003, p. 108)

In sum, what (MIR), (SS), (SD) say is that syntax cannot be overcome. The role they assign to syntax directly clashes with (PGS), which is not only a central Fregean principle but also the principle that makes room for any kind of conceptual analysis that sets apart linguistic form from logical form: a methodology that essentially defines the philosophy of logic and the philosophy of language from Frege up until the present days. An excessive attachment to syntax is responsible for many scholastic debates in analytic philosophy, and hindered Frege and Russell from envisaging a way out of Russell’s paradox, as I will discuss in Chap. 7. In particular, if syntax should be respected in the sense that (SD) claims, neither the Fregean account of quantifiers nor the Russellian theory of descriptions would have gone through. Meeting Frege ‘half-way’ means respecting his intention and avoiding artificial problems that obscure the rigour and depth of his take on concepts, thoughts, and the language we employ to express them.

3.4 Two-factor Semantics and the Meaning of Identity Extralinguistic abstract reality is only accessible via linguistic systems and only by the use of linguistic systems can agents perform specific actions (assertions and inferences, among them) that involve abstract entities such as numbers and propositions. Complex reality calls for complex systems and complex ways of representing. This is something that Frege acknowledged from the beginning. In the Begriffsschrift, §8, he introduced a double-factor approach to the meaning of expressions that flank the identity sign. Identity claims, i.e. those in which names are combined ‘by means of the sign of identity of content’, express the ‘circumstance that two names have the same content’ (Frege loc. cit.). This characterisation of the role of the identity sign is the origin of the split of the ordinary notion of meaning into sense (Sinn) and

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meaning (Bedeutung).1 The distinction between these two types of meaning was intended for names, but for the sake of simplicity and theoretical homogeneity, Frege artificially extended it to unsaturated expressions too. This explains why he says ‘scarcely anything at all […] of the senses expressed by incomplete names (save for their “unsaturation”)’ (Furth, 1964, p. xxvi). It is a reasonable hypothesis that when he introduced this distinction, he only had names in mind: ordinary names in the Begriffsschrift and also complete sentences from (Frege, 1892b) on. The identity sign thus produces a bifurcation in the meaning of ordinary names. The only other context in which the bifurcation between sense and meaning is useful concerns sentences in the context of indirect speech. Because of Frege’s commitment to (PPP) and (PCont), the analysis of subordinate clauses begins with the analysis of the complex sentence that provides their context: We are thus led to consider subordinate sentences or clauses. These occur as parts of a sentence complex, which is, from the logical standpoint, likewise a sentence ⏤a main sentence. But here we meet the question whether it is also true of the subordinate sentence that its meaning is a truth value. Of indirect speech we already know the opposite. (Frege, 1892b, p. 165)

Functions and their contribution to the proposition expressed can be identified in the sentences in which they occur. But the contribution of functions derives from the complete sense of the sentence that, in turn, is inferentially individuated. Thus, what the sense of functional terms might be is just an intra-systemic technical question with almost no semantic relevance. And it may very well be that this question had no answer. In his Introduction to (Frege, 1893), Furth affirms that the main purpose of the distinction between Sinn and Bedeutung is to keep senses out of Frege’s main project and prevent them from obstructing ‘the development of the theory at the level of denotation’ (Frege op. cit., p. xxvi). This explanation is reasonable if we only consider the foundation of arithmetics, but it loses credibility when our focus turns to Frege’s semantics, for which this distinction is essential. I have mentioned the analysis of identity claims as the origin of the two-factor theory of meaning that led to the distinction between sense and meaning. The analysis of identity displays several features that deserve attention. Identity of content, ‘equality’, is the topic with which Frege begins (Frege, 1892b). There he asks whether identity is a relation at all and, if it is, whether its terms are signs or objects. He refers back to the Begriffsschrift in order to insist that this relationship is one between signs of a certain kind. If it were between objects, then a = a and a = b would not differ, when in fact the first sentence can be known a priori, whereas the second one could also be a posteriori: ‘The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries’ (Frege, 1892b, p.  157). Nevertheless, identity is not a relation between  The standard translations of Frege’s Bedeutung in English are reference and denotation. Both of these English terms are misleading. I will stick to the most natural translation of Bedeutung, i.e. meaning, which is also the option taken by Brian McGuinness in Frege’s Collected Papers (Frege, 1984). 1

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signs as objects, but rather between signs inasmuch as they signify something in a certain way. If it were merely between signs as objects, it would be ‘arbitrary’. Thus, identity is a complex relationship that helps state the circumstance that two signs mean the same object under different modes of presentation: Let a, b, c, be the lines connecting the vertices of a triangle with the midpoints of the opposite sides. The point of intersection of a and b is then the same as the point of intersection of b and c. So we have different designations for the same point, and these names (‘point of intersection of a and b’ and ‘point of intersection of b and c’) likewise indicate the mode of presentation. (Frege op. cit., p. 158)

Identity statements thus ‘contain actual knowledge’ (Frege loc. cit.). The explanation in the Begriffsschrift is similar: Whereas in other contexts signs are merely representatives of their content, so that every combination into which they enter expresses only a relation between their respective contents, they suddenly display their own self when they are combined by means of the sign of the identity of content; for it expresses the circumstance that two names have the same content. (Frege, 1879, §8)

After offering an example from geometry, Frege continues: ‘the same content can be completely determined in different ways; but that in a particular case two ways of determining it really yield the same result is the content of a judgement (loc. cit.). He concludes: ‘In that case the judgement that has the identity of content as its object is synthetic in the Kantian sense’ (loc. cit.). In both works, Begriffsschrift (Frege, 1879) and ‘On Sense and Meaning’ (Frege, 1892b), Frege stresses that the content of an identity claim is not trivial or analytic, but rather produces genuine knowledge. Later on, in (Frege, 1893), when he explains in which aspects his Grundgesetze differs from his Begriffsschrift, he explicitly mentions identity in the following terms: The improvements may be mentioned here briefly. The primitive signs used in Begriffsschrift occur here also, with one exception. Instead of the three parallel lines I have adopted the ordinary sign of equality, since I have been persuaded myself that it has in arithmetic precisely the meaning that I wish to symbolize. (Frege, 1893, p. 6)

There is no reference to any change of mind in his conception of identity, as nevertheless has been the customary interpretation of Frege’s development. The alleged differences in his view of identity is one of the issues that most commentators on Frege’s philosophy2 have chosen in order to illustrate two distinct periods in Frege’s philosophy, the first period practically restricted to the Begriffsschrift, and the second period beginning around 1890. According to this widespread interpretation, Frege changed his mind about identity, rejecting the metalinguistic interpretation of identity that he allegedly defended in his first work and instead embracing an objectual interpretation in which identity is understood as a relation between objects. The objectual interpretation would inaugurate his mature period and, in part,  Bell (1990), Dummett (1973), Hacker and Baker (1984), Kripke (1980), Linsky (1977), Russell and Whitehead (1927), Salmon (1986), and Wiggins (1965) all defend Frege’s evolution from a metalinguistic to an objectual view of identity. 2

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triggered the distinction between sense and meaning (see, for instance, Salmon, 1986, p. 52). There is nevertheless no textual evidence of any significant change of mind in Frege’s writings. In fact, there is no evidence of any substantial disruption in his philosophical development that could justify distinguishing between two clearly defined periods in his career. Dummett contributed to disseminating this idea of the two Freges, a young Frege placing inference at the core of his system and a mature Frege replacing inference with truth (Dummett, 1973, p.  362): an evolution that Dummett considers unfortunate for the development of logic (see also Brandom, 2000, p. 51). The focus on truth instead of inference is indeed an unfortunate step in the history of logic in the past century, but it is one that Frege did not take. The blame should rather be placed on the semanticist approach to logic inaugurated by Tarski and consolidated by the progress made in model theory. It is true that the distinction between meaning and sense, in these specific terms, occurred around 1890. Nevertheless, for the case of what we now called ‘singular terms’, i.e. proper names and definite descriptions, it was basically contained in the distinction between content and the mode of determining it that Frege introduced in his (Frege, 1879, §9) to deal with identity. What happened from 1890 onwards was the extension of the category of what he called ‘names’, i.e. saturated expressions of any kind, to include also complete sentences. Once this extension was made and sentences became names, the same semantic apparatus that worked for the standard names was adapted to sentences. And thus, what in Begriffsschrift was called ‘judgeable content’ splits into sense and meaning, i.e. into thought and truth-value. For traditional singular terms, the Begriffsschrift’s content and mode of determination became meaning and sense, respectively. The real revolution that Frege carried out in (Frege, 1892b) was the application of mathematical terminology and its analytical apparatus to the study of natural language. Frege’s project brought together the analysis of natural and mathematical language, on the one hand, and a universal vehicle for presenting and carrying out inferences, on the other. The paper ‘On Sense and Meaning’ is to the philosophy of language what the Begriffsschrift is to contemporary logic. This revolution has not yet to be completely understood.

3.5 Special Notions Some aspects of the Fregean revolution are better understood than others, though. The semantics of truth and logical notions are among the aspects that have passed almost untouched among scholars.3 And nevertheless, Frege’s comments on them reveal profound insights into the subtleties of language. Once well understood,  Many authors have discussed Frege’s notion of truth (see, for instance, Burge, 1986; Greimann, 2007). Nevertheless, the challenge that the meaning of the truth predicate poses for the standard interpretation of Fregean semantics, with its exclusive categories of saturated and unsaturated terms, all contributing to the conceptual content, has hardly been spotted. 3

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these comments can help to unravel some of the most resilient problems of the philosophy of logic and of language. Logical terms and propositional-attitude verbs are technically functions (concepts), and the judgement and content strokes are also expressions in need of completion. Nevertheless, we would miss an essential clue to Frege’s general semantics if we tried to fit these expressions and what they represent into the schemes function-word/name and function/object. Frege’s semantic system is an outstanding example of simplicity and elegance partly due to its homogeneity, but Frege recognises that not all the expressions needed in the analysis of natural and mathematical languages fit so smoothly into this elegant template. The concept of truth is not an ordinary concept; logical functions are not ordinary functions. Neither the former nor the latter contribute with a concept to the thought expressed by the combination of the expressions in their arguments, and they are not parts of thoughts. Following Frege’s suggestions, it is possible to develop an understanding of truth and logical notions which takes us closer to what agents do with them, and allows an appropriate semantic characterisation, alternative to the standard Tarskian approaches. Neither the semantic theory of truth, developed by Tarski in 1936, nor his invariantist characterisation of logical notions, presented in his paper of 1986, say anything about the ordinary and scientific use of truth nor of the semantic and pragmatic behaviour of logical terms. Frege’s approaches, by contrast, were much more earthbound. The analysis of logical notions will be undertaken in Chaps. 4–6. A comparison between Tarski’s and Frege’s approaches to truth is the topic of Chap. 9. In the next three subsections, I deal with some of his deepest semantic/pragmatic insights in order to expose the most striking details of Frege’s revolution.

3.5.1 The Judgement Stroke Let us begin with the judgement stroke. Immediately after introducing the distinction between variable and constant signs, Frege introduces a sign to express that a judgement, an assertion, i.e. the explicit acknowledgement of the truth of a content, has been performed (Frege, 1879, §1). This sign is the judgement stroke, ‘∣—’. The significance of this opening, in what is generally regarded as the seminal work of contemporary logic, has not been stressed enough. Why, one might ask, would a mathematician, in describing his proposal for a logically correct representation method for inferences, begin with non-mathematical notions such as judgement and assertion? This observation about the importance of assertion and judgement might be dismissed as belonging to Frege’s youthful, immature, period. Frege’s texts, nevertheless, prevent us from taking this way out. From the Begriffsschrift to his Logical Investigations (Frege, 1918–1919a, b, 1923–26), Frege remains faithful to his former account of assertion, the primacy of actions, and the importance of distinguishing actions from their results. In the standard interpretation of Frege’s work, the judgement stroke is an oddity. Russell and Wittgenstein criticised it for what they considered to be its merely psychological role (Russell, 1903, §478; Wittgenstein, 1914–16/1998, p. 95). Not all

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commentators have accepted this psychological interpretation (Greimann, 2014; Pedriali, 2017; van der Schaar, 2017; Forero-Mora & Frápolli, 2021), but its profound significance and the clues it gives for understanding Frege’s approach to logic are still unrecognised. If the Begriffsschrift is only read as a work of quantification theory, then the judgement stroke does not fit into the project. Nevertheless, (Macbeth, 2005) provides enough reasons for us to doubt that Frege’s first work should be interpreted as having such a narrow scope. An assertion is something that an agent does. It is the public manifestation of the acknowledgement of the truth of a thought (Frege, 1918–1919a, p.  356). Logic begins with assertion, because assertion is the only way that agents have at their disposal to bring up propositions, which are the premises, the steps, and the conclusions of inferences. Premises and axioms in proofs are asserted contents in Frege’s system. This feature of his system provoked a sour debate with Hilbert about the nature of axioms (see Frege, 1980, p. 39). Surely, by asserting or negating propositions, agents neither create nor destroy them (Frege, 1918–1919b, p.  381–2). Propositions are objective entities, not ideas in individual minds (Frege, 1918– 1919a, p. 363). But for those objective entities to be effective, their truth has to be first acknowledged and then put forward (Frege, 1918–1919a, p. 356). Assertions and judgements are actions; they are acts of asserting and acts of judging (Frege, 1918–1919b, p.  381, n. 13), respectively. In those acts, propositions, which are those entities ‘for which the question of truth can arise at all’ (Frege op. cit., p. 353), are not built up out of previous and independent concepts and objects but are instead recognised and endorsed by agents who take responsibility for a content and present it as true. Assertion and judgement are primitive notions in Frege’s system: they can be characterised but not defined—‘we must not forget that not everything can be defined’ (Frege, 1918–1919b, p. 381). When the primitive nature of some notions is overlooked, we run the risk that we ‘fasten upon inessential accessories, and thus start the inquiry on a wrong track at the very outset’ (Frege loc. cit.). And at this point, Frege goes on to insist that ‘this is certainly what has happened to many people, who have tried to explain what a judgement is and soon have hit upon compositeness’ (Frege loc. cit.). The enterprise of identifying parts in the content of judgements thus leads us to ‘fasten upon inessential accessories’, and derives from a failure to recognise their primitive nature. Asserting and judging are not the only essential actions that Frege speaks of. A third conspicuous one is the action that an agent performs by using a sign to denote an object. In the Grundgesetze he calls this action ‘designation’: ‘I further say a name expresses its sense and denotes its denotation. I designate with the name that which it denotes’ (Frege, 1893, p.  35). Hence, as the former paragraphs show, Frege’s logic is a logic with a subject, even though this does not preclude it from being a method of representing and analysing objective abstract reality. Including the subject in the picture does not deliver the system into the grip of psychologism.

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3.5.2 The Predicables ‘Is True’ and ‘Is a Fact’ Assertion and the judgement stroke lead us to Frege’s treatment of truth. Before proceeding with this topic, it is essential to bear in mind the distinction between the notion of truth and the two truth values, the True and the False. They are essentially connected, since attributing the True or the False to a sentence as its meaning (Bedeutung) is the technical way that Frege finds to say that, in logic and science, sentences are true or false. Nevertheless, in his system, the True and the False are objects and, in some sense they have remained so in the tradition after Frege. Lukasiewicz introduced them in his three-valued and multi-valued calculi, Wittgenstein included them in order to explain the bipolarity of propositions, and so they have since become the customary way of representing the meaning of logical constants. Few scholars seem to have followed Frege in his bold move of treating them as genuine objects, but their role in semantics has remained basically unchanged. The expression ‘is true’, by contrast, is not a name and cannot name an object. The alternative category is that of a function, a concept in this case. Nevertheless, if truth is a concept, it is a very unusual one, since its standard position of being attached to a grammatical subject does not add any information to the content expressed by the subject itself. Its role is exactly the same as that of the judgement stroke in the Begriffsschrift, which Frege reads as ‘is a fact’, a very suggestive reading with extraordinary philosophical consequences. I will further develop this point in Chaps. 9 and 10. For both notions, ‘is a fact’ and ‘is true’, Frege vindicates some kind of semantic exceptionality. In the Grundgesetze he says of the judgement stroke that it cannot be ‘reckon(ed) neither among the names nor among the marks; it is a sign of its own special kind’ (Frege, 1893, p. 82). Concerning the content of the word ‘true’, he claims it to be ‘sui generis and indefinable’ (Frege, 1918–1919a, p. 353). Sui generis or not, what is clear is that these expressions are semantically and syntactically analogous, ‘is a fact’ and ‘is true’ are both grammatical predicables that do not add anything to the information contained in their grammatical subject. The only semantico-­pragmatic role reserved for them is that of expressing that the content in the subject is asserted, and thus they are ways of going from the sense expressed by a sentence to its meaning (truth-value). If both terms perform exactly the same task, then either they are not sui generis, and there is a semantic category which includes them both, or else they express one and the same concept. Either option leads to the same conclusion: that there are significant terms with a semantic role to play that nevertheless do not have either Sinn or Bedeutung in the standard Fregean sense. The semantic and pragmatic assimilation of the roles played by ‘is a fact’ and ‘is true’ supports the common intuition to which correspondentist approaches to truth respond. Thus, the source of the obviousness that accompanies correspondentism is not that sentences reflect an extralinguistic reality with which they share some kind of structure, which is quite debatable as Strawson’s brilliant comment on Austin’s correspondentist proposal exposes:

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With what type of state-of-affairs (chunk of reality) is the sentence ‘The cat is not on the mat’ correlated by conventions of description? With a mat simpliciter? With a dog on a mat? With a cat up a tree? (Strawson, 1950a, p. 21)

The source rather belongs to grammar: that ‘is a fact’ and ‘is true’ are interchangeable in most contexts because they share their logico-semantic job. Only stylistic reasons prevent their complete interchangeability (see Frápolli, 2013, chapter 2). That something is true if, and only if, it is a fact says something about grammar, but nothing at all about the world. For this reason, the basic intuition behind correspondence theories cannot be negated without oddity; the actual difficulty begins when theorists attempt to add conceptual flesh to this meagre linguistic remark. The Principle of Grammar Superseding (PGS), introduced in Chap. 2, enables the acknowledgement of the peculiarity of some grammatical predicates, and permits us, probably for the first time, to tackle semantically sophisticated notions that resist standard analysis. This was the case with quantifiers (see Chap. 4), and with ‘is a fact’: We can imagine a language in which the proposition ‘Archimedes perished at the capture of Syracuse’ would be expressed thus: ‘The violent death of Archimedes at the capture of Syracuse is a fact’. To be sure, one can distinguish between subject and predicate here, too, if one wishes to do so, but the subject contains the whole content, and the predicate serves only to turn the content into a judgement. Such a language would have only a single predicate for all judgements, namely, ‘is a fact’. We see that there cannot be any question here of subject and predicate in the ordinary sense. Our ideography is a language of this sort, and in it the sign ⊢ is the common predicate for all judgements. (Frege, 1879, §3)

Consequential as the rejection of the logical analysis in terms of subject and predicate is, this text goes further than that and suggests that the sentential structure does not correspond to any alleged organisation of the concepts involved. A grammatical predicate in a sentence does not need to express any substantial aspect of the sentence’s content. It can be idle in terms of the assertoric content expressed, albeit not devoid of an essential role, which can be syntactic and/or pragmatic. For ‘is a fact’ Frege reserves two very specific roles. The first belongs to syntax and it is the role of restoring sentencehood. The two expressions, ‘Archimedes perished at the capture of Syracuse’ and ‘The violent death of Archimedes at the capture of Syracuse’, share their content; they carry the same information. The syntactic difference between them is nevertheless patent. The first expression is a complete sentence, whereas the second one is a complex singular term. Syntax, semantics and pragmatics are more closely related than we usually acknowledge. The possible roles an expression can play are usually seen in syntax. Singular terms denote and refer, but they do not express true or false contents. Expressions other than complete sentences can be neither premises nor conclusions in arguments, and can be neither true nor false. Thus, the syntactic role of ‘is a fact’ has the semantic effect of allowing the content in its grammatical subject to stand in inferential relations with other contents, and to fill the gap after the that-clause that is attached to certain verbs. The second specific role of ‘is a fact’ is pragmatic: by attaching it to a singular expression with propositional content, the agent acknowledges its truth. Not all singular terms carry propositional content, though: ‘The violent death of Archimedes at the

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capture of Syracuse’ does, whereas ‘my husband’s mother’ does not. For this reason, Frege restricts the type of terms that can accompany the judgement stroke (Frege, 1879, §2). The identification of the judgement stroke with Frege’s notion of truth has sometimes been defended (see, for instance, Greimann, 2000). I will at this point remain neutral about it, but I do want to stress that Frege grants similar irrelevance in terms of assertoric content to ‘is true’: ‘the sense of the word “true” is such that it does not make any essential contribution to the thought’, he says in ‘My Basic Logical Insights’, a passage that I have reproduced in the previous chapter. He continues: This may lead us to think that the word ‘true’ has no sense at all. But in that case a sentence in which ‘true’ occurred as a predicate would have no sense either. All one can say is: the word ‘true’ has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate. (Frege, 1915, p. 251)

In this passage, Frege hints to two senses of ‘sense’: semantic value and assertoric content. The term ‘true’ is a meaningful word with a semantic value; otherwise, the whole sentence in which it occurs would be senseless. This sense of ‘sense’ connects it to linguistic items and is governed by compositionality. Nevertheless, when it is said that the sense of ‘true’ does not contribute to the thought, it is the assertoric content of the asserted sentence that goes under the spotlight. If the sense of ‘true’ does not contribute to the thought, what might be the role that it performs? Frege adds that the role of truth indicates the essence of logic, and compares it with the roles of ‘good’ in ethics and ‘beautiful’ in aesthetics (Frege loc. cit., 1918–1919a, p.  351). This comparison of the meaning of ‘is true’ with the meanings of ‘is beautiful’ and ‘is good’ has deep significance. His central criticism of the interpretation of truth as correspondence consists of asking what could possibly be those aspects of reality that language corresponds with. But even once pinpointed, the issue is not settled, for we could always ask for any one of them whether it is really true that these aspects are present, and thus we could continue ad  infinitum (Frege, 1918–1919a, p.  353). This argument is strikingly similar to Moore’s Open Question Argument, which Gibbard calls the ‘“What’s at issue?” Argument’ (Gibbard, 2012, p. 9). Truth is a remarkable notion that makes ‘the impossible possible: it allows what corresponds to the assertor force to assume the form of a contribution to the thought’ (Frege loc. cit.). When ‘is true’ occurs in a sentence, the nature of the speech act as an assertion becomes explicit (Brandom, 1984, chapter 5; Frápolli, 2013, chapter 4, Strawson, 1950a). Nevertheless, the occurrence of ‘is true’ is neither necessary nor sufficient for a linguistic act to be an assertion. It is not necessary, since ordinary sentences can be asserted without any occurrence of a truth term. The role of truth is merely to make explicit that an assertion has been performed, even though it can be performed without its help. It is not sufficient, since in the absence of certain other circumstances, the mere occurrence of ‘is true’ does not turn a non-assertive act into an assertion. An actor onstage can include the predicate ‘is true’ in their allocutions without the act being a genuine assertion (but see Chap. 10). The performance of the act requires the agent to speak seriously, i.e. it requires the agent’s

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intention to perform an assertion. The connection between assertion and truth makes clear that the mere uttering of words is not enough, and that something else needs to be done by the agent. And this extra step belongs to pragmatics and relates to the speaker’s attitudes towards the contents of her speech acts. Frege was perfectly aware of this pragmatic issue. Further semantico-pragmatic remarks speak to the semantic peculiarity that Frege attributes to ‘is true’, some of which are also shared by ‘is good’ and ‘is beautiful’. The first explains that the ‘meaning of the word “true” is spelled out in the laws of truth’ (Frege, 1918–1919a, p. 352). The laws of truth are not the laws of thinking, but the laws of thought. Frege does not develop this idea, but some illumination could be provided by a second remark that he makes earlier in the same paper: that from ‘the laws of truth there follow prescriptions about asserting, thinking, judging and referring’ (Frege op. cit., p. 351). Both of these remarks belong to ‘Thoughts’, one of the last works Frege published, and consequently cannot be dismissed as pertaining to a youthful period or to an immature way of approaching the issue. Nevertheless, they clash with the standard explanation of the semantic contribution of an expression in terms of sense and meaning. The meaning of ‘true’ is displayed in certain rules from which something about the acts of asserting and inferring follows. Frege does not say this, but one possibility is to understand these remarks along the lines of the later speech act theory, attending to pragmatic features in general. Truth would thus be the norm of assertion, as represented by the quality maxim in Grice’s work: (do not assert what is not true), would define what it is to make a judgement (the acknowledgment of the truth of a content by an agent), and would help to characterise inferences: whatever follows from a true content is true. These ideas would have to wait several decades to be explicitly and thoroughly formulated, but Frege points in this direction rather than that of the extreme syntacticism that characterises contemporary logic’s dominant path. I will further pursue the meaning of truth and Frege’s approach to this in Chaps. 9 and 10. Let us conclude this subsection with a comment on ‘is a fact’. This predicate is the ordinary reading of the judgement stroke that Frege introduces in the Begriffsschrift, §2. Like ‘is true’, it does not contribute any component to the judgeable content, since all substantial information is placed in the grammatical subject of the sentence. Also like ‘is true’, it permits a transition from the sense of a sentence to its meaning, and shows that an assertion is being performed. This connection between truth and fact is explicitly stated in (Frege, 1918–1919a, p. 386): For what I have called thoughts stand in the closest connection with truth. What I acknowledge as true, I judge to be true quite apart from my acknowledging its truth or even thinking about it. That someone thinks it has nothing to do with the truth of a thought. ‘Facts, facts, facts’ cries the scientist if he wants to bring home the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true.

Facts are those contents whose truth is acknowledged in acts of judgement, and which are put forward in acts of assertion. What is essential to understanding the semantics of both of these predicates is that they are deeply connected and that their meaning bears no relation to configurations of objects or state-of-affairs. There are

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true propositions dealing with numbers, planets, emotions and Olympic gods. It is true that I love my children, and thus that I love my children is a fact. It is true that four times two is eight, and thus that four times two is eight is a fact. And this circumstance does not force us to posit any pseudo-physical realm in which love and numbers can reside, as tables and coffee mugs reside in the world. Facts are a subclass of propositions, and as such, they are abstract and objective entities. Many years later, Strawson (Strawson, 1950a) insisted on a point which is already present in Frege’s thought: In the same sense of ‘about’, we talk about facts; as when we say ‘I am alarmed by the fact that kitchen expenditure has risen by 50 per cent. in the last year’. But whereas ‘fact’ in such usages is linked with a ‘that’-clause (or connected no less obviously with ‘statement’ as when we ‘take down the facts’ or hand someone the facts on a sheet of paper), ‘situation’ and ‘state of affairs’ stand by themselves, states of affairs are said to have a beginning and an end, and so on. […] Being alarmed by a fact is not like being frightened by a shadow. (Strawson op. cit., p. 8)

3.5.3 Implicatures and Presuppositions Subordinate clauses also defy the standard interpretation of Frege’s semantics. Their semantic complexity is not wholly accounted for by the expedient of making their customary sense their meaning and granting them an indirect sense. This move works for some subordinate clauses but not for others. Definite descriptions are among the clauses that call for an alternative explanation: ‘We now come to other subordinate clauses, in which the words do have their customary meaning without however a thought occurring as sense and a truth-value as meaning’ (Frege, 1892a, p.  168). His example is ‘Whoever discovered the elliptic form of the planetary orbits’ in ‘Whoever discovered the elliptic form of the planetary orbits died in misery’. The description does not have a complete thought as its content, Frege contends, since if it did this complete thought could be independently expressed. To establish this claim, Frege offers an explanation that avant la lettre is a complete rejection of Russell’s theory of descriptions: One might object that the sense of the whole does contain a thought as part, viz. that there was somebody who first discovered the elliptic form of the planetary orbits; for whoever takes that whole to be true cannot deny this part. This is undoubtedly so; but only because otherwise the dependent clause ‘whoever discovered the elliptic form of the planetary orbits’ would have nothing to mean. If anything is asserted there is always an obvious presupposition that the simple or compound proper names used have meaning. (Frege loc. cit., my italics)

Frege insists that the presupposition holds for ‘simple or compound names’, for ‘Whoever discovered the elliptic form of the planetary orbits’ and also for ‘Kepler’. Both are saturated expressions with uniform semantic behaviour. For the project of a logically correct language in which inferences can be faultlessly carried out, natural languages are not suitable, because they contain expressions that do not

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designate an object unless some sentence is true. Frege explains that for these expressions to have a designation, ‘the truth of some sentence is a prerequisite’ (Frege op. cit., p. 169). This refinement of Frege’s approach leads him to suggest an interpretation of descriptions close to the pragmatist proposal later issued by Strawson in ‘On Referring’ (1950b). In this paper, Strawson, without mentioning Frege, also rejects Russell’s interpretation of descriptions in the latter’s ‘On Denoting’ (Russell, 1905). Recall that under the Russellian interpretation, definite descriptions, i.e. expressions such as ‘The king of France’, are incomplete symbols. They do not contribute an identifiable component to the thought expressed by the sentences in which they occur, since they are not singular terms from a logical point of view. Frege, Russell and Strawson coincide in the view that sentences with non-denoting definite expressions are nevertheless meaningful. The sentence ‘The king of France is wise’ cannot be said to be nonsensical, nevertheless one of its terms lacks reference and so cannot be used to mention or refer to anything. To deal with these cases, Russell puts forward his now classic theory of descriptions, whose main thesis is that sentences that include definite descriptions have as part of their contents implicit existential and unicity claims. Thus, ‘The king of France is wise’ conveys the information that there is one King of France and only one. When Russell wrote ‘On Denoting’ this information was false, as it was when Strawson wrote his criticism of Russell’s view, and as it still is today. The general implication of Russell’s approach for the analysis of sentences with non-denoting parts is that they are always false. By contrast, Frege and Strawson rely on actual language users’ intuitions to reject this outcome. Presented with sentences of this kind, Strawson contends, speakers would deny that they are false, but also that they are true (Strawson op.  cit., p.  330). Speakers would rather say that ‘the question of whether [the] statement was true or false simply didn’t arise, because there was no such person as the king of France’ (Strawson loc. cit.). And then he issues a pragmatist explanation close to contemporary expressivism: ‘And this brings out the point that if a man seriously uttered the sentence, his uttering it would in some sense be evidence that he believed that there was a king of France’ (Strawson loc. cit.). By using this sentence, speakers do not say that they believe that there is a king of France, as in Russell’s analysis, but instead show that there being a King of France is one of their beliefs. The information that there is a King of France is a presupposition of many of the sentences that include this description. Unlike Frege, Strawson does not use the term ‘presupposition’; he instead says that the use of a sentence of this kind ‘implies’ the existential claim, although ‘this is a very special and odd sense of “imply”. “Implies” in this sense is certainly not equivalent to “entails” (or “logically implies”)’ (Strawson loc. cit.). The type of presupposition that Frege and Strawson seem to have in mind is pragmatic and not semantic. Presupposition in this case is not a relation between two propositions or thoughts. It is something that involves the speaker and her beliefs. The notion of pragmatic presupposition in the twentieth century points to Stalnaker’s work (1973, 1974), and it was clearly prefigured in (Frege, 1892b) and (Strawson, 1950b).

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The analysis of subordinate clauses also introduces the pragmatist possibility that the conjunction of some sentences, one main sentence together with some subordinate ones, could express more independent thoughts than the sentences uttered. This happens when the conjunction of the sentences suggests a further thought that is not directly expressed by any of them. Frege’s example is ‘Napoleon, who recognized the danger to his right flank, himself led his guards against the enemy position’ (Frege, 1892b, p. 174). In this complex sentence, three thoughts are conveyed. The first one, expressed by the first sentence, is that Napoleon recognised the danger to his right flank. The second one, expressed by the second sentence, is that Napoleon led his guards against the enemy position. But a third thought is conveyed by the form in which these two sentences are conjoined. This third thought is that Napoleon’s acknowledgement of the situation made him lead his guards against the enemy position. Frege discusses whether this third thought is part of the sense of the complex sentence, i.e. ‘whether this thought is just slightly suggested or really expressed’ (Frege loc. cit.). The third thought is not ‘really expressed’, Frege argues, because the complex sentence could still be true even if this thought were false. The situation is thus the following. The truth of the complex clause requires the truth of the two first thoughts, as always happens with conjunctions. But the third thought could be false—Napoleon’s recognition of the situation not being the reason for the movement of his guards—and yet the complex sentence could still be true. Frege thus concludes that ‘the subsidiary thought should not be understood as part of the sense’ (Frege loc. cit.). This discussion of the connection between thoughts and sentential clauses brings up the difference between actually expressing a thought and merely suggesting it. In more contemporary terminology, this is the difference between saying something and pragmatically conveying it, a topic that Grice put on the table with his 1975 paper ‘Logic and Conversation’. Grice called the suggested thought an ‘implicature’, which can be taken back without affecting the truth of what is said. This is the cancellability feature of conversational implicatures that Grice explicitly stated, and which Frege also recognised. For Grice, the implicature is not part of what is said, of the proposition expressed (Grice op. cit., p. 58). For Frege, the suggested thought is not part of the sense of the complex sentence—two alternative ways of pointing to the same pragmatic phenomenon, of which Frege was completely aware. Further pragmatic hints are appreciated in Frege’s take on logical constants and quantifiers, to which I turn in the next chapter.

References Bell, D. (1990). How ‘Russellian’ was Frege? Mind, 99, 267–277. Blackburn, S. (1999). Think: A compelling introduction to philosophy. Oxford University Press. Brandom, R. (1984). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press. Brandom, R. (2000). Articulating reasons. An Introduction to Inferentialism. Harvard University Press.

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Burge, T. (1986). Frege on truth. In L.  Haaparanta & J.  Hintikka (Eds.), Frege synthesized. Essays on the philosophical and foundational work of Gottlob Frege (pp. 97–154). D. Reidel Publishing Company. Carnap, R. (1937). Logical syntax of language (The international library of philosophy). Routledge. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. Dummett, M. (1973). Frege. Philosophy of language. Haper and Row Publishers. Dummett, M. (1991). Frege: Philosophy of mathematics. Harvard University Press. Field, H. (1984). Platonism for cheap? Crispin Wright on Frege’s context principle. Canadian Journal of Philosophy, 14, 637–662. Forero-Mora, J.  A., & Frápolli, M.  J. (2021). Show me. Tractarian non-descriptivism. Teorema XL, 2, 63–81. Frápolli, M. J. (2013). The nature of truth. Springer. Frápolli, M. J. (2015). Non-representational mathematical realism. Theoria, 30(3), 331–348. Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jean van Heijenoort (1967). From Frege to Gödel. A source book in mathematical logic, 1879–1931. Harvard University Press, 1–82. Frege, G. (1880/1). Boole’s logical calculus and the concept-script. In G.  Frege (1979), Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach. Basil Blackwell, 9–46. Frege, G. (1882). Boole’s logical formula-language and my concept-script. In G. Frege (1979), Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach. Basil Blackwell, 47–52. Frege, G. (1884/1953). The foundations of arithmetic. A logic-mathematical enquiry into the concept of number. Translated by J. L. Austin. Second Revised Edition. Harper Torchbooks/The Science Library, Harper & Brothers. Frege, G. (1891). Function and concept. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 137–156. Frege, G. (1892a). Concept and object. In G.  Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 182–194. Frege, G. (1892b). On sense and meaning. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 157–177. Frege, G. (1893/1964). The basic laws of arithmetic. Exposition of the system. Translated and Edited, with an Introduction by Montgomery Furth. University of California Press. Frege, G. (1915). My basic logical insights. In G. Frege (1979), Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach. Basil Blackwell, 251–252. Frege, G. (1918–1919a). Thoughts. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 351–372. Frege, G. (1918–1919b). Negation. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 373–389. Frege, G. (1923–26). Compound Thoughts. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Oxford, Basil Blackwell, 390–406. Frege, G. (1980). Philosophical and mathematical correspondence. Blackwell Publishers. Furth, M. (1964). Introduction to Frege’s the basic laws of arithmetic (Exposition of the system) (pp. i–lx). University of California Press. Gibbard, A. (2012). Meaning and normativity. Oxford University Press. Greimann, D. (2000). The judgement-stroke as a truth-operator: A new interpretation of the logical form of sentences in Frege’s scientific language. Erkenntnis (1975-), 52(2), 213–238. Retrieved September 28, 2020, from http://www.jstor.org/stable/200129 Greimann, D. (2007). Essays on Frege’s conception of truth: 75 Grazer Philosophischen Studien. Greimann, D. (2014). Frege on truth, Assertoric force and the essence of logic. History and Philosophy of Logic, 35(3), 272–288. Hacker, P.  M. S. (2001). Wittgenstein: Connections and controversies. Clarendon. Available at Oxford Scholarship Online: April 2005.

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Hacker, P. M. S., & Baker, G. P. (1984). Frege: Logical excavations. Oxford University Press. Kripke, S. (1980). Naming and necessity. Harvard University Press. Linnebø, Ø (2008). Compositionality and Frege’s Context Principle. Ms., invited talk given at the CSMN Workshop on Reference (Oslo 2007) and London Logic and Metaphysics Forum (November 2007). Linsky, L. (1977). Names and descriptions. University of Chicago Press. Macbeth, D. (2005). Frege’s logic. Harvard University Press. (kindle edition). MacBride, F. (2003). Speaking with shadows: A study of neo-logicism. British Journal for the Philosophy of Science, 54(1), 104–163. Pedriali, W. (2017). The logical significance of assertion. Frege on the essence of logic. Journal for the History of Analytical Philosophy, 5(8), 1–22. Russell, B. (1903). The principles of mathematics. Cambridge University Press. Russell, B. (1905). On denoting. Mind, New Series, 14(56), 479–493. Russell, B., & Whitehead, A. N. (1927). Principia mathematica. Cambridge University Press. Salmon, N. (1986). Frege’s puzzle. Schröder, E. (1880). Gottlob Frege, Begriffsschrift. Zeitschrift für Mathematik und Physik, 25, 81–94. Sluga, H. (1987). Frege against the Boleans. Notre Dame Journal of Formal Logic, 28(1), 80–89. Stalnaker, R. (1973). Presuppositions. The Journal of Philosophical Logic, 2, 447–457. Stalnaker, R. (1974). Pragmatic presuppositions. In M. Munitz & P. Unger (Eds.), Semantics and philosophy (pp. 197–214). New York University Press. Strawson, P.  F. (1950a/2013). Truth. Proceedings of the Aristotelian Society. The virtual Issue 1, 1–23. Strawson, P. F. (1950b). On referring. Mind, New Series, 59(235), 320–344. Van der Schaar, M. (2017). Frege on judgement and the judging agent. Mind, 127(505), 225–250. Van Heijenoort, J. (1967). Logic as calculus and logic as language. Synthese, 17(1), 324–330. Vilkko, R. (1998). The reception of Frege’s Begriffsschrift. Historia Mathematica, 25, 412–422. Wiggins, D. (1965). Identity statements. In R. J. Butler (Ed.), Analytic philosophy (2nd ed., pp. 40–71). Basil Blackwell. Wittgentein, L. (1914–16/1998). Notebooks 1914–1916. Blackwell Publishers.

Part II

Logical Constants

Chapter 4

Implying, Precluding, and Quantifying Over: Frege’s Logical Expressivism

Abstract  There are 16 bivalent binary truth functions, but only two fundamental logical relations between propositions: implying and precluding. In addition, some higher-level concepts express the scope and nature of inferential relations. In this chapter, I will explain Frege’s treatment of these two notions, which are the semantic and pragmatic support for conditionality and negation. I will also offer an explanation of Frege’s semantic account of universal and existential quantifiers, and the relations between them. The richness and originality of Frege’s semantics manifest in his treatment of the role of logical notions and related expressions. Frege’s logic is nothing more (and nothing less) than his semantics for logical terms together with an appropriate method for representing inferential transitions. The deductive system that converts a language into a calculus is the representation of the commitments and entitlements that individuate concepts and propositional contents. The very notation that Frege introduces shows that logical notions do not affect judgeable contents, which systematically occur as arguments of the former. In this chapter, I will argue that Frege defended an expressivist approach to the meaning of logical notions. Frege’s insights at this point can help in the as-yet unfinished task of giving a satisfactory semantics for logical notions. I consider Frege’s views and suggestions entirely correct. Keywords  Bipolar · Conditional · Conjunction · Disjunction · Expressivism [expressivist] · Frege-Geach argument · Implying · Incompatibility · Falsehood · Negation

4.1 Logical Expressivism As it happens with many terms of art in philosophy, ‘expressivism’ is polysemous and applies to a vast array of semantic positions. Beyond logical theory, it usually suggests the internalist semantic view that some philosophers, specifically Odgen and Richards, Stevenson, and Ayer, put forward in the first half of the twentieth century to explain the meaning of ethical terms (Odgen & Richards, 1923; Stevenson, 1937; Ayer, 1936). For a word to be meaningful, logical positivism required it to © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_4

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refer (or be reducible to reference) to observable entities. Evaluative terms, of which ‘good’ is the paradigm, do not fit into this picture, and yet, to catalogue them as ‘meaningless’ would be highly counterintuitive. These terms, the early expressivists assumed, did not possess descriptive meaning, but instead had work to do in the expression of certain attitudes held by speakers towards particular contents. They possessed what these philosophers called ‘expressive’ meaning because their role was to give voice to certain feelings and mental states instead of describing how things are. Following the early expressivists, metaethics has profusely appealed to the expressivist intuition, albeit with the many additions made by the various authors who have refined, complemented, and extended it in order to amend some of the obvious flaws in the original position. The most troublesome semantic outcome of some versions of ethical expressivism is that sentences with ethical (or, in general, evaluative) terms do not express propositions. They are instead treated as similar to interjections or exclamations, in that they serve to display emotions with no propositional content. In particular, evaluative sentences, such as ‘this is good’ or ‘this is wrong’, can be neither true nor false; and this implies that they cannot fill the argument places of truth-functional terms. This consequence affects their logical properties since, being unable to be true or false, evaluative sentences could neither be part of inferences nor be connected or modified by logical constants. Evaluative sentences (‘Torturing animals is wrong’, ‘Protecting children at risk of social exclusion is good’) could not occur in the antecedents of conditionals or be negated. This semantic consequence of expressivism is the core of what is known as the ‘Frege-­ Geach Argument’ against non-descriptivism, a criticism sometimes also known as the ‘embedding problem’ and, when it involves negation, the ‘negation problem’. The former label, by far the most common one, refers to a short argument that the British philosopher Peter Geach included in his paper ‘Ascriptivism’: There is a theory that to say ‘what the policeman said is true’ is not to describe or characterize what the policeman said but to corroborate it; and a theory that to say ‘it is bad to get drunk’ is not to describe or characterize drunkenness but to condemn it. There is a radical flaw in this whole pattern of philosophizing […] for that would mean that arguments of the pattern ‘if x is true (if w is bad), then p; but x is true (w is bad); ergo p’ contained a fallacy of equivocation, whereas they are in fact clearly valid. (Geach, 1960, p. 222)

Geach mentioned in this text the Fregean distinction between predicating and asserting—and hence the ‘Frege-Geach Argument’—but any deeper reason to attribute this argument to Frege escapes me. The Frege-Geach Argument exposes the contrast between radical versions of expressivism, wherein evaluative sentences are devoid of propositional content, and the sound intuitions of competent speakers who would systematically accept as well-formed and meaningful complex sentences such as (a) and (b), (a) If torturing animals is wrong, then bullfighting should be forbidden. (b) Torturing animals is not wrong.

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in which evaluative sentences combine successfully with other sentences, and their overall contents are adjusted by logical constants, the conditional and negation in these particular cases (see Frápolli, 2019 for a discussion). Logical expressivism, the kind of semantic position that—as I have mentioned in previous chapters—is usually attributed to the early Wittgenstein, and, as I will argue, Frege also favoured, at least in some of his works, is not of the internal, mentalistic kind that Geach criticises. The two senses of ‘expressivism’, the internal sense applied in the realm of metaethics, and the external sense that concerns logical terms and other higher-level notions, should be carefully kept apart. This is not to say that the internalist version has never been applied to the meaning of logical constants (see, for instance, Besson, 2019), but this is not the kind of semantic theory that is detectable in Frege, or which Wittgenstein explicitly defended. The kind of logical expressivism that Frege manifested in some of his writings1 matches precisely what Wittgenstein explicitly claimed in the Tractatus (4.0312), i.e., that logical constants do not represent. Neither Frege nor Wittgenstein understood logical terms as either referential or descriptive devices; they were in both cases seen as instruments for expressing particular relations between propositional contents, and sometimes specific attitudes of support or rejection held by agents towards them. Brandom also defends a kind of expressivism that is alien to the internalist brand commonly found in metaethics. His expressivism, whose proximal precursors are Sellars and Frege, stresses that certain concepts, logical and modal concepts among them, serve the purpose of making explicit as assertions what is implicit in practices. Regarding logical vocabulary, its expressive role consists in its provision of a ‘distinctive set of tools for saying something that cannot otherwise be made explicit’ (Brandom, 2000, p. 19). As the paradigm of logical vocabulary with an expressive function, Brandom picks the conditional: The use of conditionals makes explicit as the content of a claim, and so something one can say, the endorsement of an inference —an attitude one could otherwise manifest only by what one does. (Brandom op. cit., p. 175)

Other higher-level vocabularies perform similar tasks. Terms such as ‘believes’ and ‘claims’ make doxastic commitment explicit (Brandom loc. cit.); modal terms, on the other hand, help express commitments to certain inferential relations—consequence and preclusion—which are subjunctively robust (Brandom, 2014, p. 171). As I have previously pointed out, I contend with Brandom that Frege was a logical expressivist, and that his expressivism goes further than covering logical notions. My defence of Frege’s expressivism clashes with two widely assumed interpretations of Frege’s theory. On the one hand, his binary and systematic approach to semantics and, on the other, the unsurmountable distance between his semantics and Wittgenstein’s. Both of these assumptions are debatable: neither does the pair sense

 As I will explain in the next section, Begriffsschrift explains the meaning of the conditional stroke (§5) and negation (§7) on expressivist lines. Some of Frege’s Logical Investigations, where he deals with generality (Frege, 1923) and negation (Frege, 1918–9b), also suggest this expressivist approach. 1

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and meaning apply to every kind of term in Frege’s semantics, as we have seen in the previous chapter, nor are Wittgenstein’s views in the Tractatus incompatible with Frege’s logical theory. In this chapter, I will offer evidence of Frege’s expressivist approach to the meaning of certain notions, and hence of his refusal to apply to them his standard analysis in terms of sense and meaning. The many coincidences between Frege and Wittgenstein will not be argued for directly, but they will be exposed whenever the occasion arises.

4.2 The Conditional and Negation Let us begin with the analysis of logical notions. Logical terms represent relations between propositions. Given any two propositions, there are only three possible scenarios: that one follows from the other; that they are incompatible; and that they are compatible, with neither of them following from the other. Compatibility without consequence has no relevance for inferences since inferences require some kind of connection between propositions in terms of commitments, and compatibility is instead the expression of indifference. With compatibility off the table, only consequence and incompatibility remain as possible relations, whose linguistic representations are, respectively, the conditional and negation. Once (PPP), (PII), and (OI) are assumed, the connection between propositions and the basic logical relations between them—consequence and incompatibility—reveals itself as going in both directions. Consequence and incompatibility are the basic logical notions and, at the same time, they serve as witnesses of the presence of propositions, since propositions are those items that can stand in these relations. This complex intuition is a mark of pragmatism that is systematically detectable in authors of the last 200 years with pragmatist sensibilities, from Hegel to Brandom (Brandom, 2019, pp. 2–3, 9). I do not want to suggest that Frege had a fully elaborated account of the meaning of logical constants. He did not. Nevertheless, an implicit acknowledgement of their unique nature, which he stresses in the way in which they are represented, pops up now and then in his writings. One example is his defence of his bidimensional writing, as opposed to Peano’s notation: [I]n tabular lists (…) the two-dimensional expanse is utilised to achieve perspicuity. In much the same way I am trying to do this in my conceptual notation. I attain a clear articulation of the sentence by writing the individual clauses — e.g. consequent and antecedents — one beneath the other, and to the left, by means of combination of strokes, I exhibit the logical relation which binds the whole together. I mention this because efforts are now being made to squeeze each formula on to one line. In the Peano conceptual notation the presentation of formulas upon a single line has apparently been accomplished in principle. To me this seems a gratuitous renunciation of one of the main advantages of the written over the spoken. After all, the convenience of the typesetter is certainly not the summum bonum. (Frege, 1897, p. 236)

Macbeth offers a detailed explanation of the sense in which Fregean formulae are similar to tables, and why they are more perspicuous from a logical point of view.

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Given Frege’s assumption that equivalent sentences express the same thought, the bidimensional representation shows the mutual relations between the components of the thought that can subsequently be rendered in different equivalent unidimensional formulae (Macbeth, 2005, §2.2). The analysis of logical constants is not disconnected from the overall conception of the role of logic. As will be evident in what follows, understanding the meaning of logical notions requires keeping apart the level of thought—of judgeable contents—from the level of their expression. It also requires distinguishing between the representation of judgeable contents and the expression of their inferential connections. The ‘inner nature’ of thoughts grounds the objective logical relations between them (Frege, 1879, p. 5), whose explicit representation is achieved by the introduction of logical terms. The semantics of logical terms requires the concurrence of these two levels—the level of content and the level of its expression—and the formalist practice of disregarding the former, as if the only concern of logic were the linguistic surface, makes it impossible to give a complete characterisation of logic’s essential notions. Frege’s first mention of the interest of his conceptual writing for correctly identifying the role of logical constants occurs in the Preface of his Begriffsschrift, where he claims that the ‘demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all and so forth, deserves attention’ (Frege op. cit., p. 7). Of these, only if, not, and all are primitive. How they are represented in Frege’s conceptual writing, where logical terms do not belong to any of the two kinds of signs introduced in §1, i.e. variables and constants, might give us a clue about their semantic specificity. Frege represents the conditional with a combination of vertical and horizontal strokes. The following is a representation of the conditional ‘If B then A’: A B

Negation, on the other hand, is represented by a short vertical stroke attached below the content stroke. The following is a representation of the negation of the content A: A

The similarity between these representations of conditional and negation, on the one hand, and assertion as represented by the judgement stroke, on the other, suggests that logical notions also codify the (possible) performance of special kinds of acts. In an article in which he compares Boole’s logical calculus with his conceptual writing, Frege acknowledges that he ‘deviate(s) from the usual practice in drawing a distinction between judgement and content of possible judgements’ and then lists, and comments on the functions of, the content-stroke, the judgement-stroke, the negation-stroke and the conditional-stroke (Frege, 1880–1, p.  11). In the

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Begriffsschrift, Frege’s verbal explanation of the meanings of these ‘strokes’ also supports this interpretation. Concerning two judgeable contents, A and B, there are four possible situations: that both are affirmed, that both are denied, that A is affirmed and B denied, and that B is affirmed and A denied. The conditional formula A B

represents the situation that it cannot happen that B is affirmed and A denied (Frege, 1879, §5). Significantly, Frege does not use the words ‘true’ and ‘false’ in this context but refers instead to acts of affirming and denying. A conditional judgement thus conveys the information that the antecedent cannot be affirmed without thereby affirming the consequent, even though the thoughts in the antecedent and the consequent are not themselves asserted (Frege, 1880–1, p. 11, 1918–19b, p. 375). The conditional judgement then indicates the commitment to the consequent that is acquired by an agent who asserts the antecedent. An alternative explanation, given in ‘On Sense and Meaning’, is that a ‘hypothetical thought establishes a reciprocal relationship between two thoughts’ (Frege, 1892b, p. 171). When the antecedent and consequent clauses do not include ‘indefinite indicators’ (Frege, loc. cit.), as in ‘If the moon is in quadrature with the sun, the moon appears as a semicircle’ (Frege, 1879, §5), the conditional judgement expresses three thoughts: the (higher-level) conditional thought, the thought in the antecedent, and the thought in the consequent. When, by contrast, they include pronouns or variables of any kind that connect the information in the two clauses, there is a single thought expressed by the conditional judgement, since the connected clauses are incomplete and thus unsuited to expressing thoughts by themselves. An example of this kind is: ‘If a number is less than 1 and greater than 0, its square is less than 1 and greater than 0’ (Frege, 1892b, p. 171). The conditional, adverbial, and noun clauses are, in this case, the instruments for building complex names, which is what conditional sentences are. This analysis of conditional sentences prompts the semantic question of the contribution of these auxiliary expressions (conditionals and other kinds of adverbs) to the conditional thought. And the answer that Frege offers shows that he considers them to be semantically ‘peculiar’: Subsidiary clauses beginning with ‘although’ also express complete thoughts. This conjunction actually has no sense and does not change the sense of the clause but only illuminates it in a peculiar fashion. (Frege, 1892b, p. 172)

And he adds in a footnote: ‘Similarly in the case of “but”, “yet”’ (loc. cit., n. 16). Thus, these terms that conjoin sentences escape the binary template of sense and meaning, without being nonsensical or empty. They perform a specific semantic (or might be tempted to say ‘pragmatic’) task other than expressing a sense or referring to a reference. Logical constants belong to this auxiliary category. The analysis of negation, another primitive logical notion in the Begriffsschrift, proceeds along similar lines. Negation sometimes indicates the dual act of the act of assertion, but this is not the sense that primarily interests Frege, who prefers not to

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distinguish two kinds of judging, one negative and one positive (Frege, 1918-9b, pp. 382–3). As a logical notion, negation applies to the content of a possible judgement, and not to the act of judging. But even so, Frege calls it an ‘adjunct’ to, and not an ingredient of, a content that is not ‘advanced as a judgement’ (Frege, 1879, §4). The way of representing negation in the Begriffsschrift—after the judgment stroke, below the content stroke, and before the representation of the judgeable content—indicates that it neither modifies the judgement stroke, which would mean the rejection of the act itself, nor forms part of the judgeable content. Its function lies somewhere in-between. This early characterisation does not change in later works: People speak of affirmative and negative judgments: even Kant does so. Translated into my terminology, this would be a distinction between affirmative and negative thoughts. For logic at any rate such a distinction is wholly unnecessary: its ground must be sought outside logic. I know of no logical principle whose verbal expression makes it necessary, or even preferable, to use these terms. In any science in which it is a question of conformity to laws, the thing that we must always ask is: What technical expressions are necessary or at least useful, in order to give precise expression to the laws of science? What does not stand this test cometh of evil. What is more, it is by no means easy to state what is a negative judgement (thought). Consider the sentences ‘Christ is immortal’, ‘Christ lives for ever’, ‘Christ is not immortal’, ‘Christ is mortal’, ‘Christ does not live forever’. Now which of the thoughts we have here is affirmative, which negative? (Frege, 1918–9b, p. 380)

A similar insight is found in the Tractatus (4.0621): ‘That, however, the signs “p” and “~p” can say the same thing is important, for it shows that the sign“~” corresponds to nothing in reality.’ The notion of ‘negative thought’ performs no logical role. If we take seriously the inferential individuation of thoughts that Frege defends in the Begriffsschrift, §3, the two sentences ‘Christ is not mortal’ and ‘Christ lives forever’ express the same thought. Everything that follows from the former, with the addition of auxiliary premises, follows from the latter as well, once it is combined with the same set of premises. Thus, whereas some sentences are negative, the occurrence of negation in them does not make the corresponding thought ‘indubitably negative’ (Frege op. cit., p. 380). Even so, negation is not idle. The sense of a sentence in which a negation occurs includes the sense of the negative particle (Frege loc. cit., p. 382). The occurrence of negation in the sentence, as happens with the occurrence of any other term, must have an effect on the thought expressed since it changes the inferential properties of the information conveyed. Nevertheless, this does not make the thought negative. ‘Negative thought’, as much as ‘positive thought’, is the result of a category mistake. Being negative and being affirmative, Frege conjectures, do not seem to apply to propositions, but rather to their expressions. Other classical classificatory predicates, such as categorical, apodictic, disjunctive, and hypothetical, face the same fate since they possess ‘only grammatical significance’ (Frege, 1879, §4), and thus do not ‘affect the conceptual content of the judgement’. The whole essay ‘Compound Thoughts’ (Frege, 1923–26), one of Frege’s Logical Investigations, is an elaboration of the distinction between contents and the multiple ways of expressing them.

