Language, Logic, and the Liar-Paradox 9783957430342, 3957430348

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Pleitz · Logic, Language, and the Liar Paradox

Martin Pleitz

Logic, Language, and the Liar Paradox

mentis MÜNSTER

Einbandabbildung: Sandzeichnung mit kreisförmigem Pfeil (Foto: Martin Pleitz)

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.dnb.de abrufbar. D6: Philosophische Dissertation, Universität Münster

© 2018 mentis Verlag GmbH Eisenbahnstraße 11, 48143 Münster, Germany www.mentis.de Alle Rechte vorbehalten. Dieses Werk sowie einzelne Teile desselben sind urheberrechtlich geschützt. Jede Verwertung in anderen als den gesetzlich zulässigen Fällen ist ohne vorherige Zustimmung des Verlages nicht zulässig. Einbandgestaltung: Anna Braungart, Tübingen Wissenschaftlicher Satz: satz&sonders GmbH, Münster (www.satzundsonders.de) ISBN 978-3-95743-034-2 (Print) ISBN 978-3-95743-849-2 (E-Book)

In memory of Rosemarie Rheinwald.

Preface

In this book I discuss the Liar paradox and propose a novel solution to it. The Liar paradox arises in the following way: We consider a sentence which says of itself that it is false, or a sentence which says of itself that it is not true. If such self-referential sentences exist (and examples such as ‘This sentence is false’ and ‘The sentence on the whiteboard in room 101 is not true’ which is written on said whiteboard certainly suggest this), then our logic and standard notion of truth force a contradiction upon us: The Liar sentence is true and false! Or even: The Liar sentence is true and not true! What has gone wrong? Must we revise our notion of truth and our logic? Or can we dispel the common conviction that there are such self-referential sentences? Most people today take the first way because they think that the second way is blocked. But this is the route I have taken. Here is a brief report of where the road has taken me: By comparing the Liar reasoning in formal and informal logic (in part I), I have shown that any adequate discussion of the Liar paradox needs to construe the expressions involved as meaningful, even in cases where they belong to a language of logic. For this reason, I have turned to the theory of meaning (in part II). Based on a detailed overview of the semantics of singular terms, I have given a comprehensive list of the (purported) means for constructing self-referential sentences. I have also shown that even in the mediated sense in which the method of Gödelization does allow to construct sentences that attribute certain features (like unprovability) to themselves, it cannot be used to construct Liar sentences – despite a common misconception. After these extensive preparations, I have developed my own account of the metaphysics and semantics of self-referential languages (in part III). According to it, meaningful expressions are distinct from their syntactic bases and from their physical bases, and they exist only relative to contexts. By recourse to several semantico-metaphysical arguments I was able to show that in this dynamic setting, an object can be referred to only after it has started to exist. Hence self-reference of the kind that leads to the Liar paradox cannot occur – despite our initial impression. As this blocks the paradoxical reasoning at an early point, we are freed from the need to revise our notion of truth and our logic in a major way. And as this solution is contextualist, it evades the expressibility problems of other proposals. My main aim in writing about the Liar paradox was of course to convince the experts working in the field – philosophers and logicians who are worried about the paradox. But due to the particular character of my proposal, the investigation also needed to involve three more general questions. The first two of these belong to philosophical logic and the philosophy of language, respectively: What is the character of a language of logic? How does semantics interact with the metaphysics of expressions?

8

Preface

The third question is probably more surprising in a study about the Liar paradox, because it belongs to the philosophy of time: How can tensed reality be represented from within? Only on the basis of answers to these more general questions was I able to give my particular answer to the more specific question: How can the Liar paradox be solved? In other words, I have combined insights from philosophical logic, semantics, the metaphysics of expressions, and the tensed theory of time when I developed my extended argument for the conclusion that the circular reference needed in the Liar paradox cannot occur. Therefore this book should be of some interest also to philosophers who do not themselves work on the Liar paradox. For these reasons, the present study has grown into a long text. As the first chapter gives an introduction to the problem and explains the plan of the book, this preface need not contain anything more by way of preview. But let me say something about how you can read the book. Obviously, an ideal reader would read all of it, and then once more! But I am of course aware that few people will be able and willing to allot that much time to working through the results of my work. Therefore I have written part III, The Metaphysics of Expressions and the Liar Paradox, in a way that should enable you to assess the argument for my proposed solution on the basis of reading only this last part of the study, following back the references to parts I and II as needed. What is more, I would not discourage an impatient reader who is already an expert on these matters from jumping right to chapter 13, Subsequentism Solves the Liar Paradox: Reading only this last chapter would probably allow you to understand the basics of my proposal – although not to criticize it fairly.

Acknowledgements First, I would like to express my gratitude to some of the people who are important to me for reasons not directly connected to the project of writing this book. I am grateful to my parents for their love and their trust in my abilities, and I thank Simon, Dario, Jennie, Roman, Tobi, Pete, Deni, Arda, Fynn, and Daniel for making my life more beautiful. Next, I would like to thank members of the Department of Philosophy at the University of Münster for their support, especially during a difficult phase after the untimely death of Rosemarie Rheinwald, who was my Ph.D. supervisor and employer then. First and foremost I want to mention Niko Strobach and Sibille Mischer, and in this regard I would also like to thank Peter Rohs, Oliver Scholz, Reinold Schmücker, and Kurt Bayertz, as well as Ansgar Seide, Eva Jung, Dirk Franken, and Christa Runtenberg. I am grateful to the University of Münster for awarding its Dissertation Prize to the Ph.D. thesis that this book is based on. Many people have helped me write this book – by discussing the topic with me, by reading and commenting on preliminary drafts of the text, by encouraging me,

Preface

9

and by assisting me with other tasks during the more intense phases of writing. I am grateful to all of them! Most important were discussions with Johannes Korbmacher at the outset of the writing process, the steady support by Niko Strobach throughout, and discussions with Tobias Martin towards the end. Simon Dickel helped me greatly during some crucial months in the winter of 2011/12. In these regards I would also like to thank Mathieu Beirlaen, Andreas Bruns, Chris Scambler, Daniel Boyd, Kit Fine, Toff Fromme, Severi Hämäri, Ulf Hlobil, Andreas Hüttemann, Katrin Huxel, Roman Klauser, Vojtˇech Kolman, Nikola Kompa, ErnstWilhelm Krekeler, Roland Mümken, Christian Nimtz, Philipp Offermann, Vincent Peluce, Graham Priest, Lena Raezke, Stephen Read, Dave Ripley, Peter Rohs, Raja Rosenhagen, Stefan Roski, Liam Ryan, Sebastian Schmoranzer, Lionel Shapiro, Stewart Shapiro, Ralf Schindler, Ansgar Seide, Hartley Slater, Linda Supik, Vítˇezslav Sˇvejdar, Allard Tamminga, Alex Thinius, Heinrich Wansing, Chrischi Wolf, Maiko Yamamori, and Elia Zardini. Last but not least, I greatly appreciate my philosophy teachers. A long time ago, there were Rainer Schmidt and Marcus Willaschek. I would like to thank Rainer for getting me started and Marcus for encouraging me to go on. Today, I have the great fortune to find myself staying for a while with Graham Priest and Kit Fine in New York. I would like to thank Graham for an important conversation in Berlin and Kit for an important conversation in Oxford, and both of them for their generous support in many matters. The teacher, however, who the project of writing this book is most closely connected to is Rosemarie Rheinwald, who I continue to admire for her clarity, honesty, and kindness. When she fell ill, I re-read her book on the semantic paradoxes, but she died before I could tell her my ideas about it. I dedicate this book to her memory.

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

PART I

LOGIC AND THE LIAR PARADOX . . . . . . . . . . . . . . .

15

Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

The Liar and Its Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The concept of a paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Liar paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liar cycles and Yablo’s paradox . . . . . . . . . . . . . . . . . . . . . . . Church’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curry’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The history of the Liar paradox . . . . . . . . . . . . . . . . . . . . . . . Set theory and Cantor’s diagonal argument . . . . . . . . . . . . . . Semantic and set theoretic paradoxes . . . . . . . . . . . . . . . . . . . Just a joke? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The question of formality . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 22 25 27 28 32 34 38 41 43

Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Informal Logic and the Liar Reasoning . . . . . . . . . . . . . Informal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The naïve truth principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Truth, falsity, and negation . . . . . . . . . . . . . . . . . . . . . . . . . . . Informal variants of the basic Liar reasoning . . . . . . . . . . . . . The extended Liar reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . The phenomenon of Revenge . . . . . . . . . . . . . . . . . . . . . . . . . The Reflective problem (a. k. a. the Son of the Liar) . . . . . . . . The Problem of Expressibility . . . . . . . . . . . . . . . . . . . . . . . . Recurrence of the Reflective problem . . . . . . . . . . . . . . . . . . . Revenge, Reflection, Expressibility . . . . . . . . . . . . . . . . . . . . .

45 46 48 50 53 57 60 61 62 64 65

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Formal Logic and Tarski’s Legacy . . . . . . . . . . . . . . . . . . Formal logic: Language and calculus . . . . . . . . . . . . . . . . . . . . The Tarskian truth schema . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The background theory of a language . . . . . . . . . . . . . . . . . . . Formal variants of the basic Liar reasoning . . . . . . . . . . . . . . . The Diagonal Lemma and Tarski’s Theorem . . . . . . . . . . . . . . Tarski’s condition of paradoxicality . . . . . . . . . . . . . . . . . . . . Contagious inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metaphysical inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . .

67 68 78 85 88 91 98 104 106 110

12

Table of Contents

PART II

SEMANTICS AND THE LIAR PARADOX . . . . . . . . . .

117

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Meaning, Semantics, and Reductionism . . . . . . . . . . . . . . ‘Semantics’ and the theory of meaning . . . . . . . . . . . . . . . . . . A Fregean framework for semantics and metaphysics . . . . . . . Theoretical semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensional semantics (a. k. a. classical model theory) . . . . . . . Intensional semantics (a. k. a. possible world semantics) . . . . . Rigidity and direct reference in intensional semantics . . . . . . . Two-dimensional semantics . . . . . . . . . . . . . . . . . . . . . . . . . . Two reductionist programs . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 120 122 127 131 135 136 140 153

Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Singular Terms and the Spectrum of Liar Sentences . . . Reference and self-reference . . . . . . . . . . . . . . . . . . . . . . . . . . Names, descriptions, and indexicals . . . . . . . . . . . . . . . . . . . . ‘This sentence’ is sententially indexical. . . . . . . . . . . . . . . . . . The full spectrum of Liar sentences . . . . . . . . . . . . . . . . . . . . Quotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quotation and the irreflexivity of proper parthood . . . . . . . . Formal quotation and the well-foundedness of syntax . . . . . .

159 159 161 178 179 183 191 192

Chapter 6 6.1

Is There a Gödelian Liar Sentence? . . . . . . . . . . . . . . . . The Quinean and the Smullyanesque variant of the Liar paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gödelian construction of a self-referential sentence . . . . The semantics of Gödelian reference . . . . . . . . . . . . . . . . . . Is there a Gödelian variant of the Liar paradox? . . . . . . . . . .

.

203

. . . .

204 212 220 228

THE METAPHYSICS OF EXPRESSIONS AND THE LIAR PARADOX . . . . . . . . . . . . . . . . . . . . . .

233

How Can the Liar Paradox be Solved? . . . . . . . . . . . . . . . What we have learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Desiderata for a solution to the Liar paradox . . . . . . . . . . . . . Reductio ad absurdum and paradox . . . . . . . . . . . . . . . . . . . . The Liar paradox is as yet unsolved. . . . . . . . . . . . . . . . . . . . . The spectrum of modern approaches to the Liar paradox . . . . The limits of semantic contextualism . . . . . . . . . . . . . . . . . . .

235 235 238 242 244 246 251

6.2 6.3 6.4

PART III Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6

Table of Contents

Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

13

The Changing World of Expressions . . . . . . . . . . . . . . . . Semantic sentences as primary truth bearers . . . . . . . . . . . . . . The distinctness of an expression and its basis . . . . . . . . . . . . Grounding, ontological dependence, and coincidence . . . . . . Social ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressions as qua objects . . . . . . . . . . . . . . . . . . . . . . . . . . . An example by Kit Fine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The contextualist ontology of expressions and the internal stance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 257 263 265 269 270 274

Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12

Language in a Changing World . . . . . . . . . . . . . . . . . . . . Tense logic and temporal context-sensitivity . . . . . . . . . . . . . . Future objects: Contextual restrictions on reference . . . . . . . . Past objects: Identifiability without existence . . . . . . . . . . . . . The modal logic of existence and identifiability . . . . . . . . . . . Future objects: Irreducibly conceptual talk . . . . . . . . . . . . . . . The logic of objectual and conceptual quantification . . . . . . . The irreducibility of sense and of concepts . . . . . . . . . . . . . . . Irreducible sense in the semantics of singular terms . . . . . . . . ‘The first child born a hundred years hence’ . . . . . . . . . . . . . . Reference, grounding, dependence, and accessibility . . . . . . . The flexibility of language . . . . . . . . . . . . . . . . . . . . . . . . . . . Metaphysico-Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

281 282 284 286 287 290 292 296 297 300 303 313 314

Chapter 10 10.1 10.2 10.3 10.4 10.5 10.6

Changeable Truth in Language and World . . . . . . . . . . A dual framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The role of the intensional semantics . . . . . . . . . . . . . . . . . . . The role of the two-dimensional semantics . . . . . . . . . . . . . . . The Tarskian equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of the dual framework . . . . . . . . . . . . . . . . . . . . . . Criteria for truth and falsity in terms of two variable domains . . . . . . . . . . . . . . . . . . . . . . The two domains in terms of existence and identifiability . . . Criteria for truth and falsity in terms of existence and identifiability . . . . . . . . . . . . . . . . . . The Generalized Truth Equivalence . . . . . . . . . . . . . . . . . . . . Different kinds of truth value gaps . . . . . . . . . . . . . . . . . . . . . Solution to the problem of existence without identifiability . . . . . . . . . . . . . . . . . . . . . Kaplan’s puzzle and the status of our solution . . . . . . . . . . . .

315 316 319 326 332 338

10.7 10.8 10.9 10.10 10.11 10.12

276

340 343 346 348 358 365 371

14

Table of Contents

Chapter 11 A Self-Referential Language as Its Own Changing World I: Subsequentist Metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overlap between language and world . . . . . . . . . . . . . . . . . . . 11.2 Self-referential expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Some weird sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Names due to circular baptism? . . . . . . . . . . . . . . . . . . . . . . . 11.5 Names due to circular definition? . . . . . . . . . . . . . . . . . . . . . . 11.6 The problem of ontological co-dependence . . . . . . . . . . . . . . 11.7 The problem of metaphysical ill-foundedness . . . . . . . . . . . . . 11.8 From well-foundedness to subsequence . . . . . . . . . . . . . . . . . 11.9 The problem of referential stickiness . . . . . . . . . . . . . . . . . . . 11.10 The problem of semantic magicality . . . . . . . . . . . . . . . . . . . . 11.11 ‘This sentence’ exhibits the four problems, too. . . . . . . . . . . . 11.12 The principle of subsequence for directly referential expressions . . . . . . . . . . . . . . . . . . . . . 11.13 Drawing the line in the right place . . . . . . . . . . . . . . . . . . . . . 11.14 A social contract about ontology . . . . . . . . . . . . . . . . . . . . . .

410 416 420

Chapter 12 A Self-Referential Language as Its Own Changing World II: Subsequentist Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Some weird sentences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Objects that describe themselves? . . . . . . . . . . . . . . . . . . . . . . 12.3 From non-existence to non-identifiability . . . . . . . . . . . . . . . 12.4 The aim of semantic uniformity . . . . . . . . . . . . . . . . . . . . . . . 12.5 The problem of referential entanglement . . . . . . . . . . . . . . . . 12.6 ‘This sentence’ exhibits referential entanglement, too. . . . . . . 12.7 The principle of subsequence for identifiability . . . . . . . . . . . 12.8 Assigning contexts to inscriptions . . . . . . . . . . . . . . . . . . . . . 12.9 Subsequentism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 426 430 434 443 449 457 461 481 497

Chapter 13 13.1 13.2 13.3 13.4 13.5 13.6

Subsequentism Solves the Liar Paradox . . . . . . . . . . . . . . Subsequentism and how it solves the Liar paradox . . . . . . . . . The Liar paradox in the dual framework of chapter 10 . . . . . . A plethora of paradoxical sentences? . . . . . . . . . . . . . . . . . . . Other paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parting with tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

503 504 526 545 573 580 621

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

631

375 376 379 383 386 388 390 392 395 398 399 402

PART I

LOGIC AND THE LIAR PARADOX

Raymond Smullyan: “This joke is not funny!”

Chapter 1

The Liar and Its Kind

Rosemarie Rheinwald: “Dabei hängt dem Ergebnis häufig etwas Unerwartetes, Überraschendes an. Das Gefühl, das eine Paradoxie auslöst, ist ein Gefühl der Verwirrung.” 1

The Liar paradox poses a serious challenge for logic, and it prompts us to think deeply about language – about the meaning of linguistic expressions as well as about their metaphysical nature. In fact, paying close attention to how the metaphysical nature of expressions interacts with their meaning will allow us to solve the paradox. Or so I will argue in the present study about Logic, Language, and the Liar Paradox. The study is divided into three parts, corresponding to the three philosophical subdisciplines of logic, semantics (which is concerned with the meaning of expressions), and metaphysics – in particular, the metaphysics of expressions (which is concerned with the metaphysical nature of expressions). The aim of the first part is to look at the Liar paradox from the viewpoint of logic, in order to assess how severe a problem it is, and of what kind. The present chapter introduces the Liar paradox and uses it as an occasion to clarify the notion of a paradox. The following chapters 2 and 3 will present the reasoning of the Liar paradox in detail from the angle first of informal logic and then of formal logic. In the second part of the study we will look at the Liar paradox from the viewpoint of semantics, and in the third part we will look at it from the viewpoint of the metaphysics of expressions, and solve it. In this first chapter we will, after a look at the concept of a paradox in general (in section 1.1), present the Liar paradox (in section 1.2) and some of its close relatives (in sections 1.3 and 1.4). Then we will turn to the history of the Liar paradox: After a brief remark about its pre-modern phase (in section 1.6), we will concern ourselves with set theory and Cantor’s diagonal argument (in section 1.7), which constitutes the starting point of the modern history of the Liar paradox, and led to interest in a larger group of paradoxes (in section 1.8). We will conclude the chapter with a quick review of some typical opinions about how problematic the Liar paradox is, which will raise the question whether it looks daunting only to those philosophers who prefer to work in a formal way (in sections 1.9 and 1.10).

1

“And often, the result is unexpected and surprising. The feeling that is evoked by a paradox is a feeling of confusion.” (Rheinwald 1988, 9; my translation) The motto for part I is taken from Smullyan 2006, 152.

18

Chapter 1 The Liar and Its Kind

1.1 The concept of a paradox Before turning to the Liar paradox, we should characterize the concept of a paradox in general. Willard Van Orman Quine asks: “May we say [. . .] that a paradox is just any conclusion that at first sounds absurd but has an argument to sustain it?” 2 And with Quine, we can give an affirmative answer. Loosely speaking, any surprising and counterintuitive result of a given piece of seemingly sound reasoning can be called a ‘paradox’. 3 We speak, for instance, of the paradox of Achilles (which is about the runner Achilles who cannot overtake the turtle although he is faster) 4 and, in connection with the theory of special relativity, we speak of the twin paradox (which is about two people who are of markedly different age although they were born on the same day). Clearly, either the paradoxical reasoning must be rejected despite its seeming soundness (as in the case of Achilles) or the paradoxical result must be accepted despite its initial counterintuitivity (as in the case of the twins). That is, a paradox is a problem that calls for a solution. Now, in order to speak less loosely, the notions of seemingly sound reasoning and of a surprising and counterintuitive result should be made more precise. This is standardly done in logical terms, by representing a piece of reasoning as a collection of premises which form an argument, the conclusion of which stands in for the result of the reasoning. On this basis we can define the notions that we will work with: (Def. Paradox)

A paradox is an argument that appears to be valid from premises that appear to be true to a conclusion that appears to be unacceptable. 5

(Def. Solution)

A solution to a given paradox is an explanation 6 that dispels some of the appearances that are mentioned in (Def. Paradox) as illusory.

(Def. Antinomy) An antinomy is a paradox with a contradictory conclusion. Some remarks on these definitions and the reasons for adopting them. With regard to the definition of a paradox (and consequently that of an antinomy), there is some terminological leeway. 7 Often, it is not the argument as a whole but only its conclusion that is called a ‘paradox’ (or an ‘antinomy’). 8 But an unacceptable (or a contradictory) statement that stands alone is just that, unaccept-

2 3 4 5

6 7 8

Quine 1976a, 1. Cf. Clark 2007, 151–154 and Sainsbury 1988, 1ff. Cf. Salmon 1970 and Rheinwald 2012[1993]. This is just a slight terminological variation on a definition given by Greg Restall: “A paradox is a seemingly valid argument from seemingly true premises to a conclusion that is seemingly unacceptable” (Restall 1993, 281). More will be said about explanatoriness in sections 7.2 and 7.3. I would like to thank Severi Hämäri for alerting me to this point. Quine, Sainsbury, and Rheinwald by a ‘paradox’ mean the conclusion of a paradoxical argument. Cf. Quine 1976a, 1; Sainsbury 1988, 1; and Rheinwald 1988, 9.

1.1 The concept of a paradox

19

able (or contradictory). To constitute a paradox, it must be backed by an argument. Hence, we should prefer the usage according to which a paradox is an argument, of course with a paradoxical conclusion. 9 Further reasons for preferring this usage will emerge later. 10 The boundary between antinomies and paradoxes that are not antinomies is less clear – or rather, less stable – than might be expected. To be sure, the paradigmatic formulations of the Liar paradox and Russell’s paradox are antinomies. 11 And equally clearly, the standard version of the paradox of Achilles and the twin paradox are not antinomies, because the paradox of Achilles is standardly construed as an argument with a non-contradictory conclusion that is false (that Achilles cannot overtake the turtle) and the twin paradox is standardly construed as an argument with a non-contradictory conclusion that seems to be false (that twins can be of markedly different age). But it is possible in both cases to generate an antinomic variant by adding the negation of the usual conclusion as a further premise. Zeno, for instance, had he argued differently, might have concluded that Achilles can and cannot overtake the turtle. And even the Liar paradox has a non-antinomic variant which concludes with the negation of one of the usual premises. 12 In fact, every antinomy can be transformed into a paradox that is not an antinomy and vice versa, in such a way that the transformation preserves the validity of the argument. 13 The reason for the instability of the status of being an antinomy is that the individuation of paradoxes is rather coarse-grained, because the same paradox can come in the form of different arguments. 14 To belong to the same paradox, these variants of course have to form a close-knit 15 family. (The possibility of reformulating a paradox by moving on to an argument that is in a sense equivalent constitutes another reason to let arguments and not their conclusions be the range of application for the notion of a paradox.) It seems to be a common impression that antinomies are problems of a more severe nature than non-antinomic paradoxes, presumably because being contradictory is seen as a worse predicament for a claim or sentence to be in than simply being false. But unacceptable is unacceptable: It is by no means the case that a theory (say) with a false consequence is somehow more acceptable than an otherwise similar

9

10 11 12 13

14 15

Mackie and Restall by a ‘paradox’ mean an argument with a paradoxical conclusion. Cf. Mackie 1973, 238; and Restall 1993, 281. Cf. sections 1.4 and 1.5. Cf. sections 1.2, 1.6, and 1.8. To anticipate, cf. steps (1) through (10) of Informal Variant 5 of the basic Liar reasoning in section 2.4. In any not too non-standard logic, a valid argument from the premise that p and a collection Γ of other premises to a contradiction can be recombined to a valid argument from the other premises Γ to the conclusion that it is not the case that p. And a valid argument from a collection Γ of premises to the conclusion that p can be recombined to a valid argument from those premises and the additional premise that it is not the case that p to a contradictory conclusion. To anticipate, we will look at several distinct variants of the basic Liar reasoning in section 2.4. We should not try to be very precise on this point; paradoxology should not be carried too far. Paradoxes, after all, are mistakes and intimately tied to what must be false appearances, so that they make for an evasive subject.

20

Chapter 1 The Liar and Its Kind

theory with a contradictory consequence – in both cases there is a problem that must be solved. 16 This brings us to a question which will later turn out to have an interesting answer: What is it that makes a conclusion of a paradox appear to be unacceptable? We have seen two clear cases, the appearance of falsity and the appearance of contradictoriness. But are these all the options? Later on we will see that they are not, because there are arguments with a true conclusion which strike us as paradoxical. 17 (Here it should be enough to remind us of the paradoxes of material implication: Because of the classical understanding of implication, every claim – and hence every true claim – is implied by any contradictory claim. If we want to classify each one of the arguments of this form as paradoxical, we already have found many paradoxes with a true conclusion. 18) * Now let us turn to the relation between a paradox and its solution. Mark Sainsbury is of course right when he observes, with regard to a definition of a paradox similar to the above (Def. Paradox): “Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable.” 19

This means nothing else than that a paradox must be solved, and it motivates a definition of a paradox’s solution like the above (Def. Solution), which makes explicit its close connection to the definition of a paradox, insofar as it requires of a solution that it removes “some of the appearances that are mentioned in (Def. Paradox)”. This close connection also is a reason to adopt the above (Def. Antinomy), which is a slight deviation from the following classical definition: A classical antinomy is an argument that appears to be valid from premises that appear to be true to a contradictory conclusion. 20 The difference to the notion of an antinomy in our sense becomes evident when we note that (Def. Antinomy) and (Def. Paradox) together entail the following principle: An antinomy is an argument that appears to be valid from premises that appear to be true to a contradictory conclusion that appears to be unacceptable. As every contradictory conclusion is a contradictory conclusion that appears to be unacceptable 21 (and, of course, vice versa), an argument is a classical antinomy just in case it is an antinomy in the above sense. But this extensional equivalence 16

17 18 19 20 21

And of course, a difference in severity would not go well with the possibility just noted of moving to and fro between the status of being an antinomy and that of being a non-antinomic paradox. Cf. sections 1.4 and 1.5. Cf. Sorensen 2005, 105f. Sainsbury 1988, 1. Cf. Haack 1978, 138; Rheinwald 1988, 9. Presuming that even dialetheists, i. e., logicians who accept some contradictions (Priest 2006b, 1; cf. Berto /Priest 2010), will concede that every contradiction at least appears to be unacceptable.

1.1 The concept of a paradox

21

notwithstanding, there is an important difference on the conceptual level, because the classical notion of an antinomy leads naturally to a notion of an antinomy’s solution that is not up to date. Given the classical definition of an antinomy, it is only natural to parallel the above definition of a solution to a paradox and characterize the corresponding notion of a solution to an antinomy by requiring that it dispel some of the appearances mentioned in the classical definition as illusory. Thus, a solution to an antinomy in our sense will consist either in showing the argument to be not valid, or in arguing against the truth of some of its premises, or in giving reasons for its contradictory conclusion to be acceptable, after all. By contrast, the solution to a classical antinomy will consist either in showing the argument to be not valid, or in arguing against the truth of some of its premises – but it is no option to give reasons for the acceptability of the contradictory conclusion. Of course, most people (including myself) will have no problem with this, deeming any contradiction to be unacceptable, period. But by now there is an important group of proposed solutions to the Liar paradox (and related paradoxes) which are based on just that – arguing for the acceptability of some contradictions. 22 So we should not exclude these proposals from the range of acceptable candidates for a solution by the very definitions of a paradox and of an antinomy. * Let us conclude the present section with some clarifactory remarks about the appearances that play a central role in the definition of a paradox (and therefore also in the connected definitions of a paradox’s solution and of an antinomy). Firstly, the appearances should be shared as a package in the sense that it is (in general) to the same people that the premises seem true, the argument seems valid and the conclusion seems unacceptable. Otherwise, any conflict of intuitions among some philosophers would have to be characterized as a paradox. 23 Secondly, the appearances should be intersubjective in the weak sense of being widely shared. 24 If, e. g., the only one who finds some conclusion unacceptable is me, then the corresponding argument is certainly no paradox. Thirdly, there is an element of historicity because the status of being a paradox is changeable. Some paradoxes have, after all, been solved! When enough people change their minds, some of the premises might lose their appearance of truth, the reasoning might lose its appearance of validity, or the conclusion might lose its appearance of unacceptability. It has often been argued, e. g., that the paradox of Achilles today is a paradox in no more than name. 25 There seems to be some tension between the aspects of (weak) intersubjectivity and of historicity because historicity precludes diachronic intersubjectivity. But 22

23 24

25

We will have a brief look at the approach to the Liar paradox of Graham Priest’s and other paraconsistent logicians in section 7.5. I would like to thank Roland Mümken for a conversation which alerted me to this point. “Paradoxes are fun. In most cases, they are easy to state and immediately provoke one into trying to ‘solve’ them.” (Sainsbury 1988, 1) The matter is contentious. While a majority think that the Achilles paradox is solved, there remains a minority of dissenters. See the anthology Salmon 1970 and compare the different assessments in Rheinwald 1988, 11 and Rheinwald 2012[1993], 27.

22

Chapter 1 The Liar and Its Kind

the tension is only apparent. A paradox is usually connected (via the seemingly sound reasoning behind the paradoxical conclusion) to some theory. And, as of now, every theory has been generally accepted only during some phase in history; but during that phase it did have an intersubjective character (because it was generally accepted). What we also find here is a kind of democratic element (in a loose sense of the word): The status of being an (unsolved) paradox is connected to what the majority of people (or more likely, a majority of experts) think at a time. E. g., the main reason for the contemporary near-consensus that the paradox of Achilles has been solved arguably is the impact that mathematical progress had on how the majority of people (that is, the majority of the people who care) today think about infinity. 26

1.2 The Liar paradox The Liar paradox in its typical modern variants that are discussed in philosophy and logic involves a Liar sentence, i. e., a sentence which says of itself that it is false or which says of itself that it is not true. 27 For reasons that will become clear later, 28 a sentence that says of itself that it is false is called a simple Liar sentence and a sentence that says of itself that it is not true is called a strengthened Liar sentence. Unsurprisingly, the Liar paradox is indeed a paradox: When we, with respect to some Liar sentence, ask whether it is true, straightforward reasoning from plausible principles about truth and falsity will lead to a contradiction. Here is one typical way of arguing.

26

27

28

From a philosophy of science point of view, paradoxes have often been praised for the incentive they provide for scientific progress, as foolproof signs of a serious crisis (cf. Bromand 2001, 11f.). I do not want to dispute this as an historical observation about the role paradoxes play in modern sciences. But, pace some philosophers and historians of science, this feature surely is not essential to the concept of a paradox. To anticipate section 1.6, in medieval times paradoxes like the Liar paradox were discussed extensively, but without a sense of serious crisis. In contrast to all of its contemporary academic variants, many popular (e. g., Clark 2007, 112) and most ´ ancient (cf. Bochenski 1956, 151f. and Brendel 1992, 21f.) variants of the Liar paradox are formulated not in terms of falsity or untruth, but of lying. They concern a person who calls himself or herself a liar or a sentence that says of itself that it is a lie, e. g., ‘What I am saying now is a lie’. Now roughly, to lie is to say something false (or untrue) with an intention to deceive (cf. Künne 2013, 24ff. for an extensive discussion; cf. Shibles 1988 for a diverging opinion). The element of deception, however, plays no role in any of the usual variants of the Liar reasoning. Thus the term ‘Liar paradox’ strictly speaking is a misnomer. For this reason, Wolfgang Künne proposes to change the name of the paradox to “the antinomy of falsity” or “f-antinomy” (Künne 2013, 39; my translation). But such attempts at language reform seem hopeless to me. Talk about “the Liar” is well-entrenched in the logic community. And this may be more than a slight terminological carelessness, for as long as the paradox appears so difficult to solve, the connotation of malign intentions feels apt, and is also in keeping with talk about the “Revenge” of the Liar (cf. section 2.6). Cf. section 2.5.

1.2 The Liar paradox

23

(Step 1) Let us say that we are given a simple Liar sentence, i. e., a sentence which says of itself that it is false. Now we ask: Is it true? Is it false? (Step 2) We reason: Suppose that our simple Liar sentence is true. Surely, if a sentence is true then what it means is the case. Hence our simple Liar sentence must be false. It follows that if our simple Liar sentence is true then it is false. (Step 3) We reason: Suppose that our simple Liar sentence is false. Surely, if what a sentence means is the case then the sentence is true. Hence our simple Liar sentence must be true. It follows that if our simple Liar sentence is false then it is true. (Step 4) In sum, our simple Liar sentence is true if and only if it is false. This rather intuitive way of starting the Liar reasoning might suggest that our Liar sentence moves back and forth between truth and falsity, being true at a first stage, hence false at a second stage, hence true at a third stage, hence false at a fourth stage, and so on ad infinitum. And that would be odd, especially as we could not appeal to any element external to the Liar sentence to explain these changes. 29 But the situation is even worse. On the usual logical understanding of ‘if . . . then . . . ’-constructions, there is no leeway for any temporal (or otherwise contextual) variation. If we give a temporal reading to a sentence of the form ‘if p then q’ at all, then it has to be ‘if p, then simultaneously q’. Similarly, we would have to understand the claim we just have argued for as meaning that our simple Liar sentence is true if and only if it simultaneously is false. So we are well advised to set aside again any temporal or contextual understanding and turn back to the Liar reasoning. For we are not finished yet, because only given certain assumptions about how truth relates to falsity will the (simultaneous) truth and falsity of a sentence be contradictory. 30 (Step 5) If we add the further plausible assumption that every sentence is true or false, we can infer: 31 Our simple Liar sentence is true and false. (Step 6) But, surely, no sentence is both true and false. Hence, we can infer: Our simple Liar sentence is true and false and it is not true and false. And that is as contradictory as it gets. We can see that the essential ingredients of the basic 32 Liar reasoning are principles about truth and falsity, rules of logic, and a fact about language. More specifically, 29

30 31 32

We can appeal to an external aspect, the flow of time, in the case of a sentence like ‘Now it is night’, which does seem to switch its truth value two times a day (Hegel 1988, 71). Cf. also the remark after Informal Variant 1 in section 2.4. To anticipate, we use the rule of reasoning by cases; cf. section 2.1. The reasoning about a Liar sentence that leads to a contradiction is basic in so far as it will turn out that there also is an extended Liar reasoning, which unfolds only after a first attempt at solving the paradox has been made. Cf. sections 2.5 through 2.10.

24

Chapter 1 The Liar and Its Kind

we have appealed to the principle that a sentence is true if and only if what it means is the case in (Step 2) and (Step 3) of the above presentation 33 and to the principles that truth and falsity are exhaustive and exclusive in (Step 5) and (Step 6). 34 We have employed certain logical rules in (Step 2) through (Step 6). Both the principles and the rules are in accordance with classical logic (and many other logical systems). 35 The fact about language that is essential to the Liar reasoning is presupposed already in (Step 1). It is that there are indeed self-referential sentences and, in particular, that there are Liar sentences. Prima facie, there are many candidates for singular terms that can be used to form self-referential sentences. Here are some of them in action, i. e., occurring in sentences that are plausible candidates for Liar sentences: 36 (1) ‘This sentence is false.’ (2) ‘Larry is false’, under the assumption that it is named ‘Larry’. (3) ‘(3) is false.’ (4) ‘The fourth sentence that is mentioned in the list of five sentences in section 1.2 of Logic, Language, and the Liar Paradox by Martin Pleitz is false.’ (5) ‘The only sentence written on the whiteboard in room 101 at t0 is false’, under the assumption that that sentence is the only sentence written on the whiteboard in room 101 at t0. Each one of the five sentences mentioned here seems to say of itself that it is false, i. e., each one seems to be a simple Liar sentence. Examples like these and many others have convinced almost everyone currently working in the field of philosophical logic that at least some devices of sentential self-reference are unproblematic. Therefore the relevance of the Liar paradox is seen today by most people to lie in the fact that it calls into question principles about truth and falsity we commonly endorse, or logical rules we commonly employ, or both. For the time being I will refrain from challenging this near-consensus of contemporary logicians. But let me note at the outset that my own proposal for a solution to the Liar paradox will consist in an argument to the conclusion that – despite appearances – the required kind of self-reference is not possible, after all. 37 Because of this aim, we will take a different way than many current presentations when discussing the other essential ingredients of the Liar paradox, our notion of truth and the logic of our reasoning. For those current paradox-solvers who are convinced that there is the kind of self-reference required for the Liar paradox, a discussion of truth and logic must be revisionary, i. e., it must concern how the principles about truth or the rules of logic should be changed to block the paradoxical 33 34 35 36 37

Cf. section 2.2. Cf. section 2.3. Cf. sections 2.1 and 3.3. For better readability, I often omit the period when mentioning a single sentence. In chapter 5 we will investigate more thoroughly the different ways sentential self-reference seems to be achievable; in chapter 6 we will deal with the widespread conviction that the technique of Gödelization allows to form a formal Liar sentence; and in part III, we will argue against the possibility of sentential self-reference of the sort needed to construct Liar sentences.

1.3 Liar cycles and Yablo’s paradox

25

arguments. These ways of dealing with paradox have developed into a rich field of inquiry. 38 In contrast, we do not need to concern ourselves overmuch with logical revision in the present study about the Liar paradox. There might of course be good reasons to explicate 39 our notion of truth in a non-standard way or to explicate our reasoning in a way that deviates from classical logic. But if I am right, the Liar paradox and its relatives are not among these reasons. On our way to show that, we will of course look at current explications of truth and logic – but not with a view of revising them, but rather to ask what they are, and how they relate to each other and to our understanding of language. As these questions are interesting in their own right, our aim in the present study really is twofold: We want to use the Liar paradox as an occasion to think about the nature of logic and the nature of language, and we want to solve it.

1.3 Liar cycles and Yablo’s paradox In the last section we had a first glance at the variety of options for singular terms which seem to allow the construction of self-referential sentences. Now, we will turn to systems of more than one sentence which achieve Liar-like results: the finite loops used to form Liar cycles and the infinite sequences which are the root of Yablo’s paradox. A Liar cycle is a construction of two or more, but finitely many, sentences which refer to each other in a way that leads to a situation as paradoxical as that engendered by a Liar sentence. An example medieval logicians were fond of concerns Plato and Socrates, who speak simultaneously: Plato: ‘What Socrates is saying right now is true.’

Socrates: ‘What Plato is saying right now is false.’

There are Liar cycles of any number of sentences, as the following example of four sentences illustrates: (1) ‘(2) is true.’ (2) ‘(3) is true.’ (3) ‘(4) is true.’ (4) ‘(1) is false.’ 38

39

Cf. section 7.5 for a concise overview of the spectrum of approaches to the Liar paradox, including approaches of logical revision. For the notion of explication, cf. Carnap 1956[1947], 7f. Carnap writes: “The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication for the earlier concept; this earlier concept, or sometimes the term used for it, is called the explicandum; and the new concept, or its term, is called the explicatum of the old one.” (Carnap 1956[1947], 7f.) In Quine’s concise formulation, to explicate is to “devise a substitute, clear and couched in terms of our liking, that fills the functions” (Quine 1960, 259).