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Contents are individuated by their inferential connections, disregarding the grammatical structure of the sentences. Two formulae such as ‘A v B’ and ‘¬(¬A & ¬B)’, when the variable letters are systematically and consistently substituted by ordinary sentences, represent the same thought. These two logical forms are equivalent, and the thought expressed by their instances is, in itself, neither negative nor disjunctive. Now, how we choose to linguistically represent a thought depends on the aims of the communicative or inferential acts in which we are immersed. Still, while the way of expressing a content as disjunctive, negative, etc., may be left to the speaker, some properties of the contents expressed are independent of such communicative aims. That the two contents Christ is mortal and Christ lives forever cannot be simultaneously asserted is not for the speaker to decide. These two sentences express incompatible thoughts, and incompatibility is an objective relation between thoughts, based on their inferential properties. Negation, by contrast, is an optional feature that ‘belongs with the expression’ of thought (Frege, 1918–9b, p. 384). It belongs to language and, as (PGS) establishes, ‘languages are unreliable on logical questions’ (Frege op. cit., p. 381). Negation, at the linguistic level, corresponds to incompatibility at the level of content. The possibility for negation, which is a monadic function, to represent incompatibility, which is a binary relation, rests on the essentially dual nature of thoughts. Frege explicitly states the dual nature of thoughts in his discussion of the role of negation: Thus for every thought, there is a contradictory thought, we acknowledge the falsity of a thought by admitting the truth of its contradictory. The sentence that expresses the contradictory thought is formed from the expression of the original thought by means of a negative word. (Frege, 1918–19b, p. 385)

Propositions, thoughts, are not only individuated by the thoughts that follow from them but also by the thoughts that are incompatible with them. In ‘Logic’, one of the works published in the collected Posthumous Writings, Frege insists on the same insight: The content of any truth is ‘a content of possible judgement’, but so too is the opposite content. This opposition or conflict is such that we automatically reject one limb as false when we accept the other as true, and conversely. The rejection of the one and the acceptance of the other are one and the same. (Frege, 1879–91, p. 8)

The Wittgensteinian insight that elementary propositions divide the logical space into two parts captures the same intuition (Ammereller & Fischer, 2004, p. 6). In the Tractatus (2.11), Wittgenstein writes: ‘The picture presents the facts in logical space, the existence and non-existence of atomic facts. The picture is a model of reality.’ Pictures thus represent the whole of logical space. Wittgenstein also suggests that in any proposition, all reality is given, the part that agrees with it and the part that does not: The proposition determines reality to this extent, that one only needs to say ‘yes’ or ‘no’ to it to make it agree with reality. Reality must therefore be completely described by the proposition. (Tractatus, 4.023)

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And in ‘Notes on Logic’, we read: Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition. (Wittgenstein, 1914/1998, p. 52)

Most logicians and philosophers of logic share this intuition. The content of a sentence is inversely proportional to the amount of information it precludes. A conspicuous example is Popper’s claim: ‘The empirical content of a statement p is the class of its potential falsifiers’ (Popper, 1935, p. 103). The role of negation is then to identify which one of the two poles is accepted, and which is rejected. It also allows the expression of contents in a suitable way appropriate for triggering some inferential moves, such as those codified by the modus ponens or the disjunctive syllogism. The latter rule is a suitable candidate for representing negation’s mixed nature as syntactically monadic and semantically binary. Understood against the background of the essentially bipolar character of propositions, negation emerges as an intricate function that defies the standard classifications. Its semantics derives from incompatibility, and its import cannot be completely disclosed unless propositional bipolarity is acknowledged. On the other hand, it serves to discard one of the opposing contents. Hence, from a semantic as much as from a pragmatic perspective, it represents a binary relation. And nevertheless, it is syntactically monadic, i.e. it yields a well-formed sentence when applied to one well-formed sentence. Negation’s semantic complexity explains its resistance to being defined within the framework of proof-theoretic semantics, in which the meaning of logical terms has to be given by the rules that govern their introduction and elimination. In possible worlds semantics, nevertheless, negation can be characterised in terms of compatibility and incompatibility, as a quantifier over possible worlds (Berto, 2015).

4.3 Negation, Incompatibility, Falsehood Negation, incompatibility and falsehood belong to the same semantic family and are inter-definable. Standard and non-standard logical calculi include (different types of) negation among their constants, and logicians are willing to consider this the basic notion from which incompatibility and falsity derive. Nevertheless, this familiar picture blurs some essential category differences between them, whereby interdefinability is the ladder for moving up and down between the classical layers of discourse analysis. Before proceeding to characterise these three notions, a word about interdefinability is in order. Formalist logical theory, unlike for instance the intuitionist approach, has made extensive use of the interdefinability of logical terms. There are reasons for that since it is of some methodological value to keep primitive notions to a minimum in science. Nevertheless, understanding the divergencies in meaning

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between basic concepts is as vital as acknowledging their similarities. And in the context of the formalist approach to logic, the focus has been so dramatically narrowed that interdefinability offers hardly any illumination of the essential characteristics of the interconnected notions. An example is offered by disjunction and conjunction. The algebraic approach to the meaning of logical constants tells us that conjunction and disjunction can be mutually defined using negation. Within the realm of algebra, intersection and union are operations of the same type: both binary functions by means of which some sets can be constructed from some others. The analogous situation in the case of logic requires the substitution of sets by truth values, and thus the analogy works if truth values are the sole aspect of meaning to be considered. This perspective reduces logic to extensions and severs any relation with inferential acts. The interdefinability of disjunction and conjunction obscures some relevant differences such as for instance the fact that conjunction, unlike disjunction, is an instrument for forming complex concepts out of simpler ones. Their algebraic counterparts, union and intersection, are instruments for the formation of complex sets. Nevertheless, disjunction does not build up disjunctive concepts, although conjunction does build up concepts that are conjunctive. Whereas ‘being a kind politician’ can be argued to be a single concept which involves the characteristics—in Frege’s sense—of kindness and of being a politician, there is something artificial in the parallel claim that ‘being either a politician or a physician’ is genuinely a single concept. One reason for setting apart conjunction in this sense comes from linguistics. Some linguists with a pragmatist orientation (see, for instance, Schiffrin, 1986, 1987; Blakemore, 1987; Carston, 2002) have lent support to an interpretation of conjunction known as ‘the single processing unit’ approach. This approach finds its natural background in relevance theory, as initiated by the seminal work of Sperber and Wilson (1986) and applies primarily to conjoined sentences. Nevertheless, its extension to conjunctive concepts is straightforward. As Carston explains, following Blakemore: when a speaker produces an explicit conjunction, it is that complex conjoined proposition that carries the presumption of optimal relevance and not the constituent propositions (the conjuncts) individually. (Carston, 2002, p. 243)

At the syntactic level, this approach implies that conjoined subjects work as syntactic units and therefore take verbs in the singular. Carston (op. cit., p. 244) offers the following three examples: (1) Friends, whose [kindness and encouragement] has… (2) My [hope and wish] is … (3) That [John had an affair and Mary left him] is a sad fact.

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Nothing similar happens with disjunction. Disjunction neither combines the information in its disjuncts as a semantic whole nor bears any ‘presumption of optimal relevance’. I will come back to the meaning of conjunction in Chap. 6. Quantifiers offer another consequential example of the inadequacy of the procedure of interdefinability for establishing semantic connections. We all know that the two standard quantifiers, the universal and the existential, can be mutually defined via negation. Nevertheless, neither their roles nor their formal properties match. Whereas the existential quantifier, in Frege’s standard interpretation, is a monadic higher-level function, most uses of the universal quantifier are binary. I will develop this point further in this chapter. Undoubtedly, it is a welcome simplification that calculi do not need two standard quantifiers as primitives, or that with negation and another ‘logical constant’ (conditional, disjunction, conjunction) the other truth-­functions can be defined. But it would be a serious mistake to think that these formal characterisations are the last word on the meaning of logical constants, or even that they place their analysis on a promising path. The history of the discipline shows otherwise. I now resume the topic of this section. Negation, incompatibility, and falsity perform markedly different functions with contrasting roles. Negation is a monadic function from n-adic (n  ≥  0) predicables/concepts to n-adic (n  ≥  0) predicables/ concepts. 0-adic predicables, the limit case of the range, are just propositions. The move of including propositions and predicables/concepts in the same logical category is completely alien to Fregean semantics, which sharply distinguishes between saturated and non-saturated expressions and entities. Predicables and concepts belong to the category of unsaturated items, whereas propositions, being the senses of compound names, are saturated entities. Nevertheless, in defining some logical constants, the move of dealing in a single stroke with concepts and propositions simplifies the picture. In any case, we should keep in mind—in this case, and many others—that propositions and concepts are in themselves neither saturated nor unsaturated, nor do they possess gaps of any kind. These classical characterisations are metaphorical, and they should only be assessed by their success in explaining the significance and behaviour of the expressions concerned. Incompatibility, by contrast, is primarily a binary semantic relation between full-­ blown propositional contents, i.e., between 0-adic predicables. Finally, the meaning of falsehood derives from the use that speakers make of falsehood-ascriptions, by means of which they reject particular propositional contents. Falsehood-ascriptions are the elementary sentences in which the concept falsehood displays its potential. The idea of identifying an elementary sentence in order to discover the meaning of a word is due to Carnap: [T]he syntax of the word must be fixed, I.e., the mode of its occurrence in the simplest sentence form in which it is capable of occurring; we call this sentence form its elementary sentence. (Carnap, 1932, p. 62)

And the method for discovering the meaning of an elementary sentence, S, is revealed by answering the question of ‘[w]hat sentences is S deducible from, and what sentences are deducible from S’ (Carnap loc. cit.). Carnap thus puts forward a

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genuinely pragmatist and openly inferentialist strategy. On the one hand, the meanings of words have to be sought in the most basic sentences in which they perform their task, a suggestion analogous to Frege’s (PCont). On the other, the meanings of elementary sentences derive from their deductive connections with other sentences, which is closely related to (PII) at the linguistic level. Examples of falsehood-ascriptions are ‘What she said is false’, ‘p is false’ and ‘It is false that p’. In these three cases, there is a proposition that is rejected, of which falsehood is predicated, or to which falsehood is attributed. The result is the same: the target proposition can neither be stated nor used as a premise in our inferential acts, since we don’t assume it as (contextually) settled knowledge. Attributing falsehood to a proposition does not change its semantic or logical properties; it does not add any particular component to the proposition either. Falsehood-ascriptions express the speaker’s attitude towards certain propositions and indicate how they should be taken. I will deal with falsity in Chap. 10 when the meaning of truth is explained. (PPP) implies that, of the three notions concerned, negation cannot be the basic one. One might think that pragmatism should place falsehood-ascriptions at the foundational level, since ascribing is something that agents do in assertions, and assertions are the essential discursive act. Nevertheless, falsehood-ascriptions are something more than mere assertions. They are the explicit rejection of a content, and hence the implicit endorsement of its complement. Falsehood, like truth, is put to work in second-level acts, i.e., acts that already presuppose that an assertion, actual or virtual, has taken place. For this reason, I will consider incompatibility, the immediate semantic relation that rests on the dual nature of propositions, to be the primitive notion from which the other two derive. Being primitive, incompatibility cannot be properly defined. Nevertheless, besides explicit definition, there are other ways of becoming familiar with a concept’s meaning. Calling attention to its behaviour using examples is one of them. For sure, the task of illustrating incompatibility will bring along some references to negation and falsehood. No wonder, since this is precisely what one should expect of members of the same semantic family. Let us have a try. Incompatible propositions are those that cannot be asserted together. If Boris claims that Brexit will make Britain great again and Keir says that Brexit will lead Britain to disaster, then neither of them can endorse the other’s claim without modifying some of their attitudes towards some of their beliefs. The assertion of incompatible contents by two agents in a given context produces a particular kind of disagreement that MacFarlane calls ‘noncotenability’ (MacFarlane, 2014, section 6.2, especially p. 121, n. 5). Faced with two incompatible contents, a falsity ascription rules out one of them. And the role of negation is to expose the rejected content. The different roles of these three notions attach them to different levels. Negation is a syntactic component of the sentence that works at the level of the expression of thoughts, while incompatibility is a relation that earns its living at the semantic level. Finally, falsity ascriptions work at the pragmatic level, in which agents remove certain contents from the set of usable thoughts.

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The reaction of classical semanticists to the picture I have presented so far is that it is, at best, upside down. Negation, they would explain, is defined in terms of truth and falsity; falsity, in turn, is a semantic characteristic of contents that derives from how the world is (or is not), and incompatibility can only be defined in terms of truth and falsehood, or alternatively in terms of negation. This explanation could even be traced back to Frege, or at least to a superficial reading of Frege’s works, in which the True and the False are primitive objects. Logical constants, negation among them, are defined in terms of truth values using the standard truth tables, and incompatibility is understood as contradiction, which is defined on truth values and negation. In Chap. 2 I offered some reasons to doubt the standard interpretation of the place of the objects the True and the False in Frege’s system, and I will not insist on this point. The difficulties that logicians encounter in giving a symmetric definition of negation in proof-theoretic semantics suggest that the standard picture cannot be correct. Negation poses serious difficulties to any unified narrative about logical constants, being the sole monadic connective that takes complete sentences as arguments. But such difficulties not only come from negation. Quantifiers, on the one hand, and identity, on the other, also preclude the simplicity of any general approach (see Frápolli, 2012). Negation is particularly damaging, because it also occurs in propositional calculus, whereas quantifiers and identity are only needed when the calculi at issue have resources for representing individuals. Nevertheless, there is no direct method for either introducing or eliminating negation, for obvious semantic reasons. Kürbis discusses at length the difficulties that negation, in particular, and modal operators, in general, exhibit in meeting the requirements of proof-theoretic semantics (Kürbis, 2015a, b, 2019). His general diagnosis is that negation cannot be defined within this framework (Kürbis, 2015b, p. 1, 2019, p. 5). He concludes that, besides truth, some ‘negative’ factor has to be included as a semantic primitive (Kürbis, 2019, p. 122). And he favours falsity to play this role. Logicians’ resistance to incompatibility does not rest on solid grounds, as I will show, and might have very unwelcome consequences since it might backfire against logical consequence, which is the other primitive logical relation. I will review Kürbis’ arguments for rejecting the primary status of incompatibility. Kürbis elaborates on the reservations that Dummett and Prawitz (Dummett, 1973, 1991; Prawitz, 2006), the two most prominent advocates of proof-theoretic semantics, express about the possibilities of characterising negation within this framework. They both concede that the rules for negation in the framework they propose are defective and that the meaning of negation cannot be given by setting up clear formal criteria for its introduction and elimination. Nevertheless, Kürbis departs from these authors in their identification of classical logic and classical negation as responsible for this situation. In his opinion, intuitionistic logic and ‘intuitionistic negation [are] in the same boat’ (Kürbis, 2015a, p. 10). He also rejects the alternative of taking incompatibility as primitive, which he discusses in Tennant’s and Brandom’s versions. I agree with the general insight that a negative pole is required. In fact, the negative pole occurs hand in hand with the positive one in incompatibility and the dual

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nature of propositions that derives from Frege. I also accept the essence of Kürbis’s criticism of Dummett’s and Prawitz’s way out, although, as I see it, Kürbis’s arguments against incompatibility are misleading. I will focus on the most consequential three of them. Kürbis’s first objection derives from the alleged metaphysical obscurity of incompatibility. Metaphysicians do not have a clear explanation, the objection goes, of whether and why sentences such as ‘this is green’ and ‘this is red’, and also ‘being here’ and ‘being over there’ are actually incompatible. There is no complete metaphysical account of incompatibility that can determine whether, for instance, finding beetroot delicious is incompatible with being Nils Kürbis. On this point, Kürbis is undoubtedly right, as he is when he claims that it would not be ‘desirable to make the definition of negation dependent on the outcomes of arcane debates in metaphysics’ (Kürbis, 2015a). Fortunately, there is no need for this. Understanding incompatibility requires neither a method for identifying all particular incompatibilities nor any particularly sophisticated metaphysical treatment of this notion. For a pragmatist, it is enough that some cases are recognised when we are presented with them, and to know that, in clear cases of incompatibility, a speaker cannot assert both contents at the same time. If the semantic intuition of propositional duality, promoted by Frege and Wittgenstein, contains a grain of truth, then any competent speaker surely has a good mastery of this practical task. In any case, if Kürbis’s first objection were well founded, then logic (and semantics) would be in big trouble, since what happens with incompatibility is no different from what happens with consequence. The dual nature of propositions implies that consequence and incompatibility are either mastered together or discarded together since they cannot be learned independently of each other. As happens with incompatibility, understanding consequence does not require knowing whether from the proposition that someone is Nils Kürbis it follows that he does not like beetroot. In general, there is no need for any effective method for determining whether any two propositions stand in a relation of consequence in order to master this notion. Practical mastery is enough, and this mastery follows from the acknowledgment of some among the many commitments that we acquire in the practice of assertion. In both cases, any speaker with practical mastery of assertion—and this means any speaker at all—knows that two incompatible contents cannot be asserted together, and that if one content follows from another, then the latter cannot be asserted while the former is rejected. Even more revealing of the general framework that rejects taking incompatibility as primitive is the second argument. In (Kürbis, 2019), the reason to reject incompatibility rests on semantic grounds: incompatibility, the argument goes, cannot be primitive because it is a relation that is ‘essentially tied to the particular content of atomic sentences’ (Kürbis, op. cit., p. 173). To this, I do not have anything to say, other than that he is right. This objection frames the situation precisely: the rejection of taking incompatibility as primitive clashes with the assumption that logical constants primarily affect formal structures. They do not. As (PPP) states, propositions are the primary bearers of logical relations, and propositions are neither

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mathematical skeletons nor uninterpreted formulae, but instead judgeable contents put forward in assertive acts. The resistance that many logicians display to making logic rest on practices and intentions usually derives from the unjustified assumption that, beyond formalism, the relevant notions cannot be given precise characterisations. Nevertheless, there is no particular difficulty in characterising consequence (entailment, implication) and incompatibility in pragmatic terms, i.e., in terms that include agents and their commitments: [Implication1] A proposition p1 implies a proposition p2 if, and only if, by endorsing p1 the agent implicitly endorses p2. [Implication2] A proposition p1 implies a proposition p2 if, and only if, the epistemic ground that entitles an agent to assert p1 entitles the agent to assert p2. [Implication3] A proposition p1 implies a proposition p2 if, and only if, the commitments an agent acquires by asserting p1 include the endorsement of p2. None of these adds anything substantial to the Fregean characterisation of conditionality.2 Incompatibility can be characterised along similar lines: [Incompatibility] Two propositional contents p1 and p2 are incompatible if, and only if, endorsing p1 precludes endorsing p2. The third objection I will comment on also reveals a substantial point. Kürbis insists on the generalised assumption that incompatibility is understood via negation: ‘p and q are incompatible if, and only if p and q cannot be both true’ (Kürbis, 2015a, p. 724). Undoubtedly, the quoted sentence is a perfectly fine definition of incompatibility, but it does not prove incompatibility’s dependence on negation. That no characterisation of incompatibility can be offered that does not include any mention of either negation or falsehood (the negation of truth) is what one should expect from the fact that negation is the expression of incompatibility, since negation is the instrument that languages incorporate in order to make incompatibility visible. Exclusive disjunction is a good candidate for representing semantic incompatibility. The standard decision to make inclusive disjunction the basic sense of the notion and to define the exclusive version via negation is understandable only because propositional contents have been removed from calculi and substituted by variables. If the full-fledged judgeable contents were displayed, then disjunction would be naturally interpreted as incompatibility (generally or contextually asserted, semantically or epistemically understood).

 In the Begriffsschrift, §5. One might retort at this point that the conditional and implication are different notions. In one case, the relation between the two propositions involved is contingent, and in the other it is necessary. This point will be discussed in the Chap. 6. But here it suffices to say that conditional terms (‘if’, ‘when’) are natural language tools for expressing implication (in a broad sense). 2

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Let us take stock. Conditional and negation express some operations, relations, or attitudes towards propositional contents. These notions escape the strictures of the analysis in terms of sense and meaning. Truth, some conjunctions (‘although’), discourse markers (‘but’), and the sign of assertion, which is read ‘is a fact’, are also special from a semantic viewpoint. The meaning of these terms is expressive, and thus neither referential nor descriptive. The expressive approach does not preclude analysing all of them as functions of some kind (see, for instance, Frege, 1891, p. 154). When their arguments are complete sentences, they are first-level functions in Frege’s work, since Frege assimilates sentences with proper names. When their arguments are n-adic predicables (n > 0), as in those cases in which the arguments of conditionals do not express complete thoughts, or when negation serves to build up the complement of a given concept (mortal versus immortal, for instance), they are higher-level. I will not dispute that sometimes in his work Frege is interested in offering a homogenous semantics for all notions. His two-factor semantics was a huge advance over the previous approaches and earned him the title of the father of the philosophy of language. Yet still, ignoring the expressive function reserved for a set of terms particularly dear to logicians and philosophers would not do justice to the depth of his knowledge about language and thought. Surely, for the representation of inferences and proofs in arithmetic and geometry all these subtleties are quite irrelevant, but not so for the analysis of natural language, which is the natural locus in which humans carry out inferences. The next family of expressions which receives an innovative, expressive treatment in Frege’s work is that of quantifiers, which are the topic of the next section.

4.4 Expressions of Quantity and Relations Between Concepts Monadic quantifiers are higher-level properties of concepts and express the size of their extensions. In addition, there are higher-level relations between concepts that express inferential and algebraic connections between concepts and between their extensions. Existence and numbers, as properties of concepts, are monadic devices. Expressions of generality admit monadic and binary interpretations. It has been customary in logic to reduce the expressions of quantity with primary logical relevance to two: the universal quantifier (‘all’, ‘every’), and the existential or particular quantifier (‘some’, ‘there is’, ‘exists’). Syllogistic also included the negative duals of the universal and particular quantifiers, ‘no’ and ‘not all’, which in contemporary logic are compounds of the two basic quantifiers plus negation. Syllogistic classified natural language structures and some current proposals have followed a similar path. Mostowski put forward a theory of quantifiers that extended the theory of classical quantification to other expressions of quantity. A few years later, Lindström developed Mostowski’s proposal (Mostowski, 1957; Lindström, 1966). The Lindström-Mostowski characterisation, which was purely mathematical, was later on adapted by Montague in order to provide a formal semantics for natural languages (Montague, 1973). With these ingredients, Barwise

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and Cooper offered for the first time an approach to the semantics of natural language quantifiers completely within the framework established by the theory of generalised quantifiers (Barwise & Cooper, 1981). The theory deems as quantifiers all syntactic structures of the form ‘Det + Noun’, and thus understands quantifiers as a syntactical category. Neither Syllogistic nor the theory of generalised quantifiers follow (PGS), the principle that inspires Frege’s approach and also ours. For this reason, it lies beyond my interest at this point, which concerns contents and not their expressions. Frege did not see the need to include non-standard quantifiers in his conceptual language since, for the purpose of the Begriffsschrift, they do not have any function to accomplish. The perspicuous representation of inferences in arithmetic could be completely achieved by the explicit representation of generality and a derivative representation of existence via negation. One of the original insights that Frege introduced, though, was the representation of existential judgements as particular judgements, an advantage over Boole’s system that Frege stresses (see Frege, 1880–1, p.  14), and which Schröder recognises in his review of Frege’s book (Schröder, 1880; see also Sluga, 1987, p. 82). As I will discuss later in this section, existential and particular judgements, even when they express the same thought, do not identify the same ‘ingredients’ in it. They represent alternative analyses of the same content. When the thought is presented as existential instantiation, it is the size of the concept’s extension that is highlighted; when it is presented as a particular judgement, partial overlap is the relation that is emphasised. Thus, Frege’s remark on the possibility of representing particular judgements as existential ones is further evidence of his conviction that contents can be represented by different linguistic vehicles with divergent analyses. Frege’s take on quantifiers presupposes (PGS). His analysis goes beyond grammar to classify a specific kind of function that applies to other functions (Frege, 1879, §§11–12, 1884, § 54, 1892a, p. 187). (PGS) was a huge advance in the understanding of the semantics of quantity expressions. Logic, it must be recalled, is not linguistics. The logician’s interest lies in concepts, not in words; even if the use of words is the first step towards understanding concepts, something that Frege repeats once and again. In ‘Logical Generality’ (Frege, 1923), he expresses what is probably his latest word on this topic, which, because of the nuanced position that he puts forward, deserves to be quoted at length: Language may appear to offer a way out, for, on the one hand, its sentences can be perceived by the senses, and, on the other, they express thoughts. As a vehicle for the expression of thoughts, language must model itself upon what happens at the level of thought. So we may hope that we can use it as a bridge from the perceptible to the imperceptible. Once we have come to an understanding of what happens at the linguistic level, we may find it easier to go on and apply what we have understood to what holds at the level of thought ⎯to what is mirrored in language […]. Also, the use of language requires caution. We should not overlook the deep gulf that yet separates the level of language from that of the thought, and which imposes certain limits on the mutual correspondence of the two levels. (Frege, 1923, p. 259)

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Language is essential to the analysis of thought since thought is imperceptible by the senses. But the revolution that Frege’s logic brought about over Syllogistic relies more on the acknowledging of the dissimilarities between the level of thought and the level of language than on their similarities. The ‘level of language’ conveys the syntax of natural languages, but also—and this is crucial—the syntactic level of artificial languages, including Frege’s conceptual writing and the language of mathematics. Frege’s artificial language is apt to represent inferentially relevant aspects of the expression of thought, which it models in its syntax. But this should not lead us to forget that the target of logical analysis is what is expressed and not its expression. The linguistic level is only the vehicle. Fregean quantifiers are higher-level concepts; they are not attached to objects but to other concepts, and syntactically they modify predicates: ‘the ‘words “all”, “any”, “no”, “some”, are prefixed to concept-words’ (Frege, 1892a, p. 187). Concepts, the meaning of ‘concept-words’, are, among other things, classificatory and inferential devices (see, for instance, Brandom, 2014, p. 146). Frege recognises the classificatory role of concepts as tools for ‘collecting together’ (Frege, 1884, §48) different objects. Different concepts can organise differently the same reality, even though this does not mean that different numbers can be attached to the same thing (Frege loc. cit.). Among the infinitely many expressions of quantity, there are two limit cases, the minimal case in which we say of a concept that it applies to no object, i.e. the empty case, and the maximal case in which we say of a concept that it applies to all objects (of a certain kind). Existence is merely the negation of the empty case. The content of existential judgements is precisely that under the concept that is the quantifier’s argument there falls at least one object (Frege, 1879, §12, n. 15). The universal and the existential quantifiers are both characterised in the Begriffsschrift, although only the universal quantifier is introduced as primitive (Frege, 1879, §11) and represented by a specific configuration of signs. Macbeth notes that ‘[t]here is no simple sign in Frege’s logic for the existential quantifier, and it seems never even occur to him that he could treat the existential quantifier as the primitive sign for generality and then define the universal quantifier in terms of it’ (Macbeth, 2005, position 102). This remark is extremely significant and supports Macbeth’s refusal to read the Begriffsschrift as primarily a proposal in quantificational logic. It also adds weight to my rejection of the existential quantifier as a logical constant, as will be discussed in Chap. 6. The introduction of the universal quantifier assumes that the expression of a judgement,   u  , can always be analysed as a function of one of its terms, which in Frege’s conceptual writing is represented by a German letter, ‘ u ′ for instance. To represent generality, Frege draws a small cavity in the content stroke over which the German letter is written: ( )

This formula now means that, whatever is substituted by the German letter in the formula on the right, the resulting formula expresses a fact. Two aspects of the Begriffsschrift treatment of generality have to be stressed. The first is that, at this

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point, Frege does not offer a definition but instead a way of representing generality. The second is that, as it is explained, generality is a monadic device. Along with his description of generality, Frege displays the connections between generality and existence and introduces binary quantifiers. As he says, some combinations of negation and generality, such as for instance Not everything that is A is B, express existence, i.e. they express the thought that there is at least one thing that is A and is not B. In positive, general thoughts such as If something has property X, it also has property P, ‘Every X is a P’ or ‘All X are P’ (Frege loc. cit., §12), are ‘the way in which causal connections are expressed’ (Frege loc. cit.). To the analysis of ‘causal connections’, i.e. laws, as hypothetical thoughts, Frege devotes his paper ‘Logical Generality’ (Frege, 1923/1979). The representation of causal connections is not the only role of the universal quantifier, even if the analysis of the logical form of scientific laws is undoubtedly an important application of the conceptual writing. What sentences of the form ‘All X are P’ represent is the subordination of one concept to another (Frege, 1884, §47; 1893, §12). This view is vindicated in ‘Concept and Object’, where he insists that his view is ‘essentially the same’ and remarks that in ‘universal and particular affirmative and negative sentences, we are expressing relations between concepts; we use these words to indicate the special kind of relation’ (Frege, 1892a, p.  187). The words to which he refers are ‘all’, ‘any’, ‘no’, and ‘some’, i.e. the standard quantifiers, which are here understood as binary. The binary interpretations of general judgements, ‘All As are Bs’, and of particular judgments, ‘Some As are Bs’, are straightforward if we think of them as inclusion and partial overlap of their extensions respectively. The existential reading of particular judgements is a different story. Existence, as the negation of the ‘number nought’ (Frege, 1884, §53), is instantiation. And instantiation is the property that a concept has when something falls under it, or, in extensional terms, the property of a concept whose extension is not empty. So understood, existence is a monadic higher-level property, although its arguments need not be simple. That there are some As that are also Bs means that the extension of the single but complex concept ‘A&B’ is not empty. Conjunction, as we have seen, serves to build up a complex concept out of two others. This role of conjunction is perspicuously represented in Boole’s algebra, in which conjunction corresponds to the intersection of two sets. It is this intersection, which is a single set, which is said not to be empty. Thus, the occurrence of two terms or two concept-words in a quantified sentence does not guarantee the binary nature of the quantifier. Partial overlap and instantiation are different concepts; the former one is a binary relation, whereas the latter is a monadic second-level property. Frege does not elaborate further on the issue of the relation between monadic and binary quantifiers, and the text of 1892 just quoted is his only general remark on an interpretation of standard quantifiers as relations. The issue is relevant, though, since it shows that it is neither the case that inter-definition always points to a single notion, nor that alternative representations of thoughts identify ‘components’ that are independent of how they are represented. The situation of existence and partial overlap is similar to the situation of defining conjunction with disjunction and negation. In the latter case, the thought expressed cannot be said to be either negative or disjunctive; in the former case the thought

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expressed cannot be said to be either existential or particular. Interdefinability and alternative ways of representing provide more information about the representation systems than about the represented reality. Generality also admits of monadic and dyadic versions that highlight different features. The essence of monadic generality, first introduced in (Frege, 1879, §11), is explicitly re-stated in ‘Logical Generality’: The value a law has for our knowledge rests on the fact that it comprises many ⏤indeed, infinitely many⏤particular facts as special cases. We profit from our knowledge of a law by gathering from it a wealth of particular pieces of information, using the inference from the general to the particular, for which of course a mental act⏤that of inferring⏤is still always required. Anyone who knows how to draw such an inference has also grasped what is meant by generality in the sense of the word intended here. (Frege, 1923, p. 258)

The meaning of generality is given by the use of the mental act of inferring, which allows us to reach particular pieces of information that derived from some general information. This explanation can rightly be characterised as inferential. The elimination rule for generalisation, which sanctions as safe the transition from ‘All individuals have the property F’ (∀xFx) to ‘Some particular individual is F’ (Fa), for any individual, gives the meaning of generality. This text uncovers an aspect of Frege’s view that is closer to proof-theoretical semantics than to truth-conditional semantics, closer to Gentzen, Prawitz or Dummett than to Tarski. The argument of the monadic quantifier is the whole predicable, single or complex, (some of) whose free variables the quantifier binds. In the Fregean representation of generality, ( )

the argument of the quantifier is the whole formula that stands on its right-­ hand side. The monadic quantifier expresses the licensing of a ‘vertical’ move from the general to the particular, which allows the discharge of information. This vertical move is what happens when the quantifier is eliminated. In hypothetical compound judgements, i.e. in judgements of the form ‘All Fs are Gs’, the binary quantifier licenses the application of the second concept to any object to which the first concept can be applied. This ‘horizontal’ move from the application of the one concept to the application of the other rests on the subordination of the first concept to the second expressed by the quantifier (Frege, 1884, §47, 1893, §12) which is represented by the conditional stroke. Only general transitions from one concept to another, i.e. laws, are genuine hypotheticals. It is the combination of the conditional stroke and the general quantifier that represents hypothetical thoughts (Macbeth, 2005, positions 335–395). Before concluding this section, I will make some comments about unrestricted quantification and the need to introduce quantifiers explicitly. Both of these issues are related to the notion of scope. Frege’s quantifiers are broadly considered to be unrestricted devices, in the sense that the information expressed by quantified sentences applies with the widest scope. It is debatable, nevertheless, whether unrestricted quantification makes any sense at all. In the binary reading of generality, the

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first concept delimits the scope of the contextual application of the second. In the representation of causal laws, the occurrence of a conditional and the interpretation that Frege gives to it guarantee that the antecedent restricts the scope of the judgement to those objects covered by the concept in the antecedent. In existential judgements, the information given is that the extension of the concept that is the argument of the quantifier is not empty, and the quantifier is contextually restricted. The topic of unrestricted quantification has deservedly received much discussion among philosophers of language,3 but as a theoretical issue it does not apply to ordinary communication, where context specifies the limits of the concepts used. Scope is also relevant to the question of whether explicit quantifiers are necessary in a correct conceptual writing. It might be thought that for representing generality and instantiation, the appropriate kind of variables would be all that is needed, and that explicitly including a quantifier would be redundant. In fact, Frege characterises variables as the expression of generality in (Frege, 1879, §1). Nevertheless, he is aware that, as important as indicating generality is, so is marking its limits. Thus, in representing generality by means of the German letters, not only the particular letter is relevant, but also the precise placement of the concavity that contains the letter, which can affect only a part of the judgeable content. Quantifiers are thus binding devices that also indicate the logical priority of some operators over some others. A clear distinction between binding and logical priority in the function that quantifiers perform had to wait until Hintikka’s perceptive paper (Hintikka, 1997), but the explicit inclusion of quantifiers in the Begriffsschrift shows that Frege had at least some well-oriented intuitions towards the needs of a correct conceptual writing that avoids scope ambiguities. Now, let us take stock. Generality and existence are higher-level concepts whose arguments are also concepts. General and existential judgements do not say anything about objects. The information given by them attaches to concepts and can belong to different categories. In existential judgements, on the one hand, the information conveyed is that the size of the extension of a concept, which can be built up out of other concepts, is not empty. Particular judgements indicate partial overlap between the extensions of the two concepts involved and, although the thought expressed can also be represented by explicitly existential sentences, existence and partial overlap are different notions. General judgements, such as Whales are mammals, license inferential moves in two different directions. When the quantifier affects the complete formula on its right-hand side, it permits the assertion of any of its instances. When the quantifier is interpreted as a relation between two concepts or, alternatively, between their extensions, it represents the subordination of concepts, in the first case, or the inclusion of sets, in the second one. It is important to note that these two interpretations, i.e. the interpretations that authorise the vertical and horizontal moves, are not incompatible and can be applied to the same judgement in the same context. Both interpretations view the quantifier as granting an inference, in the first case from the general thought to its particular cases; in the

 For an up-to-date survey, see e.g. (Stanley & Szabó, 2003).

3

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second case from the application of the first concept to the application of the concept in which the former is included.

References Ammereller, E., & Fischer, E. (2004). Wittgenstein at work. Method in the philosophical investigations. Routledge. Ayer, A. (1936). Language, truth, and logic. Victor Gollanz Ltd. Barwise, J., & Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4(2), 159–219. Berto, F. (2015). A modality called ‘negation’. Mind, 124(495), 761–793. Besson, C. (2019). Logical expressivism and Carroll’s regress. In M. J. Frápolli (Ed.), Expressivisms, knowledge and truth (Royal Institute of Philosophy Supplement 86) (pp. 35–62). Cambridge University Press. Blakemore, D. (1987). Semantic constraints on relevance. Blackwell. Brandom, R. (2000). Articulating reasons. An introduction to inferentialism. Harvard University Press. Brandom, R. (2014). Analytic pragmatism, expressivism, and modality. The 2014 Nordic pragmatism lectures. www.nordprag.org Brandom, R. (2019). A spirit of trust. A reading of Hege’s phenomenology. The Belknap Press of Harvard University Press. Carnap, R. (1932). The elimination of metaphysics through logical analysis of language. Erkenntnis, 60–81. Carston, R. (2002). Thoughts and utterances. The pragmatics of explicit communication. Blackwell Publishing. Dummett, M. (1973). Frege (Philosophy of language). Haper and Row Publishers. Dummett, M. (1991). The logical basis of metaphysics. Harvard University Press. Frápolli, M.  J. (2012). ¿Qué son las constantes lógicas? Crítica. Revista Hispanoamericana de Filosofía, 44(132), 65–99. Frápolli, M. J. (2019). Propositions first. Biting Geachs bullet. In Expressivisms, knowledge and truth. Royal Institute of Philosophy supplement 86. Edited by María J. Frápolli. Cambridge University Press, 87–110. Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jean van Heijenoort (1967). From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, 1–82. Frege, G. (1880–1). Booles logical Calculus and the Concept-script. In G.  Frege (1979), Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach. Basil Blackwell, 9–46. Frege, G. (1884). The Foundations of Arithmetic. A logic-mathematical enquiry into the concept of number. Translated by J. L. Austin. Second Revised Edition. New York, Harper Torchbooks / The Science Library, Harper & Brothers. Frege, G. (1891). Function and Concept. In G. Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Oxford, Basil Blackwell, 137–156. Frege, G. (1892a). Concept and Object. In G.  Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 182–194. Frege, G. (1892b). On Sense and Meaning. In G. Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 157–177. Frege, G. (1893–1903/2013). Basic laws of arithmetic. Volumes I and II. Oxford University Press.

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Frege, G. (1897). On Mr. Peano’s conceptual notation and my own. In G. Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 234–248. Frege, G. (1918–19b). Negation. In G. Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 373–389. Frege, G. (1923). Logical generality. In G. Frege (1979), Posthumous writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach. Basil Blackwell, 258–262. Frege, G. (1923–26). Compound thoughts. In G. Frege (1984), Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell, 390–406. Geach, P. T. (1960). Ascriptivism. The Philosophical Review, 69(2), 221–225. Hintikka, J. (1997). No scope for scope? Linguistics and Philosophy, 20(5), 515–544. Kürbis, N. (2015a). Proof-theoretic semantics, a problem with negation and prospects for modality. Journal of Philosophical Logic, 44(6), 713–727. Kürbis, N. (2015b). What is wrong with classical negation? Grazer Philosophische Studien, 92(1), 51–86. Kürbis, N. (2019). Proof and falsity: A logical Investigation. Cambridge University Press. Lindström, P. (1966). First order predicate logic with generalized quantifiers. Theoria, 32(3), 186–195. Macbeth, D. (2005). Frege’s logic. Harvard University Press (kindle edition). MacFarlane, J. (2014). Assessment sensitivity: Relative truth and its applications. Oxford University Press. Montague, R. (1973). The proper treatment of quantification in ordinary English. In P. Suppes, J. Moravcsik, & J. Hintikka (Eds.), Approaches to natural language (pp. 221–242). Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematica, 44, 12–36. Odgen, C. K., & Richards, I. A. (1923). The meaning of meaning. Harcourt Brace & Jovanovich. Popper, K. (1935/2002). The logic of scientific discovery. Routledge Classics. Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148, 507–524. Schiffrin, D. (1986). Functions of and in discourse. Journal of Pragmatics, 10, 41–66. Schiffrin, D. (1987). Discourse markers. Cambridge University Press. Schröder, E. (1880). Gottlob Frege, Begriffsschrift. Zeitschrift für Mathematik und Physik, 25, 81–94. Sluga, H. (1987). Frege against the Boleans. Notre Dame Journal of Formal Logic, 28(1), 80–98. Sperber, D., & Wilson, D. (1986). Relevance: Communication and cognition. Basil Blackwell. Stanley, J., & Szabó, Z.  G. (2003). On quantifier domain restriction. Mind and Language, 15, 219–261. Stevenson, C. (1937). The emotive meaning of ethical terms. Mind, 46, 14–31. Wittgenstein, L. (1914/1998). Notebooks 1914–1916. Blackwell Publishers.

Chapter 5

Lessons from Inferentialism and Invariantism

Abstract  In this chapter, I explain what is at issue in the debate on the meaning of the logical constants, exposing some weaknesses of the standard way in which logicians approach this subject. I present and discuss the two families of proposals that have been most successful: invariantism, which derives from (Tarski. History Philos Logic, 7, 143–154, 1986), and inferentialism, which derives from (Gentzen. Am Philosoph Quart 1(4), 288–306 1935/1964) and whose more philosophical aspects have been developed by Dummett, Hacking and Prawitz (Dummett M (1973) Frege. Philosophy of Language. New  York, Harper and Row Publishers; Dummett M (1991) The logical basis of metaphysics. Cambridge, MA., Harvard University Press; Hacking. J Philos 76, 285–319 (1979); Prawitz. Synthese 148, 507–524 (2006)). I show that neither of these two approaches gets the set of logical constants right. Both have been accused of overgenerating, i.e. including notions that intuitively are not logical constants, and of undergenerating, i.e. leaving out of the picture some notions that clearly are logical constants. The constraints that their supporters have proposed to make them extensionally adequate point in the direction of taking on board the pragmatic roles that these notions play in communication. The last section of this chapter is devoted to explaining how a pragmatist account helps us to overcome the difficulties that more formalist approaches present, be they purely syntactic or semantically guided. Keywords  Analytically valid · Conservative [conservativeness, conservativity] · Erlangen · Gentzen · Inferentialism · Insubstantive · Invariantism [invariantist] · Logical constant · Subformula · Tarski, tonk

5.1 What Is the Issue with Logical Constants? The practical purposes that speakers have in mind when they use logical constants are inseparable from the kind of enterprise that logic is. I made a similar statement at the end of Chap. 2, where I connected it with the pragmatist approach to logic that must guide our inquiry into the meaning of logical terms. Most logicians recognise the connection between the definition of logical constants and the demarcation of © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_5

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logic as a discipline (see, for instance, Tarski, 1986, p. 145; Ferferman, 1999, p. 33; Dutilh Novaes, 2014, p. 81). Tarski directly acknowledged this link when he offered his characterisation of logical notions in the spirit of the Erlangen programme, designed to demarcate particular disciplines by defining their central concepts. Gómez-Torrente (2002) seems to be an exception to this general consensus, offering a more nuanced understanding of the connection between these two topics, but I will argue that there is no actual disagreement on this point and that any disparity is merely apparent. Gómez-Torrente explicitly detaches these two issues and correctly emphasises that everybody who works in the realm of logical theory already possesses a clear understanding of the essence of logic, even without any definition of logical constants. Everybody sees that logic is the science of inference, consequence, and validity (see Gómez-Torrente, 2002, pp. 2–3), and on this point, I can only agree. Hintikka and Sandu push the connection between logic as a discipline and the notion of validity even further, asserting that ‘the typical form the theory of any part of logic’ typically adopts is a ‘set of rules of inference’. They add: ‘Rules of inference are often thought of as the alpha and omega of logic’ (Hintikka & Sandu, 2007, p. 13). But even those authors who do not place logical constants at the core of the debate about the nature of logic end up drawing attention to the distinction between inferences whose validity is formal and those whose validity rests on the meaning of non-logical notions. The characterisation of the kind of validity that has interested mainstream logicians requires a sharp distinction between logically valid arguments and ‘analytically valid‘ones (Hintikka & Sandu, 2007 op. cit., p. 16), and this distinction, in turn, requires discriminating between logical and non-logical concepts or terms. Some examples will illustrate this distinction. The following argument, (1), is analytically valid: (1) Victoria is a woman├ Victoria is a human being. Its validity rests on the content of the terms ‘woman’ and ‘human being’. The standard conception of logic takes (1) to be an enthymeme, i.e. an incomplete argument with an implicit premise that is responsible for its validity. It falls short of being logically valid, since it lacks the feature of formality (see Gómez-Torrente, 2007, p. 179). Its status as a logically valid argument is restored when the missing premise—the premise in which the logical terms occur—is explicitly given, as in (2): (2) All women are human beings, Victoria is a woman ├ Victoria is a human being. The standard logical form of the first premise of (2) includes a general quantifier and a conditional. If made explicit, (2) becomes (3), (3) For all x, x is a woman → x is a human being, Victoria is a woman ├ Victoria is a human being.

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In (3) everything is finally in order: it is formally valid since all arguments of the same form are equally valid. The problem of logical constants, according to Gómez-Torrente, is ‘the problem of demarcating in some principle-based, non-arbitrary-looking way the set of expressions that logic should deal with as directly responsible for the logical correctness of arguments‘(Gómez-Torrente, 2002, pp. 2–3). This claim is by no means exceptional; indeed, it is the standard way in which logicians formulate the problem. And nevertheless, as happens all too often in the philosophy of logic, this claim is either trivial or misleading. The task of determining the class of terms responsible for the logical validity of arguments can hardly get started without identifying the class of logical terms since logically valid arguments are those whose validity rests on the logical constants contained in them. As I will explain below, this characterisation does not get us any closer to a delimitation of logical constants. The formalist approach to logic, of which Tarski is the most prominent figure, has problems in freeing itself from the circularity that threatens its basic notions. Logical notions are an example of this, and logical consequence, as Etchemendy has shown, is another one (see Etchemendy, 1990). In Chap. 9 I will explain that the Tarskian definition of truth suffers a similar weakness. Gómez-Torrente gives voice to the type of circularity I am referring to: The best analyses of the notion of logically valid argument available to us are directly inspired by the feature of formality […] and use the notion of logical constant. […] We know that this type of analysis is quite good because, when we choose a certain group of particular words that intuitively seem logical to us, and we test these analyses, the results are good. (Gómez-Torrente, 2007, pp. 182–3).

The ‘particular words’ we choose are those that yield the expected results. Thus, it is because we know where we want to get to that we adapt our starting point so as to produce the expected results. The circularity involved in this characterisation is patent in the standard substitutional/interpretational method for defining validity that I will discuss below. This is the sense in which the remark is trivial. The sense in which it is misleading derives from the fact that it might suggest that logical constants are responsible for the ‘correctness’ of arguments. (1) is valid in the semantic, immediate sense that its premise cannot be true and its conclusion false. It even meets one of the features that is associated with logical constants, i.e. modality (Gómez-Torrente, 2007, p.  179). Thus, its validity does not rest on the meaning of logical constants, which are absent from (1). Analytically valid arguments can be converted into logically valid ones just by connecting their relevant terms using conditionals and then generalising the result, as in (2). Adding the resulting formula as a premise completes the trick of converting analytically valid arguments into instances of logically valid structures. In terms of content, the ‘hidden’ premise is idle, and thus it is irrelevant to the validity of the inference as well. Logically valid arguments only expose the validity of the analytically valid arguments which are their source. Brandom puts it this way: Should inferentialist explanations begin with inferences pertaining to propositional form or those pertaining to propositional content? One important consideration is that the notion of

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formally valid inferences is definable in a natural way from that of materially correct ones, while there is no converse route. (Brandom, 2000, p. 55).

One might reject this point, arguing that the connection between analytically valid and logically valid arguments just explained is not universally accepted. This is right. But it is not confined to the ‘extreme’ pragmatism defended by Brandom either. The intuition that the correctness of logically valid arguments derives from the correctness of some analytically valid arguments is represented in the conservativeness constraint assumed in proof-theoretical approaches, for instance. I will develop this argument below. Logical constants do not add anything substantial to establishing the connections between premises and conclusions, which actually rest on the meaning of the substantive concepts involved. In this sense, there is no danger in saying that analytically valid arguments are valid by virtue of the logic of the terms involved. It is not dangerous if we keep in mind that logical concepts still retain their peculiarity and need to be defined. For classical logicians, speaking of the logic of ordinary terms is distressing. Gómez-Torrente (2002) and Haack (2005), p. 67) showed an uneasiness with what they consider a recent fashion. Gómez-Torrente, for instance, says: To be sure, there is a common tendency, especially in recent times, to speak as if every expression had a ‘logic‘. From this point of view there is no problem of logical constants, since there is no real demarcation of the mentioned sort, let alone a principle-based one. (Gómez-Torrente op. cit., p. 2. n. 1).

The first sentence is right; there is such a tendency, particularly among pragmatists. Nevertheless, this does not make the problem of logical constants go away. What does follow is that the standard method for distinguishing between logical and non-­ logical terms falls short of producing the right demarcation. The (correct) idea that non-logical terms have a ‘logic‘is an automatic consequence of the Tarski-like method of defining logical truth and logical validity. As I mentioned in Chap. 2, the ‘substitutional/interpretational’ method defines logical truths on ordinary true sentences, and validity on arguments that merely preserve truth (Etchemendy, 1983, p. 326). For this method to be applicable, we first have to fix a select set of terms and treat the rest as variables. Once the fixed terms have been picked out, it is possible to define an associated class related to any true sentence and to any truth-preserving argument. An argument is truth-preserving if either any of its premises is false or its conclusion is true. The sentences in the associated class of a given sentence are the result of substituting, in the original sentence, its variable terms by some others of the same category. In the same way, we can build up the associated class for particular truth-preserving arguments. Now we say that a true sentence is logically true if and only if all sentences in its associated class are true. If the fixed terms are logical terms, the procedure allows us to define logically true sentences and logically valid arguments as follows: a true sentence is a truth of logic if, and only if, all sentences in its associated class are true. A truth-­ preserving argument is logically valid if, and only if, all arguments in its associated class preserve truth. This method works beautifully when the class of fixed terms

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coincides with the class of accepted logical constants: ‘no’, ‘if’, ‘and’, ‘all’, ‘some’. In this case, the procedure yields the ‘correct’ outcomes, i.e. it classifies as logically true those sentences that we intuitively consider to deserve this label, and likewise for logically valid arguments. It explains why ‘The Queen is a human being, or the Queen is not a human being’ is a logical truth. If we keep fixed ‘or’ and ‘no’ and consider the rest as variable, then we can obtain by substitution infinitely many sentences, all of them invariably true: ‘The Queen is a radical, or the Queen is not a radical’, ‘The butler is the murderer, or the butler is not the murderer’, ‘The rabbit is in its hole, or the rabbit is not in its hole’, etc. Dividing terms into fixed and variable, with the construction of associated classes, permits the treatment of sentences with a similar structure that is given by the fixed terms in them. Logical truths owe their truth to certain structural features, since all the members of their associated class, which are structurally similar, are also true. Logically valid arguments, in turn, owe their validity to the structural features that they share with all members of their associated class. The question that naturally arises is what would happen if the set of fixed terms contained terms that intuitively are non-logical. Applied to truth-preserving arguments, the quick answer is that the procedure of substitution (or interpretation) of the variables would yield all those arguments which are truth-preserving in virtue of the meaning of the terms kept fixed. It would give their ‘logic‘. As an illustration, let us consider the following truth-preserving argument, (4): (4) Grass is green ├ Grass is coloured. Imagine now that ‘green’ and ‘coloured’ belong to the class of fixed terms, while ‘grass’ is variable. Then the following argument (5), also truth-preserving, would belong to its associated class, (5) Water is green ├ water is coloured. With different substitutions of the variable term, we get all the arguments that are truth-preserving in virtue of the meanings of ‘green’ and ‘coloured’. All these arguments would have a similar structure defined by a set of fixed terms. This is a very specific sense in which we can say that validity is a formal property. Philosophers have used this procedure to account for the inferential behaviour of philosophically significant concepts, such as, for instance, epistemic concepts. The logic of epistemic terms emerges when we keep fixed ‘know’ and ‘believe’ in truth-preserving arguments that contain them essentially. In particular, we learn that (6), (6) S knows that p ├ S believes that p, always generates truth-preserving arguments of ‘epistemic logic‘. There is nothing wrong in organising groups of arguments with a shared structure, whose validity derives from the meaning of some particular terms, and there is nothing wrong in

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considering the whole procedure as giving their ‘logic‘. What is wrong is to think that this extension of the term ‘logic‘to these other words in the language would settle the problem of logical constants. On the contrary, it shows how difficult it is to characterise logical constants and to discriminate them from ordinary terms. It also shows that, without some independent approach to logical constants, the Tarskian substitutional/interpretational method of defining validity does not hit the target: something that Tarski knew, and a situation that also compromises the formalist approach to logic. More recently, Etchemendy and Brandon have explained that the substitutional/interpretational method shows that some inferential relations are formal (Etchemendy, 1983; Etchemendy, 1990; Brandom, 1984, p.  104; Brandom, 2000, p. 55). Nevertheless, they insist that, without any explanation of which terms should be considered logical terms, this method is silent about the notions of logical truth and logical validity. Thus, it does not illuminate in which sense logical constants are logical, even if it shows why they are constants. The only promising approach to the problem of logical constants is to start their characterisation by taking into account the role they play in communication. Gómez-­ Torrente points out some pragmatist features that any characterisation of logical constants should include and defends them in an extended debate with Gila Sher (see, for instance, Sher, 2003). I sympathise with Gómez-Torrente’s pragmatist perspective, which I believe can be extended and deepened by embracing a complete pragmatist paradigm. Doing this would, in particular, require (i) a change in the identification of the target of the analysis⏤Gómez-Torrente still focuses on terms instead of concepts—and (ii) a change in the smallest unit of meaning and the bearers of logical properties⏤Gómez-Torrente focuses on the meanings of terms in isolation instead of on their contributions to arguments, globally considered. Both requirements come down to (PPP). As we will see, Gómez-Torrente’s suggestions are already detectable in Frege’s writings. But before discussing the pragmatist alternatives, let us take a look at some shared intuitions about the meanings of logical notions and related terms, and also comment on the main families of theories that, in the past century, have undertaken the task of defining them.

5.2 Analytically Valid Arguments The key to the meaning of logical constants should be sought in the different reasons that classical logicians offer to motivate the distinction between analytically valid and logically valid arguments. These reasons undoubtedly touch upon essential features of the meanings of such concepts. However, the proposals put forward to solve difficult cases, together with the constraints imposed on ‘genuine’ logical notions, reveal that something has gone wrong from the beginning. In particular, most of the discussions could have been avoided if we had assumed that logical notions are a specific kind of concept, in contrast with words, terms, and symbols, whose arguments are propositions, in contrast with subpropositional or subsentential items. In a nutshell, if (PPP) had been taken seriously.

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Let us go back to the two arguments (1) and (3): (1) Victoria is a woman ├ Victoria is a human being. (3) For all x, x is a woman → x is a human being, Victoria is s woman ├ Victoria is a human being. Both are truth-preserving, i.e. it does not happen that the premises are true and the conclusion false. In both cases, the truth of the conclusion necessarily follows from the truth of the premises. This is the property of modality that logicians attach to logical validity. Other truth-preserving arguments do not meet this constraint, even if they are materially correct. Brandom (2000), p. 52) offers (7) as an example, (7) Lightning is seen now, thunder will be heard soon. (7) is not analytically valid either, although its conclusion follows from the premise with a kind of necessity that we might call ‘physical’. I will concentrate on analytically valid arguments to show that the reasons for their validity are virtually identical to the reasons we have for granting validity to logically valid arguments. The only difference is the set of concepts or terms that are responsible for the validity of the arguments at issue, which in the former case are non-logical terms, and in the latter case logical terms. In both cases, some concepts are responsible for their validity. The necessity that connects the conclusion in (3) with the two premises from which it follows is the same necessity that connects the conclusion in (1) with its premise, i.e. semantic or conceptual necessity. Logicians go further and insist that (1) falls short of possessing logical validity because it lacks the property of formality. Thus, the connection of the conclusion with the premises in (3) will be said to be logically necessary, whereas the connection of the conclusion of (1) with its premise is not of this kind. If we dig deeper and try to say why these two kinds of necessity are different, the final classical answer is that all arguments of the form of (3), unlike arguments of the form of (1), are truth-preserving. But this answer cannot be the last word. I argued in the previous section that any appeal to ‘form’ rests on the identification of a set of fixed terms. Thus, the final answer should be that all arguments of the form of (3) in which the general quantifier and the conditional are fixed are truth-preserving. And now we have reached the end of the issue, which is also the starting point: logically valid arguments are valid in virtue of the meaning of certain terms, i.e. logical terms. Thus, a motivated answer has to explain in what sense the meaning of logical terms is specific and distinct from the meaning of the terms on which the validity of analytically valid arguments rests, that is, we end up with the problem of logical constants. On this point, there is a widely shared intuition that will help: that logical terms do not express substantive concepts, i.e. concepts that essentially occur in propositions and determine their topic. According to this intuition, logical notions are ‘syncategorematic’, ‘topic-neutral’, ‘procedural’, or they function as punctuation signs (Tractatus 5.4611, Došen, 1989). These labels represent different theoretical

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answers to the insubstantivity intuition, which is basically correct. A few words about each one of them is in order. ‘Syncategoremata’ is the medieval term that, in its syntactic reading, names the class of terms that cannot be either grammatical subjects or grammatical predicates on their own. On its semantic reading, these are terms whose function is to modify the meaning of categorematic terms (see for instance Klima, 2006, p.  353; and Dutilh Novaes, 2014, p.  402, n. 35). Frege and Wittgenstein expressed a similar intuition towards logical constants (Frege, 1879; Wittgenstein, 1922). In the Begriffsschrift (p. 1, but also for instance in §5), Frege confers to them the task of expressing connections between the different judgeable contents that constitute an inference. In the Tractatus, Wittgenstein characterises them as essentially non-­ representational (Tractatus 4. 0312). Ryle, in turn, uses the expression ‘topic-­ neutrality’ to describe the wider class of expressions to which logical constants belong (Ryle, 1954, p. 99). However, he does not consider this feature enough to define them. The expression ‘topic-neutral’, nevertheless, has been successful and is now commonly associated with logical constants. Linguists have also dealt with the general class of terms that do not represent a substantive concept: terms such as ‘so’, ‘therefore’, ‘nevertheless’, ‘but’, ‘and’, etc. They call these terms ‘discourse markers’ (Blakemore, 2002; Schiffrin, 1987), although other labels such as ‘inferential markers’ and ‘pragmatic markers’ have also been used (see for instance Fraser, 1990). Relevance theorists have reserved the label ‘procedural meaning’ for the kind of non-truth conditional meaning that they attribute to this kind of term (Blakemore, 2002, pp. 89ff.; Blakemore, 2011, p. 3538). The intuition that all these authors share about the absence of any compositional contribution to what is said might also be called ‘semantic irrelevance’. Besides logical terms and some clear instances of inference markers, linguists confer procedural meaning to terms belonging to many other linguistic categories: adverbs (‘certainly’, ‘allegedly’, ‘frankly’), interjections (‘wow’, ‘ouch’), quantifiers (‘every’), propositional attitude verbs (‘I think’, ‘I believe’), expressives (‘damn’, ‘that bastard’), pronouns and indexicals (‘I’, ‘this’) and many others (Schiffrin, 1987). For our topic, the interest of this hospitable attitude lies in the possibility of a common treatment for standard logical constants and other categories —alethic and epistemic modalities, temporal and deontic locutions, etc.—which have interested logicians and philosophers as well. But in this chapter, I will only deal with the classical problem of logical constants. The two main families of approaches that have fleshed out the insubstantivity intuition are inferentialism and invariantism. I will discuss these in the following sections.

5.3 Inferentialist Approaches Inferentialist approaches mostly derive from (Gentzen, 1935). Here is an explicit statement of Gentzen-like inferentialism:

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One basic idea of all these works may be that a logical constant is a constant that can be introduced, characterized, or defined in a certain way. What way? My answer is about the same as Kneale’s: a logical constant is a constant that can be introduced by operational rules like those of Gentzen. (Hacking, 1979, p. 303).

These approaches deal with topic-neutrality by characterising logical constants as those terms whose meaning is exhausted once their introduction and elimination rules are given. In natural deduction calculi, the less problematic logical constants are introduced and eliminated using rules, and there is nothing more to their meaning than the moves permitted by them. The simplest case is conjunction, which is the connective whose introduction follows the rule (&-Intr), (&-Intr) A, B ├ A&B, and whose elimination follows the rule (&-Elim), (&-Elim) A&B ├ A, B. This style is recognisable in practically all logic textbooks we use in the classroom, where we have almost completely abandoned the axiomatic presentation. There is something deeply intuitive in inferentialist approaches and their way of showing that logical notions do not represent substantive concepts that could define or alter the subject-matter of a discourse or proof. In contrast with the job usually attributed to ordinary concepts—being mechanisms to classify items in the world or being the essential ingredients of propositions—logical constants merely represent connections between propositional contents: permissions and prohibitions to go from certain assertions to certain others. Some versions of inferentialism explicitly involve a pragmatist hint in their identification of the bearers of logical properties that is virtually identical to Frege’s view in the Begriffsschrift. An example of such Fregean pragmatic inferentialism is Martin-Löf’s position: So that is what I shall talk about, eventually, but, first of all, I shall have to say something about, on the one hand, the things that the logical operations operate on, which we normally call propositions and propositional functions, and on the other hand, the things that the logical laws, by which I mean the rules of inference, operate on, which we normally call assertions. We must remember that, even if a logical inference, for instance, a conjunction introduction is written.



A••• B A& B

which is the way in which we would normally. Write it, it does not take us from the propositions A and B to the proposition A&B. Rather, it takes us from the affirmation of A and the affirmation of B to the affirmation of A&B, which we may make explicit, using Frege’s notation, by writing it. — A • • • —B

—A& B



Instead. It is always made explicit in this way by Frege in his writings, and in Principia, for instance. (Martin-Löf, 1996, pp. 11–12).

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Nevertheless, intuitive as it is, inferentialist approaches do not live up to their expectations. This happens for several reasons, some of them philosophical and others technical, on which I will comment in turn. The first reason is that their characterisation is prone to overgeneration. The challenge comes this time from the field of inferentialist semantics, which applies the same proof-theoretical-like method to ordinary notions. The second reason concerns the fact that the mere combination of introduction and elimination rules, even if they meet the formal constraints that in principle would solve the overgeneration issue just mentioned, does not preclude the production of monsters, as the case of ‘tonk‘shows. The third reason touches upon a serious technical difficulty in proof-theoretical calculi, i.e. that not all expressions that intuitively are logical constants admit of a purely inferentialist treatment. This shows that the inferentialist method also undergenerates. I mentioned this issue in the previous chapter in relation to negation, and I will not insist on it here. But it is crucial to keep in mind that proof-theoretical semantics does not offer any simple explanation of negation as a logical constant (see, for instance, Kürbis, 2015a, b; Kürbis, 2019), and this is a most serious drawback. In the rest of this section, I will elaborate a bit on the two first reasons. Inferential semantics, inferential role semantics, or conceptual role semantics (I understand these three expressions as synonyms) applies the inferentialist method to terms and concepts other than logical notions. A term such as ‘human’ can be defined by making explicit the circumstances of its correct application and elimination. Some rules for ‘human’ might look as follows: (‘Human’-Intro) x is a woman ├ x is human. (‘Human’-Elim) x is human ├ x is homo sapiens. Thus, the possibility of being defined by rules of this kind is not enough to include a term (a concept) in the class of logical notions. Inferentialists deal with this situation by constraining the set of permissible rules, allowing only what they call ‘purely’ inferential rules. The subformula property, which was suggested by Gentzen in 1934, is one of those constraints (see Hacking, 1979, pp.  303–304; Gómez-Torrente, 2002, p. 26, Gómez-Torrente, 2007, p. 194). An inferential rule satisfies the subformula property if the inputs and outputs of the introduction and elimination rules share some formula that has to be a subformula of the output of an introduction rule, or a subformula of the input of the elimination rule. The case of conjunction will illuminate this requirement. In (&-Intro) above, the result of introducing the constant, ‘A&B’, shares one of its subsentences, either A or B, with any one of the sentences to which it applies. In (&-Elim), the result is one of the subsentences of the complex sentence, ‘A&B’, to which it applies. This constraint rules out cases such as ‘human’ and, in general, all inferential rules that govern the introduction and elimination of ordinary concepts. Being precise, it rules out those rules that apply to isolated concepts. In other words, this restriction requires the inputs and outputs of purely inferential rules to be sentences or propositions, i.e. items that can be premises and conclusions, and which can be asserted and negated. A question that naturally arises is how it is possible that something that for Frege and Wittgenstein (and for everybody else) was so obvious, i.e. that logical

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constants connect propositions, has had to be reinserted in the debate via such a technical and sophisticated discussion as that concerning the issue of purely inferential rules. According to the view I am promoting here, the answer to this question is straightforward: by switching the bearers of logical properties from propositions to formal items, we have lost sight of the sense of the whole enterprise. However, the most damaging argument against the inferentialist approach relates to the second reason and was put forward by Prior in 1960. The meaning of connectives cannot be exhausted by their governing rules, Prior argued, because this would have the effect of making every inference analytically valid (Prior, 1960, pp. 38–39). To illustrate the point, he introduced the pseudo-connective ‘tonk‘with the following rules, (Tonk-Intr) and (Tonk-Elim): (Tonk-Intr) A ├ A tonk B. (Tonk-Elim) A tonk B ├ B. Notice that (Tonk-Intr) and (Tonk-Elim) meet the subformula constraint. Prior‘s argument was intended to knock down the inferentialist proposal and, even if not completely successful, it had a huge effect, forcing inferentialism to reinforce its position by the introduction of further restrictions to permissible rules. Belnap led the way out by spotting tonk’s main flaw. As he saw it, the rules for ‘tonk‘ignored the ‘antecedently given context of deducibility’ (Belnap, 1962, p.  131). Hacking followed a similar path without mentioning Belnap, and in the course of explaining his point that logical constants should be defined using rules ‘like those of Gentzen‘, he says: My answer is that the operational rules introducing a constant should (i) have the sub formula property, and (ii) be conservative with respect to the basic facts of deducibility. (Hacking, 1979, p. 304).

So, the trouble with tonk is that it is not ‘conservative‘, i.e. that it produces an extension of the original set of valid inferences that includes A├ B, for arbitrary A and B (Belnap, 1962 op. cit., p. 132). Every genuine connective, according to Belnap and Hacking, has to maintain unaltered the set of valid inferences that was accepted before its introduction. The case of ‘tonk’ will be discussed in more detail in Chap. 7. Conservativity is a very revealing property to ask of a logical constant since it makes it clear that logical constants do not produce new arguments. At most, they help to present valid arguments as arguments, but the logically valid arguments in which a connective occurs essentially had to already be valid before the connective’s introduction. This is the import of the conservativeness constraint: an indisputable fact, which is by no means new. Frege was aware that his conceptual writing did not establish any new truths (Frege, 1879, p. 13, see also Chap. 2, Sect. 2.4). Conservative rules do not introduce substantive concepts or propositions, i.e. concepts with a content that adds something to the concepts and propositions that are already in use. This is again the insubstantiveness intuition and a central part of what logical expressivism defends. Conservativeness and the subformula constraint are two steps in the direction of repairing the wrong outcomes of the right inferentialist insight. Their combination

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amounts to the claim that logical constants represent inferential connections between propositions, without adding anything either to the propositions thus connected, or to the group of valid inferences. In all this, we have not travelled an inch further from Frege’s expressivism and (PPP). On the contrary, the discussion in the files of inferentialism shows that we have walked several yards backwards. Overlooking the fact that the items involved in logical practices are full-fledged propositions, we have encountered artificial problems that can only be solved by reinserting in ‘syntactic’ (the subformula property) and ‘semantic’ (conservativeness) modes what was manifest in the ‘pragmatic‘mode. I will come back to the pragmatic mode at the end of this chapter. But before that, it will also be instructive to discuss the other big family of proposals on logical constants, i.e. the invariantist family.

5.4 The Erlangen Programme In the final paragraphs of his paper on logical consequence, Tarski acknowledges that characterising the notion of ‘following logically’ requires a distinction between logical and extra-logical terms. Without it, not only the definition of logical consequence but also the definition of other central notions in the philosophy of logic such as analyticity, contradiction, and tautology would be compromised (Tarski, 1936, pp. 59–61). Thirty years had to lapse before he provided his characterisation, in a lecture that he gave at Bedford College, at the University of London, in 1966. Then another twenty years went by before the text was posthumously published by J.  Corcoran in 1986 with the title ‘What Are Logical Notions?’ (Tarski, 1986). Tarski put forward an extension of Klein’s Erlangen Programme for geometry in order to characterise logical constants and thus define the limits of logic. What is known as the ‘Erlangen Programme’ is a general proposal for unifying the different branches of geometry that Felix Klein defended in 1872 at the senate of the Friedrich Alexander University in Erlangen in order to enter the philosophical faculty. The problem, as he put it, was the following: Given a manifoldness and a group of transformations of the same; to develop the theory of invariants relating to that group. This is the general problem, and it comprehends not alone ordinary geometry, but also and in particular the more recent geometrical theories which we propose to discuss, and the different methods of treating manifoldness of n dimensions. Particular stress is laid upon the fact that the choice of the group of transformations to be joined is quite arbitrary, and that consequently all methods of treatment satisfying our general condition are in this sense of equal value (Klein, 1872/1893, p. 218).

In the Erlangen Programme, Tarski saw a method that might give the appropriate results applied to the definition of logical notions. The overall idea that motivated Tarski‘s characterisation was his understanding of logical notions as objects of the most general kind, which matched his understanding of logic as the most general of the sciences. In his lecture and subsequent text on logical notions, Tarski declines to deal with this latter issue at length, but still gives some clues which perfectly fit his

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approach to logical notions. I repeat here a passage that I already quoted in Chap. 2, Sect. 2.5, above: I shall not discuss the general question ‘What is logic?’ I take logic to be a science, a system of true sentences, and the sentences contain terms denoting certain notions, logical notions. I shall be concerned here with only one aspect of the problem, the problem of logical notions, but not for instance with the problem of logical truths. (Tarski, 1986, p. 145).

This passage eloquently expresses Tarski‘s understanding of logic as one of the sciences. Logic is a system of true sentences, part of whose concern is the problem of logical truths. According to Etchemendy, this was not Tarski‘s view of logic all the way through. In ‘Tarski on Truth and Logical Consequence’, Etchemendy contrasts Tarski‘s take on logic with what he calls the ‘Frege-Russell’ view. The Tarski-­ Etchemendy approach takes consequence to be the ‘primary subject of logic‘(Etchemendy, 1988, p. 74), and understands logic as a metadiscipline whose concern is the deductive sciences in general (Etchemendy loc. Cit., p.  75). The Frege-Russell approach, by contrast, is the more traditional view of logic as concerned with ‘a particular body of truths: logical truths‘(Etchemendy loc. Cit., p. 74). As we saw in Chap. 2, attributing to Frege the vision of logic as a particular system of substantive truths is debatable. In fact, what their respective definitions of logical constants reveal is the mirror image of the one that Etchemendy assumes: Tarski becomes an absolutist about logic, which is considered as the queen of sciences, whereas Frege relegates it to the role of assistant. If, by contrast, our focus is on the definition of logical consequence, then Etchemendy is right to attribute to Tarski the view of logic as belonging to the metatheory of sciences. The view of logic as auxiliary and relational, focused on consequence and not on truth, perfectly fits the strategy taken by the formalist programme in logic that originates in Hilbert. As can be seen in the correspondence between Frege and Hilbert about the nature of axioms (Frege, 1980, pp. 31–51), Frege took the traditional side and understood axioms as representing basic truths whose expression should not include any non-defined sign. Contrariwise, Hilbert defended the contextual definition of geometrical terms and the priority of axioms, which were true by fiat. Nevertheless, it cannot be forgotten that, in this exchange, Frege and Hilbert discussed the foundations of geometry and not the nature of logic. By the time Tarski wrote on the notion of logical consequence, Gentzen had written his ‘Investigation into Logical Deduction’, which was the origin of the natural deduction approach to logic. Tarski‘s and Gentzen‘s views at that time were closer to Hilbert’s general conception than to the position that Frege took in his correspondence with Hilbert. For this reason, the analysis of logical notions given by Tarski in 1966 is surprising. Still, his position at this time is clear: logic is the most general among the sciences, and logical notions are the most general invariant objects of a certain kind. If anything, Frege’s and Tarski‘s internal tensions about truth and logic show that for many decades, and still today, the nature and limits of logic are far from clear. What, by contrast, is crystal clear are the connections between the nature of logic and the definition of logical notions. On this point, it can be safely said that

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the views of Frege in Begriffsschrift and Tarski in ‘What Are Logical Notions?’ are as far apart as two views can be. Tarski begins his paper by explaining Klein’s method, whose key notion is that of one-one transformation of the (geometrical) space onto itself. Briefly, the method proceeds as follows. In Euclidean geometry, the movement of a rigid body can be seen as produced by a function that assigns to each point that is occupied by the object at the beginning of the process a corresponding point in the space occupied by the object at the end of the process. The invariance in the distance between points through functions of this type defines the motion of rigid bodies. This idea can be exported to other notions that define other branches of geometry. Tarski then explains that some notions of Euclidean geometry are not only invariant under motions, but also under the functions known as ‘similarity transformations’ (Tarski, 1986, p. 147). Similarity transformations do not keep distance invariant, since they ‘allow’ geometrical objects to expand or shrink in all directions. Klein saw, Tarski acknowledges, that invariance under similarity transformations defines metric geometry (Tarski op. cit., p. 148). This process can go on. For instance, if the notion that is kept invariant is the mutual linear position of points, then we obtain affine geometry (Tarski loc. Cit.); if the notion that is preserved is connectedness, then topology is defined (Tarski op. cit., p. 149), and so on. At this point, Tarski proposes to consider the class of all one-one transformations, extending the permissible domains and ranges to cover not only the geometrical space, but also the universe of discourse, and the ‘world’ (Tarski op. cit., p. 149). He then asks: ‘What will be the science which deals with the notions invariant under this widest class of transformations?’, and he answers: Here we will have very few notions, all of a very general character. I suggest that they are logical notions, that we call a notion ‘logical’ is it is invariant under all possible one-one transformations of the world onto itself. (Tarski op. cit., p.150).

The basic idea is then that, of all the sciences that can be defined using one-one transformations, i.e. whose central notions are invariant under some bijective functions, logic is the one that possesses the highest degree of generality. Bonnay explains that, so far, invariance is the only notion that allows a boundary between logical and non-logical symbols which is pure, local, and intrinsic (Bonnay, 2014). It is pure because the characterization of logical terms is produced exclusively in semantic terms, it is local because the characterisation of a term as logical is done by considering only the semantic properties of the term concerned. It is intrinsic because the characterisation does not use any contrasting or comparative class of terms. From a formalist perspective, it might well be that these three properties pick out invariantism as the best position, as opposed to approaches that—like Quine’s, Carnap’s, and Feferman’s— appeal to alternative aspects of logical constants. Nevertheless, if our aim is to make sense of logic and inferential practices, then the essential question to be answered is why it is desirable for an approach to logical constants to possess these characteristics. And Bonnay is aware of this (Bonnay, 2014 op. cit., pp. 57–59). What do these formal properties teach us about the nature of logic? Without an answer to this question, the merits of invariantism

5.5  Invariant Terms of Logic

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are hardly evaluable; unless, again, our assessment is made against the background of a particular view about the nature of logic. This is Bonnay’s option, and the option followed by all invariantists after Tarski. Thus, after assuming that invariantism is extensionally correct—something else that is debatable—Bonnay offers two reasons to endorse invariantism. The first reason, which he finds in (Sher, 1991), is that invariantism ‘reflects the formality of logical notions’. The second reason rests on Tarski‘s own defence of logic as the most general of all sciences. I have already made some comments on the formality of logic in the previous chapters and will issue some more in the next chapter. Concerning the second reason, we arrive back at our starting point: the nature of logic, which is undoubtedly the main motive that led Tarski to apply the Erlangen programme to the definition of logical constants.

5.5 Invariant Terms of Logic In the process of adapting the Erlangen programme, Tarski abandons the realm of geometry, in which the arguments and values of functions are points in space and substitutes the geometrical space with the universe of discourse, or with the ‘world’ (Tarski op. cit., p. 149). This move, although by no means new, completely blurs the profound differences between those sciences that deal with formal objects, sets and structures and those theoretical enterprises whose concern is concepts and propositions. The properties of points and structures are not the properties of concepts and propositions. These properties are not, in any case, what places propositions at the centre of our discursive life. But let us now resume the issue of applying the Erlangen programme to logic. Tarski hastens to assure us that his invariantist notions are the standard logical constants and guarantee the extensional adequacy of his characterisation: I am not going to formulate the result in a very exact way, but the essence of it is just what I have said. Every notion defined in Principia Mathematica, and for that matter in any other familiar system of logic, is invariant under every one-one transformation of the ‘world’ or ‘universe of discourse’ onto itself. (Tarski loc. Cit.)

To be precise, this is not completely correct. The editor of the paper, John Corcoran, adds a footnote explaining that for the invariantist procedure to be successfully applied to the standard logical notions, it is necessary to transform them into notions of the type hierarchy. Thus, the standard truth values, the True and the False, have to be understood as the universal and empty classes, respectively, in order to secure an appropriate characterisation of the standard truth functions and quantifiers (Tarski loc. Cit.). It is important for my general argument to mention that Corcoran notes the discomfort that this redefinition of logical notions in terms of sets and sets of sets produces among philosophers. Tarski‘s adaptation of the Erlangen programme uses the simplest version of the type hierarchy: that in which individuals constitute the lowest type, and from the lowest level up, each level is defined on the previous one. The next level

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immediately after the level of individuals is the level of classes of individuals; the next one is the level of classes of classes and so on. Tarski also considers binary relations between individuals and mentions other n-adic relations (n > 2), although he does not go into the details. The same happens with properties of classes and relations between classes: they are mentioned but not developed. According to those who support it, a very welcome outcome of invariantism is that there is no logical object at the lowest level: no object that remains invariant through all bijective functions between individuals. This is as it should be, since logical notions are topic-neutral and insubstantive, and cannot be used to discriminate among individuals. At the level of classes of individuals, only two objects are invariant, the universal class and the empty class: in both cases they and their images are identical. If, instead of classes of individuals, we consider binary relations between individuals, then the possibilities at this point are four: the general relation that every individual has with any other, the empty relation in which no two individuals stand, identity and difference. When we travel up in the hierarchy to consider classes of classes (Tarski talks of ‘properties’ of classes), the sole logical objects are cardinalities, since the only properties of classes that are stable across transformations are sizes, i.e. the number of members in them. Tarski also mentions logical relations between classes: inclusion, overlap, disjointness, and others. It is illuminating to notice that when classes are seen as extensions of properties, the logicality of the objects obtained by the invariantist procedure becomes more intuitive. And yet the question remains, what is the connection between invariance and logicality? At this point, Tarski draws a startling conclusion about the essence of logic: This result seems to me rather interesting because in the nineteenth century there were discussions about whether our logic is the logic of extensions or the logic of intensions. It was said many times, especially by mathematical logicians, that our logic is really a logic of extensions. This means that two notions cannot be logically distinguished if they have the same extension, even if their intensions are different. As it is usually put, we cannot logically distinguish properties from classes. Now in the light of our suggestion it turns out that our logic is even less than a logic of extension, it is a logic of number, of numerical relations. (Tarski op. cit., p. 151).

This result should not be surprising if the method followed is an extension of the Erlangen programme; what is surprising is the enthusiast endorsement that Tarski gave to it. Tarski‘s definition of logical consequence can hardly be reconciled with his claim that our logic is a logic of number. Thus, it seems that when we talk of logic, we are talking about the sizes of some classes. The extensional approach to logic can easily accommodate this result. Still, much more argument is needed to give an intuitively acceptable interpretation to the claim that, when we draw inferences, we are doing something that is mostly concerned with the size of certain set-theoretical objects. The logic of number is, by contrast, essentially related to mathematics. And nevertheless, invariantism provides no reasons for deciding on the logicist programme in either way. Tarski argues that, given that mathematics is reducible to set theory, the question of logicism can be reformulated as asking whether set theory is a part

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of logic, or even more precisely, whether the membership relation is a logical relation. Depending on which one of the two methods of developing set theory one chooses—the Principia Mathematica method, or the method developed by Zermelo, von Neumann and Bernays—the answer will be one or the other. In Principia Mathematica, which assumes type theory, the answer is affirmative. In standard Zermelo set theory, the answer is negative, since there is no type hierarchy, and the membership relation is a primitive relation between individuals. Thus, Tarski‘s definition of logical terms, one might think, leaves the debate over logicism, which was Frege’s general programme, underdetermined. In a general sense, this is the case. But there is a more profound sense in which it does not leave it underdetermined but rather untouched. Frege’s logic was not the logic of number, even though his greatest logical achievement, i.e. the correct analysis of quantifiers, was accomplished along with his analysis of the notion of number. Quantifiers are higher-level concepts that express properties and relations between concepts. Frege’s logic (and semantics) offers the tools for analysing a particular group of higher-level predicables and relations, and it does so from a perspective that is more qualitative and inferential than quantitative. Frege’s logicism was a rejection of Kant’s approach to arithmetic and a defence of the conceptual character of inference, which is a relation that holds between judgeable contents. If the question of logicism in Tarski‘s approach to logic depends on the version of set theory that one favours, then this is evidence that Tarski‘s focus is on representation systems, and not on the contents that can be represented by them. Tarski‘s paper proceeds smoothly in proposing an exquisite, limpid and inclusive characterisation, which extends an already successful method of characterising abstracts objects to the realm of logic. This is the surface. Deep down, his proposal sets logic in a path that leads to specific answers to substantive philosophical questions: a proposal that seems to clash with some of his former views on the nature of logic and the consequence relation.

5.6 A Pragmatist Excursus Inferentialism overgenerates and undergenerates. And despite Tarski‘s defence of the extensional adequacy of his method, invariantism presents similar shortcomings (Gómez-Torrente, 2007, pp. 190–192; Dutilh Novaes, 2014). The common strategies for fixing extensional inadequacies are in any case insufficient, since extensional adequacy falls short of what is needed when our aim is to define the boundaries of logic. As it is common with many philosophically loaded notions, there is a mismatch between the unproblematic identification of the core cases, such as conditional and negation, and the mastery of their common use, on the one hand, and the difficulties they present to be given appropriate semantic treatments, on the other. There is an explanation that derives from the situation in the philosophy of language, which has only recently developed the resources to deal with higher-level concepts properly.

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The simplest tasks that we do with language are referring to, i.e. pointing at, and describing the reality around us. Pointing at and mimicking are the origin of human linguistic activity (see Tomasello, 2008, pp. 57–108). But possessing a language, in the human sense of ‘language’, is not limited to the mastery of these tasks. The kind of semantic theory that is suited to dealing with descriptive discourse is relatively unsophisticated, and in one way or another, all theories of language propose reasonable narratives that explain the functioning of communication at this basic level. By contrast, logical terms neither refer nor describe: they instead represent functional concepts whose arguments are propositions and concepts, all of them abstract entities. Their entry into language and thought results from our dealing with other concepts and their semantics is less straightforward since they do not contribute any isolable component to the propositional content of our linguistic acts. Classical logical terms express either operations with concepts, as in the case of conjunction, or relations between contents, as in the case of the conditional. These relations can be normative or factual. In semantic theories that do not have the resources to explain non-representational kinds of meaning, the alternatives are either conferring to the offending terms no meaning at all—or at most some kind of emotional scent—or else forcing them into the representational template, which stirs up metaphysical issues about the ‘objects’ referred to and the kind of referential relation that is proper to them. Only semantic theories that are sophisticated enough to make room for the diverse communicative contributions of terms and concepts can safely and fruitfully undertake the analysis of logical notions. And this also applies to the analysis of other higher-level concepts, such as alethic and epistemic modalities (truth, knowledge) and evaluative notions (good, correct, bad). A significant portion of logical and metaphysical debates derive not from philosophical sophistication, but from semantic shallowness. Only semantic pluralism has any chance of success when our aim is to explain the meaning of terms and concepts that define the most abstract levels of our linguistic and conceptual activity. The intuitions about logical notions that all proposals purport to explain are similar: the difference between these proposals stands in the methods they use to account for these intuitions. These intuitions are that logical terms do not represent, that they are insubstantive and that they have a role to play in deductions and inferences. The connection with actual inferences is harder to trace in the case of invariantism, as we have seen. Still, Tarski assumes that the universe on which the bijective functions are defined is the universe of discourse, i.e. the realm in which concepts combine to help rational agents establish certain truths. Inferentialism, by contrast, is closer to Frege’s views, which make the role of logical terms depend on the outcome of certain acts of rational agents, particularly assertion. In spite of all the efforts and thousands of pages devoted to discussing the meaning of logical constants, most authors have the bitter impression that there is no agreed-upon characterisation of the meaning of these notions on which the other logical concepts essentially depend. Tarski expressed his pessimistic feelings about the topic at the end of (Tarski, 1936). Others, from completely different backgrounds, have more or less explicitly acknowledged that this issue might not reach

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a positive resolution, and that the closest we will get to it is the elaboration of an open-ended list, as Frege did (Frege, 1879, p. 7), and many others have done since (Carnap, 1937, pp. 18–19; Quine, 1951, p. 20; Quine, 1970, pp., 1, 22–24; Hacking, 1979, p. 287). This pessimistic feeling is widespread among pragmatists. Warmbrõd, for instance, begins his paper on the topic by saying that there ‘is as yet no settled consensus as to what makes a term a logical constant or even as to which terms should be recognized as having this status’ (Warmbrod, 1999, p.  503). Gómez-­ Torrente expresses his scepticism about the possibility of giving an extensionally correct characterisation based only on mathematical or semantic features (Gómez-­ Torrente, 2002, p. 34). He proposes to include in this characterisation some other aspects that reflect what agents use logical constants for, and to this end, he identifies the features of broad applicability and inferential relevance (Gómez-Torrente, 2007, pp.  199–200). Both of these features have been explicitly or implicitly accepted by almost everybody working on the topic. For Frege, the inferential relevance of the logical constants rests on their role as a means to represent safe transitions between judgeable contents. This characterisation already included the feature of broad applicability, since there was no further constraint on the intervening judgeable contents, over and above their status as judgeable contents. I agree with the pragmatist positions on the diagnosis of this situation, i.e. that most of the proposals are seriously faulty. I also agree with the general positive suggestion, i.e. that mathematical and semantic characterisations have to be completed by the addition of aspects related to their role in the rational activities of human beings. I nevertheless disagree about the lines along which these formal characterisations should be implemented. This point will be properly discussed in the next chapter. I cannot agree with the general assumption that pragmatic approaches are necessarily vague. If the role of logical constants is clear, then the boundaries between logical uses and non-logical uses of specific expressions should also be clear. Surely, if their role is described as ‘inferential relevance’, without further qualifications, then this division cannot be clear-cut. Relevance is a highly context-­ dependent notion that includes in its application criteria a slot for specifying a purpose, an aim: ‘relevant for what?’. And inferential is very general. Every concept possesses inferential relevance, as the discussion about inferentialism above has shown. The characterisation proposed in Begriffsschrift, for instance, is not vague. A different issue is whether it precisely delimits the set of notions that have traditionally been considered to be logical notions. But there are two different topics here: the first is extensional adequacy, and the second is philosophical correction. And without this second aspect, which involves the provision of criteria, extensional adequacy is meaningless. If we do not know what we are looking for, then we can hardly say whether a set contains everything that it should contain and nothing else. Pragmatic approaches do not need to be vague, even though they can produce different boundaries for different criteria. Depending on how strictly we understand the idea of ‘inferential relevance’, notions such as conjunction or identity would fall inside or outside the boundaries. This does not mean that the issue is arbitrary or imprecise, or that pragmatist approaches are incompatible with syntactic and

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semantic precision. A precise pragmatist account of the meaning of logical constants will be the aim of the next chapter.

References Belnap, N. (1962). Tonk, plonk and plink. Analysis, 22(6), 130–134. Blakemore, D. (2002). Relevance and linguistic meaning. Cambridge University Press. Blakemore, D. (2011). On the descriptive ineffability of expressive meaning. Journal of Pragmatics, 43(14), 3537–3550. Bonnay, D. (2014). Logical constants, or how to use invariance in order to complete the explication of logical consequence. Philosophy Compass, 9(1), 54–65. Brandom, R. (1984). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press. Brandom, R. (2000). Articulating reasons. Cambridge, Mass., Harvard University Press. Carnap, R. (1937). Logical syntax of language. The International Library of Philosophy. Došen, K. (1989). Logical constants as punctuation Marks. Notre Dame Journal of Formal Logic, 30(3), 362–381. Dummett, M. (1973). Frege. New York, Harper and Row Publishers. Dummett, M. (1991). The logical basis of metaphysics. Harvard University Press. Dutilh Novaes, C. (2014). The Undergeneration of permutation invariance as a criterion for logicality. Erkenntnis, 79(1), 81–97. Etchemendy, J. (1983). The doctrine of logic as form. Linguistic and Philosophy, 6, 319–334. Etchemendy, J. (1988). Tarski on truth and logical consequence. The Journal of Symbolic Logic, 52(1), 51–79. Etchemendy, J. (1990). The concept of logical consequence. Mass.: Harvard University Press. Ferferman, S. (1999). Logic, logics, Logicism. Notre Dame Journal of Formal Logic, 40(1), 31–54. Fraser, B. (1990). An approach to discourse markers. Journal of Pragmatics, 4(3), 383–398. Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In Jean van Heijenoort (1967), from Frege to Gödel. A source book in mathematical logic, 1879–1931 (pp. 1–82). Harvard University Press. Frege, G. (1980). Philosophical and mathematical correspondence. Blackwell Publishers. Gentzen, G. (1935). Investigation into logical deduction. American Philosophical Quarterly, 1(4), 288–306. Gómez-Torrente, M. (2002). The problem of logical constants. Bulletin of Symbolic Logic, 8, 1–37. Gómez-Torrente, M. (2007). Constantes Lógicas. In M. J. Frápolli (Ed.), Filosofía de la Lógica (pp. 179–205). Madrid. Haack, S. (2005). Formal philosophy? A Plea for Pluralism. Five Questions on Formal Philosophy. VIP Press, Vincent Henricks. Hacking, I. (1979). What is logic? Journal of Philosophy, 76, 285–319. Hintikka, J., & Sandu, G. (2007). What is logic? In D.  Jaquette (Ed.), Philosophy of logic (pp. 13–39). North-Holland. Klein, F. (1872/1893). A comparative review of recent researches in geometry. Bulletin of the New York Mathematical Society, 2(1892–1893), 215–249. Klima, G. (2006). In K. Brown (Ed.), Syncategoremata. Elsevierʼs Encyclopedia of language and linguistic (Vol. 12, 2nd ed., pp. 353–356). Elsevier. Kürbis, N. (2015a). Proof-theoretic semantics, a problem with negation and prospects for modality. Journal of Philosophical Logic, 44(6), 713–727. Kürbis, N. (2015b). What is wrong with classical negation? Grazer Philosophische Studien, 92(1), 51–86. Kürbis, N. (2019). Proof and falsity: A logical investigation. Cambridge University Press.

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Martin-Löf, P. (1996). On the meanings of logical constants and the justifications of the logical laws. Nordic Journal of Philosophical Logic, 1(1), 11–60. Prawitz, D. (2006). Meaning approached via proofs. Synthese, 148, 507–524. Prior, A. (1960). The roundabout inference-ticket. Analysis, 21, 38–39. Quine, W. V. O. (1951). Two dogmas of empiricism. The Philosophical Review, 60, 20–43. Quine, W. V. O. (1970). Philosophy of logic. Harvard University Press. Ryle, G. (1954). Dilemmas. Cambridge University Press. Schiffrin, D. (1987). Discourse markers. Cambridge University Press. Sher, G. (1991). The bounds of logic. MIT Press. Sher, G. (2003). A characterization of logical constants is possible. Theoria, 18(2), 189–198. Tarski, A. (1936). On the concept of following logically. Translation from the polish and German by Magda Stroiska and David Hitchcock. History and Philosophy of Logic, 23(3), 155–196. Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7, 143–154. Tomasello, M. (2008). Origins of human communication. A Bradford Book. The MIT Press. Warmbrod, K. (1999). Logical constants. Mind, 108(431), 503–538. Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. In C.  K. Ogden (Ed.), Prepared with assistance from G. E. Moore, F. P. Ramsey, Wilhelm Ostwald, and Wittgenstein. Routledge & Kegan Paul. Logisch-Philosophische Abhandlung (1st ed., p. 14). Annalen der Naturphilosophie.

Chapter 6

The Inference-Marker View of Logical Notions: What a Pragmatist Proposal Looks Like

Abstract  In this chapter, I discuss an informed pragmatist proposal for characterising the class of logical constants, which I call ‘the inference-marker view’. It includes syntactic, semantic, and pragmatic aspects, all of them essential to the task that ultimately defines logical terms as expressive devices. Logical notions are not objects, nor do they refer to objects. Rather they are relational expressions whose meaning conveys some kind of movement between their arguments. The meaning of the relevant terms that represent logical notions linguistically has to reflect the dynamic function that they perform. The inference-marker view is inspired by Frege’s Begriffsschrift. The complex role of logical notions requires them to be binary higher-level predicables with propositions as their arguments. Being binary and higher-level are the syntactic aspects of logical terms. Concerning their semantics, the proposal stresses that logical terms do not stand for substantive concepts. Logical terms are expressive tools for bringing inferential commitments into the open. They are used to mark the presence of an inferential link between two propositions, or between one set of propositions and another proposition that follows from them. Alternatively, they express a blockage or a veto. The inference-marker view is in complete harmony with Brandom’s expressivism. The specific label that I use—‘inference-marker view’—is intended to make the role of logical constants more perspicuous, drawing on some contemporary linguistic positions in order to understand the role of these terms in language (or the role of these notions in our conceptual system). Keywords  Adversative · Brandom · Conditional · Expressive [expressivism] · Higher level [higher-level] · Logical constant · Membership · Negation · Pragmatist [pragmatism] · Predicable

6.1 The Proposal It is time now to take stock, to revisit what we have learned about logic and logical terms in the previous chapters and make a positive proposal. We have accepted some Fregean principles: The Principle of Propositional Priority (PPP), which © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_6

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makes propositions the bearers of logical properties, and its complement, the Principle of Grammar Superseding (PGS), which rejects the reliability of grammar for logical issues. These two principles imply that logical notions are properties of—and/or relations between—complete propositions, and therefore apply neither to isolated concepts nor to linguistic items of any kind. On the pragmatist view in which Frege’s approach participates, logical notions represent inferential links rather than substantive concepts affecting contents and meanings. ‘Substantive’ stands opposed to ‘expressive’. We might discuss whether logical notions are actually concepts or not. If we think of concepts as abstracted from propositions, then logical notions do not hit the mark. If, by contrast, concepts are understood as the senses of functions, then they can be included in this category. Nothing philosophically relevant hangs on this terminological choice. The specific function that logical terms perform has been acknowledged by philosophers, but also by linguists, who include them in the general class of discourse markers. Nevertheless, syntactic and semantic characterisations of logical constants fall short of getting the demarcation right and yielding the correct distinction between genuine logical constants, such as ‘if’ and ‘therefore’, and other markers such as ‘moreover’ or ‘anyway’, which we would not like to take on board. These characterisations usually overgenerate, although, as we saw in the previous chapter, they sometimes undergenerate as well. The adjustment of the general kind of terms that meet the syntactic and semantic characterisations to the specific character of logical terms requires adding the pragmatic role that these terms perform. Some authors have mentioned the inferential relevance of logical terms (Gómez-­ Torrente, 2002, Gómez-Torrente, 2007; see the previous chapter), but merely insisting on their inferential relevance is still too general. Frege offers a more accurate view by attributing to logical constants the role of marking inferential transitions. Following his suggestion, I have called my proposal the ‘inference-marker view’ (IMV) (Frápolli, 2012; Frápolli & Assimakopoulos, 2012). The Fregean intuition that sees logical terms as markers has been stressed by Brandom in his expressivist proposal (see Chap. 4), which not only identifies logical terms correctly but also offers an explanation of their functioning against the background of a general inferentialist approach to meaning, also of Fregean inspiration. The provisional characterisation I propose, (IMV), the discussion of which is the topic of the present chapter, is faithful to the pragmatist approach to meaning I assume, as well as to the Fregean view of logic that I have endorsed in the previous chapters. It goes as follows: (IMV) (The Inference-Marker View) Logical constants are binary higher-level predicables that have 0-adic predicables as their arguments. The terms that represent them linguistically neither name nor describe. Their function is rather to show inferential connections between their arguments. A question that comes up every time we deal with logical notions is whether we are talking about linguistic items, i.e. signs and words, or about the notions, concepts or rules that these terms purport to represent. The pragmatist answer is that

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these two levels—the level of words and the level of their meanings—do not make sense in isolation from each other. Logical notions are not merely syntactic items infused with mathematical or structural features. To be included in the class of logical terms, the terms concerned must represent operations or functions that speakers actually perform through their use. It is this pragmatic aspect that determines their membership in this category, although their syntactic and semantic features are necessary conditions, essential for them to be fit for the job. Sometimes entities (terms) that ‘occur’ in propositions (sentences) are said to stand in logical relations. In fact, some of the standard logical terms significantly intervene in the building of complex concepts and expressions. Some examples are the occurrences of conjunctions and disjunctions between conceptual arguments. I will discuss this point as the chapter progresses but let me say in advance that these cases are not a serious threat. Quantifiers, by contrast, present a genuine challenge to (IMV), since being the kind of concept that defines the Fregean revolution in logic, they do not have complete propositions as their arguments. If we want quantifiers on board—and this is not a trivial debate—we might weaken (IMV) to obtain (IMV)weak: (IMV)weak Logical constants are binary higher-level predicables whose arguments are n-adic propositional functions (n ≥ 0). They don’t name any kind of entity, nor do they describe any aspects of the world, but are natural language devices for making inferential relations between concepts and propositional contents explicit. In the traditional classificatory terms that I no longer consider illuminating, it can be said that (IMV) and (IMV)weak involve syntactic, semantic, and pragmatic claims. At the level of (logical) syntax, logical constants are higher-level functions. From a semantic point of view, the terms that express logical concepts do not name or describe. Their pragmatic function is rather that of allowing speakers to signal the presence of an inference, either to endorse it or to reject it, or to merely analyse its logical features. The addition of the pragmatic level is crucial: in the first place, because linguistic practices and the purposes of speakers are the phenomena to be explained. Which are the logical constants in particular artificial systems is trivial. The non-trivial discussion that logical constants put on the table concerns why logicians have chosen these particular natural language terms to be formally represented as logical constants in the logical languages that they develop. In the second place, the pragmatic aspect is needed to downsize the class of non-­ substantive terms to the specific class of logical terms, since the most repeated criticism against traditional proposals is that they offer at most necessary conditions. There are different kinds of terms that are insubstantive, syncategorematic, or topic-­ neutral. The meaning of many different terms can be given by rules, and many are functions that do not discriminate between individuals and are higher-level too. Nevertheless, we would not like to include prepositions, interjections, adverbs, or propositional attitude verbs in the class of logical terms. The function they perform—the use speakers give to them—plays a part in their characterisation and provides the required restriction for selecting the right set of notions.

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I will provisionally stick to (IMV), to see how far we can get with the strong version of the proposal. Its syntactic claim will be dealt with in the first place. The term ‘predicable’ stems from (Geach, 1962) and (Williams, 1976), and highlights the distinction between uses in which concepts are effectively attributed to objects and uses in which concepts are not assigned to anything. ‘Predicable’ is generic, whereas ‘predicate’ refers to tokens with a specific kind of use. Logical constants are higher-level predicables, i.e. functions, whose arguments are 0-adic predicables, i.e. propositions. The characterisation of propositions as 0-adic predicables is due to Peirce and abundantly used by C. J. F. Williams. In Frege’s theory, propositions and n-adic (n > 0) concepts belong to opposing categories. Propositions are saturated entities, whereas concepts are unsaturated (see, for instance, Frege, 1892a). Nevertheless, the strategy of beginning with propositions and defining concepts by removing some ‘ingredients’ from them, leaving an empty space that can be filled by a different term or concept, is an essential strategy of Frege’s semantics. It is used, among other places, in (Frege, 1879, §9) and (Frege, 1884, §70), as well as in (Frege, 1892b, p.  162), to discriminate between sense and meaning in sentences. Thus, even in Fregean terms, propositions can be understood as those concepts that we obtain from propositions—the actual or possible contents of assertive acts—by the procedure of removing 0 ingredients from them. If we stick to the Fregean option of explaining the difference using the chemical term, metaphorically used, “saturation”, then propositions and concepts fall into different semantic categories. This should not be a problem. The analytical apparatuses that we apply to the analysis of a specific area of discourse or reality affect the outcome we obtain. But analytic tools is not what is to be explained; tools are tools for understanding something which is independent of them. Compatible or not with Frege’s view, understanding propositions as 0-adic concepts has the benefit of allowing a unified treatment of high-level predicables: a category that includes functions of propositions, but also functions of n-adic concepts (n > 0) (see Williams, 1992b). The syntactic characterisation of logical notions as functions of 0-adic predicables is not arbitrary. It accounts for the fact that logical transitions obtain between propositions, which are the bearers of logical relations. This is what follows from placing validity at centre stage and understanding logic as assistant and not as queen. The subformula property introduced by Gentzen and Hacking is the way found by inferentialism to express the same intuition (see Chap. 5). It also answers to the intuition that logical constants do not belong to the content of inferences represented by propositions in premises and conclusions, but are instead ‘outside’ them, marking connections and acting somehow as punctuation signs. Logical constants, understood as higher-level predicables, also give precise expression to the intuitions of the insubstantivity and formality of logic, as discussed in Chaps. 5 and 3 respectively. The syntactic characterisation, nevertheless, is not enough to demarcate the class of logical notions. There are many expressions in language that work as functions of propositions without qualifying for this category (see Frápolli & Villanueva, 2012).

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The semantic claim, in turn, captures those aspects of insubstantivity to which expressivism, in its negative thesis, points out, i.e. that logical constants do not contribute to what is said. All logical constants possess the expressive meaning of marking transitions, although the kind of transition that they mark differs from case to case. This is the (general) positive thesis of expressivism that is captured by the pragmatic claim which I will discuss below. I have already mentioned the challenge that quantifiers pose to (IMV). And not only quantifiers. Identity and conjunction do not qualify as logical constants either, although for different reasons. By contrast, disjunction and adversative conjunctions are serious candidates for this category. But before discussing the scope of and potential objections to this definition, let me state the terms of my commitment to (IMV). (IMV) is not an ad hoc proposal. It has not been developed to fit a rigidly predetermined set of expressions or abstracted to cover the more general aspects of a selected group of terms, although it has been guided by some pre-theoretical intuitions that include reference to certain expressions. This kind of virtuous circularity is unavoidable in any inquiry about how language works, which has to proceed following some sort of reflective equilibrium. A philosophical account of logical constants that aspires to any depth has to begin by establishing what kind of enterprise logic is. Against this background, it has to ask for the pragmatic significance of the terms that have a rooted history as logical terms, i.e. quantifiers, truth-­ functional connectives, and possibly modal and epistemic operators. On the other hand, it should not only produce an (intuitively) adequate set of notions but—and this is most important—explain why some notions qualify for the task that they are meant to perform according to our favoured approach to logic, and why some others do not. Thus, it should permit the discussion of new possible cases. The direction of an enlightening inquiry has to run contrary to Quine’s ‘list view’, or Russell’s enumeration in The Principles of Mathematics. For what is at stake is not a formal characterisation with extensional success, which in any case is an impossible task since there is no general agreement about the set of logical constants, but rather a general philosophical understanding of the reasons that support the selection. Understanding, rather than demarcating, is the pragmatist’s aim. Demarcation, if anything, is a consequence of an in-depth comprehension of the meaning and role of certain terms. What I submit is not an explanation of what makes these terms logically interesting, but rather the cluster of features that makes them logical terms. If I am right, then we should count as logical notions any (uses of) terms that fit the definition and only these. As we will see, the result of this procedure will be that some of our dearest ‘logical’ notions are not logical notions after all. Conjunction, identity in any one of its versions, the set-theoretical relation of membership, quantifiers (either binary or monadic), modal, temporal, and epistemic operators: none of them fill the bill according to (IMV), even though they have some identifiable similarities with notions that do. This way we will be able to identify a network of notions, all connected by certain features (syntactic, semantic or pragmatic), and all relevant to inferences in one way or another, but not all of them logical terms in a strict sense.

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By granting the possibility of toying with the three theses in it, (IMV) offers a positive and a negative insight. That the standard list of logical constants does not capture a homogeneous category is the negative insight. The positive insight is that logicians have got it right if we look at the big picture since, by identifying certain terms as logical, they have hit on terms with at least some principled connections with the core class of genuine logical terms. The acknowledgement that there are different subsets of logically interesting expressions, all of them with specific functions to perform, is a positive result that broadens our understanding of many specific tasks surrounding the practice of drawing inferences. The alternative method of subsuming all expressions with some logical relevance under a single category defined by a single, formal or structural, criterion gains generality at the price of omitting crucial information. The pragmatist way of looking at the issue is neither vague nor arbitrary, but rich and informative. This is a way of placing specific terms in more and more general linguistic and conceptual categories, paying attention to speakers’ discursive behaviour. At this point, I borrow Brandom’s words about the extension he intends for the project of analysis and the expressive role of logic: ‘I want to show how pragmatism can be turned from a pessimistic, even nihilistic, counsel of theoretical despair into a definite, substantive, progressive, and promising program in the philosophy of language’ (Brandom, 2008, pp. 31–2). My analysis of logical terms is part of this programme. What we pragmatists discuss is the status of certain actual concepts that are put to work by real people in real communicative exchanges. The project of understanding non-standard expressions of Twin English as used by silicon-based creatures, or ingenious logicians’ latest witty examples, will not be my concern here. This point will be taken up in the next chapter. On the contrary, I am committed to the project of finding a philosophical account that sheds light on the meaning and use of certain concepts that are actually involved in our inferential practices.

6.2 Some Consequences of (IMV) The import of (IMV) is clearer in the following table of higher-level predicables (Fig. 6.1). Most expressions in (Fig. 6.1) have, at some point in the history of logic, been considered logical constants, and for good reasons. All of them are like genuine logical terms in some respect, even though only a small subset ticks all the boxes to qualify as such. According to (IMV), only negation, adversative conjunctions, disjunction, the conditional, the bi-conditional, implication and equivalence are logical constants. The binary status of logical constants is an essential feature of the category. Because they signal inferential connections, logical constants need to acknowledge two poles between which the inferential transitions move. As I defended in Chap. 4, negation is syntactically monadic but semantically binary and, for this reason, it qualifies as a logical constant according (IMV). (IMV)weak also takes

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A. Monadic

B. Binary

1. Functions of nadic predicables (n > 0)

Negation; monadic quantifiers

Conjunction; binary quantifiers; higher-level identity; reflexivity

2. Functions of 0adic predicables

Negation; epistemic, modal, and temporal operators; truth; propositional attitude verbs

Conjunction; adversative conjunction; disjunction; the conditional; the biconditional; implication; equivalence

3. Terms with All the operators listed in A.1 expressive meaning and A. 2 4. Terms with inferential significance as inference markers

Negation and some uses of truth1

All the operators listed in B.1 and B. 2 Disjunction; adversative conjunctions; the conditional; the biconditional; implication; equivalence; variable hypotheticals (combinations of the conditional with standard and nonstandard quantifiers)

Fig. 6.1  Higher-level predicables a The truth operator can, in specific circumstances, have the force of an adversative conjunction. See Sect. 10.5 in Chap. 10

binary quantifiers on board, i.e. subordination and partial overlap. That logical notions are functions, i.e. unsaturated expressions and concepts, rules out truth-­ values, and the universal and empty classes, which are considered logical constants by most invariantists (see Chap. 5). Because the items whose inferential connections they mark are n-adic concepts (n ≥ 0), logical constants have to be higher-­ level. This feature rules out the first-order relation that is commonly known as ‘identity’, which is represented in first-order calculus by the sign ‘=’, and also the set-theoretical relation of membership, represented by ‘∈’. Following C.  F. J. Williams, I interpret ‘=’ as co-referentiality, a relation that involves mentioned expressions, and which is instrumental to the substitution rule that applies to terms in sentences. Co-referentiality is not a relation between propositions. It is a metalinguistic relation between terms that permits a proposition to be recognised under different modes of presentation. The two sentences ‘Mark Twain is an American writer’ and ‘Samuel Clemens is an American writer’ share their truth conditions and inferential potential, but this fact is only explicit once we know that Mark Twain is Samuel Clemens, i.e. that the two names ‘Mark Twain’ and ‘Samuel Clemens’ name one and the same individual (Williams, 1989, 1992a). Co-referentially (‘=’) is the sole first-order binary relator traditionally included in

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the set of logical constants, although some authors have resisted this received view (see, for instance, Peacocke, 1976; Warmbrod, 1999). There are some reasons that explain why it has been counted among the logical constants, though. Identity, or co-referentiality, shares with genuine logical constants its expressive type of meaning and, although it is not an inferential marker, it has essential inferential utility, if only behind the scenes. The expressive role of the semantic notion of reference, and hence of co-­ referentiality, has been explained at length by Brandon (Brandom, 1994, chapter five; Brandom, 2000, chapter four). As happens with genuine logical constants, co-­ referentiality does not represent a propositional component but rather shows that two terms are interchangeable (see Frege, 1879, §8). Williams identified a higher-­ level identity operator that converts n-adic predicables into (n-1)-adic predicables. Higher-level identity corresponds to the operation of reflexivity in natural language that converts transitive verbs into intransitive verbs and is one of the operators that Quine included in his proposal for combinatorial logic (Geach op. cit., pp. 161ff.; Williams, 1989, pp. 48ff.; Quine, 1960a). Like their first-level cousin, higher-level identity and reflexivity still fall short of being logical constants. They don’t have propositions as arguments, and although they are higher-level and have expressive meaning, they don’t possess the required inferential significance. Higher-level identity deserves attention as a powerful logico-semantic tool. Williams has proposed interpreting the notion of truth as an instance of higher-level identity with propositions as arguments (Williams, 1989, chapter five). This move captures the intuition, developed by identity theories of truth, that truth involves identity between something which stands at the linguistic level and something that is non-linguistic. But even with propositions as arguments, truth lacks the required pragmatic role as an inference marker (but see footnote 1). Similarly, the membership relator is not a logical constant according to (IMV). The debate over its status as a logical constant has a long history and accompanies the fate of logicism, understood as a claim about the reducibility of mathematics to logic, during the early decades of the last century. Tarski, as mentioned in Chap. 5, made its status depend on the version of the set theory used to characterise it (Tarski, 1986, p. 152). Nowadays, it is generally accepted that membership is not a logical relation (Gómez-Torrente, 2002, p.  17). What membership represents in set-­ theoretical terms is predication, which is the representation of the relation that Frege understood as holding between an object and the concept under which it falls. Returning to Fig. 6.1, two pairs of contrasting categories in it deserve a more detailed commentary. One pair is functions of n-adic concepts (n > 0) versus functions of complete propositions. The other is the distinction between monadic and binary operators, which produces a division in the class of quantifiers. We will consider these in turn in the next two subsections.