26

Chapter 1 The Liar and Its Kind

The purpose of examples like these – besides the fun they seem to stir in some people – is to show that sentential self-reference is not necessary for the Liar paradox (nor for paradoxes much like it). Some have even denied that the notion of the referential circularity involved in Liar cycles is intelligible. 40 But as we already have an understanding of what it is for a sentence to refer to an object, 41 it is easy to define what it is for a sentence to refer* to an object, namely to be connected by an appropriate referential chain of sentences to the object. (In logical jargon, the relation of reference* is the transitive closure of the relation of reference.) And clearly, every sentence in a Liar cycle refers* to itself. A more severe test for the claim that self-reference or at least some kind of circularity is at the root of Liar-like paradoxes is Yablo’s paradox. 42 It concerns an infinite sequence of sentences, each one saying that all the following ones are false. One example would be the following Yablo sequence: S1 =def ‘For every i greater than 1, Si is false.’ S2 =def ‘For every i greater than 2, Si is false.’ S3 =def ‘For every i greater than 3, Si is false.’ ... Another Yablo sequence makes use only of indexicals: ‘Every sentence in this sequence that is later than this one is false.’ ‘Every sentence in this sequence that is later than this one is false.’ ‘Every sentence in this sequence that is later than this one is false.’ ... Like Liar sentences, Yablo sequences produce paradox. For suppose that, for some number n, the nth sentence is true. Then all following sentences must be false. But then the n+1th sentence would have to be true, and hence true and false. As this is a contradiction, the nth sentence is false. And as this argument works for any number n, we can use it to show that every sentence in the Yablo sequence is false, and, in particular, that every sentence that follows the first is false. But then the first sentence must be true, hence true and false, which is a contradiction. Although some people have tried to argue that there is some hidden circularity at work in Yablo’s paradox, this is far from obvious – and the matter is highly contentious. 43 As in the case of Liar sentences, it seems to be difficult to deny that there are Liar cycles and Yablo sequences. Therefore the paradoxes that they engender are 40 41

42 43

Cf. Leitgeb 2002. To anticipate, we will say that a sentence refers to an object if and only if some singular term that is used in that sentence refers to that object. Cf. section 5.1, where we will say more about the relata of the relation of reference, primary and derivative. Yablo 1993. There is a small but lively debate about whether Yablo’s paradox is circular in some sense; cf. Priest 1997, Sorensen 1998, Beall 2001, Leitgeb 2002, Cook 2004a & 2004b, Ketland 2005, and Yablo 2006. For a book-length study about Yablo’s paradox, cf. Cook 2014.

1.4 Church’s paradox

27

important additions to the Liar paradox, which put further pressure on our common theories about truth and logic.

1.4 Church’s paradox Due to an early Christian’s derogatory remark about the inhabitants of Crete being documented in the Bible, a certain variant of the Liar paradox has become popular according to which a Cretan utters the sentence ‘All Cretans are liars’. 44 As that particular Cretan was identified as Epimenides of Knossos, this variant became known as “the paradox of Epimenides”. 45 To have a chance of being a starting point of a variant of the usual basic Liar reasoning, the utterance of Epimenides must be understood to mean that every sentence uttered by a Cretan is a lie – or rather, false. But that is not enough. For note that in the highly probable event that some other sentence uttered by a Cretan is true, the utterance of Epimenides, understood in this way, is not contradictory but just false. This has often been pointed out; for instance by Quine, who chided: “Actually, the paradox of Epimenides is untidy; there are loopholes. Perhaps some Cretans were liars, notably Epimenides, and others were not; perhaps Epimenides was a liar who occasionally told the truth; either way it turns out that the contradiction vanishes.” 46

Therefore, for the reasoning to be paradoxical in the usual way, the slightly slanderous 47 assumption is needed that all other sentences uttered by a Cretan (including all other sentences uttered by Epimenides) are false. As many people have treated the problem generated by the utterance of Epimenides simply as a variant of the Liar paradox, a principle of charity demands that we think of them as adding the slanderous assumption, albeit tacitly. Thus, the Epimenides paradox is the paradox that starts from the two assumptions, that, firstly, there is a sentence which means that all sentence of some sort are false, which is of that sort, and that, secondly, all other sentences of that sort are false. 48 Treating the paradox of Epimenides as a contingent – and improbable – variant of the Liar paradox does not mean to belittle its paradoxicality. But Alonzo Church has observed that even without making the improbable extra assumption we are prone to fall into a deep problem when we ponder the utterance of Epimenides. Had not Arthur Prior and John Mackie taken it up later, Church’s observation might well have gone unnoticed, 49 because he formulated it in only two sentences of a brief (but devastating) review of a text by Alexandre Koyré about the Liar paradox. There Church draws attention to a consequence of Epimenides making his utterance . . . 44

45 46 47 48 49

Cf. Anderson 1978[1970], 1ff. – It is unknown since when the paradox of Epimenides is well-known; it was not discussed in the medieval insolubilia literature (Spade /Read 2009, section 1.2). Cf. Anderson 1978[1970], 2f., as well as the more recent and more detailed Künne 2013, 48–66. Quine 1976a, 6. – See also, e. g., Anderson 1978[1970], 3; Haack 1978, 136. We see that the anti-Cretan sentiment of the Biblical passage seeps through even to the logical level. For an explicit characterization, cf. Rheinwald 1988, 18f. Church 1946; Prior 1961; and Mackie 1973, 276ff.

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Chapter 1 The Liar and Its Kind

“[. . .] which, though not outright antinomy, might well be classed as paradox. Namely, without factual information about other statements by Cretans, it has been proved by pure logic (so it seems) that some other statement by a Cretan, not the famous statement of Epimenides, must once have been true.” 50

For suppose that some Cretan in fact utters ‘Every sentence uttered by a Cretan is false’. Then we can turn the Liar reasoning of the paradox of Epimenides into a reductio argument for the claim that there is another, true, sentence uttered by a Cretan. We need only assume for reductio that all other Cretan utterances are false, follow the steps of the usual Liar reasoning 51 to infer a contradictory claim, and conclude that there must be some Cretan utterance that is true. Even in the very probable case that this conclusion is true, it leads to a further conclusion that is obviously unacceptable: The reasoning establishes a necessary connection between two distinct contingent states of affairs (for if there is the utterance of Epimenides, then it must be the case that there is some true Cretan utterance). In the words of Mackie, this “would violate Hume’s principle”; 52 and it should fly in the face even of those who are less skeptical of modality than Hume because it contradicts some plausible principles about linguistic utterances, especially regarding the high degree of compatibility between distinct possible utterances. Let us call this unacceptable reductio argument Church’s paradox – to distinguish it from the more Liar-like paradox of Epimenides. While the paradox of Epimenides rests on Epimenides’s utterance contingently producing an antinomy, Church’s paradox shows that the same utterance necessarily produces a paradox, albeit a non-antinomic one. As noted before, 53 its not being an antinomy need not make Church’s paradox a problem less severe than the Liar paradox. (We will have another look at non-antinomic paradoxes in the next section.)

1.5 Curry’s paradox There is a relative of the Liar paradox that is problematic and interesting in its own right. It is Curry’s paradox and involves what we can call a Curry sentence. One specific example for a Curry sentence is Curry Snow, which means that if Curry Snow is true, then snow is white. Surprisingly, reasoning about this sentence that appeals only to the same principle about truth that we have used in the Liar reasoning 54 and employs a few uncontentious logical rules allows to infer that snow is white. Even more surprisingly, we can in a similar way infer any arbitrary claim, be it true, false, or even contradictory, when we start from a Curry sentence with a consequent that expresses that claim. 50 51 52 53 54

Church 1946, 131. E. g., (Step 1) through (Step 6) in section 1.2. Mackie 1973, 276. Cf. section 1.1. More specifically, in Curry’s paradox we use the principle that says that a sentence is true if and only if what it says is the case, but we need not appeal to the principles that say that truth and falsity are exhaustive and exclusive.

1.5 Curry’s paradox

29

Here is the reasoning involved in this inference for the case of the specific Curry sentence Curry Snow: 55 (Step 1)

Let us say that we are given a specific Curry sentence, which says of itself that if it is true then snow is white. Let us call that sentence ‘Curry Snow’.

(Step 2)

We reason: Suppose that Curry Snow is true. Surely, if a sentence is true then what it means is the case. Hence if Curry Snow is true, then snow is white. Now, as we have supposed that Curry Snow is true, it follows that snow is indeed white. So, we have used the assumption that Curry Snow is true to show that snow is white. In other words, we have shown: If Curry Snow is true, then snow is white.

(Step 3)

But that is exactly what Curry Snow means! And surely, if what a sentence means is the case, then that sentence is true. Hence Curry Snow must be true.

(Step 4)

In (Step 2) we have shown that if Curry Snow is true, then snow is white; in (Step 3) we have shown that Curry Snow is true. Taking these two claims together, we can infer: Snow is white.

We can see that, besides the existence of our Curry sentence, nothing is involved in the Curry reasoning but the principle that a sentence is true if and only if what it means is the case and a few logical rules. In particular, these are the rules which govern the logical behavior of classical conditionals and, under the classical understanding, of ‘if . . . then . . .’-constructions of natural language. 56 From the form of the Curry reasoning presented above it is obvious that any sentence can take the place of the sentence ‘snow is white’ as the consequent of a Curry sentence, e. g., the sentence ‘snow is not white’ or even the sentence ‘snow is white and snow is not white’. Many presentations concern a claim that is more surprising than the one used here, more often than not a contradictory or false claim – “for effect”, as Jc Beall says 57. This can turn the Curry reasoning into, 55

56 57

Anticipating some material from chapter 2, the reasoning can be given more concisely: (1) Curry Snow means that if Curry Snow is true, then snow is white.

meaning of the Curry sentence

(2) Curry Snow is true.

assumption for conditional proof

(3) If Curry Snow is true then snow is white.

(1), (2), naïve truth principle

(4) Snow is white. (5) If Curry Snow is true then snow is white.

(2), (3), modus ponens (2)–(4), conditional proof

(6) Curry Snow is true.

(5), (1), naïve truth principle

(7) Snow is white. (5), (6), modus ponens To anticipate, these are the rules of modus ponens and conditional proof. Cf. sections 2.1 and 3.3. Beall 2009, section 2.3.

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say, an easy proof that God exists. 58 As variants of Curry’s paradox can be used to show any arbitrary claim, it has been seen by some as a problem even more severe than the Liar paradox. Anil Gupta and Nuel Belnap, e. g., state about the instances of the principle that a sentence is true if and only if what it says is the case that “they are inconsistent not only on a few isolated points; as Curry’s paradox shows, they are thoroughly inconsistent”. 59 Of course, given the logical rule of ex falso according to which any contradiction entails any arbitrary claim, 60 the Liar paradox in the end is no less inconsistent than Curry’s paradox, because applying the ex falso rule to the contradiction that is inferred in the basic Liar reasoning also allows to infer any arbitrary claim and thus both paradoxes lead to the most severe form of inconsistency, triviality. But Curry’s paradox allows to reach triviality more directly than the Liar paradox. And as the ex falso rule is not needed in the Curry reasoning, there are logical systems according to which the Liar paradox allows to infer only some contradictory claims whereas Curry’s paradox still allows to infer every contradictory claim. Therefore Curry’s paradox constitutes a serious extra problem for those approaches to the Liar paradox that are based on rejecting the ex falso rule. * But, evidence of seriousness notwithstanding, it is prima facie questionable whether Curry’s paradox fits the notion of a paradox we work with. 61 We have defined a paradox as an argument that appears to be valid from premises that appear to be true to a conclusion that appears to be unacceptable; and an antinomy as a paradox with a contradictory conclusion. 62 The reason for doubting that Curry’s paradox satisfies this condition is that in cases like the above example, there seems to be nothing unacceptable about the conclusion of the Curry reasoning. For, as we all know, snow is white! It is of course possible to regiment the meaning of the label ‘Curry’s paradox’ such that it applies to a whole bundle of Curry arguments, starting from a collection of different Curry sentences, which together will allow to infer contradictory claims. Then the conclusion of Curry’s paradox will indeed be a contradiction. 63 – But although this bundling strategy of construing Curry’s paradox as an antinomy fits well with the indications of seriousness we saw before, it fails to explain the feeling of paradoxicality or absurdity we have even in the case of those single Curry arguments which end with something as uncontentious as the claim that snow is white. For is it not strange to infer from a meaning fact and a principle about truth, using only logical rules, a claim that is empirically falsifiable? 58 59 60 61 62 63

Cf. Prior 1955, 177f. Gupta /Belnap 1993, 15. Cf. section 2.1. I would like to thank Elia Zardini for alerting me to this point. Cf. also Pleitz 2015a, 239f. Cf. section 1.1. Also, we can of course start with a Curry sentence with an unacceptable consequent, and if we want to make sure we land in a paradox in the strict sense, with a Curry sentence Curry Contradiction, which can be translated as ‘If Curry Contradiction is true then (snow is white and snow is not white)’. But this reaction is unsatisfactory because it does not explain why every instance of Curry’s paradox feels paradoxical.

1.5 Curry’s paradox

31

And the feeling of paradoxicality should stay with us even when we regard those Curry arguments which allow to infer an instance of a logical law, starting for example from a Curry sentence that means that if it is true then (snow is white if and only if snow is white). The real problem about these Curry arguments does not lie in their conclusion (as such, or in isolation) being unacceptable, but in the fact that an acceptable conclusion is reached way too easily. As Roy Sorensen observes: “The paradox can be in how you prove something rather than in what you prove. This point causes indigestion for those who say that all paradoxes feature unacceptable conclusions. Their accounts are too narrow.” 64

Although I do agree with Sorensen that the conclusions of some paradoxical arguments are by themselves acceptable, I dare dissent from his diagnosis of indigestion. There is a way to achieve harmony between the intuitive paradoxicality of even those Curry arguments with harmless conclusions and the notion of a paradox – by being liberal concerning the meaning of the term ‘unacceptable conclusion’. We need only say that the conclusion of an argument can be unacceptable in two ways: firstly, by the claim that is the conclusion being unacceptable in itself (usually because it is contradictory or obviously false), and secondly, by the claim that is the conclusion being unacceptable as the conclusion of the given argument. If a conclusion is unacceptable in this second way, then although the conclusion surely does follow, given the premises and the logical rules that are used, it nonetheless intuitively does not seem to follow. And this is what is paradoxical about the above Curry argument for the claim that snow is white, and about the Curry argument for the claim that if snow is white then snow is white. 65 * A brief remark on the history (and name) of Curry’s paradox: In 1942, it was stated in a set theoretic formulation by Haskell Curry (hence the name) who used it to show that there can be paradox without negation. 66 In 1955, the paradox was restated in a meta-mathematical paper by Martin Hugo Löb, as a comment on his proof of what has become known as Löb’s theorem. 67 (The reasoning in the proof of Löb’s theorem bears a far-reaching similarity to the Curry reasoning, but Löb does not mention Curry in his text. One or the other of these two facts probably is the reason for Curry’s paradox sometimes going under the name of ‘Löb’s paradox’ 68.) Also in 1955, Curry’s paradox made its way from mathematical logic to less formal parts 64 65

66 67 68

Sorensen 2005, 106. Sorensen made the observation quoted above not with regard to Curry’s paradox, but with regard to the paradoxes of material and of strict implication (Sorensen 2005, 105f.). The present strategy, according to which the unacceptability of the conclusion of a paradoxical argument need not be intrinsic to the claim that is the conclusion, obviously can be transferred from Curry’s paradox to the paradoxes of implication. It explains, e. g., why the argument from the claim that one is prime and not prime to the claim that Socrates is famous can be classified as paradoxical despite the truth of its conclusion. Curry 1942. Löb 1955. E. g., Barwise/Etchemendy 1987, 23. Cf. Boolos /Burgess/Jeffrey 2002, 237f., where an instance of the Curry reasoning is used to motivate the proof of Löb’s theorem, but Curry is not mentioned.

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of philosophy, when Peter Geach and Arthur Prior explained it in (somewhat) less technical terms than Curry and Löb. 69 In recent years, Curry’s paradox has received growing attention due to its importance in discussions about those approaches to the Liar paradox that depart from classical logic. 70 Curry sentences, in contrast to Liar sentences, are a modern invention. It has been pointed out by Stephen Read, however, that there are forerunners among medieval paradoxes of validity. 71 Corresponding to the Curry sentence used above, we need only consider the following self-referential argument: This argument is valid. Hence, snow is white. Simple reasoning about this argument will allow to infer that snow is white.

1.6 The history of the Liar paradox Some problems have been of interest to philosophers during all of European and Western history, but the Liar paradox was discussed only in three phases which had long pauses and few intertextual connections 72 between them. The ancient phase had its most intense period in the fourth and third century BCE, but it continued long after that. 73 The medieval phase started in the twelfth century, had an especially fruitful period during the fourteenth century, and continued on beyond the fifteenth century. 74 And our ongoing modern phase of discussing the Liar paradox started only at the end of the 19th century. 75 The ancient history of the Liar paradox begins with its invention – or discovery 76 –, which is generally attributed to Eubulides the Megarian. 77 While Plato did

69 70 71 72

73 74 75

76

77

Geach 1955; Prior 1955. Cf., e. g., Beall 2009 and the literature mentioned there. Cf. Read 1979. Paul Vincent Spade and Stephen Read show that there have been surprisingly few links between the ancient and the medieval phase (Spade/Read 2009, section 1). And although there is a considerable amount of contemporary interest in medieval approaches to the Liar paradox, this must be seen as a re-discovery, because (to anticipate what we will see at the end of this section) the roots of modern renewed interest in the Liar paradox do not lie in a philosophical tradition but in new developments in mathematics that occurred at the end of the 19th century. ´ Spade /Read 2009, section 1. Cf. Bochenski 1956, 150–153, as well as Künne 2013, 119ff. ´ Spade /Read 2009. Cf. Bochenski 1956, 277–292 and Spade 1975. ´ Much of the pre-modern material on the Liar paradox is collected in Rüstow 1910, Bochenski 1956, and Spade 1975. (This material (as usual) belongs solely to European philosophy; research into Arab treatments of the Liar paradox is only beginning; cf. Alwisha/Sanson 2009.) Overviews of the medieval period are provided by Dutilh Novaes 2008, Spade /Read 2009, and some passages in Sorensen 2005. While even someone with only slight platonist inclinations will hold that a proof is out there to be discovered, it must be doubtful that a paradox, although it also is an argument, can be accorded more independence from thinking beings than the artifacts they create. The reason for this is the important part played by appearances for the existence of a paradox (cf. section 1.1). ´ Bochenski 1956, 151; Spade /Read 2009, section 1.

1.6 The history of the Liar paradox

33

not discuss it, 78 Aristotle proposed a solution, 79 and Chrysippus and others are reported to have produced copious writings about it (most of which are lost). 80 But in spite of this productivity, and pace Philetas of Kos (who according to an often told anecdote worried so much about the Liar paradox that he lost his night’s sleep and in the end also his life), 81 it seems that in general no great systematic relevance was attributed to the Liar paradox even during the heyday of ancient discussions. Later on, it was seen as no more than a curiosity, meriting only brief remarks from Latin authors like Cicero and Seneca. 82 Medieval European philosophers discussed the Liar paradox extensively as one of a group of problems they called “insolubilia” but by no means deemed to be unsolvable 83. They produced an amazing plurality of approaches. Most of these can be seen as precursors of contemporary approaches; 84 and some are of great systematic interest from a contemporary perspective: Arthur Prior turned to John Buridan’s approach for inspiration, 85 and Stephen Read has done much to articulate and defend the subtle approach of Thomas Bradwardine. 86 But the huge amount of work medieval philosophers spent on the Liar paradox should not be seen as a sign that they were greatly troubled by it. This is pointed out by Paul Vincent Spade: “the medievals did not seem to have had any ‘crisis mentality’ about these paradoxes. Although they wrote a great deal about them, there is no hint that they thought the paradoxes were crucial test cases against which their whole logic and semantics might fail. [. . .] the medievals did not draw great theoretical lessons from the insolubles. They did not seem to think the paradoxes showed anything very deep or important about the nature of language and its expressive capacity”. 87

And Catarina Dutilh Novaes, who otherwise is more cautious in her assessment than Spade, 88 observes: “As Barwise and Etchemendy put it, ‘the significance of a paradox is never the paradox itself, but what it is a symptom of’ [. . .]; the medieval authors, by contrast, were mostly interested in the paradoxes themselves, as particularly difficult logical puzzles.” 89

An even greater air of unconcern characterizes two discussions of the Liar paradox that were produced in the mid 19th century, by Bernard Bolzano and Charles 78 79 80 81 82

´ Bochenski 1956, 151. Aristotle, Sophistical Refutations, 25, 180a27-b7. Cf. Brendel 1992, 23f. ´ Bochenski 1956, 151; Spade /Read 2009, section 1. ´ Bochenski 1956, 151; Spade /Read 2009, section 1. Cicero, e. g., does little more than pose the question: “A man says that he is lying. Is what he says true or false?” (Cicero, Prior Analytics, II, 96)

83 84 85 86 87 88 89

Cf. Sorensen 2003, 88f. Spade /Read 2009, section 5. Cf. Simmons 1993, 83ff. and especially Dutilh Novaes 2008. Prior 1976. Cf. Müller 2002, 137–140. Read 2002, 2006, 2008, and 2009; and cf. the other contributions in Rahman /Tulenheimo /Genot 2008. Spade 1982, 253. Dutilh Novaes 2008, 228. Dutilh Novaes 2008, 228; cf. Barwise /Etchemendy 1987, 4.

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Sanders Peirce. Neither is longer than three pages, and Bolzano concludes his piece: “But enough of this sophistry!” 90 The interest of both modern philosophers seems to have been kindled by some medieval treatise on insolubles; Bolzano reacts to Savonarola and Peirce to Paul of Venice. Their contributions are probably nothing more than late echoes of the medieval debate; Bolzano and Peirce are neither proponents nor precursors of the modern discussion about the Liar paradox. The modern phase in the history of the Liar paradox begins in the late 19th century with Georg Cantor’s invention of set theory and in particular with his diagonal argument, which is a method of proof that led Bertrand Russell to discover his famous paradox of the set of all sets that are not an element of themselves. 91 Soon, further paradoxes were discovered – some of them are like Russell’s paradox insofar as they involve set theoretical notions, others are like the Liar paradox insofar as they involve semantic notions. In the early years of the 20th century, the set theoretic paradoxes were recognized as a serious threat to the newly laid foundations of mathematics; 92 and in the 1930s, Alfred Tarski showed that the Liar paradox could be a serious threat to the new formal science of semantics. 93 These foundational problems explain the modern “crisis mentality”; and in their wake there was new interest in the Liar paradox 94 and its history. 95 But the connection between Cantor’s diagonal argument and the Liar paradox is not only historical but also systematic. The diagonal argument not only caused renewed interest in the Liar paradox, it also bears a resemblance to it because it involves a construction that is structurally similar to a Liar sentence. We will study this structural similarity in the next section (1.7), and then we will turn to the larger group of paradoxes which the Liar paradox belongs to (in section 1.8).

1.7 Set theory and Cantor’s diagonal argument Cantor used his diagonal argument to prove some important theorems of set theory which concern the size of sets with infinitely many elements. But before we come to his proof, I would like to make some brief and opinionated remarks about the notion of a set, about set theory, and about set theoretical parlance. The opinionated part does not concern set theory itself (which as a mathematical subject is beyond philosophical reproach), but the practice common among analytic philosophers to try and use set theoretical parlance as a kind of lingua franca and the corresponding inflationary use of the notion of a set in analytic metaphysics. 90

91 92

93 94 95

Bolzano 1978, 24–26; Peirce Collected Papers 5.340; my translation. The original text is: “Doch schon genug von dieser Spitzfindigkeit!” Bolzano’s discussion of the Liar paradox is part of a letter he wrote in 1848; Peirce’s discussion was originally published in 1868. Bolzano’s approach is discussed in Brendel 1992, 42 and Künne 2013, 71ff. Russell 1903, 101. Cf., e. g., Deiser 2002, 183ff; Ebbinghaus 1994, 7ff.; Tiles 1989, 114ff.; Potter 2004, 25ff.; and especially Rheinwald 1988, 282ff. Cf. sections 3.6 and 3.7. Russell 1967[1908], 153ff.; Whitehead /Russell 1910, 63ff.; and cf. Russell 1985[1959], 59ff. Cf., e. g., Rüstow 1910.

1.7 Set theory and Cantor’s diagonal argument

35

It is important that we distinguish set theory, which is part of modern mathematics, and the metaphysical notion of a set, which is among its origins (at least historically). The metaphysical notion entails that a set is a collection of objects that is itself conceived as an object. According to Cantor, it collects these objects “into a whole”, and according to Felix Hausdorff, it collects them “into a new object”. 96 To see which metaphysical commitments are connected to this notion of a set, it is helpful to contrast it with the notion of a concept and the notion of a plurality. In contrast to a concept, of which we may well want to deny that it is a special kind of object, 97 a set clearly is an object. 98 And a set is a single object, in contrast to a plurality like a flock of birds (of which we say ‘they are flying’ rather than ‘it is flying’). By talking of the set of some objects one thus commits to the claim that for the collection of objects in question there exists an object, namely the set which has just those objects as its elements. It is this metaphysical notion of a set that was at the origin of mathematical set theories, 99 which were formulated in an informal way first and foremost by Cantor (but also by Richard Dedekind and others) and later given axiomatic treatments by Ernst Zermelo, Abraham Fraenkel, John von Neumann, and others. 100 In mathematics, set theory plays a dual role, because it both is a subdiscipline among other mathematical subjects (distinct, e. g., from geometry, analysis, and algebra), and its ontology and language are used often as a background metaphysics and a lingua franca for the formulation of all of mathematics (now including the subdisciplines of geometry, analysis, and algebra). 101 Set theoretical parlance has become a widespread dialect also in analytic philosophy, where some people would not shy away from construing a flock of birds as a (flying) set. Aside from the scientific glamour associated with set theory, this may well be due to reservations regarding pluralities. But although a majority of analytic philosophers view pluralities with suspicion, there are some who have made good use of the concept in important contexts. 102 In his rant “Against Set Theory”, Peter Simons reports the misuse of set theory in philosophy and speaks forcefully against it. 103 However, although I am sympathetic to his critique and prefer pluralities myself, I here do not want to commit myself in the matter. Instead, I will try to adopt a neutral jargon in this study: Aside from 96 97 98

99

100 101 102

103

Cantor 1895, quoted in Deiser 2002, 15, and Hausdorff 1914, 1. Cf. section 4.2, where we will endorse the Fregean claim that concepts are not objects. In addition to their objecthood, the extensionality of sets is a further feature which sets them off from concepts: A set x is identical to a set y if and only if they have the same elements, but on many construals concepts need not be equivalent if the same objects fall under them. Thus even axiomatic set theory primarily is not to be understood in an algebraic way, i. e., as a formal theory which is about whatever satisfies its axioms, but in an assertory way, i. e., as being about a specific subject matter. Cf. Shapiro 2005, 67; and sections 3.1 and 4.2. Cf., e. g., Ebbinghaus 1994, 7ff. In recent years, category theory has been a new contender for this role; cf. Awodey 2010. E. g., Peter Simons uses them to do justice to the metaphysics of ordinary objects, e. g., of the constitution of an orchestra out of musicians (Simons 1987, 144ff.) and George Boolos employs pluralities to give a non-standard interpretation of second order logic (cf. Boolos 1995, 54ff.). Simons 2005.

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set theory itself (which is of course about sets and their elements), we will, where others would speak Settish, talk about a collection and the objects which are among it. This talk can be spelled out either in terms of a set and its elements or in terms of a plurality and the many single objects which constitute it. * That being said, let us turn to some basic notions of the theory of sets which are needed to present Cantor’s proof. 104 We write ‘x ∈ y’ to express that the object x is an element of the set y. We say that a set x is a subset of a set y if and only if every object that is an element of x is an element of y. The set of all subsets of a set x is called the powerset of x; we write ‘P (x)’. To be able to speak about the size of infinite sets, we need to extrapolate our ordinary notion of the size of a collection, which concerns finite collections, to the notion of cardinality, which concerns not only finite but also infinite sets. Inspired by the everyday operation of counting, the sets x and y are said to have the same cardinality if and only if there is a bijective function from x to y (i. e., intuitively speaking, each element of x can be paired off with an element of y in such a way that no elements of x or y remain single). 105 A set x is said to be of (properly) smaller cardinality than a set y if and only if there is a bijection from x to a subset of y and there is no bijection from x to y. Now, what Cantor showed with the diagonal argument – Cantor’s Theorem – is that, even in the infinite case, 106 every set has a smaller cardinality than its powerset. Specifically, the set of natural numbers has a smaller cardinality than the set of sets of natural numbers. And as the technical notion of cardinality explicates and extends our intuitive notion of the size of a collection, we can say more simply that any set has more subsets than elements and that there are more sets of numbers than numbers. In view of what Cantor’s theorem says we should be able to distinguish different infinite cardinalities. We say that a set is enumerable if and only if it has the same cardinality as some subset of the natural numbers. Thus we can distinguish between sets which are finite (and hence enumerable), sets which are enumerably infinite, and sets which are indenumerable, i. e., infinite and not enumerable. 107 The proof of Cantor’s Theorem goes like this: Let f be an injective function from a set m to its powerset P (m). We show that f is not surjective, i. e., that there is a y ∈ P (m) such that there is no x ∈ m with f(x) = y. For consider the set y which has as elements all and only those objects z 104

105

106

107

Cf. Ebbinghaus 1994; Deiser 2002; and Smullyan /Fitting 2010 for set theory, and cf. Tiles 1989 and Potter 2004 for the history and philosophy of set theory. A function f from a set m to a set n is surjective if and only if for every y ∈ n there is an x ∈ m with f(x) = y; a function f is injective if and only if for every x, y ∈ m, if f(x) = f(y) then x = y; and a function f is bijective if and only if it is surjective and injective. In the infinite case, some of our ordinary intuitions about the size of collections fail. Notably, an infinite set is of the same cardinality as some of its proper subsets; e. g., there is a bijection between the natural numbers and the even numbers. Therefore Cantor’s result is in fact more surprising than it might seem to someone acquainted only with finite collections. Cf., e. g., Smith 2007, 13f.

1.7 Set theory and Cantor’s diagonal argument

37

such that (i) z ∈ m and (ii) z 6∈ f(z). Then y ∈ P (m) because of (i). But because of (ii) there can be no x ∈ m such that f(x) = y. For assume (for reductio) that there is an x with f(x) = y. Is x ∈ y? If x ∈ y, then, because of f(x) = y and (ii), x 6∈ y. And if x 6∈ y, then, again because of f(x) = y and (ii), x ∈ y. Either way, there is a contradiction. Hence there is no such x; and therefore f is not surjective. Since m and f were arbitrary, we have shown that no injective function from a set to its powerset is surjective. 108

But there was nothing obviously diagonal about that. To bring out what is diagonal about this method of proof, we will now turn to a special variant of the argument that concerns the set of natural numbers (i. e., of the numbers 0, 1, 2, 3, and so on). – Are there as many sets of natural numbers as there are natural numbers? A negative answer is already entailed by Cantor’s Theorem, but it can also be justified more directly. If there were as many sets of natural numbers as natural numbers, then we could give an (infinite) list of these sets like this: m0, m1, m2, m3, etc. And for any set of natural numbers m, we can specify its element by answering an (infinite) list of questions: Is 0 ∈ m? Is 1 ∈ m? Is 2 ∈ m? Is 3 ∈ m? Etc. In other words, we can specify any set of natural numbers by giving an infinite sequence of ‘yes’s and ‘no’s. 109 Thus we can specify any list of sets of natural numbers by an (infinite) table that has ‘yes’s and ‘no’s as its entries. To give an example, let m0 be the set of all odd numbers, m1 be set that has zero as its only element, m2 the set of all even numbers and m3 the set of all prime numbers. We can present these bits of information in the following (infinite) table: m0 m1 m2 m3 etc.

Is 0 ∈ m?

Is 1 ∈ m?

Is 2 ∈ m?

Is 3 ∈ m?

etc.

no yes no no ...

yes no no no ...

no no yes yes ...

yes no no yes ...

... ... ... ... ...

Now we can specify a set that cannot occur at any position of our list. For take the diagonal sequence (here: ‘no, no, yes, yes, . . .’) and invert it by changing each ‘yes’ into a ‘no’ and each ‘no’ into a ‘yes’. The resulting inverse-diagonal sequence (here: ‘yes, yes, no, no . . .’) specifies a set of natural numbers that cannot occur in the list of sets of natural numbers we started with. For assume that it occupies the nth position in the list. Then the sequence of ‘yes’s and ‘no’s at its nth place could neither have a ‘yes’ nor a ‘no’, on pain of contradiction. Therefore the set of sets of natural numbers is not listable – it is not of the same cardinality as the natural numbers. 110 108

109 110

For expositions of the diagonal argument, cf. Boolos /Burgess/Jeffrey 2002, 16ff., or any other textbook introducing mathematical logic. Simmons 1993 presents a whole theory of diagonal arguments, already with a view to their similarity to the reasoning of the Liar paradox. In technical jargon, this amounts to specifying a set of natural numbers via its characteristic function. Note that this argument, which has an obviously diagonal character, is a special case of the proof we saw above for Cantor’s theorem. Listing the sets m0, m1, m2, m3, etc. amounts to specifying a function

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This presentation of the above proof of Cantor’s Theorem does not only show what is diagonal about it, but it can also illustrate the resemblance between the diagonal argument and the Liar paradox. The resemblance concerns not the arguments themselves, but the objects reasoned about, because it is the specification of a set by the inverse-diagonal sequence that has a similar structure as a Liar sentence: Something points back at itself in a negative way. A Liar sentence points back at itself by way of reference, and obviously in a negative way because it ascribes falsity or untruth. This general characteristic of the Liar paradox has sometimes been noted. Bradwardine defines an insoluble as a certain kind of “difficult paralogism” that arises “from some [speech] act’s reflection on itself with a privative determination”; 111 Mackie says that a Liar sentence “depend[s] inversely on itself” 112. Similarly, the inverse-diagonal sequence points back at itself insofar as it is a diagonal sequence, and in a negative way insofar as it is inverse. 113 This shows how Cantor’s diagonal argument is more than the historical origin of modern interest in the Liar paradox – it is systematically similar on a deep level, because Cantor transferred the structure of privative reflection into a mathematical setting.

1.8 Semantic and set theoretic paradoxes The Liar paradox is one member of a larger group of paradoxes that are often seen as forming two different families, nowadays called the semantic paradoxes and the set theoretic paradoxes. 114 Here is a double list of important paradoxes belonging to the two families:

111

112 113

114

semantic paradoxes:

set theoretic paradoxes:

the Liar paradox Curry’s paradox Berry’s paradox Grelling’s paradox

Russell’s paradox Cantor’s paradox Burali-Forti’s paradox Richard’s paradox Mirimanoff’s paradox

from the natural numbers to sets of natural numbers, which corresponds to the function f. Inverting the diagonal corresponds to condition (ii). And the set specified by the inverse-diagonal sequence corresponds to the set y. Spade 1975, 106; the English translation is from Spade/Read 2009, section 5; the addition “[speech]” is by Spade and Read; the italics are mine. Mackie 1973, 256. It is not the set specified by the inverse-diagonal sequence that points back at itself (sets do not point to anything), but rather the sequence itself. And the sequence does its back-pointing only in virtue of the particular listing we work with. This need not be seen as a structural difference to the Liar paradox, because the reflective nature of a Liar sentence is also grounded in its position within a larger framework: It can only refer to itself in virtue of which meanings attach to its terms. The distinction was first made (in other terms) in Peano 1906 and Ramsey 1990[1925], 183f. Some overviews and discussions are: Kneale/Kneale 1962, 652–657; Rheinwald 1988, 14f.; Brendel 1992, 45– 54; Priest 1994 and 2002, 141–143; and Deiser 2002, 183–194.

1.8 Semantic and set theoretic paradoxes

39

Before we turn to the question of what distinguishes the families of semantic and of set theoretic paradoxes, let us look briefly at the last two semantic paradoxes and the first two set theoretic paradoxes of the double list. 115 Berry’s paradox concerns a description like ‘the least natural number that cannot be described in under thirteen words’. The paradoxical reasoning goes like this: There are only finitely many descriptions of a given length, but infinitely many natural numbers, so there must be natural numbers that cannot be described in under thirteen words. The least of them satisfies the description ‘the least natural number that cannot be described in under thirteen words’. But then it can be described in under thirteen words; a contradiction. Grelling’s paradox concerns the predicate ‘. . . is heterological’, which is satisfied by all and only those predicates that do no satisfy themselves. The predicate ‘. . . is heterological’ seems to be unproblematic as long as it is applied to other predicates; we can say for instance that while ‘monosyllabic’ is heterological, ‘polysyllabic’ is not. But the predicate makes trouble when we try to apply it to itself; ‘. . . is heterological’ is heterological if and only if ‘. . . is heterological’ is not heterological; a contradiction. Russell’s paradox concerns the set of all sets that are not an element of themselves, r. The notion of not being an element of itself seems to be unproblematic as long as it is applied to other sets; we can say for instance that while the set of teacups is not an element of itself, the set of abstract objects is. But it makes trouble when it is applied to the set r itself. r’s definition entails that every set must satisfy the following condition: x ∈ r if and only if x 6∈ x. But as r is a set, this condition must also hold of r. Thus, r ∈ r if and only if r 6∈ r; a contradiction. Cantor’s paradox concerns the set of all sets, v. Due to v’s containing all sets, every subset of v must be an element of v, so that v is its own powerset. Because of Cantor’s Theorem, v does not have the same cardinality as itself – but every set has the same cardinality as itself. A contradiction. The set theoretic paradoxes do not occur in the systems of set theory that were axiomatized in the first third of the 20th century (which are what will be found in a standard textbook on the mathematical subdiscipline of set theory). 116 Their place is in naïve set theory, which is a theory of sets based on the assumption, often called the naïve comprehension principle, that for every concept there is a set of all and only those objects that fall under that concept. If, in particular, we use some basic concept of set theory itself to define a set, paradox is wont to follow. For example, Russell’s paradox is based on the notion of elementhood and Cantor’s paradox is based on the notion of sethood. It is usually said that what distinguishes the two families of paradoxes is that, besides logic, set theoretic paradoxes involve only set theoretic notions and semantic 115

116

The Liar paradox is of course discussed in enough places of this study, and Curry’s paradox will be presented in section 1.5. Burali-Forti’s, Richard’s, and Mirimanoff’s paradox involve notions that are too technical for a presentation here. Two important axiomatic set theories which have shown no sign of paradox so far are ZermeloFraenkel set theory (ZF or ZFC) and the Neumann-Bernays-Gödel theory of sets and classes (NBG); cf. Ebbinghaus 1994, especially 7–14; Deiser 2002, 183ff.; and Smullyan /Fitting 2010, especially 11–14.