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6.2.1 Concepts and Propositions Negation, conjunction, and disjunction, all have combinatorial uses. The arguments of these connectives are either complete sentences or predicative expressions. Consider the following examples: (1) John is happy and John is rich. John is happy and rich. (2) 3 is odd or 3 is even. 3 is either odd or even. In Chap. 4 I mentioned the contrast between conjunction and disjunction and denied that disjunction is a concept-builder, although it undoubtedly is a predicable-­builder. This issue is debatable, though, and the algebraic counterparts of conjunction and disjunction—intersection and union respectively—provide arguments for a unified interpretation. The unified interpretation would apply, in any case, to the sense of disjunction that logicians call ‘inclusive’. Inclusive disjunction could be seen as a mechanism for building concepts, whose extensions would be the logical sum of the extensions of the concepts in the disjunction. And some concepts are indeed disjunctive. For instance, ‘Iberian’ is an adjective that means ‘of or relating to Iberia’. Thus, an individual falls under the concept Iberian if and only if it falls under any of the following concepts: Spanish, Portuguese, Andorran or Gibraltarian. Exclusive disjunction, by contrast, usually represents an alternative—a dilemma— and, although standard logic blurs the distinction between inclusive and exclusive disjunctions, they correspond to different concepts. Formally, the two examples (1) and (2) have similar logico-semantic ingredients—a singular term, two first-order predicates, and either conjunction or disjunction—and the same structure. It makes no difference in extensional contexts whether we build the sentence after the period by putting together two complete sentences and then elide one of the repeated subjects, or by building first a conjunctive or disjunctive predicable and then filling its only argument-place with a singular term. It is easier to visualise the idea using a formal language. Let ‘P(…)’ and ‘Q(…)’ be monadic predicables, and ‘a’ an individual constant. The two biographies of example (1) would be (1.1) and (1. 2): (1.1) P (…); Q (…); a; Pa; Qa; Pa & Qa; (1.2) P (…), Q (…); P (…) & Q (…); P&Q (…); a; P&Q (a), The two biographies of example (2) would be (2.1) and (2.2): (2.1) P (…); Q (…); a; Pa; Qa; Pa v Qa; (2.2) P (…), Q (…); P (…) v Q (…); PvQ (…); a; P v Q (a). (1.1) and (2.1) are cases in which the arguments of conjunction and disjunction are complete sentences. In (1.2) and (2.2), by contrast, conjunction and disjunction are

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used to build complex predicables out of simple ones,1 and only the complex predicables are ‘predicated’ of the single argument, the individual constant. The two itineraries are truth-conditionally equivalent. Negation is also sometimes used as a former of negative predicables. Quine, for instance, includes, among the logical particles, three negative words, ‘no’, ‘un-’, and ‘not’ (Quine, 1951, p. 23). In English, the content of the three sentences ‘John is unhappy’, ‘John is not happy’ and ‘It is not the case that John is happy’ is one and the same and can be reached by three alternative itineraries. The first is by attaching a monadic complex predicate, ‘… is unhappy’, built from the simple predicable, ‘happy’, and the negative prefix, ‘un-’, to a proper name. In the second, it is the result of attaching to a ‘positive’ sentence, ‘John is happy’, a negative particle, to obtain ‘John is not happy’. The third is the result of prefixing the ‘positive’ sentence, ‘John is happy’, with a negative operator, ‘it is not the case that’, which yields ‘It is not the case that John is happy’. Carston, from a different perspective, discusses the pragmatics of negation and distinguishes two significant aspects that have to be considered in explaining how negative sentences work. The two aspects are scope and what she calls the ‘representation distinction’ (Carston, 2002, p. 266), which she considers a version of the use-mention dichotomy. Here I borrow her examples: (3) (4) (5) (6)

We didn’t see hippopotamuses. But we did see the rhinoceroses. We didn’t see the hippopotamuses. We saw the hippopotami. She is not pleased with the outcome. She’s angry that it didn’t go her way. She’s not pleased with the outcome. She’s thrilled to bits. (Carston op. cit., p. 269)

In examples (4) and (6), negation does not negate contents, but ways of expressing them. Negation corrects (and thus rejects) either grammatical aspects or the accuracy of the rendering of some thoughts. It can also be used to correct pronunciation and virtually any aspect of linguistic communication. When negation, conjunction, and disjunction are used to construct complex predicables, or to reject some features of how a thought is expressed, as in Carston’s negation cases, the terms concerned do not work as inference markers. If they only performed these ‘linguistic’ jobs, then they shouldn’t be counted as logical constants. But at least negation and disjunction have instances in which, besides being functions of complete propositions, they possess the required inferential import. Quine’s reflexivity operator, ‘REF’ (Quine, 1960a), and Williams’s higher-level identity operator, ‘≡’ (Williams, 1989, pp. 52ff.), are—like negation, conjunction, and disjunction in some of their uses—predicable-building devices that fall short of being logical constants.

 The steps from ‘P(…) & Q(…)’ and ‘P(…) v Q(…)’ to ‘P&Q(…)’ and ‘PvQ(…)’, respectively, are far from trivial, they require the addition of the higher-level identity operator that converts n-adic predicables into (n-1)-adic operators (see C.J. F. Williams, 1989, ch. 3, for an explanation). 1

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Conjunction deserves some separate attention. Even if it is often used as a paradigm case of logical constanthood because of its simplicity, it is not a logical constant according to (IMV). This result should not be surprising. The job of conjunction is mostly syntactic, combinatorial, as has been discussed in Chap. 4. It serves to unify in a package information that was previously scattered. Paradigmatically, it is used to combine in a single unit the complex predicables that are sometimes arguments of the existential quantifier. If we sometimes wonder why in first-order languages existence is systematically combined with conjunction and generality with the conditional, here there is an explanation. The existential quantifier, understood as instantiation, is monadic. Its argument can be either a simple predicable such as ‘being a number’, as in ‘numbers exist’, or a complex predicable such as ‘being a tame tiger’, as in ‘tame tigers exist’. In this latter case, in order to comply with the syntactic requirements, the two single predicables ‘being a tiger’ and ‘being tame’, have to be converted into a single one in order to be the argument of monadic existence. Generality, by contrast, has a natural reading as concept subordination, which is a binary relation. Conjunction also serves to combine different sentences into a unique conjunctive one that can fill the argument slots in conditionals. That a certain proposition 𝛾 follows from a group of certain others—𝛤 or {1, 𝛾2, 𝛾3…, 𝛾n}—is represented in the metatheory of first-order logic either as (7) or as (8): (7) 𝛤 ⊢𝛾. (8) { 𝛾1, 𝛾2, 𝛾3…, 𝛾n} ⊢ 𝛾n+1. But its rendering into natural languages usually uses conditionals. Only by bonding the information in 𝛤 or {1, 𝛾2, 𝛾3…, 𝛾n} into a single sentence of the form represented in the antecedent of (9), (9) 𝛾1& 𝛾2& 𝛾3…& 𝛾n → 𝛾n+1, will we have a well-formed sentence. The indispensable syntactic role of conjunction contrasts with its semantic and pragmatic dispensability (see Carston, 2002, chapter three). There is no information, either semantic or pragmatic, that the conjunction of two sentences, p & q, conveys that the simple juxtaposition of p and q does not. Nor even its temporal and causal implicatures. Consider the following examples: (10) Victoria fell down the stairs. She got hurt. (11) The customers protested. She was fired. (12) The Sun rose over the horizon. The air warmed up.

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(10) suggests a causal link, (11) succession of events, and (12) simultaneity. The insertion of conjunction between the two juxtaposed sentences maintains unaltered their pragmatic effects. Thus, conjunction is a purely syntactic device that, even if with expressive meaning, as its semantic irrelevance shows, does not possess any specific inferential import.

6.2.2 Monadic and Binary Operators The second relevant distinction I mentioned above holds between monadic and binary operators. Negation aside, no monadic operator has inferential significance as an inference marker. Thus, no monadic concept other than negation is a logical constant according to (IMV). There is a reason for removing monadic concepts from the list, although at this moment I can only offer a metaphor. Inferential transitions are movements represented by principles and rules of inference that permit the passage from a certain piece of information to another. Thus, logical constants have to possess some kind of dynamic meaning. Transitions, movements and permissions all require a starting point and a point of arrival; and these points are somehow represented by the two arguments of binary operators. The movement allowed by logical notions also has a dual counterpart, i.e. the right to veto. Together with inferential permissions, natural languages need some ways of expressing refusals to accept propositions and inferential transitions, i.e. some means of vetoing propositions, rejecting them as premises of possible inferences, and also of blocking or suspending inferential tickets. Negation plays this role, which is its inferential task. I will come back to negation in the next section. Quantifiers are a further kind of function that systematically appear in standard lists of logical terms. Modal and epistemic operators are also included in the lists of extended logics of the appropriate kinds. None of these is a logical constant, according to (IMV). That some quantifier-like expressions do not qualify as logical constants, even in the most classical approaches to the topic, is no news. Different versions of invariantism refuse to apply this label to specific types of quantifiers. Non-standard quantifiers are not invariant according to Tarski’s original plan, and in the version defended by Casanovas (2007), the first-order universal quantifier is not a logical constant (Gómez-Torrente, 2007, pp.  190ff.). Besides, as mentioned in Chap. 4, even Frege left the existential quantifier out of the group of basic logical notions (Macbeth, 2005, position 102). The question that comes up at this point is whether standard quantifiers are monadic or binary devices, and the answer is not straightforward. Frege accepts both options in relation to the existential and the universal quantifier. The universal quantifier in the Begriffsschrift is compatible with both monadic and binary interpretations. A formula such as a

 (a)

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expresses that the result of substituting the German letter, the variable, in the formula on the right-hand side of the content stroke is always a fact (Frege, 1879, §11). The judgement stroke that precedes the formula indicates that the content is asserted and that the speaker puts it forward as true. Then, according to the Fregean explanation, the assertion of a quantified proposition authorises the speaker to the assertion of any of its instances as a fact. Asserting a quantified proposition is a way of asserting all of its instances since quantifiers are devices for carrying a possibly infinite amount of information, which is partially downloaded every time the quantifier is cashed out. Understanding universally quantified propositions—variable hypotheticals—as packages of information displays an aspect of the inferential import of quantifiers: their ‘vertical’ inferential import. The meaning of the universal quantifier is captured when one understands the inferential relation between the quantified proposition and its non-quantified instances. Frege supports this interpretation, as we saw in Chap. 4, when in 1923 explains that to understand generality is to understand how to draw an inference from the general to the particular (Frege, 1923, p. 258). In The Foundations of Arithmetic, the existential quantifier is interpreted as monadic—the negation of the number nought—and the universal quantifier as binary—the subordination of one concept to another. Quantifiers are properties of concepts, not characteristics of objects (Frege, 1884, §53). For this reason, quantifiers express something—sometimes positive, sometimes negative—about the extensions of concepts. The negative information that existence conveys is that the extension of its argument is not empty. Even in cases in which an existentially quantified sentence includes two predicables—as in ‘There are decent rich people’—the correct way of understanding the sentence is as involving a conjunctive predicable as the only argument of the quantifier. The existential sentence then ‘says’ that the complex predicable is instantiated or, in other words, that the intersection of the extensions of the two concepts involved includes at least one element. By contrast, the universal quantifier is interpreted in the Grundlagen and the Grundgesetze as a binary device that represents a relation between two concepts, or between their extensions. Being binary, it is a better candidate for logical constanthood than its existential counterpart. Nevertheless, conceptual subordination does not seem to be a further logical constant. I will come back to this issue in Sect. 6.4 below. One might think that the existential quantifier also admits binary uses. As explained in Chap. 4, Frege seems to accept this view in relation to the interpretation of ‘some’ (Frege, 1892a, p. 187), and he counts as an advantage of his conceptual writing over Boole’s calculus its unified treatment of existential and particular judgements (Frege, 1880–1881, p. 14). Examples (13) and (14) will serve to elaborate this point: (13) There are extremist Europeans, (14) Some Europeans are extremist.

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The information conveyed by (13) is that something falls under the conjunctive predicable. (14), by contrast, might be interpreted along inferential lines, as giving information about transitions between the two concepts involved. Of them, it would say that not all applications of ‘being European’ license the application of ‘being moderate’ (or ‘not being extremist’). In this explanation, the inferential import of (14) would stem from an implicit negation that vetoes some transitions. And then, the implicit negation would be the notion that carries the inferential weight. According to (PII), (13) and (14) express the same thought. Nevertheless, in its chosen representation, (13) identifies a monadic function that indicates instantiation, and (14) a binary relation that indicates partial overlap. Neither of these concepts are actually there in the thought. The thought is rather an unstructured item whose explicit import can be rendered by different functions with different statuses. Modal, epistemic and temporal operators, in turn, are higher-level monadic functions whose arguments are propositions,2 introduced in each case by a that-clause, and don’t possess representational meaning. Their lack of representational meaning derives from their status as circumstance-shifting operators. The notion of a circumstance-­shifting operator is a well-established one in the contemporary philosophy of language. It stems from (Kaplan, 1977) and (Lewis, 1980) and is extensively used by Recanati (Recanati, 2007, chapter one; Recanati, 2010, chapter six). Its definition requires distinguishing between the context in which a sentence is used and the circumstance in which its content is evaluated. The context of use of an utterance determines what is said by it. If I utter the words (15). (15) My daughter is in Spain now, the proposition thereby expressed is that a specific individual is in a specific place at a specific time. The circumstance of evaluation, on the other hand, is the situation from which you assess the proposition for truth and falsehood. ‘Now’ is an indexical, but it can also be seen as a circumstance-shifting operator, albeit one that indicates that the contexts of utterance and of evaluation coincide. But this is not always so, and there are some expressions in language that, unlike ‘now’, serve to inform us that the proposition expressed in a given context has to be evaluated in a different one. ‘Fifteen years ago’, ‘next week’, ‘Joan believes that’, and ‘it is possible that’ are examples of circumstance-shifting operators of this kind. Examples (16)–(19):

 The received view makes epistemic operators, knowledge and belief, standard first-level relations between an agent and an object (Moore, 1953). Nevertheless, the many different semantic and epistemic difficulties caused by the relational interpretation made authors such as Wittgenstein (1922), §5.5421, Prior et al. (1971), p. 16ff., Quine (1960b), p. 216, and Recanati (2000), 19ff., among many others, reject the view and embrace something similar to the view I am defending here. In Villanueva (2006), there is a comprehensive and detailed defence of epistemic operators as higher-level, and we have also argued for the view in Frápolli and Villanueva (2012). Recently, Trueman (2018) has taken over the view as the ‘prenective view of propositional content’. 2

6.3 Inferential Significance

(16) (17) (18) (19)

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Fifteen years ago, I was in Helsinki. Next week my daughter will come to visit. Joan believes that Santa lives in the North Pole. It is possible that he moves out.

all express propositions that have to be evaluated in relation to circumstances other than the ones in which they were uttered. In which circumstances is something that the higher-level operator in them shows. What is said in (16) is true if and only if the content of ‘I am in Helsinki’, uttered now, was true fifteen years ago, and similarly for the rest. What is relevant here is that the operator itself does not belong to the proposition expressed. Its meaning is expressive and not representational, for it indicates a mental or imaginary move to be made by the hearer in order for her to evaluate the proposition expressed. Nevertheless, although they are evaluation aids, these operators are not inference markers, and as a consequence, they should not be counted as logical constants. The non-representational meaning of modal, epistemic and temporal operators also follows from their alternative interpretation as quantifier-like devices (Kratzer, 1981, 1991; Kroeger, 2020).

6.3 Inferential Significance Inferential significance, or inferential relevance, is a substantial part of my explanation of logical constants. It is time to clarify it. According to inferential semantics, every concept has inferential significance in one way or another, since the inferential connections between the propositions of which the concept is part determine the concept’s content. Thus, the mere mention of inferential relevance does not set logical notions apart from the rest (see Chap. 5). What is characteristic of logical words is that their inferential significance makes them inference markers. Logical words are a subclass of the general class of discourse markers, whose specific role is to put on display inferential links between propositions. They are linguistic devices for presenting material inferences as explicit inferences. This is the role that Frege conferred to them in the Begriffsschrift and the core of Brandom’s logical expressivism. Logical expressivism is the ‘view that the expressive role that distinguishes logical vocabulary is to make explicit the inferential relations that articulate the semantic contents of the concepts expressed by the use of ordinary, nonlogical vocabulary’ (Brandom, 2018, p. 119). This Brandomian kind of expressivism applies to normative expressions in general. Epistemic, modal and logical notions are linguistic devices that say what is done by the use of some terms. Epistemic and deontic terms reveal commitments and entitlements on the part of speakers. The same goes for logical terms, except their specific role is to bring into the open inferential connections between propositions, which also display (derivative) inferential connections between concepts.

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The Principle of Propositional Priority and inferential semantics place assertion at the centre of our discursive lives. Nothing short of assertion bears pragmatic force and hence can count as a move in a linguistic game. Brandom takes from Frege the inferential individuation of propositions and from Sellars the special kind of rationality that places assertion at its centre. The kind of linguistic game of which assertion is the essential move is the game of ‘giving and asking for reasons’ (Brandom, 1994, p. 189). Besides, the inferential approach to propositions that Frege assumes in the Begriffsschrift takes the content of an act of assertion—i.e. what an agent says by it, that to which she is entitled and becomes committed to—to be an inferential node. This node is defined by those material inferences endorsed by the speaker in which the content occurs either as one of the premises or as the conclusion. Thus, assertions are complex packages of information inferentially organised, of which only a small part is explicitly displayed. Their complexity can nevertheless be made explicit by presenting them as parts of propositional networks whose nodes are signalled by logical notions: basically, the conditional and negation. Making this complexity explicit means disclosing the agent’s reasons to assert particular contents, and also the contents to which the agent becomes bound by her assertion. This is precisely the task for whose performance agents use the conditional and negation. In the case of the conditional, positive transitions are displayed by converting those materially correct inferences that support the assertion into logically valid inferences that instantiate modus ponens. In the case of negation, the situation is the exact complementary. Commitment to particular contents is the rejection of others (see Chap. 4). The normative meaning of logical constants can later be represented, if we wish, by a set of rules, as in proof-theoretical semantics. What is important to keep in mind is that the order of the explanation is not that logical constants are those terms whose meaning can be displayed as a set of rules, but rather the reverse. Sets of rules are in this case informative because the meaning of the terms involved is inferential. But we know that this is not enough. The aspect of their meaning that defines logical terms consists in their role as devices for explicitly representing inferential transitions. Being higher-level expressive devices is necessary, but only the addition of their pragmatic role makes the list sufficient as well. (IMV) is an elaboration of Frege’s view in Begriffsschrift, in which the different aspects involved in a correct characterisation of logical constants have been disclosed, the form that these aspects take specified, and the complexity of the whole enterprise made explicit. The label I use, the ‘inference-marker view’, reveals the intra-story of the pragmatist approach to logical words and pays tribute to the work of linguists. At this point, not only Frege must be mentioned. Pragmatic approaches, in general, have found appealing the idea of understanding logical notions more as rules than as substantive chunks of information. Ramsey explained the role of universally quantified propositions, ‘variable hypotheticals’ as they were then called, not as ‘judgments but rules for judging’ (1929, p. 429). Nomological generalisations, Ramsey’s variable hypotheticals, were characterised by Ryle as ‘an inferenceticket (a seasonal ticket) that enables its holder […] to pass from one statement to another, to offer explanations of given facts and to make states of things through the manipulation of what is found as existing or occurring’ (Ryle, 1949/2009, p. 105).

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And classical pragmatism, from Peirce to contemporary proof-­theoretical semantics, has acknowledged a contrast between premises that provide substantive information and principles of inference that allow the transition from premises to conclusions (see, for instance, Prior, 1976, p. 125). This is also one of the lessons of Carroll’s fable (Carroll, 1895), which Brandom uses to illustrate his expressivism (see, for instance, Brandom, 1994, p. 100; Brandom, 2018, p.  121). Where Brandom’s expressivism speaks of commitments and entitlements, the inference-­ marker view focuses on permissions and vetoes, insisting not only on the normative aspects of the meanings of logical terms but also on their dynamic type of meaning. Now, let us take a closer look at the specific notions that, according to (IMV), qualify as logical.

6.4 Genuine Logical Notions Truth-preserving transitions from propositions to propositions are paradigmatically presented as inferences by the use of conditionals and some other discourse markers such as ‘therefore’. Peirce, the most prominent of American pragmatists, considered the relation represented by the particle ‘ergo’ to be the primary logical relation (Peirce, 1932, 3.440), and he was right. But ‘ergo’ has to be complemented by a device for representing vetoes or blockages of inference,3 to complete the basic inferential apparatus of a language. In other words, the conditional and negation represent the core of logic. It is no wonder, then, that the basic inference rules that all calculi include, and that axiomatic presentations include almost exclusively, are the modus ponens and its complement, the modus tollens. Implication, the relation expressed by ‘therefore’ and ‘ergo’, is arguably stronger than the material conditional. It is the relation that holds between the premises and the conclusion of a valid argument and involves the modal feature of necessity. This is the standard story. Nevertheless, going beyond logical theory and back to inferential practices, conditional and implication are the same kind of relation. Explained in terms of commitments, a speaker becomes committed to the conclusion of an argument whose premises she accepts, and to the consequent of a conditional, if the antecedent has already been endorsed. At a syntactic level, implication represents deducibility, which is a metalinguistic relation that holds between sentences or formulae in a calculus. Thus, my discussion here could also be accused of being ‘conceived in sin, the sin of confusing use with mention’, as Quine said of modal logic (Quine, 1962, p. 177). Nevertheless, the use-mention distinction is of very little interest in real life, where we hardly ever deal with isolated syntactic items. Use and mention, object language and metalanguage, type hierarchies and so on, are ad hoc devices introduced in logical theory and semantics to overcome the paradoxes and complications

 Kürbis is completely right to require some negative factor as a semantic primitive (Kürbis, 2019, p.122). See the discussion on negation, falsehood, and incompatibility in Chap. 4, §3. 3

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produced by the previous estrangement of signs from their meanings and the real practice of rational agents. In actual linguistic exchanges, misunderstandings occur as well as misuses and failures, but there are no paradoxes in the philosophers’ sense (see Camós, 2011). Nobody bothers about the Liar paradox in real life exchanges, not even St. Paul in his epistle to Titus, since all of us use the truth apparatus of our languages extensively and proficiently. The set of all sets that do not belong to themselves should not have stopped any sensible discussion in set theory, and the least number not nameable in less than nineteen syllables does not play any role in number theory. I will discuss this issue in Chap. 7. Introducing distinctions, such as the Vicious Principle (Russell, 1908. p.  237) or language hierarchies like the one defined in Russell’s theory of types (op. cit., pp.  236ff.) is not the pragmatist’s way out. Pragmatists take care of difficulties by rethinking the background, and the job of paradoxes is precisely to alert us to the presence of faulty assumptions. This being said, the distinctions and principles that have been mentioned have surely helped to clarify some details, discussions and mistakes. But they do not deserve the central, usually paralysing, role credited to them by logic and semantics in the past century. The first genuine logical notion that I will mention is negation. I adopt the Fregean view of negation as the visible side of incompatibility. As has been explained in Chap. 4, negation is monadic from a syntactic viewpoint, although it is semantically binary. Propositions or thoughts, as dual entities, are individuated by the information that follows from them and by the information they exclude (Frege 1918–1919b; Tractatus 5.5151). In themselves, they are neither negative nor positive, but they can be linguistically represented in positive or negative sentences, i.e. sentences with negative words, depending on certain contextual interests. The assertion of a negative sentence implies the rejection of the content of its non-­negative part. Negative words in natural languages perform different roles. Incompatibility is one of them, but there are others. Because (IMV) focuses on the concepts expressed, and not on the terms used to express them, the status of the logical constant can be attributed to specific uses of negative terms and not to the complete class of negative words. What is relevant to the logical nature of negation is the semantic status of incompatibility that it sometimes codifies, and not the syntactic features of negation as a term. Negation is a logical constant when it is used to reject a complete proposition or the attribution of a property to an object. When it is used as a predicable-former, it does not have any specific logical role. A negative trait is implicit in the meaning of other notions whose logical status has also deserved attention. Disjunction, specifically in its exclusive version, is one of them. The class of adversative conjunctions, of which ‘but’ is a conspicuous example, is another. The status of disjunction as a logical constant is reflected in the disjunctive syllogism. Examples of disjunctive sentences are (20) and (21), (20) The candidates have to be proficient in either Spanish or English. (21) The President will be either a Republican or a Democrat.

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In both cases, the negation of one disjunct implies the alternative one. This might be enough to account for its inferential meaning as an inference marker. Nevertheless, in (20) disjunction is explicable without artifice as a predicate-builder. This explanation does not seem to be appropriate for (21), where disjunction actually marks alternative inferential paths. For these reasons, disjunction in (21) is a stronger candidate for logical constanthood than disjunction in (20). When disjunction is understood as inclusive, its non-logical interpretation as a predicable-builder is more natural. Exclusive disjunction, by contrast, when it codifies a dilemma, could be counted among the logical constants. That the default interpretation in classical logic is inclusive is a further sign that something does not work as it should. In any case, the potential status of disjunction as a logical constant derives from its negative import, as made explicit by the disjunctive syllogism. Something similar happens with ‘but’ and the other adversative conjunctions. Conjunction is not a logical constant—this is clear already—but adversative conjunctions, ‘but’ among them, could be. They indicate a contrast between one conjunct and something that might seem to follow from it (Blakemore, 1989; Frápolli & Villanueva, 2007). There are reasons, then, to assume that adversative conjunctions convey some information as inferential markers that standard conjunction does not. At this point, it makes no difference whether only negation is deemed a genuine logical constant, and disjunction and adversative conjunctions are merely viewed as expressions that involve a negative trait that is responsible for their inferential import, or whether all of them are taken on board as logical constants in their own right. The second indisputable logical constant, according to (IMV), is the conditional. Conditionals are logical terms par excellence. Even so, as it happens with negation, not all occurrences of ‘if…then’ in natural languages express inferential transitions. Given their more natural interpretations, the following occurrences of conditional words in examples (22)–(24) do not qualify as logical constants, (22) If it is raining, it’s raining heavily. (23) If the President is saying the truth, I am the Pope of Rome. (24) If he hadn’t killed the President, somebody else would have. By contrast, basic inferential uses of the conditional are represented by the following examples, (25) and (26), (25) If Joan is in the classroom, the lecture has not finished yet. (26) If the car is red all over, it is not green. Note that the consequent clauses in (25) and (26) do not express complete propositions, since part of their content refers back to some piece of information in the antecedent. This is by no means exceptional. Conditional sentences often include pronouns in anaphoric uses, to which quantifiers binding them are frequently attached. (26) would be a hypothetical thought in Frege’s sense, and the occurrence of ‘if…then’ would be justified because it can be generalised.

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These uses are arguments for adopting (IMV)weak rather than (IMV) as our preferred characterisation. Binary quantifiers provide a further reason in favour of (IMV)weak. In formulae with binary quantifiers, the logical role of the conditional reduces to a device for distinguishing the two arguments involved. Consider example (27), its standard first-order translation (28), and the latter’s natural language reading of (28), (29), (27) All human beings are mortal. (28) ∀x (Hx → Mx). (29) If something is a human being, then it is mortal. Notice that no explicit conditional occurs in (27). The conditional ‘emerges’ in its standard translations to a first-order languages and its natural language readings, and thus a question naturally arises: is the conditional actually at work at a deeper level in (27), in a way that its translations uncover? For a pragmatist, this is a strange question. What could ‘actually’ possibly mean here? If we stick to the Fregean interpretation of thoughts as non-structured, then the answer should be that thoughts do not ‘actually’ include any specific concepts, even though their inferential potential can be made explicit by different equivalent representations. As I explained in Chap. 4, I endorse Frege’s interpretation of universal binary quantifiers as indicating concept subordination. The two concepts human being and mortal belong to the same category, although the first one is subordinated to the second. In normative terms, what this means is that those circumstances that allow the application of the first concept to some item likewise allow the application of the second concept to the same item. The negative complementary also holds, i.e. applying the first concept is incompatible with withholding the second concept from the same item. In terms of the extension of concepts, i.e. of sets and classes, a variable hypothetical indicates that the extension of the first concept is a subset of the extension of the second one. Even charitably accepting the metaphor that the true structure of discourse emerges in its translation into logically correct languages, my answer to the aforementioned question would be negative. The two concepts involved in variable hypotheticals and their extensions maintain a principled inferential connection that the conditional helps to reveal, and whose foundation could be semantic, physical, or metaphysical. But there is no conditional at work on which the meaning of the binary quantifier rests. The situation instead seems to be the opposite; subordination has certain inferential effects, among them that there is a sequence of implications from the application of a concept to an object to the application of all concepts to which the first one is subordinated. These implications can be made explicit by means of conditionals that, at this point, work as logical constants, making explicit certain inferential features of subordination that were previously implicit. Thus, as Frege saw, subordination is more basic than implication. Conditionals do not explain the meaning of binary quantification; it is rather the subordination between concepts expressed by binary quantifiers that gives the meaning of quantified conditional

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sentences. Frege uses the conditional stroke to represent both: conditionals (a relation between two complete judgeable contents) and subordination (a relation between two concepts). Macbeth argues that only when the clauses in the antecedent and consequent have an indefinite sign, i.e. a variable, can Frege’s conditional stroke be adequately translated by ‘if…then’. Translating as a conditional the combination of two complete thoughts sounds odd, except in one case: when we do not know whether the thoughts in the antecedent and the consequent should be affirmed or denied, as Frege says in the Begriffsschrift §5 (Macbeth op. cit., position 357). Thus, Frege thinks, the import of the ‘connective’ ‘if…then’ is not truth-conditional but inferential. This point is relevant to seeing how misleading is the understanding of Frege’s logical notions along the lines of the traditional truth-tables account, which even van Heijenoort highlights in his preface to the Begriffsschrift. The versatility of the conditional stroke is a reason for Frege’s bidimensional notation since Frege’s notation allows the codification of relations between concepts and judgeable contents that can be differently rendered in the standard linear notation of quantification theory. But the point is that only those uses of the conditional stroke that are general, i.e. that express concept subordination, have genuine ‘logical’ significance. As Macbeth points out, Frege’s conditional stroke, although in itself somewhat unintuitive as a sign for the primitive logical relation, is justified, according to Frege, by its role in the foundation of genuine (that is, generalized) hypotheticals, which express the subordination of one concept to another. (Macbeth, 2005, position 351)

If we now recall that Fregean logical relations are our semantic relations, as Coffa says (Coffa, 1991, p. 64), Frege’s intuition at this point becomes clear. Subordination is a semantic primitive, and conditionals represent transitions between complete judgeable contents. This explains why, according to (IMV), subordination is not a logical relation in the sense in which I am using ‘logic’ in this book. Also note that his treatment of subordination as a semantic primitive suggests that Frege’s semantics, far from being primary truth-conditional, shares intuitions with semantic inferentialism. There is nothing wrong with translating (27) as (28) and (29). Conditionals display some features of binary quantifiers and make explicit the different commitments undertaken by the assertion of general contents. What would be wrong is to assume that conditionals are semantically primitive. They are not. As expressive devices, logical constants cannot be primitive. What is primitive, in a specific sense, are the two basic logical relations between propositions and concepts: consequence and incompatibility. And these also derive from the correct material inferences that we endorse, founded in turn in our forms of life. Conditionals and negation, as logical constants, represent these inferences as grounded in the relations of consequence and incompatibility as inferences. (28) is the standard translation of (27) to a first-order language. Other formal systems would yield different results. At this point, the algebraic and set-theoretical translations of universally quantified sentences are more perspicuous than the first-­ order languages of classical logic. The translation of (27) as (30),

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(30) H ⊆ M, where ‘H’ is the extension of the concept human being and ‘M’ is the extension of the concept mortal, displays the job of the universal quantifier in a less misleading way. Should we then include ‘subordination’ or ‘subset’ as a further logical constant? Again, the answer is negative. Neither subordination nor subset complies with the requirements for qualifying as logical terms, according to (IMV), because neither of them represents a relation between propositions. Subordination, as a relation between n-adic (n > 0) concepts, might count as a logical constant according to (IMV)weak if its use conferred upon it some dynamic meaning. The subset relation would not qualify even in the weaker version. As such, subordination and subset do not represent transitions but rather structures. These structures are crucial to the inferential meaning of the concepts involved; all concepts are related to other concepts by relations of subordination and incompatibility, which are the relations on which logical transitions rest. Nevertheless, logical terms belong to the linguistic apparatus that explicitly represents those relations, since their function is not to show subordination and incompatibility, but rather to say that these relations obtain. Because such-and-such concepts are subordinated one to the other, conditionals can be used to display the inferential connection between the propositional contents that involved them. Because the application of such-and-­ such concepts is incompatible with the application of certain others, negation can be used to display this circumstance. In sum, (IMV) deems as logical constants some uses of negation, some uses of disjunction and adversative conjunctions, and some uses of conditionals. (IMV)weak would also take on board conditionals whose clauses are not complete propositions, even if they are represented by complete sentences, and subordination, whose arguments are concepts. Neither identity nor conjunction is a logical constant, and this negative result extends to monadic functions of propositions. Nevertheless, all these notions represent relations that support inferential transitions or, as it happens with identity, auxiliary rules with a definite role in inferences. (IMV) gives the result that one would expect. Its merit is that it provides reasons that explain why some notions, unlike some others, and some uses of particular notions, unlike some others, represent logical notions. As we have seen, the pragmatic level is crucial for discriminating among uses, whereas the syntactic and semantic levels offer the first provisional perimeter. None of the subtleties that we have discussed in this chapter could have been approached, had artificial languages been selected as our theoretical playground.

References Blakemore, D. (1989). Denial and contrast: A relevance theoretic analysis of ‘but’. Linguistic and Philosophy, 12(1), 15–37. Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press.

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Brandom, R. (2000). Articulating reasons. An introduction to Inferentialism. Harvard University Press. Brandom, R. (2008). Between saying and doing: Towards and analytic pragmatism. Oxford University Press. Brandom, R. (2018). Inferentialism, normative pragmatism, and metalinguistic Expressivism. 2018 John Dewey Lectures at the Dewey Center of Fudan University. Camós, F. (2011). Sinsentidos: Un análisis pragmático de los fracasos comunicativos. Editorial Académica Española. Carroll, L. (1895). What the tortoise said to Achilles. Mind, New Series 4(14), 278–280. Carston, R. (2002). Thoughts and utterances. The Pragmatics of Explicit Communication. Blackwell Publishing. Casanovas, E. (2007). Logical operations and invariance. Journal of Philosophical Logic, 36, 33–60. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. Frápolli, M.  J. (2012). ¿Qué son las constantes lógicas? Crítica. Revista Hispanoamericana de Filosofía, 44(132), 65–99. Frápolli, M. J., & Assimakopoulos, S. (2012). Redefining logical constants as inferential markers. The Linguistic Review., 29(4), 625–641. Frápolli, M.  J., & Villanueva, N. (2007). Inference markers and conventional Implicatures. Teorema, 26, 125–140. Frápolli, M. J., & Villanueva, N. (2012). Minimal expressivism. Dialectica, 66(4), 471–487. Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. In J. van Heijenoort (Ed.), (pp. 1–82). Frege, G. (1880/1881). Boole’s logical Calculus and the Concept-script. In G. Frege (1979). In G. Frege (1979). Posthumous Writings. Edited by Hans Hermes, Friedrich Kambarte, Friedrich Kaulbach (pp. 9–46). Basil Blackwell. Frege, G. (1884/1953). The foundations of arithmetic. A logic-mathematical enquiry into the concept of number. (J.  L. Austin, Trans.). Second Revised Ed. Harper Torchbooks/The Science Library, Harper & Brothers. Frege, G. (1892a). Concept and object. In G.  Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 182–194). Basil Blackwell. Frege, G. (1892b). On sense and meaning. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 157–177). Basil Blackwell. Frege, G. (1918–19a). Thoughts. In Frege (1984), (pp. 351–372). Frege, G. (1918–19b). Negation. In Frege (1984), (pp. 373–389). Frege, G. (1923). Compound Thoughts. In Frege (1984), (pp. 390–406). Frege, G. (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness. Basil Blackwell. Geach, P. (1962). Reference and generality: An examination of some medieval and modern theories. Cornell University Press. Gómez-Torrente, M. (2002). The problem of logical constants. Bulletin of Symbolic Logic, 8, 1–37. Gómez-Torrente, M. (2007). Constantes Lógicas. In M. J. Frápolli (Ed.), Filosofía de la Lógica (pp. 179–205). Tecnos. Kaplan, D. (1977/1989). Demonstratives: An essay on the semantics, logic, metaphysics and epistemology of demonstratives and other Indexicals. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes from Kaplan. Oxford University Press. Kratzer, A. (1981). The notional category of modality. In H.  J. Eikmeyer & H.  Rieser (Eds.), Words, worlds, and contexts. New approaches in word semantics (pp. 38–74). de Gruyter. Kratzer, A. (1991). Modality. In A. von Stechow & D. Wunderlich (Eds.), Semantics: An international handbook of contemporary research (pp. 639–650). de Gruyter. Kroeger, P. (2020). Modality as quantification over possible worlds. (2020, August 12). Retrieved April 9, 2021, from https://human.libretexts.org/@go/page/65866 Kürbis, N. (2019). Proof and falsity: A logical Investigation. Cambridge University Press.

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Lewis, D. (1980). Index, context, and content. In S. Kanger & S. Öhman (Eds.), Philosophy and grammar (pp. 79–100). D. Reidel Publishing Company. Macbeth, D. (2005). Frege’s logic. Harvard University Press (kindle edition). Moore, G. E. (1953). Some Main problems of philosophy. Allen & Unwin. Peacocke, C. (1976). What is a logical constant? Journal of Philosophy, 73(9), 221–231. Peirce, C.  S. (1932). Collected papers of Charles Sanders Peirce. Edited by Hartshorne, C., & Weiss, P., Volume II. Elements of logic. Harvard University Press. Prior, A. (1976). What is logic? Papers in logic and ethics. University of Massachusetts Press. Prior, A. N., Geach, P. T., et al. (1971). Object of thought. Clarendon Press. Quine, W. V. O. (1951). Two dogmas of empiricism. The Philosophical Review, 60, 20–43. Quine, W. V. O. (1960a). Variables explained away. Proceedings of the American Philosophical Society, 104(3), 343–347. Quine, W. V. O. (1960b). Word and object. The MIT Press. Quine, W. V. (1962). Reply to professor Marcus. In Quine, 1966, 177–186. Ramsey, F. P. (1929/1991). The nature of truth. In N. Rescher & U. Majer (Eds.), On truth: Original manuscript materials (1927–1929) from the Ramsey collection at the University of Pittsburgh (pp. 6–24). Kluwer Academic Publishers. Recanati, F. (2000). Oratio obliqua, oratio recta. An essay on metarepresentation. The MIT Press. §§. Recanati, F. (2007). Perspectival thought: A plea for (Moderate) relativism. Clarendon Press. Recanati, F. (2010). Truth-conditional pragmatics. Oxford University Press. Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262. Ryle, G. (1949/2009). The concept of mind (60th Anniversary ed.). Taylor and Francis e-Library. Tarski, A. (1986). What are logical notions? History and Philosophy of Logic, 7, 143–154. Trueman, R. (2018). The Prenective view of propositional content. Synthese, 195, 1799–1825. Villanueva, N. (2006). Ascriptions with an attitude. A Study on Belief Reports. Dissertation. Editorial Universidad de Granada. Warmbrod, K. (1999). Logical constants. Mind, 108(431), 503–538. Williams, C. J. F. (1976). What is truth? Cambridge University Press. Williams, C. J. F. (1989). What is identity? Oxford University Press. Williams, C. J. F. (1992a). Being, identity and truth. Oxford University Press. Williams, C. J. F. (1992b). Towards a unified theory of higher-level predication. The Philosophical Quarterly, 42(169), 449–464. Wittgenstein, L. (1922). Tractatus logico-philosophicus. Routledge & Kegan Paul.

Part III

Further Applications of Propositional Priority

Chapter 7

Grue, Tonk, and Russell’s Paradox: What Follows from the Principle of Propositional Priority?

Abstract  In this chapter, the structural connections between three paradoxes— Goodman’s ‘grue’, Prior’s ‘tonk’, and Russell’s—are traced. It is argued that none of them arises in a context in which the Principle of Propositional Priority holds. To derive them, a strong hypothesis is needed: that all syntactically well-formed expressions express genuine concepts that are significant regardless of the tasks that speakers (or theorists) use them to perform. These three paradoxes have been chosen for their relevance to the topic of this book. The interest of Goodman’s ‘grue’ is the introduction of two new analytical tools, the notions of projectibility as applied to hypotheses, and the notion of entrenchment as applied to terms, which neatly pinpoint what happens with the artificial introduction of new terms. Prior’s ‘tonk’ is relevant to understanding what is at issue with logical constants and, finally, our discussion of Russell’s paradox is intended to offer a further argument for the validity of Frege’s unitary project from the Begriffsschrift to the Grundgesetze. It also shows the potential of (PA), (PPP) and (PGS) for the foundation of arithmetic. In the pragmatist perspective I am developing in this book, concepts have to be identified from propositions and propositions are identified as the (actual or virtual) results of successful discursive actions. Languages, either natural or artificial, only produce senseful concepts and contents against the background of discursive practices, be they ordinary or scientific. Goodman, Prior and Frege had the instruments to tackle these paradoxes and push their proposals further, in a better and more fruitful perspective on language and science. Keywords  Compositional [compositionality] · Context · Entrenchment [entrenched] · Goodman · Grue · Russell’s paradox · Prior · Projectibility [projectible] · Tonk · Wittgenstein

7.1 Paradoxes Paradoxes are a philosophical topic in themselves. For Russell they were logicians’ crucial experiments:

© Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_7

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A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science. (Russell, 1905, pp. 484–5)

Paradoxes confront philosophers with the limits of their proposals (Priest, 1995; Frápolli, 1996), although, unlike some crucial experiments in science, they do not open up a clear path to be taken. In mathematics and logic, paradoxes usually show that a particular term is not well defined or, in general, that it is defective. Russell (1908), while engaged in promoting mathematical logic, described some of the paradoxes that have been most discussed by logicians and philosophers. The Liar, the paradox that allegedly he found in Frege’s Grundgesetze (known as Russell’s paradox), and Burali-Forti’s paradox of the greatest ordinal were listed there, among others. Later on, Goodman introduced the ‘grue’ paradox in order to show the limits of induction (Goodman, 1955), and Prior used the flawed connective ‘tonk’ to expose the weaknesses of inferential explanations of logical constants (Prior, 1960). Set-theoretical and semantic paradoxes alike rest on some kind of circularity, and hence Russell’s vicious-circle principle (Russell op.  cit., p.  237). Nevertheless, circularity is not always ‘vicious’, as Gödel’s incompleteness theorem shows. Moreover, not all paradoxes are circular: neither ‘tonk’ nor ‘grue’ involves any kind of self-reference, and thus such relatively simple and radical solutions as the ones suggested by Russell do not apply. But if not circularity, then what is it that all paradoxes have in common, if anything? As a pragmatist, I pinpoint the source of the problem in the fabricated character that all of them present, and which should make philosophers and logicians consider the limits and scope of the procedure of mental experiments. Neither ‘the set of all sets that do not belong to themselves’ nor ‘tonk’ nor ‘grue’ represent a concept in the pragmatist sense of the term. For these terms are not used, as opposed to mentioned, in the making of successful assertions. There are different answers to the question of what concepts are. In a class-room context, it would be safe to respond that concepts are the meanings of predicates and tools for classifying the world.1 Even if incomplete, this answer is basically correct. But philosophers and logicians have acquired a sort of generalised disrespect for language, too often thinking of it as something that can be manipulated at will. Possibly, the abundance of artificial languages built up over centuries, with many varied purposes, is partly responsible for this unjustified attitude towards the linguistic/conceptual medium that we live in, and which makes us humans, ‘sapients’ in Brandom’s terms (Brandom, 1994, 2000). In exactly the same sense in which there is an essential distinction between ‘logic’, i.e. the inferential apparatus of our conceptual and linguistic systems, and ‘logics’, i.e. formal proposals for the inferential organisation of particular sets of terms and notions, there is an essential distinction between ‘language’ and ‘languages’. When philosophers of language  What concepts are has become a very technical philosophical issue. See, for instance, Peacocke (1992) and Prinz (2002) for two alternative proposals about this topic. For my purposes, nothing so elaborated is needed. 1

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use the former term, they do not refer to something over which we individually exert any kind of direct influence. The language that defines what we are is an evolutionary product that also harbours significant artificial extensions, as represented by the languages of particular sciences. For terms and concepts, old and new, to be significant, i.e. to belong to the system, they have to be supported by what Wittgenstein called a ‘form of life’ (Wittgenstein, 1953b, §§241–242). Words become meaningful when they are put to work by rational agents for specific purposes that are socially supported. Predicates, in the relevant sense in which I am using this term in the present context, are not simply unsaturated syntactic or formal items that, together with a singular term of any kind, form syntactically well-formed sentences or formulae. They rather must be expressions that occur in sentences that speakers use to say something. That concepts are identifiable ‘ingredients’ of propositions is an alternative reformulation of the same insight. Classic paradoxes such as those I have mentioned above violate the pragmatist principle of propositional priority, which also requires concepts to be inserted in complete propositions individuated by their inferential connections. The artificial forging of predicate-like terms is not enough to produce a concept, and this common practice that we see in Russell, but also in Prior, exposes the dramatic consequences of having dismantled the essential unity of the linguistic instrument and the rational life of agents. In this chapter, I discuss some well-known paradoxes in order to argue that, from a pragmatist viewpoint, they are (or should be) innocuous. Their usually paralysing effect produced by them on logic and philosophy is a consequence of some kind of ‘forgetting the origin’ derived from the formalist/linguistic approach to logic and semantics.

7.2 Goodman’s ‘Grue’ Nelson Goodman discussed what he called a ‘new riddle of induction’ (Goodman, 1955, pp. 59ff.) that, in its novel terms, consists in understanding (i) what counts as a confirming instance of (ii) which particular hypothesis. As the formulation just given makes explicit, the riddle has two distinguishable parts. One of them asks what kind of hypothesis has the appropriate status for being confirmable by new evidence. In other words, it focuses on the distinction between those universal sentences or formulae whose positive new instances add epistemic strength to them, and those that do not obtain further support by what seem to be positive cases. For instance, the fact that this piece of copper in front of me conducts electricity reinforces (1), (1) All metals conduct electricity. By contrast, my 21-year-old son entering a room in which some of his classmates are gathered would not reinforce the general truth of (2),

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(2) All the people in the room are younger than 50. (1) and (2) illustrate the time-worn distinction between nomological and accidental generalisations or, alternatively, the distinction between law-like hypotheses and hypotheses that do not involve any (scientific) laws. This distinction, as Goodman rightly acknowledges, cannot be given in syntactic terms (Goodman op.  cit., pp. 72–73). That the distinction involves reference to the use we make of the hypotheticals involved is no news. Ramsey, for instance, dealt with this topic in (Ramsey, 1928) and (Ramsey, 1929), offering different explanations each time, but stressing in both cases the pragmatic function of universals of law, as he called nomological generalisations. His most radical pragmatist interpretation led him to claim that nomological generalisations do not codify genuine propositions (Ramsey, 1929, p. 146). The second aspect of the new riddle of induction focuses on the opposite direction, the direction that goes from the evidence to the hypothesis and asks which one of the infinitely many hypotheses compatible with the evidence is actually confirmed by it. This issue has essential points in common with Wittgenstein’s rule-­ following paradox (see Wittgenstein, 1953b, §§185, 198, 201, 210), and leads us to confront semantic and pragmatic challenges like those we encounter in the practice of applying concepts. In this context, Goodman introduces the neologism ‘grue’, a term that applies to green objects before time t and to anything blue afterwards. Let us fix t at some particular instant in the near future. Consider now the problem that ‘grue’ raises concerning inductive generalisation. We know that all the emeralds that humans have dealt with so far are green. This being so, which one of the following two law-like generalisations, (3) and (4), are supported by the evidence that we currently have about emeralds? (3) All emeralds are green. (4) All emeralds are grue. From a formal point of view, (3) and (4) are equally supported, yet they predict incompatible possibilities. According to (3), we should expect all emeralds encountered after time t to be green, whereas (4) predicts them to be blue. These aspects of the new riddle show that syntactic and formal answers will not do. What is interesting to note is that, outside the philosophical discussion, we all feel that (3) is the reasonable hypothesis to be supported. The relation between evidence and inductive generalisation was one of the most pressing problems of the philosophy of science in the first half of the twentieth century. Goodman’s treatment of the new riddle takes an unfamiliar perspective on the old discussion and, to mark this, he introduces the new terms ‘projectibility’ and ‘entrenchment’. The first aspect of the new riddle mentioned above enquiries about which hypotheses are projectible. The analysis of the second aspect seeks to

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understand which terms are (better) entrenched. Projectibility and entrenchment, as Goodman understands them, are pragmatic notions that cannot be completely defined with any set of necessary and sufficient conditions. Goodman offers very little information to characterise projectability, and much less to explain entrenchment. Proper analyses of these notions would involve paying attention to past uses of terms without blocking the possibility of introducing new terms, which would suffocate scientific practice altogether (Goodman op. cit., p. 97). Entrenched terms usually occur in projectible hypotheses: this is the intuition. Goodman does not take sides on the relative dependence of the two notions—i.e. on whether projectible hypotheses make the terms in them entrenched, or whether it is rather entrenched terms that make the hypotheses in which they occur projectible—and acknowledges that this decision might vary when these notions are applied to different kinds of terms: To begin with, what I am primarily suggesting is that the superior entrenchment of the predicate projected is in these cases a sufficient even if not a necessary indication of projectibility; and I am not much concerned with whether the entrenchment or the projectibility come first. But even if the question is taken as a genetic one, the objection seems to me ill-founded. In the case of new predicates, indeed, the legitimacy of any projection has to be decided on the basis of their relationship to older predicates; and whether the new ones will come to be frequently projected depends upon such decisions. But in the case of our main stock of well-worn predicates, I submit that the judgment of projectibility has derived from the habitual projection, rather than the habitual projection from the judgment of projectibility. The reason why only the right predicates happen so luckily to have become well entrenched is just that the well entrenched predicates have thereby become the right ones. (Goodman op. cit., p. 98)

According to the Principle of Propositional Priority, projectibility is nevertheless prior to entrenchment. It is because we say what we say using such-and-such concepts that the concepts at issue become integrated into the system. Concepts cannot arise in isolation from propositions and assertive practices. Even novel concepts in scientific theories or social discourses—concepts such as dark matter or epistemic injustice—enter the system as means of saying something that was previously harder to say, if expressible at all, and of identifying realities that were undoubtedly harder to detect. Sayings precede concepts. It is true, though, that in real-life practices in which we benefit from a mature conceptual/linguistic system put to work by evolved rational creatures in their social medium, no harm is done by focusing on concepts in order to stress—as Goodman does in the text just quoted—their connection with ‘older predicates’. And I agree that the genetic question is somehow ill-­ formed. Concepts and propositions develop together: concepts occur in propositions and propositions involve concepts. There is no need to point out an original instant of genetic priority of one item over the other. Even if Goodman points in the right direction and puts into practice the right insights, some of his terminological options, mostly derived from the positivist paradigm that dominated a substantial part of the philosophy of science of the past century, are potentially damaging. In particular, Goodman’s concern with terms instead of concepts raises some artificial difficulties, which his pragmatist inclinations, fortunately, manage to neutralise. Beneath the standard positivist

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terminology, then, Goodman’s interest in transcending the level of the linguistic surface is easy to detect. In particular, even when he asks about the behaviour of terms that occur in scientific hypotheses, he takes the precaution of stressing that isolated terms are not the target of his analysis, but rather a term together with all other terms that are ‘coextensive with it’ (Goodman op.  cit., p.  95), which is an (extensional) way of saying that he is actually concerned with concepts and not with purely linguistic items. Entrenchment is not an extensional property. Well-entrenched terms are not merely familiar predicates (Goodman op. cit., p. 97). At this point too, the positivist terminology enforces difficulties that otherwise would not have arisen. Goodman has to insist that his intention cannot be ‘ruling unfamiliar predicates out of court’ and appeals to the fact that ‘[a]n entirely unfamiliar predicate may be very well entrenched (…) if predicates coextensive with it have often been projected’ (Goodman loc. cit.). Discussing concepts instead would have spared him the clarification at this point. If the Fregean principles of Grammar Superseding and Propositional Priority are taken seriously, then the relevant philosophical issue at stake here is not whether ‘grue’ is a well-entrenched predicate, but whether ‘grue’ stands for a genuine concept. And this, in turn, has to be pursued as an aspect of the more basic question of whether propositions involving concepts expressible by ‘grue’ have been or would be effectively put to work. Predicates as linguistic items, as opposed to concepts, can be artificially created. We only have to stipulate that ‘grue’ is a monadic predicate that applies to objects of such-and-such kind and the trick is done. An entirely different issue will be whether it represents a bona fide concept: something that meets the conditions for concepthood. These conditions are not arbitrary, and thus stipulation falls short of providing the basis for a new concept to emerge. This diagnosis applies to ordinary concepts, mostly products of the natural development of the evolutionary instrument that is language, as much as it does to technical notions that are artificially introduced into the system. In all cases, something is a concept if and only if it contributes to the inferential nodes that propositions consist in. In Goodman’s example, ‘green’ expresses a concept because it has been effectively used by a community of rational beings in discursive practices. By contrast, ‘grue’ does not have those credentials. It might acquire them, though. But this would require a community to adopt it as part of its communicative repertoire. It would require being supported by a ‘form of life’, as Wittgenstein suggested (Wittgenstein loc. cit.; Hunter, 1968; Hacker, 2015). Similarly, projectible hypotheses are those that have actually been used for a particular purpose. In the context of Goodman’s discussion, they should have been used for the purpose of making predictions. Goodman has the instruments to completely embark on the pragmatist paradigm that I am defending, and in the text just quoted he even claims that ‘the judgment of projectibility […] derive(s) from the habitual projection’ and not the other way around. Hypotheses that are actually projected, i.e. that are actually used to make predictions, become projectible and their terms become entrenched, and this is not something that can be either artificially arranged or explained in purely syntactic and semantic terms, disconnected

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from the real practices of real agents. Without actual use, signs and terms are lifeless instruments (see Wittgenstein, 1953a, p. 4). Whether luck is the cause of their actually entering the system is a debatable issue, which calls for elaboration of what is meant by ‘luck’ here. In any case, it is not a question of casual coincidence, but rather a result of past history that also involves, like any evolutionary process, a certain amount of arbitrariness. Only projectible hypotheses are nomological generalisations that are reinforced by new evidence, and only well-entrenched terms occur in projectible hypotheses. And this is so because projectible hypotheses support counterfactual conditionals (Goodman op. cit., pp. 121–2). This is a completely pragmatic reformulation of the classical issue that Goodman also addressed in pragmatic terms. The received view of language—a view in which syntax, semantics, and pragmatics are separate realms with a logical and chronological ordering among them— has to explain why ‘green’, as opposed to ‘grue’, expresses a genuine concept, even though we all see, and Goodman acknowledges, that it is (3) above and not (4), that is the hypothesis that is projectible. This challenge, which seems unsurmountable for the received view, is the price we pay for the positivist account of language that has dominated the past century, and which treats language and logic as resting on formal, soulless systems. Predicates such as ‘green’ and ‘blue’ are ‘well-behaved’, as Goodman acknowledges (Goodman op. loc., pp. 79ff.), whereas ‘grue’ is not. But ‘being well-behaved’ is not something that admits of a formal characterisation. He says: Of course, one may ask why we need worry about such unfamiliar predicates as ‘grue’ or about accidental hypotheses in general, since we are unlikely to use them in making predictions. If our definition works for such hypotheses as are normally employed, isn’t that all we need? In a sense, yes; but only in the sense that we need no definition, no theory of induction, and no philosophy of knowledge at all. We get along well enough without them in daily life and in scientific research. But if we seek a theory at all, we cannot excuse gross anomalies resulting from a proposed theory by pleading that we can avoid them in practice. (Goodman op. cit., p. 80)

Goodman is right, philosophers ask more from theories. We seek to understand and explain phenomena and not merely to be able to describe them. Nevertheless, practices are at the base of the phenomena to be explained, and if our best efforts do not succeed in explaining them, then this might be a reason to reconsider our background theoretical assumptions. One of these assumptions, which distorts the discussion of ‘grue’ and the other paradoxes addressed in this chapter, is that language can be freely modified and created, and that the artificial creation of syntactically irreproachable terms magically confers conceptual life on them. Discursive, conceptual practices confer life on terms, as Wittgenstein explained, practices that begin with complete linguistic actions. Predicates that have not been used in ‘habitual projections’ do not express real concepts, and this is independent of their flawless syntactic and semantic features. Goodman’s pragmatist commitment is evident in the following text: If I am at all correct, then, the roots of inductive validity are to be found in our use of language. A valid prediction is, admittedly, one that is in agreement with past regularities in

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what has been observed; but the difficulty has always been to say what constitutes such agreement. The suggestion I have been developing here is that such agreement with ­regularities in what has been observed is a function of our linguistic practices. (Goodman op. cit., p. 121)

The received view of language, which pushes him to deal with terms instead of concepts, and allows the practice of creating words anew, burdens his treatment of the new riddle and raises artificial problems that would not have emerged, had his pragmatism been more heartfelt.