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Chapter 1 The Liar and Its Kind

paradoxes involve only semantic notions, i. e., notions like truth, satisfaction, reference, and description. 117 And to be sure, the Liar paradox involves the notion of truth, Grelling’s paradox the notion of satisfaction, and Berry’s paradox the notion of description. But there has been considerable dispute about whether this criterion provides a justification for seeing these paradoxes as forming two distinct and essentially different families. An historical reason for seeing an essential difference is that there is a consensus among set theoreticians that the set theoretic paradoxes are solved, while there is no such consensus among the people who work on semantics (and especially not among those who work on the theory of truth) about a solution to the semantic paradoxes. The main systematic reason for seeing an essential difference is, naturally, that the notions appealed to in the above criterion belong to essentially different disciplines, set theory and semantics. 118 But this has been contested. Graham Priest, e. g., points out that after the formalization of considerable parts of semantics it has become difficult to disentangle as a discipline from set theory. 119 And I would like to argue that, conversely, set theory is difficult to disentangle from semantics, because the naïve comprehension principle postulates the existence of a set as the extension of every meaningful predicate, and the meaningfulness of predicates of course falls into the province of semantics. A further systematic reason for the underlying similarity between the paradoxes of both families, by contrast, is positive. For the paradoxes presented above, it is not too difficult to see a common structure, namely the structure of privative reflection that was described in the last section (1.7). 120 And every one of the paradoxes in the above double list at least involves a structure of reflection, or, to use Russell’s term, “circularity”; 121 therefore they and their relatives are sometimes called “the paradoxes of self-reference”. 122

117

118

119 120

121 122

Tarski 1956[1936], 401; Tarski 1944, 345. Cf. Künne 2003, 176–180; Sher 2005, 150. – Roughly, a notion is semantic in this (Tarskian) sense if and only if it is in some way about a connection between a language and the world represented by it. Cf. section 4.1 for more on why these notions are called semantic. Elke Brendel sees the problem of expressibility as distinctive of the semantic paradoxes, thus giving a further systematic reason for seeing an essential difference to set theoretic paradoxes (cf. Brendel 1992, 45–54). The problem of expressibility is, to anticipate section 2.8, the problem that key notions employed in a solution to a paradox turn out to be inexpressible in the paradox-free language or theory the paradox-solver tries to develop. But there are also problems of expressibility for important solutions to the set theoretic paradoxes. Frederic Brenton Fitch has pointed out that the sentences of Russell’s theory of types themselves violate type restrictions (Fitch 1964); and Mirimanoff’s paradox concerns the set of all well-founded sets, which is well-founded if and only if it is not well-founded (Deiser 2002, 185ff.), where well-foundedness is a key notion in all classical solutions to the set theoretic paradoxes (cf. Bromand 2001, 127ff.). Priest 2002, 142. This may be less obvious for Cantor’s paradox as it is described here, but the structure of privative reflection also occurs in the diagonal construction that is used in the proof of Cantor’s theorem (cf. section 1.7) which is an essential part of the reasoning behind Cantor’s paradox. Whitehead /Russell 1910, 39ff. Cf., e. g., Priest 1994, 25 and Restall 1993, 279.

1.9 Just a joke?

41

My own sympathies lie with the similarity view, but I cannot give a conclusive argument here. 123 The question of whether there are essential differences between semantic and set theoretic paradoxes is far too difficult to settle. An important reason for this difficulty is that probably no one can answer the question independently from his or her approach to these paradoxes. 124

1.9 Just a joke? People who are into neither logic nor analytic philosophy understand the reasoning of the Liar paradox easily, but in general they do not see it as a problem of any philosophical depth. 125 This phenomenon stands in an odd tension to the fact that the philosophical literature on the Liar paradox of the last one hundred years is truly vast. 126 So, how serious is the philosophical problem posed by the Liar paradox? Let us hear answers by contemporary philosophers who have worked on it, starting with two extreme positions. Dorothy Grover is relaxed: “it is not easy to persuade others that we need to ‘resolve the paradox of the liar’; or that analysis of the liar may reveal crucial insights. Non-philosophers may grant there is a curious puzzle, but it is a ‘don’t care’ puzzle. How, and where, is the liar so crucial to our understanding? I will now argue that there is something right in the naïve reaction of unconcern.” 127

Vann McGee, on the other side of the spectrum, is dead serious: “There are scarcely any philosophical problems of greater urgency than the liar paradox, for there are scarcely any concepts more central to our philosophical understanding than the concept of truth. [. . .] The liar antinomy and the closely related antinomies involving reference show us, quite unmistakably, that our present way of thinking about truth and reference is inconsistent. Unless we can devise new ways of thinking about truth and reference we shall not have even the beginning of a satisfactory understanding of human language.” 128

In a similar vein, Jon Barwise and John Etchemendy move swiftly from considering a relaxed attitude like Grover’s to adopting an attitude nearly as earnest as McGee’s: 123 124

125

126

127 128

But cf. Pleitz 2015a. On an historical note: Frank Plumpton Ramsey, whose 1925 text is the locus classicus for the distinction between semantic and set theoretic paradoxes (Ramsey 1990[1925], 183f.), drew attention to it in order to defend his favorite approach to the set theoretic paradoxes (i. e., Russell & Whitehead’s) against the criticism that it did not solve the semantic paradoxes (Ramsey 1990[1925], 191). I was able to collect anecdotal evidence for this observation in conversations with friends and acquaintances about the topic of my research. This vastness is attested to, for instance, by the following four anthologies which appeared in the last decade: Beall 2003; Priest /Beall /Armour-Garb 2004; Beall /Armour-Garb 2005; Beall 2007a. Cf. also sections 7.4 and 7.5. Grover 2005, 177. McGee 1991, vii.

42

Chapter 1 The Liar and Its Kind

“On first encounter, it’s hard not to consider assertions of this sort [i. e., Liar sentences] as jokes, hardly matters of serious intellectual inquiry. But when one’s subject matter involves the notion of truth in a central way, for example when studying the semantic properties of a language, the jokes take on a new air of seriousness: they become genuine paradoxes. And one of the important lessons of twentieth century science, in fields as diverse as set theory, physics, and semantics, is that paradox matters. [. . .] a paradox demonstrates that our understanding of some basic concept or cluster of concepts is crucially flawed, that the concepts break down in limiting cases. And although the limiting cases may strike us as odd or unlikely, or even amusing, the flaw itself is a feature of the concepts, not the limiting cases that bring it to the fore. If the concepts are important ones, this is no laughing matter.” 129

Why are the reactions of philosophers to the Liar paradox so diverse? I think that we will find some clues for an answer in a frank observation James Cargile makes about his fellow philosophers and logicians. “Semantic paradoxes are very commonly regarded by philosophers as trifling problems. Of course they are taken seriously by logicians, but most philosophers are not logicians, and to this majority, the paradoxes are ranked intellectually about on a par with party games or newspaper ‘brain teasers’. Handed a card with ‘The statement on the other side is false’ on one side and ‘The statement on the other side is true’ on the other, even an intellectually serious person may consider an amused smile an appropriate response. One way of getting the problems treated as serious is to treat them mathematically. The introduction of formal symbolism tends to have a sobering effect, and since it is not generally understood, it makes it less embarrassing to be overheard discussing these problems.” 130

If Cargile is right, taking the Liar paradox seriously is somehow connected to taking a mathematical or formal 131 approach in philosophy. And indeed, while philosophers of all persuasions think of the notion of truth as important, 132 the philosophers who are troubled that the notion of truth might be compromised by the consequences of the Liar paradox tend to be just those who like to incorporate mathematical or formal methods in their work. 133 129 130 131

132

133

Barwise /Etchemendy 1987, 4. Cargile 1979, 235. The notion of formality employed here is basically the vulgar one according to which a theory is formal if its formulation contains lots of formal symbols. The vulgar notion of formality works well enough in describing the day-to-day interactions of philosophers. Other notions of formality, which are more important systematically, can be derived from the concept of a formal language that we will characterize in section 3.1. A case in point is Wolfgang Künne, who in his monumental monograph about theories of truth says about the debate about the Liar paradox in the late 20th century: “As I had to confess already in the preface to this book, I have nothing enlightening to say about, let alone to contribute to this debate. So I quickly, and somewhat shamefacedly, move on [. . .]” (Künne 2003, 203). Ten years later, however, he did present a (less monumental) study on the Liar paradox (Künne 2013). To anticipate, this claim can be substantiated by noting that the proponents of the meaninglessness solution adopt the no problem attitude and in general decline using formal methods while the proponents

1.10 The question of formality

43

Let us tentatively (and only for the descriptive purposes of the present section) distinguish between formal and non-formal philosophers. Of course, on a systematic level any distinction between formal and non-formal philosophy would be difficult to uphold. But we are concerned here only with the attitudes contemporary working philosophers have to their everyday work. And there a formal and a non-formal fraction can indeed be made out – perhaps a late echo of the division of early 20th century analytic philosophers into philosopher-logicians (like the earlier Wittgenstein) and ordinary language philosophers (like the later Wittgenstein), perhaps no more than a reflection of personal tastes and talents. In these terms we can say, roughly, that where non-formal philosophers say ‘no problem’, formal philosophers cry ‘no joke!’ And usually, they do not listen much to each other. The lack of communication between the two fractions is another important aspect of the situation characterized by Cargile. Volker Halbach describes the situation in the case of philosophical theories of truth: “When we look at the literature by analytic philosophers on the topic of truth, it is difficult to resist the impression, not only that very heterogeneous projects and programs are pursued under the same label, but also that two completely independent areas of research have developed. Each has its own key literature, its own prominent and trend-setting protagonists, its own school and paradigms. Mostly, the people working in one area will not quote the contributions of the other area and – so we can surmise – will not even be aware of them.” 134

So the diversity of philosophical reactions to the Liar paradox can plausibly be explained by the nearly conflict-free co-existence of two camps of philosophers, which is nearly conflict-free because intellectual conflict presupposes communication.

1.10 The question of formality Let us not contribute further to this sorry state of affairs. I propose that we adopt and if possible defend the conciliatory hypothesis that probably there is some justification both for the no-joke and for the no-problem attitude. In order to do justice to both sides I set myself a twofold aim in writing this study: First, in parts I and II (especially in chapters 2 through 5), I want to convince non-formal philosophers (and everyone else) that the Liar paradox is indeed a serious problem. But then, in part III, I want to convince formal philosophers (and everyone else) that the Liar paradox in the end of the day will turn out to have been not much more than a joke, by proposing a solution that is meant to literally explain away the whole problem. Because of the (loose) correlation we found in the preceding section between taking the Liar paradox seriously and belonging to the camp of formal philosophers, we will also present the Liar reasoning in a non-standard way in the following two chapters: They are devoted to a presentation of the basic Liar reasoning which deals

134

of what we will call approaches of sophisticated surgery and approaches of palliative care adopt the no joke attitude and enjoy using formal methods. Cf. section 7.5. Halbach 2005, 229; my translation.

44

Chapter 1 The Liar and Its Kind

with informal and formal variants separately. Our discussion will be guided by what can be called the question of formality, which we can pose here only in the rough form of the following collection of sub-questions: How formal are the informal variants of the Liar paradox? How informal are its formal variants? Are the formal variants more serious than the informal ones, or vice versa? What is formality, anyway? In chapter 2 we will present the basic Liar reasoning in an informal way; in chapter 3 we will turn to the notion of formality and present the basic Liar reasoning in a formal way. The methodological device of splitting up informal and formal variants of the Liar paradox upon two chapters while keeping an eye on the guiding question of formality will enable us to understand better the serious problem posed by the Liar paradox, and to see that its seriousness is not bound up to one particular way of doing philosophy.

Chapter 2

Informal Logic and the Liar Reasoning

Arthur Prior: “Logic is commonly thought of as having something to do with argument, in fact as being the systematic discrimination of good arguments from bad; and, as a first approximation, this will do.” 135

There are many good introductions to the Liar paradox. 136 We can therefore allow ourselves to give a non-standard presentation of the Liar reasoning, splitting it up into a non-formal presentation in this chapter and a formal presentation in the next chapter. This will not only allow us to get a deeper understanding of the seriousness of the problem posed by the Liar paradox, 137 contrasting the formal approach to logic which by now is common to an informal approach that is no less warranted (in sections 2.1 and 3.1) and comparing different ways of explicating basic intuitions about truth (in sections 2.2 and 3.2, and in sections 2.3 and 3.3) will also help us answer the question of formality. 138 Along the way we will establish some terminology and introduce some notions which will be important later on. The present chapter starts with a list of some rules and laws of informal logic (in section 2.1), followed by a formulation of some principles about truth and falsity that are essential to the basic Liar reasoning (in sections 2.2 and 2.3). The central part of the chapter is a comparison of a number of informal variants of the basic Liar reasoning (in section 2.4), which will show that the paradox does by no means presuppose formal logic and is not connected essentially to classical logic. After that we will look at how the Liar reasoning typically goes on to unfold after a first attempt has been made to thwart the problem (in sections 2.5 through 2.10). Although this presentation of the extended Liar reasoning plays no part in the contrast between informal and formal variants of the Liar paradox that is the overall aim of chapters 2 and 3, it is best given at the end of the present chapter because an informal presentation of these matters will suffice for our purposes. 135 136

137 138

Prior 1962[1955], 1. Introductions to the Liar paradox can be found as single contributions (Quine 1976a; Visser 2004; Dowden 2010; Beall/Glanzberg 2014), as chapters in books which introduce the philosophy of logic (Haack 1978, 135–151; Read 1995, 148–172; Priest 2000, 31–37), in books which introduce the theory of truth (Soames 1999; Burgess /Burgess 2011), as chapters of books which are devoted to a wider range of paradoxes (Cook 2001, 30–61; Sainsbury 1988, 114ff.; Rescher 2001, 199ff.; Priest 2002, 141–155; Sorensen 2005 (cf. the entry in the index on page 387); Clark 2007, 112–119), as introductory chapters in anthologies devoted to the Liar paradox (Martin 1984b; Beall 2007b), and as introductory chapters of monographs specialized on the Liar paradox or a larger group of paradoxes (Barwise /Etchemendy 1987, 3–25; Rheinwald 1988, 9–56; McGee 1991; Brendel 1992, 3–17; Simmons 1993, 1–19; Gupta / Belnap 1993, 1–32; Bromand 2001, 11–36; Priest 2006a, 19–27; Field 2008, 1–19). Cf. sections 1.9 and 1.10. Cf. the end of section 1.10.

46

Chapter 2 Informal Logic and the Liar Reasoning

As we have already given one informal presentation of the Liar paradox in the first chapter, 139 a word of explanation is in order before we set out to give a second one. The two informal presentations differ in their aim and are aimed at a different (imagined) audience. In the previous chapter, our aim was to introduce the Liar paradox and its main ingredients in a thoroughly unchallenging way, without going into any depth when describing those ingredients. 140 Although the actual readers of this study will likely be familiar with logic, the presentation in the introductory first chapter was directed at an imagined audience of un-initiated readers. In the present chapter 2, in contrast, our (official) aim is to convince an audience that is familiar with the usual more formal ways of presenting the Liar reasoning that all of that reasoning can already be done in an informal way, as well as to remind this audience of how robust the Liar reasoning is. 141 Thus there is no redundancy in giving another informal presentation.

2.1 Informal logic Logic is concerned with what makes an argument valid, i. e., with the rules and laws of good reasoning. Formal logic explicates good reasoning by giving a complete specification of a formal language and a detailed description of the inferential relations between sentences of that formal language. 142 By informal logic we shall mean a specification of good reasoning from within the language we actually reason in (or in a language that is akin to the language we reason in). That language is not formal, at least not in the sense of formal logic that will be explained in the next chapter. 143 But it may well be regimented, for instance by the stipulation that the word ‘and’ is to be used unambiguously as connecting sentences in a way that disregards their order as well as any temporal or causal connotations. And it may even contain formal symbols, but these will be nothing more than abbreviations of logical expressions of our language (usually regimented ones). E. g., in the symbolically augmented English we use in this study, ‘=’

abbreviates ‘is identical to’,

‘=def’ abbreviates ‘is by definition identical to’, ‘⇒’ abbreviates ‘if . . . then’, and ‘⇔’ abbreviates ‘if and only if’. Here is a list of rules of informal logic that will play a role later: 144 139 140

141

142 143 144

Cf. section 1.2. The logic ingredients will be discussed in chapters 2 and 3, and the language ingredient in parts II and III. The occasional reader who is truly uninitiated but interested is of course invited to read on! Most of the more difficult material will be explained in a self-contained way. Cf. chapter 3 and in particular sections 3.1 through 3.3. Cf. section 3.1. We present these rules of informal logic in a non-standard way here, phrased in a temporalized way and not mentioning but using sentences. Further, to distinguish informal from formal logic, we say

2.1 Informal logic conjunction introduction:

If (at some step of our reasoning) we have inferred that p and that q, then we can (go on to) infer that p and q.

modus ponens:

If we have inferred that if p then q and we have inferred that p, then we can infer that q.

conditional proof :

If from the assumption that p we have inferred that q, we can infer that if p then q.

transitivity of ‘if and only if’:

If we have inferred that p if and only if q and that q if and only if r, then we can infer that p if and only if r.

substitution of equivalents:

If we have inferred that p if and only if q and we have inferred something that includes the claim that p, then we can substitute the claim that q for the claim that p in it.

reasoning by cases:

If we have inferred that p or q, that if p then r, and that if q then r, then we can infer that r.

consequentia mirabilis:

If from the assumption that p we have inferred that it is not the case that p, then we can infer that it is not the case that p.

double negation elimination:

If we have inferred that it is not the case that it is not the case that p, then we can infer that p.

reductio ad absurdum:

If from the assumption that it is not the case that p we can infer a contradiction (e. g., that q and it is not the case that q), then we can infer that p.

ex falso sequitur quodlibet:

If we have inferred that p and it is not the case that p, then we can infer that q (for any arbitrary claim that q).

De Morgan:

If we have inferred that it is not the case that p and that it is not the case that q, then we can infer that it is not the case that p or that q.

47

A logical law can be conceived as the limit case of a logical rule; that is, as a logical rule without any precondition of the form ‘if we have inferred . . .’. The following two logical laws play important roles in connection with the Liar paradox: 145 (Excluded Middle)

We can infer that p or it is not the case that p.

(Non-Contradiction) We can infer that it is not the case that (p and it is not the case that p).

145

‘infer’ instead of ‘deduce’ or ‘derive’, ‘logical rule’ instead of ‘inference rule’, and ‘logical law’ instead of ‘theorem’ or ‘valid formula’. Some readers might nevertheless be reminded of the inferential rules of a natural deduction variant of the deductive system of classical logic (cf. section 3.1). But that is due mainly to the success of natural deduction at emulating real reasoning. There are also important differences, because we here do not give a complete list of rules meant to specify the meanings of the connectives, some of our rules would have the status of derived rules in a natural deduction system, and all that only modulo formalization, i. e., the translation into a formal language. Cf. section 2.3.

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Chapter 2 Informal Logic and the Liar Reasoning

Each one of these rules (and laws) of informal logic corresponds to an inference rule (or theorem) of classical propositional logic. 146 And that is as it should be. Classical propositional logic is, after all, considered by many to be (an important part of) a codification of our best reasoning – in other words, it is a promising candidate for (an important part of) an explication of informal logic. Others disagree, for a variety of reasons. 147 However, we need not take a stand on this matter here. What is important for our presentation of the Liar paradox is only that rejecting one or more of these rules amounts to departing from classical logic.

2.2 The naïve truth principle The principles of truth and falsity that are appealed to in the basic Liar reasoning are of two kinds. Some are about the relation of truths and falsehoods to what is the case (and will be discussed in this section); others are about the relation between truth and falsity (and will be discussed in the next section). The principles about how truths and falsehoods relate to what is the case that play a role in the Liar paradox are often traced back to Aristotle’s famous statement about truth and falsity: 148 “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.” 149

This is open to different interpretations, especially as it is not clear what “what is” and “what is not” mean. But very probably Aristotle’s dictum is meant to sanction claims like the following: To say that Socrates is famous is true, because Socrates is famous. As Plato has not been forgotten, it would be false to say that Plato has been forgotten. And so on. A formulation that sounds more precise (at least to modern ears) is given by Alfred Tarski: “a true sentence is one which says that the state of affairs is so and so, and the state of affairs indeed is so and so”. 150

This is easily supplemented 151 by a corresponding principle about falsity: 146 147 148

149

150 151

Cf. section 3.3. For overviews of non-classical logic, cf. Haack 1978 and Priest 2008. Less famously, Plato had already formulated a similar principle: “a true proposition says that which is, and a false proposition says that which is not” (Plato, Cratylus 385b; cf. Sophistes 240e 10–241 a1). Aristotle Metaphysics, Γ 7: 1011b 26–7. Künne aptly characterizes this English version of the famous principle of Aristotle as “stunningly monosyllabic” (Künne 2003; 95). Tarski 1956[1935], 155. It is common to use the second half of Aristotle’s famous statement as an inspiration for a principle about how a truth relates to what is case. But the fact that its first half provides an equally good inspiration for a corresponding principle about how a falsehood relates to what is the case is often overlooked. Graham Priest, for example, defines falsity not directly as a mismatch with what is the case, but indirectly as truth of negation (Priest 2006a, 64). Thus he (knowingly) bars the way to a noncircular definition of negation in terms of truth and falsity: “It would seem that falsity and negation can

2.2 The naïve truth principle

49

A false sentence is one which says that the state of affairs is so and so, and the state of affairs is not so and so. The most important difference between these two formulations of the principles about truths and falsehoods is that while in Aristotle’s statement, truth applies to saying (perhaps to what is said, perhaps to the act of saying something), for Tarski the objects truth applies to are sentences. E. g., the sentence ‘Socrates is famous’ is true because it says that Socrates is famous and Socrates is famous. 152 Of course, for Tarski the principle about truth in the quoted formulation was merely a starting-point on the way to (and a motivation of) the by now famous and canonical Tarskian truth schema. Endorsing the Tarskian truth schema amounts to endorsing every claim that can be obtained from the form ‘the sentence s is true if and only if p’ by substituting for the symbol ‘s’ a singular term that refers to a sentence and for the symbol ‘p’ a translation of that sentence into the language we use. 153 There are several important differences between Tarski’s first formulation of the Aristotelian principle and the Tarskian truth schema. For the time being (i. e., for the rest of this chapter), 154 we will mention only one of them, namely that the unclear ‘and’ of the quote has rightly been sharpened to ‘if and only if’. Our reason for postponing the further discussion and use of the Tarskian truth schema to the next chapter (cf. sections 3.2 through 3.7) is that in the present chapter we want to present the Liar paradox in a non-technical fashion, in order to bring out the high level of generality of the problem it confronts us with. For that purpose, the Tarskian truth schema would be of dubious use, and a formulation of the truth principle and the falsity principle in terms of what a sentence means is entirely sufficient. Tarski himself might have agreed – after all, he commented on the above formulation of the truth principle in a not altogether unfavorable way: “From the point of view of formal correctness, clarity, and freedom from ambiguity of expressions occurring in it, the above formulation leaves much to be desired. Nevertheless, its intuitive meaning and general intention seem to be quite clear and intelligible.” 155

Let us now look at the formulations of the truth principle and the falsity principle that we will work with when presenting a variety of informal variants of the Liar reasoning. They are the following:

152

153 154 155

be defined in terms of each other, but neither can be defined without the other. [. . . ] The situation is a common enough one in philosophy: we are faced with a circle of interdefinable terms, and in this case one of very small radius.” (Priest 2006a, 64) Given what below we will call the naïve falsity principle, the definitional circle can be considerably expanded, which will allow us to give the (standard) definition of choice negation and exclusion negation in terms of truth and falsity in section 2.3. In section 8.1, we will argue that (meaningful) sentences are the primary truth bearers, but as of now that is not important. Here, sentences can be understood as default truth bearers. It is easy to formulate corresponding principles for propositions, sentences-in-a-context, and the like. Tarski 1956[1935], 187f. Cf. section 3.2 for a discussion of Tarski’s exact formulation of the truth schema. Tarski 1956[1935], 155.

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Chapter 2 Informal Logic and the Liar Reasoning

(NT) A sentence is true if and only if what it means is the case. (NF) A sentence is false if and only if what it means is not the case. We will call the first one the naïve truth principle and the second one the naïve falsity principle (hence the letter ‘N’ in ‘(NT)’ and ‘(NF)’). The reason for speaking of naïveté here is that in paradox studies, a notion is called ‘naïve’ if it is highly plausible but leads to paradox. 156 It is called ‘naïve’ because it is based on some innocent intuition that is as yet unspoiled by the awareness of paradox. And we have already seen that the naïve truth principle is very intuitive, but will allow to derive a contradiction when added to a few further assumptions. 157 In order to enable us to say more exactly at which step of some variant of the Liar reasoning which principle is appealed to, it will sometimes be helpful to split up the principles (NT) and (NF) into the following four principles: 158 (NT⇒) If a sentence is true then what it means is the case. (NT⇐) If what a sentence means is the case then it is true. (NF⇒) If a sentence is false then what it means is not the case. (NF⇐) If what a sentence means is not the case then it is false. It is interesting to note that, without looking at Tarski’s own approach to the Liar paradox which is commonly seen as the locus classicus of the distinction between an object language and a meta-language, 159 we have already come to a point where that distinction is important. In every case where language is talked about, we can distinguish between the object language which is talked about and the metalanguage which is used to talk about it. Although object language and meta-language often will be distinct (e. g., when we talk in English about the German language 160), this need not be the case (e. g., in an English grammar textbook). With regard to the naïve principles about truth and falsity presented here, the distinction between object language and meta-language is important because someone who asserts one of these principles thereby is talking about sentences. The same is not as unambiguously the case with the original Aristotelian formulation of the principles.

2.3 Truth, falsity, and negation We turn now from principles about how truths and falsehoods relate to what is the case to principles about how truth and falsity relate to each other: (Exhaustiveness) Every sentence is true or false. (Exclusiveness) 156 157 158

159 160

No sentence is both true and false.

We have already seen this in our brief encounter with naïve set theory in section 1.8. Cf. section 1.2. Thus we can say, e. g., that in the informal Liar reasoning presented in section 1.2, the principle (NT⇒) is used in (Step 2) and the principle (NT⇐) is used in (Step 3). Cf. sections 3.4 through 3.7. Just as well, we can talk in German about the English language – which goes to show that the notions of object language and meta-language are relational.

2.3 Truth, falsity, and negation

51

It has become customary to call a sentence a truth value 161 gap if and only if it is neither true nor false and to call a sentence a truth value glut (or dialetheia) if and only if it is both true and false. In these terms, the principle of (Exhaustiveness) says that there are no gaps and the principle of (Exclusiveness) says that there are no gluts. The two principles are independent of each other. The conjunction of the principles of (Exhaustiveness) and (Exclusiveness) is equivalent to the classical principle of bivalence: (Bivalence) Every sentence is either true or false (but not both). Splitting up the principle of (Bivalence) into the principles of (Exhaustiveness) and (Exclusiveness) has the advantage, again, that it allows to say more exactly which principle is appealed to at which step of some variant of the Liar reasoning. 162 Another reason for splitting up the classical principle of (Bivalence) is that whereas some approaches to the Liar paradox deny the principle of (Exhaustiveness), other approaches deny the principle of (Exclusiveness). 163 It would be misleading to group these together under the label of ‘non-bivalent solutions’ because there is an established tradition of calling all and only approaches which deny (Exhaustiveness) ‘nonbivalent’. 164 A likely explanation for this terminological practice is that the early propounders of these non-bivalent solutions did not think of giving up the other half of the principle of bivalence as an option to be taken seriously. Historically, the denial of (Exclusiveness) came later than the denial of (Exhaustiveness). 165 Related to but distinct from the two principles about how truth relates to falsity are the following two logical laws (mentioned already in section 2.1): 166 (Excluded Middle)

p or it is not the case that p.

(Non-Contradiction) It is not the case that (p and it is not the case that p). The similarities are obvious: logical law: principle about truth & falsity: 161

162

163

164 165

166

no underdetermination:

no overdetermination:

(Excluded Middle)

(Non-Contradiction)

(Exhaustiveness)

(Exclusiveness)

We will say more about the notion of a truth value in section 4.2. To anticipate, we will understand the piece of logical jargon ‘Sentence s has truth value The True (The False)’ as no more than a complicated way of saying ‘Sentence s is true (false)’. E. g., we can now say that, in the informal Liar reasoning presented in section 1.2, in (Step 5) we appeal to the principle of (Exhaustiveness) and in (Step 6) we appeal to the principle of (Exclusiveness). There are other approaches which deny both the principle of (Exhaustiveness) and the principle of (Exclusiveness), but that is not important here. Cf., e. g., Rheinwald 1988, 39. While Bochvar gave a solution of the Liar paradox based on rejecting (Exhaustiveness) as early as 1938, Priest proposed to solve the Liar paradox by rejecting (Exclusiveness) only in 1979. A terminological remark: While the label ‘law of excluded middle’ is used consistently for the logical law of (Excluded Middle), and the label ‘law of non-contradiction’ is sometimes used for the logical law of (Non-Contradiction), the latter label is also used quite often for the principle of (Exclusiveness); e. g., Priest 1984, 154; Berto /Priest 2010, introductory remarks; and Beall 2004, 3; also cf. Grim 2004, 49–51.

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Chapter 2 Informal Logic and the Liar Reasoning

Given classical notions of negation and of validity, the principle of (Exhaustiveness) holds if and only if the law of (Excluded Middle) is valid and the principle of (Exclusiveness) holds if and only if the law of (Non-Contradiction) is valid. 167 But then, the whole point of explicitly stating the four principles is to open up options for moving away from classical logic. And there are non-classical logical systems which have been proposed to solve the Liar paradox where (Exhaustiveness) fails but the (Law of Excluded Middle) is valid 168 and other non-classical logics where (Exclusiveness) fails but the (Law of Non-Contradiction) is valid. 169 * Once we play with the idea of giving up the (Exhaustiveness) of truth and falsity, it is advisable to follow the customary practice of distinguishing two notions of negation, which are given by the following principles: (Choice Negation) The choice negation of some sentence is true if and only if that sentence is false. (Exclusion Negation) The exclusion negation of some sentence is true if and only if that sentence is false or it is neither true nor false. 170 What the two notions of negation have in common can be expressed by the following minimal principle about negation: (Neg⇐) If a sentence is false then its negation is true. The difference between choice negation and exclusion negation comes out in a context where (Exhaustiveness) does not hold so that there are gaps. While the choice negation of a gappy sentence is itself gappy, its exclusion negation is true. The distinction between choice negation and exclusion negation will be helpful in the discussion of a strengthened Liar sentence, because a strengthened Liar sentence says of itself that it is not true. 171 Here, it allows to illustrate the fact that (Exhaustiveness) and (Excluded Middle) need not coincide, for even given that truth and falsity are not exhaustive, any sentence of the form ‘p or not p’ must be true when ‘not’ expresses exclusion negation. 172

167

168 169 170 171 172

What is more: By De Morgan’s law, the (Law of Excluded Middle) and the (Law of Non-Contradiction) are themselves classically equivalent. Cf., e. g., van Fraassen 1966, 493. Cf., e. g., Priest 1984, 154. Cf., e. g., Martin 1978[1970]c, 92; and Donnellan 1978[1970], 113ff. We will discuss a strengthened Liar sentence in section 2.5. A further presupposition is that a disjunction is true if one of its disjuncts is true.

2.4 Informal variants of the basic Liar reasoning

53

2.4 Informal variants of the basic Liar reasoning Now, after some logical rules and some principles about truth and falsity have been stated, it is possible to present the Liar reasoning in more detail than in section 1.2. To illustrate the considerable logical robustness of the Liar reasoning, we will look at several variants. In each variant we start from a fact about language, namely, the existence of a simple Liar sentence. We can make the corresponding claim like this: (Existence of a Simple Liar Sentence) There is a sentence that means that it is false. Let us call the simple Liar sentence that this existence claim is about ‘λ’. Then we can appeal to the following fact about λ in our reasoning: 173 (Meaning of λ)

λ means that λ is false. A remark about this way of bringing a Liar sentence onto the stage. Contrary to current custom, we do not display any specific Liar sentence, and we do not need to do so in order to give a complete presentation of the Liar reasoning. Of course, there appear to be several ways of introducing a specific Liar sentence. E. g., by definition: Larry =def ‘Larry is false.’ 174 This identity statement would provide another starting point for the Liar reasoning because under some plausible assumptions it entails the general claim (Existence of a Simple Liar Sentence). But it is based on much more specific claims, because the Liar sentence Larry of this example depends on the admissibility of circular definitions and the self-referential names they appear to introduce. In contrast, our present method of introducing a Liar sentence is as general as possible. It does not lay down anything about the singular term that is used in the Liar sentence and thus is not prejudiced towards any one of the particular devices for achieving self-reference, like a name, a description, or an indexical. We will deal with all of these in due course, 175 but here we see it as an advantage that we can separate the discussion of means for achieving the self-reference out from the presentation of the Liar reasoning. We are now standing at the common starting point of all the ways the informal reasoning of the Liar paradox can take. Let us begin laying out that variety of variants by formulating the Liar reasoning of section 1.2 in a more explicit way: 173

174

175

In the usual terms of predicate logic, the move from (Existence of a Liar Sentence) to (Meaning of λ) is warranted by the rule of existential instantiation. This appears to be the way that Jc Beall and Michael Glanzberg go when they write (Beall /Glanzberg 2014, section 1.1): “FLiar: FLiar is false.” The different devices for achieving self-reference are presented in detail in part II, and some of their differences will play a role in the approach outlined in part III. In particular, an overview of all singular terms that may be used in self-referential constructions is to be found in section 5.4, and a discussion of the admissibility of self-referential names like ‘Larry’ in chapter 11.

54

Chapter 2 Informal Logic and the Liar Reasoning Informal Variant 1 of the basic Liar reasoning (1) λ means that λ is false.

(Meaning of λ)

(2)

assumption for conditional proof 176

(3) (4) (5) (6) (7) (8) (9) (10)

λ is true. λ is false. If λ is true then λ is false. λ is false. λ is true. If λ is false then λ is true. λ is true if and only if λ is false. λ is true or λ is false. λ is true and λ is false.

(1), (2), (NT⇒) (2)–(3), conditional proof assumption for conditional proof (1), (5), (NT⇐) (5)–(6), conditional proof (4), (7), conjunction introduction (Exhaustiveness) for the case of λ (8), (9), reasoning by cases & conjunction introduction

(11) It is not the case that λ is true and λ is false.

(Exclusiveness) for the case of λ

(12) λ is true and λ is false and it is not the case that λ is true and λ is false.

(10), (11), conjunction introduction

An impatient reader will probably think that an unacceptable conclusion has been reached as early as at step (8), or at least at step (10). But then he or she will be led by an intuitive understanding of the notions of truth and falsity. Neither a claim of the form that a is F if and only if a is G nor a claim of the form that a is F and a is G is per se contradictory – only if being F and being G stand in a certain relation: if they are incompatible. For the notions of being true and being false, this relation of incompatibility is described by the principle of (Exclusiveness). Only at step (12) a contradictory result has been established explicitly. 177 * The first eight steps of Informal Variant 1 are meant to capture the ping-pong quality of our intuitive reasoning about a Liar sentence (if true then false, if false then true, and so on). 178 But given the naïve truth principle in its ‘if and only if’-formulation (NT), this part of the reasoning can be abridged considerably:

176

177

178

When presenting an argument as in the table above, we use indentation to indicate a sub-argument which depends on some assumption. An alternative way from the result at step (8) to an explicit contradiction is to appeal to the following logical rule, that is classically acceptable: If we have inferred a biconditional claim of the form that p if and only if q, then we can go on to infer the disjunctive claim that (p and q) or (not p and not q). This allows to infer from the result at step (8) that λ is true and false or λ is neither true nor false. But the first conjunct contradicts (Exclusiveness) and the second conjunct contradicts (Exhaustiveness). Cf. the remark about the non-temporal nature of the reasoning in section 1.2.

2.4 Informal variants of the basic Liar reasoning

55

Informal Variant 2 of the basic Liar reasoning (1) λ means that λ is false.

(Meaning of λ)

(2) λ is true if and only if what λ means is the case.

(NT) for the case of λ

(3) λ is true if and only if λ is false.

(1), (2), transitivity of ‘if and only if’

From here we reach a contradiction as before, i. e., as in steps (8) through (12) of Informal Variant 1. In the first two variants of the Liar reasoning, we appeal to the naïve principles about how a truth relates to what is the case ((NT⇒) and (NT⇐) in Informal Variant 1 and (NT) in Informal Variant 2. We can go another way, and argue from one principle about truths and one about falsehoods ((NT⇒) and (NF⇒)): Informal Variant 3 of the basic Liar reasoning (1) λ means that λ is false.

(Meaning of λ)

(2)

assumption for conditional proof

(3) (4) (5) (6) (7) (8) (9)

λ is true. λ is false. If λ is true then λ is false. λ is false. λ is not false. λ is true. If λ is false then λ is true. λ is true if and only if λ is false.

(1), (2), (NT⇒) (2)–(3), conditional proof assumption for conditional proof (1), (5), (NF⇒) (6), (Exhaustiveness) (5)–(7), conditional proof (4), (8), conjunction introduction

From here we reach a contradiction as before, i. e., as in steps (8) through (12) of Informal Variant 1. It is also possible to change the overall structure of the reasoning. In Informal Variant 1 and 3, the first part of the reasoning consists of two branches, which together have the result that the Liar sentence is true if and only if it is false, which is shown to be contradictory in a second part of the reasoning. By contrast, in the following variant a contradiction is proved in each branch of the first part of the reasoning and these are then put together in the second part of the reasoning (with the help of the law of (Excluded Middle) and the rule of reasoning by cases): Informal Variant 4 of the basic Liar reasoning 179 (1) λ means that λ is false.

(Meaning of λ)

(2)

assumption for conditional proof

(3) (4) (5)

179

λ is true. λ is false. λ is not true. λ is true and λ is not true.

Cf. Beall /Glanzberg 2014, subsection 2.3.3.

(1), (2), (NT⇒) (3), (Exclusiveness) (2), (4), conjunction introduction

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Chapter 2 Informal Logic and the Liar Reasoning (6) If λ is true then λ is true and λ is not true.

(2)–(5), conditional proof

(7)

assumption for conditional proof

(8) (9) (10) (11) (12) (13)

λ is not true. λ is false. λ is true. λ is true and λ is not true. If λ is not true then λ is true and λ is not true. λ is true or λ is not true. λ is true and λ is not true.

(7), (Exhaustiveness) (8), (NT⇐) (7), (9), conjunction introduction (7)–(10), conditional proof (Excluded Middle) for the case of λ (6), (11), (12), reasoning by cases

Up to this point it might seem as though the basic Liar reasoning depends essentially on the rule of conditional proof. But in its stead we can also use the rule of consequentia mirabilis: Informal Variant 5 of the basic Liar reasoning 180 (1) λ means that λ is false.

(Meaning of λ)

(2)

assumption for consequentia mirabilis

(3) (4)

λ is true. λ is false. λ is not true.

(1), (2), (NT⇒) (3), (Exclusiveness)

(5) λ is not true.

(2)–(4), consequentia mirabilis

(6)

assumption for consequentia mirabilis

(7)

λ is false. λ is not false.

(1), (6), (NF⇒)

(8) λ is not false.

(6)–(7), consequentia mirabilis

(9) λ is not true and λ is not false.

(5), (8), conjunction introduction

(10) It is not the case that λ is true or λ is false.

(9), De Morgan

(11) λ is true or λ is false.

(Exhaustiveness) for the case of λ

(12) λ is true or λ is false and it is not the case that λ is true or λ is false.

(10), (11), conjunction introduction

An interesting point about Informal Variant 5 is that the result of the first part of the reasoning (here, steps (1) through (9)) is not that the Liar sentence is true if and only if it is false, but that it is neither. This of course contradicts the principle of (Exhaustiveness), so that Informal Variant 5 like the variants before ends in a contradiction. But, as we shall see later, this point can also be taken as a clue for a first attempt to solve the Liar paradox. 181 * 180 181

Cf. Prior 1961, 16 (for steps (1) through (9)). Cf. sections 2.5 through 2.8.

2.5 The extended Liar reasoning

57

We have now seen enough variants to illustrate the basic Liar reasoning. Our reason for presenting a wide spectrum of diverse variants was that we wanted to show that the basic Liar reasoning is logically robust in the sense that there is lots of room for variation of the logical rules that are used in the reasoning. It should have become evident that solving the Liar paradox will not be a matter merely of deciding on which logical rule or law of classical logic to drop. Of course there are also similarities between all variants. In particular, in each variant we have appealed to some of the naïve principles about truths and falsehoods and to both the principle of the (Exhaustiveness) and of the (Exclusiveness) of truth and falsity. Another important similarity concerns the question of formality that was posed in section 1.10, and in particular the sub-question: How formal are the informal variants of the Liar paradox? What we have seen here is that the Liar paradox can be presented in a way that is perspicuous but entirely informal – all our claims, including the principles about truth and falsity, were formulated in natural language, and the reasoning was governed by informal logic. But since informal logic as it was presented here is easily formalized by classical logic, and the principles about truth and falsity are clear enough to be amenable to formalization, we also have made plausible that this informal reasoning is less far off from a formal treatment than both some formal and some non-formal philosophers might think. 182

2.5 The extended Liar reasoning We have seen how informal reasoning about a simple Liar sentence leads to a contradiction like its being true and not true, or like its being true-or-false as well as not true-or-false. 183 With regard to this we have spoken of the basic Liar reasoning because, when that point has been reached, the story of the Liar paradox tends to go on as soon as an attempt is made to solve it. This is what we will be concerned with in the rest of this chapter. Rather than take an historical orientation, we will describe in an abstract (or, systematic) way how the problem of the Liar paradox unfolds after the basic Liar reasoning, because it turns out that after the first attempt at solving the paradox has been made, typically new problems will crop up. As so many attempts have been made to solve the paradox constituted by the basic Liar reasoning, it is possible to describe some common features of the dialectic which tends to unfold when a solution to the Liar paradox has been proposed. We will call this dialectic the extended Liar reasoning, and we will present it in a simplified and purely systematic fashion, i. e., without trying to do justice to the intricacies of any proposal which has actually been made. 184 So let us see how the story goes on. In our reasoning about a simple Liar sentence λ, which ascribes falsity to itself, we have used the naïve truth principle (and 182 183 184

We will of course say more about the formalization of the Liar reasoning; cf. chapter 3. Cf. Informal Variant 4 and 5 in section 2.4, respectively. We will give an overview of contemporary approaches to the Liar paradox in section 7.5, but it will also be too brief to deal with the details of any approach in an adequate way.

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sometimes also its companion principle about falsity) and the principles of (Exhaustiveness) and (Exclusiveness) to infer a contradiction. 185 After thus taking stock, a paradox solver may well go on to reason thus: As we have reached an unacceptable result, we want to give up one of our premises. It is perhaps most natural to give up the (Exhaustiveness) of truth and falsity. After all, other sentences than λ have been taken to be neither true nor false, e. g., ‘The present king of France is bald’ 186 and ‘There will be a sea-battle tomorrow’ 187. So let us stick to the (Exclusiveness) of truth and falsity but give up their (Exhaustiveness) – we henceforth will tolerate truth value gaps! After this change of mind, let us take a fresh look at the last informal variant we discussed previously – specifically, at its first nine steps: Informal Variant 5 of the basic Liar reasoning with a simple Liar sentence (1) λ means that λ is false.

(Meaning of λ)

(2)

assumption for consequentia mirabilis

(3) (4)

λ is true. λ is false. λ is not true.

(1), (2), (NT⇒) (3), (Exclusiveness)

(5) λ is not true.

(2)–(4), consequentia mirabilis

(6)

assumption for consequentia mirabilis

(7)

λ is false. λ is not false.

(1), (6), (NF⇒)

(8) λ is not false.

(6)–(7), consequentia mirabilis

(9) λ is not true and λ is not false.

(5), (8), conjunction introduction

As we have decided to tolerate truth value gaps, we can now accept the conclusion in line (9); without the principle of (Exhaustiveness), the subsequent reasoning of Informal Variant 5 will not go through. The sentence λ has thus turned out to be unproblematic, after all. We have reached what might be called the simple truth value gap solution. It is called simple because the only thing we have done is to give up the principle of (Exhaustiveness) – full of hope that this will suffice for a solution. But it does not suffice. For, if there are simple Liar sentences like λ, then there will surely also be strengthened Liar sentences. Let us have a look a one of them. We appear to be able to introduce it in the same way we introduced its simpler sibling in the previous section: We start from a fact about language, namely, the existence of a strengthened Liar sentence. We can make the corresponding claim like this: (Existence of a Strengthened Liar Sentence) There is a sentence that means that it is not true. Let us call the strengthened Liar sentence that this existence claim is about ‘Λ’. Then we can appeal to the following fact about Λ in our reasoning: 185 186 187

Cf., especially, section 2.4. Cf., e. g., Strawson 1950. Cf., e. g. Łukasiewicz 1930 about Aristotle De Interpretatione, 18a 33.

2.5 The extended Liar reasoning

59

(Meaning of Λ) Λ means that Λ is not true.

Now we can argue as follows: 188 Variant 1 of the basic Liar reasoning with a strengthened Liar sentence (1)

Λ means that Λ is not true.

(Meaning of Λ)

(2)

Λ is true if and only if what Λ means is the case.

(NT) for the case of Λ

(3)

Λ is true if and only if Λ is not true.

(1), (2), transitivity of ‘if and only if’

(4)

Λ is true or Λ is not true.

(Excluded Middle) for the claim that Λ is true

(5)

Λ is not true.

(3), (4), reasoning by cases

(6)

Λ is true.

(3), (4), reasoning by cases

(7)

Λ is true and Λ is not true.

(5), (6), conjunction introduction

As we did not need to appeal to the principle of (Exhaustiveness), the strengthened Liar sentence Λ does indeed constitute a problem for the simple gap solution. The proponent of the simple gap solution might now react by jettisoning, in addition to the (Exhaustiveness) of truth and falsity, an item of logic, namely the law of (Excluded Middle), 189 which in the above variant is needed to justify the application of the rule of reasoning by cases in steps (5) and (6). Intuitionistic logic, which after all is a perfectly respectable weakening of classical logic, does not have the law of (Excluded Middle). We might call this new attempt of the paradox solver the simple paracomplete solution. 190 But it turns out that the law of (Non-Contradiction) – which is a law not only of classical but also of intuitionistic logic – can take the place of the law of (Excluded Middle) in the reasoning about a strengthened Liar sentence: 191 Variant 2 of the basic Liar reasoning with a strengthened Liar sentence

188 189 190 191

192

(1)

Λ means that Λ is not true.

(Meaning of Λ)

(2)

Λ is true if and only if Λ is not true.

(1), (NT) 192

(3)

It is not the case that Λ is true and Λ is not true.

(Non-Contradiction) for the claim that Λ is true

The first three steps are similar to Informal Variant 2 of the basic Liar reasoning in section 2.4. For the relation of (Exhaustiveness) to (Excluded Middle), cf. section 2.3. With a nod to Hartry Field and Jc Beall. The idea of using the law of (Non-Contradiction) in this specific way, i. e., as in steps (3) through (7), is due to Field 2008, 8f., fn. 8. But much earlier, Solomon Feferman pointed out that most people fail to point out that there are intuitionistically acceptable variants of the Liar reasoning (Feferman 1984, 244f.). The first two steps of Variant 2 compress the first three steps of Variant 1.

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Chapter 2 Informal Logic and the Liar Reasoning (4)

It is not the case that Λ is true and Λ is true.

(2), (3), substitution of equivalents

(5)

Λ is not true.

(4), propositional logic

(6)

Λ is true.

(2), (5), substitution of equivalents

(7)

Λ is true and Λ is not true.

(5), (6), conjunction introduction

Note that, at least in its first part, the paradoxical reasoning about Λ is similar in structure to the paradoxical reasoning about λ. The main point of difference is that the role that being false plays in the reasoning about λ is played by being not true in the reasoning about Λ. Switching from falsity to untruth allows, in the second part of the reasoning, to infer a contradiction without appealing to (Exhaustiveness) or (Excluded Middle). Thus giving up the (Exhaustiveness) of truth and falsity and the law of (Excluded Middle) solves the paradox engendered by λ but not the paradox engendered by Λ. That is why Liar sentences like λ, which ascribe falsity, are called simple, and Liar sentences like Λ, which ascribe untruth, are called strengthened. 193 The simple gap approach and the simple paracomplete approach fail to solve the Liar paradox because they dissolve the paradoxicality only in the special case where the Liar sentence is simple.

2.6 The phenomenon of Revenge The fact that the simple gap solution is shown to fail by introducing a strengthened Liar sentence is a special case of what is called the Revenge of the Liar. 194 More generally, the Revenge phenomenon is that after a specific solution has been proposed there shows up a close relative of the original paradox which defies that solution. The new paradox usually starts with a relative of the original Liar sentence, often called a Revenge sentence, which is custom-made to fit (or rather, not to fit) what is characteristic of the solution that has been proposed. That this is also the case with regard to the strengthened Liar sentence which above was the downfall of the simple gap solution can be brought out by noting that (within the framework of that solution) being not true is equivalent to being false or being gappy. Instead of Λ we might as well have worked with Λ0 which means that Λ0 is false or gappy. 195 And in view of what Λ0 says it is no great surprise that it constitutes a problem for the gap solution. 196 * 193 194

195

196

Cf. section 1.2. Cf., e. g., Beall 2007b, 1ff. and the contributions in Beall 2007a. Some speak about “the extended Liar paradox”; e. g., Priest 2006a, 15ff. Despite the equivalence of the notions of being not true and being false or gappy, the sentences Λ and Λ0 are not synonymous because they say of distinct objects that they fall under these notions, namely of the two sentences Λ and Λ0 . This strategy can be generalized to deliver Revenge sentences for a broad range of a solutions in the following way: Any solution which is based on the claim that something is wrong with the Liar sentence

2.7 The Reflective problem (a. k. a. the Son of the Liar)

61

Now, if the proponents of the simple gap solution (or the simple paracomplete solution) do not want to give up after encountering their first Revenge sentence, they will probably refine the gap approach (or the paracomplete approach) by further weakening the logic, adopting complicated principles about when a sentence is true, and so on. Let us imagine that thus a sophisticated gap solution is reached which blocks the paradoxical reasoning even in the case of a strengthened Liar sentence. According to the sophisticated gap solution, the strengthened Liar sentence falls into the same gap between being true and being false as the simple Liar sentence does according to the simple gap solution.

2.7 The Reflective problem (a. k. a. the Son of the Liar) But now the paradox solver encounters a new problem. She claims that the strengthened Liar sentence is not true and not false, and hence that the strengthened Liar sentence is not true. But that is exactly what the strengthened Liar sentence seems to say! So it would appear that the strengthened Liar comes out as true after all, in which case the paradox would of course recur. This, however, need not be so because the complicated principle about truth which is part of the sophisticated gap solution might well be such that it entails that the Tarskian truth schema does not apply to the strengthened Liar sentence. But even then a serious problem will remain, because in that case there will be two sentences which appear to mean the same, but one of them (the strengthened Liar sentence) is not true while the other (the paradox solver’s sentence about the strengthened Liar sentence) is true. In many cases, these two sentences, which differ in truth value, will be orthographically indistinguishable. A nice illustration is provided by what has been called the Line Liar: 197 1 The sentence on line 1 is not true. 2 The sentence on line 1 is not true. A proponent of a sophisticated gap solution will take the sentence on line 1 to be not true and the sentence on line 2 to be true. But they are occurrences of the same sentence! Although there is no outright contradiction here, something really strange

197

can be construed as classifying the Liar sentence as belonging to the Rest, which is that category of sentences which are not eligible to bear a truth value because something is wrong with them. Now a sentence that can be translated as saying of itself that it is false or belongs to the Rest will be a Revenge sentence. (Cf. Priest 2006a, 23f.) In general, Revenge sentences can get rather complicated. E. g., tailored to an approach that is based on the idea that distinct tokens of a sentence can differ in truth value, the following Revenge sentence has been proposed (Weir 2000, 160): “Some token of any sentence equivalent to ‘All tokens of the first sentence containing ‘equivalent’ in this article are untrue’ is true.” To my knowledge, this way of presenting what we will in a moment call the Reflective problem was first given by Haim Gaifman and the name ‘Line Liar’ is due to Alan Weir (Gaifman 1992, 223; Weir 2000, 158). The Reflective problem already plays a role in the seminal contributions by Charles Parsons and Tyler Burge (Parsons 1974a, 388f. and Burge 1979, 178f.). See also Sainsbury 1995, 122ff. (particularly 126); Rheinwald 1988, 53–56; and Simmons 2003, 230.

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seems to be going on. Usually, occurrences or tokens of a single sentence can differ in truth value only if some expression which is part of the sentence is indexical or otherwise context-sensitive – but prima facie no semantic part of ‘The sentence on line 1 is not true’ is essentially 198 context-sensitive. The problem that we have just described for a sophisticated gap solution is often called ‘the Son of the Liar’. 199 I prefer the label ‘the Reflective problem’ (or ‘the problem of Reflection’), which is meant to be mnemonic of the “reflective step” 200 which is essential to the reasoning that leads to the problem. Although the Reflective problem is serious – because to solve it the paradox solver must explain why synonymous sentences differ in truth value – it must be noted that it is not as severe a problem as the contradictory result of the basic Liar reasoning or the phenomenon of Revenge. Its result appears to be (very) uncomfortable, but it does not appear to be unacceptable. Hence we need not see it as another paradox. 201

2.8 The Problem of Expressibility The Reflective problem can be construed as a special case of a more general problem, which is called the problem of Expressibility. It results from an aspect that is inherent to many (probably most) approaches to the Liar paradox, the requirement that the paradox-free object language be distinct from the meta-language of the paradox solver. 202 A paradox solver runs into what has been criticized as the Expressibility problem when she describes a paradox-free object language in a meta-language that is richer in the sense that it must allow to express some notion which (on pain of Revenge) is inexpressible in the object language. An expressive difference like this is considered by many to be problematical because they have a strong intuition that at least natural languages do have the expressive resources to enable talk about themselves. In other words: There is an Expressibility problem because we have the strong intuition that a language can be its own meta-language, so that what usually is called semantic closure 203 is possible, at least in the natural language that is the basis of our language of use. In the words of Christopher Gauker: “[. . .] the deep challenge [. . . ] is not [. . .] to construct a language in which there are no paradoxical sentences. The deep challenge is to construct such a language 198

199

200

201 202 203

Although the expression ‘line 1’ arguably is context-sensitive, it can easily be expanded to a contextfree expression without changing what is characteristic of the Line Liar. E. g., c.f. Rheinwald 1988, 53ff. – Terminology is not at all uniform. It is especially confusing that Burge (Burge 1979, 173) and some who follow him (e. g., Gaifman 1992, 223ff.; Sheard 1994, 1047) refer to the Reflective problem as “the strengthened Liar paradox”; ignoring the at least equally well established usage according to which the term ‘strengthened Liar sentence’ refers to a sentence which ascribes untruth to itself. This terminology is inspired by Christopher Gauker, who speaks of “stepping back” and about reflexivity; cf. Gauker 2006, 393ff. Cf. section 1.1. For the distinction of object language and meta-language, cf. section 2.2. Cf. section 3.7.

2.8 The Problem of Expressibility

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without imposing unacceptable limitations on the semantic properties that can be expressed in that language. In particular, we must not deny ourselves the ability to talk about the semantic properties of our own language that we can in fact talk about.” 204

The Reflective problem which the sophisticated gap solution is confronted with is a case of the Expressibility problem because there is something which the paradoxfree language cannot express but which the paradox solver needs to express, namely that the Liar sentence is not true. This can be brought out by turning our attention to the notion of negation that is involved. For proponents of the sophisticated gap solution, all Liar sentences are gappy. They can achieve this by understanding the negation in any strengthened Liar sentence as choice negation. It had, from the vantage point of the sophisticated gap solution, better not be understood as exclusion negation. 205 (A sentence that says of some sentence that it is not true, where ‘not’ is meant in the sense of exclusion negation, is true if the sentence it is about is gappy. 206 But as the strengthened Liar sentence is a sentence which says of the strengthened Liar sentence that it is not true, this precludes the strengthened Liar sentence from being gappy.) But it is natural for the paradox solver to express the gappiness of the strengthened Liar sentence by saying that it is not true, using exclusion negation. The point is made succinctly by Keith Donnellan when he criticizes the sophisticated gap solution of Robert Martin, which is based on a theory of category distinctions: 207 “Martin seems to me to leave himself no way of saying what, after all, constitutes the central idea in the theory of category differences: that when a particular predicate is applied to a subject outside the range of application of the predicate, the result is not true and not false. If he had two kinds of negation, he could express this by ‘It is not the case that F(a)’, where the negation is exclusion negation.” 208

So, the sophisticated gap solution can stick to its claim that strengthened Liar sentences are gappy by excluding exclusion negation (pun intended) for the paradoxfree object language, but to do that it needs the notion of exclusion on the metalevel (as witnessed by the word ‘excluding’ in the pun formula). Hence the sophisticated gap solution, in its attempt to thwart Revenge, runs into the Expressibility problem. 204 205 206

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Gauker 2006, 394. For the notions of choice negation and exclusion negation, cf. section 2.3. This is so, independent of questions about how the truth value of a truth ascribing sentence depends on the truth value of the sentence it is about. The main question is whether a sentence which ascribes truth to some sentence, which is gappy, is (a) also gappy or (b) false (cf. the distinction between the “weak” and the “strong” conception of truth in Yablo 1985, 301). To have a chance of consistently ascribing gappiness to a strengthened Liar sentence, the proponent of the sophisticated gap solution has to adopt the evaluation strategy (a). While this strategy is in accordance with the intuition that a sentence that ascribes truth should have the same truth value as the sentence it is about, it stands in some tension to the naïve principles about truth and falsity (cf. section 2.3). But this point is not important here. Martin 1978[1970]c; especially 96. Martin’s proposal is criticized by Donnellan 1978[1970]; especially 113ff. Donnellan 1978[1970], 115; notation changed.

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2.9 Recurrence of the Reflective problem Let us imagine that now, in view of these problems, our paradox solver finally turns her back on all gap solutions and adopts what we will call the simple meaninglessness solution. It is based on the claim that the (purported) Liar sentence is not meaningful, which amounts to rejecting the claim that a Liar sentence means that it is false (untrue), which was the starting point of all the variants we presented of the basic Liar reasoning, and entailed by the claim of the existence of a Liar sentence. 209 Hence, it blocks the Liar reasoning at the very beginning, much earlier than the gap solution. 210 (The meaninglessness solution that is sketched here is simple because – among other things it does not do – it gives no explanation for the meaninglessness of Liar sentences. 211) The characteristic difference between gap and meaninglessness solutions can be brought out after noting that there are two ways for something to be neither true nor false: One way is to be a candidate for being true or being false but for some reason to be neither (examples that would be accepted by many are sentences that appear to be about the present king of France or about a future sea-battle), another way is not to be a candidate for being true or being false in the first place, like trees and thunderstorms. It is natural to allow only meaningful declarative sentences as candidates for being true or being false, 212 and to group meaningless sentences with trees and thunderstorms. Only a meaningful declarative sentence that is neither true nor false can properly be called a truth value gap. By contrast, and to speak figuratively, a meaningless sentence is less than a gap, or an immaterial gap – albeit also neither true nor false 213. This accounts for the main advantage that the meaninglessness solution has over the gap solution. While both agree on the untruth of any (purported) Liar sentence, it appears that only the meaninglessness solution can guarantee that its untruth will not (at some later step of the paradoxical reasoning) entail its truth. But it can be argued that even the meaninglessness solution is beset by the Reflective problem, at least by a variant of it. Recall the Line Liar: 214 1 The sentence on line 1 is not true. 2 The sentence on line 1 is not true. According to the meaninglessness solution, the sentence on line 1 is meaningless and hence not true. But, given any one of the usual explanations for the meaninglessness of the sentence on line 1, there is no reason why the sentence on line 2 should also be 209 210

211 212 213 214

Cf. the first lines of the informal variants in section 2.4 and in section 2.5. The gap solution accepts all steps of the Liar reasoning up to the claim that the Liar sentence is true if and only if it is false (untrue) – that is, all steps of a variant of the Liar reasoning that follows the standard sequence of inferential steps, i. e., in variants 1, 2, 3, or 5 in section 2.4 and the two variants in section 2.5. For reference to variants of the meaninglessness solution that are not simple, cf. section 7.5. Cf. section 8.1. At least when the negation of the language we use is exclusion negation! The exact terminology may vary here: Depending on the approach in question, we might need to replace the word ‘sentence’ by phrases like ‘sentence-like object’ or ‘proto-sentence’ in both lines.

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meaningless. Hence the proponent of the meaninglessness solution has to construe it as meaningful, and, in view of what her claim about the sentence on line 1 entails, as true. It is this difference between the orthographically equivalent sentences on line 1 and 2, that one is meaningless and not true and the other is meaningful and true, which at least resembles the Reflective problem the sophisticated gap solution was confronted with. But here, the problem is not as severe. While for the gap solution, the Reflective problem was that we have an untrue sentence which means the same as a true sentence; for the meaninglessness solution, the true sentence only appears to mean the same as the untrue sentence, because it looks the same in the sense of being orthographically indistinguishable, which in the absence of essentially context-dependent expressions usually guarantees synonymy. 215

2.10 Revenge, Reflection, Expressibility As the presentation of the dialectic between paradox and proposed solutions in sections 2.5 through 2.9 went hither and thither, we should sum up quickly what we have learned from it. In the last five sections, we have encountered the bare bones of several strategies of dealing with the Liar paradox – to wit, what we have called the simple gap solution, the simple paracomplete solution, the sophisticated gap solution, and the simple meaninglessness solution. Because of brevity, little justice has been done here to any of the attempts which have actually been made to argue in these directions. But in view of the systematic aims of the present chapter, details would have been distracting. It makes little sense to try and evaluate a proposal for solving a problem when the problem has not yet been fully described. And in fact the main purpose of the last five sections has been to describe some serious extra problems which tend to show up so persistently when first attempts at a solution are made that we should actually view them as part of the (larger) problem comprised by the Liar paradox. They are the following. The phenomenon of Revenge is the phenomenon that, after a specific proposal for a solution has been given, a new paradox can be found that is not solved by the original proposal. Often the new paradox is engendered by a Revenge sentence, i. e., a sentence tailored to the tools of the proposal at hand. Strengthened Liar sentences are the Revenge sentences that a gap solution has to deal with. The problem of Reflection (a. k. a. the Son of the Liar) is that for many a solution which in some way rejects every (purported) Liar sentence on the level of the paradox-free object language, there will be a similar sentence, often on the level of the paradox-solving meta-language, that is asserted by the solution and that is very similar to the rejected sentence. (Both the rejection and the similarity can take different forms: For the gap solution, rejecting is assigning a gap and the similarity is synonymy; for the meaninglessness solution, rejecting is claiming to be meaningless and the similarity is orthographical equivalence.) 215

Here I disagree with Rheinwald, who argues that a solution which does not accept the (purported) Liar sentence as a candidate for being true or being false is confronted with the Reflective problem in the same way as the gap solution. Cf. Rheinwald 1988, 53–56; especially 55.

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The problem of Expressibility is a more general problem that grows out of the problems of Revenge and Reflection. It occurs when the paradox-free language that is designed by a paradox solver does not allow to express all notions that are essential to that solution. One case of the problem of Expressibility is the Reflective problem as it occurs for the gap solution: The object language cannot express the particular notion of not being true that is needed to describe the status of the Liar sentence itself.

Chapter 3

Formal Logic and Tarski’s Legacy

Gottlob Frege: “Ich wollte nicht eine abstracte Logik in Formeln darstellen, sondern einen Inhalt durch geschriebene Zeichen in genauerer und übersichtlicherer Weise zum Ausdruck bringen, als es durch Worte möglich ist.” 216 Alfred Tarski: “It remains perhaps to add that we are not interested here in ‘formal’ languages and sciences in one special sense of the word ‘formal’, namely sciences to the signs and expressions of which no material sense is attached. For such sciences the problem here discussed [i. e., the definition of truth] has no relevance, it is not even meaningful.” 217

We now come to the second part of our dual presentation of informal and formal variants of the basic Liar reasoning, which will concern the Liar paradox in a formal setting. The seminal text is Tarski’s monograph “The Concept of Truth in Formalized Languages”, 218 which arguably is the origin for all later formal work on the Liar paradox and the concept of truth. It will be our point of departure in almost every section of the present chapter; and confrontation with the Tarskian legacy will prove especially fruitful with regard to the question of formality. 219 In accordance with the common project of the preceding and the present chapter, we will see to it that the first sections of chapter 3 as far as possible are parallel to the first four sections of chapter 2. In both chapters, we first describe logic in general (in sections 2.1 and 3.1), then characterize how truths relate to the world (in sections 2.2 and 3.2), go on to present some particular logical rules and laws that play a role in the Liar reasoning (in sections 2.3 and 3.3), and conclude the parallel 216

217 218

219

“I did not intend to present an abstract logic in formulas, but to express a content in written signs and thus more precisely and clearly than it is possible to do in words.” (Frege 1998[1882], 97f.; my translation) Tarski 1956[1935], 166. After publishing a preliminary version of his monograph on truth in Polish in 1933, Tarski published a longer German version of it in 1935. It is the basis for the English version Tarski 1956[1935], with which we will work here. Cf. Tarski’s bibliographical note, Tarski 1956[1935], 152. Among the extensive literature on Tarski’s theory of truth and the role the Liar paradox plays in it, I have found the following to be particularly valuable: Feferman/Feferman 2004, 109–123, is an accessible and authoritative “step-by-step exposition” of Tarski’s theory; and Soames 1999, 67–162, is a careful discussion of both the theory and its relevance. See also Burgess/Burgess 2011, 16ff. and Künne 2003, 175–225.

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part by discussing the basic Liar reasoning (in sections 2.4 and 3.5). From there onwards, the two chapters take different paths. The present chapter 3 is much longer than the corresponding passage in its companion chapter 2. For this there are three reasons: Firstly, extra care is needed in the exposition of my preferred conceptions of a formal language, formal logic, and formal theories, because – even though in the 1930s Tarski still shared them – they are non-standard from a contemporary point of view (in section 3.1); and this non-standardness calls for a detailed inspection of different ways of formulating the Tarskian truth schema (in section 3.2). Secondly, the presentation of formal variants of the basic Liar reasoning needs more preparatory work than that of its informal variants, because here we can no longer rely on our intuitive understanding (in sections 3.1 through 3.4). Thirdly, after some formal variants of the basic Liar reasoning have been presented and duly compared to their informal counterparts (in section 3.5), we for both historical and systematic reasons need to take a look at the role the Liar-like reasoning plays in the proof of Tarski’s Theorem of the undefinability of truth (in section 3.6) and at the condition of paradoxicality Tarski formulated as the lesson he drew from putting the Liar paradox into a formal setting (in section 3.7). The tools developed for these purposes will then allow us to conclude the chapter by making a non-Tarskian and (as far as I know) novel point about how severe the problem posed by the Liar paradox is (in sections 3.8 and 3.9).

3.1 Formal logic: Language and calculus What distinguishes formal from informal logic? As we said at the beginning of the last chapter, formal logic explicates good reasoning by giving a complete description of a formal language and of the inferential relations between sentences of that language. 220 Before we can fruitfully investigate what is distinctive of formal logic, we therefore need to ask: What is distinctive of a formal language, as contrasted with natural languages? 221 In this section we will first give a preliminary answer by interpreting a remark of Tarski’s about the notion of a formal language, and characterize the corresponding notions of formal logic and of a formal theory. Then we will distinguish two quite different ways in which these notions of a formal language, of formal logic, and of formal theories can be made more specific, which we will dub the semanticist and the syntacticist understanding, and refer to their historical root, which is the debate about logic as language versus logic as calculus. Later in this chapter, the distinction between semanticism and syntacticism will play an important role in the investigation of what is characteristic of formal variants of the basic Liar rea-

220 221

Cf. section 2.1. In the following brief characterization of formality, we cannot hope to do full justice to all uses of the term ‘formal’. For a general discussion and an overview of the broad range of conceptions of formality, cf., e. g., Rheinwald 1984, 23ff. Cf. also MacFarlane 2000 and in particular Dutilh Novaes 2012.

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soning and of what distinguishes them from informal variants. Much later in the study, it will again prove crucial, because an endorsement of the semanticist understanding of language will be one of the first steps of my own approach to the Liar paradox. 222 * In his monograph about truth, Tarski characterizes formal languages 223 thus: “These [i. e., formal languages] can be roughly characterized as artificially constructed languages in which the sense of every expression is unambiguously determined by its form. Without attempting a completely exhaustive and precise definition, which is a matter of considerable difficulty, I draw attention here to some properties which all the formalized languages possess: (α) for each of these languages a list or description is given in structural terms of all the signs with which the expressions of the language are formed; (β) among all possible expressions which can be formed with these signs those called sentences are distinguished by means of purely structural properties.” 224

One important condition that Tarski mentions here is that a formal language is completely specified on the syntactic level, i. e., that it is unambiguously clear which strings of letters are expressions and to which syntactic category a given expression belongs. This is usually achieved by a recursive definition of the expressions of the language, which starts with a complete list of the syntactic atoms and then explains how, step-by-step, all further expressions can be formed from them by concatenation. Another characteristic of formal languages that is mentioned by Tarski and others is their artificiality. It must be meant here in a way that allows the contrast with natural languages, although they, too, are human artifacts. It is likely that the artificiality of formal language in this sense of a characteristic feature that is not shared by natural languages is a precondition of the complete specification that is required of them, given the unsystematic way in which natural languages evolve and grow. What I find most important in the above quote is Tarski’s observation that, in a formal language, “the sense of every expression is unambiguously determined by its form”. What Tarski means by the “form” of an expression very probably is its physical shape, i. e., the geometrical shape of an inscription. 225 But, as we can learn from Nelson Goodman and others, signs are never determined by their physical shape alone. E. g., even in the shared context of a notational system of the Latin script, the same geometrical shape may be read as an inscription of the lowercase letter ‘a’ in one context and in another context as an inscription of the lowercase 222 223

224 225

Cf. part III and chapter 8 in particular; and cf. already chapter 6. A terminological remark: Where we say “formal language”, Tarski in his monograph says “formalized language”. Tarski 1956[1935], 165. We restrict our attention here to written languages. In the case of spoken natural language, the physical shape of an expression is the sound-shape of an utterance. But in a discussion of formality, spoken language can be put to the side, because formal languages are modeled on written language (e. g., Frege 2002f[1882], 72–74 and Łukasiewicz 1957, 15; cf. Krämer 2003).

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letter ‘d’. 226 We can put to the side the difficult question of how shape and context together determine a sign, and move on directly to the alphabetical and the syntactic level. The alphabetical shape of an expression can be specified by saying what string of letters it is; the syntactic shape of an expression is its alphabetical shape together with its syntactic structure, i. e., with what in natural language would be its grammatical structure. In these terms, Tarski’s observation can be reformulated as the claim that in a formal language, the meaning of an expression is determined by its syntactic shape. E. g., whereas we can easily read off the one and only intended meaning from the arithmetical expression ‘6 > 5’, the English expression ‘He is going to the bank’ by its syntactic shape alone is far from determining a single meaning, because of the context-sensitivity of the anaphoric ‘he’, the indexicality of the tensed predicate ‘is going’, and the ambiguity of the noun ‘bank’. Thus Tarski’s observation entails among other things that in a formal language, there are none of the elements of context-sensitivity, indexicality, and ambiguity which are ubiquitous in natural language. It will be helpful to give a second reformulation of Tarski’s observation, now in terms of syntactic and semantic form. By the syntactic form of an expression we mean an inner structural aspect that may be shared by some expressions which are of different syntactic shape. 227 E. g., as ‘a’ and ‘b’ are different letters and thus differ syntactically, ‘a > m’ and ‘b > m’ differ in syntactic shape, but they nevertheless have the same syntactic form. 228 On the semantic level, there is a notion corresponding to the notion of syntactic form, which similarly is reached by abstracting away from some features of particular expressions. From the meaning of a particular expression we can move on to its semantic form, by which we mean nothing else than what is generally known as its logical form, i. e., those aspects of a meaningful expression which are relevant for the inferential relations it stands in. 229 E. g., as ‘Alice’ and ‘Bruce’ are names of different people and hence different in meaning, ‘Alice is faster than the messenger’ and ‘Bruce is faster than the messenger’ are not synonymous, but they nevertheless have the same semantic form (e. g., each one entails that someone is faster than the messenger and that there is a messenger). – In these terms we can say that in a formal language, syntactic form and semantic form coincide always. And this is distinctive of formal languages! Just compare the formal language expression ‘¬∃x x > m’, which clearly differs in syntactic form from ‘a > m’, 230 to the natural language expression ‘Nobody is faster than the messenger’, which has the same grammatical structure and thus is similar in syntactic form to ‘Alice is faster than the messenger’. 231 As we can think of the syntactic form

226 227

228

229 230

231

Goodman 1976, 130ff.; cf. Scholz 2004, 103 & 112. Our characterization is intentionally left unspecific here, because in different contexts different aspects of syntactic shape will be counted as belonging to syntactic form. These expressions are meant to belong to some standard language of Arithmetic, meaning that the number a is greater than the number m and that the number b is greater than the number m. Cf. Müller 2002, 94ff. and especially 95. These expressions are meant to belong to some language of arithmetic, with their standard meaning that no number is greater than the number m and that the number a is greater than the number m. Cf. Carroll 1971, 198ff. For a similar example cf. Frege 1988[1884], 45.

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of an expression as being directly visible in the relevant contexts, we can also say that each well-formed expression of a formal language wears its logical form on its face. 232 * We can now characterize the general notion of formal logic by saying that any particular system of formal logic consists of a formal language together with a deductive system, i. e., a specification of the inferential relations between the sentences of that formal language. A deductive system is meant to explicate the general notion of proof which we grasp intuitively. Corresponding to each intuitively correct particular proof, it specifies a deduction, i. e., a list of sentences of the formal language which starts with some formal sentences that correspond to the premises of the proof and ends with a formal sentence that corresponds to its conclusion. Because a deductive system is meant to explicate the notion of proof, it must meet certain requirements: Any deduction consists of only finitely many formal sentences, and each sentence in the list is required to stand in certain relations to some sentences prior to it in the list which can be specified by recourse to their syntactic shape alone and which can be checked in an algorithmic way. 233 It is worth emphasizing that a deductive system operates on the level of syntactic shape alone. This explains the metaphor that each sentence of a formal language wears its logical form on its face. The logical form of an expression corresponds to the inferential relations it stands in, the face of an expression just is its syntactic shape, and these two are connected by the deductive system because it is purely syntactic. We say that a sentence φ can be deduced from a collection of sentences Γ if and only if there is a deduction which lists the sentences of Γ as premises, which has no other premises, and which ends with the sentence φ as the conclusion. The sentences that can be deduced without having to use any other sentences to deduce them from are called the deducible (or provable) sentences or the theorems of our logical system. We can express both deductive relations and theoremhood with the help of the symbol ‘`’. We write, e. g., ‘‘a > m’ ` ‘∃x x > m’’ to say that the sentence ‘∃x x > m’ 232

233

This coincidence of form does not go down to the alphabetical level – alphabetical form and syntactic form need not coincide in a formal language. But there must be unique readability, i. e., syntactic shape must be determined by alphabetical shape. And there is a strong tendency to minimize the number of letters that make up a syntactically atomic expression, using for example single lowercase letters as constants and single uppercase letters as predicates in typical languages of quantified logic (cf. section 3.3). In natural language, by contrast, syntactic form and alphabetical form coincide only in exceptional cases like the indefinite article ‘a’ and the indexical singular term ‘I’. It is much more common that the alphabetical parts of a natural language expression are neither syntactic nor semantic parts of it. To give some examples: The expression ‘it’ is an alphabetical and a syntactic, but not a semantic part of the sentence ‘It is raining’, because otherwise that sentence would absurdly entail that some object satisfies the monadic predicate ‘is raining’. And the expressions ‘man’, ‘hat’, and ‘tan’ are alphabetical, but neither syntactic nor semantic parts of the proper name ‘Manhattan’. But then again, Lorin Sklamberg of The Klezmatics shows how a semantic dissection of the name ‘Manhattan’ is possible when he sings: “I met a man in a hat with a tan [. . . ] I met a Manhattan man.” (The Klezmatics 2002, “Man In A Hat”, Jews with Horns, Rounder, CD) Cf., e. g., Enderton 2001, 109f.

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can be deduced from the sentence ‘a > m’ and, e. g., ‘` ‘a = a’’ to say that the sentence ‘a = a’ is a theorem. 234 (The symbol ‘`’ is not an expression of the formal language we deal with, but an augmentation of the language we use to talk about it.) 235 There are two different approaches to giving the deductive system of a formal logic, depending on whether the focus is put on logical laws or on logical rules. In the classic (Frege /Hilbert-style) axiomatic approach, first some sentences are designated as theorems, the axioms of our system, and then only a few rules of deduction (modus ponens and some rule of substitution, say) are designated as admissible. From there, in a sometimes tedious process, further sentences can be shown to be theorems (and further rules can be shown to be admissible). But in many cases one can also take an approach of natural deduction (or give a sequent calculus), that specifies only rules of deduction, which then allow to show sentences to be theorems. 236 The result of adding to a system of formal logic further axioms which we understand to have non-logical content is called a formal theory. The notion of a formal theory is meant to explicate the notion of a scientific theory, with the axioms corresponding to the basic principles of that theory. Once a specific collection of axioms has been specified, a formal theory is similar to a system of formal logic in that the syntactic shape of a sentence suffices to check whether it is one of the theorems of the theory. An example of a formal theory is Peano Arithmetic, which is the result of adding the Peano axioms to first-order predicate logic; and each one of the formal set theories that were mentioned in section 1.8 is the result of adding a certain collection of axioms to first-order predicate logic. 237 * After this overview of the notions of a deductive system and a formal theory, we should mention briefly that there is another way to characterize and study a logic besides giving a deductive system, which differs markedly insofar as it takes an external perspective: formal semantics or model theory. We will look at it in detail when we come to the chapter on semantics. 238 Very roughly speaking, doing model 234

235

236 237

238

In contemporary logic texts it is customary to omit quote marks when expressions of a formal language are mentioned, writing in the present example ‘a > m ` ∃x x > m’ and ‘` a = a’. We do not follow this practice because it obscures that signs like ‘`’ belong to the meta-level, and also for the more specific reason that in an investigation of whether and to what extent it is possible for a language to speak about its own expressions it is important to be explicit about whether an expression is used or mentioned. Cf. section 5.5. In order to describe the general case, we anticipate some notation which will be explained later: We write pΓ ` ϕq to say that the sentence ϕ can be inferred from the collection of sentences Γ and p` ϕq to say that the sentence ϕ is a theorem. (Like ‘`’, the symbols ‘ϕ’, ‘ψ’, and ‘p. . . q’ are not expressions of the formal language we deal with, but augmentations of the language we use to talk about it.) There typically are also some notational differences on the meta-language level. As the distinction between a formal theory and a logical system depends on how we draw the line between logical and non-logical content, it will not always be clear whether our formal system is a logic or just some theory. E. g., while most people speak of ‘the (formal) theory of truth’, some will go so far and use the honorific ‘the logic of truth’. But for present purposes nothing important hangs on the question whether a given system is a logic or a theory. Cf. chapter 4 and section 4.4 in particular.