7.3 Prior’s ‘Tonk’ To expose what he thought were the weaknesses of the inferential approach to logical constants, Prior introduced his much-discussed connective ‘tonk’ (Prior, 1960). I have already discussed the tonk-rules in Chap. 5, but repeat them here for the present argument’s sake: (Tonk-Intr) A ├ A tonk B (Tonk-Elim) A tonk B ├ B. In calculi with ‘tonk’, any pair of formulae would form a valid argument. The paper is an ironic piece of work, written in a humorous style, in which Prior builds up a case against the notion of analytically valid inference, as previously defended by Popper, Kneale, Strawson and Hare. The enormous influence that this paper has exerted in the debate on the meaning of logical constants contrasts with the weakness of the argument it offers. To say the least, Prior’s argument is insufficient to establish his point, and the recognition obtained by this logical joke is, I contend, utterly undeserved. On the one hand, the formulation of the philosophical target makes it difficult to understand the reasons that moved Prior to lead such a frontal opposition to it. On the other hand, the proposal of the connective ‘tonk’ misses the target entirely. I will comment on these two aspects in turn. Prior mounts his argument against the thesis that ‘there are inferences whose validity arises solely from the meanings of certain expressions occurring in them’ (Prior, op. cit., p. 38). Let us call this thesis ‘Inferential Validity’, (Inf. Val.), with the following two versions, one linguistic and the other conceptual, (Inf. Val.)ling The validity of inferences and the meaning of some distinguished terms occurring in their expression are essentially related.2

 I have deliberately used the non-committal term ‘related’ in order to allow different versions, some stronger than others. In particular, in order to make my point as general as possible, I want to leave open the two possibilities that I will mentioned later in the rest of this section: that validity depends on the meaning of terms, and that the meaning of terms depends on the inferences in which they occur. 2

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(Inf. Val.)con At least some of the concepts involved in valid inferences are essentially related to their validity. It is difficult to see how these trivial statements could be disputed. The validity of the inference that goes from (5) to (6): (5) Victoria is a woman. (6) Victoria is a human being. rests on the concepts woman and human being. All inferences that can be obtained by fixing these two concepts in the appropriate way would be valid as well. The validity of the inference that proceeds from (7) and (8) to (9), (7) If Joan is a musician, then he is a vegan. (8) Joan is a musician. (9) Joan is a vegan. rests on the meaning of the conditional. Holding the conditional fixed and substituting consistently and systematically the sentences in the argument will result in valid inferences. The inference from (5) to (6) is what we called in Chap. 5 an ‘analytically valid’ inference, while the inference from (7) and (8) to (9) is also logically valid, since its validity rests on the meaning of some privileged term: a logical constant. Prior’s formulation blurs the difference between these two kinds of inference, and thus obscures the role of logical constants, although it is clear from his argument that his targets are inferential proposals of logical constanthood. There are two alternative ways of explaining how validity and meaning are connected. The first assumes that the meaning of terms pre-exists their possibilities for their engaging in inferences; the second rests on the idea that the correctness of some inferences determines the content of the concepts involved. The first alternative is the standard view that contends that the inference from (7) and (8) to (9) is valid because ‘if’ means if. The second alternative amounts to saying that speakers use ‘if’ to show the validity of certain inferences. This latter option is Brandom’s expressive view of the conditional (see Brandom, 2000, pp. 60), and follows from (IMV), the view that was defended in Chap. 6. The expressive approach to logical constants finds its most comfortable accommodation in the realm of inferential semantics, although it is compatible with alternative proposals about the meaning of ordinary terms. In both ways of approaching meaning, meanings and inferences are inextricably connected. Thus, to understand Prior’s criticism accurately, some standard and non-­standard implicit assumptions have to be added, some of them so unusual that not even the most radical formalists have ever proposed them.

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The standard formalist view of logic centres the discussion of logical constants on what happens inside particular logical calculi, or in particular formal systems. The meanings of connectives are given in calculi either by the axioms or by the rules of inference that govern their use. Haack calls the debate about the instances responsible for the meaning of logical terms the thesis of ‘meaning-variance’ (Haack, 1978, p.  31). The two possibilities are, first, that connectives in calculi become completely characterised solely by explicit transformation rules internal to them and, second, that characterising them also requires appealing to properties and features to be determined outside the calculi. This discussion is rather artificial, though, since some connection with what rational agents ideally do when they are involved in inferential activities is required for a function to be a logical connective, and for a calculus to be a logical calculus. As Haack explains, there are 16 different bivalent truth-functions for any two arguments, and not all of them have ever been selected as logical connectives, even though they share all the classical logical connectives’ formal qualifications (Haack op.  cit., p.  28). The Sheffer stroke (Russell & Whitehead, 1927, p. xvi ff.) and the arrow (Haack, op. cit., p. 29) have been formally proposed as connectives on elegance grounds, both of them being functionally complete, i.e. able to represent any bivalent binary truth function. But irrespectively of their formal properties, they have not enjoyed the logicians’ favour and only remain as curiosities. The reason, which is clear, should make us think about the scope of the formalist approach: logicians have not adopted them because they do not have informal counterparts, either in science or in ordinary discourse. In more pragmatist terms, they have not been favoured as logical constants because speakers do not use them in their actual inferential practices. In the terms discussed in the previous section, they are not entrenched. And because they do not belong to the actual conceptual system that rational agents actually employ, they do not stand for genuine concepts. Syntactic systems undoubtedly include signs whose properties are completely specified by internal rules, but we would not call them ‘logical calculi’ merely on the basis of their intrinsic properties. In the demarcation debate, concerning which of the calculi that have ever been proposed as logics deserve to be considered logical calculi, Haack makes the very reasonable proposal that only formal systems some of whose interpretations seek to represent inferential relations between truth-­ bearers should be so classified: The claim of a formal system to be a logic depends, I think, upon its having an interpretation according to which it can be seen as aspiring to embody canons of valid arguments. (Haack, 1978, p. 3)

Haack’s judicious approach takes logical calculi to be interpreted systems and takes the connectives in them to stand for natural language expressions that speakers use in their ordinary and scientific inferences. Surely, natural language connectives and their formal counterparts do not need to share their meanings completely. In fact, whether formal and informal connectives coincide in meaning, or to what

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extent this is so, is a classic topic of pragmatics (see, for instance, Grice, 1975; Carston, 2002, chapters three and four), a topic in which the words outside calculi and their formal renderings have to be kept in mind and compared to each other. The conjoined consideration of formal and informal expedients is something that Prior’s analysis lacks. His criticism of inferential accounts of connectives assumes a version of inferentialism in which no other constraint is needed on the rules that define connectives, beyond some very minor syntactic restrictions, such as the subformula constraint that I mentioned in Chap. 5. The toy inferentialism that Prior criticises would also accept that merely by fulfilling those syntactic conditions, the inferences in which these connectives occur would be ‘valid’. No one has ever made such a proposal, however. This weakness of Prior’s argument is exposed in Stevenson’s response. For an inference to be analytically valid in virtue of the meaning of the connectives in it, Stevenson claims, the rules that govern those connectives have to be sound, i.e. they cannot produce falsity from truth: The crucial point to be noted is this: in order to completely justify an inference we must appeal to a sound rule of inference. A complete justification of an inference has two parts: we must first validate the inference by submitting it under a rule, and secondly we must vindicate the rule itself by showing that it is a sound rule. A deductive rule is sound if and only if it permits only valid inferences, an inference being valid in this sense if and only if it is such that when the premises are true the conclusion must be true. (Stevenson, 1961, pp. 125–6)

Thus, Stevenson contends that characterising an inference as analytically valid requires the meaning of the connectives in it to be given in terms that guarantee the soundness of the transitions in which these connectives are involved. In his comment on Prior’s paper, Stevenson follows the first option that I have mentioned above: that meaning precedes the possibility of entering into propositions and inferences. In his view, truth tables serve the function of characterising connectives completely before they can do their job (Stevenson op. cit. p. 126). A more sophisticated proposal, which still assumes that meaning pre-exists inferences, is put forward by Belnap, who sees rules, and not truth-tables, as the relevant instances on which the meaning of connectives is grounded. In his explanation, Belnap explicitly assumes a version of (PPP): It seems plain that throughout the whole texture of philosophy one can distinguish two modes of explanation: the analytic mode, which tends to explain wholes in terms of their parts, and the synthetic mode, which explains parts in terms of the wholes or contexts in which they occur. In logic, the analytic mode would be represented by Aristotle, who commences with terms as the ultimate atoms, explains propositions or judgments by means of these, syllogisms by means of propositions which go to make them up, and finally ends with the notion of a science as a tissue of syllogisms. The analytic mode is also represented by the contemporary logician who first explains the meaning of complex sentences, by means of truth tables, as a function of their parts, and then proceeds to give an account of correct inference in terms of the sentences occurring therein. (Belnap, 1962, p. 130)

Like Kneale, Popper, Gentzen and Wittgenstein before him, Belnap favours the synthetic mode. Connectives are defined in terms of rules of inference, but rules of inference are nevertheless not enough; the background against which inferences are

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made is also relevant. By appealing to the knowledge shared by the agents that carry out inferences, Belnap places his contribution in the pragmatist tradition: It seems to me that the key to a solution lies in observing that even on the synthetic view, we are not defining our connective ab initio, but rather in terms of an antecedently given context of deductibility, concerning which we have some definite notions. By that I mean that before arriving at the problem of characterizing connectives, we have already made some assumptions about the nature of deducibility. (Belnap, op. cit., p. 131)

Prior’s argument does not survive Stevenson’s and Belnap’s criticisms. Maybe Belnap is right to call Prior’s ‘The Runabout Inference’ a ‘gem’, and to compare it with Carroll’s ‘What the Tortoise said to Achilles’ (Belnap loc. cit., n. 2). What is true is that it has had a similar effect, spurring debate and forcing inferentialism to work out better proposals. But this indisputable status, due to the actual attention it has received, should not obscure the fact that neither ‘tonk’ nor anything like it has been seriously assumed by logicians, and for good reasons. These reasons have not been completely disclosed, and sometimes they have been masked by ever more sophisticated constraints that usually seem ad hoc. But, I contend, the actual reason is felt deep down by all participants in the debate: ‘tonk’ has no use, and thus that it does not represent a concept.

7.4 Russell’s Paradox The foregoing analyses of ‘grue’ and ‘tonk’ can be replicated for the case of the most consequential contradiction in logic and set theory: Russell’s paradox. Some dissimilarities between them should be mentioned, though. Goodman’s ‘grue’ contrasts with ‘green’ and ‘blue’, two ordinary natural language predicates. With ‘tonk’, a new formal connective is introduced, besides ‘and’ and ‘or’. Connectives, even if formally defined by rules, maintain some ties with ordinary terms that at least guide their formal definitions and whose absence, in the case of ‘tonk’, motivated Stevenson’s and Belnap’s responses. Russell’s paradox, by contrast, refers to a procedure that occurs within a specific system: the system that Frege developed in his Grundgesetze. In fact, as Macbeth notes, Russell discovered the paradox first in his own system for a logic of relations. After failing to find a way out there, he turned to Frege’s Grundgesetze for help, only to discover that the paradox was reproduced in Frege’s system as well (Macbeth, 2005, positions 145–153). Peano and Russell did not completely understand Frege’s project. Frege accepted Russell’s argument, which rested on completely different grounds from those the Grundgesetze was built on. The influence of Peano’s and Russell’s interpretations has been huge, with the effect that the project that Frege started in the Begriffsschrift has been almost completely wiped out from logical theory and its philosophy. A mathematical system is not a natural language. The semantic and pragmatic complexities of natural language sentences cannot be compared with the mechanisms that confer meaning on formulae in mathematical systems; formulae and

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natural language sentences are only analogous from a syntactic point of view. Thus, the temptation to think of mathematical systems as watertight is difficult to resist. And there is a grain of truth in this feeling. Natural languages are gigantic systems rooted in brains, social practices and social and natural evolution. Any move in them requires enormous effort and enough time for any novelty to smoothly fit in. Mathematical systems, on the contrary, can be created from scratch, and mathematicians enjoy the freedom that they lack as ordinary speakers of a language. But even so, mathematical systems have to make sense. Russell stated his paradox in semi-­ formal terms, and its hugely damaging influence rests, at least in part, on the fact that it was intelligible. Even the most abstruse mathematical notions have to earn their living in formulae and proofs that are able to be effectively put to work. This is not to say that mathematical theories need to have a use in physics or some connection to everyday life. Most mathematical theories have no such connections. Still, their terms have to have a principled function within the theory they belong to, and they have to be able to occur significantly in formulae that effectively play the roles of premises, steps, and conclusions in proofs. Russell communicated his discovery to Frege in a letter of 1902, in which he expressed doubts about Frege’s assumption that functions can be taken as the variable part of a judgement: On functions in particular (sect. 9 of your Conceptual Notation) I have been led independently to the same views even in detail. I have encountered a difficulty only on one point. You assert (p. 17) that a function could also constitute the indefinite element. This is what I used to believe, but this view now seems to me dubious because of the following contradiction: Let w be the predicate of being a predicate which cannot be predicated of itself. Can w be predicated of itself? From either answer follows its contradictory. We must therefore conclude that w is not a predicate. Likewise, there is no class (as a whole) of those classes which, as wholes, are not members of themselves. From this I conclude that under certain circumstances a definable set does not form a class. (Russell to Frege, 16/6/1902; Frege, 1980, pp. 130–1).

Frege harboured some doubts about the status of the extensions of concepts, which he sometimes considered to just be concepts (Frege, 1892; Burge, 1984, p. 16); although, as Burge notes, the reasons for his suspicions are not straightforward (Burge op.  cit., passim). Even so, the contradiction startled Frege ‘beyond words’ (Frege op.  cit., p.  132) and, although he corrected Russell’s formulation, there is no doubt that Frege granted the contradiction the utmost significance. The formulation that Frege proposed is: ‘A concept is predicated of its own extension’ since, he explains, predicates are first-level functions whose arguments have to be objects (Frege op. cit., pp. 132–3). Russell’s paradox, Frege thought, jeopardised any ‘possible foundation of arithmetic as such’ and showed that ‘the transformation of the generality of an identity into an identity of ranges of values (sect. 9 of [his] Basic Laws) is not always permissible, that [his] law V (sect. 20, p. 36) is false, and that [his] explanations in section 31 do not suffice to secure a meaning for [his] combinations of signs in all cases’ (Frege op. cit., pp. 132–3). I quote here paragraph §9 of the Grundgesetze:

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If   a     a  is the True, then by our earlier stipulation (§3) we can also say that the function φ(ξ) has the same course-of-values as the function ψ(ξ); i.e., we can transform the generality of an identity into an identity of courses-of-values and vice versa. This possibility must be regarded as a law of logic, a law that is invariably employed, even if tacitly, whenever discourse is carried on about extensions of concepts. The whole Leibniz-­ Boole calculus of logic rests upon it. One might perhaps regard this transformation as unimportant or even as dispensable. As against this, I recall the fact that in my Grundlagen der Arithmetik I defined a Number as the extension of a concept, and indicated then that negative, irrational, in short all numbers were to be defined as extensions of concepts. (Frege, 1893/1964, pp. 43–44)

My purpose in this section is to explore whether Frege had the conceptual and philosophical tools to answer Russell’s doubts without ascribing the paradox the power to virtually destroy his whole project. And I contend that he had. The answer could have proceeded by appealing to two central features of Frege’s logic/semantics. The first one, that for something to be a logical concept, it has to have sharp boundaries, i.e., it must be always possible to decide whether a particular object falls under it. Frege insisted that concepts have no history, but that it is rather our grasping of them that evolves from less to more clarity (Frege, 1884, p. vii; Burge, 1984, p. 7). The job of mathematicians and logicians is to make progressively more accurate our understanding of mathematical and logical concepts. But concepts in themselves are neither vague nor indeterminate. If Frege actually held such a view of concepts, then it would have given him a powerful tool against Russell’s objection. The expression ‘being a concept that is predicated of its own extension’ does not have sharp limits, as the paradox shows. And in fact, Russell proposed the way out: ‘Can w be predicated of itself? From either answer follows its contradictory. We must therefore conclude that w is not a predicate.’ Only it is a predicate, if by ‘predicate’ we mean an unsaturated expression. But it does not express a concept, because its limits cannot be determined, and this is purely a question of logic/ semantics. Frege was perfectly aware of the difference between contents and their expression, and thus of the distance that lies between proposing an expression and determining a content for it. The paradox does not challenge his courses of values, or Law V. It merely shows that not every predicative expression has a meaning. As I mentioned at the beginning of this chapter, paradoxes provide evidence that something is wrong, but they do not force the solution one way or the other since there are always alternative ways of tackling them. It is astonishing how a poor underlying semantics can block otherwise promising philosophical options. The simplest way of accommodating the Liar paradox is not by rejecting the notion of truth, but rather by rejecting the thesis that isolated sentences are truth bearers. I will discuss this issue later, in Chap. 10. The simplest way of accommodating Russell’s paradox is not by destroying Frege’s logico-semantic project, but rather by denying that any linguistic combination with a predicative form expresses a concept. And the only way to determine whether newly introduced expressions are meaningful or not is to see how they behave in worked-out systems. This remark leads us to the second feature of Frege’s project that could have gotten him off Russell’s hook, i.e. Frege’s (PCont).

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I agree with Burge when, while discussing Frege’s criticisms of sets as groups of objects, he says: I think that behind the objection lies Frege’s assumption of a context principle according to which the justification for postulating abstract entities, such as sets, extensions, numbers, concepts and the like, derives from their role in providing denotations or meanings (Bedeutungen—in the earlier writings, contents, Inhalte) for components of sentences that express true thoughts. Frege held that one could think about such objects only through predication or nominalizations of predication; one could not justify the postulation of abstract entities by making lists. (Burge op. cit., p. 20)

I will give arguments to show that Frege never took back the context principle and, although he sometimes expresses himself as if he had, he never gave too much weight to the intuition that inspires the principle of compositionality, at least not in a strong version in which terms are given in ‘lists’ and precede in the logical order to sentences and thoughts expressed by them. The context principle would have advised us to assess the strength of Russell’s paradox by considering the offending predicate at work. But before discussing whether (PCont) is detectable in the Grundgesetze, let us see what follows from Russell’s communication in the letter quoted above. Russell’s doubt relates to the possibility of interpreting functions as variables that are able to fill the argument place of predicates (or concepts, in Frege’s correction). If ‘being a predicate that cannot be predicated of itself’ is contradictory, this means, Russell thinks, that it is not a predicate. The analogous claim is that there is no class ‘(as a whole)’ corresponding to this predicate (Russell to Frege 16/6/1902; Frege, 1980, p. 131). Here there are four separable issues at work. The first one is whether a particular ‘combination of signs’, as Frege says, is a predicate. A second issue focuses on whether every predicate expresses a concept, i.e. whether every accepted combination of signs has a sense. The third topic points in one direction of Law V of the Grundgesetze, i.e. whether for every concept there is a class, ‘as a whole’, which is the concept’s extension. Finally, there is the converse issue, i.e., whether every class as a whole can be captured by a concept. As Coffa notes, the third issue is raised by Russell’s paradox, and the fourth was confronted by Zermelo’s Axiom of Choice (Coffa, 1991, p. 114). The fourth issue was also a source of discomfort for Frege, who was never at ease with the idea of sets as self-standing logical objects. Frege’s extensions depended on the concepts of which they were extensions, and he gave priority to intensions over extensions (Burge op. cit., pp. 19–21). The question that the first two issues raise is the validity of the principles of Context and Propositional Priority and directly affects the debate over whether propositions are structured entities. As I have defended in Chap. 2, my contention is that Frege considered propositions to be essentially unstructured, even if he sometimes expresses himself in ways that might suggest otherwise (see Pérez-Navarro, 2020). Their unstructured nature directly follows from the inferentialist method of propositional individuation. Now, if (PCont) and (PPP) maintain their validity in the Grundgesetze, then issues three and four should be reformulated accordingly.

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The first issue belongs to grammar. From a grammatical viewpoint, what predicates are is a question that has to be answered by appealing to the rules of language, and this affects natural and artificial languages—such as Begriffsschrift—alike. In the case of natural languages, it is the responsibility of linguists to solve it; in the case of artificial languages, it depends on stipulation: on the vocabulary and the formation rules that govern the well-formedness of the expressions in a particular system. The second issue, i.e., whether all predicates express a concept, has a definite answer in Frege’s semantics, where concepts are the meanings of grammatical predicates (Frege, 1892, p. 183, n. 1) and all predicates have to have meaning. But this answer applies to the predicates that belong to the language, not to all possible combinations of words with a predicative appearance. From (PCont), it follows that the meanings of such combinations have to be investigated in the context of a sentence and determined by the complete meaning of the sentence concerned. This principle, which is explicit in Frege’s early works, still exerts its influence in his mature period. In a letter to Jourdain written around 1914, he says: I do not believe that we can dispense with the sense of a name in logic; for a proposition must have a sense if it is to be useful. But a proposition consists of parts which must somehow contribute to the expression of the sense of the proposition, so they themselves must somehow have a sense. […] The possibility of our understanding propositions which we had never heard before rests evidently on this, that we construct the sense of a proposition out of parts that correspond to the words. If we find the same word in two propositions e. g., ‘Etna’, then we also recognize something common to the corresponding thoughts, something corresponding to this word. (Frege, 1980, p. 79)

One might think that this text is contrary to the thesis I want to establish. In particular, it can be read as arguing that propositions have parts which correspond to words, and that these words are independently meaningful. Nevertheless, I believe that understanding it as evidence of (PComp) in Frege’s mature work, together with a rejection of (PCont), would be wrong. First of all, ‘proposition’ in this text corresponds to the German ‘Satz’, as can be seen in Frege’s original letter (Frege, 1976, p. 127). I have mentioned in Chap. 1 that ‘sentence’ would be a more accurate translation. And sentences are made of words, even if thoughts are not made up of independent concepts. Frege relates the issue of the meaning of words with the possibility of understanding sentences that we have not heard before. In these cases, familiar words guide the identification of the senses of the new sentences in which they occur—and by doing it they guide the identification of the thought expressed. But this fact does not invalidate (PCont). As Frege explains in the text just quoted, we recognise that something corresponds to a word, ‘Etna’ in this case, by seeing what it is that the thoughts expressed by sentences in which it occurs have in common. Thus, it is the thoughts expressed by these sentences that ultimately give the word its meaning: a meaning that can subsequently help us to understand new sentences’ senses. Words have references only because they are parts of meaningful sentences. This is Linnebø’s explanation of the scope of the context principle, which he considers a metasemantic principle that regulates the process by which the reference of names is determined. The concern of semantics, by contrast, is how the meaning of

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complex expressions depends on the meaning of their constituents. As mentioned in Chap. 2, Frege also points to this semantic aspect, when he explains that the ‘parts’ that contribute to the expression of the sense of the ‘proposition’ ‘must somehow have a sense’. Compositionality thus belongs to semantics (Linnebø, 2008, §6.2, p. 24), whereas (PCont) belongs to the meta-semantics. This being so, context and compositionality do not conflict. Frege’s alleged support of some mild version of (PComp) is not an argument against his steady endorsement of (PCont). But the text just quoted goes even further: besides ‘proposition’, sentence, Frege also mentions the ‘corresponding thought’. As explained in Chap. 2, neither context nor compositionality, as linguistic principles, apply to thoughts, i.e. neither apply to the kind of abstract entity that can be also called ‘assertoric content’. Since the same thought can be expressed by different sentences, with different meanings (or semantic values), thoughts do not replicate sentential structures. They are made of parts only in a metaphorical sense (Frege, 1918–1919, p. 386). Penco notes the patent tension between the combinatorial, compositional explanation of the meaning of complex expressions in Frege’s writings and his support of the thesis that the same thought can be expressed by different equivalent sentences. According to Penco, Frege adopted two different kinds of ‘sense’, one that corresponds to the combinations of the meanings, i.e. semantic values, of the parts of sentences, and another that is inferentially individualised. The two senses of ‘sense’ are thus the following: senses as semantic values, which are properties of sentences, to which compositionality applies and are essentially linked to linguistic systems, and senses as judgeable or assertoric contents, which are independent of the linguistic vehicles used to express them. The tension between these two senses can only be resolved by a more nuanced analysis that allows for different levels and aims from which meanings and contents can be analysed, and which contemporary philosophy of language has acknowledged, as mentioned in Chap. 2. Frege does not offer such a refined analysis, but there is little doubt that he was aware of this tension, and that he oscillated between these two kinds of sense that relate to different theoretical concerns (Penco, 2003, esp. §4). The questions of which of these two senses is more fundamental, and of whether they are mutually independent, need not distract us. Thoughts, judgeable contents or assertoric contents, inferentially individuated, do not mirror grammar, even if senses as semantic values might admit of a compositional interpretation. This is what (PGS), an essential principle that defines Frege’s approach to logical analysis, states. Sentences, on the other hand, are thought’s linguistic presentations, and only in them are names referential. For this reason, (PCont) ‘plays an essential role in Frege’s logicist account of mathematics’ (Linnebø, op. cit., p. 1). Linnebø convincingly argues that the context principle ‘retains an important role in the Grundgesetze’, in spite of some ‘appearances to the contrary’ (Linnebø, 2019, p. 90). These appearances rest on some features of Frege’s mature semantics that were absent by the time he wrote the Grundlagen. There are three reasons, Linnebø claims, that justify the doubts cast on the relevance of this principle in the Grundgesetze. The first reason is Frege’s interpretation of sentences as names. The second reason is Frege’s mature support of the principle of compositionality. Frege’s

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rejection of contextual definitions is the third reason. Linnebø discusses these and concludes that ‘the principle survives into the Grundgesetze, where it retains a central role in what is essentially the same explanatory project’ (Linnebø op.  cit., p. 112). In what follows, I will review and extend Linnebø’s arguments. In his response to the first difficulty, i.e. Frege’s interpretation of sentences as proper names, Linnebø presents textual evidence of two versions of (PCont) in the Grundgesetze. The first version echoes the better-known occurrences of the principle in the Grundlagen: ‘one can ask after reference only where signs are components of propositions expressing thoughts’ (Frege, 1903/2013, §97; Linnebø op. cit., p. 100). But there is a fresh version of the principle, the ‘generalized context principle’ (Linnebø, loc. cit.), which offers a criterion for signs to refer in the following terms: A proper name has reference if, whenever it fills the argument places of a referential name of a first-level function with one argument, the resulting proper name has a reference, and if the name of a first-level function with one argument which results from the relevant proper name’s filling the ξ-argument-places of a referential name of a first-level function with two argument places, always has a reference, and if the same also holds for the ζ-argument-places. (Frege, 1893/2013, I, §29; Linnebø, loc. cit.)

As these texts make clear, and Linnebø explains, the principle was at most reformulated and adapted, but never abandoned. Moreover, one can agree that the move of understanding sentences, which are complex signs, as names, which are semantically simple, is a strange one that is only explicable by Frege’s interest in extensions instead of on intensions in the context of the foundations of arithmetic. But this move cannot be used against (PCont) and in favour of an alleged (PComp) since understanding sentences as names conflicts with compositionality at least as much as it conflicts with contextual characterisations. The second reason that Linnebø discusses is Frege’s conversion to a compositional semantics in his mature period. Linnebø follows this common lead and explains that the two principles could be seen as incompatible only if (PCont) were taken to preclude that ‘individual words can be units of significance’, something that Linnebø had previously rejected, borrowing Stalnaker’s distinction between semantics and metasemantics, as we have seen. Words can retain their individual meanings even if their semantic contributions can only be explained in relation to complete sentences. Linnebø assumes that Frege had compositional inclinations but does not offer any textual evidence of a formulation of (PComp) in Frege’s work. Nevertheless, there are strong reasons to sustain that Frege neither endorsed anything that could be read as a compositionality principle for thoughts understood as assertoric contents, nor supported any intuition that had this principle as a consequence (Pelletier, 2001, p. 105). Thoughts, as inferentially individuated entities that are able to be expressed by structurally divergent sentences, are not compositional. By contrast, sentences and their semantic values may well fall under the scope of (PComp). In any case, either because compositionality and context can be made compatible in the sense explained by Linnebø, or because (as Pelletier argues) compositionality has no clear presence in Frege’s project, Frege’s mature adherence to (PCont) can be safely assumed.

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Linnebø considers a third reason in Frege’s rejection of creative definitions in the Grundgesetze, §146. Frege defends his own procedure of characterising value ranges while rejecting formalist and ‘creationist’ ways of incorporating new entities. If Frege rejects creative definitions, the argument would go, then this would mean that he also rejects his own previously adopted procedure of introducing new notions via definitions by abstraction. Nevertheless, the fact that Frege even acknowledges an apparent conflict here, and tries to offer an explanation, ‘undermine(s) the thought that Frege’s criticism of creative definitions must have forced him to reject the context principle’ (Linnebø op. cit., p. 105). This is Linnebø’s argument. Hallett also gives an argument for the compatibility of Frege’s methods for defining abstract objects and his rejection of mathematical creation. Commenting on the same §146, Hallett introduces the distinction between creating and recognising. Some statements, he thinks, are better understood as ‘elucidations’ that ‘point to’ objects that, because they are not composite, cannot be strictly defined (Hallett, 2019, p. 317). Thus, as Hallett sees it, Frege’s methods and his rejection of creation do not conflict with each other. Frege’s defence of his characterisations over creative definitions would emphasise the difference between ontological and epistemological perspectives: two perspectives that Frege clearly distinguished. This is Hallet’s argument. My argument will be different and more general. Linnebø’s and Hallet’s arguments are respectively directed at showing the prevalence of a context principle in Frege’s Grundgesetze and its compatibility with his criticism of creative definitions. I agree with them. Nevertheless, I want to insist on two further points. The first is the tension between inferential semantics and a representationalist, atomist, and compositional view of language that takes empirical discourse as its model. The effect that the paradox had on Frege shows that he was struggling between these two models. The second point is that, for the paradox to be effective, Frege should have had to give up not only (PCont)—which, as Linnebø and Hallett have shown, he did not—but also (PGS), which would blow away his whole approach to logic and language. The apparent tension between creation and characterisation is nothing new. Cantor experienced the same situation concerning his transfinite numbers, towards which he confessed to having felt ‘forced’ (Cantor, 1932/1962, p. 175). Adequate definitions—those that produce the right consequences and allow establish the right connections between different concepts—do not create their objects. But this does not mean that it is specific objects that have been discovered if by ‘object’ we mean some entity that pre-exists the theory, as the American continent pre-existed its ‘discovery’ by Christopher Columbus. The dichotomy of creation versus discovery is a trap that derives from a representationalist way of understanding meaning (Frápolli, 2015). The thought expressed and the information acquired do not need to be interpreted by strictly following the grammatical surface. That something like (RT), (RT) (The Recarving Thesis): f (α) = f (β) ↔ α~β

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has often been interpreted as presupposing independent objects to which the terms flanking the identity sign refer is a consequence of taking syntax as a rigid guide, which goes against the Fregean (PGS). The non-objectualist way of interpreting (RT) that I favour, and which I attribute to Frege, are sometimes interpreted as a ‘deflationist’ reading of (RT). Deflationism can be defended by accusing the opposing side of inflationism. I have previously followed this strategy against ‘substantive’ theories of truth (Frápolli, 2013, p. 11), and against the alleged risk of ‘creeping minimalism’ applied to higher-level notions (Frápolli, 2019, pp.  108–9). Nevertheless, here I will follow a different path. In Chap. 3, we saw two principles adduced against a deflationist interpretation of (RT). The two principles, which I reformulate here, are (MIR) and (SS): (MIR) (Meanings Involve Referents): Any characterisation of the meaning of a sentence S that contains a referential occurrence of a singular term, a, must make use of a itself or some co-referring term. (SS) (Surface Syntax): The two sides of an abstraction principle have the syntactic and semantic form that they appear to have.

Both of these principles advise us to interpret names referentially wherever they occur. Regarding principles like the ones just mentioned, one might wonder what a ‘referential occurrence’ of a singular term might possibly mean in this context. Philosophers have some clear insights about how to understand referential uses of terms that ‘points to’ physical objects, and whose role is to anchor discourse to a particular context—among them demonstratives, indexicals and de re locutions. Nevertheless, translating the discourse of reference to the realm of abstract objects, unless it is a façon de parler, is an abuse of language. And—as in the case of ‘proposition’ and ‘Satz’—at this point too, some translations of Frege’s terms have nudged some debates in the wrong direction. The systematic translation of ‘bedeuten’ by ‘refer’ and of ‘Bedeutung’ by ‘reference’ in Frege’s works may have reinforced the similarities between the empirical and the abstract realms, making it harder to understand Frege’s position about the relation of numbers and other abstract entities to their linguistic modes of expression. Syntacticist principles such as (MIR) and (SS) place abstract discourse in an impossible position, forcing its interpretation along the lines of empirical discourse, opening the door to naive Platonism, and neutralising the benefits that derive from (RT) and, in general, from (PGS). As I have discussed in previous chapters, a grammatical constraint is not only anti-Fregean, but also jeopardises some of the best findings of the twentieth-century philosophy of logic and of language. (RT) can be naturally interpreted as explaining the meaning of certain identities, showing that what superficially seem to be claims about objects are actually equivalence claims among sentences, whose interpretation is metaphysically and semantically less committed. Frege claimed that ‘creating proper is not available to the mathematician, or at least, that it is tied to conditions that make it worthless’ (Frege, 1903, §146; Linnebø, 2019, p. 105; Hallett, 2019, p. 317). One may feel forced to acknowledge certain truths, as Cantor said. Frege, like Cantor, spoke of abstract objects as self-standing entities. Abstract entities, unlike subjective products of our imagination, oppose

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‘epistemic resistance’ (Frápolli op. cit., p. 337). As we saw in Chap. 3, self-standing entities differ from the fantasies of the individual mind in that they do not need an owner (Frege, 1918–1919, p. 362). They are objective, they belong to a ‘third realm’ distinct from the world of external things and the world of our mental lives (Frege op. cit., p. 363). But the third realm is not a place, it is a category. Physical and abstract objects are similar in some respects and dissimilar in others, and both are similar to and dissimilar from psychological entities in different ways. Epistemic resistance does not presuppose independence from the rational life of human beings. Frege’s rejection of creative definitions is a vindication of the objectivity of the enterprise he is committed to. Objectivity is not Platonism, and the geographical references used by him and others—such as Cantor or Gödel—should not confuse us. Frege’s procedure does not create objects; it only characterises certain notions by some of their properties and relations in a system, and this procedure requires seeing names in the context of complete sentences. Once the appropriate version of (RT) is implemented, Frege’s strategy entitles us to speak of objects of a certain kind. This is a huge advantage of Frege’s method, which only produces dysfunctions when combined with (MIR), (SS) or suchlike, which reinstall—by the back door—a semantic approach that is poorly suited even to empirical discourse, and that, when applied to abstract discourse, is fatal. What lesson can we draw from our previous discussion of Frege’s (PCont) and the scope of Russell’s paradox? I cannot see why, for the purpose of logicism, Frege needed the unreasonable principle that for any predicate-like expression there is an object that corresponds to it as its extension. For a term to be significant, it has to be put to work in sentences that express thoughts and refer to a truth value, i.e. that are either true or false. Alternatively, it has to produce significant sentences when substituted with other terms in previously significant sentences. This is the generalised context principle stated above. Only by identifying a concept in a thought, or by testing a term in a bigger context, is there a guarantee for the term to be meaningful and for something to be a concept. Here I cannot resist quoting Burge in his explanation of Frege’s extensions of concepts: Thus logical analysis was not separable from the acquisition of logico-mathematical knowledge. Frege thought that one attained insight into the relevant concepts or senses only through developing a theory and seeing it work. This rather pragmatic emphasis on the interdependence of theory and understanding is an integral part of Frege’s rationalist conception. (Burge, 1984, p. 33)

I can only agree. In Chap. 5, I argued that certain constraints, introduced to avoid unwanted consequences of the inferentialist approach to logical constants, pointed towards a view of them that respects (PPP). In the same spirit, the axiom schema of specification (separation), in Zermelo-Fraenkel set theory, by requiring for a set to exist that it be a subset of an already existent set, fleshes out, in set-theoretical formalist fashion, the semantic intuition that I have defended in this chapter: that a meaningful term is always a part of meaningful sentences and its meaning an ‘ingredient’ of possible propositions, which are the outcomes of successful assertions.

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Frege oscillates between two general paradigms: an atomist and compositional paradigm, inherited from the mathematical and philosophical context that he was schooled in, and an organic and contextual paradigm that he advanced from his first writings and that placed him within the semantic tradition to which Kant and Bolzano also belonged. He is no different from other theorists who inaugurate a fresh approach to a certain realm. Russell’s paradox struck him hard but, all things considered, it was not as damaging as it seemed. In fact, it would have been a great opportunity for him to develop his many sound insights about the complexities of language further in the right direction.

7.5 Taking Stock To resume Russell’s analogy between paradoxes and experiments, the puzzles commented on in this chapter suggest that something is deeply wrong with a view of language that allows purely combinatorial rules to be able to produce meaningful terms and meaningful sentences by fiat. Yet this view is so deeply rooted that even philosophers with pragmatist sympathies fall prey to it. Language is a living system on which particular speakers have very little influence. It is common within the received view to think of language as ‘conventional’, and there is a sense in which this characterisation is correct. But ‘conventional’ does not mean arbitrary. (PA) marks the origin of discursive actions. (PPP) and (PII) fix an approach to propositions in which they are inferential nodes, from which concepts can be abstracted. And logical connectives are devices that speakers have at hand to represent arguments as arguments. There is no connective ‘tonk’, besides ‘and’ and ‘no’; there is no concept grue, besides green and blue and for good reasons. The practice of designing artificial languages has possibly deluded us into believing that language can be freely manipulated. Ordinary concepts are tested by their role in practices belonging to particular forms of life, and scientific concepts are tested by their role in the theories to which they belong. There is no Vulcan, ‘phlogiston’ does not stand for any stuff, ‘the biggest prime number’ is not satisfied by any number, and ‘the class of all classes that do not belong to themselves’ does not describe a class. These well-known examples do not challenge astronomy, chemistry, arithmetic, or set theory, though. They simply do not do any job. By contrast, Cantor’s transfinite numbers have proved to be genuine numbers, ‘Higgs bosson’ is a meaningful term of particle physics, dark matter and dark energy are concepts (tentatively) accepted by contemporary astrophysicists, agoraphobia is a well-established concept in psychology, aporophobia is progressively gaining support in ethics and political philosophy, and two novel concepts have burst onto the political and social scene with great force in recent years: post-truth and fake news. We will see how they fare in these contexts. In any case, the way in which all of them were introduced, how their relations to other concepts were tested, and the reasons why some of them have finally been discarded while others remain, provide invaluable information about how language works.

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References Belnap, N. D. (1962). Tonk, plonk, and plink. Analysis, 22(6), 130–134. Brandom, R. (1994). Making it explicit: Reasoning, representing, and discursive commitment. Harvard University Press. Brandom, R. (2000). Articulating reasons. An Introduction to Inferentialism. Harvard University Press. Burge, T. (1984). Frege on extensions of concepts, from 1884 to 1903. The Philosophical Review, 93(1), 3–34. Cantor, G. (1932/1962). Gesamelte Abhandlungen mathematischen und philosophischen Inhalt. Georg Olms. Carston, R. (2002). Thoughts and utterances. The pragmatics of explicit communication. Blackwell Publishing. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. Frápolli, M. J. (1996). Review of Graham Priest, beyond the limits of thought. Modern Logic, 6(4), 437–439. Frápolli, M. J. (2013). The nature of truth. An updated approach to the meaning of truth ascriptions. Springer. Frápolli, M. J. (2015). Non-representational mathematical realism. Theoria, 30(3), 331–348. Frápolli, M.  J. (2019). Propositions first. Biting Geach’s bullet. In M.  J. Frápolli (Ed.), Expressivisms, knowledge and truth (Royal Institute of philosophy supplement 86). Cambridge University Press. Frege, G. (1884). The foundations of arithmetic. A logic-mathematical enquiry into the concept of number. Translated by J. L. Austin. Second Revised Edition. New York: Harper Torchbooks / The Science Library, Harper & Brothers. Frege, G. (1892). Concept and object. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 182–194). Basil Blackwell. Frege, G. (1893–1903/2013). Basic Laws of arithmetic (Volumes I and II). Oxford University Press. Frege, G. (1893/1964). The basic Laws of arithmetic. Exposition of the system. Translated and Edited, with an Introduction by Montgomery Furth. University of California Press. Frege, G. (1893/2013). Gottlob Frege: Basic Laws of arithmetic. Edited by Philip A. Ebert and Marcus Rossberg. Oxford University Press. Frege, G. (1918–1919). Negation. In G. Frege (1984). Collected papers on mathematics, logic, and philosophy. Edited by Brian McGuinness (pp. 373–389). Basil Blackwell. Frege, G. (1976). Wissenschaftlicher Briefwechsel. G.  Gabriel et  al. (Ed.). Felix Meiner Verlag Hamburg. Frege, G. (1980). Philosophical and mathematical correspondence. Basil Blackwell. Goodman, N. (1955). Fact, fiction, and forecast (2nd ed.). The Bobbs-Merrill Company, 1965. Grice, H.  P. (1975). Logic and conversation. Syntax and semantics 3: Speech acts, Cole et  al. (pp. 41–58). Haack, S. (1978). Philosophy of logics. Cambridge University Press. Hacker, P. M. S. (2015). Forms of life. Nordic Wittgenstein Review, Special Issue 2015 Wittgenstein and Forms of Life. Hallett, M. (2019). Frege on creation. In P. Ebert, & M. Rossberg, (pp. 285–324). Hunter, J. F. M. (1968). “Forms of life” in Wittgenstein’s philosophical investigations. American Philosophical Quarterly, 5(1968), 233–243. Linnebø, Ø. (2008). Compositionality and Frege’s Context Principle. Ms., invited talk given at the CSMN Workshop on Reference (Oslo 2007) and London Logic and Metaphysics Forum (November 2007). Linnebø, Ø. (2019). The context principle in Frege’s Grundgesetze. In P. Ebert & M. Rossberg (Eds.), (pp. 90–114). Macbeth, D. (2005). Frege’s logic. Harvard University Press (kindle edition). Peacocke, C. (1992). A study of concepts. A Bradford Book, the MIT Press.

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Pelletier, F.  J. (2001). Did Frege believed Frege’s principle? Journal of Logic, Language and Information, 10(1), 87–114. Penco, C. (2003). Frege: Two theses, two senses. History and Philosophy of Logic, 24(2), 87–109. Pérez-Navarro, E. (2020). Are Frege’s thoughts Fregean propositions? Grazer Philosophische Studien, 97(2), 223–244. Priest, G. (1995). Beyond the limits of thought. Cambridge University Press. Prinz, J. (2002). Furnishing the mind: Concepts and their perceptual basis. The MIT Press. Prior, A. N. (1960). The runabout inference-ticket. Analysis, 21(2), 38–39. Ramsey, F. P. (1928). Law and causality. In D. H. Mellor (Ed.) (1990). F. P. Ramsey: Philosophical Papers (pp. 140–144). Cambridge University Press. Ramsey, F. P. (1929). General propositions and causality. In D. H. Mellor (Ed.) (1990). F. P. Ramsey: Philosophical Papers (pp. 145–163). Cambridge University Press. Russell, B. (1905). On Denoting. Mind. New Series,, 14(56), 479–493. Russell, B. (1908). Mathematical logic as based on the theory of types. American Journal of Mathematics, 30(3), 222–262. Russell, B., & Whitehead, A. N. (1927). Principia Mathematica (Vol. I, 2nd ed.). First published 1910). Cambridge University Press. Stevenson, J. T. (1961). Roundabout the runabout inference-ticket. Analysis, 21(6), 124–128. Wittgenstein, L. (1953a). The blue and Brown books. Harper. Wittgenstein, L. (1953b). Philosophical investigations. (G. E. M. Anscombe, Trans.). Blackwell Publishing Ltd.

Chapter 8

Visual Arguments: What Is at Issue in the Multimodality Debate?

Abstract  Informal Logic and Argumentation Theory claim to accept a sense of ‘argument’ that is broader than the sense used in classical logic and mainstream philosophy of language. In this allegedly more general sense, pictures, sounds, gestures, maps, etc. can be premises of arguments, even if conclusions are usually seen as propositional. In this chapter, I argue that multimodality is a well-known phenomenon among linguists and among mathematicians. Relevance Theorists and Truth-Conditional Pragmatists have discussed the impact of gestures and body language in communication and the essential ineffability of what they call ‘expressives’. Mathematicians and philosophers of mathematics, in turn, have discussed the relevance of figures and diagrams in proofs. I will argue that perceptible items do not have logical properties. Logical properties are properties of a certain kind of abstract entities whose identification follows the organic intuition, (OI). Argumentation Theory has not provided arguments that advised a revision of Frege’s view that judgeable contents are the sole concern of logic. Still, the multimodality debate brings to the fore significant issues related to how information flows; among these issues, the insight that different representation modes open up different affordances and are differentially suitable for particular communicative aims. Keywords  Affordance · Argument [argumentation] · Bearer · Expressive · Effable [effability][ineffable · Ineffability] · Mode · Multimodal [multimodality] · Picture · Vehicle · Visual

8.1 Multiple Modes Are there multimodal arguments? Argumentation Theory and Informal Logic have put this issue on the table. Not only sets of propositions or sentences and actions in which propositions are defended and justified by other propositions are arguments and argumentations. Pictures and sounds can also possess these statuses, or at least they can be means by which legitimate premises in arguments are expressed. This is

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the claim. Here is an illustration due to Groarke, one of the more active defenders of multimodality: Imagine that we are standing in line at the Amsterdam airport, Schiphol, having debated the question whether the foremost proponent of pragma-dialectics, Frans van Eemeren, is in Brazil. I have adamantly maintained that he is still in Amsterdam, and you have disagreed. In the wake of our unresolved disagreement, I happen to see him standing in a line in front of us. In attempting to settle our disagreement silently, I open a book and display a photo of van Eemeren, nudge you with my elbow, and point at the elegantly dressed man a few meters in front of you, raising my eyebrows as I do. You look and nod resignedly when I ask you ‘Was I right?’ (Groarke, 2015, p. 136).

And now the question. Is this exchange of information between the two characters in the story an argument? A different question is whether an argument can be extracted from this situation. This latter question has an obvious positive answer, but it is not what is at stake in the debate about multimodality; what is at stake is whether entities other than propositions can bear logical properties and relations. Multimodality concerns the possibility of arguments some of whose elements are not linguistically articulated. The issue that multimodality raises is which items can be propositionally meaningful. It is essential in this discussion to distinguish between argumentations and arguments. Argumentations are actions in which a claim, explicit or implicit, is supported by the evidence in its favour. This evidence does not need to be conceptual. Feelings, the expression of emotions, gestures, etc. can perform a relevant role in the communication of the message (see Chap. 1 for more details). By contrast, arguments—in the sense in which I use this term—are the objective results of argumentations. Arguments, in the pragmatist approach I defend, are composed items whose ‘parts’ are pieces of information that are able to bear truth, i.e. propositions and judgeable contents. Strictly speaking, neither arguments nor their elements are primarily linguistic. Still, the most straightforward way of presenting them is by using some linguistic system. To this way of presenting the debate, some have objected that not all parts of an argument need to be propositional and that this is precisely what is at issue. And they are right. The propositionality of the elements of arguments is what some authors challenge, as we will see. Discussing whether physical means other than sentences can carry propositional information would be a milder version of the multimodality debate. This issue would require awareness of the distinction between propositional contents and ways of representing them, a distinction that some defenders of multimodality seem to forget (Gilbert, 1997, b, 2014; Groarke, 2015, 2020; Roque, 2015). My view should be clear at this point. As I have argued in previous chapters, propositions are, by definition, those items that possess the appropriate logical properties, which in this case comes down to the possibility of establishing relations of consequence and incompatibility with other items of the same category. This follows from the organic intuition (OI) since propositions are the items that bear propositional properties, independently of the ‘vehicle’ we use to represent them. Thus, in line with what I have defended so far and leaving aside superficial appearances, only conceptual contents, judgeable contents, or propositions, are steps in

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arguments. And this is so by definition because being premises or conclusions of arguments is one of the properties that identify propositions. Still, multimodality deserves close attention. Discussing the nature of the elements of arguments involves some of the central topics of the philosophy of language and logic. In particular, the discussion leads us to treat the issue of the bearers of logical properties: a question that is central to the comprehension of the aim of logic, to the definition of logical constants, and, as we will see in Chaps. 9 and 10, to the analysis of truth. Since a multimodal argument is an argument cast in different modes, the notions of argument and mode require attention from the beginning. As Tseronis and Forceville acknowledge (Tseronis & Forceville, 2017, p.  4), there is no clear-cut characterisation of what ‘mode’ means and, a fortiori, of what precisely the multimodality of arguments consists in. Their option is to take a ‘practical solution and consider the following as (semiotic) modes: written language, spoken language, static images, moving images, music, non-verbal sounds, gestures, gaze, and posture’ (op. cit., p.  5). Groarke includes ‘pictures, maps, sounds, diagrams, smells, video clips, and other non-verbal phenomena’ (Groarke op. cit., p. 135), and in his entry on Informal Logic in the Stanford Encyclopedia of Philosophy (Groarke, 2015), he endorses the extraordinarily broad position defended by Gilbert (1997, b, 2014) that ‘a hug, a forlorn look, or tears may count as argument’. In the introduction to the collection of essays that they edit, and which I have already mentioned, Tseronis and Forceville explain that the originality of the papers collected there ‘resides in the awareness that non-verbal argumentative discourse seldom consists of visuals alone. Consequently, the contributors to this volume are all concerned with discourses in which text and image (as well as other semiotic modes) combine to create meaning in argumentative contexts’ (Tseronis & Forceville, 2017, p.  2). The importance of acknowledging that multimodal arguments are mostly understood as mixed types, i.e. composites of linguistic and non-­ linguistic data and contributions, cannot be overlooked. The claim is, thus, that multimodal arguments include, besides conceptual (propositional) elements linguistically represented, some elements that might not belong to this category. This is debatable. Even in Groarke’s example, at the beginning of this chapter, the conclusion is a proposition, i.e. that Frans van Eemeren is in Amsterdam. Literally, the end of the argument-as-activity is the utterance of the sentence ‘Was I right?’. The connection between the interrogative sentence used and the proposition expressed is straightforward for any competent speaker, but its theoretical explanation is not always so. The explanation that connects the question and the propositions that competent speakers understand as the conclusion of the argument needs to be given in two steps. The first step is to understand that some interrogative sentences ask for a yes-or-no answer. Together with their answer, they work as genuine assertions. ‘Was I right?’—a rhetorical question with an implicit answer ‘yes’— counts as the assertion of whatever would have been said by the assertoric utterance of the sentence ‘I was right’. Frege made this point in one of his Logical Investigations, ‘Thoughts’:

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Propositional questions [i.e. yes-no questions] are a different matter. We expect to hear ‘yes’ or ‘no’. The answer ‘yes’ means the same as an assertoric sentence, for in saying ‘yes’ the speaker presents as true the thought that was already completely contained in the interrogative sentence. This is how a propositional question can be formed from any assertoric sentence (Frege 1918-19a, p. 355, my emphasis).

The thought already contained in the sentence is sometimes easily extracted in the appropriate context. Has your daughter recovered from Covid-19? The answer ‘yes’ is equivalent to the assertive use of the sentence ‘My daughter has recovered from Covid-19’. It may be noticed that a nod of the head1 would play here the same role as the adverb ‘yes’. Not all sentences exhibit their content as the propositional question in our example does. Some sentences are mere propositional variables that contextually inherit any propositional content. Of this kind are blind truth-ascriptions (see Frápolli, 2013, pp.  57ff.), such as ‘What she said is true‘, and sentences of the type that Groarke uses in his story. An utterance of ‘You are right’ pragmatically means I endorse what you have said: In order to be contentful, it requires that something has been said, in the technical sense of ‘saying’. ‘You are right’ works like ‘This is indisputable’, ‘This is true’, or ‘This is established beyond question’ (see Strawson, 1950/2013, p. 14). What, then, does the main character of Groarke’s story say by asking ‘Was I right?’, together with the expected affirmative answer? What is said, inherited from the context in which the conversation occurs, is that Frans van Eemeren is in Amsterdam. This is the second and final step of the explanation of why the conclusion in the story is propositional. I will go back to this issue in Chap. 10. It seems complicated to argue that also conclusions could be cast in modes other than linguistic modes and, in fact, this path is almost entirely unexplored in informal logic. Nevertheless, it has been defended in the realms of aesthetics and publicity, where a single picture can be either the conclusion of an aesthetic argument, or even an argument in itself, and in practical arguments,2 where actions are considered acceptable outputs of arguments. In the standard multimodality debates, nevertheless, the claim is that premises can be non-linguistic even if conclusions are usually the products of assertions. This is a genuinely new assumption. Logic and semantics, in all schools and paradigms, have always subscribed to what we might call the ‘Principle of Homogeneity’ (PH): (PH) All steps in arguments, understood as products, belong to the same logico-semantic category.

Inferential semantics has its own version of (PH), i.e. that propositions are those items that are able to work as premises and conclusions. It is inconsistent with the principles of inferential semantics and with the inferential individuation of

 Genoveva Martí drew my attention to the role of specific gestures like nods and shakes of the head to affirm and deny contents. 2  Geneoveva Martí, Francisca Pérez Carreño and Esther Romero helped me to better understand what is at issue in cases such as those mentioned. 1

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propositions that something that at one time is a premise in an argument cannot be the conclusion at another time. Rejecting (PH) raises the question of the relation between ‘heterogeneous’ items. For instance, how can a picture force one to assume a proposition with logical necessity? What kind of necessity, commitment, warrant, etc. connects pictures, which are visual items, with conclusions, which even in the realm of informal logic, are conceptual items? A similar problem occurs with epistemic and practical arguments, such as (Argument 1) and (Argument 2): (Argument 1) [water drops coming down from sky] ⊢ It’s raining. (Argument 2) It’s raining ⊢ [action of taking the umbrella]. In inferential semantics, the answer is clear. Concepts and conceptual contents maintain certain kinds of relations that make speakers commit to the acceptance of some of them once they have accepted certain others because these relations determine the content of concepts. These relations, we could say, are internal. Arguments 1 and 2 are not genuine arguments, although the relevant information given in the premise of Argument 1 and the conclusion of Argument 2 can be conceptually articulated to become propositional. The rejection of (PH) suggests that supporters of multimodality use ‘argument’ with an idiosyncratic sense, different from the standard sense used in logic and semantics. And something like this is true. Together with the two senses of ‘argument’ already mentioned, and explained in Chap. 1, i.e. as product and as action, or argument-as-object and argument-as-activity in Goodman’s terminology (Goodman, 2018), there is an alternative way of understanding arguments: arguments are claims together with the (implicit) speaker’s justification or evidence for holding the claim. Groarke, for instance, defines ‘argument’ as ‘a standpoint (a conclusion) backed by reasons (premises) offered in support of it’ (Groarke, 2015, p. 134). This is what I have called ‘argumentation’ at the beginning of this chapter and a case of a ‘looking-­ backward’ argument, as I explained in Chap. 1. And argumentations, so understood, are coherent with the way in which semantic inferentialism and pragmatism, broadly considered, understand assertion. To be successful, assertions require the speaker that she believes what she asserts and to possess reasons that justify her assertion. If challenged, the speaker should produce her reasons in the form of the premises of an argument that would lead to the content of her original assertion. All this goes no further than the two quality maxims formulated by Grice (Grice, 1975, p. 46): Maxim 1: Do not say what you believe to be false. Maxim 2: Do not say that for which you lack adequate evidence. Thus, Groarke’s characterisation implies that all genuine assertions might turn into arguments once the reasons the speaker has to make them are made explicit. Understanding assertions as arguments does not raise any unsolvable problem, although it might blur the distinction between implicit and explicit communication. This distinction is essential to any pragmatist stance on communication and has been developed by Relevance Theory and Truth Conditional Pragmatics, and, from a different perspective, is the core of inferential semantics and expressivism. According to Brandom’s expressivism, assertions are moves in the game of giving

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and asking for reasons. By asserting a content, a speaker shows her willingness to defend the content of her assertion with reasons. When these reasons are explicitly produced, we have an argument. The role of logical constants, as we saw in Chap. 6, is precisely to mark these arguments as arguments. Brandom’s logical expressivism confers on logical constants this explicitating role (Brandom, 2000, pp. 57ff.). Thus, assertions can be seen as implicit arguments, which are made explicit when the assertion is challenged, and the speaker is invited to show her cards. When premises are understood as justifications for a claim, it becomes more natural to count pictures, records, maps, and diagrams as premises. They can serve as data that the speaker adduces in favour of her claim (see Bermejo-Luque, 2011). In contrast, conclusions are not adduced; they are stated. Nevertheless, even if it is ‘more natural’, the claim that premises in arguments can be non-propositional is still questionable. Analysing what is at stake in this claim is the aim of the following sections. Before concluding this section, and since I have already mentioned Grice’s maxims, a word on the compatibility of the Gricean and Brandomian universes is in order.3 Grice’s view of communication and Brandom’s account of assertion belong to different theoretical frameworks, but they are more congenial to each other than meets the eye. My approach to assertion includes insights belonging to both positions. Grice is more specific than Brandom in saying that communicating something requires the speaker’s intention to put forward a belief, the recognition of this intention by the hearer, and the speaker’s acknowledgement that it is the recognition of her communicative intention that confers meaning in her act. It is this interplay between speaker and hearer that allows communication. In Brandom’s case, assertions are also expressions of beliefs. And here too, making an assertion has the pragmatic meaning of making a move in a linguistic game, by which one is bound by certain obligations; among others, being able to offer reasons to defend one’s claim (see Chap. 1, Sect. 1.3). Brandom’s normative statuses, i.e. commitments and entitlements, have counterparts in Grice’s quality maxims. Both Grice and Brandom understand assertion (‘saying something’) as part of a social and normative activity in which speakers and hearers are engaged. And, from a more comprehensive perspective, both projects ultimately intend to explain rational linguistic behaviour.