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theory means studying the behavior of the language by construing it as a system of objects that is mapped on another system of objects, the ontology. * At this point, we should also give a brief list of the many uses we can make of formal logic. We can: (1) use a particular logical system to explicate good reasoning, (2) use a particular logical system to enhance the language we use by symbolic augmentation (and regimentation), (3) use some sentences of a particular logical system to specify the logical form of some sentences of natural language, (4) use a particular logical system as an idealized image of an entire fragment of natural language, (5) use a particular formal theory to supplant a large fragment of natural language (usually in the sciences), and, last but not least, (6) study logical systems for their own sake. 239 In the present study, formal logic will be put to the first four uses of this list, and its fifth and sixth use will be discussed at some points. * After our overview of basic notions of formal logic, we are now in a good position to turn to an important distinction between two ways of understanding what it is. We have seen that syntax exhausts semantics on the three levels of formal language, formal logic, and formal theories: The syntactic shape of a formal expression determines its meaning, its inferential relations, and whether it belongs to a given theory. Therefore it is tempting (and sometimes beneficial) to forget all about meaning when dealing with a formal language, a formal logic, or a formal theory. And indeed, there is a syntacticist conception according to which a formal language is nothing but a system of syntactically specified expressions which are devoid of any meaning, and there are corresponding syntacticist conceptions of formal logic and of formal theories as nothing more than patterns among such purely syntactic expressions. Accordingly, a system of formal logic or a formal theory often is specified just by the collection of the purely syntactic sentences that are (or rather, which correspond to) its theorems. Syntacticism goes well with the treatment of a formal language as a mathematical object (or as a system of mathematical objects) and the corresponding mathematical treatment of formal logic and formal theories. We find an example in the following statement by Stewart Shapiro: “Strictly speaking, a formal language is a mathematical object, a recursively defined set of strings on a fixed collection of characters.” 240 239

240

Cf., e. g., Shapiro 1998, 135–137, for a discussion of (3), (4), and (5) and Priest 2006a, 73f., for an argument for the legitimacy of (4). Shapiro 1998, 135.

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Syntacticism is so near to mathematical logic because the syntactic shapes of expressions can easily be encoded by a system of abstract objects which can then be treated mathematically. 241 Such mathematical treatment would be difficult if there was any role left to play for intuitive facts about meaning. On the semanticist conception, by contrast, the expressions of a formal language do have meaning. And there are corresponding semanticist conceptions according to which a system of formal logic consists of a formal language that is understood in this semanticist way plus a deductive system that is understood as expressing principles of good reasoning (rather than just specifying a particular pattern among purely syntactic expressions), and a formal theory is understood to have a particular subject matter and its axioms are understood to be true principles that concern that subject matter. Put in another way, on the semanticist conception a formal language is a limiting case of a natural language, formal logic is a limiting case of informal logic, and a formal theory is a limiting case of a scientific theory, each one of course with the characteristics of complete specification and of syntax exhausting semantics. The conceptual distinction between what we call ‘syntacticism’ and ‘semanticism’ has played an important role in the modern history of logic and of formal science in general. Several important figures of modern logic were semanticists – among them Peano, Frege, Russell, the earlier Gödel, the earlier Tarski, 242 Quine, and, a bit later, Prior. But earlier mathematical logicians like Boole and Hilbert had already been syntacticists, and with the second half of the 20th century, syntacticism has become the mainstream in logic and formal semantics, expressed forcefully for instance by Jaako Hintikka, and today has achieved the status of a nearly unquestioned orthodoxy. 243 We cannot do full justice to the historical debates here, 244 but let us allude briefly to them. Frege contrasts his own semanticism with the syntacticist (or mathematical) orientation of Boole: “[. . . ] my purpose was not the same as Boole’s [. . .]. I did not intend to present an abstract logic in formulas, but to express a content in written signs and thus more precisely and clearly than it is possible to do in words. In fact, what I wanted to create was not a mere calculus ratiocinator, but a lingua characteristica in the sense of Leibniz, although I acknowledge this inferential calculation at least as an essential part of a concept script [Begriffsschrift, i. e., Frege’s system of formal logic].” 245

241 242

243 244

245

Tarski 1956[1935], especially 173ff.; Quine 1981[1940], 283ff.; and Corcoran et al. 1974. Wilfrid Hodges points out the surprising historical fact that Gödel and Tarski in their seminal texts of the 1930s (Gödel 1931 and Tarski 1956[1935]) had a semanticist orientation, but that the very same texts later on were perceived as classics of syntacticism. Cf. Hodges 1986, 143 and Müller 2002, 51ff. Hintikka 1997. Cf. Blackburn 2006 and Müller 2002, 50–53. Someone who does do justice to the historical debates is Thomas Müller in his monograph on the philosophical logic of Arthur Prior. Cf. Müller 2002, 29–53 & 76–140. Cf. also the seminal article van Heijenoort 1967a, and also Hodges 1986 and Hintikka 1997. Frege 1998[1882], 97f.; my translation.

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Frege’s semanticist orientation can thus be expressed by the slogan that formal logic is language and calculus – while according to syntacticism, formal logic is calculus without language. 246 Better known than Frege’s rejection of Boole’s understanding of logic is his debate with Hilbert about the correct conception of the axioms of a formal theory. According to the traditional understanding, which was inspired by the paradigmatic example of Euclid’s axiomatic treatment of geometry, the axioms of a theory are truths about the subject matter of the theory which are so basic that they entail everything the theory has to say about its subject matter (and thus are ideally suited to organize the body of knowledge constituted by that theory). By contrast, Hilbert in his new axiomatization of geometry 247 prided himself to have reached a new concept of an axiom. According to Hilbert, an axiomatic system does not have a particular subject matter – rather, it can be satisfied by any system of objects, i. e., by any collection of objects that stand in some relations, 248 if the language of the theory is understood to be about these objects and their relations and the objects do stand in the relations as the axioms say they do under that understanding of the language. Hilbert’s geometrical axioms, for example, need not be understood to be about points and straight lines in Euclidean space, but can also be understood to be about set theoretical constructions from real numbers. Frege dissented vigorously and defended the traditional conception of an axiom, 249 for instance when he said with regard to the axioms of geometry: “I call axioms propositions that are true, but cannot be proved because our knowledge of them flows from an entirely different epistemic source than the logical one, which may be called spatial intuition.” 250

Hilbert, in addition to his conviction that the axioms of a theory do not have a particular subject matter, claimed that they constitute definitions of the concepts involved. The geometrical axioms, for instance, should be understood as laying down what it is to be a point and what it is to be a line. Frege objected that axioms under Hilbert’s understanding fall short of being satisfactory definitions of the concepts involved in them because they give no criterion that decides which objects fall under those concepts, using the example that Hilbert’s geometrical axioms cannot tell us whether Frege’s pocket watch is a point. 251 According to a common complaint, Frege simply did not understand what Hilbert said about axioms; 252 and an improved variant of Hilbert’s claim that axioms are implicit definitions is held by many today, as is witnessed by structuralism in the philosophy of mathematics. 253 246

247 248 249 250 251 252 253

This slogan of course presupposes an emphatic conception of language, that is, in the case of formal languages, a semanticist understanding. Cf. Hilbert 1899. Cf. Shapiro 1997, 152ff. and especially 161ff. and Shapiro 2005, especially 63ff. Cf. Hodges 1986. Cf. Frege 1976, 62. Frege 1976, 63; my translation. Frege 1976, 73. E. g., Hintikka 1997, 110. Cf. Müller 2002, 38. Cf., e. g., Resnik 1981 and Shapiro 1997. The root of structuralism is Benacerraf 1983.

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In philosophy of mathematics, there is an important special case of the distinction between Hilbert’s and Frege’s understanding of the axioms of a formal theory, which has been called the distinction between the assertory and the algebraic understanding of a mathematical theory. 254 Understanding a mathematical theory in an assertory way means to understand it as having a specific subject matter, in the sense in which before the 19th century geometry was understood as a science of space, and in which arithmetic even today is usually thought of as a theory of the natural numbers. 255 Understanding a mathematical theory in an algebraic way means to understand it as being about anything that satisfies its axioms, as is the case with the namegiving mathematical theory, algebra. Structuralists hold that all theories of current mathematics are algebraic – or at least, most parts of it. 256 Hilbert’s conception of axioms and the algebraic understanding of a mathematical theory can be subsumed under syntacticism because for a system of axioms to have variable subject matter (and also for them to constitute an implicit definition of a structure) the non-logical expressions of the language the axioms belong to must be understood in the syntacticist way. Correspondingly, Frege’s conception of axioms and the assertory understanding of a mathematical theory is semanticist because understanding a system of axioms to be truths about a particular subject matter presupposes that even the non-logical expressions of the language the axioms belong to are understood in the semanticist way. 257 This connection between the understanding of formal theories and of formal languages leads us on to a third area where the distinction between semanticism and syntacticism is important, the debate about whether semantics 258 should be understood in a universalist or in a model theoretic way. Speaking figuratively, according to the universalist understanding of semantics language is a universal medium which we cannot step outside of even when we are working within the theory of meaning. Hence semantics must ultimately rely on intuitive interpretation, i. e., on translation into the language we use to do semantics. By contrast, and in an equally figurative way, on the model theoretic understanding of semantics we can, at least for purposes of theoretical semantics, step outside of language and specify its semantics by describing how the expressions of the language relate to the world – that is, which systems of objects are models of the language. 259 Independently of the issue of syntacticism versus semanticism, it must be noted that there is an important element of universalist semantics that at least prima facie has no correlate in model theoretic semantics, namely intuitive interpretation. In 254 255

256

257 258

259

Shapiro 2005, 67. Cf. Pleitz 2010a, 214. As mentioned already in section 1.7, set theory is assertory as well when it is meant as a theory about the metaphysical notion of a set. Stewart Shapiro, himself an important proponent of structuralism, doubts whether this claim holds across the board, because he finds it difficult to construe those mathematical theories which are used to show something about mathematics itself – meta-mathematics, set theory – as algebraic. Cf. Shapiro 2005, especially 61; and cf. Pleitz 2010a, 213ff. In this context, Wilfrid Hodges speaks of “Frege-Peano languages” (Hodges 1986, 143). In chapter 4 we will deal at length with theoretical semantics; and some of the notions which are important here will be given a more detailed presentation in section 4.3. Cf., e. g., Hintikka 1988 and 1997. Cf. sections 4.3 and 4.4.

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model theoretic semantics, all models are equal, and hence a syntacticist cannot check a formal language, a system of formal logic, or a formal theory against the world in the same way as a semanticist, who is allowed to appeal to an intuitive understanding of the language involved. As the element of intuitive interpretation is missing in syntacticism, a syntacticist seems to be at a disadvantage compared to a semanticist when it comes to most of the uses of formal logic listed above. 260 (How, to give an example, is a syntacticist to argue that some formal theory is an adequate explication of a given scientific theory, given that he cannot appeal to an intuitive understanding of that formal theory?) But intuitive facts about meaning can be accommodated within model theoretic semantics when an important element of semantic theory is added: Designating one of the many models as the intended model (or in general, one of the many interpretations as the intended interpretation) allows to re-forge the link between a formal language and the meanings we grasp intuitively, which the syntacticist had severed before. So a syntacticist can say that her explication of some scientific theory is successful when the intended model of her formal theory coincides with the subject matter of the scientific theory. Peano arithmetic, to make the example more specific, is indeed a formalized theory of arithmetic when the natural numbers are designated as its intended model. 261 * In view of the fact that the proponents of the contemporary syntacticist mainstream often trace back their tradition to Tarski’s work on truth, 262 it is interesting to note how unambiguously he still sided with the semanticist side when he wrote his monograph on truth in the 1930s: “It remains perhaps to add that we are not interested here in ‘formal’ languages and sciences in one special sense of the word ‘formal’, namely sciences to the signs and expressions of which no material sense is attached. For such sciences the problem here discussed [i. e., the definition of truth] has no relevance, it is not even meaningful. We shall always ascribe quite concrete and, for us, intelligible meanings to the signs which occur in the languages we shall consider. The expressions which we call sentences still remain sentences after the signs which occur in them have been translated into colloquial language. The sentences which are distinguished as axioms seem to us to be materially true, and in choosing rules of inference we are

260

261

In fact, every one but the sixth use listed above presupposes that the formal language involved is understood as meaningful (and as having the meaning we intuitively assign to it). We cannot, for instance, check whether classical logic provides an adequate explication of informal logic without knowing that the symbol ‘¬’ is understood to mean ‘It is not the case that . . . ’ (cf. section 3.3). As we have mentioned four different debates, an overview will be helpful to bring out the parallels: formal language:

semanticist conception

syntacticist conception

cf. Hodges 1986

formal logic:

“logic as language”

“logic as calculus”

cf. van Heijenoort 1967a

formal theories:

Frege about axioms

Hilbert about axioms

cf. Müller 2002, 37–53

mathematical theories: assertory understanding algebraic understanding cf. Shapiro 2005 262

semantics: universalist semantics Again, cf. Hodges 1986, 135ff.

model theoretic semantics cf. Hintikka 1997

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Chapter 3 Formal Logic and Tarski’s Legacy always guided by the principle that when such rules are applied to true sentences the sentences obtained by their use should also be true.” 263

Why does Tarski think that the definition of truth is not relevant for a formal language under the syntacticist understanding? He does not go into the matter, but a likely reason is that under the syntacticist understanding, no sentence is true or false simpliciter, but each sentence will only be true relative to some models and false relative to others. For it is not in virtue of its syntactic shape alone, but in virtue of its meaning that a true sentence is true and a false sentence false. 264 As we have seen, grasp of meaning can be accommodated within syntacticism (by designating the intended interpretation). It would therefore be wrong to think that an investigation of truth can proceed only under a semanticist understanding of a formal language (as Tarski suggests in the above quote). Rather, as the syntacticist can mirror everything the semanticist can do by working with the notion of an intended interpretation, Tarski’s point can be put in the following way: An investigation of truth in some formal language must either proceed under a semanticist understanding of the language in question, or, if it proceeds under a syntacticist understanding, must also specify the intended interpretation of the language in question. But observe also how it is the syntacticist how has to take a detour. 265 * Within the scope of our presentation of the Liar paradox (here in part I) and to describe the background semantics of this study (in part II), we will use the conceptual distinction between syntacticism and semanticism, but need not decide whether the semanticist or the syntacticist understanding is the right one. But when we come to my proposal for an approach to the Liar paradox in part III, we will argue in favor of the semanticist understanding, and endorse it. 266

3.2 The Tarskian truth schema Recall the naïve truth principle from section 2.2: (NT) A sentence is true if and only if what it means is the case. As we said there, Tarski explicated this principle by his well-known truth schema. In the present section, we will give a general formulation and discuss some special 263 264

265

266

Tarski 1956[1935], 166f.; footnote omitted. The fact that truth attaches to meaning is reflected both in the naïve truth principle, which mentions meaning explicitly, and in the Tarskian truth schema, via the notion of translation. Cf. 2.2 and 3.2. Anticipating my own endorsement of semanticism in chapter 8, I would like to point out a further consideration in its favor: How can the syntacticist justify her choice of one interpretation rather than another as the intended one – if not by an intuitive understanding of the language she is investigating? It would seem that in the end of the day there must be some guidance by a semanticist understanding of the language in question, if only implicitly. E. g., how can the syntacticist justify with regard to a language of arithmetic that the numerals ‘1’, ‘2’, ‘3’, . . . refer to the natural numbers 1, 2, 3, . . . under the intended interpretation, and not to the negative integers –1, –2, –3, . . . ? Cf. section 8.1, which can be understood as a plea for semanticism.

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variants, as well as alternative formulations. We will pay considerably more attention to the details of phrasing than with regard to the naïve truth principle, for the two interconnected reasons that we want to do justice to the schematic character of the Tarskian explication and that we are now looking at the Liar paradox from the angle of formal languages. We start by displaying the general variant that we will use: (Tarskian Truth Schema) Asserting 267 the (Tarskian Truth Schema) amounts to asserting every sentence that can be obtained from the sentential form 268 ‘s is true if and only if p’ by substituting for the symbol ‘s’ a singular term that refers to a sentence and for the symbol ‘p’ a translation of that sentence into the language we use. 269 A sentence that can be obtained from the schema is often called a T-sentence or, when it belongs to a formal language, a T-biconditional. This is the most general form of the Tarskian truth schema; we will discuss some special variants shortly. But let us first note two important differences between the (Tarskian Truth Schema) and the naïve truth principle. Firstly, the notion of a sentence meaning something is spelled out in terms of translation in the schema. And secondly, the generality of the locution ‘every sentence’ is realized by its schematic form. Both differences result from a second reflective step that explicating the naïve truth principle in the Tarskian schematic form amounts to: We no longer speak directly about the sentences of some object language, 270 giving a condition for their being true, but we specify some sentences of a meta-language, which express a corresponding criterion for the truth of object language sentences, and so we have moved to the meta-meta-level. The languages of these three different levels need not be wholly distinct; they can, e. g., overlap in the object language. But using three distinct languages will be helpful in giving a perspicuous example. In this example, a language of arithmetic is the object language and English as usual is our language 267

268

269

We use ‘assertion’ here as an umbrella term to cover theoremhood, truth, endorsement, and kindred notions, because the (Tarskian Truth Schema) can be implemented into a theory of truth in a number of different ways. Tarski appears to follow a similar practice when he uses the unspecific locution “can be asserted” (Tarski 1944, 344). The expression ‘s is true if and only if p’ is not a sentence, because the letter ‘s’ is not a name and the letter ‘p’ is not a sentence. Neither is the expression an open sentence (as in a language of quantified logic, cf. section 3.3) because neither ‘s’ nor ‘p’ is a variable here. ‘s’ and ‘p’ are nothing but placeholders, similar to the dots and the dash in ‘. . . is true if and only if —’. Cf. Corcoran 2006, 219–222. Tarski 1956[1935], 155f. (italics in the original): “The general scheme of this kind of sentence can be depicted in the following way: (2) x is a true sentence if and only if p.

270

In order to obtain concrete definitions we substitute in the place of the symbol ‘p’ in this scheme any sentence, and in the place of ‘x’ any individual name of this sentence.” In section 2.2 we did not note yet that we have to distinguish between object language and metalanguage already in the case of the naïve truth principle.

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of use; and to illustrate that when we work with the (Tarskian Truth Schema) there is a third level intermediate between object language and language of use, German is used to formulate a theory of truth for the language of arithmetic. Now we can compare: Asserting the naïve truth principle would lead us to assert the claim that ‘2 + 2 = 4’ is true if and only if two plus two equals four, because ‘2 + 2 = 4’ means that two plus two equals four. Asserting the (Tarskian Truth Schema) will lead us to assert the German sentence ‘‘2 +2 = 4’ ist genau dann wahr wenn Zwei plus Zwei gleich Vier ist’, because ‘2 + 2 = 4’ can be translated into German as ‘Zwei plus Zwei ist gleich Vier’. Tarski gives his official formulation of the truth schema as part of a desideratum that he thinks any theory of truth must meet, the celebrated “convention T”. According to it, a theory of truth should contain as a theorem every sentence that can be obtained from the sentential form ‘s is true if and only if p’ “by substituting for the symbol ‘s’ a structural-descriptive name of any sentence of the language in question and for the symbol ‘p’ the expression which forms the translation of this sentence into the metalanguage”. 271

Never mind for the moment what exactly Tarski requires of a structural-descriptive name; for now, we only need to know that the class of singular terms that are to be substituted for the symbol ‘s’ can be restricted to singular terms of a given kind, at least when we can be sure that for every sentence that is of interest to us there will be a singular term of that kind that refers to it. One such restriction of the class of singular terms is to quotation expressions, and the special variant of the (Tarskian Truth Schema) that can be obtained by this restriction is the Homophonic truth schema. (Homophonic Truth Schema) Asserting the (Homophonic Truth Schema) amounts to asserting every sentence that can be obtained from the sentential form ‘s is true if and only if p’ by substituting for the symbol ‘s’ a sentence that is enclosed within quote marks and the same sentence (without the quote marks) for the symbol ‘p’. Here, the object language must be a part of the meta-language that the sentences obtained from the schema (the T-sentences) are to belong to, because these sentences would otherwise be ungrammatical. The (Homophonic Truth Schema) is a special case of the general form, the (Tarskian Truth Schema), because a sentence enclosed within quote marks is a singular term that refers to that sentence 272 and (as long as we stay within the scope of the same language and ignore phenomena of contextsensitivity) every sentence can be seen as a trivial, homophonic translation of itself. One instance that the (Homophonic Truth Schema) delivers is likely to be the most 271 272

Tarski 1956[1935], 187f.; notation altered. Cf. section 5.5. Here we only need to know that a quotation expression refers to whatever is between the quote marks.

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famous example of a T-sentence: ‘‘Snow is white’ is true if and only if snow is white.’ 273 The Homophonic truth schema is often formulated in the following way: Asserting the Homophonic truth schema amounts to asserting every sentence that can be obtained from the sentential form ‘‘p’ is true if and only if p’ by substituting for both occurrences of the symbol ‘p’ a sentence of the language we are concerned with. 274 Strictly speaking, this formulation can be misleading because, with regard to the first three symbols of the sentential form (the letter ‘p’ between quote marks), the particular semantics of quotation 275 might lead us to forget that the letter ‘p’ is meant as a placeholder, so that we would end up ascribing truth not to some sentence but to that letter. 276 We should rather use ‘‘. . . ’ is true if and only if . . .’ as the sentential form, laying down that both occurrences of three dots are to be filled with the same sentence, if we want to make visible the homophonic aspect of the (Homophonic Truth Schema). Even more misleading is a variant of the (Tarskian Truth Schema) which appears to resemble the (Homophonic Truth Schema) closely, but differs crucially from it, at least in the general case. It can be found in some contemporary logic texts 277 and looks like this: For every sentence ϕ, we should assert pTrue(hϕi) ↔ ϕq. 278 This is usually accompanied by the explanation that ‘h . . . i’ is a standard name operator, i. e., that phαiq is to be understood as a meta-level abbreviation of the standard name 279 of the expression α (which of course presupposes that there is a standard 273

274 275 276 277

278

279

Contrast ‘‘Schnee ist weiß’ if and only if snow is white’, which is an instance of the (Tarskian Truth Schema) in its general form, but not of the (Homophonic Truth Schema). Cf. Tarski 1969, 105; cf. Corcoran 2006, 226. Cf. section 5.5. This point is made already by Tarski; cf. Tarski 1956[1935], 159f. E. g., Priest 2006a, 17; Field 2008, 28; and, using a different notation for the standard name operator, Brendel 1992, 63; We use ‘p. . .q’ as Quine corners, i. e., as a device of referring to a plurality of expressions. A typical Quine corner expression will contain symbols of an object language as well as meta-language variables that range over expressions of the object language. For a particular assignment of values to these meta-variables, the Quine corner expression is understood to refer to the expression which is the concatenation of the object language expressions between the Quine corners and the object language expressions which are the values of the meta-variables between the Quine corners – in the right order, of course. Thus the above Quine corner expression ‘pTrue(hϕi) ↔ ϕq’ can be read as the description ‘the expression which results from concatenating the predicate ‘True’, the left bracket ‘(’, the standard name of ϕ, the right bracket ‘)’, the biconditional ‘↔’, and ϕ’. The seminal text on Quine corners is Quine 1981[1940], 33–37. A standard name of some object need not be a (proper) name of that object – it is a singular term of any kind which is designated as the standard device for referring to that object. E. g., in a given context the standard name of an expression can be its quotation.

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name for every expression in the range of the variable ‘α’). This formulation of the truth schema is so misleading because the standard name operator ‘h . . . i’ looks as if it belongs to the same language as the predicate ‘True(. . .)’ and the connective ‘↔’, that is, it looks like a device of the meta-language that the theory of truth is formulated in. But that is not what it is (rather, it is a meta-meta-language device for finding some meta-language expression). What is more, it looks like a pair of quote marks, which it most definitely is not 280 (rather, it will help to find some singular term of the meta-language which does not need to be a quotation expression at all). 281 * Let us put this into the context of our guiding question of formality. 282 Is the (Tarskian Truth Schema) in some sense intrinsically formal? In other words, is there some deep connection between working with formal languages and employing the (Tarskian Truth Schema)? The history of the schema – and its formulation by Tarski in his seminal monograph about truth in formal 283 languages – might suggest an affirmative answer. And, to be sure, to deliver the exactness that usually is expected from it, the (Tarskian Truth Schema) might appear to need formal languages: It must be clear beforehand which expressions are sentences of the object language and it must be guaranteed that each of them is referred to by a singular term of the meta-language, and that presupposes a complete specification of all expressions, which can be given only for formal languages. So, for maximum exactness, both the object language that the theory of truth is about and the meta-language that the theory of truth is formulated in should be formal. 284 But maximum exactness is not always what 280 281

282 283

284

Cf. sections 5.5 and 6.3. The variants of the Tarskian truth schema we present above all specify theorems of a formal theory of truth. But the Tarskian truth schema can also be formulated in rule form, and this can make a difference in non-classical contexts. Today, some speak of the rule of truth introduction (from a sentence ϕ we can infer the sentence pTrue(hϕi)q) and the rule of truth elimination (from the sentence pTrue(hϕi)q we can infer the sentence ϕ); cf. Maudlin 2004, 4ff. and especially 10ff. This is often seen as giving the Tarskian truth schema the form of the natural deduction rules which (in an inferentialist way) determine the semantics of the predicate ‘True(. . .)’ (or rather, of the weird pseudo-expression ‘True(h. . .i)’, which is no predicate because it is not completed by a singular term but by a sentence, but which also is no sentential operator because it is not even an expression, in view of ‘h. . .i’ belonging to a higher language level than ‘True(. . . )’). The same rules are used under different names by Jc Beall, who quips “truth is often thought to play Capture and Release” (Beall 2007b, 1); cf. Beall /Glanzberg 2014, section 2.2. Another way of giving the Tarskian truth schema in rule form is to require the intersubstitutability of any sentence ϕ and a sentence that ascribes truth to ϕ (Beall /Glanzberg 2014, section 4.1). Cf. section 1.10. A terminological remark: In his monograph, Tarski uses “formalized language” for a formal language according to the semanticist construal and appears to use “formal language” for a formal language according to the syntacticist construal (Tarski 1956[1935], 166f.; I would like to thank Stefan Roski for drawing my attention to this). I prefer the pair ‘semanticist’/ ‘syntacticist’ to the pair ‘formalized’/ ‘formal’ because the former two expressions are more informative and less liable to be confused than the latter two. The meta-meta-language we use to speak or write about the formal theory of truth can be a natural language. And it usually is – for who would want to write a paper about a formal language in a formal language?

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83

is best. The (Tarskian Truth Schema) works well in a natural language setting, as is witnessed by the usual examples for T-sentences (‘‘Schnee ist weiß’ is true if and only if snow is white’, and so on). In practice – when we are concerned with a specific problem like the basic Liar reasoning –, we know well enough which expression is a sentence and whether some singular term of the right sort refers to it. The (Tarskian Truth Schema) is no more difficult to formulate in or to apply to a natural language than the naïve truth principle, which also did not strike us as being too vague to work with. It is another question altogether where the (Tarskian Truth Schema) is needed. And in this respect its connection is indeed closer to formal languages than to natural languages, because there are feats that in some formal languages can be accomplished only with a schema, while in natural language schemata are always dispensable. Often, when in a given formal language quantification over some sort of items is impossible, it can be approximated by a schema, thus in effect mimicking quantification over the said sort of items on a meta-level by quantification over expressions of the object language. E. g., as we cannot quantify over properties in first order logic, the axiom of induction of second order arithmetic has to be formulated as the schema of induction in first order arithmetic, approximating object language quantification over properties by meta-level quantification over predicates. 285 In natural language, by contrast, we will always find a way of expressing some generality, and thus can always avoid the stylistically cumbersome device of a schema. This brings us to an interesting question. For assuming that Tarski’s interest in the notion of truth initially was not restricted to formal languages, it is not evident why he formulated his principle about truth as a schema. Here is a suggestion for an answer: In the naïve truth principle, we implicitly quantify over states of affairs. A natural way of making this metaphysical commitment explicit is to formulate the naïve truth principle in the following way: For every sentence, there is a unique state of affairs associated with what the sentence means, and the sentence is true if and only if that state of affairs obtains. Although this general statement can easily be formalized in classical logic, Tarski the logician is likely to have felt uncomfortable about mixing metaphysics into the theory of truth and Tarski the physicalist 286 would probably have shied away from endorsing the particular metaphysical theory about states of affairs and their relation to the meaning of sentences that stands behind the above variant of the naïve truth principle. By his move to the meta-level, he was able to forgo all talk about a sentence having a certain meaning that is associated with a certain state of affairs and replace it by talk involving nothing more than a name and a translation of the sentence in question. But this meta-linguistic way of explicating the naïve truth principle does not generalize easily, because it is not possible in standard logical systems to 285 286

Cf., e. g., Boolos /Burgess /Jeffrey 2002, 283 and Smith 2007, 189. Cf. Tarski 1956[1936], 401–408; especially 406; cf. Soames 1999, 107ff.; Künne 2003, 191; and FrostArnold 2004.

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quantify into sentence position, 287 which would correspond to generalizing over states of affairs. Therefore a schema is needed, which can be formulated on the metameta-level. * We now turn back to the question whether and in how far the (Tarskian Truth Schema) is of a formal nature. There is a common impression that the (Tarskian Truth Schema), though perhaps not intrinsically formal, can at least be formalized in a natural way. Tarski expresses this impression clearly in the following footnote to convention T: “If we wished to subject the metalanguage and the metatheory expressed in it to the process of formalization, then the exact specification of the meaning of various expressions which occur in the convention T would present no great difficulties, e. g. the expression ‘formally correct definition of the given symbol’, ‘structuraldescriptive name of a given expression of the language studied’, ‘the translation of a given sentence (of the language studied) into the metalanguage’. After unimportant modifications of its formulation the convention itself would then become a normal definition belonging to the metatheory.” 288

We have no reason to call into question this claim with respect to the special variant of the (Tarskian Truth Schema) that is part of convention T, 289 but I contend that Tarski’s claim is wrong when it is applied to the general form, the (Tarskian Truth Schema), in which a non-trivial notion of translation plays an essential role. To formalize the notion of translation, it is not enough to use some two-place predicate of a formal language, e. g., ‘Trans(x, y)’, with the intention of expressing the same as by ‘x is a translation of y’. Rather, we would need to have an algorithmic method of checking with recourse to nothing but the syntactic shape of two expressions whether one is a translation of the other. But a method like that is not in general available. Although the syntactic shape of a formal expression determines its meaning in the context of the formal language which that expression belongs to, 290 this is not so in the absolute sense that would be needed in order to facilitate translation from one language to another. Given two distinct formal languages of the same cardinality, there are many different bijections between their expressions

287

We would want to express in our formal language something like the following: For every sentence s (of the object language), there is a sentence S (of our language), such that s means that S and s is true if and only if S.

288 289

290

Here, there is quantification into sentence position by means of the second variable ‘S’, which is why the quantifying phrase ‘there is a sentence S’ cannot be formalized by the quantifier of (first order) classical logic. In contrast, the first variable ‘s’ is used only for quantification into singular term position, which is what the quantifiers and variables of (first order) classical logic are for. Tarski 1956[1935], 188. In the context of convention T, Tarski is right because there he requires that each T-sentence refers to the object language sentence it is about by a structural-descriptive name, and a structural-descriptive name by its syntactic shape alone determines the expression it refers to (quotation expressions hence are a special case of structural-descriptive names); cf. Tarski 1956[1935], 167. Cf. section 3.1.

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and hence many different candidates for a translation between them. Syntax does not exhaust semantics in an absolute sense, but can do so only relative to a specified interpretation: Meaning is not that easy to get rid off. 291 An interpretation is of course easy to specify in the limiting case of trivial translation, i. e., when (part of) one language is translated into itself according to the principle that every expression is its own translation. Therefore nothing stands in the way of formalizing the (Homophonic Truth Schema), according to which every well-formed substitution-instance of ‘‘. . .’ is true if and only if . . . ’ is a theorem of the theory of truth, because we can see the (Homophonic Truth Schema) as that special case of the general form, the (Tarskian Truth Schema), where every sentence is referred to by its quotation expression and translated trivially. The process of trivial translation can be formalized because it can be described by recourse to its syntactic side alone: To get the trivial translation of the sentence we refer to by a quotation expression, we need to do no more than drop a quote mark on each side of the quotation expression we are concerned with. 292 But in the general case, when we are concerned with non-trivial translation, syntax does not in general exhaust semantics, and therefore the general form, the (Tarskian Truth Schema), is in the same boat as the naïve truth principle as far as the question of formalizability is concerned. To sum up, the perhaps surprising result of our considerations about whether the (Tarskian Truth Schema) is intrinsically formal is that, in contrast to its homophonic variant, the (Tarskian Truth Schema) cannot in general be formalized. 293 This result will be helpful in section 3.5 when we will ask what makes a variant of the basic Liar reasoning that involves the truth schema formal.

3.3 Classical logic Due to the historical development, and likely for good systematic reasons, one logical system has been singled out as the standard against which all other logical systems are to be measured, classical logic. 294 It is often presented in two stages, giving first classical propositional logic, and then extending it to classical quantified logic, i. e., to first-order predicate logic, with or without a logic of identity. The system of classical logic that we will sketch here and use later is a first-order predicate logic with identity. The alphabet of its language consists of brackets ‘(’ 291

292 293

Similar misgivings about Tarski’s tendency to minimize the role meaning had to play in his theory of ´ ´ truth were expressed already in the 1930s by his friend Marja Kokoszynska (Kokoszynska 1936a and 1936b; cf. Künne 2003, 182). Cf. section 5.5. This is why the common formulation of the Tarskian truth schema mentioned above, ‘For every sentence ϕ, we should assert pTrue(hϕi) ↔ ϕq’,

294

might lead to some harmful misunderstandings: It makes the general form of the Tarskian truth schema look like its homophonic special variant. For some this may mean little more than to acknowledge that classical logic is what is taught in introductory logic courses of both the mathematics and the philosophy departments. But even so, it is the common point of departure from where we can embark on various non-classical paths, or towards extensions that enrich the system but remain classical in spirit, like higher order logic or modal logic.

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and ‘)’, sentential operators (or connectives) of one or two places ‘¬’, ‘∧’, ‘∨’, ‘→’, ‘↔’, quantifiers ‘∀’ and ‘∃’, letters that stand in for sentences (e. g., ‘p’, ‘q’, ‘r’, . . .), the two-place predicate symbol ‘=’, predicates of one or more places (e. g., ‘F(x)’, ‘G(x)’, ‘H(x, y)’, . . .), and two kinds of (primitive) terms, namely constants (e. g., ‘a’, ‘b’, ‘c’ . . . ) and variables (e. g., ‘x’, ‘y’, ‘z’, . . . ). 295 In some cases there will also be functors (e. g., ‘f(x)’, ‘g(x, y)’, . . . ). All strings of such symbols are expressions of the language, and its well-formed expressions are the terms and the well-formed formulas. They can be specified in the following recursive way: Constants and variables are terms. If α is a term and pT(x)q is a monadic functor, then pT(α)q is a term, and similarly for functors of two or more places. 296 A predicate letter or the symbol ‘=’ that is supplemented by the appropriate number of terms is a formula. E. g., if α is a term and pΦ(x)q is a monadic predicate, then pΦ(α)q, i. e., the result of substituting the term α for (all free occurrences of) the variable ‘x’ in pΦ(x)q, is a formula; and if α and β are terms, then pα = βq is a formula. If ϕ is a formula, then p(¬ϕ)q is a formula. If ϕ and ψ are formulae and • is a two-place sentential operator, then p(ϕ • ψ)q is a formula. If ξ is a variable and ϕ is a formula, then p∀ξ ϕq and p∃ξ ϕq are formulae. Nothing else is a formula. A formula with neither sentential operators nor quantifiers is called atomic. A variable is called bound if it is in the scope of a quantifier; otherwise it is called free (this can be made precise by another recursive definition). A formula is called open if it contains a free variable; otherwise it is called a sentence. We present these definitions here chiefly to give an example of how the complete specification of expressions that Tarski found characteristic of a formal language can be supplied by a recursive definition and to illustrate our use of Latin letters as expressions of a formal object language and of Greek letters as meta-language variables. *

295

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In this sketch, we suppress some of the usual details: Some connectives can be defined in terms of others; one quantifier can be defined in terms of the other and the negation symbol ‘¬’; pα = βq abbreviates p=(α, β)q; and sentence letters can be defined as zero-place predicates. And we have declined from specifying how we are to generate infinitely many sentence letters, predicates, constants, and variables – that is, how the respective three dots are to be filled. This can for instance be done by subscripting numerals. We use the term ‘functor’ to mean a formal symbol which if applied to a term yields a term; in this we follow Carnap 2002[1937], 14. There is a more general notion, to be found in earlier work by Polish logicians, according to which a functor is any formal symbol which applied to an expression of some kind delivers an expression, possibly of a different kind; e. g., a predicate can be construed as a sentenceforming functor which applies to terms (cf. Tarski 1956[1935], 161, fn. 1). A functor in Carnap’s sense is a special case of a functor in the Polish sense: a term-forming functor that applies to terms.