8.2 Non-linguistic Aspects of Linguistic Communication When the question arises as to whether multimodality is possible, several answers surface, some trivial, some reasonable, some others dubious, and none of them particularly new. There is no news that non-conceptual factors contribute, in some way or another, either to the communicated message or to the success of the

 Cristina Corredor raised this objection. I am grateful to her for drawing my attention to this important issue of the compatibility of different theoretical frameworks. 3

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communicative act. Linguists in the orbit of Relevance Theory have stressed this phenomenon with varying degrees of intensity (see, for instance, Carston, 2002, Chap. 2; De Brabanter, 2010). Carston is particularly explicit: [T]he domain of pragmatics is a natural class of environmental phenomena, that of ostensive (=communicative) stimuli; verbal utterances are the central case, but not the only one, and they themselves are frequently accompanied by other ostensive gestures of the face, hands, voice, etc. all of which have to be interpreted together if one is to correctly infer what is being communicated (Carston, 2002, p. 129).

Sounds, vocal stimuli (≠verbal stimuli) and gestures, all help the correct interpretation of an utterance. Expressions with procedural meaning, such as discourse markers, and those terms that linguists called ‘expressives’ (see, for instance, Potts, 2007) facilitate communication without putting forward substantive contributions to the message conveyed. As this point is hardly debatable, I will not pursue it further. A different question is whether non-linguistic modes occur essentially in arguments, or even whether non-linguistic modes can constitute arguments on their own. This question points to the distinction between component items and merely concurrent ones (Clark, 1996, p. 178). Even with our focus thus narrowed, there are still at least three different theoretical issues that should be distinguished. The first one is whether non-linguistic actions are always or mostly produced with genuine communicative intentions. De Brabanter (De Brabanter, 2010, p.201) contrasts the following two examples, (1) and (2), which he borrows from (Clark & Gerrig, 1990) and (Clark, 1996), respectively: (1) Herb! [points to Eve] + [puts an imaginary camera to his eyes and clicks the shutter]. (2) Someboby ‘placing a candy wrapper in a litter basket […] indicate(s) that it is waste’ (De Brabanter, 2010 op. cit., p. 202). In (1), the agent draws Herb’s attention to the fact that Eve is taking a picture. Example (2) is of a different kind and poses the question of the limits of understanding actions as being moved by authentic communicative intentions. These two examples belong to different categories. Even if we can eventually extract information from any action, and for that matter, from any situation whatsoever, I follow the main Gricean lesson that communication is a matter of recognising intentions. Thus, (1) is a genuine communicative act whereas (2) is not, even if there is something that we can learn from the behaviour of a person throwing away a candy wrapper. This point is relevant to the analysis of multimodality in argumentation theory. Some supporters of visual and auditive arguments seem to assume that almost everything is, in principle, suitable for being a premise in a multimodal argument. This general claim has a right reading and a wrong one. The right reading is that almost everything can be intentionally used to convey information, information that can subsequently be conceptually represented. The wrong reading is to assume that it is the physical carrier of information that contributes to the argument. In fact, once

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the message has been conceptually articulated, the vehicle of representation is irrelevant to the nature of the argument as product. The second theoretical issue that must be identified in the discussion of multimodality is whether gestures, pictures and sounds can stand for parts of sentences. De Brabanter uses the following example, (3), (3) I didn’t see the [IMITATION OF FRIGHTENING GRUMPINESS] woman today; will she be back this week? For linguists, but not for us, the debate here concerns whether the gestures and sounds used to convey the idea of grumpiness are part of the sentence, i.e. whether they can be included as parts of a syntactically articulated system. The translation of this point to the debate in argumentation theory is whether, as it happens with the words in (3), the gestures and sounds make a conceptual contribution. If they do, then what is said by (3), with all its complexity, is apt to interact with other propositions in arguments. If they do not, then what is said would be unaffected by the non-linguistic items. These items would add to the colour but not to the core of the speech act. Whether non-sentential expressions can be used as assertions is the third theoretical issue. By ‘non-sentential’, linguists usually mean subsentential, but we can also add pictures, sounds and diagrams. Consider the following example, (4), which De Brabanter (De Brabanter, 2010, p. 204) borrows from Carston (Carston, 2002, p. 130): [I]t’s breakfast time and, coming into the kitchen, I see my companion searching around in the lower reaches of a cupboard; knowing his breakfast habits, I guess that he’s looking for a jar of marmalade and I utter:

[(4)] On the top shelf There are two different angles from which this issue can be approached. We can begin with the syntax and semantics of linguistic (and non-linguistic) items and ask whether their production has the pragmatic significance of an assertion. Or else we can place ourselves at the pragmatic level and see whether a speaker has performed an assertion by uttering some words (or by producing pictures, maps, etc.). The first way is the semantic approach. The second one is the pragmatic approach. In the semantic approach, we are interested in whether the tool has the appropriate properties, i.e. whether it is a syntactically and semantically well-formed declarative sentence and possesses a complete sense. In the pragmatic approach, by contrast, we assess whether the act is an assertion, and this means asking whether the speaker has put it forward as true, whether the audience has understood it as making some move in the relevant linguistic game, whether it has added information to the common ground, whether the speaker is committed by the outcome of her act, whether she is entitled to make it and is in a position as to adduce reasons in its favour, etc.

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Regarding example (4), ‘On the top shelf’, De Brabanter discusses the two perspectives, both present in the literature. There are those who—like De Brabanter himself, Carston, and Stainton (Carston, 2002; Stainton, 2005)—consider (4) to be a semantically and pragmatically complete act and those who—like Jason Stanley and Peter Ludlow (Ludlow, 2005; Stanley, 2000)—even while assuming that (4) in this example has illocutionary force, still consider this utterance to be ‘an instance of syntactic ellipsis’ (De Brabanter, 2010, p. 204). From our pragmatic perspective that follows (OI), in which pragmatics answers to semantics, speakers engaged in the speech act have the last word. If by uttering ‘On the top shelf’, the speaker can be confronted about the truth or falsehood of her utterance or her utterance can be contested by somebody saying (5), or (6), (5) I disagree. (6) I have already looked there, and it is empty, Then (4) is an assertion, and the speaker has expressed a proposition. This point carries over to the discussion on multimodality. If the effect of a particular item, produced or adduced in an argumentative action with the right communicative intention, is what one should expect from a proposition, then that item has propositional content, no matter the nature of its physical basis. Some might worry that (OI) gives rise to an account of propositions in which ‘anything goes’, but this would be inaccurate. The Fregean pragmatist perspective that follows (OI) is undoubtedly more comprehensive than the semantic view that focuses on the characteristics of particular representation systems, i.e. languages, and the features of the linguistic items involved in linguistic actions. Nevertheless, the pragmatist perspective is less charitable than some approaches that, as we have seen, would grant inferential significance to virtually any kind of representation (see, for instance, Gilbert, 1997, b; Groarke, 2015). One of the examples that Groarke uses to support his claim that some pictures are truly arguments is Fig. 8.1. Which argument could Fig. 8.1. possibly codify? Without a fair amount of contextual information, the range of possible arguments is huge. For instance, we might want to establish that the shape and size of the shadows that are projected on surfaces vary according to the time of the day and the day of the year. In this context, we might translate the pictures in Fig. 8.1 as the following argument: Argument 3  Premise 1: The shape of the shadow in the picture on the left is identical to the shape of the shadow in the picture on the right. Conclusion: Both pictures were taken at a similar time and season. A different possibility is that we wanted to prove that, since some works ceased— for instance, because of the coronavirus pandemic—they have not been resumed. The argument that Fig. 8.1 would allegedly codify would then be something like the following:

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Fig. 8.1  Mars surface allegedly showing ice sheets

Argument 4  Premise 1: The picture taken on day 1 shows that the workers have abandoned the ditch. Premise 2: The picture taken on day 2 shows that there are no workers in the ditch. Premise 3: (similarly with another picture). Premise 4: (similarly with another picture). Etc. Conclusion: The works have not been resumed. Neither of these is the argument that Groarke reads into Fig.  8.1, however. Instead, he explains that the pictures were taken on Mars’s surface, and that what can be seen in them is that the white patches in the picture on the right are smaller than those in the picture on the left. Allegedly, this is evidence that the patches are ice, which has evaporated during the time that has lapsed between one picture and the next. The conclusion of the argument is thus that there is water on Mars. The argument is the following (Groarke, 2020): Argument 5  (Visual) Premise: First photograph of the dig. (Visual) Premise: Second photograph of the dig. (Verbal) Premise: “The most plausible way to explain the changes we see in the photographs is by postulating the evaporation of water ice.” Inference Indicator: “We can conclude that...” Conclusion: “...there is water on the planet Mars. Why are arguments 3 and 4 less appropriate for representing the visual argument in Fig. 8.1? There is no reason other than contextual information, and the intention of the agents who produced the pictures to obtain a particular effect. Figure 8.1 will have different effects depending on the way we describe it and the aspects we highlight. I will come back to this point in section four. Describing and highlighting aspects is the job of concepts. By conceptually codifying a picture, we cast the information it gives in the appropriate form for it to have

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logical effects. This is not to deny that the pictures in Fig. 8.1 offer crucial evidence in favour of the claim that there is water on Mars. Evidence is, nevertheless, an epistemic notion. This evidence helps agents develop the belief that these patches are ice. This belief has a propositional content, represented by the that-clause, and it is this propositional content that acts as a premise—surely among others—for the conclusion that there is water on Mars. Pictures can support claims, but only the propositional contents of claims can be rejected, negated, disagreed with, or else commit the speaker to subsequent claims. In other words, pictures can induce inferences, but only conceptual contents can be parts of arguments-as-objects.

8.3 Sentences, Pictures, and Relational Linguistic Pragmatism We have to deal with the phenomenon of a variety of different messages connected to a unique physical item—a picture, a drawing, a diagram, a sentence—and also with individual messages expressed via different media. A common way of doing this is to distinguish between the physical support—the ‘carrier’ or the ‘vehicle’—, and the content—the message proper. The ‘vehicle’ metaphor is frequent in linguistics and in the discussions of the role of visual intuition in mathematical thinking in the philosophy of mathematics (see, for instance, Giardino & Greenberg, 2015, §1). As happens with every metaphor, those of message and carrier, content and container, thought and vehicle include aspects or layers that usefully represent relations and structural properties in the realm that the metaphor purports to illuminate, as well as aspects that do not have counterparts in the target realm, being so potentially misleading. Romero and Soria, for instance, speak of a ‘partial mapping from a conceptual domain into another’ (Romero & Soria, 2014, p.  490, my emphasis). Being clear about which aspects of the metaphor of vehicle and message are appropriate, and which ones are deceptive is crucial for understanding how language works and making an informed assessment of the possibility of multimodal argumentation. In the philosophy of language, the distinction becomes between sentences plus their linguistic meanings, and the assertoric content that is contextually conveyed. It is a basic assumption of all pragmatic approaches that sentences in themselves do not ‘say’ anything, i.e., that for a sentence to express semantically evaluable content it has to be used by an agent with a communicative intention since the linguistic meaning attached to a sentence falls short of expressing a proposition (see, for instance, Romero & Soria, 2019, p. 52). The analogous claim, as applied to multimodality, is that pictures, sounds, etc. are the carriers, and that only the carried contents have logical properties. That linguistic meaning falls short of being propositional admits of two different interpretations, both endorsed by prominent researchers working in the areas of influence of Relevance Theory and Truth-Conditional Pragmatics. The first interpretation has it that linguistic meaning only provides an ‘incomplete’ item, more

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like a framework or a structure than a judgeable content. According to this interpretation, linguistic meanings become full-fledged contents, able to be true or false, once the information for fleshing out the structure is obtained from the context of utterance (see Carston, 2002, p. 17). This, in turn, is explained in two different ways by different authors. Some philosophers argue that the contextual enrichment of linguistic structures proceeds by analogy with what happens with indexicals. What this means is that there is some linguistic indication in the logical form of the sentences concerned that gets saturated by information from the context. This is the view of indexicalists such as Stanley (see, for instance, Stanley, 2000). The alternative view holds that the contribution of the context is unarticulated, i.e. that it is not linguistically codified in the logical form. This is Recanati’s position (see, for instance, Recanati, 2002). When the contextual enrichment is not linguistically demanded by a ‘gap’ in the structure, the situation is not that the sentence does not express a complete content, but rather that it might not express the content intended by the speaker. Nevertheless, and this is the second interpretation, there is a more radical way of understanding the claim that sentences are not bearers of truth and logical relations. In this case, the thesis would not be that sentences plus their linguistic meanings need completion to become truth-bearers, but rather that they do not pertain to the category of truth-bearers, i.e. that they do not belong to the domain of those concepts or relations such as truth and logical relations. I will take this point again later, in Chap. 10 but for the present discussion this issue is also essential. That sentences are not primary truth-bearers is a constant claim among pragmatists, including classical pragmatists such as Ramsey, Grice and Strawson. Truth, consequence, and incompatibility are properties of abstract entities or groups of them, and not of their linguistic representations. An isolated token of a sentence like (7), (7) I am not going to attend the meeting today. does not express a proposition. Appearances notwithstanding, neither (8) does, (8) Winston Churchill was a conservative politician. At this point, the only difference between sentences in which indexicals occur and eternal sentences is that the latter suggest context-types or scripts that help the hearer to interpret them. The former, by contrast, are more open, i.e. they are compatible with a wider variety of situations. But the difference is only one of degree. To see this, remember Searle’s famous example: Suppose I go into the restaurant and order a meal. Suppose I say, speaking literally, ‘Bring me a steak with fried potatoes.’ Even though the utterance is meant and understood literally, the number of possible misinterpretations is strictly limitless. I take it for granted that they will not deliver the meal to my house, or to my place of work. I take it for granted that the steak will not be encased in concrete, or petrified. It will not be stuffed into my pockets or

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spread over my head. But none of these assumptions was made explicit in the literal utterance. The temptation is to think that I could make them fully explicit by simply adding them as further restrictions, making my original order more precise. But that is also a mistake. First, it is a mistake because there is no limit to the number of additions I would have to make to the original order to block possible misinterpretations, and second, each of the additions is itself subject to different interpretations (Searle, 1992, p. 180).

There is an enormous amount of implicit information that we assume to be able to understand even the most apparently harmless sentence. Thus, the content intended by the speaker and retrieved by the hearer involves not only the sentence and its linguistic meaning but also contextual information and information provided by the background knowledge that we share in virtue of our common ways of life. In the linguistic kind of pragmatism that Wittgenstein and Brandom favour, which I endorse, there is no priority of either language or thought, both being inseparable aspects of the same phenomenon. Thoughts are essentially claimable, which is what the principle of effability states and claims and assertions are essentially contentful. Significant primarily is the combination of the sign, its meaning, and the particular use an agent makes of it to convey her communicative intentions to an audience in a context that also provides relevant information. The sign cannot be severed from its use. This insight is traditionally attributed to Wittgenstein. In the Tractatus 3.323, Wittgenstein explains: In the language of everyday life it very often happens that the same word signifies in two different ways—and therefore belongs to two different symbols—or that two words, which signify in different ways, are apparently applied in the same way in the proposition. Thus the word ‘is’ appears as the copula, as the sign of equality, and as the expression of existence; ‘to exist’ as an intransitive verb as ‘to go’; ‘identical’ as an adjective; we speak of something but also of the fact of something happening. (In the proposition ‘Green is green’ ⎯where the first word is a proper name as the last an adjective⎯ these words have not merely different meanings but they are different symbols.).

The same point is made in Philosophical Grammar, written during the years 1931–4: When I think in language, there aren’t meanings going through my mind in addition to the verbal expressions; the language is itself the vehicle of thought (Wittgenstein, 1974, p. 161 [345CP]).

This much is clear and must be kept in mind, so as not to overestimate the scope of the metaphor of carrier and vehicle. Nevertheless, nothing of what I have said so far contradicts the fact that the same thought can be cast in different languages and modes, by different agents or by one agent at different times. By the utterance of two different sentences, (9) and (10), (9) My daughter is at school today. (10) Your daughter was at school yesterday. Two agents might express the same proposition. That two equivalent, yet different, sentences express the same thought is a consequence of inferential semantics. The Fregean example of the Greeks and the

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Persians at Plataea is well known, but maybe it is not so well known that Frege’s ‘Compound Thoughts’ (Frege, 1923–26) is a defence of the same intuition. Sentences and formulae can be equivalent, thoughts cannot. There are no equivalent thoughts (Frápolli & Villanueva, 2016). It is unclear whether uninterpreted formulae express Fregean thoughts, but if they do, then the thought represented by ¬(A v B) is the thought represented by ¬A &  ¬ B. To sum up, despite all its limitations, the metaphor of content and vehicle is a useful one. If we could not make sense of the distinction between what is communicated and how we communicate it, ordinary activities such as translations between languages, for instance, would remain unexplained.

8.4 Affordances The discussion of the previous section raises the question of which vehicles are more appropriate for which tasks. Even within particular linguistic systems, different aims call for different categories of terms. For instance, singular terms are essentially instruments for referring and identifying individuals, physical or abstract, whereas sentences are the natural instrument for making assertions. By a sentence such as (11), (11) Boris Johnson was the Prime Minister who accomplished Brexit. An agent can say something evaluable in terms of truth and falsehood. By a singular term such as (12), (12) The Prime Minister who accomplished Brexit. An agent can only identify or describe a person. Frege was aware of this difference, which he introduced in §2 of the Begriffsschrift as the difference between what the content stroke indicates and what is indicated by the judgement stroke. I will come back to this point in Chap. 10. The Wittgensteinian toolbox metaphor points towards this pragmatist insight: Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screw-driver, a rule, a glue-pot, nails and screws. —The functions of words are as diverse as the functions of these objects (And in both cases there are similarities). Of course, what confuses us is the uniform appearance of words when we hear them spoken or meet them in script and print. For their application is not presented to us so clearly. Especially when we are doing philosophy! (Wittgenstein, 1953, §11).

Even if Wittgenstein’s concern at this point was the traps that language lays for philosophy, his insight is applicable to our discussion here, i.e., that not all words signify in the same way, and that there is a plurality of ways in which words are used to accomplish communicative tasks.

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How does all this discussion apply to multimodality? Directly, since the semantic pluralism that derives from Wittgenstein‘s toolbox metaphor can be extended to cover different modes, which open different affordances. Very often we hear that ‘a picture is worth a thousand words’, but this saying should be completed by the answer to the question ‘Worth for what?’. The picture of a dead Syrian child on a European beach surely affected public opinion more deeply than merely reading the news in a newspaper. But for the European Union to implement policies to avoid more repetitions of this tragedy in the Mediterranean Sea, it is undoubtedly more informative to receive the data, the political causes, the humanitarian needs of Syrian refugees, etc., and this can only be transmitted linguistically. For the two pictures in Fig. 8.1 to be evidence of water on Mars, a large amount of conceptual work is required. This requires the knowledge of these pictures being of Mars’s surface, of the appearance and composition of ice, of what happens to ice when the temperature rises, etc. With this information on board, these pictures can be adduced as epistemic support for certain premises that lead to the intended conclusion. Pictures are like sentences in that they are physical objects. Type-sentences are certainly abstract objects, but they cannot be used for any purpose without turning into perceptible items. To this, it might be retorted that the same happens with propositions or judgeable contents: the only access we have to them is via some physical instantiation. Propositions can be neither seen nor heard, i.e. they always need a ‘vehicle’. But unlike what happens in the relationship between sentence-types and their tokens, the same proposition can be expressed by different tokens belonging to different types, and different propositions can be expressed by different tokens of the same type. The distinction between syntactic structure plus linguistic meaning, on the one hand, and content, or the propositions expressed, or what is said, on the other, generalises for any kind of representation system. Even if it must be treated with caution, the metaphor of carrier and message is illuminating. The carrier is the container into which we pour the message, which in some sense acquires the container’s shape. That the message as such does not have any specific grammatical form is one of the correct insights that the metaphor conveys. The Fregean Principle of Inferential Individuation (PII) points in that direction. Different semiotic modes can be used to convey the same message. In some cases, the possibility is straightforward, as, for instance, in oral and written modes. Spoken language and its counterpart, written language, have a similar capacity to codify information, each with specific properties that make it advisable to use one or the other for particular purposes. In some other cases, nevertheless, transmodality is at least arguable. Are pictures and sentences entirely intertranslatable? This does not seem to be the case, and this negative answer raises a series of essential questions in the philosophy of language and the theory of communication. I borrow the use that Tseronis and Forceville make of the technical term ‘affordance’ to refer to the diverse possibilities that different semiotic modes provide:

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But as multimodal analysts would argue, each semiotic system has its own specific affordances, which can be shown to be more suitable for one function than for another. At the same time, each semiotic system consists of a number of levels where choices can be made […], each with a different meaning potential (Tseronis & Forceville, 2017, p. 9).

The term ‘affordance‘is extensively used in the realms of ecological psychology, philosophy of biology and cognitive science (de Pinedo García, 2020; Heras-­ Escribano, 2019) to explain agents’ complex relationships with their surroundings. As Heras-Escribano claims, ‘[a]ffordances are the possibilities for acting in our environment: objects of a certain size are graspable, floors are walkable, obstacles are avoidable, etc.’ (Heras-Escribano, 2020, p. 30). What is significant about the use of this term is that it stresses that agents are essential parts of their environments, and that cognition is essentially a relational issue. Applying the philosophy of affordances to the discussion of multimodality is not entirely straightforward, though. Still, it gives flesh to the intuition that different means of conveying messages open different possibilities of interaction, including different communicative effects, which may not be central to the message, but undoubtedly affect its impact on the audience. Forceville has successfully applied Sperber and Wilson’s Relevance Theory (RT) to visual argumentation, with suggestive results. As the saying mentioned at the beginning of this section indicates, a picture might after all be better than a sentence for some purposes. The central tenet of (RT), as Forceville explains it, is ‘that a communicator cannot help but presume to be optimally relevant to her addressee. For a message to be relevant to a given addressee, that message must have an “effect” on the sum total of knowledge, beliefs, and emotions […] of that addressee’ (Forceville, 2014, p. 54). Forceville has company on this point, since applying (RT) to the realm of visual advertising is not uncommon (see, for instance, Pinar-Sanz, 2013; Xu & Zhou, 2013). Speakers choose not only which messages to convey, but also by which means, to obtain maximal effect with minimum cognitive effort on the part of their audience. Since part of the message expressed relates to emotions (see Forceville op. cit., pp. 59–60; Tomasello, 2008, Chap. 3), for some purposes pictures and sounds help to achieve the intended effect more efficiently than articulated language.

8.5 Ineffability and Conceptual Articulation We communicate emotions, and the recognition of emotions helps the communicated message to get through. This point is not in question. A different issue is whether emotions belong to the proposition, i.e. whether they affect what is said. Frege rejected this, and most pragmatists follow his example. But everybody accepts that the debate is consequential and that it has prompted much-discussed issues such as the ineffability of expressives and the nature of non-conceptual content. Potts, in his seminal work, identifies descriptive ineffability (DI) as one of the features that define expressives:

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[DI] Descriptive ineffability: Speakers are never fully satisfied when they paraphrase expressive content using descriptive, i.e. nonexpressive, terms (Potts, 2007, p. 166).

Consider, for instance, Potts’ examples, (13) and (14), (13) Whenever I pour wine, the damn bottle drops, (14) That bastard Kresge was late for work yesterday (Potts, op. cit., p. 171, my emphasis). Any competent speaker understands the roles of ‘damn’ and ‘that bastard’ in them, but it is also clear that there is no easy way of translating them into words. This difficulty receives illumination from Brandom’s expressivism. Apart from the specific kind of expressive role that Brandom attributes to normative notions, logical constants among them—a role that helps us to say what otherwise could only be done, and which allows cases of knowing-how to be turned into cases of knowing-­that— Brandom grants a more basic role to expression. Expression, and not representation, is, in his view, the genus to which the conceptual belongs (see Chap. 1, Sect. 1.2 above). The central role of concepts—the central activity of the conceptual life that defines us as a species—consists in the possibility of translating our doings linguistically. Thus, we make explicit what was implicit in practices. The central tenet of Brandom’s expressivism is that making explicit is essentially an issue of concept application. Making explicit something that was previously implicit is not bringing out what was inside us; it is not expressing feelings or giving vent to emotions. This is the classical understanding of expressive meaning that was defended by the logical positivists, and which has survived in metaethics. Brandom’s version understands the process of making explicit as the process of casting experience into concepts, i.e. the process of becoming conceptually aware of our doings: [A]s is suggested by this characterization of a pragmatist form of expressivism, in the cases of most interest in the present context, the notion of explicitness will be a conceptual one. The process of explicitation is to be the process of applying concepts: conceptualizing some subject matter (Brandom, 2000, p. 8).

Thus, to make something explicit is to make it conceptually articulated, and to be conceptually articulated is to be linguistically articulated. At this point, Brandom follows an essential pragmatist thread: that language and thought—the linguistic and the conceptual—cannot be made sense of independently of each other (see Chap. 3, Sect. 3.2): The line of thought pursued here is in this sense a relational linguistic approach to the conceptual. Concept use is treated as an essentially linguistic affair. Claiming and believing are two sides of one coin —not in the sense that every belief must be asserted nor that every assertion must express a belief, but in the sense that neither the activity of believing nor that of asserting can be made sense of independently of the other, and that their conceptual contents are essentially, and not just accidentally, capable of being the contents indifferently of both claims and beliefs (Brandom, op. cit., p. 6).

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Conceptual contents are thus essentially expressible, essentially effable. The immediate consequence is that those aspects of communication that cannot be linguistically processed do not belong to the category of the conceptual. Linguists broadly assume the effability of the conceptual. Carston, for instance, endorses a weak effability principle with the following formulation: (FPE) (First Principle of Effability): Each proposition or thought can be expressed (=conveyed) by some utterance of some sentence in any language (Carston, 2002, p. 33)

Of which she says that it ‘does not raise too many problems’ (loc. Cit.). Thus, the contrast between what is effable and what is ineffable is the contrast between what is conceptual and what is not. If the message codified in pictures is not linguistically translatable, then it is not conceptual, and, if it is not conceptual, then it cannot bear the properties that define the category, i.e. inferential properties. The principle of effability, as I understand it, does not imply that sentences or utterances are, by themselves, suitable for expressing any thought. That is, the principle does not go against the semantic underdetermination of sentences. What it claims is that conceptual contents are essentially expressible, even if what is said on every occasion necessarily incorporates contextual factors. A further consequence of effability is that natural languages are all equally appropriate for expressing thoughts. Even if we must sometimes hear that philosophy, or science for that matter, is better pursued in such-and-such a language in contrast with some other, what follows from the principle of effability is that all thoughts can be expressed in any language, i.e. that speakers have at their disposal an appropriate instrument for communicating their thoughts, and that no conceptual content can be said to be inexpressible. The conclusion is even more straightforward if the Chomskyan hypothesis of innatism is taken on board. But there is no need to follow such a radical path to accepting that human thought and human languages perfectly fit each other and that any human being using her mother tongue can express any thoughts that she is capable of entertaining. Defenders of multimodality are aware of the complexities of the step that goes from seeing a picture, participating in an argumentative action, or having an experience, on the one hand, to organising it as an argument, i.e. to dressing the argument, on the other: In analyzing acts of arguing and the arguments they forward, one of the first tasks we face is the identification of an argument’s premises and conclusions. Doing so is sometimes called ‘dressing’ (or ‘standardizing’) the argument […]. Dressing an argument is a more complex matter when we are faced with many real-life arguments, for their premises and conclusions may be unclear or open to debate. Identifying them may require that we decide how to deal with digressions; with rewordings of (roughly) the same claim; with implicit (or ‘hidden’) premises or conclusions; with rhetorical questions and other rhetorical devices; and with incomplete, vague or ambiguous claims and utterances (Groarke, 2015, p. 13).

The term ‘dressing’, which Groarke borrows from (Woods, 1995), resonates with the insight that language is the clothing of thought, an insight that is found in Frege (see for instance Frege, 1918-19a, p. 354) and in Wittgenstein (Tractatus 4.002).

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In the dressing of arguments, we identify premises and conclusions. This identification means fixing the conceptual terms that support the argument, making explicit the inferential relations, choosing between relevant and irrelevant information and, in the end, interpreting the non-linguistic data. Once the message is linguistically cast, i.e. conceptually fixed, its inferential properties become fixed too. ‘Messages’ that are not fixed conceptually, like those transmitted by images, are richer because they can evoke different feelings Frege’s logic. Logic is a transversal proposal that applies to the foundations of arithmetic and any other discipline that deals with judgeable contents. Frege rejected the formalist shift produced by the work of Peano, Boole and Schroeder, and continued by Hilbert and Tarski, to install an approach in which propositional contents were logic‘s only concern. The author defends logic as immanent to our conceptual system and the bearers of logical properties as judgeable contents (propositions) instead of uninterpreted sentences in formal systems. This approach will allow a definition of logical constants, logical consequence, and truth that not only correctly identifies the syntactic and semantic properties of these terms but also connect them with the use agents give to them in science and ordinary communication., sensations, and attitudes; but they are also poorer because they do not possess logical properties. In casting messages into words there are thus gains and losses; they lose in colour and emotional richness what they gain in precision and logical connections.

8.6 Visual Thinking in Mathematics Before concluding this chapter, let me make some comments on the use of visual arguments in mathematics. This topic, with all its complexities, would surely deserve a chapter by itself, but, from a structural point of view, the theses involved, the arguments adduced, and the conclusions arrived at do not essentially differ from what I have discussed in this chapter focusing on what happens in argumentation theory, linguistics, and the philosophy of language. It is customary among mathematicians to distinguish between geometrical and algebraic thinking (Giaquinto, 2007, pp. 240ff.). The two mathematical areas that are probably most easily related to visual argumentation are geometry and topology. Still, arithmetic and analysis have also vindicated the relevance of visual thinking to their aims (Giaquinto op. cit., pp. 163ff.). There is little doubt that children access calculation multimodally, using, for instance, finger-counting (Giaquinto, op. cit., p. 121). In addition, diagrams and geometrical proofs help us to understand the connection between certain claims, axioms, theorems, and certain others, and contribute to the simplicity of proofs (Cain, 2019). Mathematical practice, on the other hand, may involve mathematical and visual intuitions (Giardino, 2010). Nevertheless, relevant as all these discussions are to the comprehension of mathematical thinking, the concurrence of alternative sources of information is not what is at stake when we enquire about visual thinking in mathematics. In mathematical thinking, as much as in linguistic thinking, the relevant point comes down to deciding whether visual

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aspects are instrumental, and their role mainly epistemological, or else whether they contribute essential steps in proofs and essential information in theories. The methods by which some knowledge is acquired, or some connections understood, are not particularly informative about the nature of the knowledge concerned. This latter claim points to the time-honoured distinction between the context of discovery and the context of justification, which was central to the philosophy of science of the last century (see Hoyningen-Huene, P.  Hoyningen-Huene, 1987 for a survey). Even Frege uses this distinction in the Preface of the Begriffsschrift: In apprehending a scientific truth we pass, as a rule, through various degrees of certitude. Perhaps first conjectured on the basis of an insufficient number of particular cases, a general proposition comes to be more and more securely established by being connected with other truths through chains of inferences, whether consequences are derived from it that are confirmed in some other way or whether, conversely, it is seen to be a consequence of propositions already established. Hence we can inquire, on the one hand, how we have gradually arrived at a given proposition and, on the other, how we can finally provide it with the most secure foundation. The first question may have to be answered differently for different persons; the second is more definite, and the answer to it is connected with the inner nature of the proposition considered (Frege op. cit., p. 5).

As Frege claims, agents arrive at a proposition by many varied ways. Still, this variability in access does not affect a content’s connections with the contents from which it follows, or with the contents that follow from it. And it is these connections that make up a proof. Thus, the epistemological and logical aspects are separable. Propositions can be directly justified, or else justified indirectly by justifying their antecedents and consequences, but the web of their logical relations only depends on their ‘inner nature’. Logical connections and the inner nature of propositions are two sides of the same coin. Translation into a system such as the Begriffsschrift, whose aim is to explicitly represent conceptual structures, displays those aspects from which particular inferential relations derive. This way of putting things may suggest that conceptual structure is something fixed that precedes inferential connections, but to draw such a conclusion would be wrong. Judgeable contents are what they are because they belong to specific inferential networks. This is what follows from the Principle of Inferential Individuation, (PII), which I stated in Chap. 2. Translation only makes patent how these connections serve the task of identifying certain structural features. The possibility of translation into a conceptual writing implies that the information so codified is conceptual, propositional in this case. This procedure is analogous to the ‘dressing of arguments’ that I have mentioned in the previous section. The translation of a visual proof into a conceptual representation system—its conceptual dressing—shows that the analogical and perceptible aspects of the proof were ‘peripheral’, to use Giaquinto’s phrase. Which part of a mathematical argument is conceptual, and which part is essentially visual or spatial, is part of what is at issue in the discussion of Frege’s logicism. Logicism is often understood as a particular position about the nature of the membership relation and the status of the existential axioms of set theory, basically the Axiom of Infinity and the Axiom of Choice. This approach characterises what

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has been called ‘Russell’s logicism’ (Klement, 2019), and basically boils down to the question of whether mathematics can be reduced to the system in the Principia Mathematica or, in general, to set theory. Thus stated, logicism rests on the prior identification of logic with set theory. Because of this identification, many logicians have understood that Gödel’s incompleteness theorem (Gödel, 1931) proved logicism to be false (see, for instance, Klement, 2013). Nevertheless, Russell’s logicism is not Frege’s logicist project. Frege was concerned with the grounds on which arithmetical proofs rest and his answer in The Foundation of Arithmetic was that those grounds are conceptual. This is what is meant by the claim that mathematics can be reduced to logic. To get a good grasp of the scope of Frege’s project, it is essential to recall that Frege’s logic is not a formal enterprise. Recall at this point Coffa’s remark: ‘logic’ means semantics in Frege’s writings (Coffa, 1991, p. 64). Logicism, then, is the claim that mathematics rests on semantic relations, i.e. on relations between concepts, and that the perceptible, visual, or intuitive aspects that might concur in proofs do not occur there essentially. In the case of arithmetic, Frege would thus reject essentially multimodal arguments. His approach to geometry is a different story. Frege understood Hilbert’s project in geometry as parallel to his own project in arithmetic (Frege, 1980, p. 43) and, to a large extent, rejected it. As his writings on the foundations of geometry (Frege, 1903, 1906) and his correspondence with Hilbert make clear (Frege, 1980, pp. 31–52), his logicist approach to arithmetic is not extensible to geometry. But in arithmetic, Frege defended the essentially conceptual nature of numerical equations, uncontaminated by visual modes.

8.7 Some Conclusions What exactly is the challenge that multimodality poses for logical theory? Its defenders sometimes sell it as if it were a new step in understanding how assertions, inferences, and arguments work. That speakers receive information from multiple channels cast in multiple modes is no news. Contextual factors, such as assumptions, situation components, the reference of indexicals, gestures, etc. affect the message in multiple ways. Thus, some claims put forward by the defenders of multimodal arguments can easily be accommodated by linguists and philosophers of language. The problematic point is whether non-conceptual information possesses logical properties; in other words, whether items other than propositions can be elements in an argument. The feeling that this claim challenges the classical studies of language and argumentation rests on the failure to distinguish between the channels by means of which information travels and the nature of the information itself. ‘Information’ is a general term that covers both conceptual and non-conceptual information. Assertions, i.e. acts in which speakers utter declarative sentences belonging to articulated languages, present the information as conceptual. Acts of assertion can be accompanied by gestures, pictures, sounds, etc. that add a different kind of information that, in some cases, helps the audience to correctly identify the

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conceptual message. But neither sentences understood as linguistic items, nor images, maps, figures, and diagrams belong to arguments. It is what is said by using them that can act as premises and conclusions. Sentences, in a very specific sense, take precedence over pictures, since sentences, with their linguistic meanings, are the only means of making explicit the conceptual articulation of information. Against the pragmatist background that derives from (OI), the question of whether non-propositional information can be part of arguments does not make any sense. From (OI) it follows that if something can be a premise in an argument, then it is propositional since what it is to be propositional is defined by its properties. If, on the other hand, the question is whether pictures can convey propositional information, then the answer must be given in two steps. The first step is to recognise that pictures give information, albeit information that is unprocessed. The second step consists in its conceptual processing, and this can only be done linguistically. Acknowledgements  The following colleagues have made useful comments on a previous version of this chapter: Lilian Bermejo, María Cerezo, Juan José Colomina, Cristina Corredor, Manuel Li z, Juan Mamberti, Genoveva Martí, Andrei Moldovan, Francisca Pérez Carreño, Esther Romero, Alejandro Secades, Marga Vázquez, and Antonio Yuste. To all of them, I am profoundly grateful.

References Bermejo-Luque, L. (2011). Giving reasons. A linguistic-pragmatic approach to argumentation theory. Springer. Brandom, R. (2000). Articulating reasons. An Introduction to Inferentialism. Harvard University Press. Cain, A.  J. (2019). Visual thinking and simplicity of proofs. Philosophical Transactions of the Royal Society A, 377, 20180032. Carston, R. (2002). Linguistic meaning, communicated meaning and cognitive pragmatics. Mind and Language: Special Issue on Pragmatics and Cognitive Science, 17(1), 127–148. Clark, H. H. (1996). Using Language. Cambridge University Press. Clark, H. H., & Gerrig, R. J. (1990). Quotations as demonstrations. Language, 66, 764–805. Coffa, A. (1991). The semantic tradition from Kant to Carnap. Cambridge University Press. De Brabanter, P. (2010). In E. Romero & B. Soria (Eds.), Uttering sentences made up of words and gestures (pp. 183–198). de Pinedo García, M. (2020). Ecological psychology and Enactivism: A normative way out from ontological dilemmas. Frontiers in Psychology, 11, 1637. Forceville, C. (2014). In D. Machin (Ed.), Relevance theory as model for analyzing visual and multimodal communication (Vol. 2014, pp. 51–70). Frápolli, M. J. (2013). The nature of truth. Springer. Frápolli, M. J., & Villanueva, N. (2016). Pragmatism. Propositional priority and the organic model of propositional individuation. Disputatio, VIII(43), 203–217. Frege, G. (1903). On the Foundation of Geometry: First Series. In G. Frege (Ed.), Collected papers on mathematics, logic, and philosophy (pp. 273–284). Basil Blackwell. Frege, G. (1906). On the Foundation of Geometry: Second Series. In G. Frege (Ed.), Collected papers on mathematics, logic, and philosophy (pp. 293–340). Edited by Brian McGuinness. Oxford, Basil Blackwell. Frege, G. (1918–19). Thoughts. In G. Frege (Ed.), Collected papers on mathematics, logic, and philosophy (pp. 351–372). Edited by Brian McGuinness. Oxford. Basil Blackwell.

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Frege, G. (1923). Compound Thoughts. In G. Frege (Ed.), Collected papers on mathematics, logic, and philosophy (pp. 390–406). Edited by Brian McGuinness. Basil Blackwell. Frege, G. (1980). Philosophical and mathematical correspondence. Blackwell Publishers. Giaquinto, M. (2007). Visual thinking in mathematics. Oxford University Press. Giardino, V. (2010). Intuition and visualization in mathematical problem solving. Topoi, 29(1), 29–39. Giardino, V., & Greenberg, G. (2015). Introduction: Varieties of iconicity. Review of Philosophy and Psychology, 6(1), 1–25. Gilbert, M. (1997). Coalescent Argumentation. Lawrence Erlbaum Associates. Gilbert, M. (2014). Arguing with people. Broadview Press. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik Physik, 38: 173–198. English translation in Jean van Heijenoort, 1967, 596–616. Goodman, J. (2018). On defining ‘Argument’. Argumentation, 32(4), 589–602. Grice, H.  P. (1975). Logic and conversation. Syntax and semantics 3: Speech acts, Cole et  al., pp. 41–58. Groarke, L. (2015). Going multimodal: What is a mode of arguing and why does it matter? Argumentation, 29, 133–155. Groarke, L. (2020). Informal logic. The Stanford Encyclopedia of philosophy (spring 2020 edition), Edward N. Zalta (ed.), Heras-Escribano, M. (2019). The philosophy of affordances. New Directions in Philosophy and Cognitive Science. Heras-Escribano, M. (2020). The evolutionary role of affordances: Ecological psychology, niche construction, and natural selection. Biology and Philosophy, 35(2), 1–27. Hoyningen-Huene, P. (1987). Context of discovery and context of justification. Studies in History and Philosophy of Science, 18(4), 501–515. Klement, K. C. (2013). Neo-logicism and Russell’s Logicism. The Journal of Bertrand Russell Studies. The Bertrand Russell Research Centre, McMaster U. N. S, 32, 127–159. Klement, K. C. (2019). Russell’s Logicism. In Russell, Wahl (ed.). The Bloomsbury companion to Bertrand Russell. : Bloomsbury, 2019, 151–178. Ludlow, P. (2005). A note on alleged cases of nonsentential assertion. In R. Elugardo & R. Stainton (Eds.), Ellipsis and nonsentential speech (pp. 95–108). Springer. Pinar-Sanz, M.  J. (2013). Relevance theory and political advertising. A case study. European Journal of Humour Research, 1(2), 10–23. Potts, C. (2007). The expressive dimension. Theoretical linguistics, 33(2), 165–198. Recanati, F. (2002). Unarticulated constituents. Linguistics and Philosophy, 25(3), 299–345. Romero, E., & Soria, B. (2014). Relevance theory and metaphor. Linguagem am(Dis)curso, 14(3), 489–509. Romero, E., & Soria, B. (2019). Semantic content and compositional context-sensitivity. Theoria, 34(1), 51–71. Roque, G. (2015). Should visual arguments be propositional in order to be arguments? Argumentation, 29, 177–195. Searle, J. (1992). The rediscovery of mind. The MIT Press. Stainton, R. (2005). Defense of non-sentential assertion. In Z. G. In Szabó (Ed.), Semantics versus Pragmatics (pp. 383–457). Clarendon Press. Stanley, J. (2000). Context and logical form. Linguistics and Philosophy, 23, 391–434. Strawson, P.  F. (1950/2013). Truth. Proceedings of the Aristotelian society. The Virtual Issue, 1, 1–23. Tomasello, M. (2008). Origins of human communication. A Bradford Book. Tseronis, A., & Forceville, C. (2017). Argumentation and rhetoric in visual and multimodal communication. In A. Tseronis & C. Forceville (Eds.), Multimodal Argumentation and Rhetoric in Media Genres (pp. 1–24). John Benjamins Publishing Company.

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Chapter 9

Truth and Satisfaction: Frege Versus Tarski

Abstract  In this chapter I discuss the philosophical presuppositions and consequences of Tarski’s and Frege’s approaches to truth. Tarski’s is the most successful theory of truth ever proposed. Nevertheless, there are serious doubts about the actual effect of its technical details and, above all, about its philosophical significance. Despite starting with similar pre-theoretical intuitions, Frege and Tarski developed conflicting positions. Thus, neither at this point nor, in general, to any central topic in the philosophy of logic, is there a ‘Frege-Tarski approach’. I will argue that truth cannot be reached from satisfaction, which was Tarski’s strategy, while the opposite strategy of explaining satisfaction in terms of truth, which was Frege’s strategy, presents no difficulty. All the weaknesses of Tarski’s position result from his mistaken identification of sentences as the bearers of truth. Because of this, truth became linked to particular linguistic systems, with the unwelcome consequence of an infinite proliferation of truth predicates. Frege, by contrast, placed judgeable contents at the centre of the picture and explained truth via assertion. Truth is simple and thus indefinable, but not uncharacterisable. Assertion, in turn, is the most basic linguistic action, and is thus also ‘primitive’ and immediately understood. Frege’s view of truth derives from the principles I discussed in Chap. 2, i.e. the Principle of Assertion (PI), the Principle of Context (PCont), the Principle of Inferential Individuation (PII), and the Principle of Propositional Priority (PPP). In relation to the truth predicate ‘is true’, his proposal is expressive, as it is in relation to the analogous predicate ‘is a fact’. The semantic status of ‘is true’ has been confused with an alleged redundancy theory of truth, which Frege never supported. Keywords  Assertion [asserted] · Convention · Correspondence · Deflationism · Physicalism · Redundancy · Satisfaction · Tarski · Translation · Truth

9.1 The Scope of Tarski’s Proposal Tarski’s semantic definition of truth is the most successful proposal for the meaning of truth in the last century, at least among logicians, semanticists, and philosophers of language. Etchemendy goes further in deeming Tarski’s definition as ranking © Springer Nature Switzerland AG 2023 M. J. Frápolli, The Priority of Propositions. A Pragmatist Philosophy of Logic, Synthese Library 470, https://doi.org/10.1007/978-3-031-25229-7_9

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‘among the most influential works in both logic and philosophy of the twentieth century’ (Etchemendy, 1988, p.  51). Considering how analytic philosophy has evolved, it is no exaggeration to say that the combination of Tarski’s work on truth (Tarski, 1935), logical consequence (Tarski, 1936) and logical notions (Tarski, 1986) has shaped the subsequent development of the philosophy of logic and the philosophy of language until the present day. Without diminishing the merits that Tarski may have in other fields of knowledge, his work on the foundation of logic and semantics has had the unhappy effect of completing the process of sweeping away Frege’s project, which is practically undetectable in mainstream logic and philosophy of language. And this is so in spite of the profusion with which Frege’s works are cited in the literature. This situation is particularly evident in the discussion of truth. Tarski’s semantic theory is a rejection of all that was worthy in Frege’s approach to truth. Both authors share the same basic intuitions, as could not be otherwise in competent speakers, in particular, that truth does not work as an ordinary predicate and that, in some of its uses, it can in practice be eliminated without affecting the content or truth conditions of particular utterances. But the foundations on which they account for the role of truth could not be more divergent. Frege’s knowledge of how language works reached a depth and subtlety unparalleled by any previous or subsequent classic author, including Wittgenstein. Tarski, by contrast, had no interest in the language in which we communicate, which he considered contradictory, vague, imprecise, and unable to host notions of a degree of complexity similar to the complexity of truth and logical notions (see, for instance, Tarski, 1944, p. 355). Even if he wanted his approach to truth to define the ordinary notion or, at least, some of its aspects (Tarski, 1944, p. 356), his contempt for natural language made him turn his mind to artificial languages, mathematically specified, and restrict his definitions of the basic notions of logic and semantics to them, giving rise to formal semantics and model theory (although see Etchemendy, 1988, p. 52). Two features of Tarski’s proposal show the distance between his view and Frege’s and prevented Tarski’s truth from receiving acceptable treatment. These two features are, first, making sentences, i.e. linguistic items, its primary bearers. The second is making truth rest on satisfaction. The mistaken identification of truth-bearers has produced endless discussions about the Liar paradox, and an almost unbearable succession of complex proposals for avoiding it (see, for instance, Kripke, 1975), which in practice have led to no illumination. The most damaging result is the generalised feeling that truth is unreliable, obscure, contradictory, fraught with difficulties, and philosophically untreatable. Fortunately, the main consequence of understanding truth as a limit case of satisfaction is less harmful. Truth (and assertion) are logically prior to satisfaction, and the opposite direction promoted by Tarski is merely an illusion, of which there is evidence even in Tarski’s own explanations. The pragmatist background that supports my arguments in this book offers alternatives to these two features that characterise Tarski’s work on truth, and both derive from the Principle of Propositional Priority. In the case of truth-bearers, the contrast is between what is said by a speaker who uses a particular sentence and the sentence itself. Sentences, as I have discussed in previous chapters and specifically

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in Chap. 8, are merely the vehicles of content, as the contents that are expressed by using them are the items to which truth and falsity are attributed. As mentioned above, Tarski’s definition of truth rests on the common intuitions that any competent speaker possesses about this notion. In fact, (Tarski, 1944), especially in the second part where he explains some consequences of his definition and answers some objections and misunderstandings, offers a perfectly reasonable account of how this notion works. He even vindicates Aristotle’s explanation in Metaphysics, Γ 6 1011b25: ‘To say of what it is that it is not, or of what is not, that it is, is false, while to say of what is that it is, and of what is not that it is not, is true’ (Tarski, 1944, pp. 342–3; Tarski, 1969, p. 63). And yet, in the process of allegedly making these intuitions more precise, the proposal progressively departs from them and becomes a series of technical claims whose application to the ordinary use of the notion is hard to see (see Frápolli, 2013, pp. 86–106). One might be tempted to respond to this latter claim by appealing to an alleged ‘philosophical’ or ‘scientific’ sense of truth that would contrast with the ordinary, everyday sense. There is no such thing, and even if there were, it would not be of any value. Again, at this point, Tarski would agree with the rejection of any ‘philosophical’ sense (Tarski, 1944, p. 361 and n. 30). Truth, as I will show in the next chapter, is univocal, its significance and role being stable across contexts. What scientists do by appealing to truth is exactly what we do in everyday life by appealing to the same notion. Even if Tarski accepted (infinitely) many truth predicates (Tarski, 1944, p. 355), he did not consider ‘truth’ to be equivocal. The function of truth, even in Tarski’s view, is constant through all the uses of its many homophonic terms. In my view, this means that we have a unique concept. The task of philosophers is to explain the pragmatic, semantic, and syntactic features of the terms used to express it; to account for it in the fullest possible way, without turning it into a different notion. This is the challenge. Beginning with more or less confused intuitions, in the hope of ending up with technical proposals, sometimes highly artificial, is respectable philosophical and scientific practice. It is what linguists and philosophers of language do every day. Nevertheless, for this practice to accomplish its job, the resulting technical proposals have to be able to trace back connections between, first, the roles the notions in question actually fulfil and, second, the technical aspects of the proposals. This procedure is an instance of reflective equilibrium. Successful examples of this practice are the various accounts of demonstratives and indexicals, from Kaplan to Recanati (Kaplan, 1977; Recanati, 2000), and their application to the prosentential theory of truth (Williams, 1976, 1995; Brandom, 2009). As I see it, Tarski’s view of truth does not belong to the group of successful proposals. Even today, and despite the thousands of voices that praise Tarski’s definition, there are many others that (i) give reasons to doubt whether Tarski completed his own project (see, for instance, Field, 1972; Etchemendy, 1988), (ii) still enquire about which kind of theory he proposed, and (iii) debate whether it is philosophically illuminating at all.

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9.2 Physicalism and the Unity of Science I will begin by commenting on (i). The elimination of semantic notions in the definition of truth was one of Tarski’s main purposes (Tarski, 1935, p. 153; Raatikainen, p. 5). If meaning, even in the tame version of sameness of meaning, is taken to be a semantic notion —and what else could it be?— then there are reasons to doubt that Tarski actually accomplished his purpose (see Davidson, 1990; Field, 1972; Soames, 1984). But, as Raatikainen notes, only satisfaction, denotation, truth and definability were semantic notions for Tarski (Raatikainen, 2007, p. 104).Translation (from the object language into the metalanguage) is not, and this is the only notion that Tarski apparently needs. This is nothing to object to if ‘translation’ is understood as a blind mapping of one sign system into another (see Raatikainen, 2008 for more details). But if translation is anything more than blind mapping, then it is difficult to leave it out of the group of semantic notions. To understand translation in the way required by Tarski’s project, we must seriously depart from the common usage in which ‘translating’ from one language to another means saying in the target language what has been said in the initial language. When we learn a language, we learn ‘how to use its sentences to perform the same propositions that others do’ (Soames, 2020, p.  296). Correctness of translation requires comparing the intertranslated sentences with something external to the two linguistic systems concerned if only to be sure that the two expressions say the same or perform the same function. Nevertheless, acknowledging the external part would make translation a semantic notion in Tarski’s own sense: By semantics we mean the part of logic that, loosely speaking, discusses the relations between linguistic objects (such as sentences) and what is expressed by these objects. (Tarski, 1969, p. 63)

I will come back to this point below. The debate over the success of Tarski’s project has almost completely neglected a further aspect that I consider essential, i.e. the issue of reductionism. Tarski intends his definition of truth to comply with the positivist ideal of the unity of science (Tarski, 1956, p. 406). He did not elaborate on this commitment but, Raatikainen notes (Raatikainen, 2008, p. 292), it can hardly be a coincidence that Tarski mentioned the unity of science in his lecture on truth, while Neurath and Carnap spoke of physicalism and the unity of science in the presentation of their new scientific philosophy, all around 1935. Field sees Tarski’s project as failing to comply with the physicalist condition required by positivism (Field, 1972). Raatikainen, by contrast, argues that Tarski succeeded in this respect since his kind of physicalism only required ‘reducing all statements with reference to non-observable to statements in a language of everyday observable entities and properties’ (Raatikainen, 2008, p.  299). The fate of Tarski’s proposal does not concern me here. Either way, the physicalist constraint is marginal in Tarski’s proposal. But physicalism as part of the positivist ideal is of the utmost importance to understanding the history of logic and semantics. Let me make some quick remarks.

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Positivism rejects the legitimacy of abstract entities—concepts and propositions among them—unless they can be reduced to expressions defined by reference to observable items. In this framework, as the history of science and philosophy of the past century has shown again and again, not only is philosophy compromised but also mature natural science and mathematics. The indispensability argument in favour of mathematical entities (see, for instance, Field, 1980) is, when analysed with some distance, a mere acknowledgement of defeat. The problem of defining theoretical terms was the Achilles heel of the positivist philosophy of science. For the philosophy of logic in all its short history, and for the philosophy of language of the first half of the twentieth century, scepticism about abstract entities was very damaging. The reduction of normative concepts such as good to some observable properties, be they feelings or any scientifically testable property, already denounced by Stevenson (Stevenson, 1937), has also proved to be unattainable. To comply with the positivist ideal, the strategy in logic was to swap the bearers of logical properties from propositions and concepts to sentences, formulae or terms, with the results that we all now know. And this strategy, once it starts, cannot be stopped and devours all theoretical enterprise since not only concepts and propositions are abstract entities; numbers, type-expressions, and type-sentences also belong to the group of suspicious items. Now I go back to Tarski. Tarski and Frege alike identified certain objects as the bearers of truth. In Tarski’s case, sentences are the entities said to be true or false. In Frege’s case, judgeable contents are the entities to which truth applies. The choice of truth-bearers produces technical complications for Tarski that do not affect Frege. One of these is the proliferation of truth predicates in Tarski’s proposal. The immediate consequence of making truth a property of sentences, which always belong to particular languages, is that the definition of truth becomes linked to particular languages too, and thus the ordinary predicate ‘is true’ splits up into a potentially infinite number of predicates of the form ‘true-in-L’, for a specified L. The multiplication of truth-predicates deriving from Tarski’s treatment has been widely discussed and found to be a problem for reconciling Tarski’s view with the ordinary notion. The disparity between Tarski’s truth and the ordinary notion can be seen simply by considering what happens in translations, in everyday situations and in philosophical and scientific contexts. Tarski wanted his truth definition to fit Aristotle’s claim in Metaphysics Γ 6 1011b25, presumably because he considered it true. One might wonder whether Tarski could read any Greek. Surely, Tarski, like all of us, relied on good translations. A good translation should preserve a content through different ways of expressing it. It should render the content expressed in the original language into the target language while maintaining its semantic and logical properties, truth among them. If truth is a property of sentences, then much more argumentation is needed to explain how these properties of sentences are transferred from certain sentences to some others that belong to a different language, without any intermediate step involving the contents that the sentences in both languages correctly represent. Good translations preserve truth because they preserve contents— this is what a good translation is meant to do—and truth is a property of contents. For the defenders of sentences as the bearers of truth, this is an unsurmountable

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difficulty if—as Tarski intended—they want to keep talking about the ordinary notion (or about a technical one with certain connections to the ordinary notion). If sentences were the bearers of truth, then arguments in French or Spanish could not be combined with arguments in English. Given that their particular sentences would only be true-in-French, true-in-Spanish, or true-in-English, their combination into a single argument would irremediably produce a fallacy of equivocation. I believe that Aristotle’s dictum is true, even though I have never read it in its original Greek formulation. A scientific truth, say the mechanism of natural selection for evolution and all its consequences, is true in English and remains true when translated into Spanish. In our globalised world, we are all familiar with debates in which arguments and counterarguments are expressed in different languages. In scientific journals and books, there are no prohibitions against reference lists including items in different languages, although there might be some ideological bias in favour of items written in English. In philosophical discussions, we accept the truth of what Hegel, Cantor, Frege, or Wittgenstein said sometimes without being able to read it in the original German. Tarski wrote his famous paper on truth in Polish. It is hard to argue that the truth of what he said in Polish is a different matter from the truth contained in the subsequent translations of his works into German and English. The same applies to formal languages, and to the usual style of combining formal and natural languages in mathematical and philosophical proofs. If truth were a property of sentences, then validity, a notion defined on truth and necessity, would become a property of sets of sentences. This is the standard approach to these notions in the formalist view of logic. Thus, a valid argument in German might turn out to be invalid if translated into English; and a formal argument in, say, Polish notation, might change its logical properties when recast into the standard language of the predicate calculus. In Chap. 1, I used Goodman’s text about Anselm’s ontological argument to make this point (Goodman, 2018, n. 5). Frege, writing in German, rejected Anselm’s argument, originally in Latin, as invalid. Would Frege have had to read it in Latin to be sure that his diagnosis was correct? No doubt, trained philosophers have answers to this objection. We have limitless talents for defending the most untenable theses with the most sophisticated arguments. But, outside classrooms and academic journals, it is hard to argue that truth is a property of sentences and that validity is a property of sets of sentences. Pragmatists’ loyalties lie with ordinary intuitions, and it is these intuitions that need to be theoretically fleshed out. Sentences as primary truth-bearers do not pass the pragmatist test. Unlike propositions, sentences and formulae, given that they are extensional entities, make it easier to reassure oneself that one is not dealing with ‘creatures of darkness’, as Quine calls non-extensional entities (Quine, 1956, p. 180). The standard argument in favour of sentences over propositions rests on the relative simplicity of sentences’ identity conditions. Identity conditions for sentences are unambiguous, the argument goes, whereas identity conditions for propositions are allegedly highly unstable. A closer look at this matter reveals, nevertheless, that things are not so straightforward. Identity conditions for physical entities have puzzled metaphysicians from ancient times until the present day. It is enough to recall the classical paradox of Theseus’s ship, or the profound difficulties surrounding

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personal identity (see, for instance, Parfit, 1971). By contrast, reliable criteria for propositional identity are not uncommon. Prior offered a criterion along the lines of (OI) in (Prior, 1963), and Frege’s or Brandom’s inferentialism provides a neat answer to this issue too. Still, for positivists concerned with the unity of science, the suspiciousness of concepts and propositions seems to stem more from their nature as abstract entities and their ontological status than from their identity conditions. And in this respect, sentences are not better placed than propositions; since, in the sense that a theory of truth requires, sentences are still abstract entities and thus incompatible with the physicalist ideal. Sentences, in any sense of interest to science and philosophy, are neither ephemeral sound waves nor observable traces on a surface. If they were so, then Tarski’s truth predicate should be defined not only for every language or language level, but also for every uttered instance. Abstract entities cannot be reduced to physical entities; numbers are not groups of pebbles; sentences are not groups of sounds. Normative properties, such as good and truth, cannot be reduced to properties of physical objects either. And this is not a contingent fact, or a challenge for natural science, but a consequence of the kind of concepts that they are. Identifying the bearers of truth with sentences is not only a source of difficulty but—and this is the relevant point—it goes against our most entrenched intuitions. This is nothing new in the philosophical world, in which we all too often insist on defending the indefensible and on rejecting what is plainly right—usually as a ‘cheap’ solution to allegedly profound philosophical challenges.