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For formulations of the deductive system of first-order predicate logic with identity, we can allude to the great number of good logic textbooks. 297 What is important here is only that all the rules and laws that play a role in the basic Liar reasoning are indeed part of classical logic. That is, corresponding to each of the logical laws and logical rules of informal logic that were listed in section 2.3, there is a theorem or a deductive rule of classical logic. Under a semanticist understanding of the language of classical logic, this correspondence can be checked directly; and under a syntacticist understanding of the language of classical logic, it can be checked once the intended reading of the logical symbols of that language has been specified. 298 * For the formal Liar reasoning, we will need only one further deductive rule; it has no counterpart in the rules of informal logic so far presented. It corresponds to what in metaphysics is called Leibniz’s law, or the indiscernibility of identicals, and can be formulated in the form of the following deductive rule of the indiscernibility of identicals: For terms α and β and a formula pΦ(x)q that is open in the variable ‘x’, we can use the premise pα = βq to deduce pΦ(α) ↔ Φ(β)q. E. g., from ‘a = b’, we can deduce ‘F(a) ↔ F(b)’. 299 (The indiscernibility of identicals could also be formulated in the form of a rule that states the intersubstitutivity of co-extensional terms. 300) For the variants of the basic Liar reasoning we will look at shortly, 301 we will also need to know that the two principles from section 2.3 that express the (Exhaustiveness) and the (Exclusiveness) of truth and falsity hold in classical logic. Though no big surprise, this is not self-evident at the present point – given that, while we are by now acquainted with the language and deductive system of classical logic, we officially do not know much yet about its formal semantics (or model theory). But to anticipate section 4.4, the intuitive understanding of the language of classical logic leads very naturally to a particular theory about the standard formal semantics of classical logic – extensional semantics – which among other things tells us which sentences are true and which false. And according to these standard formal semantics of classical logic, every sentence is either true or false, and none is both. 302 * Logical systems where the (Exhaustiveness) or (Exclusiveness) of truth and falsity are not given are non-classical insofar as they will not sanction the same inferences as classical logic does. Non-classical logics of this kind are used in many approaches to

297

298 299 300 301 302

To give two specific examples, a system of axioms is presented in Enderton 2001, 109–128; especially 112; and a sequent calculus is given in Ebbinghaus /Flum /Thomas 1992, 71–85. Cf. the concluding paragraphs of section 3.1. Cf. Enderton 2001, 112. The rule can be derived by modus ponens from Enderton’s axiom 6. Cf. Ebbinghaus /Flum /Thomas 1992, 81. Cf. section 3.5. In section 2.3 we distinguished exclusion negation and choice negation. Classical negation (‘¬’) switches truth and falsity and thus agrees with both exclusion and choice negation as long as it works on a sentence that is true or false. And in view of the principle of (Exhaustiveness) holding classically, the difference between exclusion and choice negation vanishes in classical logic.

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the Liar paradox. 303 We should note that aside from these departures from classical logic, there are also extensions, many of which by now also merit the label ‘classical’, namely modal logic, tense logic, and modalized tense logic. The languages of these logics can be specified by recursive definitions like the above with clauses added for some additional sentential operators, which capture the meaning of ‘necessarily’, ‘possibly’, ‘it will be the case that’, ‘it was the case that’, ‘it will one day hence necessarily be the case that’, ‘it will one day hence possibly be the case that’, and so on. The deductive system of these logics will contain extra axioms that govern the behavior of these sentential operators.

3.4 The background theory of a language The lesson Tarski drew from the Liar paradox, which will be the topic of the remaining sections of this chapter, is often phrased like this: Every semantically closed language is inconsistent. 304 But how is it even possible to apply the notion of inconsistency to a language? Recall from section 3.1 that a formal language is just a collection of expressions that meets certain requirements, a system of formal logic is a formal language plus a deductive system, and a formal theory is a formal language plus a deductive system (the background logic of the theory) plus a particular collection of axioms. Now, formal theories present themselves as the primary bearers of consistency or inconsistency, because it is natural to define: (Syntactic Inconsistency of a Theory) A theory is (syntactically) inconsistent if and only if there is some sentence such that both it and its negation are theorems of the theory; otherwise the theory is (syntactically) consistent. In any formal language that can express negation, by contrast, there will always be some sentence such that its negation is a sentence of the language, too. 305 And that is as it should be, because in this way the language allows to formulate a wide spectrum of theories, some of them contradicting others. In the words of Scott Soames: “Theories are defective if they contain both a sentence and its negation; languages are defective if they fail to contain any such pair.” 306 It is therefore not possible to transfer the definition of inconsistency from theories to languages in a simple way and retain a notion that can be used to make non-trivial (i. e., informative) statements. So, how should we construe the Tarskian talk of inconsistent languages? 307 303

304 305 306 307

Our own approach will have a logic that can be characterized as restricting the principle of (Exhaustiveness); cf. chapters 10 and 11. The notion of semantic closure is not yet important at this point; it will be explained in section 3.7. For all but the weirdest formal languages, this is in fact the case for every sentence of the language. Soames 1999, 63. Our main interest in asking this question is not exegetical but systematic. We are not primarily asking what Tarski meant by certain untypically unclear passages, but we want to present the lesson he drew from the Liar paradox in a clear way that is still Tarskian in spirit.

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Let us first have a look at where in the Tarskian train of thought the claim of the inconsistency of a certain language occurs. Tarski starts with a given language, which is held fixed during all of his reasoning, formulates a claim in it that entails that there is a Liar sentence 308 and an instance of his truth schema for that Liar sentence, and then uses classical logic to infer or deduce in that language a contradiction from these assumptions. 309 Only after all that has happened does Tarski conclude that the language in question is inconsistent. What he has shown in this way is that some theory that includes a claim that entails that there is a Liar sentence as well as the T-sentence for that Liar sentence is an inconsistent theory. Is he justified in summing up this result by saying that the language in question is inconsistent? For note that other theories that can be formulated in the same language may well be consistent. I think that the best way to make sense of this situation is to acknowledge that for any given language, there is a unique background theory that typically is formulated in that language and encodes important information about that language. Then we can say that a language is inconsistent in a derivative way if and only if its background theory is. 310 (Inconsistency of a Formal Language) A formal language L is inconsistent if and only if its formal background theory TL , i. e., the formal theory that typically is formulated in the formal language L and encodes important information about the formal language L, is an inconsistent theory. This will leave much room for debate in many special cases, because we have not specified which bits of information about a language are important. A collection of sentences which have a fair chance of being theorems of the background theory TL are the analytic truths of the language L, i. e., those sentences of L that are true in virtue of their meaning. In a Tarskian context, when we are dealing with languages which contain their own truth predicate, clearly at least the analytic truths about truth should be theorems of TL . As these presumably include the T-sentences for the sentences of L, this amounts to requiring that the background theory satisfies convention T. But we probably have good specific reason for the background theory to include some sentences that are not analytically true, because it is likely that a claim that entails the existence of a Liar sentence, e. g., is not analytic. And there is a good general reason against all theorems of the background theory being analytically true, that goes back to an observation made by Hans Herzberger:

308 309 310

Or, as we shall see in section 3.6, a counterpart of a Liar sentence. We will see formal variants of this Tarskian reasoning in detail in sections 3.5 and 3.6. There is textual evidence that Tarski thought that each formal language is associated with a specific formal theory, because he repeatedly says that in order to characterize some language, we must specify its expressions as well as a deductive system and a collection of axioms (Tarski 1956[1935], 166; Tarski 1944, 346). This is also noted by Soames: “Tarski [. . . ] sometimes did conflate languages and theories, indicating, for example, that in order to specify a language one must specify its axioms, rules of inference, and theorems.” (Soames, 1999, 63)

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As every analytical truth is true and truth guarantees consistency, 311 no collection of analytical truths can be inconsistent. 312 But as our notion of a background theory is meant to elucidate the notion of an inconsistent language, it would make little sense for us to ensure its consistency by definition. So we can only say at this point that it is plausible that the background theory of a language, in addition to (at least some of) the analytic truths of the language, includes some sentences about the language which, though not analytically true, also encode important bits of information about the language. 313 With a view to our question of formality, we should note that the present construal of what it is for a language to be inconsistent – despite the notion of the inconsistency of a theory it is based on being syntactic in nature – works best on the basis of a semanticist construal of the language in question, 314 because to know which sentences are true in virtue of their meaning and with regard to other important bits of information about the language, we need to appeal to the meaning of its expressions. 315 Besides its role in clarifying what kind of inconsistency Tarski saw as flowing from the Liar paradox, our notion of the background theory of a language will be helpful with respect to the logic and some further assumptions that will be needed in the Liar reasoning. As any formal theory includes a deductive system, specifying a background theory also is specifying a particular logical system, the background logic. In the Tarskian setting, this is of course classical logic; and it was not addressed explicitly because there were no serious doubts in Tarski’s time about classical logic being the one and only deductive system to codify correct reasoning. 316 As the principles of (Exhaustiveness) and (Exclusiveness) hold of classical logic, 317 and as the languages we will be concerned with in connection with the Liar reasoning allow to talk about truth and falsity, we will want to have something like the following two theorems in the formal background theory of each one of these languages: 311

312 313

314 315

316 317

In a paraconsistent setting, truth does not guarantee consistency. But Herzberger wrote before paraconsistent logics were considered serious options. And perhaps more importantly in the present context, Tarski did not consider seriously any non-classical logic (cf., e. g., Tarski 1944, 349). Cf. Herzberger 1965; Herzberger 1967; Church 1968a. There is an alternative view of how the Tarskian talk of the inconsistency of a language should be construed, that is endorsed by Scott Soames, who gives credit to Nathan Salmon for suggesting it (Soames 1999, 53f. & 62–64). According to Soames, we should say that a language is inconsistent if and only if there is an inconsistent collection of its sentences each one of which is “true in the language” (Soames 1999, 54). But what does this mean? The notion of being true in a language cannot be equivalent to the notion of being a true sentence of the language, because then every language would be consistent by definition because of Herzberger’s consideration, which Soames knows of (Soames 1999, 63). But when being true in a language does not entail being true – what then is there to keep it from collapsing into the notion of being a sentence of the language that expresses an important bit of information about the language, i. e., our notion of being a theorem of the background theory? Cf. section 3.1. The same goes for the alternative proposal of Salmon and Soames. For how would they know which sentences are “true in” a language, if they had only its syntactic side at their disposal? Cf. Tarski 1944, 348f. Cf. section 3.3.

3.5 Formal variants of the basic Liar reasoning

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(Theorem of Exhaustiveness) ∀x (True(x) ∨ False(x)) (Theorem of Exclusiveness)

∀x ¬(True(x) ∧ False(x))

The formal predicates ‘True(x)’ and ‘False(x)’ are of course meant to express truth and falsity (or, to have the collection of true sentences and the collection of false sentences as their respective intended interpretation). We think of the quantifier as restricted to the collection of all sentences. 318

3.5 Formal variants of the basic Liar reasoning Now, as all their essential ingredients have been introduced, we get to present and discuss formal variants of the Liar paradox. In this section we will look at some variants of the basic Liar reasoning that start from a formal Liar sentence – a sentence that belongs to the same formal language of classical logic that is used for the deduction that formalizes the Liar reasoning. With our question of formality in mind, 319 we will try to answer the following questions along the way: Is the Liar reasoning completely formalized in these variants? And how do these variants differ from the informal variants of the basic Liar reasoning that were presented in section 2.4? One more preliminary remark before we begin: Most 320 informal variants of the Liar reasoning that were presented in section 2.4 had two stages: First, it was inferred that the Liar sentence is true if and only if it is false, and then it was shown that this ‘if and only if’-statement in conjunction with the principles of (Exhaustiveness) and (Exclusiveness) entails an explicit contradiction. We have seen that these principles have formal counterparts, the (Theorem of Exhaustiveness) and the (Theorem of Exclusiveness), 321 and it is clear that in the setting of classical logic these theorems will do a similar job: From a biconditional of the right form, e. g., ‘True(al) ↔ False(al)’, they will allow to deduce a contradictory sentence, e. g., ‘True(al) ∧ False(al) ∧ ¬(True(al) ∧ False(al))’. Therefore we will follow through each deduction that is a formal variant of the basic Liar reasoning only up to the point where a biconditional pTrue(α) ↔ False(α)q has been deduced, where α is a term that refers to the respective Liar sentence. Let us start with a prototype for our formal variants, i. e., an informal variant of the basic Liar reasoning that, in contrast to the informal variants of the last chapter, makes use not of the naïve truth principle (NT) but of an informal variant of the (Tarskian Truth Schema). It starts from a natural language Liar sentence that achieves self-reference with the help of a schematic description – ‘the F’ – which 318

We could express this restriction formally with the help of a predicate ‘Sentence(x)’ that is satisfied by just those objects in the domain of the theory that are sentences: ∀x (Sentence(x) → True(x) ∨ False(x))

319 320 321

∀x (Sentence(x) → ¬(True(x) ∧ False(x))) Cf. section 1.10. That is, every one except variant 5. Cf. section 3.3.

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can for instance be a description that identifies a particular sentence by giving its position in a printed text. 322 Informal Prototype (1) The F = ‘The F is false’.

existence of a Liar sentence with a description

(2) The F is true ⇔ ‘The F is false’ is true. from (1), by the indiscernibility of identicals 323 (3) ‘The F is false’ is true ⇔ the F is false. the (Homophonic Truth Schema), applied to ‘The F is false’ (4) The F is true ⇔ the F is false.

from (2) and (3), by the transitivity of ‘⇔’ (‘if and only if’)

Now, to transform this piece of informal reasoning into a deduction, we have to substitute logical symbols for logical words, and the formal predicates ‘True(x)’ and ‘False(x)’ for the natural language predicates ‘. . . is true’ and ‘. . . is false’. Furthermore we need to enrich our formal language with the description-forming iotaoperator ‘ ’, which allows to formalize the natural language locution ‘the F’ as ‘ x F(x)’, 324 and with (formal!) quote marks (‘«. . .»’): 325

ι

ι

Formal Variant 1 (description / quote marks) existence of a Liar sentence with a description

ι

(2)

True( x F(x)) ↔ True(«False( x F(x))»)

from (1), by the indiscernibility of identicals

(3)

True(«False( x F(x))») ↔ False( x F(x))

the (Homophonic Truth Schema), applied to the sentence ‘False( x F(x))’

(4)

True( x F(x)) ↔ False( x F(x))

from (2) and (3), by the transitivity of ‘↔’

ι

ι

ι

ι ι

ι

ι ι

322

x F(x) = «False( x F(x))»

(1)

Given the focus of this chapter, it is fitting to present some examples of this way of constructing a selfreferential sentence that can be found in printed texts by Tarski: “The sentence printed on this page, line 5 from the top is not a true sentence” (Tarski 1956[1935], 158, l. 5) “The sentence printed in this paper on p. 347, l. 31, is not true.” (Tarski 1944, 347, l. 31)

323

324

325

“The sentence printed in red on page 65 of the June 1969 issue of Scientific American is false.” (Tarski 1969, 65; printed in red). Using the intersubstitutability of co-extensional terms instead of the indiscernibility of identicals would allow us to abridge the reasoning by one step: (1) The F = ‘The F is false’. (2) ‘The F is false’ is true ⇔ the F is false.

existence of a Liar sentence with a description the (Homophonic Truth Schema), applied to ‘The F is false’

(3) The F is true ⇔ the F is false.

(1), (2), substitution of co-extensional terms

With regard to its role in a formal variant of the Liar reasoning, it is not important whether the description operator is understood in a Fregean way as a further primitive symbol of our formal language or as an abbreviation that can be analyzed in the Russellian way. For the two different ways of understanding descriptions, cf. section 5.2, especially subsection 5.2.4. Cf. section 5.5.

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Our formal language has three kinds of terms: constants, descriptions, and quotation expressions. Only the first two kinds allow to construct self-referential sentences. 326 We have seen a formal Liar sentence with a description, so we should complete the picture and also present a deduction that starts from a formal Liar sentence with a constant: 327 Formal Variant 2 (constant / quote marks) (1)

al = «False(al)»

(2)

True(al) ↔ True(«False(al)»)

(3)

True(«False(al)») ↔ False(al)

(4)

True(al) ↔ False(al)

existence of a Liar sentence with a constant from (1), by the indiscernibility of identicals the (Homophonic Truth Schema), applied to the sentence ‘False(al)’ from (2) and (3), by the transitivity of ‘↔’

Formal Variants 1 and 2 are formal deductions. As each step is covered by a deductive rule of classical logic, it is clear in both cases that the fourth line can be deduced form the first and third line. But what is the status of the fourth lines – do the deductions show that they are theorems of the background formal theory that encodes the most important facts about the formal language we are concerned with here? The answer depends on the status of those lines that are not deduced from other lines, the lines that correspond to the premises of a piece of informal reasoning. So, if the first and third lines were theorems of the formal background theory, then the fourth lines would be theorems, too, and our deductions would show that the Liar paradox proves the background formal theory inconsistent. With regard to the third lines, the matter is clear: They are theorems of the background theory because they are instances of the (Homophonic Truth Schema). As we have seen in section 3.2, the homophonic variant of the (Tarskian Truth Schema) is fully formalizable. That is evident in the case at hand, because for each third line, it can be checked by recourse to its syntactic shape alone that it is an instance of the sentential form ‘True(«. . .») ↔ . . . ’. 328 But what about the first lines? Why should they be theorems of the background formal theory? The identity statement ‘al = «False(al)»’ is true if and only if ‘al’ and ‘«False(al)»’ have the same extension, that is, if and only if ‘al’ refers to the sentence ‘False(al)’. The identity statement ‘ x F(x) = «False( x F(x))»’ is true if and only if the sentence ‘False( x F(x))’ is the unique object that falls under the concept expressed by the predicate ‘F’. In those cases where these particular identity statements about our formal language are true, should we just add them as axioms to the background theory? There are some reasons for a negative answer: Our

ι

ι

ι

326

327

328

Here we anticipate a result that will be established in section 5.6: No sentence can achieve self-reference with the help of quotation alone. Briefly, the argument is that to quote itself a sentence would have to be a proper part of itself, which is impossible. We will in fact stick to Liar sentences with a constant from now on, because they are syntactically and semantically simpler. Here, the sentential form used in the (Homophonic Truth Schema) must of course belong to the formal language.

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background theory is meant to encode important facts about the formal language, and it is doubtful that the identity statements are important (apart from their role in the formal Liar reasoning, of course). Also, as an axiom system should be finite, we cannot in general go and add all particular statements like these to the background theory in a principled way. But be that as it may – what is clear is that, in contrast to the T-biconditionals in the third lines, the identity statements in the first lines cannot in general be seen by their syntactic shape alone to be theorems. 329 As yet, our result is inconclusive: Although we have seen that, given our background formal theory, the problematic biconditionals can be deduced from the respective first lines, we are unsure whether they are theorems because we are doubtful about the theoremhood of the identity statements in the first lines. Thus we cannot be sure to have shown the inconsistency of the formal language that our Liar sentences and the deduction are formulated in. There appears to be an informal residue in the formal variants of the Liar reasoning which precludes a purely formal argument for inconsistency. * Maybe the root of the inconclusiveness is the use of formal quote marks in the respective identity statement? Quote marks are by no means available in every formal language, and we might want to look for an alternative way of formulating the identity fact at the beginning of the Liar reasoning. So let us see what happens when we generalize and exchange quote marks for the standard name operator! This will provide further illustration for the point 330 that this device can be misleading, it will allow to point out a surprising similarity between some of the formal variants and the informal variants of the Liar reasoning – and it will prepare the way for the next section, where we will look at the common way of formulating the Liar paradox in a formal setting (which also makes use of the standard name operator). So let us drop quote marks from our formal language, and transform Formal Variant 2 by exchanging each occurrence of a pair of quote marks for the standard name operator. The result again concerns a Liar sentence with a constant: Schema for Formal Variants A (constant/standard name operator)

329

(1) al = hFalse(al)i

existence of a Liar sentence with a constant

(2) True(al) ↔ True(hFalse(al)i)

from (1), by the indiscernibility of identicals

(3) True(hFalse(al)i) ↔ False(al)

the (Tarskian Truth Schema), which applies because the sentence ‘False(al)’ is referred to by the singular term phFalse(al)iq and translates homophonically as ‘False(al)’

(4) True(al) ↔ False(al)

from (2) and (3), by the transitivity of ‘↔’

Under certain circumstances and for a cleverly chosen predicate ‘G’, the formal sentence ‘False( x G(x))’ might be the unique object that falls under the concept expressed by the predicate ‘G’, so that in a sense the syntactic shape of the identity statement ‘ x G(x) = «False( x G(x))»’ would be enough to justify its being a theorem. The standard way to achieve this is by Gödelization. But it is prudent to postpone this issue; cf. section 3.6 and chapter 6. Cf. section 3.2.

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ι

ι

330

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What is displayed in the above table is not a particular deduction, because the standard name operator ‘h . . . i’ does not belong to the formal language. Given a particular standard name operator – a particular function that maps each expression α to a term phαiq that refers to it –, Schema A will of course have a particular deduction as an instance. However, as long as the standard name operator remains unspecified, Schema A is nothing more than a recipe for constructing a plurality of deductions – which, as we will see, can differ significantly. But before turning to that let us have a look at line (1) of Schema A. What we see there tells us that the constant ‘al’ and the standard name of the sentence ‘False(al)’ (whatever term that may be) are co-extensional, 331 so that we can gather that the sentence ‘False(al)’ does refer to itself via the constant ‘al’ (which is why all instances of Schema A concern the same formal Liar sentence as Formal Variant 2). Note that this information is not in general expressed on the level of the formal object language. Only when the standard name of the Liar sentence is its quotation do we obtain a sentence of the object language (‘al = «False(al)»’, which we know as line (1) of Formal Variant 2) that makes visible that the Liar sentence refers to itself with the help of the constant ‘al’. But when the standard name of the Liar sentence is, e. g., the constant ‘bl’ or the description ‘ x F(x)’, the sentence we obtain as the first line (‘al = bl’ or ‘al = x F(x)’) on the syntactic face of it does not even reveal that it is about a sentence, much less that there is a self-referential sentence. The importance of these observations lies in the fact that the semantic information that is transported by the meta-linguistic abbreviation we see in line (1) of Schema A plays an essential role in the justification of the application of the truth schema in line (3). To see why, let us have a look at the deduction that is produced by Schema A if the standard name of ‘False(al)’ is ‘bl’:

ι

ι

Formal Variant 3 (constant/a different constant) (1)

al = bl

existence of a Liar sentence with a constant

(2)

True(al) ↔ True(bl)

from (1), by the indiscernibility of identicals

(3)

True(bl) ↔ False(al)

the (Tarskian Truth Schema), which applies because the sentence ‘False(al)’ is referred to by the singular term ‘bl’ and translates homophonically as ‘False(al)’

(4)

True(al) ↔ False(al)

from (2) and (3), by the transitivity of ‘↔’

Each step of Formal Variant 3 is covered by classical logic, so that it is a deduction, but (aside from doubts much like those voiced above about the theoremhood of the first line) it is important to note that we can no longer be sure that the third line (as in Formal Variants 1 and 2) is justified by an application of the (Homophonic Truth Schema), and therefore have to turn to the (Tarskian Truth Schema) in its general form. What is more, to justify this particular application of the general form of the (Tarskian Truth Schema) we need the information that ‘bl’ refers to a sentence that can be translated as ‘False(al)’, which is no longer visible in the first line (or anywhere else in the deduction), because the identity statement ‘al = bl’ would only 331

What we see in line (1) does not entail that ‘al’ is the standard name of ‘False(al)’.

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give the needed information if conjoined with the information that ‘al’ refers to ‘False(al)’, which is neither expressed nor entailed nor made visible by any of the formal sentences of the middle column. The result of Formal Variant 3 therefore rests on a semantic fact that has no counterpart in that formal deduction. The contrast between of Formal Variant 3 and 2 illustrates the more general fact that Schema A relies heavily on the device of the standard name operator, which in line (1) of Schema A, by way of meta-linguistic abbreviation, encodes the information that the constant ‘al’ refers to the sentence ‘False(al)’ (and thus to a sentence that trivially can be translated as the sentence ‘False(al)’), which is needed to justify the application of the (Tarskian Truth Schema). The use of the standard name operator in Schema A hides the fact that external semantic information is needed to justify the instance of the (Tarskian Truth Schema) that is part of the deduction. This is illustrated nicely by the limiting case of Schema A where the standard name of the Liar sentence ‘False(al)’ is the particular constant ‘al’ itself: Formal Variant 4 (constant/the same constant) (1)

a l = al

the logic of identity

(2)

True(al) ↔ True(al)

from (1), by the indiscernibility of identicals

(3)

True(al) ↔ False(al)

the (Tarskian Truth Schema), which applies because the sentence ‘False(al)’ is referred to by the singular term ‘al’ and translates homophonically as ‘False(al)’

(4)

True(al) ↔ False(al)

from (2) and (3), by the transitivity of ‘↔’

As it is displayed here, Formal Variant 4 is strikingly redundant. Line (1) is not needed to deduce line (2), which is a theorem of propositional logic, and line (2) in turn is not needed because we do need not go on to line (4) after we have reached line (3), because line (3) is identical to line (4). Formal Variant 4 boils down to a one step deduction: Formal Variant 40 (one line formulation) (1)

True(al) ↔ False(al)

the (Tarskian Truth Schema), which applies because the sentence ‘False(al)’ is referred to by the singular term ‘al’ and translates homophonically as ‘False(al)’

Here, all the deductive work is done by the (Tarskian Truth Schema) in its general form. Its application again presupposes a non-trivial semantic fact external to the deduction, namely that the constant ‘al’ refers to a sentence that can be translated as ‘False(al)’. 332 In its brevity variant 40 is a good illustration of an important similarity between deductions that depend essentially on the (Tarskian Truth Schema) in its general form and the informal Liar reasoning that depends essentially on the naïve truth principle. The Formal Variants 3, 4, and 40 are similar to the informal Liar reasoning of section 2.4 in that we do not get to see the Liar sentence. In Formal Variant 3, e. g., 332

This is the same effect as in Formal Variant 3, but here it is more obvious.

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the only information pertinent to the application of the (Tarskian Truth Schema) is that some sentence is referred to by the constant ‘bl’ and can be translated as ‘False(al)’, however that sentence may be worded. The similarity goes deeper. This becomes obvious when we compare the more general form of Formal Variant 40 to the crucial step of the informal reasoning (which is presented here in a formulation similar to its formal counterpart): 333 Formal Variant 40 (one line formulation) (1)

True(al) ↔ False(al)

the (Tarskian Truth Schema), given that the singular term ‘al’ refers to a sentence that can be translated as ‘False(al)’

Informal Liar reasoning (one line formulation) (1)

λ is true ⇔ λ is false.

the naïve truth principle (NT), given that λ means that λ is false.

We see here that the difference between the naïve truth principle and the Tarskian truth schema is really not that big when the latter is taken in its general form. The role that in the formal deduction is played by the fact that the singular term ‘al’ refers to a sentence that can be translated as ‘False(al)’ is played in the informal reasoning by the fact that λ means that λ is false – which stands in an intimate connection to the fact that the term ‘λ’ refers to a sentence that can be translated as ‘λ is false’. Both the naïve truth principle and the Tarskian truth schema depend for their application on a non-trivial semantic fact. Thus there is a non-formalizable residue not only in the naïve truth principle, but also in its purportedly more precise counterpart, the Tarskian truth schema. * This brings us back to our guiding question of formality, 334 and, as it is time to take stock of what has been achieved in chapters 2 and 3 so far, also to a summarizing comparison between informal and formal variants of the basic Liar reasoning. When we compare the variants of the basic Liar reasoning that were discussed in sections 2.4 and 3.5, we see that basically there are three groups: firstly, the informal variants (Informal Variants 1 through 5 in section 2.4); secondly, those formal variants that depend essentially on the (Tarskian Truth Schema) in its general form (Formal Variants 3 and 4, and many other instances of Schema A); and thirdly, those formal variants that make use of the (Homophonic Truth Schema) in particular (Formal Variants 1 and 2). Thus, the three groups are given in a rough order of an increasing degree of formality. We have observed in the last chapter that the informal variants, although formulated in natural language and governed by informal logic, are not so far off from their formal counterparts insofar as they are easily formalized. 335 And we have seen just now that those formal variants that depend on the general form of the (Tarskian Truth Schema) have an essential informal 333 334 335

Cf. Informal Variant 2 in section 2.4. Cf. section 1.10. Cf. section 2.4.

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residue because they need a non-trivial semantic fact that must remain external to the deduction for the justification of one deductive step. So – what about the formal variants that belong to the most formal third group (Formal Variants 1 and 2)? As they make use of the (Tarskian Truth Schema) in its special homophonic form, we seem to have reason to suppose that they, at least, are fully formal. For in its homophonic variant the truth schema is definitely more on the formal side than in its general form, because homophonic translation can be formalized – in contrast to translation in the general case 336, which is rather similar to its counterpart in the naïve truth principle, the relation of meaning that. However, our discussion of Formal Variants 1 and 2 ended with an inconclusive result, insofar as we remained in doubt over the theoremhood of the identity statements in the first lines. By now we can see that what each one of those identity statements expresses corresponds to the non-trivial semantic facts that in Formal Variants 3 and 4 had to remain external to the deductions, namely that the respective (putative) Liar sentence does refer to itself. As this semantic fact can be expressed in a formal language that contains quote marks, the Liar reasoning is formalized to a higher degree in Formal Variants 1 and 2 than in Formal Variants 3 and 4. But we cannot be sure to have reached full formality, because it is doubtful whether the sentence that expresses the semantic fact has the status of a theorem of the background theory about our formal language. In other words, it is doubtful that this last semantic fact can fully be reduced to syntax, and so there would appear to remain an informal residue even in the most formal of the formal variants of the Liar reasoning. To sum up the comparison, in chapters 2 and 3 we wanted to present and discuss the basic Liar reasoning guided by the question of formality. What has emerged is far from the clear division between the non-formal and the formal that seemed to open up in section 1.9: The informal is easily formalized, but an informal residue appears to remain in the formal. In view of our objective of reconciling the formal and the non-formal stance, 337 this preliminary result need not be unwelcome.

3.6 The Diagonal Lemma and Tarski’s Theorem We come out of the last section still unsure whether any of the variants of the Liar paradox that are formulated in a formal language is fully formal. Therefore it is interesting to note that there is a proof that under certain circumstances, a very Liarlike paradox would unfold in a wide range of formal languages. The proof goes back to Tarski’s monograph on truth and employs a technique that at about the same time had been developed independently and in more detail by Kurt Gödel. What we are talking about is of course Tarski’s Theorem of the undefinability of truth, which is based on the Diagonal Lemma. What the Diagonal Lemma contributes to the transferal of the Liar paradox into a formal setting is a formal counterpart (or analogue) of sentential self-referentiality, which does not depend on any external semantic information. A full explanation 336 337

Cf. section 3.2. Cf. section 1.10.

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of the Diagonal Lemma and of its proof would not fit the objective of the present section, which is to conclude the discussion of formal variants of the Liar paradox by showing whether and in how far the Diagonal Lemma enables us to take the final step towards a full formalization of the Liar reasoning. For this objective, it is enough to give a rough statement of it: 338 (Diagonal Lemma) 339 For a formal theory T that fulfills certain fairly weak requirements, and under certain restrictions with regard to the standard name operator ‘h . . . i’, the following is the case: For every formula pΦ(x)q that is open in one variable (here, ‘x’), there is a sentence, ϕΦ, such that `T pϕΦ ↔ Φ(hϕΦi)q. 340 The sentence ϕΦ that is delivered by the (Diagonal Lemma) is a formal counterpart of a self-referential sentence because, speaking loosely, it can be understood as saying of itself that it falls under the concept that is expressed by the formal predicate pΦ(x)q. 341 But strictly speaking, this is not true. The (Diagonal Lemma) tells us at the most that ϕΦ is provably equivalent to a sentence that refers to ϕΦ. And even that is not always the case, because the operator ‘h . . . i’ in many cases will satisfy the restrictions mentioned in the Lemma without delivering a term that refers to the 338

339

340

341

The Diagonal Lemma and its role in the proof of Tarski’s Theorem (as well as its role in the proof of Gödel’s Incompleteness Theorems) are explained in great detail in any textbook on mathematical logic, cf., e. g., Enderton 2001, 182–281; Boolos /Burgess/Jeffrey 2002, 221–232; Smith 2007, 169–182; Ebbinghaus /Flum/Thomas 1992, 173–217, and in a number of monographs devoted to the subject, cf., e. g., the classic Stegmüller 1957[1968] & 1959[1970]; Smullyan 1992; and Smith 2007. There are also a lot of books which try to explain the methods that are employed to reach the Diagonal Lemma, especially with regard to their role in Gödel’s proof of his Incompleteness Theorems, to a more general public, in particular Nagel /Newman 2005[1958]; Smullyan 1987; Franzén 2005; Berto 2009; and Hofstadter 1985 & 2007. The Diagonal Lemma is similar to Cantor’s Diagonal Argument (hence the name) insofar as both concern an object that points back at itself in some way; and in the most famous applications of the Diagonal Lemma (i. e., in the proofs of Gödel’s Incompleteness Theorems and of Tarski’s Theorem) the sentence it delivers can even be said to point back at itself in a negative way because the predicate it is formed from correlates to a concept that has a negative aspect (unprovability or falsity / untruth). Cf. section 1.7. The Diagonal Lemma is sometimes called a “fixed point lemma”, because the characterization of the sentence ϕ it delivers (by ‘` pϕ ↔ Φ(hϕi)q’) is reminiscent of the characterization of a fixed point of a mathematical function, which is, for a function f, any object x such that f(x) = x. But there are important differences: Firstly, there is the grammatical issue that while ϕ and pΦ(hϕi)q are sentences, ‘f(x)’ and ‘x’ are singular terms; secondly, the provability of a biconditional is not at all the same as an identity statement; and thirdly, even if we ignored the first two points, the Diagonal Lemma would not (as is the common suggestion) give us a fixed point of the function expressed by the predicate Φ, but only of whatever is expressed by the weird expression pΦ(h. . . i)q, which does not even belong to the formal language. In a formal language, we can regard every open formula as a (complex) predicate; cf. section 4.2. The concept of being an even number, e. g., in the usual languages of arithmetic is not expressed by a primitive predicate (i. e., there is no predicate letter with the meaning ‘. . . is an even number’), but by the open formula ‘∃y 2y = x’ (as well as by infinitely many other open formulas, e. g., ‘x = 2 ∨ (x > 3 ∧ ∃y 2y = x)’).

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expression that is enclosed within it, but will rather deliver a term that refers to a code of that expression. 342 The last point can be illustrated in the context of the arithmetization of syntax, which is also called Gödelization, because it was used first by Gödel in his seminal 1931 paper on the incompleteness of certain formal theories. We start with a formal theory of arithmetic – say, Peano Arithmetic – and in a systematic way associate each expression of the formal language of that theory with a natural number, called its Gödel number. 343 Then a sentence of that language can be understood in two ways; firstly, as a statement about natural numbers, and secondly, via the code that is constituted by the Gödel numbers, as a statement about expressions of the formal language it is couched in. And it turns out that certain concepts that apply to expressions of the formal language can also be understood to be expressed in a similar indirect way by some open formulas of our language of arithmetic which usually would be understood to express concepts that apply to natural numbers. 344 In order to give a simple example, let us suppose that our Gödel numbering is strange enough so that an expression will be well-formed just in case its Gödel number is even. Then the formal sentence p∃y 2y = nq, where pnq is a term of the formal language that refers to the natural number n, can be understood as saying indirectly that the expression with the Gödel number n is well-formed – at least when certain further requirements are met. 345 In this setting, we can see the sentential self-reference is never given in a direct way, because what the (Diagonal Lemma) gives us, for some predicate of natural numbers, is a sentence that is equivalent to a sentence that says of the Gödel number of the first sentence that it satisfies that predicate of natural numbers. We can illustrate the indirectness of the self-reference that is achievable by way of the (Diagonal Lemma) in our simple example, where an expression is well-formed just in case it is provable of its Gödel number that it is even. When we instantiate the (Diagonal Lemma) with a particular formal predicate that expresses the concept of being even (say, ‘∃y 2y = x’), it will deliver a sentence, ϕeven, which is provably equivalent to a sentence, say ‘∃y 2y = 1010 + 8’, 346 such that 1010 + 8 is the Gödel number of ϕeven. In this situation, because of the nature of our particular Gödel numbering, we do seem 342 343 344 345

346

For a further discussion of the question whether we should count this as reference, cf. section 6.3. We will say more about the requirement of systematicity in section 6.2. We will say more about this in sections 6.2 and 6.3. It is not enough that a certain formal sentence makes a statement about natural numbers that is correlated to the intended statement about expressions, because theoremhood also comes into play. In our example, for each well-formed expression the sentence that says of its Gödel number that it is even must be a theorem and for each ill-formed expression the sentence that says of its Gödel number that it is not even must be a theorem. The details do not matter at the present point (but cf. section 6.2). But we should keep in mind that to use Gödelization as a way of encoding statements about expressions in statements about numbers it is not enough to map the expressions of our formal language on natural numbers, because it is also crucial which sentences can be shown to be theorems of the formal theory of arithmetic we employ. I would like to thank Vítˇezslav Sˇvejdar for alerting me to the importance of this point. It will be of crucial importance in our discussion of whether there can be a Gödelian Liar sentence in section 6.4. We follow the common practice of underlining numerals (like ‘1’, ‘2’, ‘3’, and so on) and descriptive terms that refer to a natural number (like ‘1010 + 8’) to make explicit that the formal language term that

3.6 The Diagonal Lemma and Tarski’s Theorem

101

to be justified in asserting that the sentence ϕeven indirectly says of itself that it is well-formed – but only given a considerable amount of charity. Is this really self-reference? Does it allow to construct a sentence that, in some robust sense of the word, ascribes falsity or untruth to itself? These questions are interesting and highly relevant for our study of the Liar paradox. But it is prudent to postpone their discussion until after we have given the broad strokes of a theory of meaning and reference in chapters 4 and 5. We will return to this matter in chapter 6. Here we should only add that, however our answer might turn out with regard to the subtle semantics of Gödelian constructions, even the rather indirect form of circularity that clearly characterizes them suffices to prove the powerful limitative results of Gödel and (as we will see in a moment) of Tarski. * On our way to a variant of the proof of Tarski’s Theorem of the undefinability of truth, we will first look at a formal deduction that is meant to parallel the presentation of formal variants of the basic Liar reasoning in section 3.5. Let us presuppose that in our formal language there are open formulas ‘True(x)’ and ‘False(x)’ 347 which are meant to express truth and falsity, i. e., which in the background theory are governed by the theorems which are delivered by the (Tarskian Truth Schema), and which conform to the (Theorem of Exhaustiveness) and the (Theorem of Exclusiveness). Then if we feed the open formula ‘False(x)’ into the (Diagonal Lemma), it will deliver a formal sentence ϕFalse which in the indirect way described above says of itself that it is false. In that regard, ϕFalse is a counterpart of a Liar sentence. But we cannot be sure that it actually is a Liar sentence according to our notion 348, because we do not know whether ϕFalse refers to itself. 349 Anyway, it can certainly be employed to the usual effect in a counterpart of the Liar reasoning: 350 Schema for Formal Variants B (Diagonal Lemma/standard name operator) (1) ϕFalse ↔ False(hϕFalsei)

347

348 349

350

result of the (Diagonal Lemma) for the open formula ‘False(x)’

refers to a natural number in general is distinct from the singular term of natural language that on that occasion is used to refer to the same number. Although in some cases these open formulas might be primitive predicates, they could also be quite complex. We write ‘True(x)’ and ‘False(x)’ only for better readability. Cf. section 1.2. We know only that the biconditional pϕFalse ↔ False(hϕFalsei)q is a theorem, but theoremhood of a biconditional need not coincide with synonymy. E. g., the formal sentences ‘1 + 1 = 2 ↔ 2 + 3 = 5’ and ‘1 = 0 ↔ 5 = 7’ are theorems of formal arithmetic, but surely the sides of these biconditionals are not synonymous, as they are about different numbers. In addition to the element of indirectness due to the Diagonal Lemma delivering no more than a biconditional, there is the further element of indirectness due to the fact that the standard name operator as used in the Diagonal Lemma need not deliver a term that refers to the enclosed expression, but maybe delivers only a term that refers to a code thereof. This is again a schema (cf. Schema A in section 3.5), because we have not specified the standard name operator. But note that, in contrast to its use in Schema A, the standard name operator in Schema B must fulfill certain requirements for the Diagonal Lemma to be applicable.

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(2) True(hϕFalsei) ↔ ϕFalse

the (Tarskian Truth Schema), applied to the sentence ϕFalse

(3) True(hϕFalsei) ↔ False(hϕFalsei)

from (2) and (3), by the transitivity of ‘↔’

The role played here by the sentence ϕFalse that is delivered by the (Diagonal Lemma) is clearly the same as the role played by a Liar sentence in the formal variants of the Liar reasoning. But in contrast to when we discussed those formal variants that revealed the self-referentiality of the Liar sentence (Formal Variants 1 and 2) in section 3.5, we can now be sure of the theoremhood of the first line: It follows directly from the (Diagonal Lemma). What about the second line, though? We have seen that the (Tarskian Truth Schema) when it is not in its homophonic form is not in general amenable to formalization. 351 And in the present case we cannot apply the (Homophonic Truth Schema) because the standard name phϕFalseiq need not be a quotation expression of ϕFalse. But it turns out that the requirements we put on the standard name operator for the (Diagonal Lemma) to be applicable (which in the context of Gödelization are guaranteed by certain requirements that a Gödel numbering must meet 352) allow for a formalization of this special case of the standard name variant of the (Tarskian Truth Schema), which tells us that each instance of pTrue(hϕi) ↔ ϕq is a theorem. So we can be sure that line (2) is a theorem, too, and thus that the deduction presented above is as problematic as possible from a formal point of view. It shows that our formal background theory (after applying the (Theorem of Exhaustiveness) and the (Theorem of Exclusiveness) to line (3)) will prove a contradictory sentence. There is a way of showing in a fully formal way that a formal language is inconsistent if it includes predicates of truth and falsity (in the sense described above). * Now that we have seen how the (Diagonal Lemma) allows to complete the formalization of the Liar reasoning (or a piece of reasoning that is very similar), let us turn to its role in the proof of Tarski’s Theorem, which settles the question of the definability of truth in a negative way. According to Tarski’s convention T, if a formal theory defines truth, then there is some open formula pΘ(x)q in the language of the theory such that every instance of pΘ(hϕi) ↔ ϕq is a theorem of that theory. Employing the notion of the background theory of a language, 353 we can explain what it is for a language L1 to (derivatively) define the truth predicate of some language L2 in the following way: A language L1 defines the truth predicate of a language L2 if and only if there is some open formula pΘ(x)q in the language L1 such that for every sentence ϕ of the language L2 the sentence pΘ(hϕi) ↔ ϕq is a theorem of the background theory of language L1, i. e., TL1 .

351 352 353

Cf. section 3.2. Cf. section 6.2. Cf. section 3.4.

3.6 The Diagonal Lemma and Tarski’s Theorem

103

We give this formulation for languages L1 and L2 in the interest of generality. Cases where the languages are distinct will come up only later, 354 because in the proof of Tarski’s Theorem only the case where L1 = L2 will be relevant. Let us therefore note what the two language notion entails for the one language case: If a language L were to define its own truth predicate, then there would be some open formula pΘ(x)q in L such that for every sentence ϕ of L the sentence pΘ(hϕi) ↔ ϕq is a theorem of the background theory TL . Now, a piece of reasoning much like that of Schema B allows to show that a language that defines its own truth predicate is inconsistent. We want to present the proof of Tarski’s Theorem here as a piece of informal reasoning, which at one point makes use of a formal deduction. This can be brought out by writing the argument in the following form, which groups together the formal deduction as a single step of the informal reasoning (steps (2.1) through (2.4)): The Structure of the Proof of Tarski’s Theorem of the Undefinability of Truth (1) There is an open formula of the formal object language L, pΘ(x)q, which defines the notion of truth for L (i. e., the formal background theory TL allows to deduce the (Tarskian Truth Schema) with respect to the open formula hΘ(x)i).

assumption for conditional proof

(2) (2.1) ` pϕUntrue ↔ ¬Θ(hϕUntruei)q

the (Diagonal Lemma), for the open formula p¬Θ(x)q 355

(2.2) ` pΘ(hϕUntruei) ↔ ϕUntrueq

(1) and the (Tarskian Truth Schema), applied to the sentence ϕUntrue

(2.3) ` pΘ(hϕUntruei) ↔ ¬Θ(hϕUntruei)q

(2.1) and (2.2), by the transitivity of ‘↔’ (2.3), (Excluded Middle) and reasoning by cases for ‘∨’

(2.4) ` pΘ(hϕUntruei) ∧ ¬Θ(hϕUntruei)q (3) The background theory TL is inconsistent, i. e., 356 the formal language L is inconsistent.

354 355

356

from (2.4)

(4) If the formal language L defines a truth predicate, then it is inconsistent.

(1) through (3), conditional proof

(5) Any formal language (that is similar to L in its expressive resources) that allows to define its own truth predicate is inconsistent.

(6), by universal generalization (L was arbitrary, except for its expressive resources)

Cf. section 3.8. In contrast to all variants of the basic Liar reasoning, we follow Tarski here in so far as the sentence ϕUntrue delivered by the Diagonal Lemma is the counterpart not of a simple but of a strengthened Liar sentence. By working with an ascription of untruth rather than an ascription of falsity Tarski needed to presuppose only convention T (that is, the (Tarskian Truth Schema)) in the proof of the undefinability of truth, but not the (Theorem of Exhaustiveness) and the (Theorem of Exclusiveness). We have seen in section 2.5 how an explicit contradiction can be derived in the absence of these theorems when we start from a strengthened Liar sentence. Cf. section 3.4.

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Observe that the main argument presented here is in hypothetical mood; but note also that it is not a reductio but a conditional proof. Although line (5) is backed up by an argument that starts from a hypothetical assumption (in line (1)) and leads on to a contradiction (in step (2.4)), a contradiction is asserted at no step of the hypothetical proof that makes up lines (1), (2), and (3). The reason for this is that the contradictory object language sentence is not used but only mentioned (in step (2.4)); the formal deduction that shows the contradiction to be a theorem is talked about (in steps (2.1) through (2.4)), but it is not a part of the informal reasoning. The observation that the Liar reasoning stands here in a hypothetical mood will play a crucial role in our study of (purported) Gödelian Liar sentences in chapter 6 – but for now, our main interest is what Tarski made of the result of the reasoning of which the hypothetical proof is only a subpart. Tarski’s Theorem says that a broad range of formal languages do not allow to define their own truth predicate. It is entailed by what is stated in line (5), because a definition of truth (or any other notion) that leads to the inconsistency of the language it is formulated in surely cannot be “formally correct” and “materially adequate”, 357 and thus cannot fulfill Tarski’s famous two desiderata for any successful definition. 358

3.7 Tarski’s condition of paradoxicality The lesson Tarski famously drew from the Liar paradox is that a consistent language cannot be semantically closed. Let us learn this lesson by paraphrasing a seminal passage of Tarski’s in the terms of the present chapter. In his 1944 paper on the semantic conception of truth, Tarski sums up the assumptions needed in the basic Liar reasoning: “If we now analyze the assumptions which lead to the antinomy of the liar, we notice the following: (I) We have implicitly assumed that the language in which the antinomy is constructed contains, in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term ‘true’ referring to sentences of this language; we have also assumed that all sentences which determine the adequate usage of this term can be asserted in the language. A language with these properties will be called ‘semantically closed’. (II) We have assumed that in this language the ordinary laws of logic hold. (III) We have assumed that we can formulate and assert in our language an empirical premise such as the statement (2) which has occurred in our argument.” 359 357 358

359

Tarski 1956[1935], 152 & 187f.; cf. Tarski 1944, 341. Otherwise any language with an inconsistent background theory, a classical background logic, and the resources to refer to some particular collection of sentences would define truth for that collection of sentences, because in a classical setting an inconsistent theory is trivial in the sense that every sentence is provable in it, and so the (Tarskian Truth Schema) would be (vacuously) satisfied by any old formula that is open in a variable ranging over the sentences in question. Tarski 1944, 348.

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By “sentences which determine the adequate usage of [. . .] the term ‘true’”, Tarski means the T-sentences. Tarski’s unclear phrase “can be asserted in the language” can be given a precise meaning with the help of our notion of the background theory of a language: A sentence can be asserted in a language if and only if it is a theorem of the background theory of that language. Thus we glean the following definition of semantic closure from condition (I): (Semantic Closure) A language is semantically closed if and only if it contains, for each one of its expressions, a term that refers to it, as well as semantic predicates, among them the predicate or open formula like ‘True(. . . )’ which expresses the notion of truth in the sense that every instance of the (Tarskian Truth Schema) for ‘True(. . .)’ is a theorem of the background theory of the language. What we have made explicit here is that semantic closure is similar to inconsistency (and consistency) insofar as it involves not only the language in question, but its background theory, too. Tarski’s talk of a language having a certain feature should really be construed to be about the language taken together with its background theory. Condition (II) means that the background logic is classical. In order to understand the third condition, we need to add that what Tarski calls “statement (2)” in his text is of the same form as the sentence ‘The F = ‘The F is not true’’, 360 i. e., it expresses an identity statement that entails the existence of a Liar sentence. Thus condition (III) means that such an identity statement is not only expressed by some sentence of the language, but that that sentence is provable in the background theory. This last point – the theoremhood of the sentence that entails the existence of a Liar sentence – is what we had some doubts about when discussing the formal variants of the Liar reasoning, 361 but which turned out to be not really needed because the (Diagonal Lemma) secures the existence of a theorem of the background theory that can play a similar role in the formal Liar reasoning as the identity statement. 362 In the light of these considerations it is interesting that Tarski notes, shortly after the passage quoted above, that, although conditions (I) and (II) are both essential, (III) is not really needed for the Liar reasoning. 363 Now, as condition (I) expresses semantic closure and Tarski thinks that condition (II) is beyond doubt and that condition (III) is not essential, it is clear that according to Tarski the Liar reasoning establishes the following Tarskian condition of paradoxicality: Every semantically closed language is inconsistent. Both the notion of semantic closure and the notion of inconsistency need to be spelled out here in terms of the background theory of the language. 360

361 362 363

Thus it is like the identity statement in the first step of the Informal Prototype of the formal Liar reasoning in section 3.5 (the only difference is that Tarski works with a strengthened Liar sentence). Cf. section 3.5. Cf. section 3.6. Tarski 1944, 348f.

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Tarski reacted to this situation with his famous approach to the Liar paradox that involves a hierarchy of meta-languages: Each language can contain ascriptions of truth only for languages lower in the hierarchy, so that none of them is semantically is closed.

3.8 Contagious inconsistency We have seen that according to Tarski the lesson of the Liar paradox is that every semantically closed language is inconsistent. In the two concluding sections of this chapter, it will turn out that matters are (or appear to be) even worse. We will see that, in a sense that will become clear shortly, the inconsistency of the Liar paradox is contagious (in the present section) and that any Liar sentence is metaphysically inconsistent (in section 3.9). 364 In contrast to many of the variants of the Liar reasoning discussed in the preceding sections of this chapter, 365 the reasoning that will show that the inconsistency of the Liar paradox is contagious and that any Liar sentence is metaphysically inconsistent will not be purely formal. It nevertheless is best discussed here, because it builds directly on what we have learned in earlier parts of the chapter about the role of the (Tarskian Truth Schema) in the Liar reasoning and because it will strengthen Tarski’s condition of paradoxicality. And in this way, we will also be able to provide additional evidence for the central claim of chapter 2, that the seriousness of the Liar paradox is not bound up with taking the formal stance. * The key observation that gives the clue for the reasoning in this section is that in every variant of the basic Liar reasoning, the Liar sentence itself is not used but only mentioned. To be sure, a translation of the Liar sentence is used in the reasoning (as one side of an ‘if and only if’-statement or biconditional), and in the homophonic case the Liar sentence is translated as itself and thus is used in the reasoning. But translation does not need to be homophonic; and when it is not, it suffices to mention the Liar sentence, and it need not even be quoted or otherwise displayed within the paradoxical reasoning. 366 This opens up the way for a variant of the Liar paradox that involves two languages, such that the Liar sentence belongs to the first language and the reasoning about it is formulated in the second language. Here is an example: 364

365 366

Although the presentation of the basic Liar reasoning in chapters 2 and 3 is untypical insofar as it is structured by the question of formality, pretty much everything we have seen so far is common lore. By contrast, the points made in the present section about the contagious inconsistency of the Liar paradox and in the next section about the metaphysical inconsistency of any Liar sentence are novel, as far as I know. Cf. sections 3.5 and 3.6. This is witnessed by all variants of the Liar reasoning presented in section 2.4, and by Formal Variants 3 and 4 and (although here it is obscured by the use of the standard name operator) in Schema A, which are presented in section 3.5.

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Two-Language Variant 1 of the basic Liar reasoning (1) In the Classical Tibetan language there is a sentence λCT that can be translated into English as ‘λCT is false’. (2) λCT is true if and only if λCT is false.

(3) λCT is true and false and λCT is not true and false.

existence of a Liar sentence

the (Tarskian Truth Schema), which applies because (1) guarantees that ‘λCT’ refers to a sentence that can be translated into English as ‘λCT is false’ (2), (Exhaustiveness), (Exclusiveness)

In order to embed this reasoning in an interesting scenario, we can tell a story like the following: An eager logician has taken to heart the Tarskian lesson that, because of the Liar paradox, every semantically closed language is inconsistent. In his naïveté and optimism, he has turned it around, and believes every language which is not semantically closed to be free of the Liar paradox. So he goes on to reform the English language he uses by eliminating from it all devices of sentential self-reference and, to be on the safe side, restricts its truth predicate to sentences of languages that are more ancient than English. For a while these measures allow him to calm his fear of contradiction because the language he uses is not semantically closed. But then something happens that shatters his naïvely optimistic Tarskian beliefs. For he meets an old friend, an archaeologist and philologist, who in an old temple ruin in the Himalayas had found a pottery shard with a mysterious inscription. Because of her knowledge of Classical Tibetan, she understands it as saying that it is false. Uncertain of the relevance of this, she asks him for his professional opinion as a logician, whether that ancient sentence is true or false, and what else may be the case with it: 367

He can neither read the inscription nor pronounce it aloud; in fact, the only thing the philologist has told him is that it ascribes falsity to itself. So he pronounces that, yes, what we have before us is a Tibetan Liar sentence. When he sits down to study it, he is happy at first that he can do his friend a favor without breaking the careful strictures he had established for his language of use. Soon he is doubtful whether he will be able to give a decisive answer, because he does not know enough about Classical Tibetan to judge whether 367

I would like to thank Roman Klauser for supplying me with a translation of ‘This sentence is false’ into Classical Tibetan.

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it allows to reason about truth and falsity in the way essential to the Liar paradox. He thinks it is likely, however, for how could the respective string of symbols be a predicate of falsity if it was not governed by the usual principles within the language? Suddenly, when he had already convinced himself that he could go on and report his findings, he realizes with a shock the deep trouble he is in. A moment’s reflection (along the lines of the above Two-Language Variant 1 of the basic Liar reasoning) has forced him to see the following: However things may stand with the favorite ancient language of his philologist friend, Classical Tibetan, his own report will without doubt have to be inconsistent. Sure, the regimented fragment of English that he speaks may still be consistent (its background theory still would not entail any contradiction, because the principles that govern the truth and falsity predicate of Classical Tibetan do not belong to the background theory of English). But his mini theory about Classical Tibetan that is couched in the words of this cleared up variant of English does entail a contradiction. He will have to tell his friend that the inscription on the shard of pottery is true and false and not true and false. As he sees no warrant for rejecting the mini theory, that is a problem not so much for her, but for him. And that makes him very uncomfortable, indeed. 368 The novel aspect of this two-language variant of the Liar reasoning is not the contradiction that is likely to occur in Classical Tibetan – that is a problem we are already familiar with in another guise –, but the contradiction that occurs in the logician’s cleared-up English. The crucial observation is that the fragment of English that is used in Two-Language Variant 1 does not need to be semantically closed, because it is compatible with the scenario that the reasoning is conducted in a regimented fragment of English that does not allow to form self-referential sentences and has a truth predicate that is applicable only to sentences of languages other than English. Let us generalize from the Two-Language Variant 1 of our example: Two-Language Variant 2 of the basic Liar reasoning (1) In some language, L1, there is a sentence, λ1, existence of a Liar sentence that can be translated into another language, L2, as ‘λ1 is false’. (2) λ1 is true if and only if λ1 is false. the (Tarskian Truth Schema), which applies because (1) guarantees that ‘λ1’ refers to a sentence that translates into our language L2 as ‘λ1 is false’ (3) λ1 is true and false and λ1 is not true and false.

368

(2), (Exhaustiveness), (Exclusiveness)

There is an analogous observation by Arthur Prior, albeit with less flourishes and in a single-language scenario: “Some paradoxical statements are, on the face of it, awkward for the propounder only, while

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When we think of the reasoning which here is displayed in English 369 to be formulated in the language L2 (including the quoted expression ‘λ1 is false’), we have reached the general case of a two-language variant of the basic Liar reasoning. 370 Now let us step back and have a look at what we need to assume to reason in this way. For the reasoning, we need the language L1 to contain its own falsity predicate (or negation and its own truth predicate) and some device to construct self-referential sentences. We further need the language L2 to contain singular terms that refer to the expressions of L1, a truth and a falsity predicate that are applicable to sentences of L1, and we need to assume that in the language L2, we “can assert” (to use Tarski’s words) the instances of the truth schema and of the falsity schema for the sentences of L1, i. e., that the instances of the schemata are theorems of our mini theory about L1 – which, it must be emphasized, is formulated not in L1, but in L2. Thus, we separate the Liar sentence from the Liar reasoning, by dividing them up between the two languages L1 and L2. Although it is likely that L1 is semantically closed and hence inconsistent, in accordance with Tarski’s condition of paradoxicality, nothing in the paradoxical reasoning demands of the language L2 in which we reason that it is semantically closed; it need not satisfy Tarski’s condition of paradoxicality. But although L2 on its own could therefore be perfectly harmless, we are forced to assert a contradiction as soon as we use it to reason about the Liar sentence in L1. The result is a strengthening of the condition of paradoxicality as it was formulated by Tarski. Every Liar sentence gives rise to contagious inconsistency, in the following sense: (Contagious Inconsistency) If there is a Liar sentence in any language, then every language that allows to talk about it is inconsistent. Here, requiring a language to allow to talk about a certain sentence means that the language both allows to refer to that sentence and to reason about whether it is true (in the sense described just now). In present terms, Tarski’s condition of paradoxicality is, by contrast, that every language that contains a Liar sentence and allows to talk about it is inconsistent.

369

370

some are also awkward for the looker-on. The Eubulidean version of the Liar paradox is of the second sort – if a man says ‘What I am now saying is false’, not only he himself but we who look on seem forced to say contradictory things (that his statement must be true because even if it were false it would be true, and that it must be false because even if it were true it would be false).” (Prior 1961, 16) This is done not only for the reader’s sake, but also because there simply is no way of displaying the expressions of an arbitrary language. It does not help here to relativize truth and falsity to languages, as is often done in the Tarskian tradition. In that case we need only start with a Liar sentence that can be translated into L2 as ‘λ10 is false relative to L1’. Then a variant of the truth schema that is relativized to L1 will let us infer from the premise in line (1) that λ10 is true relative to L1 if and only if λ10 is false relative to L1; and variants of the principles of (Exhaustiveness) and (Exclusiveness) that concern notions of truth and falsity that are relativized to L1 will let us infer that λ10 is true and false relative to L1 and λ10 is not true and false relative to L1 – which is no less contradictory than what is inferred in line (3) of Two-Language Variant 2.

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Thus (Contagious Inconsistency) is strictly stronger than Tarski’s condition of paradoxicality. And we see that, even when we refrain from trying to squeeze everything into a single language, and are very careful with the language we use, relinquishing all devices of self-reference, we will remain in danger of running into the Liar paradox as long as any Liar sentence is around somewhere. Why did Tarski not draw this more general lesson from the Liar paradox? Did he not see this? It is more likely that he was not interested in such two-language scenarios. Although we can only speculate about his reasons, one thing is quite clear: He very probably did not want to work with those variants of his truth schema that cannot be formalized, and the variant of the (Tarskian Truth Schema) that figures in the two-language reasoning is not formalizable because it depends crucially on non-homophonic translation. 371 For, if the translation from L1 to L2 was homophonic, then L1 would be part of L2 (or, if the translation of the Liar sentence was homophonic, then at least L1 and L2 would overlap in a fragment that includes the Liar sentence), and we would be back at a one-language variant of the Liar reasoning. In connection with this attempt at an explanation we can observe, with a view to our guiding question of formality: 372 Going informal has not lessened the problem of the Liar paradox; 373 rather, the essentially informal reasoning of twolanguage variants forces us to acknowledge that the problem constituted by the Liar paradox is even greater than we might think when taking the formal stance.

3.9 Metaphysical inconsistency There is one last bit that I would like to add to complete our endeavor of fathoming the problem posed by the Liar paradox. We have seen in the preceding section how the contradiction engendered by a Liar sentence cannot be contained in the language that it belongs to, but spreads to every language that allows to reason about it. Let us now turn to the Liar sentence in the center of this turmoil, and try to study it in isolation. In order to see what exactly can go wrong, paradox-wise, with a single sentence, it will be helpful to distinguish some notions of contradictoriness and inconsistency. A sentence is a contradiction if and only if it has the form ‘p and not p’, where ‘p’ is replaced by any sentence. A sentence is self-contradictory if and only if its negation can be inferred from it. We have already laid down that a theory is inconsistent if and only if a contradiction is among its theorems. 374 As we can treat a given sentence as the limit case of a theory if we understand it as its sole axiom, it is plausible to construe a self-contradictory sentence as a limit case of an inconsistent theory. What are the corresponding semantic notions? In classical logic and model theoretic terms, a contradiction is unsatisfiable, which means in our semanticist terms that no re-interpretation of its non-logical parts will lead us to a true sentence. The 371 372 373 374

For the connection between translation and the formalizability of the truth schema, cf. section 3.2. Cf. section 1.10. Cf. section 1.9. Cf. section 3.4.

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same goes for self-contradictory sentences and the more general case of inconsistent theories. But beyond classical logic, it is less clear that unsatisfiability models an intuitive notion of contradictoriness. When the (Exhaustiveness) of truth and falsity fails, a sentence that cannot be made true will seem less daunting. And when the (Exclusiveness) of truth and falsity fails, classically unsatisfiable sentences can be true – if they are false, too. Which brings us to an impeccable semantic explication of contradictoriness for the non-classical context: the notion of a dialetheia, i. e., of a sentence that is true and false. 375 Let us now use these notions of contradictoriness and inconsistency to characterize Liar sentences – of course under the assumption, for now, that every step of the Liar reasoning does indeed go through. The Liar sentences we have seen so far obviously are not contradictions because they are atomic sentences. 376 Within a semantically closed setting, a Liar sentence is self-contradictory, but this does not need to be so in general. We were not entirely sure, e. g., that we have to construe the background theory of Classical Tibetan as strong enough to allow to infer the Classical Tibetan Liar sentence’s negation from it. Even within a classical setting, we cannot in general show that a Liar sentence is unsatisfiable, because in almost every case it will be easy to (re-)interpret the singular term that refers back to the Liar sentence to refer to a false (or untrue) sentence, so that the Liar sentence on that (re-)interpretation would come out as true even if we held the meaning of the falsity predicate (or of the untruth predicate) fixed. 377 But if we give up the principle of (Exclusiveness), then there are still variants of the Liar reasoning that can be turned into an argument for the conclusion that the Liar sentence is a dialetheia, so that it arguably falls at least under this non-classical semantic notion of contradictoriness. * Note that this characterization of a Liar sentence – as self-contradictory in a semantically closed setting and as a dialetheia in certain non-classical contexts – is still language-bound: Given a certain language together with its background theory, or given a certain background logic, the Liar sentence has a certain feature. To those uncomfortable with that feature, this might suggest leaving behind the language or the logic. But as with all real world problems, not talking about the Liar paradox will not make it go away. To see this, we need a notion of inconsistency that is more worldly. Or to spill out this figurative way of speaking, what we need is the following metaphysical notion of inconsistency, which is applicable not only to a sentence within a certain language, logic, or theory, but to any object whatsoever. An object is inconsistent in a metaphysical way if and only if it both falls under a certain concept and does not fall under that concept. 375 376

377

Cf. section 2.3. We should be careful and refrain from making the general claim that no Liar sentence is a syntactic contradiction, because a syntactic contradiction that ascribes falsity to itself can be constructed, e. g., ‘This conjunction is false and this conjunction is not false’. An exception might be a Liar sentence that uses a regimented variant of the sentential indexical ‘this sentence’, because if there were reasons to count that indexical term among the logical expressions, this would forestall any (re-)interpretation of the Liar sentence ‘This sentence is false’.

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Here the usual Aristotelian strictures about contradictoriness are required; that is, the definition of metaphysical inconsistency concerns falling and not falling under the same concept all other things being equal, in the same respect, at the same time, and so on. 378 It is a delicate matter to give an example for the metaphysical notion of inconsistency, because many people will be fast to cry that there just cannot be any inconsistent objects (and I agree). As some notions are necessarily empty, there is room at least for inconsistent concepts, e. g., for the notion of being a round square. But while round squares would qualify as inconsistent objects, they are not to be found among actual objects or possibilia. Talk of instances of inconsistent concepts does not really help illustrate what it would mean to encounter a particular inconsistent object. As so often when trying to talk coherently about what does not exist, it is good to resort to semantic ascent, here to stories. For if we hold on to our conviction that inconsistent objects cannot exist, our best chance of finding a concrete example is in dialetheist fiction. 379 Although the genre is not exactly thriving yet, there is a short story by Graham Priest, called “Sylvan’s Box”. Priest describes in it how he found, in the house of his late friend Richard Sylvan (a. k. a. Routley), between a pile of Sylvan’s papers devoted to paraconsistent logic and another pile devoted to Meinongian metaphysics, a box labeled “Impossible Object”. It was properly so called, because it was empty and at the same time it contained a little figurine, which was “carved of wood, Chinese influence, south-east Asian maybe”. 380 Priest is careful to describe his perception of the inconsistency of the box in a way that prevents the reader from dismissing it as a mere trick of the senses: “Gently, I reopened the box and gazed inside. One cannot explain to a congenitally blind person what the colour red looks like. Similarly, it is impossible to explain what the perception of a contradiction, naked and brazen, is like. Sometimes, when one travels on a train, it is possible to experience a strange sensation. One’s kinaesthetic senses say that one is stationary; but gazing out of the window says that one is moving. Phenomenologically, one experiences what stationary motion is like. Looking in the box was something like that: the experience was one of occupied emptiness. But unlike the train, this was no illusion. The box was really empty and occupied at the same time. The sense of touch confirmed this.” 381

Priest’s purpose in telling his dialetheist story is not to argue for the existence of dialetheias, but rather to draw a weaker and more specific moral, namely that our usual reasoning – as exemplified by the reader’s reactions to the story – is compatible 378 379

380 381

Cf. Aristotle, Metaphysics Γ, 3 1005b 19–23. A theory is dialetheist if and only if it says that there are dialetheias, i. e., sentences that are both true and false (cf. Priest /Berto 2013). Let us call a fictional text dialetheist if and only if it is meant to be inconsistent. Here, the locutions “says that” and “meant to be” are important to distinguish a dialetheist theory or fiction from any old inconsistent theory or fiction. We would not want to classify the theory of Frege’s Grundgesetze as dialetheist, and neither the plot of some novels and many TV shows, where inconsistency is a mere mistake that the author is likely not aware of. Priest 2006c, 128. Priest 2006c, 128.

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with paraconsistent logic. 382 But these issues are not important here, because our purpose in giving a précis of Priest’s story is merely that of illustrating the notion of an inconsistent object: An inconsistent object is something like Sylvan’s box. Together with this way of illustrating it, the metaphysical notion of inconsistency will turn out to be useful for measuring the severity of the problem constituted by the Liar paradox. * Is a Liar sentence an inconsistent object? To answer this question, we can again employ a two-language variant of the Liar reasoning. As what we want to show goes beyond (Contagious Inconsistency), it is no surprise that we need two extra assumptions. But both are highly plausible. The first extra assumption is that our own language of use is like the second language of the two-language variants insofar as it allows us to speak about the meaning and the truth or falsity of sentences that belong to other languages. Note that this does not entail that our language of use is semantically closed – we can be as careful as the eager logician of our story in the previous section. The second extra assumption brings in a new element. It is a criterion of metaphysical commitment that lays down that our language of use is transparent in the following sense: Asserting a sentence of our language of use that says that things are so-and-so just is saying that things are so-and-so. This entails that by speaking our language of use we metaphysically commit ourselves to what we assert. When you claim ‘There are unicorns’ in our language, you thereby commit yourself to an ontology that includes unicorns. 383 Our first extra assumption sanctions instantiating the language L2 in the TwoLanguage Variant 2 384 by our language of use. After some further steps of the usual basic Liar reasoning we will have to assert that the Liar sentence λ1, which belongs to some other language L1, is true and false and not true and false. In view of the second extra assumption, this commits us to the actual existence of an object – the Liar sentence λ1 – that both falls under the concept of being true and false and does not fall under the concept of being true and false, and that hence is an inconsistent object. And as λ1 was arbitrary, we must argue in a similar way about any Liar sentence. Thus any Liar sentence is an inconsistent object, just like Sylvan’s box. (Metaphysical Inconsistency) Any Liar sentence is an inconsistent object and thus inconsistent in a metaphysical way. This is bad. Just how bad it is comes out when we compare Sylvan’s box to Frege’s Basic Law V, which is the root of the inconsistency of the formal theory 382 383

384

Priest 1999 & Priest 2006c, 125–133; cf. Pietz 2010. We will not argue for these assumptions here, but we should make an important qualification: What we call ‘our language of use’ need not be our colloquial language (e. g., English as it actually spoken today); it is whatever language (or language fragment) we use in our scientific reasoning. This might be a fragment of natural language that has undergone some far-reaching reform, it might be highly regimented, or even formalized. As it possible that in the future we will find reasons for further reform of our language of use, our criterion of metaphysical commitment is incompatible with a corresponding claim of infallibility, but that is as it should be. Cf. section 3.8.

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Frege laid down in his Grundgesetze der Arithmetik. 385 On a first impression, Basic Law V of course would appear to constitute a worse problem than Sylvan’s box. After all, the discovery of the inconsistency of Basic Law V destroyed the lifework of a brilliant mathematical logician, Frege, and poses a serious obstacle to all later attempts to carry on his project of logicism. Sylvan’s box, by contrast, is a fun invention. 386 But we feel Sylvan’s box to be comparatively harmless for the sole reason that it so clearly is fictional. Just try to imagine for a moment that Priest in 1996 actually found an inconsistent object in Sylvan’s house, just like Russell in 1901 actually found an inconsistent sentence in Frege’s theory. Now it should be evident that Sylvan’s box would constitute a much more severe problem than Basic Law V, because, with all due regret over Frege’s failure to complete his important project, we can (and should) simply leave behind his inconsistent theory (and maybe look for consistent alternatives 387). But we cannot in the same sense leave behind an inconsistent object! After its discovery the “world [will] never be the same” 388 – for any of us. The keyword here is ‘world’. What we achieve by characterizing a Liar sentence as an inconsistent object is to move beyond the bounds of any language and out into the world. In order to explain this, let us quickly retrace our steps. The standard presentation of the Liar paradox (which we gave in chapters 1, 2, and in the present chapter up to section 3.7) culminates in Tarski’s condition of paradoxicality and thus construes the paradox as a problem relative to the language that the particular Liar sentence occurs in. The stronger condition (Contagious Inconsistency) that we found to hold in the last section construes the paradox as a problem relative to any language that allows to talk about the Liar sentence. In the present section we use the assumption that our language of use is transparent to show that (Metaphysical Inconsistency) holds, so that the Liar sentence is an inconsistent object. Thus we construe the Liar paradox as a problem that is not relative to any language at all. This should not be obscured by the fact that the Liar sentence, being an expression, does of course belong to some language. The point is that we now view the Liar sentence not from within any language, but understand it as an object that is out there in the world to make trouble regardless of whether we talk about it. In effect, we have shifted our perspective from semantics (or the theory of meaning) 389 385

386

387

388 389

For the present considerations, the exact content of Frege’s Basic Law V and of his formal theory in general is not important. Roughly, Frege’s formal theory is a higher order variant of quantified classical logic combined with the axioms of what nowadays would be called ‘naïve set theory’. Basic Law V encodes what has turned out to be naïve about it, namely Frege’s specific version of the strong abstraction principle (or naïve comprehension principle; cf. section 1.8) that for every open formula of the formal theory there is an object, the set of exactly those objects that are the extension of that open formula. Cf. Frege 2009[1893/1903] and Burgess 2005, 1ff. It is more than that. By writing “Sylvan’s Box”, Priest did not only provide a neat example for one of his philosophical tenets (to wit, that our ordinary reasoning might be paraconsistent), but he also found a special way of honoring his late friend and expressing his affection. Many attempts have been made in recent years to salvage consistent parts of Frege’s system; for an overview cf. Burgess 2005. Cf. Priest 2006c, 133. We will say more about this in section 4.1.

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to the metaphysics of expressions. Thus we are still concerned with how things are with the expressions of a language, but we now think of them as part of the world and treat them on a par with the other objects the language might be about. Questions like ‘What is a word?’ are now treated in the same way as questions like ‘What is a physical object?’. And in particular, the question whether a Liar sentence is true or false is now treated in the same way as the question whether a particular box does or does not contain a figurine. There are independent reasons to see the metaphysics of expressions as a valuable complement to the theory of meaning. 390 A further reason is especially relevant within the scope of the present study. In semantics, we are accustomed to a picture where a language and the world it is about are separate. In most scenarios, this is either correct or a harmless idealization. But once we are interested in a language that to some extent is self-referential, the neat division breaks down, because now the language will allow to talk about some of its expressions, which thus become part of the world the language is about. We will make much more of this idea later, 391 and more generally, paying attention to the metaphysics of expressions will be crucial for our own proposal for a solution to the Liar paradox in part III. But that is far off, because we have part II still before us where we will look at the matter from a semantic angle. It may seem ironic that what will later enable the diagnosis and therapy of the Liar paradox is the same as what here helps to complete its anamnesis, insofar as we see only from the perspective of the metaphysics of expressions that a Liar sentence is an inconsistent object. Anyway, as the inconsistency of an object, in contrast to the inconsistency of a theory or of a language, cannot be confined in any way – as it cannot be removed by merely discarding a theory or reforming a language –, our observation of the (Metaphysical Inconsistency) of the Liar paradox should trouble us deeply.

390 391

Cf., e. g., the seminal Kaplan 1990, as well as Pleitz 2007 and 2015b. Cf. part III and sections 9.12 and 11.1 in particular.

PART II

SEMANTICS AND THE LIAR PARADOX

Anil Gupta and Nuel Belnap: “When confronted with some perplexing and extraordinary phenomenon, one is tempted to focus on it and to ignore the ordinary and familiar phenomena that are related to it. However, this may not be the best strategy for achieving an understanding of the puzzling phenomenon. An analogy may help to make the point clear. Imagine that before the development of astronomy the members of a tribe observe a solar eclipse. Naturally perplexed by this extraordinary occurrence, they seek an explanation of it. Their first tendency may well be to speculate on the causes of the event. We can imagine the kinds of hypotheses that would appear natural: the wrath of gods, the swallowing of the sun by an evil demon, and what not. But it is unlikely that such speculations will yield an understanding of the eclipse. The proper method for gaining understanding here is to undertake a patient and systematic study of the ordinary and familiar behavior of the sun, the moon, and the stars. Often we come to understand the extraordinary only when we see it in terms of the ordinary. [. . . ] In order to gain a better understanding of the Liar, we need to give less attention to the paradoxes than we have given them.”

Chapter 4

Meaning, Semantics, and Reductionism

Kit Fine: “We shall follow Frege (1892) in taking there to be a basic distinction between objects and concepts. Objects are referred to by means of singular terms and concepts by means of predicates; and variables for objects and concepts are respectively taken to occupy either a nominal or a predicative position. Although concepts may correspond to objects, no concept can sensibly be said to be an object, since this would involve a grammatical confusion between a singular term and a predicate.” 392

In part I, we have assessed the problem posed by the Liar paradox and found it to be severe. We have seen that it is logically robust and that attempts at a solution invite Revenge problems; 393 and we have brought to light that the inconsistency of a Liar sentence is contagious and metaphysical, i. e., goes beyond the language of the Liar sentence and is more difficult to get rid of than a theory that might be discarded. 394 One thing in particular that we have learned is that the problem of the Liar paradox is not bound to taking a formal stance – on the contrary, the problem arises already in informal logic, 395 and its formal variants have proven difficult to separate from a recalcitrant informal residue. 396 The reason for this is that some recourse must be made in the Liar reasoning to translation or meaning. This brings us to the topic of part II, where we continue the study of the Liar paradox from the angle of semantics. Our aim is to understand how a singular term can refer to the sentence it occurs in, assuming for the moment that this is possible because this is a precondition for there to be Liar sentences. For the most part, we will proceed as if we could forget both about the problematic nature of the Liar paradox that we saw in part I and about our goal of solving it in part III. You could see this as our way of following the methodological advice given by Gupta and Belnap, of trying to understand a problem better by (initially) paying less attention to it. 397 There is also a welcome side effect, because the Liar paradox thus provides us with an occasion to think about the nature of language. In the present chapter we will introduce semantics in general. In chapter 5, we will describe the semantics of singular terms and apply it to the plurality of Liar 392

393 394 395 396 397

Fine 2008[2002], 1. Fine’s “Frege (1892)” is “Über Begriff und Gegenstand” (Frege 2002d[1892]). The motto for part II is taken from Gupta /Belnap 1993, 17. Cf. chapter 2. Cf. section 3.7 through 3.9. Cf. section 2.4. Cf. sections 3.5 and 3.6. Cf. the motto of this part.

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sentences. In the concluding chapter 6, we will turn to a more particular question of some importance, the semantics of sentences that appear to achieve self-reference by way of Gödelization. After some terminological remarks on ‘semantics’ and ‘reference’ (in section 4.1), we will start out with the sketch of a broadly Fregean framework for doing semantics (in section 4.2) and an overview of the relations between core notions of theoretical semantics (in section 4.3). We will cover a lot of well-known territory in the central part of the chapter, which outlines extensional semantics, intensional semantics, and two-dimensional semantics (in sections 4.4 through 4.6), but I hope to have found a path through it that is not too well-trodden. And anyway, we need to pick up some important tools for later use, especially from two-dimensional semantics (in section 4.6). We will conclude the chapter with a discussion of the three semantic theories which measures them against the Fregean framework sketched before (in section 4.8).