9.3 Correspondence and Deflationism As indicated by (ii) above, there has been a lot of noise about whether Tarski’s proposal was a kind of correspondence theory of truth. Popper famously argued that it was (Popper, 1962), as have many others since (see, for instance, Sher, 1999; Fernández-Moreno, 2001). Haack and Davidson, on the other hand, have defied this characterisation (Haack, 1976; Davidson, 1996). The arguments for characterising Tarski as a correspondentist are either indirect, appealing to his endorsement of Aristotle’s characterisation, or else rest on Tarski’s use of satisfaction in his definition of truth (Sher, op. cit., p. 154). This is a tricky issue since we do not have a shared fully developed definition of correspondence. There are instead many different versions, whose unique point of agreement is the utterly general intuition that truth somehow indicates how things are. This agreement concerns what some authors understand as weak correspondentism: that truth-bearers correlate with something in the world (Raatikainen, 2007, Kirkham, 2001, pp. 119ff.). Depending on what this ‘something in the world’ is taken to be, the spectrum of correspondence theories varies. The correspondentist intuition is overwhelming. If the constraint on the ‘worldly’ correlate of truth-bearers is weak enough, almost any theory of truth ever proposed would count as a version of correspondentism. Aristotle’s claim is the clearest

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example, vindicated by philosophers of all schools, from Peirce and Ramsey to Tarski and Austin, and given all kinds of interpretations (see, for instance, Malpas, 2016). Peirce argued that beliefs to which we would ‘come to were we to inquire as far as we could on that matter’ are true (Misak, 2016, p. 24), and this claim, even with its pragmatist flavour, is what his correspondence proposal comes down to. In Austin’s version, ‘[a] statement is said to be true when the historic state of affairs to which it is correlated by the demonstrative conventions is of a type with which the sentence used in making it is correlated by the descriptive conventions’ (Austin, 1950, p. 5). The connection between statements and sentences with states-of-affairs by two types of conventions has won for Austin the title of correspondentist. The trick is done by including in the definition the word ‘historic’, which is meant to bring the world. Ramsey, in turn, claimed that ‘a belief is true if it is a belief that p, and p’ (Ramsey, 1927/2001, p. 338) and feared that his proposal would be dismissed as a view of truth as correspondence (Ramsey op. cit., p. 439). All of these authors accepted the Aristotelian characterisation. The only way to make this general support of Aristotle’s claim compatible with its wildly divergent implementations is by acknowledging that Aristotle’s statement is analytic, and hence devoid of any substantive content. It cannot be rejected, since it agrees with intuitions of all kinds. There is no criticism in my words here, however. The Aristotelian characterisation is what it should be, and, although it is in need of theoretical elaboration, it pinpoints the role of truth just right. Two basic intuitions, widely accepted, suffice for a theory of truth to belong to the correspondentist family: (a) that, as Austin said, ‘it takes two to make a truth’ (Austin, 1950, p. 6, n. 13), and (b) that truth is something that, in some sense or another, is forced upon us. (a) rests on the intuition that merely by looking at a contingent sentence in isolation, or by understanding a single contingent proposition, truth cannot be determined. True and false sentences, true and false propositions, look alike; there is nothing in them that accounts for their truth. (b) rejects subjectivist proposals according to which what is true and what is false is for the subject to decide, without the ‘world’ having any say in the matter. The ‘world’ may range from the physical aspects of reality, passing through some socially agreed background, to the successful results of our actions. (b) stresses that truth is not created but acknowledged, and this holds for classical correspondentists as much as it does for pragmatists, coherentists and deflationists of any kind. The correspondentist intuition will be further explained in Chap. 10. A further point of discussion among Tarski scholars focuses on the characterisation of Tarski as a deflationist about truth. As happens with the debate over correspondentism, there is no agreed definition of what deflationism amounts to (Davidson, 1990, p.  288 and p.  296, n. 33). Thus, as I have argued elsewhere (Frápolli, 2013, pp.  10ff.), ‘deflationism’ is not a useful label, not only because there is not precise definition of it, but, more importantly, because its use is usually evaluative. Deflation suggests that there is something important that has been left out. It is sometimes added that, according to deflationism, truth has no role to play in philosophy (see, for instance, Gupta, 2001, p. 582). Some provocative slogans from supporters of non-inflationist views have fuelled the negative comments that

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are sometimes addressed against this family of proposals. Brandom, for instance, gave one of his papers on the matter the title ‘Why Truth is Not Important in Philosophy’ (Brandom, 2009). Ramsey famously claimed that ‘there is no separate problem of truth but only a linguistic muddle’ (Ramsey, 1927/2013, p.  38), and Austin declared that a ‘theory of truth is a series of truisms’ (Austin, 1950, p. 152). Appearances notwithstanding, none of these authors argued that there is nothing valuable in the debates about truth. Their claims would be better understood as contributing to their clarification. In particular, Ramsey never proposed a theory of truth as redundancy, even if his alleged redundantism is one of the few things that everybody ‘knows’ about him (Frápolli, 2011). There are undoubtedly substantial issues related to truth, although the difficulty does not lie in defining or characterising the concept of truth, but rather in identifying the practices, including the mental states and intentions, in which the discourse about truth makes sense, and its function therein. In Ramsey’s case, the notion that takes responsibility for the analysis of truth is belief; Austin and Brandom give this responsibility to the practice of assertion. And there is nothing trivial in this debate. Austin’s is a complex characterisation that makes mention of sentences and utterances, types and historic states-of-affairs, and demonstrative and descriptive conventions, as we have seen. Brandom puts forward a worked-out version of the prosentential account of truth: a rather technical proposal that is anything but trivial (Brandom, 1994, chapter five). Thus, the debate over whether Tarski’s is a deflationist proposal has no philosophical weight unless what is actually at stake is clearly identified. Tarski answered to some sound intuitions and explained them by relying on a technical system that he considered free of metaphysical and semantic adornments. A further question is whether he succeeded in his attempt. And as we have seen, many philosophers acclaim his works as ground-breaking and fruitful, whereas many others have cast serious doubts on it. I count myself among the latter.

9.4 Satisfaction Besides the intended physicalist reduction, the other aspect that will settle the question about Tarski’s success relates to his interpretation of truth as a limit case of satisfaction. The role of satisfaction in the definition of truth is responsible for the classification of Tarski’s as a ‘semantic’ theory. His work on truth is also praised as inaugurating model theory and laying a scientific foundation for semantics. (Etchemendy, 1988) is almost the only exception to this opinion. I am not concerned here with the origin of the model-theoretic approach to semantics. My concern at this point is the connection between satisfaction and truth, and my contention is that, whereas satisfaction is essential to model-theoretic semantics, it is idle in the characterisation of truth, even in Tarski’s framework. In this section, I intend to win support for the straightforward fact that truth cannot be reached from satisfaction, although satisfaction can be reached from truth.

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This follows from (PPP) and (PCont), but we do not need to embrace the pragmatist framework of this book to recognise that from satisfaction to truth there is an infinite jump, only achievable because of our previous understanding of the practice of assertion, something that every competent speaker carries with her, regardless of the philosophical theory she favours. The definition of truth on satisfaction is an illusion that has impregnated semantics since Tarski’s work of 1935. We understand satisfaction because we understand truth, not the other way around, and we understand truth because we understand assertion. This is what Tarski’s characterisation shows and is something that he explicitly recognised in some of his explanations of the right interpretation of his proposal. Tarski required any definition of truth to be formally correct and materially adequate. The requirement of formal correctness constrains the language in which the definition is given and the form of the definition itself. Languages appropriate for this task have to be completely specified and the definition cannot contain in the definiens any term that is used or presupposed in the definiendum. Moreover, in order to be materially adequate, a definition has to imply all instances of Convention T: (CT) x is true iff p (where ‘x’ is a name of p). Tarski, and everybody after him, uses the example (1), (1) ‘Snow is white’ is true if, and only if snow is white. If the set of sentences in a language were finite, a list in which (CT) were applied to all of them would count as a definition of truth for that language. Unfortunately for (CT), all languages—at least all of those of any interest for philosophy and science—have mechanisms to build an enumerable number of well-­formed sentences. For this reason, the simple definition represented by (CT) has to be modified using the semantic notion of satisfaction and the logical notion of recursion. The procedure is the standard one. The basis of recursion is the explicit indication of which objects satisfy the simplest sentential functions. Tarski’s definition uses a potentially infinite set of indexed variables, sentential functions and infinite sequences of objects, but nothing relevant hangs on these technicalities. Let us suppose that the language at issue has only one monadic predicable, ‘P’, one binary predicable, ‘R’, and three individual constants, ‘a’, ‘b’, and ‘c’. The definition of truth for this language would begin with the following list: (2) a satisfies ‘P’ iff Pa. (3) b satisfies ‘P’ iff Pb.

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c satisfies ‘P’ iff Pc. the pair  satisfies ‘R’ iff Raa. the pair  satisfies ‘R’ iff Rab. (Likewise for all other pairwise combinations of the three constants.)

That is, the definition begins with a list of the conditions under which any one of the individual constants satisfies the monadic predicable, and any pair of the individual constants satisfies the binary relation. For more complex languages, infinitely many variables are included, and the clauses are modified accordingly. As we all are familiar with this procedure, I will cut short the explanation at this point. For complex sentences, the procedure proceeds recursively. Let ‘s’ denote a sequence of objects and ‘Φ’ and ‘Θ’ sentential functions. Then: (7) A sequence s satisfies ‘~Φ’ iff s does not satisfy Φ. (8) A sequence s satisfies ‘Φ v Θ’ iff s satisfies Φ or s satisfies Θ. (9) A sequence s satisfies ‘∀xi Φ’ iff any sequence that differs from s in at most its i-th element satisfies Φ. So far, satisfaction has been defined for sentential functions, i.e. for formulae with free variables. The challenging step is to move from the satisfaction of functions to the truth of sentences that do not include free variables. Tarski’s manoeuvre is surprising: for sentences, satisfaction becomes trivial. Given a particular sentence p, either all sequences satisfy p, or none does. Sentences satisfied by all sequences are true, and those not satisfied by any are false. This way of defining truth has been among us for so long and has been so eloquently defended by many of the greatest minds of the past century that it takes a great effort to see how disappointing it is. Or maybe not, and the situation resembles the story of the emperor’s new clothes. Truth, for Tarski, is a limit case of satisfaction for those formulae without free variables. To decide whether either all sequences satisfy a particular sentence, or none does, we have to see that either the sentence can be stated, i.e. that it holds, or that it cannot. This is something immediate. All the technicalities that precede this infinite jump from the previous satisfaction clauses to the definition of truth are irrelevant. With all the complexities that the actual Tarskian definition includes, it does not take us any step further from the claim that a sentence s is true if, and only if, s says that p and p. This is exactly Ramsey’s version and, as we will see, a correct rendering of Tarski’s own interpretation of (CT). I have insisted that the original sin of Tarski’s definition, and the move that has been responsible for all the many problems that defining truth has encountered in analytic philosophy, is making sentences the bearers of truth. Apparently, in (CT) the clause on the right-hand side represents a sentence and the clause on the left-­ hand side includes a name of that sentence. This is the story. But if that were the

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case, (CT) would be unintelligible. To make sense of (CT), the clause on the right-­ hand side has to be an asserted sentence, not merely a sentence. If ‘p’ on the righthand side represented a sentence, then the objection that is sometimes addressed against Tarski’s definition, i.e. that the clause on the right-hand side is uncomplete and should be somehow completed with a further predicate, would be correct. It would be correct even in the framework defined by Tarskian semantics. In this framework, names like the permissible substitutes for “x” name objects, and sentences on the right-hand side of (CT) would have to be understood as objects of some kind. To overcome this difficulty, some critics propose to complete the right-­ hand clause with the predicate ‘is true’, but this would deprive the definition of any possible utility. The objection is well-founded, though. In fact, the definition only works if either we have a(n) (conditional) assertion of ‘p’, or we complete the sentence ‘p’ with the predicate ‘is true’, which expresses that the content represented in the sentence is asserted. This is precisely Frege’s positions with respect to non-­ asserted judgeable contents and the judgement stroke. Tarski answered this objection in (Tarski, 1944). Curiously enough, Ramsey foresaw a similar objection to his own definition. In his answer, he reminded critics of the propositional nature of the variable p, which already includes a verb. Ramsey was justified in making a move that is not open to Tarski, for Tarski understands the clause on the right as a linguistic object whereas Ramsey understands it as a full proposition, a complex compound that is essentially assertible (Ramsey, 1927/2001, p.  437). The focus on propositions makes the objection innocuous for Ramsey’s approach but fatal for Tarski’s unless we are not really talking about sentences after all, but about (conditionally) asserted propositions, in which case, the whole edifice of the semantic theory, with all its nominalist and formal flavour, would be an illusion. Tarski himself gives us reasons to think that this is so. As an answer to the aforementioned objection, he says: This new objection seems to arise from a misunderstanding concerning the nature of sentential connectives (and thus to be somehow related to that previously discussed). The author of the objection does not seem to realize that the phrase “if, and only if” (in opposition to such phrases as “are equivalent” or “is equivalent to”) expresses no relation between sentences at all since it does not combine names of sentences. (Tarski, 1944, p. 358)

Thus, he correctly claims, ‘if and only if’ expresses no relation between sentences or their names. Tarski goes on to claim that ‘the whole argument is based upon an obvious confusion between sentences and their names’ (loc. cit.). But it is not the confusion between sentences and their names that makes his definition difficult to grasp as it is stated; it is Tarski’s insistence that we are concerned with languages and linguistic items. Tarski’s approach is based on a subtle fallacy of equivocation produced by the merging of sentences with their assertion. (CT) is an unmissable clause that any correct definition of truth has to include as a consequence because it makes explicit the right connection between the use of the truth predicate and the practice of assertion. But what is required to comply with Tarski’s nominalist project, and to establish semantics as a scientific enterprise, is to get rid of such ‘confusing’ notions as propositions and assertion. Unfortunately, you cannot have your

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cake and eat it too. That (CT) has to be understood in terms of assertion is something that Tarski explicitly affirms when he deals with the scope of his proposal. By way of explaining why his proposal does not give the assertion conditions of sentences, he says: In fact, the semantic definition of truth implies nothing regarding the conditions under which a sentence like (1),

(1) Snow is white, can be asserted. It implies only that, whenever we assert or reject this sentence, we must be ready to assert or reject the correlated sentence (2), (2) The sentence ‘snow is white’ is true. (Tarski op. cit., p. 361, my emphasis) There is nothing else to be added. To conclude this section, a brief comment on two minor issues that recur again and again in relation to Tarski’s proposal will be in order. The first relates to the ‘extension’ of the truth predicate; the second to the status, necessary or contingent, of (CT). As I have mentioned at the beginning of this chapter, truth for Tarski is a first-level predicable whose arguments are sentences. In the classical positivist style of definition, the definition of a predicate should sort those items for which the predicate holds from those for which it does not. A definition of sentencehood in English would proceed by offering a grammatical criterion that sets apart English sentences from mere combinations of English words. Presented with a particular item, any grammatical theory should be able to determine whether it is an English sentence or not, even if some cases might be borderline. Likewise with other predicates of sentences. A sentence can be described as short or long, clumsy or inspired, French or Russian. Nevertheless, sentences cannot be ‘described’ as true or false, since there are no sentential features whose presence or absence could account for that predicates’ application. No test run on sentences will determine whether they are true or false. In §2 of (Tarski, 1944, p. 342), labelled ‘The extension of the term “true”’, Tarski touches upon the issue of the bearers of truth and decides in favour of sentences, and against propositions without giving any particular reason. He says: [A]s regards the term “proposition”, its meaning is notoriously a subject of lengthy disputations by various philosophers and logicians, and it seems never to have been quite clear and unambiguous. For several reasons it appears most convenient to apply the term “true” to sentences, and we shall follow this course. (Tarski, loc. cit.)

Given the logico-semantic category of truth, as it is reflected in our use of the notion, asking for the ‘extension’ of this term hardy makes any sense. Truth does not classify objects in the world. The only reasonable approach to it is indirect and does not merely consider sentences as linguistic items but, instead, them together with the contents they express and speakers’ attitudes towards them. In other words, we attribute truth to those contents that we are willing to assert and, derivatively, to the sentences that express them. And this is exactly what (CT) states. Sentences as

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bearers only add an extra layer—an idle one—to the answer. And looking at the informal explanations and defence that Tarski gave to his own position, it is clear that, beyond his technical proposal, he had the right insights about the connection between truth and assertion. A further objection that is sometimes raised against Tarski’s view deems (CT) as contingent, on the basis that ‘Snow is white’ might mean Schnee is weiss or La nieve es blanca. This objection and the standard answers given to it rest again on a confusion between sentences, their linguistic meanings, and the contents they produce in assertion. This is how Davidson formulates the objection, which he attributes to Putnam: Thus, according to Putnam, a sentence like ‘“Schnee ist weiss” is true (in German) if and only if snow is white’ ought to be a substantive truth about German, but if for the predicate ‘s is true in (German)’ we substitute a predicate defined in Tarski’s style, the apparent substantive truth becomes a truth of logic. (Davidson, 1990, p. 288)

If (CT) only involves sentences and their names, then (CT) is analytically true, or a truth of logic. And yet, as Putnam claims, (CT*) would say something substantive about German, (CT*) ‘Schnee ist weiss’ is true (in German) if and only if snow is white. What this says is that by using an instance of ‘Schnee ist weiss’, German speakers mean that snow is white. In other words, German speakers mean by ‘Schnee ist weiss’ the same proposition that English speakers express by the sentence ‘Snow is white’. In any case, the level of content, over and above the linguistic level, is unavoidable. In sum, Tarski’s technical proposal, designed to avoid semantic notions, and his choice of sentences as the bearers of truth do not work unless assertions and their contents are added to the picture. Tarski’s own words betrayed his project by showing that, in relation to truth, his position does not differ from that of any other competent speaker. In the next section, Frege’s view will be surveyed and contrasted with Tarski’s.

9.5 Frege on Truth and Judgeable Contents Frege’s approach to truth is a consequence of his semantics. Truth applies to judgeable contents, i.e. those contents that can occur as arguments of the judgement stroke. Following Frege’s Principle of Inferential Individuation, as suggested in the Begriffsschrift, §3, two different sentences can express the same judgeable content if they have the same consequences when added to the same set of premises. There is no reason why sentences sharing their contents cannot belong to different languages, which solves the issue with translation that afflicted Tarski’s approach.

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There is an essential connection between truth and assertion, which Frege explains as follows: The goal of scientific endeavour is truth. Inwardly to recognize something as true is to make a judgement, and to give expression to this judgement is to make an assertion. (Frege, 1879–1891, p. 2)

Truth is thus displayed in assertions, which are the expression of judgements. As the Principle of Assertion requires, assertion is the primitive discursive act. It is with assertion that logic and semantics begin. Frege marks this priority of assertion by his introduction of the judgement stroke at the very beginning of his first work, in §2 of the Begriffsschrift. Assertions put forward propositions, which are the basic bricks of logic, as (PPP) holds; and only when propositions are available, can concepts, i.e. functions, and objects be identified. Functions are reached at by the analysis of propositional contents (Frege, 1879, §9; Frege, 1891, pp. 140–1; Frege, 1892a, p. 162). In the expression of complete judgeable contents some parts can be substituted by some others. Sometimes we obtain different thoughts; sometimes the thought remains the same, depending on the senses of the swapped parts. Removing singular terms from sentences or formulae leaves functional expressions, which could be saturated, i.e. completed, by other singular terms. These functions are predicated of the objects named by the singular terms, and—in Tarskian terms— these objects satisfy the function if the resulting sentence is true. Truth (as well as assertion) thus precedes satisfaction in the logical order. Unlike the converse procedure of trying to reach truth from satisfaction, reaching satisfaction from truth is unproblematic. Contrasting Frege’s and Tarski’s views of truth is probably the best way of testing the profound fissure that separates their assumptions about language and logic. The received view of logic takes Frege and Tarski to belong to the same tradition that shaped the discipline in the past century. This view is repeated again and again, even though a closer look at the work of these two logicians would easily reveal otherwise. Here is an example: The latter alternative [that the laws of logic should explicitly employ the notion of truth], indeed, is just what one should expect, insofar as one thinks of Frege as the primary early impetus in the evolution of that conception of logic which received its most decisive clarifications in the work of such figures as Gödel and Tarski. (Kemp, 1995, p. 31)

I have given reasons to reject the received view. Because they were both competent speakers, Frege and Tarski shared their pre-theoretical intuitions about how truth works in everyday communication, and also about what we do when we draw inferences. Nevertheless, the theoretical apparatuses that they respectively implement differ significantly. The contrasting ways in which they approach natural language and the degree of knowledge of how it works that each shows are partly responsible for the differences in their conceptions of logic and their analyses of truth. Frege’s comprehension of language greatly outmatched the knowledge of any of his contemporary logicians, and of many of his followers, and this made it possible for him to detect phenomena that philosophers of language re-discovered only many decades after his death.

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There are a few claims that serve to condense Frege’s take on truth. The first is that truth is primitive, sui generis, and undefinable (Frege, 1918–1919, p. 353). The second is that truth is not a property. The third is that the truth predicate does not contribute to the thought (Frege, 1915, pp. 250–1). The fourth is that truth is connected with assertion, i.e. that it is a force marker (Frege, 1879–1891, p. 2; Frege, 1918–1919, p. 356; although see Chap. 10). These claims are, as I see it, completely right about truth, even though not all interpretations given to them do justice to Frege’s position or understand the role of truth correctly. I will comment on them in what follows. The common insight that connects all of them is that two sentences such as (10) and (11), (10) 5 is a prime number. (11) The thought that 5 is a prime number is true. express the same thought,1 i.e. 5 is a prime number. The initial distinction that supports this insight is between sentences and the thoughts that they help to express. It is these thoughts that are the bearers of truth; sentences are merely the vehicles used to carry them. The depth allowed by the distinction between these two levels—the level of the linguistic expression and the level of the content conveyed—is essential to any account of truth that aims to pinpoint its meaning correctly, as I will explain in Chap. 10. It is also the source of the correspondentist intuition. A failure to recognise this distinction, together with the misidentification of the truth-bearers, precludes any appropriate connection between the notion defined on these incorrect assumptions and the notion that is actually used in science and everyday life. That sentences such as (10) and (11) express the same thought has led most authors to draw the mistaken conclusion that Frege endorsed a redundancy theory. The term ‘redundancy’ is misleading. As I have argued elsewhere (Frápolli, 2013, chapter six), ‘redundancy’ conveys the impression that the notion so characterised has no role to perform, and that it can be eliminated from a linguistic system without loss. This impression is false concerning Frege’s view, and it is false concerning the meaning of truth, even if the claim that ‘It is true that p’ and ‘p’ express the same thought is a true one. The equivalence (12), (12) It is true that p if, and only if p. shows an essential feature of truth that is based on an essential insight about language and meaning, i.e. that not every meaningful expression has the role of  Heck and May have contested the identity of sense in (10) and (11), but this only shows their failure to understand the inferential individuation of judgeable contents (see Heck & May, 2020, p. 3, n. 6, and p. 4). 1

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contributing a component to the thought. This is a pragmatist insight that reinforces semantic pluralism, an insight whose defence Frege did not explicitly undertake, but which is shown in how he explains the meanings of those notions that are essential to his project, i.e. truth and logical notions (see Chap. 4). In ‘My Basic Logical Insights’, Frege explains: Knowledge of the sense of the word ‘salt’ is required for an understanding of the sentence [‘sea-water is salt’], since it makes an essential contribution to the thought […]. With the word ‘true’ the matter is quite different. If I attach it to the words ‘that sea-water is salt’ as a predicate, I likewise form a sentence that expresses a thought. For the same reason as before I put this also in the dependent form ‘that it is true that sea-water is salt’. The thought expressed in those words coincides with the sense of the sentence ‘that sea-water is salt’. So the sense of the word ‘true’ is such that it does not make any essential contribution to the thought […]. This may lead us to think that the word ‘true’ has no sense at all. But in that case a sentence in which ‘true’ occurred as a predicate would have no sense either. All one can say is: the word ‘true’ has a sense that contributes nothing to the sense of the whole sentence in which it occurs as a predicate. (Frege, 1915, pp. 251–2, my emphasis)

The predicables ‘is salt’ and ‘is true’, both having sense, work in different ways (see Chap. 2, §2). Frege’s view of meaning (in a broad sense) is nuanced enough as to acknowledge the diverse ways in which terms can be meaningful. The contrast between ‘salt’ and ‘true’ is an example. Both ‘is salt’ and ‘is true’ are predicables. Yet ‘is salt’ expresses a property, while ‘is true’ does not. But what it is to be a property? ‘Property’, unlike ‘concept’, ‘function’, ‘proper name’ and ‘object’, is not a technical term of either Frege’s semantics, his logic, or his ontology. It is sometimes used as if it were synonymous with ‘concept’, but this is not completely clear. Two ways of characterising it, both with some support in Frege’s work, immediately present themselves. First, that properties are what predicates express. According to this option, truth (and existence, on the same grounds) would be a property. Second, properties are features of objects and concepts that have representatives in thoughts. According to this option, neither truth nor existence would be properties. The term ‘property’ paradigmatically occurs in Frege’s discussion about the status of numbers. Frege insists that numbers are not properties of external things, since they are not characteristic of concepts. They are properties of some concepts, i.e. those properties that indicate how many objects fall under them. In The Foundations of Arithmetic, he says By properties which are asserted of a concept I naturally do not mean the characteristics which make up the concept. These latter are properties of the things which fall under the concept, not of the concept. Thus ‘rectangular’ is not a property of the concept ‘rectangular triangle’; but the proposition that there exists no rectangular equilateral rectilinear triangle does state a property of the concept ‘rectangular equilateral rectilinear triangle’; it assigns to it the number nought. (Frege, 1884, §53)

The same claims could have been made using ‘concept’ instead of ‘property’. Existence and number are concepts whose arguments are concepts (see also Frege, 1892b, p. 188). They are higher-level concepts or, alternatively, higher-level functions. By the same argument, he might have said that truth is a concept that applies to thoughts and which indicates that the thought is asserted. But Truth does not

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contribute a component to the thought. Whereas the sense of a name is its contribution to the thought (Frege, 1893, §32), this does not extend to predicates, which can be senseful without making any such contribution. The view of meaning that allows some terms to be meaningful without being an identifiable part of thoughts, of what is said, is what I have called ‘expressivism’ in previous chapters: an alternative to representationalism which comes along with pragmatist approaches to meaning. Connected with the rejection of truth as a property is Frege’s claim that truth is undefinable or primitive. As happened with our previous discussion of properties, the scope of the claim depends on what counts as a definition. If by ‘definition’ we mean an identity in which a particular notion is characterised in terms of necessary and sufficient conditions for its application—i.e. an identity that eventually allows for the elimination of the definiendum—then truth is not definable. In ‘On Concept and Object’, Frege explains that only composite notions can be properly defined. In this case, the issue is concepts, but his words could be applied to truth without change: Kerry contests what he calls my definition of ‘concept’. I would remark, in the first place, that my explanation is not meant as a proper definition. One cannot require that everything shall be defined, any more than one can require that a chemist shall decompose every substance. What is simple cannot be decomposed, and what is logically simple cannot have a proper definition. Now something logically simple is no more given to us at the outset than most of the chemical elements are; it is reached only by means of scientific work. (Frege, 1892b, p. 182)

And the same could have been said be said of the notion of number. If only in the context of a sentence do words have meaning, which is (PCont), then only contextual definitions are possible. For numbers, this is the main thesis of The Foundations of Arithmetic. For truth, the clue is given at the beginning of ‘Thoughts’ (1918–1919): Just as ‘beautiful’ points the ways for aesthetics and ‘good’ for ethics, so do words like ‘true’ for logic. All sciences have truth as their goal; but logic is also concerned with it in a quite different way: logic has much the same relation to truth as physics has to weight or heat. To discover truths is the task of all science; it falls to logic to discern the laws of truth. (Frege, 1918–1919, p. 351)

And a few lines later, he continues: From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring. (Frege loc. cit.)

Moore had a similar intuition about truth (Baldwin, 1997).Truth, like good, is, for Moore, a simple indefinable concept (see Moore, 1899, p. 177; Moore, 1903, §6). In this sense, ‘true’ and ‘good’ are like ‘yellow’ or ‘red’, but only in this sense. The comparison between these concepts and colours need not go any further. Trivial as it might appear, the nature of good as a simple notion has substantive philosophical consequences. Among them, that any statement in which ‘good’ occurs is synthetic, and that we are not forced to accept claims such as ‘“Pleasure is the only good” or that “The good is the desired”’ (Moore loc. cit.), on the basis of any alleged definition of ‘good’. Three decades after the publication of Moore’s work, Stevenson made a similar remark. He rejected so-called ‘interest theories’—those that place

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the meaning of good in some attitude of the subject or of a particular group— because any proposal should leave room for disagreement. In addition, he ruled out those theories according to which the verification of what is good could be done ‘solely by use of the scientific method’ (Stevenson, 1937, p. 16). The mention of the scientific method explains the core of Stevenson’s view: that the presence of good cannot be detected by the senses. Was also Moore’s view. And truth is in the same situation. Formulated in simpler terms, Frege’s, Moore’s and Stevenson’s intuition comes down to the thesis that these notions are not properties of objects, with the emphasis on property or object depending on the characterisations and views proposed. In less ambiguous terms, this means that neither truth nor good are first-level predicables (see Frápolli, 2019, pp.  97ff.). Their arguments are concepts or else thoughts or propositions, which in Frege’s and Moore’s views are complex objects with concepts as ingredients. The claim shared by Moore and Frege, that truth and good are indefinable, must be understood as a rejection of any reductionist view. Faced with the dilemma of reducing these concepts to first-order properties or else leaving them undefined, they both took the second option—the only reasonable one. But these are not the only possibilities available and, as already mentioned, ‘definition’ is not only analysing away individual notions in terms of others. Not everything can be explicitly defined, but this does not mean that notions that are not definable in an explicit definition are immediately given to us. Scientific and philosophical work is needed to understand even the simplest notions in logic and semantics. Debates over what counts as a definition are usually futile. I would be happy with the claim that truth cannot be defined, in the classical sense of ‘definition’, but not with the claim that truth is sui generis (see Chap. 10), nor with the claim that no characterisation of it is possible that completely explains its use in context. On this point, Moore and Frege would agree. That truth is not a property, or that it does not contribute to the thought, are intuitions that Frege embraced from the beginning. The judgement stroke in the Begriffsschrift represents an idle predicate that restores sentencehood to descriptions whose content is fully propositional. The violent death of Archimedes cannot play the role of a premise. It is merely judgeable content. To be in possession of its logical properties it has to be judged, i.e. asserted, and then expressed by means of a sentence such as: ‘The violent death of Archimedes is a fact’. The dummy predicate ‘is a fact’ does not add anything to the asserted content. Nevertheless, it adds something essential; it adds the assertive force and, at the linguistic level, the status of a sentence. That thee judgement stroke is Frege’s truth predicate is quite straightforward. In ‘Thoughts’, he makes the connection between facts and truth explicit: ‘Facts, facts, facts’ cries the scientist if he wants to bring home the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. (Frege, 1918–1919, p. 368)

Facts are thoughts whose truth we acknowledge in a judgement. Instead of ‘facts’ we may say ‘truths’. Science deals with truths, which are just the propositions that we endorse. The similarity between the roles performed in language by the abstract substantives ‘fact’ and ‘truth’, and by the dummy predicates ‘is a fact’ and ‘is true’, even with their stylistic differences (see Frápolli, 2013, chapter two), is the source

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of the so-called identity theory of truth (Hornsby, 1997; Dodd, 2007; see Frápolli, 2013, pp. 31–9), which is also attributed to Moore (Baldwin op. cit., p. 3) and Frege. It is also a powerful source for the correspondentist intuition that matches truths, allegedly at the level of thought or language, with facts, allegedly at the level of reality. Nevertheless, the connection between truths and facts, or between ‘is a fact’ and ‘is true’, is a grammatical connection, of logical grammar in Wittgenstein’s sense, and not a hint of any deep or mysterious metaphysical doctrine. The fourth issue mentioned at the beginning of this section—that truth has a principled connection with assertion—has also produced a great deal of confusion, even in the interpretation of Frege’s view of this matter. The actual connection is completely shown by the function of the judgement stroke. To be logically effective, a propositional content has to be asserted. In Frege’s system this is also a grammatical issue: non-asserted judgeable contents do not have the right status to act as premises. Only when they are asserted—i.e. when they are put forward as true, or alternatively, when an agent takes responsibility for them—are they apt to be logical bearers. Examples (10) and (11) above, and all Fregean explanations of this matter, show that truth is not necessary in order to make an assertion. Assertions are made by the uttering of sentences under certain rules. Truth, in the case of the complete expression of its argument (this remark will be explained in Chap. 10), is only needed when the propositional content does not occur in language displaying its fully propositional status, as in ‘the violent death of Archimedes’. In cases such as (11), the expression of truth is semantically redundant, even if it might be pragmatically required. The expressive role of truth means that one of its functions consists in saying explicitly what the agent is doing; in this case, that the agent is engaged in an act of assertion. The essential connection between truth and assertion has traditionally been contested by appealing to what happens when truth is used in fictional contexts, as on the stage in the mouths of actors. Frege’s view on this point is that truth and assertion go hand in hand, even though assertion is the grounding level. If no assertion is being performed, as it happens with actors on the stage, then the use of truth terms does not confer assertive force on the act. And this is what one should expect of an expressivist account. But this point will be discussed at some length in Chap. 10.

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Chapter 10

Truth Ascriptions as Prosentences: Further Lessons of the Principle of Propositional Priority

Abstract  Truth is not one of a kind. Philosophers have widely assumed that truth is mysterious, intractable, troublesome, or else, trivial and redundant. I argue here that none of these alternatives is true. The only sense in which truth is unique is in the treatment that philosophers give to it. On the basis of the same corpus of evidence, we make assumptions and draw conclusions about truth that we do not draw about any other notion of a similar kind. Truth ascriptions work in natural languages as propositional variables. They are prosentences. To them applies all the knowledge that linguists and philosophers of language have gathered about the general class of proforms and the specific class of indexicals. The philosophy of language has the tools to explain the meaning of truth in all its depth. The air of mystery that has surrounded this notion is due in part to the Liar paradox, and in part to a lack of understanding of the variety and plurality of ways in which expressions can be meaningful. The Liar paradox is easy to explain. And a pluralistic approach to meaning that distinguishes between linguistic meaning and the content expressed can offer a correct account of this notion that answers philosophers’ doubts. At the end of this chapter, I deal with the allegedly puzzling connection between truth and assertion. I argue that truth is a reliable mark of assertive actions, and that the apparent counterexamples that come from special contexts, such as fiction or conditionals, can be explained by the appropriate theory of meaning. I also give reasons to doubt that Frege detached truth from assertion in those contexts. Truth, like logical constants, needs for its analysis a background shaped by (PA), (PPP), (PGS), and (PII). It needs an approach that pays attention to the role of agents in the use of expressions and the difference in semantics between first-level and higher-level predicables and concepts. To understand truth, we need to adopt a theoretical approach like the one I have defended in this book, or another one with similar resources. Otherwise, truth cannot be released from the confusing playground that still makes it refractory to analysis. Keywords  Affordance · Ascription · Assertion · Blind · Exhibitive · Expressive [expressivism] · Implication · Liar paradox · Proform · Prosentence

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10.1 Why Truth Is So Elusive Truth is a never-ending source of puzzlement for philosophers. Yet nevertheless, in scientific and ordinary communicative exchanges, truth is widely used without producing any particular distortion. Philosophers have a general feeling that truth is peculiar, sui generis (Frege, 1918–1919, p. 354), or that there is something mysterious about it. There is not. Truth is only special in the treatment that philosophers give it, in two different senses. The first one relates to the philosophical impact granted to some semantic features of truth. Concerning truth, we are willing to make assumptions and draw conclusions that we would not dare to make or draw about any other notions, even on the basis of the same piece of evidence. The second is that, in their discussion of truth, philosophers seem to dwell in a bubble, harking back to the thirties of the past century, unaffected by the findings of linguistics and the philosophy of language of subsequent decades. My aim in this chapter will be to explain how truth works and how, notwithstanding the simplicity of its usage, understanding it requires putting to work some sophisticated developments in linguistics and philosophy of language that have only recently become available. It is difficult to explain to laypeople what the philosophical problem of truth is. It is difficult to explain why truth is a problem at all. The fact is that truth is mastered by any competent speaker and smoothly applied in science and in everyday life. The challenge is then to explain why truth is such a difficult topic from a philosophical perspective. ‘A theory of truth’, Austin said, ‘is a series of truisms’ (Austin, 1950, p.  4). Ramsey declared that, concerning truth, there is only a ‘linguistic muddle’ (Ramsey, 1927, p. 4). Brandom, in turn, explained ‘why truth is not important in philosophy’ (Brandom, 2009). By contrast, their positive proposals are not devoid of technical complexity. An outstanding example of this tension is Tarski’s semantic proposal, with its truistic Convention T, x is true iff p (where ‘x’ is a name of p), and its apparatus of infinite sequences, propositional functions, a hierarchy of languages and recursion. The Liar paradox seems to be the only issue that philosophers have found to justify all the hard work they have invested in truth. The Liar paradox, nevertheless, dissolves as soon as one understands that sentences are not the primary truth-bearers and that some sentences work as propositional variables. In fact, the Liar paradox provides a knock-down refutation of sentences as truth-bearers. Sentences are not even secondary truth-bearers, although they are sometimes said to be true whenever their contents are true. As Truth-Conditional Pragmatics explains, a sentence’s linguistic meaning provides, at most, a skeleton that, outside a context, falls short of being semantically evaluable (Romero & Soria, 2019, p. 52). The puzzling aspect of the Liar paradox is that Liar sentences do not say anything. This is the bad news. The good news is that nobody engaged in actual exchanges is puzzled by Liar-like sentences. Saint Paul, in the Epistle to Titus, did not show any perplexity when he said: ‘One of the Cretans, a prophet of their own, said “Cretans are always liars”’

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(Titus 1:9). He was not puzzled because in ordinary communications the mechanism of the contextual restriction of quantifiers is always at play (for a survey, see Stanley & Szabó, 2000). What philosophers sometimes present as difficulties inherent to the notion of truth—the Liar paradox included—are either cases of misfires in Austin’s sense (Austin, 1962, p. 16) or else category mistakes in Ryle’s sense (Ryle, 1938, Magidor, 2013, p. 9). Let us consider some examples. Confronted with a speech act in which sentence (1) is uttered, (1) What she said is true. the natural reaction is to ask what she had said. That something has been said to which (1) refers is a presupposition of the performed act. If the presupposition does not hold—if the description in the grammatical subject is not satisfied—then (1) does not say anything and the speech act misfires. The particular kind of misfire applicable to this situation is what Austin calls ‘misexecutions’ (Austin, op.  cit., p.  17): a failure of the procedure that precludes the act from taking place. If we agree that, by default, an utterance of (1)—because the sentence is declarative— should have had the significance of an assertion, then the failure of this presupposition impedes the assertion from being performed. There is no assertion because there is no judgeable content, no proposition, to be asserted. Exactly the same situation occurs with the Liar paradox, either in its simplest versions (2) and (3), or in more sophisticated ones, such as the ‘dialogue’ in (4): (2) This sentence is false. (3) I’m lying. (4) A: What B says is true. B: What A says is false. (2) also commits a category mistake: the kind of mistake that happens when the objects referred to by the expressions in the subject position do not belong to the domain to which the concepts in the predicate positions apply. Misfires, misexecutions, and category mistakes are frequent in actual communication. And nevertheless, the wide range of notions that at some time or another produce dysfunctions are not rejected as inherently faulty. Except in the case of truth. Concerning the alleged distortions that sentences (2), (3), and those in (4) produce, we might appeal to popular knowledge and say ‘Bad workmen always blame their tools’. Linguistic systems, and the subsystems in them, are communicative tools, with a function, a purpose, and a scope. It is the responsibility of speakers to

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meet the pragmatic conditions for their correct usage, i.e. those conditions that guarantee successful communication.

10.2 The Pragmatist Strategy: Truth Ascriptions and the Fregean Principle of Context Throughout this book, I have maintained that only propositions stand in logical relations, which are basically implication and incompatibility. Concepts are subpropositional items that are only derivatively related by those relations. The universally praised characterisation of number that Frege presented in The Foundations of Arithmetic was only possible because he stuck to (PCont) and sought the contribution of numerical expressions in numerical equations. As the history of mathematics shows, focusing on numerals, as terms, or on numbers, as objects, makes the task of understanding their role very hard. Frege’s (PCont) applies generally, to any kind of term, but its import is better appreciated in the definition of higher-level notions which by their very nature do not represent properties of objects. Philosophers’ favourites—knowledge, good, truth—belong to this category. The pragmatist strategy for dealing with higher-level notions replaces the traditional method of asking for the necessary and sufficient conditions of their application by instead looking at what agents intend to do when they use them. It enquires, for instance, as to what agents do in knowledge attributions, as in (7) and (8), (7) Boris knew that economic recovery would require courageous policies. (8) Elon knows how to make it work, as well as in moral valuations, as in (9) and (10), (9) Racial discrimination is bad. (10) Promoting equality among people is good. and in truth ascriptions, as in (1) above. The strategy of defining isolated words and concepts—the ‘analytic mode’ (Belnap, 1962, p. 130)—has encountered endless difficulties, not only in accounting for the meaning of normative concepts but also in defining theoretical terms in science. The strategy of explaining the practice of applying them—the pragmatist strategy—enhances, by contrast, our chances of completing this task. Numbers and truth are two outstanding examples.

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Pragmatists begin with practices and conclude with them. This is what ‘pragmatism’ means: that agents’ actions are what trigger the research process and are also the tribunal in front of which the results must be validated. It has been an assumption shared by all pragmatisms that only complete sentences can serve to make a move in a linguistic game. Moves are not performed by sentences, though, but by uses of them. It is the assertion of a proposition, usually expressed by a sentence,1 that has the significance of a move. Besides, assertive force can only be attached to judgeable contents, and this is why Frege considered them his only concern. What being a move means admits of different explanations, ranging from Stalnaker’s proposal that assertions modify background assumptions (Stalnaker, 1999, p. 78) to Sellars’s and Brandom’s view that they alter the score of the commitments and entitlements of those involved in the linguistic game. In any case, the metaphor is clear: assertions make the conversation go. How all this applies to the analysis of truth is straightforward. Truth cannot be properly understood without paying attention to the uses that speakers make of truth ascriptions. The expression ‘truth ascription’ is, like many others when we deal with language (see Chaps. 1 and 8), ambiguous between a semantic and a pragmatic reading. The pragmatic reading refers to the action of ascribing truth to a content. The semantic reading refers to a specific kind of sentence that is used in ascriptions in the pragmatic sense. This ambiguity suits us well, since, with the exception of some discussions about prosentences, the analysis of truth centres on what we do by the utterance of these sentences, and thus both levels, pragmatic and semantic, are relevant. Truth ascriptions are to truth what equations are to numbers. The following examples, (11), (12), (13), and (14), are truth ascriptions too: (11) (12) (13) (14)

Everything Penrose says is true. String theory is true. It is true that Brexit affects the European economy. ‘Snow is white’ is a true sentence.

As (1) and (11)–(14) show, truth ascriptions adopt different forms. For the purpose of understanding how they work, I classified them into blind and exhibitive truth ascriptions (Frápolli, 2013, pp. 57ff.). Be they blind or exhibitive, all truth ascriptions presuppose an assertion, actual or virtual, which produces the content to which they ascribe truth. They are, thus, second-level acts. In exhibitive truth ascriptions, the content of the basic assertive act is explicitly represented, allowing the content of the truth ascription to be easily retrieved. (13) and (14) are simple exhibitive

 Sometimes assertions can be made using subsentential expressions (see Chap. 8).

1

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cases. (12) is also exhibitive but its content cannot be explicitly given using a list. Nevertheless, this fact does not affect the meaning or role of the truth ascription; it only says something about what kind of artefact theories are. Even if a representation of their contents is explicit in some truth ascriptions, exhibitive ascriptions are still second-level acts. In (14), two distinguishable acts are presented together. The first act is the simple assertion (15): (15) Snow is white. The second act is the ascription of truth to its content, (16): (16) This is true. In other words, in exhibitive truth ascriptions, the speaker performs two actions, the first-level assertion that fixes the content of the ascription, and the second-level ascription, by which the speaker explicitly recognises the truth of the content. Philosophers abuse exhibitive truth ascriptions when they discuss the meaning of truth. If only exhibitive truth ascriptions are taken into account, then the complex structure of ascriptions as sentences and their second-level nature as acts become difficult to perceive. (1) and (11) are blind truth ascriptions. The content of (1) depends on what she said in the assertion to which (1) refers. Even a superficial consideration of the standard use of (1) would reveal its second-level nature. For (1), (1) What she said is true. isolated from any context, does not give any substantive information. Sure, we learn from (1) that some female individual has said something that the person who utters (1) endorses. But we have no means of identifying this content that the utterer of (1) endorses. An isolated utterance of (1) does not add anything substantial to the context or modify the operative set of background assumptions. At least in this sense, truth ascriptions are not ordinary declarative sentences. Unlike what happens with (1), the following examples (17), (18) and (19), (17) Brexit affects people’s lives. (18) Elon Musk is a controversial person. (19) Global warming is as threatening as pandemics, give information about the beliefs of the person who utters them. Even in sentences such as (17)–(19), the levels of linguistic meaning and content can be distinguished. What is said by them, their contents, is the result of enriching their linguistic meanings by adding information contextually retrieved. But their linguistic meanings are still recognisable in the contents expressed.

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Consider now what happens with blind truth ascriptions. If (1) refers to the act in which (17) was produced, then the content of the truth ascription will be Brexit affects people’s lives; if the act referred to by (1) is one in which (18) is uttered, then the content of (1) will be Elon Musk is a controversial person. Finally, if (19) is the first-level act to which (1) refers, then the utterer of (1) will be saying global warming is as threatening as pandemics. And none of these contents bears any relation to the linguistic meaning of (1). The analysis of (11) is similar, but in this case the utterer of the first-level act is identified, and its content might be unknowable if it is intended as a profession of faith regarding everything that Penrose might say. What we learn when we consider (1) in isolation is its linguistic meaning: its ‘character’ in Kaplan’s terminology (Kaplan, 1979, p.  505). The character of an expression is a function from contexts to contents (Kaplan, op.  cit., pp.  505–6), which Kaplan also calls ‘descriptive meaning’ (Kaplan, op. cit., p. 497). The word ‘content’ is ambiguous, and it is sometimes uncertain in which sense Kaplan uses it. It is ambiguous because it sometimes refers to the semantic value of a linguistic item, and sometimes to the assertoric content expressed by the use of a declarative sentence. The difference that I want to stress at this point is the difference between character or linguistic meaning, which applies to sentences as linguistic devices, and the assertoric contents that speakers put forward by means of them. Since (1) can express different assertoric contents depending on the primary act to which it is attached, its linguistic meaning does not determine the content expressed. This is a central feature that Kaplan recognises in referential expressions: Some directly referential terms, like proper names, may have no semantically relevant descriptive meaning, or at least none that is specific: that distinguishes one such term from another. Others, like the indexicals, may have a limited kind of specific descriptive meaning relevant to the features of a context of use. Still others, like ‘dthat’ terms […] may be associated with full-blown Fregean senses used to fix the referent. But in any case, the descriptive meaning of a directly referential term is no part of the propositional content. (Kaplan op. cit., p. 497)

Not only directly referential terms possess this feature. The linguistic meaning of anaphoric expressions is also irrelevant to their content. Blind truth ascriptions work as anaphoric expressions that inherit their contents from the primary acts to which they are attached. Sometimes, self-reference has been blamed as the source of the paradox, or even as the source of all set-theoretical and semantic paradoxes. (Russell, 1908) is the classical locus. But in fact, self-reference as such does not produce such distortions. There are many examples of perfectly sound self-referential sentences which can be used and have been used safely. Gödel sentences, for instance, designed to prove the incompleteness of arithmetic, are self-referential (Franzén, 2005, pp. 34–38, Nagel & Newman, 1958). Less complex self-referential sentences are (20), (21), and (22): (20) This is an English sentence. (21) This is a French sentence. (22) This sentence has less than one hundred words.

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Even safe cases such as (20)–(22) hide a double layer. Their complete expansion is (23)–(25): (23) ‘This is an English sentence’ is an English sentence. (24) ‘This is a French sentence’ is a French sentence. (25) ‘This sentence has less than one hundred words’ has less than one hundred words. (20), (22), (23) and (25) can be used to say something true, whereas what is said by (21) and (24) is false. But none of these sentences is paradoxical. Because ‘being an English sentence’, ‘being a French sentence’, and ‘having less than one hundred words’ express properties of sentences, these self-reflexive cases are not risky. Neither are those cases in which two acts are performed together. As we have seen, exhibitive truth ascriptions can be analysed as assertions of the exhibited content together with its explicit endorsement. Unlike what happens in Liar-like sentences, in exhibitive truth ascriptions one of the layers specifies the content to be assessed. By uttering (12) we endorse string theory, by uttering (13) we assert that Brexit affects the European economy. Even if sentences are not truth-bearers, some sentences—i.e. those that do not include indexicals, that are not ambiguous, etc.—indicate which contents are standardly put forward by uttering them. This is sometimes a blessing since it avoids dysfunctions, but it is also a curse since it makes it easy to forget that it is not sentences that we deem true or false. In Liar sentences, by contrast, there is no indication of how to recover a content, and thus assertion gets blocked. In sum, there are two reasons why Liar sentences are paradoxical. First, the mistaken identification of truth-­bearers that makes self-reference possible, and second, the failure to understand the second-level nature of truth ascriptions. The analysis of the kind of sentence that truth ascriptions exemplify is the topic of the next section.

10.3 Proforms Consider again sentences (1) and (15): (1) What she said is true. (15) Snow is white.

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Both are well-formed English declarative sentences, and both are suitable to be used in assertive acts. Nevertheless, there is a crucial difference between them. Listening to (15), one learns something about the mental state of the person who utters it, about her beliefs. An assertion of (15) modifies the background assumptions of a communicative exchange. Its content is put forward to be used in the context of utterance by the speaker, or by another participant. It might also be accepted by others as settled information and added to their own belief systems, which might require the elimination of some previous beliefs. A speaker who utters (15) implicitly endorses a series of material inferences that follow from it. A hearer who accepts (15) might also modify the collection of inferences to which he lends support. The situation with (1) is very different. What do we learn by listening to (1)? Only metadata. The sole information retrievable from (1) relates to formal aspects of the situations in which it is appropriate to utter it, but nothing that can be taken up and applied in other contexts. The difference between (1) and (15) parallels the difference between pronouns and names. The name ‘Boris Johnson’ refers to an identifiable person; ‘he’, by contrast, does not refer to anyone in particular and can refer to any male individual whatsoever given the appropriate context. One might retort that ‘John’ or ‘Maria’, even if they are proper names, may refer to many different persons too. This is right, but still, the mechanism by which ‘he’ and ‘Maria’ reach their references is different. All actual communicative exchanges happen in context. The participants in a conversation share some information that is necessary for the conversation to proceed. The participants in an ordinary exchange such as (27) (27) A: Is Maria coming? B: No, she went to Reading to visit her son. must be familiar with the reference of ‘Maria’. Knowledge of the name’s reference belongs to the background knowledge that could remain unchanged from one context of utterance to another. If an outsider asked: ‘Who’s Maria?’, the answer would provide an information that transcends this context. Once you know who Maria is, this information lingers, and can be applied everywhere else. Pronouns do not work this way. The reference of ‘he’ is confined to the context of utterance. In anaphoric uses of pronouns (see Branco et al., 2005, for a survey), context (or co-text) can be extended almost indefinitely, but there must always be a connection between the anaphoric head, i.e. the name or description that fixes the reference, and subsequent anaphoric uses of pronouns (and adjectives). (28) exemplifies anaphoric uses of terms:

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(28) Boris Johnson had Covid-19. After his recovery, he warmly thanked the foreign nurses that took care of him. Unfortunately, he did not change his mind about public health services or the essential role that immigrants perform. He remained faithful to his former policies. The nature of pronouns ensures that, out of a context, ‘he’ is not more closely connected to Boris Johnson than it is to Joe Biden, Aristotle, the Dalai Lama or my son. Some adverbs also work like pronouns do; ‘here’ and ‘now’ are classic examples. The adverbs ‘yes’ and ‘no’ are examples too. Generally, the place from which the speaker produces his utterances is referred to by ‘here’. There are exceptions, though. Pointing at a spot on a map, a speaker who utters (29), (29) Here is where I lost track of him. does not refer to the place from which he is talking, but rather to the place he is pointing at. The adverbs ‘yes’ and ‘no’ contextually acquire any propositional content with which they are appropriately connected. (27) above might be answered by a simple ‘yes’, and then the adverb would have the content that Maria is coming, or by a ‘no’, in which case the content would be Maria is not coming. Answering ‘yes’ or ‘no’ to the question (30): (30) Have you finished the book? would mean that I have finished the book, or that I have not finished the book, respectively, and the same happens with any yes-or-no question whatsoever—those questions that Frege called ‘propositional questions’ (Frege, 1918–1919, p. 355). Terms and expressions that work like pronouns and some adverbs do are proforms (Keizer, 2011; Demirci, 2014). These occur in all categories. In English, ‘do’ is a pro-verb, ‘she’ is a pro-noun, ‘here’ is a pro-adverb, ‘yes’ and ‘no’ are prosentences, and ‘so’ is trans-categorical. Some pragmatist approaches to communication, Truth-Conditional Pragmatics and Relevance Theory among them, argue that speakers have a direct, non-­ inferential, access to what is said in a speech act. What is said is not intentionally built up from linguistic meaning and contextually provided information; it is directly expressed and understood (Recanati, 2003, p. 11, p. 17). In fact, the level of linguistic meaning is cognitively and semantically idle (op. cit., p. 14); something to which the speaker does not have conscious access. If this is so, then we might think that, in the explanation of communication, linguistic meanings are dispensable; their interests restricted to linguistics as a theoretical postulate internal to linguistic theory. Some pragmatists have advised so, and, in fact, ‘linguistic meaning’ is not a notion that is easily found in pragmatist writings. This strategy would be mistaken, however. This notion (or some other notion equivalent to it) is crucial to understanding language’s versatility and adaptability, and it is essential to understanding how proforms work. In the philosophy of language, it was not until Kaplan’s seminal work, ‘Demonstratives’ (1979), that we learned in some detail the complex nature of these terms, although some hints about their functioning can be found since the

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beginning of the discipline (see, for instance, Frege, 1918–1919, p. 358). What is crucial to accounting for the functioning of proforms is a double-factor theory of meaning. The nature of proforms cannot be explained with just a single semantic category. In the analysis of proforms, not only the distinction between linguistic meaning and content is crucial. The grammatical category and the semantic content of an expression are related in non-trivial ways. As illustrated by my comment on example (19) above, ‘yes’ and ‘no’ express complete propositions. From a semantic point of view, they work as sentences even if they syntactically belong to the category of adverbs. Thus, ‘yes’ is grammatically an adverb and semantically a prosentence. (1), in turn, is syntactically a sentence and semantically a prosentence. Both ‘yes’ and ‘what she said is true’ are prosentences, as are all truth ascriptions. The difference is that whereas ‘yes’ and ‘no’ are simple prosentences, truth ascriptions are complex. Natural languages present a dramatic shortage of simple prosentences, which practically come down to the aforementioned adverbs ‘yes’ and ‘no’, some uses of the adverb ‘so’, and some uses of the pronouns ‘this’, ‘that’, ‘it’, ‘what’. Not much more than that. Here we have some examples of pronouns in prosentential uses: (31) He said that capitalism is the best economic system. I cannot accept it. (32) A: What did she say? B: She said that the next pandemic is imminent. Meanwhile, example (33) illustrates the prosentential use of the adverb ‘so’: (33) A: The pandemic is already under control. B: I don’t think so. Ramsey was aware of the dysfunction generated by the failure to distinguish between the syntactic status of proforms and their semantic function and connected the meaning of truth with the semantic category of prosentences: As we claim to have defined truth we ought to be able to substitute our definition for the word ‘true’ wherever it occurs. But the difficulty we have mentioned renders this impossible in ordinary language which treats what should really be called pro-sentences as if they were pronouns. The only pro-sentences admitted by ordinary language are ‘yes’ and ‘no’, which are regarded as by themselves expressing a complete sense, whereas ‘that’ and ‘what’ even when functioning as short for sentences always require to be supplied with a verb: this verb is often “is true” and this peculiarity of language gives rise to artificial problems as to the nature of truth, which disappear at once when they are expressed in logical symbolism, in which we can render “what he believed is true” by “if p was what he believed, p”. (Ramsey, 1927/2001, p. 437)

By the ‘artificial problem’, Ramsey refers to the standard criticism addressed against those views that explain the role of truth by resorting to propositional or sentential variables. I have mentioned this objection in Chap. 9, in relation to Tarski’s Convention T. The contents of beliefs are Ramsey’s primary truth-bearers. ‘A belief is true’, Ramsey says, ‘if it is a belief of p, and p’ (Ramsey, 1927/2001, p. 437). The

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difficulty arises concerning the status of the second occurrence of the variable p. The objection that he anticipates takes it to be a nominal variable, i.e. a pronoun. If this were so, then Ramsey’s characterisation would be ill-formed since conjunction requires sentences as arguments and ‘p’, allegedly a nominal variable, would not possess the appropriate grammatical status. At this point, Ramsey insists on reminding us of the difference between pronouns and prosentences. In artificial languages, nominal variables work as natural language pronouns, and propositional variables work as natural language sentential prosentences. Thus, Ramsey’s definition is perfectly well formed, since the variable in the second conjunctive clause is propositional. The objection is, nevertheless, understandable, since natural languages have a tendency to treat (semantic) prosentences as pronouns, as the prosentential uses of ‘it’, ‘that’, ‘what’, etc. show. Here is how he deals with the objection: This definition sounds odd because we do not at first realize that ‘p’ is a variable sentence and so should be regarded as containing a verb; “and p” sounds nonsense because it seems to have no verb and we are apt to supply a verb such as “is true” which would of course make nonsense of our definition by apparently reintroducing what was to be defined. But ‘p’ really contains a verb; for instance, it might be “A is B” and in this case we should end up “and A is B” which can as a matter of ordinary grammar stand well by itself. The same point exactly arises if we take, not the symbol ‘p’, but the relative pronoun which replaces it in ordinary language. Take for example “what he believed was true”. Here what he believed was, of course, something expressed by a sentence containing a verb. But when we represent it by the pronoun ‘what’ the verb which is really contained in the ‘what’ has, as a matter of language, to be supplied by “was true.” (Ramsey, op. cit., pp. 437–8)

Ramsey speaks of ‘variable sentences’, I have used the expression ‘propositional variables’. Nothing really hangs on this terminological disparity. Ramsey is thinking of the kind of syntactic item to be substituted by the variable. I focus instead on the content of the variable. But to tackle the issue raised by ‘what’, and ‘what he said’, and in his example, ‘What he said is true’, my distinction between the grammatical category of an item and its semantic behaviour proves to be useful. ‘What’ is a relative pronoun and ‘what he said’ is a description; both are singular terms from a grammatical point of view. Their status explains why it is so difficult for inattentive speakers to resist the temptation to complete the conjunctive clause in the semi-formal Ramseyian definition with a verb. But both are prosentences from a semantic point of view, since both possess propositional content or, in Ramsey’s terminology, have ‘propositional reference’ (Ramsey, 1927/2001, p. 434). The pressure we feel to complete nominal prosentences such as ‘what’ and ‘what he said’ with ‘is true’ derives from the need to restore grammaticality. The use of ‘is true’ in this kind of context is merely grammatical. The prosentences ‘what’, ‘what he said’ and ‘what he said is true’, in Ramsey’s example, point to the same propositional reference. They differ in syntactic status, though, and because of this, their affordances are different. The extension of this notion, which belongs to ecological psychology (Heras-Escribano, 2020; de Pinedo García, 2020), to the case of truth ascriptions is justified by their structural similarities. Singular terms and sentences open different possibilities for the agents to use them.

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The job that ‘is true’ performs exactly parallels what Frege says of ‘is a fact’ in the Begriffsschrift. The judgeable content of (34), (34) Archimedes died violently. can be represented by a singular term, (35), (35) The violent death of Archimedes. and can be placed as the subject of a sentence whose predicate is the empty predicable ‘is a fact’, as in (36), (36) The violent death of Archimedes is a fact. In sentences in which the grammatical subject has a propositional content, the grammatical predicate has only a syntactic function (Frege, 1879, §3). Frege and Ramsey understood that (34), (35) and (36) share the same content but express it in different formats. This might seem trivial, but some authors have rejected it (see, for instance, Austin, 1950, p. 9). The predicates ‘is a fact’ and ‘is true’ serve to the task of presenting contents with a sentential structure, and thus allow them to be asserted and combine with other sentences in complex ones. This function of the predicates ‘is a fact’ and ‘is true’ is difficult to see if they are analysed in isolation, which usually prompts the bogus problem of identifying the property that they represent. The distinction I have introduced between the grammatical status of an expression and its semantic category helps answer one of the most widespread criticisms against the prosentential account of truth. Ramsey’s definition, and in general all definitions that present the role of truth using propositional variables, the objection goes, have to face the problem of the right interpretation of propositional quantifiers. If the favoured interpretation is objectual, then the definition commits us to spooky entities, i.e. propositions. The alternative is favouring the substitutional account of quantifiers, which would interpret all quantified sentences as metalinguistic. Grover’s version of the prosentential theory (Grover, 1992) chose the substitutional option. In (Frápolli, 2013, pp. 127ff.) I called this criticism ‘the logical objection’. The objection is unwarranted, for several reasons. First, because the debate between objectual and substitutional accounts applies to the interpretation of specific artificial languages. The prosentential account, by contrast, is an attempt to explain the role of truth talk in natural languages. Surely, it is sometimes useful to illustrate features of natural languages using semi-formal enrichments of them, as Ramsey did, but the enrichment does not commit us to any particular interpretation of quantifiers in formal languages. In fact, neither the objectual nor the substitutional interpretations represent how quantifiers work in natural languages. The metalinguistic reading of the substitutional view, on the one hand, is a very bold assumption that makes quantified sentences express metalinguistic information. In the objectual view, on the other hand,

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the values of bound variables are objects in the universe of discourse. This central thesis of Quine’s philosophy of logic, assumed with very little criticism thereafter, confers on quantifiers an ontological import which they lack (see Frápolli, forthcoming). Examples of quantified sentences in which the commitment to spooky entities is unclear, to say the least, are the following: (37) There are many ways of developing a decent academic career. (38) I don’t understand some political reactions to the Covid-19 situation in Spain and Italy. It will take more arguments than those given by Quine and his followers to convince us that, by uttering (37) or (38), a speaker shows her commitment to ways of developing a decent academic career or to political reactions as basic objects of her ontology. The classical argument applies to scientific theories in canonical presentation, i.e. expressed in a first-order language, and requires complex strategies of reformulation in order to adapt sentences such as (37) and (38) to the strictures of a language that only binds nominal variables. The situation is not very different from the tasks of Medieval monks trying to fit every sentential structure into the subject-predicate template. The case of nominal prosentences nevertheless shows that the grammatical category of a term is independent of the semantic category—others would say ‘ontological’ category—of its content. A pronoun can point at a proposition, without this being any evidence that propositions are taken as individuals. The disparity between grammatical and logico-semantic categories is what (PGS) establishes. When formal theories in logic and our linguistic performances clash, the analytic reaction is to blame our practices. The pragmatist reaction is to reconsider formal theories and their background assumptions.

10.4 Pragmatism, Expressivism, and the Priority of the Proposition As the Fregean Principle of Context prescribes, the meaning of ‘true’ has to be sought in the context of a complete sentence. Only when the meaning of the sentence is fixed, is the analysis of its sub-sentential parts viable. The meaning of sentences, in turn, derives from their pragmatic significance. Except for pro-sentences, it is the proposition that a sentence systematically expresses in standard contexts that determines the sentence’s linguistic meaning. To make a similar point, Brandom uses the slogan ‘[s]emantics must answer to pragmatics’, which he explains in the following text: It is specifically propositional contents that determine these pragmatic significances, so it is specifically propositional contents that it is the task of semantic explanatory theories to attribute. Semantic contents corresponding to subsentential expressions are significant only insofar as they contribute to the determination of the sorts of semantic contents expressed by full sentences. (Brandom, 1994, p. 83)

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The consequence of (PCont) and Brandom’s pragmatist explanation for the meaning of truth come down to the realisation that ‘is true’ can only be understood as part of the meaning of sentences of the kind represented by (1), (1) What she said is true. In the logical order of explanation, propositions precede concepts and sentences precede terms. This is what (PPP) and (PCont) state. The meaning of ‘is true’ is, thus, its overall contribution to (1) and similar sentences. A restrictive understanding of ‘overall contribution’ prompts the discussion about redundancy. Is truth redundant? It depends on what is meant by ‘redundant’. I have discussed this topic at length in (Frápolli, 2013, chapter 6). Truth terms—‘true’, ‘truth’, ‘truly’—are meaningful and have a function to perform but, as Frege explained (Frege, 1915, pp. 251–2; see Chap. 9), their sense, in a sense of ‘sense’, does not contribute a new component to what is said. These terms nevertheless do affect the semantic value of the sentences in which they occur. If not contributing a conceptual component to what is said were enough to declare a notion redundant, then many of philosophers’ dearest notions would also be deemed redundant. Logical constants, and alethic and epistemic modalities, among other terms, would be redundant. Here too, the debates about truth are idiosyncratic, since philosophers draw conclusions about truth that they would not accept about other similarly complex notions on the basis of the same evidence. In (39), (40) and (41), (39) She won the lottery and bought a house in Notting Hill. (40) Necessarily, everything is identical to itself. (41) John knows that Denisovans spread throughout Eastern and Central Europe. ‘and’, ‘necessarily’ and ‘knows’ do not conceptually enrich the judgeable contents expressed. Conjunction, as we saw in Chaps. 4 and 6, does not represent an ingredient of alleged state-of-affairs. Alethic modalities indicate the scope of application of propositions, and epistemic modalities express the degree of support that speakers attribute to certain contents, but none of these modifies the judgeable core to be expressed. These higher-level notions are semantically irrelevant, if semantics is restrictedly understood as focusing on the relationship of languages with the world. I have used the feature of semantic irrelevance, as applied to higher-level notions, in (Frápolli, 2019, p. 103) and I mentioned it in relation to logical notions in Chap. 5. In a similar sense, some higher-level notions frequently used in logical theory were understood by Frege to be devoid of logical interest from the beginning of his work: The distinction between categorical, hypothetical, and disjunctive judgements seems to me to have only grammatical significance. The apodictic judgement differs from the assertory in that it suggests the existence of universal judgements from which the proposition can be inferred, while in the case of the assertory one such a suggestion is lacking. By saying that a proposition is necessary I give a hint about the grounds for my judgement. But since this does not affect the conceptual content of the judgement, the form of the apodictic judgement has no significance for us. (Frege, 1879, p. 13, his italics)

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Frege adds truth to the list that includes logical and modal terms. It is worth noting that in this text he uses particularly weak expressions, i.e. ‘suggest’ and ‘hint’, to refer to the relation of these terms with what they signify. To highlight the role of those higher-level notions, we say that the corresponding terms possess expressive meaning, as opposed to the allegedly descriptive meaning possessed by other groups of terms. I mentioned this kind of meaning in Chaps. 2 and 4. Expressive meaning and expressivism do not necessarily make reference to the private and subjective mental lives of speakers voiced by their use of particular terms. In Frege, Ramsey, Wittgenstein, Prior and Ryle, in Truth-Conditional Pragmatics and in Relevance Theory, and in some versions of contemporary expressivism (Brandom, 1994; Charlow, 2015; Frápolli, 2019; Frápolli & Villanueva, 2012; Price, 2019; Starr, 2016), ‘expressivism’ contrasts with ‘representationalism’ or ‘descriptivism’, with no particular reference made to private or subjective areas of the speakers’ activities. ‘Semantic irrelevance’ seems to be a better label for the kind of meaning proper to higher-level terms, since it does not evoke in us any subjectivist picture. Semantic pluralism is a necessary assumption in the analysis of higher-level terms. Nevertheless, the kind of pluralism that helps us to understand the role of truth is not the variety that understands the role of truth as different in different contexts, or when attached to different contents. Pluralism about truth might also be explained by saying that, even if the notion of truth is not ambiguous, it can be exemplified by different properties. This is the core of Lynch’s alethic functionalism (Lynch, 2009). Resorting to properties or, as Lynch also does, to the ‘underlying nature of truth’, adds a metaphysical layer that I do not consider to be either explicative or relevant. A truth ascription such as (1) has a constant meaning, a constant Kaplanian character, and its function is neither to point to one or to many metaphysical properties nor to represent the underlying nature of reality. It is a complex variable with a stable linguistic meaning that is a function from contexts to propositional contents. The ‘plural’ aspect is provided by the potentially infinite number of propositional contents that (1) can be used to assert.

10.5 The Prosentential Approach to Truth Truth ascriptions are proforms. This is the central thesis of the prosentential approach to truth. In the past century, the word ‘prosentence’, with the sense intended here, occurs for the first time in (Ramsey, 1927/2001), in the text quoted above. Ramsey understood truth by analogy with pronouns and propositional variables in artificial languages. A worked-out analysis of the prosentential intuition had to wait almost fifty years until the publication of D. Grover, J.  L. Camp and N. Belnap’s paper of 1975 and C. J. F. Williams’ book of 1976. In 1992, D. Grover published her book A Prosentential Theory of Truth and C. J. F. Williams published Being, Identity and Truth. These books and papers have constituted the corpus of the theory until Brandom’s expressivist analysis in (Brandom, 1994, chapter 5). I have

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defended a pragmatist development of prosententialism, in which I included some recent developments from linguistics and the philosophy of language, in (Frápolli, 2013). C. J. F. Williams (Williams, 1976, 1992, 1995) is the source of my prosententialism, which I have complemented with Ramsey’s general pragmatism. In the classical tripartite approach to the study of language, pragmatics is the foundational level. Semantics and syntax only come after the pragmatic level is secured. In (Frápolli, 2013), I followed the more traditional order of beginning with the syntax of ‘is true’, going through the semantic analysis of prosentences, and finishing with the pragmatics of truth ascriptions. Here I will take the right order. What do we do with truth ascriptions? This question has a general answer and a more specific one. Truth ascriptions perform those roles proper to their general category, i.e. to proforms. With these, speakers refer to certain items, directly or anaphorically, and quantify over their standard contents. Here are some examples of referential proforms: (42) (43) (44) (45)

He [pointing to an individual] is my son. It’s very nice here. A: How do you like Japan? B: I’ve never been there. A: She said that other pandemics are imminent. B: What she said is true.

I have described these uses as ‘referential’ but the term must be taken with caution. Referring, in the sense of ‘pointing to’, is something that speakers do by the use of indexicals. Most indexicals are singular terms, either demonstrative or pronouns. By contrast, sentences cannot be used to point to their contents; speakers express or display their contents by uttering them. Apart from referring directly, proforms can also refer anaphorically. The pronouns and adjectives in (28), the adverb in (44), and the truth ascription in (45) are anaphoric devices. Their role is to extend the effect of their anaphoric head to different contexts and co-texts, maintaining at the same time the unity of discourse. In (28), anaphoric pronouns and adjectives show that the topic has not changed. In the sentences that follow from the first one, in which the anaphoric head is introduced, anaphora guarantees that we are still talking about Boris Johnson. The anaphoric use of truth ascriptions is similar, with the addition of the pragmatic effects derived from their status as declarative sentences, able to be used in assertive acts. If after (46), uttered by speaker A, (46) A: Pandemics will become an actual risk for humanity. I respond by uttering (47), (47) What he says is true. I perform exactly the same act performed by A in the context of (46). My act is an assertion of the content that A has put forward in his assertive action. The content of my act is inherited from the content of A’s act, which functions as its anaphoric

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head. Gathering together all the information about anaphora given so far, the explanation of the Liar paradox given in Sect. 10.1 above is now complete: Liar-like sentences are anaphoric devices without anaphoric heads. They are as content-less as anaphoric pronouns that are unconnected to any names or descriptions that could provide their reference. Truth ascriptions as prosentences extend the impact of propositions. By being referred to by a truth ascription, the content of an assertion can exert its influence in contexts that are temporarily very distant from the context in which it was first produced. By uttering (46), (48) What Aristotle said about truth is true. we assert the content of Aristotle’s saying and adding it to the background assumptions that operate in the context of (48). Thus, two main actions are performed with truth ascriptions. The first one is the endorsement of a content expressed in a different actual or virtual context, and the second is the action of moving contents across contexts. What about direct reference uses? Can truth ascriptions be used as directly referential devices? As a mere question of grammar, sentences do not refer. Nevertheless, some truth ascriptions perform their roles in a way that is reminiscent of the role of pronouns in directly referential uses. This happens with exhibitive truth ascriptions. Consider again example (14): (14) ‘Snow is white’ is a true sentence. Here the content of the complete sentence is explicitly displayed in its grammatical subject. In some sense, the sentence showcases its content. This is the closest a sentence can get to a directly referential use. As well as being tools for direct and anaphoric reference, proforms play a further major role in language. As natural language variables, they are generalisation aids. Because they can inherit any content in their category, they are the appropriate tools to make general assertions. In order to either assert together or say something about a group of propositions, we need truth or some equivalent apparatus. The role of prosentences will become clearer if English is enriched by the addition of specific propositional variables, p, q, r, and variables for sets of propositions, Γ, Δ. Let us call this extension English*. Then (49) is the principle of excluded middle in English*, (49) Either p or not p. The meaning of logical consequence would be represented as (50): (50) If p follows from Γ, then necessarily, if Γ, then p. And the law of non-contradiction is (51): (51) Not at once both p and not p.

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This is the standard way in logic, where we have no need to include any explicit truth operator. Truth is not necessary because its job is performed by assertion together with the propositional variables that, as Frege said in the Begriffsschrift, §1, are expressions of generality. Given the lack of simple sentential prosentences in English, (49), (50) and (51) cannot be rendered into plain English without resorting to truth (or some equivalent device). In English, (49), (50) and (51) become (52), (53) and (54), respectively: (52) For any two propositions, either it or its negation is true. (53) If a proposition follows from a set of propositions, then necessarily, if all propositions in the set are true, then the proposition that follows from them is true. (54) A proposition and its negation cannot both be true at once. Instead of using truth, we could have used alternative formulations, using for instance assertion, as in (55): (55) Two mutually contradictory propositions cannot both be asserted at once. And there are other alternatives, as in (56): (56) If a proposition follows from a set of propositions, and all propositions in the set hold, then the proposition that follows from them holds too. The alternatives do not change anything. Nothing is gained by talking of assertion instead of truth, or by speaking of ‘holding’, ‘happening’, ‘occurring’, or of any other substitute. First, because truth and assertion are two sides of the same coin, and second because truth ascriptions are not the only prosentences in English. We have mentioned the Fregean ‘is a fact’, but there are many others, as Strawson observed (Strawson, 1950/2013, p. 65). Here are some further examples: (57) She is right. (58) I agree. (59) Things are as he says they are. Any prosentence-former will be an appropriate substitute for truth in (52), (53) and (54). All this reveals two significant consequences of prosententialism. The first is that truth is not one of a kind. Truth ascriptions are prosentences, and there are different kinds of prosentences; and prosentences are proforms, and there are different kinds of proforms. The second is that, even if truth is redundant in English*, it is not redundant in English. Proforms are not redundant. They are devices for direct and anaphoric reference and generalisation: tasks that in some artificial languages are performed by variables—nominal or sentential—and by quantifiers binding them.

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To the two pragmatic roles of truth ascriptions already mentioned—assertion and content mobility—Ramsey adds another pragmatic role for the truth operator, ‘it is true that’, which I have elsewhere called ‘reactive’ (Frápolli, 2018, p. 21). This is Ramsey’s text: Let us take three statements like this: The earth is round. It is true that the earth is round. Anyone who believes that the earth is round believes truly. It is really obvious that these statements are all equivalent, in the sense that it is not possible to affirm one of them and deny another without patent contradiction; to say, for instance, that it is true that the earth is round but that the earth is not round is plainly absurd. Now the first statement of the three does not involve the idea of truth in any way, it says simply that the earth is round. [In the second we have to prefix “It is true that” which is generally added not to alter the meaning but for what in a wide sense are reasons of style [and does not affect the meaning of the statements].2 Thus we can use it rather like ‘although’ in conceding a point but denying a supposed consequence. (Ramsey, 1927/2001, pp. 440–1)

Sometimes, Ramsey thinks, we use the semantically empty operator ‘it is true that’ to reject the connection between a content and some other contents that might be forced upon us, as if they followed from the content that is assumed to be true. As in (60): (60) Even if it is true that Phyllis is a Republican, she still supports minority rights. In this reactive use, the truth operator could be considered a logical constant in the derivative sense in which adversative conjunctions count as such, since it is used to block a foreseen unwarranted inference. Now that we are clear about the meaning of truth ascriptions and the role they play in language, we can proceed to the analysis of ‘is true’, as a sub-sentential part of some prosentences. From a syntactic point of view, ‘is true’ is a predicate. It includes a verb and, together with some names or descriptions, it forms a complete sentence. Consider the following singular terms, (61), (62), (63): (61) What she said. (62) String theory. (63) The violent death of Archimedes at the taking of Syracuse. Each of these refers to a propositional content. For instance, (61) might refer to (17) or to (46): (17) Brexit affects people’s lives. (46) Pandemics will become an actual risk for humanity.

 This text was not prepared for publication and the brackets do not match.

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Meanwhile, (62) refers to an indefinite set of propositions that are the consequences of a particular set of axioms, and (63) refers to an historic event. Even if their contents are propositional, they still cannot perform the roles reserved for expressions of propositions in natural languages. This is because they do not express propositions, they merely refer to them. (61)–(63) cannot be the arguments of logical constants or premises in inferences. If they were placed in these positions, the result would be ill-formed expressions, as in the following examples: (64) If what she said, the government should take urgent action.*. (65) String theory, and this has revolutionary consequences.*. (66) The violent death of Archimedes, therefore he could not finish his mathematical work.*. To restore well-formedness, (64)–(66) have to be modified by adding a predicate to the singular terms in them. This predicate will not add anything to the contents to which they refer. It will merely restore sentencehood. The predicate could be ‘is true’ or some other equivalent, as in (67), (68) and (69): (67) If what she said is true, then governments should take urgent action. (68) String theory is true, and this has revolutionary consequences. (69) The violent death of Archimedes was a fact, therefore he could not finish his mathematical work. (69) also illustrates the reason why Frege required the premises in an inference to be asserted. Assertion adds the required force, the commitment of the speaker, which in the Begriffsschrift was represented linguistically, i.e. explicitly, by the only predicate in the system. Borrowing Horwich’s terminology, ‘is true’ is a denominalisor, as it is ‘is a fact’ (Horwich, 1998). In Brandom’s terminology, these are prosentence-former operators (Brandom, 2009, p. 163, kindle edition). As Strawson saw, ‘[t]here is no nuance, except of style, between “That’s true” and “That’s a fact”’ (Strawson, 1950, p. 58). The grammatical equivalence between ‘is true’ and ‘is a fact’ explains the endurance of the correspondentist intuition. Correspondentism, in the weakest sense, is undeniable, because it rests on a grammatical remark. The same can be said of Aristotle’s claim: ‘To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true’. ‘What is’, ‘that is’, ‘it is’ and ‘it is not’ are presentences. Thus, not even Aristotle went any further than highlighting the functional equivalence of prosentences. Developing correspondentism as to make it a substantive theory is a different story. The two expressions, (61) and (1), (61) What she said. (1) What she said is true.

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are prosentences. (61) is a nominal prosentence and (1) a sentential one. Let us suppose that both of these refer to an utterance of (46): (46) Pandemics will become an actual risk for humanity. In this case, both have the same content, except (61) refers to it and (1) expresses it. Nevertheless, since (61) is a definite description and (1) a complete sentence, they differ in their affordances, as we saw in Sect. 10.3 above. Together with denominalisors—‘is true’, ‘is a fact’ and the like—natural languages also include nominalisors. Frege mentions ‘the circumstance that’ and ‘the proposition that’. In general, ‘that’ in that-clauses functions as a nominalisor. Quotation marks, in their logicalblock interpretation, are nominalisors too. The presence of nominalisors and denominalisors in a sentence has been a source of confusion that has lent unwarranted support to the thesis that truth is redundant. The left-hand side of Tarski’s Convention T includes both, as in (70): (70) ‘Snow is white’ is a true sentence if and only if snow is white. The false impression of redundancy is explained by the fact that on the left-hand side two devices occur that neutralise each other. Again, this situation not only happens with truth. In natural language, there are many devices that undo moves that other devices do. Some examples are ‘twice’ and ‘the half of’, and ‘the father of’ and ‘the son of’. The father of John’s son is John. Twice half of 10 is 10. Nevertheless, nobody would deem ‘the son of’ or ‘twice’ as redundant expressions merely because, when they are used together with their converse expressions, the result is just the starting point. In this case, too, we draw conclusions about truth that we do not draw about other notions that behave similarly. Before concluding this section, a word on falsity is in order. Falsity ascriptions are prosentences too. (71), (72), and (73) are examples of falsity ascriptions: (71) What she said is false. (72) ‘Snow is green’ is a false sentence. (73) That the pandemic is under control is false. These ascriptions work exactly like their positive counterparts, only in the negative. By uttering (71), a speaker rejects the content to which the description refers, and it is as if he had asserted himself the negation of the content asserted by the speaker referred to by ‘she’ in the relevant circumstance. If truth adscriptions have the role of importing contents into the context of utterance, then the role of falsity adscriptions is to veto the content referred to. A falsity adscription prevents its content from becoming part of the shared background.

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10.6 Truth and Assertion In assertion, one expresses one’s recognition of something as true. This is Frege’s view (Frege, 1879–1891, p. 2; Frege, 1918–1919, p. 356). The connection between truth and assertion has been widely acknowledged (see, for instance, Strawson, 1950, Bar-On & Simmons, 2007, Brandom, 1994, p. 231, Howat, 2018). Jager calls this the ‘initial datum’, which he formulates as follows: ‘[t]o make a statement, to assert something, is somehow to incorporate the claim that what is asserted is true’ (Jager, 1970, p.  161). The initial datum is represented in Grice’s supermaxim of quality: ‘Try to make your contribution one that is true’ (Grice, 1975, p. 46). And together with knowledge and belief, truth has been identified as one of the rules of assertion (Marsilli, 2018; Frápolli, 2018). There are different ways of understanding the connection between truth and assertion. One of them, which we might call the ‘truth version’, asks us not to assert anything unless it is justified without the possibility of error. In this interpretation, the connection is too strong and thus unattainable. If we could only assert contents for which we had a complete and definitive justification, then we would be condemned to silence. Fortunately, this is not the connection that the norm of truth expresses. The connection, in Grice’s submaxim, is normative: to assert a content is to express a commitment to its truth. The pragmatist aspect of the connection is grammatical in Wittgenstein’s sense. It amounts to saying that truth only makes sense related to acts of assertion. Obvious as it might seem, the connection between truth and assertion has seemed to be problematic, or ‘obscure’ (Jager, 1970), in part due to some remarks in Frege’s writings. The two scenarios that seem to challenge the initial datum are conditionals and fiction. Let us begin with conditionals. Truth and assertion cannot possess the essential relation that is sometimes attributed to them, the argument goes, because there are contexts in which we use the truth predicate without anything being properly asserted. Conditionals such as (74) illustrate this point: (74) If it is true that pandemics are a serious risk for humankind, then governments should take urgent action. The antecedent of (74) is not used to make an (independent) assertion. In the representation of the conditional in the Begriffsschrift, the judgement stroke covers the relation between the antecedent and the consequent. Frege explains that ‘we can make the judgement (that if B, then A) without knowing whether A and B are to be affirmed or denied’ (Frege, 1879, §5). And he is right. Brandom uses the case of the conditional to reject the pragmatist analysis of truth as a mere marker of pragmatic force (Brandom, 1994, p. 299; Brandom, 2009). The argument is that, in embedded uses of truth ascriptions such as (74) above, truth can be removed or included without affecting the force. In fact, (74) is equivalent to (75):

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(75) If pandemics are a serious risk for humankind, then governments should take urgent action. Brandom is also right to draw attention to the equivalence between (74) and (75). Nevertheless, this equivalence rests on the equivalence of the unembedded instances and does not prove that the antecedent of a conditional is not asserted. Pace Frege and Brandom—two of my heroes—the antecedents of conditionals are asserted, but their assertion is somehow deferred. This point requires the Kaplanian distinction between the context of utterance and the circumstance of evaluation. In the context of utterance, indexicals and other expressions acquire their contents. I mentioned in Chap. 8 that isolated sentences fall short of yielding a proposition. This happens in cases in which those sentences include indexicals and demonstratives and in cases in which they don’t. But in indexical sentences, the phenomenon is easier to spot. What is said by the use of a sentence is something that is constituted in the context of utterance, guided by the sentence’s linguistic meaning and completed by contextual factors. But the assertoric content of a sentence is not always evaluated in the context of utterance. Natural languages possess what Lewis called ‘circumstance-shifting operators’ (Lewis, 1980), operators whose function is to direct the hearer to a different situation, factual, counterfactual or imaginary, in which the content is evaluated. Modal, epistemic and temporal operators are of this kind. Conditionals belong to this category too. Recanati considers conditionals to be ‘world-shifting’ operators (Recanati, 2000, pp. 71–72). The situation depicted by the antecedent of a conditional might be true of the actual world, but it is not presented as such. It might well refer to another possible world different from the actual world: ‘So one appeals to a world w possibly distinct from @ [his symbol for the actual world], and one says that, with respect to that world, the situation denoted by the antecedent supports the consequent’ (Recanati, 2000, p.  72). Johan van Benthem makes the same point, albeit more poetically. A conditional, he says, ‘invites us to take a mental trip to the land of the antecedent’ (van Benthem, 1984, p. 311). Thus, in the land of the antecedent, it is asserted, and, from there, we say that the consequent follows from it, in a sense of ‘follow’ that is contextually determined. In fiction, the situation is similar. The objection that fiction might be a counterexample to the initial datum was also triggered by some remarks from Frege: And even when we do use it the properly assertoric force does not lie in it, but in the assertoric sentence-form; and where this form loses its assertoric force the word ‘true’ cannot put it back again. This happens when we are not speaking seriously. As stage thunder is only sham thunder and a stage fight only a sham fight, so stage assertion is only sham assertion. It is only acting, only fiction. When playing his part the actor is not asserting anything; nor is he lying, even if he says something of whose falsehood he is convinced. (Frege, 1918–1919, p. 356)

From this and similar texts, commentators have drawn the conclusion that Frege rejected the generalised connection between truth and assertion. I am not sure, however. What Frege says in this text is literally true: that the actor does not assert, and even if he utters sentences that he considers false, he does not lie either. The actor does not assert, but the character he is impersonating does. The character intends

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that other characters in the play take his words as true and act accordingly. The same happens with promises and commands. The character who makes a promise is bound by it, and the fellow character to whom the promise is made is right to feel betrayed if the first character does not keep his word. A character who gives an order expects it to be carried out too. This is what Predelli calls the ‘uniformity argument’: [A]ny satisfactory theory of fiction must inevitably account for inter-force relationships that are parallel to the relationships between straightforward speech acts such as assertions and requests. […] If assertives and expressives are distinct speech-act types in everyday speech, at least two illocutionary types ought thereby to be at issue also in fictional scenarios. (Predelli, 2019, p. 316)

Predelli’s uniformity argument is a re-elaboration of one of Searle’s intuitions: that the meanings of the linguistic devices by means of which we indicate force should be the same both in fiction and outside fiction. Language works in fiction exactly as it works outside fiction.3 As Byrne notes, [t]he principles that govern conversation contain the truth in fiction.’ (Byrne, 1993, p. 35) Otherwise, we would not understand plots in plays and novels. It is just that everything is sham, as Frege said. But, as he also remarked, ‘[w]e do not need to consider mock assertions in logic’ (Frege 1879–1891, p. 126). Predelli’s work belongs to the tradition initiated by Searle (1975)—broadly understood—a tradition that develops another well-known Searle’s insight: that fictional narratives are acts of pretence—and to which Recanati also belongs (Recanati op. cit., pp. 51ff., pp. 221ff.). By reading or listening to fictional stories, we engage in generalised acts of make-believe in which stories are taken as if they were true (within the restricted scope that the story defines). Fictional contexts are within the scope of an implicit operator that shifts worlds and can also shift contexts. This is seen in the reference of indexicals. Demonstratives and pronouns in fiction refer to fictional characters, and neither to the writer nor to the actors: The word ‘I’ [in the novel] does not denote the person who, in the actual context, issues the sentence (the novelist); rather, it purports to denote a character in the novel: the ‘narrator’, distinguished from the actual author. In a perfectly good sense, then, the context for those sentences is not the actual context, but an imaginary context. In that imaginary context, the speaker is on board of a spaceship, he or she has a twin brother called ‘Henry’, etc. (Recanati, op. cit., p. 171)

For Recanati, fictional contexts are meta-representations. Context shifts sever the ties between utterances and their actual context, between speech acts and the actual world. But they do not cut the links between truth and assertion, or between promises and commitments, or between commands and obedience, and this is what the uniformity argument claims. García-Carpintero argues against the approach to fiction represented by Searle and Predelli, offering an alternative explanation, the ‘Dedicated Representation

 I refer here to pieces of fiction that purport to tell a story in a reasonable standard way, not to experimental proposals with merely aesthetic or experimental purposes. The aim is to show that Frege’s claim does not reject the initial datum. 3

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View’ (García-Carpintero, 2022), to the thesis that fiction is mere pretense. This alternative consists in the proposal of a specific kind of force applied to fiction, assuming thus that fictions belong to a specific representational category on the same level as other speech acts. For the purposes of the debate on the connection between assertion and truth, taking sides in this further debate is unnecessary. All that is needed in order to see that truth and assertion are aspects of a unified phenomenon is to stick to the ‘uniformity argument’, which, although proposed within Searle’s tradition, is not specific to the pretence view. As García-Carpintero argues, his Dedicated View ‘can unproblematically account for the “parallel inter-force relationships”—when they obtain, as I agree they standardly do’. (García-Carpintero, op. cit., p. 85). What I have discussed in this section illustrates a further way in which philosophers make truth special. Nobody would reject Austin’s account of promises and commands just because actors, the real persons behind characters, are not bound by the promises they make on stage when they are off stage. A different illustration comes from the role of indexicals. The ontological status of the referents of referential expressions, indexicals and proper names, in fiction has received much attention, and philosophers discuss whether fictional characters have to be understood as Meinongian entities, as artefacts, concrete non-actual possibilia, etc. (see García-­ Carpintero, 2019a for a survey; see also Corazza & Whitsey, 2003; García-­ Carpintero, 2019b). The whole contemporary debate is shaped by a representationalist approach to meaning, the interpretation of existence as spatio-temporal location, and truth as correspondence. These assumptions cede centre stage to semantic, metaphysical and ontological issues that, in a different paradigm, would appear artificial. But in any case, referential expressions in fiction still work as referential.4 In fiction, reference might be deferred, pretended, empty, etc., but the role of referential expressions is preserved. By contrast, the connection between truth and assertion is often rejected on the basis of how we use language in fictional contexts. Let me now conclude by insisting on the principles and assumptions that have shaped my arguments throughout this book, but which have now been applied to the analysis of truth. Only insensitivity towards the philosophy of language of the second half of the twentieth century can support the thesis that truth is a unique concept. Only by disregarding well-known phenomena acknowledged by linguists and philosophers of language can the Liar paradox survive as a serious challenge to a characterisation of the function of the truth predicate and truth ascriptions. Understanding truth requires understanding that propositions take centre stage, and that truth ascriptions are special vehicles of propositional expression, endorsement and displacement. They are special because they work like indexicals. The difference between nominal prosentences and sentential prosentences, being Fregean in  Corazza and Whitsey (2003, p. 122) lists, among the possible interpretations of indexicals in fictions, the theoretical possibility of indexicals being understood ‘not as singular terms but disguised descriptions’. But they do not pursue this option, which they consider ‘revisionary’. For an eloquent defence of an asymmetry thesis in the semantics of indexicals and descriptions, see García-­ Carpintero, 2005. 4

10.6 Truth and Assertion

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origin, benefits from (PGS). The expressive meaning of the truth predicate requires a pluralist view of semantics that is granted by (PII). In sum, for the analysis of higher-level predicables and concepts, of which logical constants and truth are distinguished examples, standard representationalist semantics and classical formalist approaches offer a conceptual apparatus that is too narrow to allow a satisfactory explanation of how they work. It is not these notions, it is not truth that is to be blamed, but rather the inadequacy of those instruments and the failure of philosophers to pay attention to the theoretical possibilities that the twentieth century has offered them. And Frege, whose outstanding knowledge of language is still largely unappreciated, was the origin.

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Index

A Adversative, 129, 130, 142, 143, 146, 240 Affordances, 188–190, 196, 232, 242 Analytically valid, 104–106, 108–110, 113, 158, 159, 161 Argumentations, viii, 23–25, 47, 175, 176, 179, 181, 182, 185, 190, 193, 195, 196, 203 Arguments, vii, viii, 5, 16, 17, 19, 21, 23–25, 31–33, 35, 37, 41–43, 46, 53, 58–61, 66, 69, 70, 80, 91, 92, 94, 96–99, 104–110, 113, 117, 118, 120, 126–128, 132–138, 141, 144, 146, 158–163, 165, 167–169, 172, 175–184, 192–196, 200, 203–205, 210–212, 215, 217, 218, 232, 234, 241, 243, 245, 246 Ascriptions, 90, 196, 222–247 Asserted, 8, 12, 14, 24, 25, 41, 67, 68, 70, 72, 84, 86, 90, 92, 93, 112, 137, 191, 210, 211, 215, 217, 218, 223, 233, 239, 241–244 Assertibles, 8, 210 Assertions, 6–8, 10–15, 17, 21, 23–25, 30, 31, 37, 40–42, 56, 57, 62, 66–68, 70, 71, 81, 83, 84, 90, 92, 94, 99, 111, 120, 137, 140, 142, 145, 152, 171, 177–180, 182, 183, 187, 188, 191, 195, 196, 200, 207, 208, 210–214, 218, 223, 225, 226, 228, 229, 237–241, 243–246 Austin, J., 206, 207, 222 Austin, J.L., 223, 246 Ayer, A., 79 B Baker, G.P., 64

Barwise, J., 94 Bearers, 24, 25, 31, 33, 38, 40, 41, 43, 56, 92, 108, 111, 113, 126, 128, 164, 177, 186, 193, 200, 203–205, 209, 211, 212, 214, 218 Bell, D., 64 Belnap, N.D., 236 Bipolar, 87 Blackburn, S., 57 Blind, 10, 15, 21, 178, 202, 225–227 Bolzano, 172 Bonnay, D., 116 Brandom, R., vii–ix, 7, 8, 10, 13–15, 17, 21, 23, 25, 31–33, 35, 42, 45, 46, 49, 55, 57, 65, 70, 81, 82, 91, 96, 105, 106, 126, 130, 132, 139–141, 152, 159, 179, 180, 187, 191, 196, 205, 207, 222, 225, 234–236, 241, 243, 244 Brandon, 108 Bronzo, S., 36 Burge, T., 165 Byrne, A., 245 C Cantor, G., 169, 171 Cappelen, H., 5 Carnap, R., 13, 54, 55, 89, 116 Carroll, L., 141 Carston, R., 183 Casanovas, E., 136 Chalmers, D., 5 Chomsky, N., 7 Claimables, 8, 9, 22, 23, 187 Compositional, 31, 39, 110, 167–169, 172, 196

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252 Compositionality, 5, 34, 36, 38, 40, 70, 165, 167, 168 Conditionals, 7, 24, 25, 41, 48, 80–87, 89, 93, 94, 98, 99, 104, 105, 109, 110, 119, 120, 130, 135, 140, 141, 143–146, 157, 159, 179, 210, 243, 244 Conjunctions, 30, 48, 74, 84, 88, 89, 94, 97, 111, 112, 120, 121, 127, 129, 130, 133–136, 142, 143, 146, 232, 235, 240 Conservative, 113, 186 Conservativeness, 106, 113, 114 Conservativity, 113 Contexts, 4, 6, 10, 13, 14, 18, 30, 31, 33, 34, 48, 55, 56, 63, 64, 69, 84, 88, 90, 99, 113, 133, 138, 152–154, 156, 161, 162, 165–172, 177, 178, 183, 186, 187, 191, 194, 196, 201, 203, 216–218, 222, 224–230, 232, 234, 236–238, 242–246 Conventions, 69, 206–208, 222, 231, 242 Cooper, R., 95 Corazza, E., 246 Correspondence, 5, 9, 18, 69, 70, 95, 115, 195, 196, 205–207, 246 D Davidson, D., 205, 212 De Brabanter, P., 181–183 Deflationism, 170, 205–207 Disjunctions, 48, 88, 89, 93, 97, 127, 129, 130, 133, 134, 142, 143, 146 Dodd, J., 9 Dummett, M., 55, 64, 91, 98 E Effability, 35, 187, 192 Effable, 192 Entrenched, 155, 156, 160 Entrenchment, 154–156 Erlangen, 47, 104, 114, 115, 117, 118 Etchemendy, J., 105, 108, 115, 199 Exhibitive, 225, 226, 228, 238 Existence, 4, 5, 17–20, 33, 34, 57, 58, 86, 94–97, 99, 135, 137, 187, 215, 235, 246 Expressive, 11–14, 25 Expressives, 80, 81, 94, 110, 126, 129, 130, 132, 136, 139, 140, 145, 159, 181, 190, 191, 196, 218, 236, 245, 247 Expressivism, 10, 48, 73, 79–100, 113, 114, 129, 139, 141, 179, 180, 191, 216, 234–236

Index Expressivists, 10, 30, 80–82, 126, 218, 236 F Facts, 5, 6, 10, 11, 14, 17, 21, 25, 35, 37, 41, 44, 45, 47–49, 57, 60, 62, 63, 65, 68–72, 80, 86, 88, 91, 93, 94, 96, 98, 99, 105, 112–115, 127, 128, 131, 137, 140, 153, 156, 160, 162–164, 166, 169, 172, 178, 181, 187, 201, 207, 210, 211, 217, 218, 222, 226, 227, 230, 233, 239, 241–243 Falsehood, 40, 44, 87–94, 138, 183, 188, 244 Feferman, 116 Field, H., 202 Forceville, C., 177 Frápolli, M.J., 79–100 Frege, F., 54 Frege, G., vii, 13, 16–21, 30, 32–35, 38–41, 46, 53, 55, 56, 58, 60–62, 65, 66, 68, 70–73, 81–86, 91, 92, 95, 97, 99, 110, 112, 115, 121, 128, 140, 145, 163–166, 168, 170, 171, 192, 194, 195, 199–218, 224, 225, 230, 233, 235, 236, 239, 241–245, 247 Frege-Geach argument, 41, 80 Fricker, M., 10 Functions, 6–8, 16, 18, 33, 41, 46, 53, 58–63, 66, 68, 81, 83, 85–89, 94–96, 99, 109–111, 116–118, 120, 126–128, 130–132, 134, 136, 138, 146, 154, 158, 160, 161, 163–165, 168, 188, 190, 201, 202, 207–209, 213, 215, 218, 223, 227, 231, 233, 235–237, 242, 244, 246 G García-Carpintero, M., 245, 246 Geach, P.T., 80 Gentzen, G., 48, 98, 111–113, 115, 128, 161 Gödel, K., 195 Gómez-Torrente, M., 108, 121 Goodman, J., 23, 24, 179, 204 Goodman, N., 152–158, 162 Grammar, viii, 4, 31, 32, 38, 69, 95, 126, 156, 166, 167, 187, 196, 218, 232, 238 Grice, H.P., 41, 55, 71, 74, 161, 179, 180, 186, 196 Grice, P., 243 Groarke, L., 177, 178, 183, 184, 192 Grover, D., 233

Index Grover, D.L., 236 Grue, 151–172 H Haack, S., viii, 7, 13, 160, 205 Hacker, P.M.S., 64 Hacking, I., 128 Hanson, N.R., 9 Haslanger, S., 5, 10 Heras-Escribano, M., 190 Higher level, 4, 18, 19, 33, 34, 57, 81, 84, 89, 94, 96, 97, 99, 119, 120, 126–128, 130–132, 134, 138–140, 170, 215, 224, 235, 236, 247 Hilbert, 115 Hintikka, J., 104 Hornsby, J., 9 Horwich, P., 241 Hume’s principle, 20 I Identities, 5, 9, 15, 20, 23, 54, 61–65, 91, 118, 121, 129, 131, 132, 134, 146, 163, 164, 170, 204, 205, 216, 218, 236 Implications, 73, 93, 130, 141, 144, 224 Implicatures, 72–74, 135 Implying, 79–100 Incompatibilities, 14, 82, 86–94, 141, 142, 145, 146, 176, 186, 224 Ineffability, 190–193 Ineffable, 192 Inferential, 10, 14, 15, 17, 22, 23, 25, 30–34, 36, 45, 54, 58, 59, 69, 81, 83, 85–88, 90, 94, 96, 98, 99, 107, 108, 110, 112–114, 116, 119, 121, 126, 127, 130–132, 134, 136–141, 143–146, 152, 153, 156, 158–161, 169, 172, 178, 179, 183, 187, 189, 192–194, 212 Inferentialism, 25, 32, 55, 103–122, 128, 145, 161, 162, 179, 196, 205 Inferentialists, 25, 31, 90, 105, 110–114, 126, 165, 171 Insubstantive, 118, 120, 127 Invariantism, 103–122, 136 Invariantists, 66, 114, 117, 118, 131 J Jager, R., 243 James, W., 10

253 Judgeable contents, 16, 17, 20, 22, 31, 36, 39–41, 65, 71, 83–85, 93, 99, 110, 119, 121, 145, 167, 176, 186, 189, 193, 194, 203, 210, 212–218, 223, 225, 233, 235 Judgement stroke, 18, 31, 66–68, 70, 71, 83, 85, 137, 188, 210, 212, 213, 217, 218, 243 Julius Caesar problem, 21, 22 K Kant, 55, 172 Kaplan, D., 39, 201, 227, 230 Kauppinen, A., 4 Klein, F., 114 Kripke, S., 6, 64 Kuhn, T., 9 Kukkla, R., 5 Kürbis, N., 91, 92 L Lewis, D., 244 Liar paradox, 142, 164, 200, 222, 223, 238, 246 Linnebø, Ø, 167–169 Linsky, L., 64 Logical constants, 16, 17, 30, 34, 43, 44, 47, 48, 68, 74, 80–84, 88, 89, 91, 92, 96, 103–117, 120–122, 126–132, 134–137, 139, 140, 142–146, 152, 158–160, 171, 177, 180, 191, 193, 235, 240, 241, 247 Logics, vii, viii, 4, 5, 13, 16–25, 29–34, 37, 38, 41–49, 53–55, 58, 62, 65–68, 70, 71, 74, 83, 85–88, 91–96, 103–108, 111, 114–119, 125–130, 132, 133, 135, 136, 141–143, 145, 152, 153, 157, 160–162, 164, 166, 169, 170, 175, 177–179, 193, 195, 196, 200, 202–204, 212, 213, 215–217, 234, 239, 245 Ludlow, P., 183 Lynch, M.P., 236 M Macbeth, D., 96 MacFarlane, J., 11, 13 Manne, K., 5 Medina, 10 Membership, 119, 127, 129, 131, 132, 194 Modes, 34, 39, 61, 64, 65, 89, 114, 131, 161, 170, 175–181, 187, 189, 195, 196, 224

254 Moore, G.E., 216–218 Mostowski, A., 94 Multimodal, 175, 177, 181, 185, 190, 195, 196 Multimodality, 176–185, 189, 190, 192, 195 N Negation, 37, 41, 48, 54, 61, 80–97, 112, 119, 130, 133, 134, 136–138, 140–143, 145, 146, 239, 242 Numbers, 4, 17–22, 35, 37, 38, 54, 56, 58, 61, 62, 72, 84, 94, 96, 97, 118, 119, 135, 137, 142, 164, 165, 169, 170, 172, 186, 187, 190, 194, 203, 208, 214–216, 224, 225, 236 O Odgen, C.K., 79 Organic intuition (OI), 22, 23, 40–42, 82, 176, 183, 196, 205 P Peirce, C.S., 6, 128, 206 Penco, C., 167 Physicalism, 202–205 Pictures, 10, 33, 47, 57, 67, 80, 86, 87, 89, 91, 130, 175, 177–190, 192, 195, 196, 212, 236 Plunkett, D., 5 Popper, K., 158, 205 Pragmatics, 7, 8, 13, 14, 17, 18, 31, 33, 37, 42, 53–74, 84, 87, 90, 93, 110, 111, 114, 121, 126, 127, 129, 132, 134–136, 140, 146, 154, 155, 157, 161, 162, 171, 179–183, 185, 196, 201, 222, 224, 225, 230, 234, 236, 237, 240, 243 Pragmatism, viii, 3–25, 57, 59, 82, 90, 106, 130, 141, 158, 179, 185–188, 196, 225, 234–237 Pragmatists, vii, viii, 6, 8–15, 17, 18, 21–23, 25, 29–31, 33, 45, 49, 55, 56, 73, 74, 82, 88, 90, 92, 103, 106, 108, 111, 119–122, 125–146, 152–157, 160, 162, 172, 176, 179, 183, 186, 188, 190, 191, 196, 200, 204, 206, 208, 215, 216, 224–228, 230, 234, 235, 237, 243 Prawitz, D., 91, 98 Predelli, S., 245

Index Predicables, 19, 37, 58, 68–72, 89, 94, 98, 119, 126–128, 130–135, 137, 138, 208, 209, 211, 215, 217, 233, 247 Presuppositions, 9, 16, 21, 22, 24, 30, 35, 41, 72–74, 223 Prior, A.N., 40, 43, 113, 141, 152, 153, 158–162, 195, 205, 236 Proforms, 228–234, 236–239 Projectibility, 154–156 Projectible, 154–157 Propositional priority, 5, 16, 31, 34, 125, 140, 151–172, 196, 200, 222–247 Propositions, vii, viii, 4–9, 14, 16–25, 30–45, 47, 48, 54, 57–59, 62, 63, 67–69, 72–74, 80, 82, 85–90, 92, 93, 108, 109, 111–114, 117, 120, 126–128, 131–143, 145, 146, 153–156, 161, 165–168, 170–172, 175–179, 182, 183, 185–187, 189, 190, 192–195, 202–206, 210–213, 215, 217, 223–225, 231, 233–236, 238, 239, 241, 242, 244, 246 Prosentences, 222–247 Putnam, 13 Q Quine, W.V.O., 13, 116, 141, 204 R Raatikainen, P., 202 Ramsey, F.P., 6, 40, 140, 154, 206, 207, 209, 210, 222, 231–233, 236, 237, 240 Recanati, F., 39, 138, 201, 244, 245 Redundancy, 207, 214, 235, 242 Richards, I.A., 79 Russell, B., 35, 45, 46, 55, 64, 66, 73, 151–153, 163, 165 Russell’s paradox, 21, 62, 151–172 Ryle, G., 13, 223, 236 S Salmon, N., 64 Sandu, G., 104 Satisfaction, 199–218 Saul, J., 5 Sellars, 55 Stainton, R., 183 Stalnaker, R., 225 Stanley, J., 183

Index Stevenson, C., 79 Stevenson, C.I., 203, 216, 217 Stevenson, J.T., 161 Strawson, P., 41, 158, 196, 239, 241 Strawson, P.F., 64, 68–70, 72, 73, 178, 186 Subformula, 112–114, 128, 161 T Tarski, A., 5, 24, 33, 40, 43, 45–48, 54, 65, 66, 98, 104, 105, 108, 114–120, 132, 136, 193, 199–218, 222, 231, 242 Tonk, 112, 113, 151–172 Translations, 24, 144, 145, 166, 170, 182, 188, 194, 196, 202–204, 212 True, 4, 12, 15, 19, 20, 31, 32, 39, 40, 43, 44, 47, 48, 58, 60, 61, 63, 65, 67–74, 80, 84, 86, 87, 91, 93, 105–107, 109, 115, 117, 137, 139, 144, 155, 161, 162, 164, 165, 171, 178, 179, 182, 186, 201, 203, 204, 206, 208–218, 222, 223, 225, 226, 228, 231–235, 237–245 Truths, viii, 5, 7, 9, 11, 12, 14, 15, 17–20, 22, 25, 40–49, 54, 60, 62, 63, 65–71, 73, 74, 86, 87, 90–94, 105–109, 113, 115,

255 117, 120, 132, 138, 142, 143, 153, 160, 161, 163, 164, 170, 171, 176, 177, 179, 183, 186, 188, 193, 194, 196, 199–218, 222–247 Tseronis, A., 177 V van Benthem, J., 244 Vehicles, 11, 25, 56, 65, 95, 96, 167, 176, 182, 185, 187–189, 201, 214, 246 Visuals, 175–196 W Whitehead, A.N., 64 Wiggins, D., 64 Williams, C.F.J., 131, 132 Williams, C.J.F., viii, 4, 19, 236, 237 Williamson, T., 12 Wittgenstein, L., viii, 4, 8, 10, 16, 30, 34, 35, 42–46, 48, 49, 55–57, 66, 68, 81, 82, 86, 87, 92, 110, 112, 153, 154, 156, 157, 161, 187–189, 192, 196, 200, 204, 218, 236, 243