4.1 ‘Semantics’ and the theory of meaning When we theorize about an important concept, it is often possible to distinguish a foundational and a criterial approach. While a foundational theory of a given concept tries to explain the nature of that concept, a criterial theory tells us only, for a large number of possible cases, which items fall under the concept and which do not. There may of course be interdependencies between a foundational and a criterial theory of some notion. When we theorize about the concept of truth, to give an example we are already acquainted with, we should distinguish the deep and philosophical question ‘What is truth?’ from the more everyday question ‘What is true?’. The first question will be answered by foundational theories of truth (e. g., the correspondence theory or deflationism) while the second question will be answered by criterial theories of truth (e. g., a Tarskian recursive definition of truth or the theory outlined by Saul Kripke); 398 and it is criterial theories of truth that are the primary area of application for convention T and hence for the Tarskian truth schema. 399 When we theorize about meaning, we also should distinguish foundational from criterial approaches. A foundational theory of meaning will try to explain the nature 398

399

In practice, the boundary can be unclear. Tarski was not very decisive in his assertion that his truth schema is an explication of one of the foundational theories of truth, the correspondence theory (e. g., Tarski 1944, 342ff.; and for a balanced assessment cf. Künne 2003, 208ff.). And often proponents of another foundational theory of truth, deflationism, hold that the Tarskian truth schema, which so clearly seems to be on the criterial side, is all that is needed to describe the nature of truth (cf. Künne 2003, 225ff. and Burgess/Burgess 2011, 33ff.), so that for them the foundational and the criterial theory of truth would fall together. Cf. section 3.2. – It would of course be contrary to Tarski’s intentions to say that foundational theories of truth are independent of convention T, because a foundational theory that entails that the true sentences are not the same as the sentences that are true according to the Tarskian truth schema would surely not be acceptable to Tarski. But in the case of criterial theories of truth, meeting convention T is not only a necessary but also a sufficient condition (cf. Tarski 1956[1935], 187f.).

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of meaning, answering questions like: ‘Is meaning based on mental entities, on abstract objects, or constituted by social practice?’ and ‘What is language?’. A criterial theory of meaning, by contrast, will try to give a systematic account of the meanings of important sorts of expressions, often for a given language or family of languages. It is criterial theories of meaning that we will call semantic theories (or just semantics). Semantics will be more important in the present study than foundational theories of meaning. The semantics of singular terms will be an indispensable tool when we try to understand the workings of self-referential sentences 400 and observations about semantics will play an important role in my approach to the Liar paradox (in part III). At those points where we are concerned with the connection between semantics and metaphysics (in section 4.2 401 and in chapters 9 through 12), however, foundational theories of meaning will also be touched upon. * We should note here that many people follow Tarski 402 and use the term ‘semantics’ in a different sense, to refer solely to a theory of extensions, i. e., a theory about the reference of singular terms, the extension of predicates, the truth or falsity of sentences, and how these hang together. This was observed by Quine (who uses the term ‘reference’ to refer to the more general notion of extension): “When the cleavage between meaning and reference is properly heeded, the problems of what is loosely called semantics become separated into two provinces so fundamentally distinct as not to deserve a joint appellation at all. They may be called the theory of meaning and the theory of reference. ‘Semantics’ would be a good name for the theory of meaning, were it not for the fact that some of the best work in so-called semantics, notably Tarski’s, belongs to the theory of reference.” 403

Although there certainly is an important difference between meaning and extension, these notions are not as far apart as Quine suggests. According to most semantic theories, in a certain sense meaning determines extension (and we have already seen that the truth of a sentence depends on its meaning 404). It therefore makes good sense to say that the theory of extensions is a proper part of the corresponding semantic theory. So, contrary to Tarski and Quine we should say that semantics is the theory of meaning and of extension. 405 But we will see later that for

400 401 402 403 404 405

In particular, cf. chapters 5, 11, and 12. I here allude to the Fregean claim that concepts cannot be reduced to objects; cf. section 4.2. Tarski 1956[1936], 401; Tarski 1944, 345. Cf. Künne 2003, 176–180; Sher 2005, 150. Quine 1953/1961, 130. Cf. the naïve truth principle (NT) in section 2.2 and the (Tarskian Truth Schema) in section 3.2. In the common phrase ‘the semantic paradoxes’, the word ‘semantic’ is used in the narrow TarskiQuine sense. As the task of changing this label to something like ‘the extension theoretic paradoxes’ might be commendable but is hopeless, we are well-advised to remind ourselves that what still looks like a description may well have turned into a name. In that regard, ‘the semantic paradoxes’ are quite similar to ‘the Holy Roman Empire’.

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some people, there was indeed a compelling systematic reason to start the tradition of using the term ‘semantics’ to refer to the theory of extensions alone. 406

4.2 A Fregean framework for semantics and metaphysics In order to get clearer about the relation between the notion of meaning and the notion of extension, we will sketch a broadly Fregean framework for a semantic theory and the metaphysics behind it. It is Fregean because it takes up key elements of Frege’s philosophy of language, among them the construal of predicates and the distinction between sense and extension. It is only broadly Fregean as it is meant to leave room for refinements that are due to later insights in the philosophy of languages (the keywords here are rigid designation and direct reference). An important purpose is to introduce some terminology and to open up a spectrum for possible semantic theories, and the semantic theory (or the broad strokes thereof) that we will settle on is pretty standard from a current point of view. 407 Therefore, we will not argue at length for the exegetical and the systematic correctness of the following claims. Among the basic building blocks of a language, singular terms (names, descriptions, and indexicals 408), predicates, and the sentences that can be formed from them are the most important categories. It is a Fregean idea to start with a sentence that contains a name (or more than one name) and to understand a predicate as what remains after we take a name (or more than one name) out of the sentence. When we start, e. g., with the sentence ‘Socrates is famous’ and remove the name ‘Socrates’, we will end up with the predicate ‘x is famous’. 409 This way of expressing notationally that we understand predicates as incomplete expressions that have to be complemented by the right number of singular terms to form a sentence (by exchanging a name for a variable, here ‘x’) makes the language they belong to amenable to formalization in quantified classical logic (another important innovation of Frege’s), because the quantifiers (‘for all x’, ‘for some x’) can hook unto the unbound variable (‘x’) in the predicate. In classical logic, the most basic formal counterparts of natural language singular terms, primitive predicates, and the sentences formed from them are constants, predicate letters, and atomic sentences, but usually there will be ways to 406 407

408

409

Cf. section 4.8 and the sections preceding it. To anticipate, our claim that sense and concepts are irreducible is not standard, but in this case we will argue in more detail. Cf. sections 4.8, 9.7, and 9.8. Strictly speaking, a quotation expression is arguably neither a name, nor a description, nor an indexical and thus quotation expressions form a fourth category of singular terms. Cf. section 5.5. The identification of a predicate with (the natural language counterpart of) an open formula cannot be kept up in languages that include non-extensional operators. The open sentence ‘x is possibly broken’ is ambiguous, because it can either mean that it is possible that x satisfies the predicate ‘. . . is broken’ or that x (actually) satisfies the predicate ‘. . . is possibly broken’, i. e., ‘. . . is breakable’. In languages with non-extensional operators, predicates can be formed in an unambiguous way by abstracting on open formulas. A full formal treatment of predicate abstraction in modal languages is given by Mendelsohn / Fitting 1998, 187ff. We will ignore this complication in the following because in the context of the present study the mentioned ambiguities will play no role.

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form more complex expressions that can play the same syntactic role, and a variety of terms and open formulas will be available to formalize the singular terms, the (non-primitive) predicates, and the more complex sentences of natural language. 410 Every expression that belongs to one of the three categories of singular terms, predicates, and sentences according to Frege has a sense and an extension. 411 We will say: The extension of a singular term is the object it refers to, the extension of a predicate is the collection of just those objects that satisfy that predicate, and the extension of a sentence is its truth value. These notions of extension are semantic in the narrow Tarski-Quine sense of the word. 412 In order to move on to semantics in the wider sense of a theory of meaning, we need to turn to the corresponding notions of sense: The sense of a singular term is the way the object it refers to is given by the term (the way an object is given is also called its mode of presentation). The sense of a predicate is a concept. An object satisfies a predicate just in case it falls under the concept which is expressed by that predicate. The sense of a sentence is the proposition it expresses. 413 Some remarks of clarification. We depart here from Frege’s own account and the standard Fregean picture in at least three respects: Firstly, a collection can be construed either as a set 414 or as a plurality. 415 Secondly, we would like to find a way 410

By construing a predicate not as the single expression that is concatenated with the grammatical subject, but as an open formula, predicates can be as complex as sentences. E. g., the (simple) predicate ‘x is even’ of the mathematical fragment of natural language can be formalized in a standard language of arithmetic by the complex predicate ‘∃y x = 2y’. To form terms more complex than constants, in some languages of classical logic there are primitive functors, i. e., symbols which when applied to a term deliver a term like the successor functor of arithmetic (cf. section 3.3); and in some languages there is the iota-operator, which is a device to transform any given open formula pΦ(x)q into a description p x Φ(x)q (read: pthe x such that Φ(x)q). Cf. subsection 5.2.4. In Frege’s own words, this is the well-known distinction of “Sinn” and “Bedeutung”. Cf. section 4.1. In Frege’s words, the sense of a sentence is the Thought (“Gedanke”) that it expresses. Cf. section 1.7. Frege himself did not construe extensions as sets, but as Wertverläufe, which in his semantico-metaphysical system, however, play a role similar to sets because they are objects associated with predicates (e. g., Frege 2002b, 13; cf. Künne 2010, 225f.). I would prefer to construe a collection as a plurality and thus not as a further (singular) object in addition to the (singular) objects that are among it (as a flock of five birds is not a sixth individual flying through the air, and certainly not an abstract object like a set). Cf. Simons 2005 and cf. section 1.7. But although this kind of metaphysical parsimony is likely to be relevant to the set theoretic paradoxes (cf., e. g., Linnebo 2010), it is not important within the scope of a study of the Liar paradox. Should we choose to construe the extensions of predicates as pluralities and not as sets, then there would also be a technical problem to be overcome, because there is no empty plurality but some predicates are not satisfied by any object. In this point the construal in terms of sets is likely to be more elegant.

ι

411 412 413 414

415

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around Frege’s strange semantico-metaphysical doctrine that the truth values the True and the False are objects that sentences refer to. Although we will follow the common practice of talking about truth values as objects, we would like to be able to construe this as shorthand for talk about the concepts of truth and falsity, which in turn can often be abridged to talk that uses the corresponding predicates. E. g., we would like to be able to understand ‘The extension of the sentence ‘Snow is white’ is the True.’ not literally but as shorthand for ‘The sentence ‘Snow is white’ falls under the concept of truth.’ or for the even shorter ‘The sentence ‘Snow is white’ is true.’ We will see shortly that some care is needed with the technical implementation of this understanding. 416 Thirdly, the claim that the sense of a predicate is a concept is a departure from Frege’s official position, according to which a concept is not the sense but the extension (“Bedeutung”) of a predicate. 417 However, this way of correcting Frege’s theory is not unheard of, 418 and as Rosemarie Rheinwald points out, Frege himself sometimes talks in a similar way. 419 What does it mean to argue for something so basic as the claim that the sense of a predicate is a concept and the corresponding claim that a concept is the sense of a possible predicate? Maybe this is not much more than a definition that commits us to a particular conception of concepts. But at least, it is not an uncommon conception. Kant, for one, seems to have held something like it when he construed concepts as “predicates of possible judgments”. 420 More importantly, adopting it will enable us to smooth out an irregularity in Frege’s original theory. According to the general picture of this theory, there are three levels: An expression expresses its sense which in turn determines its extension. This holds true of a singular term, its (singular) sense, and the object it refers to, as well as of a sentence, the proposition it expresses, and its truth value. But for another sort of expressions, Frege deviates from this tripartite structure, because for a predicate, he speaks of its sense (which Frege does not say much about), the concept that is its extension 421 – and the objects that fall under that concept! 422 As we identify the concept with the sense of the predicate, we are free to construe the objects that fall 416 417 418 419 420 421

422

Cf. section 4.4. Frege 2002b and 2002d[1892]. E. g., Rheinwald 2012[1997], 154 and Kleemeier 1997, 256. Cf. Künne 2010, 219f. Frege 1903, 150 = Frege 2009[1893/1903], 448; cf. Rheinwald 2012[1997], 154. Kant 1990 [KrV], A69/B94. Cf. Künne 2010, 178f. Frege construes a concept as a function (Frege 2002b, 11), but as Frege understands a function to be an unsaturated item (Frege 2002b, 13), he cannot mean a function as construed by contemporary mathematics, which would be a set theoretic object and hence saturated (cf. Künne 2010, 197). Thus the concepts that Frege attributes as extensions to predicates differ clearly from the sets (or pluralities) that we would standardly attribute as extensions to predicates. Cf. the diagram in Frege’s letter to Edmund Husserl of May 25, 1891; Frege 1976, 96f.

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under the concept as the extension of the predicate (which we can spell out as a Fregean Wertverlauf, as a set, or as a plurality), and thus to restore the number of levels for predicates to three. It should be noted that the conceptual distinction between sense and extension does not preclude that there is overlap in certain special cases. An important semantic theory – the standard semantics of classical logic, which we will speak about shortly 423 – identifies sense and extension for every expression of the language, and can therefore properly be called extensional semantics. And even in a setting where sense is not identified with extension across the board, there might be reason to identify sense and extension for a particular kind of expressions, or rather, to construe the extension as a direct constituent of the sense for that kind of expressions. 424 This occurs in theories according to which proper names and some indexicals refer directly, 425 which therefore can be built into the present Fregean framework even though Frege himself held a different semantic theory about proper names. (The question whether extension can be a direct constituent of sense is often presented as the point of departure between a Fregean and a Russellian orientation in semantics because Russell (famously) claimed at one time that there are singular propositions, i. e., propositions which contain an object they are about as a direct constituent. 426 I nevertheless do not want to speak of the broad framework for semantics I describe in this section as being both Fregean and Russellian, mainly because a Russellian orientation in semantics usually also includes a reduction of concepts to functions 427 and thus to objects, whereas a main feature of the present framework will be the claim that concepts are not reducible to objects, which is Fregean in spirit. 428) Getting back to the traditional Fregean picture, we should list three further important principles about sense and extension: (Compositionality of Extension) The extension of a composite expression is determined by its semantic form and the extensions of its semantic parts. (Compositionality of Sense) The sense of a composite expression is determined by its semantic form and the senses of its semantic parts. (Sense Determines Extension) The sense of an expression determines its extension. There is lots of leeway for understanding these three principles because the notion of determination can be specified in different ways and because their formulations

423 424 425 426 427 428

Cf. section 4.4. It will become clear later why this caveat is necessary for us; cf. sections 4.6, 4.7, and 5.2. Cf. section 4.6. Russell in Frege 1976, 250f. and Russell 1917. Cf., e. g., Fitch /Nelson 2009 and Kaplan 1989, 496. Cf. Kaplan 1975. See below and cf. section 4.6.

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here are meant to allow for relativization to some parameter (e. g., to possible worlds or moments in time 429). The principles of compositionality explain the productivity of language users, i. e., their astonishing ability to understand a vast number of expressions. On another level, the principles of compositionality allow a theory of meaning to be systematic; some such principles are a precondition for moving from intuitive semantics to theoretical semantics. Their formulation here also illustrates the close connection between semantic theory and logic, because the semantic form that is mentioned in both principles is nothing else than logical form, 430 and the logical words (of a natural language) or the logical vocabulary (of a formal language) are the main means for composing complex expressions from simpler ones. * What distinguishes sense from extension? Frege famously argued for the theoretical need of positing sense by pointing out that a true sentence of the form ‘a = b’ usually is much more informative than the corresponding sentence of the form ‘a = a’, despite there being no difference on the level of extensions. 431 This argument from cognitive significance (Erkenntniswertargument), however, establishes only a necessary condition for the equivalence of sense – sense-equivalent expressions cannot differ in cognitive value –, which we can put alongside the necessary condition we already know from the principle (Sense Determines Extension), which entails that sense-equivalent expressions cannot differ in extension. But the notion of sense itself remains elusive. What we can do is point out that, when we depict language and its users as opposed to the world, extension is more on the world side and sense is more on the side of language and its users, 432 so that extension might appear to be more robust than sense. While extensions are just those parts of the world that correspond to some expressions of a language, their sense concerns how these parts are given to the language users by these expressions. Related to the important semantic distinction between extension and sense is the metaphysical distinction between objects and concepts, which in this form is also due to Frege. He transfers a characterization that prima facie describes singular terms and predicates to objects and concepts when he says that while objects are selfsufficient or saturated, concepts are unsaturated. 433 This figurative way of speaking maybe is not overly informative, but it clearly entails a claim that will turn out to be important for the present proposal: Concepts belong to a different kind of items than objects; and therefore no concept is an object. 434

429 430 431 432 433 434

Cf. sections 4.5 and 4.7. Cf. section 3.1. Frege 2002c[1892]. Cf. the table in section 4.3. Frege 2002d[1892]; cf. Künne 2010, 180ff., especially 191f. A remark on terminology: We follow Kit Fine in using ‘item’ as an umbrella term for both concepts and objects (Fine 2008[2002], 1ff.). For an investigation that countenances items that are not objects,

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It has to be noted that this is by no means self-evident. According to the present framework, a concept is what is expressed by a predicate. And another common candidate for what is expressed by a predicate is a property, and properties as standardly construed are abstract, but nonetheless objects. 435 However, what distinguishes a Fregean concept from an object is not that it is not concrete but that it is not saturated. In fact, the claim that no concept is an object is controversial. It is controversial not only in the sense that many people are sure that it is false, but in the more serious sense that some people will even deny that they can understand it, because (as they will say) to be an object is nothing more than to be something (‘some thing!’, they will exclaim) that can be talked about, and claiming of concepts that they are not objects is talking about them. 436 For the moment I have not much to say against this consideration, but please be patient, because later I will try to make the claim more plausible. 437 For now, we will turn back from metaphysics to semantic theory. First we will characterize theoretical semantics in general (in section 4.3), then we will describe three important semantic theories (in sections 4.4 through 4.6), and we will conclude the chapter by having a look at how the Fregean principles presented in the present section fare in these semantic theories (in section 4.8).

4.3 Theoretical semantics We already mentioned theoretical semantics in chapter 3 when we contrasted the theoretical semantics of some language with its intuitive semantics, i. e., with its translation into the language we use. 438 Now we want to discuss in some detail how the Fregean framework of sense and extension can be realized in concrete semantic theories, thereby delineating some theoretical options and fixing some terminology. The first thing to do when we want to give the theoretical semantics of a particular natural language is to map it (or more likely, to map a fragment of it) onto a formal language; this operation is often called ‘formalization’. In practice, formalization can be difficult. But in this study we will be concerned only with simple natural language sentences, the formal counterparts of which can easily be found (e. g., by mapping proper names on constants and natural language predicates on predicate letters, 439 thus formalizing the English sentence ‘Socrates is wise’ as the sentence ‘F(a)’ of classical logic).

435 436 437 438 439

the older umbrella term ‘entity’ (Carnap 1956[1947], 22f.) has been spoiled by Quine’s slogan “no entity without identity”, which presumes that entities are objects. Cf., e. g., Künne 2007, especially 15f. and 340ff. For a typical example, cf. Priest 2014, 5ff. Cf. chapter 9 and especially section 9.6. Cf. sections 3.1 and 3.3. We will also often map a natural language predicate to an open formula, but in the particular cases this will also be a straightforward matter.

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After formalization, we can go on to give the theoretical semantics of the formal language in question. We need to keep apart several aspects; therefore, it will be helpful to start with an overview. Theoretical Semantics syntacticist language

formal interpretation

formal semantic values

formal ontology

de-semantification

material interpretation

semantic materialization

ontological materialization

semanticist language

re-interpretation

material semantic values

material ontology

sense

extension

intuitive semantics language

meaning world

In this diagram, some important strands of our study are tied together (or rather, get disentangled). These are the dichotomy of language and world, the distinction between sense and extension within semantics, and the duality of the formal level and the material level (corresponding to what we previously would have called the informal level), which includes both the alternative of syntacticism and semanticism about a formal language and the corresponding alternative of a formal and a material understanding of the ontology. In itself, the diagram is meant to be neutral with regard to the alternative of the formal and the material level. But it certainly allows for partial readings. The single-minded syntacticist will look only at the upper row which represents the formal level, and will move in her account from the syntacticist language by formal interpretation to the formal semantic values, and thence to the extension in the formal ontology. In contrast, the single-minded semanticist will look only at the lower row which represents the material level, and will move in her account from the semanticist language by re-interpretation to the material semantic values, and thence to the extension in the material ontology. 440 It is quite obvious that we ourselves sympathize with semanticism, and we will endorse it when we propose a solution to the Liar paradox. 441 We do not want our semanticism to be single-minded, though, and we can see here how that is possible: In accordance with our semanticist orientation, we will prefer both the start and endpoint of a semantic theory to be on the material level, connecting the semanticist language to the material ontology. But nothing stops us from taking a detour through the formal level, thus gaining access to the insights of purely formal theories. Let us look at this in more detail. 440

441

Here it becomes obvious that the present section is a companion to section 3.1, where we introduced the alternative between syntacticism and semanticism. Cf. section 8.1.

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The language we are concerned with is a formal language that is understood as meaningful, i. e., it is a semanticist language. 442 The intuitive semantics of that language (i. e., its translation into the language we use) will give us its meaning, but in an unsystematic and pre-theoretical manner. On our way to theoretical semantics, we have decided to follow Frege in splitting up meaning into sense and extension. In a semantic theory, the sense of an expression is explicated as a theoretical object that we will call its semantic value, 443 and the extensions of subsentential expressions are specified as parts of a background ontology (as objects and collections of objects, respectively). 444 As long as we stay on the material level (the lower row in the diagram), the picture is simple enough: An expression of a semanticist language has a material semantic value that picks out an extension, which is part of the real world, here called the material ontology. 445 The picture gets more complicated when we, to reach higher levels of systematicity and generality, include a formal level (the upper level in the diagram). This opens up a second, third, and fourth route from the expressions of our semanticist language to the objects of the corresponding material ontology. All three start with the step of de-semantification, i. e. the process of abstracting away from the meanings of the formal language we are concerned with, which brings us to the syntacticist skeleton of that formal language. 446 On the second route, the next step we will take is the material interpretation of the syntacticist formal language, which attaches a material semantic value to each of its expressions. 447 The final step of the second route is identical to the final step of the first route; it leads from the material semantic value to the extension it determines within the material ontology. The third route splits up the second step of the second route, material interpretation, into two steps: By formal interpretation, a formal semantic value is attached to the

442

443

444

445

446 447

For the distinction between a semanticist and a syntacticist understanding of a formal language, cf. section 3.1. Here we deviate from a terminological tradition according to which the semantic value of an expression is what we call its extension. As we do not construe truth values as objects (cf. section 4.2), we have to refrain from explicating the extension of a sentence – its being true or being false – in an ontological manner, i. e., as an object. This is so at least within the material semantics – the common practice of formal semantics of explicating truth values as formal objects (e. g., as the natural numbers zero and one) is unproblematic because formal semantics as such does not carry the same kind of metaphysical commitment with it. In certain applications, we will want to vary the semantic values of certain (non-logical) expressions. In the case of a semanticist language, this amounts to a re-interpretation, because each expression is already understood as meaningful. The term “de-semantification” is due to Krämer 2003. Cf. also Dutilh Novaes 2012, especially 198ff. A terminological remark: Here, the term ‘interpretation’ is not used in the hermeneutical sense of making a (hidden) meaning explicit, which presupposes that what is interpreted is meaningful to begin with, but roughly in the sense of model theory or contemporary mathematical logic, where what is interpreted is an object devoid of all meaning, namely an expression of a syntacticist language. However, as a look at the other relatum of the relation of interpretation shows, our notion of interpretation is more general than the model theoretic one, because we understand interpretation as a relation between a syntacticist expression and a semantic value. The common notion of interpretation takes it to be a relation between a syntacticist expression and an extension; hence, it can be construed as the limit case of our more general notion of interpretation where semantic values are identified with extensions.

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syntacticist expression, which then is semantically materialized by correlating it with a material semantic value. 448 But (re-)materialization can also be postponed until we have reached the ontological level (represented by the rightmost column in the diagram), which is what happens when we take the fourth route. It leads from the semanticist to the syntacticist language, then via formal interpretation to a formal semantic value, which in turn determines an extension within a formal ontology (e. g., of sets); and in a final step, this (formal) object or collection of (formal) objects is ontologically materialized, whereby an extension within our material ontology is picked out. Ontological materialization correlates each object of the formal ontology with an object of the material ontology (e. g., the sets that make up the formal domain of discourse are correlated one-one with the physical objects that make up the material domain of discourse). Which of the four routes is the best way for us? This will depend on our aim when doing theoretical semantics. But if all goes well, the above diagram commutes, and each route will lead to the same result at the endpoint it shares with the other routes. The result at that endpoint – be it the material semantic value or the extension in the material ontology it determines – can be checked against our intuitions, which will provide a measure of the adequacy of our semantic theory. The check can be conducted in two entirely different ways, depending on whether our overall approach to semantics is universalist or model theoretic. Recall that according to the model theoretic approach we can do theoretical semantics by stepping outside language and we specify semantic values by describing how the expressions of the language relate to the world, whereas according to the universalist approach we cannot step outside language when we do semantics because it is the universal medium of all our intellectual activity. 449 It might appear that the neutral overview of theoretical semantics given by the above diagram is most easily understood as following the model theoretic approach to semantics, because it readily suggests a picture according to which we take a God’s eye perspective on language and its relation to the world. (Note, however, that this model theoretic understanding of theoretical semantics is more complex than classical model theory, 450 because it has an additional layer of semantic values between expressions and extensions.) But the above overview of theoretical semantics can equally well be understood in a universalist way (and that is what we ultimately will prefer). It all depends on how we construe the intuitive check of the results which in the end of the day is the only way of justifying any particular claim of our semantic theory. When we check a prediction of our theory about how language relates to the world against our intuition of how language relates to the world, we follow the model theoretic approach. When we check a prediction of our theory of how some expression of the semanticist object language translates into the language we use against our intuitive 448

449

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In the form of a slogan: While the process of de-semantification consists in subtracting meaning from a meaningful expression, the process of formal interpretation and semantic materialization consists in adding meaning to a meaningless expression. Cf. section 3.1, where we introduced the distinction between a model theoretic and a universalist orientation in semantics. Cf. Hintikka 1997, and Müller 2002, 29ff. Cf. section 4.4.

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expectations about this translation, we follow the universalist approach. And it is the latter, universalist approach that is in line with our semanticist sympathies, and we will take it in our proposal for a solution to the Liar paradox. * Another word on the importance of the distinction between the material and formal level and the plurality of options for doing theoretical semantics opened up by it. On the material level, there are the semanticist language, material semantic values, and the material ontology. These are, respectively, the formal language understood as meaningful, the theoretical entities that explicate its meaning, and the part of the world that the meaningful language really is about. (The material ontology need not be concrete, as is illustrated by the case of the formal language of arithmetic, which is about the natural numbers and hence about abstract objects.) On the formal level, there are the syntacticist language, the formal semantic values, and the formal ontology. On the side of language, we have abstracted away from its meaning, and on the side of the world, we have exchanged the subject matter of the meaningful language for an abstract or formal substitute. Conflating the material and the formal level of theoretical semantics can lead to ambiguities. Many a theoretical semanticist will describe a route from the expressions of a language via their semantic values to the ontology of the language, and at each one of the three stages is in danger of being unclear whether it is understood in a formal or a material way. Contemporary mathematical logic, to give an important example, will officially take the route that stays on the upper, formal level but lapse sometimes into a material understanding. 451 Another reason to be clear and explicit about the distinction between the material and the formal level of theoretical semantics is that metaphysical commitment is carried only by those claims of theoretical semantics that pertain to its material level. This point will become important for our own investigation when we study the interaction of semantics and metaphysics (in part III). 452

4.4 Extensional semantics (a. k. a. classical model theory) In this and the next two sections, we will sketch three important ways of giving the theoretical semantics of the formal language of classical logic (or its modal extensions). As this formal language can be used to specify the logical form of important fragments of natural language, this will provide us with some important tools that we will employ to understand the semantics of self-referential sentences. The three semantic theories – extensional semantics a. k. a. classical model theory (this section), intensional semantics a. k. a. possible worlds semantics (sections 4.5 and 4.6) and two-dimensional semantics (section 4.6) – will also serve as illustrations for some basic reductionist attitudes that are common within contemporary semantics and

451 452

Cf. Shapiro 2005. Cf. chapters 9 through 12.

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metaphysics (section 4.8) and the rejection of which will later in this study also be relevant for our proposal for a solution to the Liar paradox. 453 * The basic idea of extensional semantics is that the semantic value of an expression is nothing else than its extension (hence the name). As it identifies the meaning of any expression with the part of the world it represents, it can be characterized as a very worldly way of doing semantics. Though too simple for an adequate description of natural language, 454 extensional semantics does surprisingly well when it comes to languages of mathematics. 455 A specification of the extensional semantics of the language of classical logic is reminiscent of an introductory logic course. We will therefore not aim to be comprehensive, but look only at some exemplary clauses, sticking to monadic predicates and only some of the sentential connectives, and highlight come conceptual issues. The basic building blocks of the language that have an extension are constants and predicate letters, and in some cases sentence letters. The extension of a sentence letter, as well as the extension of any complex sentence, is one of two (formal) objects, the True and the False, which we also denote as ‘T’ and ‘F’, and that are meant to encode the concepts of being true and of being false, respectively. The extensional semantics of the quantified part of classical logic is based on the domain of discourse, i. e., the collection of objects that the language is about. The extension of a constant is an object from the domain of discourse that it refers to. The extension of a (monadic) predicate letter is the collection of those objects from the domain of discourse that satisfy it. The extension of the basic building blocks determines the extension of every well-formed expression according to criteria like the following: For a sentence ϕ, the extension of the sentence p¬ϕq is the True if and only if the extension of ϕ is the False. For sentences ϕ and ψ, the extension of the sentence pϕ ∧ ψq is the True if and only if the extension of ϕ is the True and the extension of ψ is the True. For a constant α and a monadic predicate letter Φ, the extension of the atomic sentence pΦ(α)q is the True if and only if the object that is the extension of α is among the collection that is the extension of Φ. 456

453 454 455

456

Cf. part III and in particular section 9.7. Pace Donald Davidson. For syntacticist languages (and presupposing a construal of collections as sets; cf. section 1.7), extensional semantics is nothing else but classical model theory. Recall from section 1.7 that we use ‘collection’ and ‘is among’ as umbrella terms to cover both talk about a singular object being among a plurality and about an object being an element of a set. Those who feel more comfortable with set theoretical parlance can read the right hand side of the above condition as ‘the object JαK is an element of the set JΦK’, or ‘JαK ∈ JΦK’.

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The last criterion is the most interesting: It shows how the extensions of (monadic) atomic sentences are determined by the extensions of constants and (monadic) primitive predicates. There is a common notation that abbreviates the descriptive functor ‘the extension of . . .’ with double square brackets, so that phEiq abbreviates pthe extension of the expression Eq. For a sentence ϕ, pthe extension of the sentence ϕ is the Trueq can thus be abbreviated as pJϕK = Tq. All in all, this gives us a more perspicuous way of making the same statements: For a sentence ϕ, Jp¬ϕqK = T if and only if JϕK = F.

For sentences ϕ and ψ, Jpϕ ∧ ψqK = T if and only if JϕK = T and JψK = T.

For a constant α and a monadic predicate letter Φ, JpΦ(α)qK = T if and only if JαK is among JΦK.

The present sketch of extensional semantics allows to connect back in two ways to our presentation of the Liar paradox in the preceding part: Firstly, were we to put together all of the criteria, we would get something much like a Tarskian recursive definition of truth for the language of classical logic. 457 Unsurprisingly, this recursive definition would satisfy the Tarskian truth schema, 458 if it was complemented by a natural translation of the object language into the meta-language. 459 Secondly, the extensional semantics of classical logic guarantees the (Exhaustiveness) and (Exclusiveness) of truth and falsity. In our presentation, this is implicit already in how we talk about the truth and falsity of sentences. By using the descriptive functor ‘the extension of the sentence . . .’ (or formally, by applying square brackets to a sentence) and laying down that the True and the False are the truth values, we imply that (Exhaustiveness) must hold. 460 By saying that the True and the False are two truth values, so that they are distinct, we imply in connection with the use of the descriptive functor (or the square brackets) that (Exclusiveness) must hold, too. Observe that the two principles, although they do hold, are thus not stated in a particularly explicit way, and that there is no natural way of describing a variation of the logic where they are no longer required. Thus we encounter an interesting difference between the alternative ways of attributing a certain extension to a sentence, that is important in connection with the Liar paradox. For compare the following alternative formulations: ‘The extension of the sentence s is the True (the False).’ ‘The sentence s is true (is false).’ While any formulation of some semantic criteria in the first manner of speaking already entails the principles of (Exhaustiveness) and (Exclusiveness), this is not so 457

458 459 460

Tarski himself had a more complicated account of what it is for an open formula to be satisfied by some objects (Tarski 1956[1935], 209ff.) than the one that has become standard and that we have sketched here (e. g., Burdman Feferman /Feferman 2004, 116ff.). Cf. section 3.2. E. g., the object language sentence pΦ(α)q is translated into the meta-language as pJαK is among JΦKq. This is so under the implicit presupposition that the extension of a descriptive functor is given by a function that is total, i. e., has a value for every argument.

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in the case of the second manner of speaking. Because of the use of predicates, the schematic sentences ‘the sencence s is neither true nor false’ and ‘the sencence s is both true and false’ are both grammatical, and prima facie they are not precluded from having true instances. This need not stop us from formulating extensional semantics (or later, intensional semantics and two-dimensional semantics) in the second manner, with predicates of truth and falsity. But in contrast to the first manner, we will need to add criteria for falsity when we want to give a full account, 461 and we will need to endorse the two principles explicitly 462 to ensure that all sentences are evaluated in the same classical way as by extensional semantics formulated in the first manner of speaking. Typically, the extensional semantics of classical logic are presented in the first manner of speaking; but we have used the second manner of speaking in our informal presentation of the Liar paradox (in chapter 2). This was not due to our interest in informal logic – in fact, the formulation of extensional semantics with predicates of truth and falsity is as amenable to formalization as the formulation in terms of truth values. 463 Rather, it was due to our wish to be able to vary the logic in a natural way, which in the formulation with predicates of truth and falsity is easily done by dropping the principle of (Exhaustiveness) or the principle of (Exclusiveness). The first occurs in gap approaches to the Liar paradox, the second in dialetheist approaches. So when you are thinking about the Liar paradox, you are well advised to be able to characterize these approaches, even if they are only alternatives that you do not want to adopt. 464 The formulation in terms of predicates is of course also in keeping with our preferred material understanding of theoretical semantics, according to which truth and falsity are concepts and not objects. 465 Therefore, we will in the following often eschew talk about truth values in favor of talk using predicates of truth and falsity. 461

A recursive definition of truth and falsity will also include clauses like the following: For a constant α and a monadic predicate letter Φ, the atomic sentence pΦ(α)q is false if and only if the object that is the extension of α is not among the collection that is the extension of Φ.

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For sentences ϕ and ψ, the sentence pϕ ∧ ψq is false if and only if ϕ is false or ψ is false. Alternatively, we could adopt certain requirements for predicates and lay down certain criteria for the truth and falsity of atomic sentences and the go on to show that the principles of (Exhaustiveness) and (Exclusiveness) hold for all sentences of the language. Michael Dunn has developed an alternative to the standard approach in theoretical semantics of both classical and non-classical logics, by giving up the implicit restriction that the relation between sentences and truth values is functional, i. e., right-total and left-unique (Dunn 1976). Thus the basic locutions are no longer (in our notation) ‘JϕK = T’ and ‘JϕK = F’, but ‘R(ϕ, T)’ and ‘R(ϕ, F)’, which allows for a formal representation of the four different notions of being neither true nor false, being true only, being false only, and being both true and false. Cf. Priest 2008, 142ff., and cf. Priest /Tanaka /Weber 2015, section 3.6, for a brief overview. I would like to note that although this solves the problem on a technical level, it still construes truth and falsity as objects, T and F. Why not use primitive predicates ‘True(ϕ)’ and ‘False(ϕ)’ in the place of the complex predicates ‘R(ϕ, T)’ and ‘R(ϕ, F)’? Thus truth and falsity would be construed as concepts. In fact, our own approach will at least in part be a gap approach; cf. chapters 10 and 11, as well as section 13.2, subsection “Which logic?”. Cf. section 4.2.

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4.5 Intensional semantics (a. k. a. possible world semantics) In order to extend classical logic to modal logic, we first extend its language by adding sentential operators ‘’ and ‘♦’ which express necessity and possibility. And in order to move from the extensional semantics that was adequate for classical logic to a semantic theory that is adequate for a language about modal matters, we enhance the framework of theoretical semantics by a collection of (formal) objects that will play the role of a parameter, called possible worlds, and a relation among these possible worlds, accessibility. 466 For transferring the notion of a domain of discourse to the framework of possible worlds there are two theoretical options: We can either say that there is a single domain of discourse (the constant domain approach) or that there is a domain of discourse for each possible world (the variable domain approach). 467 The basic idea for the theory of extensions for the language of modal logic is to say that an expression has an extension only in relation to a possible world. The paradigmatic clause is that, for a sentence ϕ, the sentence pϕq is true with regard to some possible world w if and only if the sentence ϕ is true with regard to every possible world that is accessible from w. The idea of relativizing extensions to possible worlds motivates intensional semantics, i. e., the semantic theory according to which the semantic value of an expression is a function that maps each possible world to the extension the expression has with regard to that world, also called the expression’s intension. 468 The semantic value of a sentence letter is a function that maps each possible world to the truth value the sentence letter has with regard to that world; the semantic value of a constant is a function that maps each possible world to the object from the domain of discourse (of that possible world) which that constant refers to with regard to that possible world; and the semantic value of a predicate letter is a function that maps each possible world to the collection of objects from the domain of discourse (of that possible world) that is such that each object that is among that collection satisfies that predicate letter with regard to that possible world. The semantic values of composite expressions are determined by criteria like the following: If ϕ is a sentence, then the sentence p¬ϕq is true with regard to a possible world w if and only if ϕ is false with regard to the possible world w.

466

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The framework of intensional semantics is connected to the names of Rudolf Carnap, Saul Kripke, and David Lewis, among others. There are many good introductions to modal logic and possible worlds semantics, e. g., Hughes /Cresswell 1996, Fitting /Mendelsohn 1998, and Humberstone 2016. Hughes /Cresswell 1996, 235ff. and especially Fitting /Mendelsohn 1998. We will look at the distinction between constant and variable domain approaches in more detail when we apply the framework of intensional logic in sections 9.4 and 9.12. Carnap 1956[1947], 23ff.; cf. Soames 2010, 50ff. – A terminological remark: The term ‘intension’ is often used for everything that pertains to meaning or sense and thus to everything that is not (merely) extensional. In contrast, we use it here as a technical term for functions from possible words to extensions (that are meant to explicate intensionality in the wider sense).

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If ϕ and ψ are sentences, then the sentence pϕ ∧ ψq is true with regard to a possible world w if and only if ϕ is true with regard to w and ψ is true with regard to the possible world w. Thus the semantic values of the logical operators like ‘¬’ and ‘∧’ correspond to operations on collections of possible worlds – in set theoretical terms, negation corresponds to a function that maps each set on its complement and conjunction corresponds to a function that maps each pair of sets on their intersection. – More complex operations on collections of possible worlds correspond to the clauses for the sentential operators of necessity and possibility; in fact, these are the only criteria of intensional semantics that make use of the relation of accessibility: If ϕ is a sentence, then the sentence pϕq is true with regard to a possible world w if and only if ϕ is true with regard to every possible world that is accessible from the possible world w. If ϕ is a sentence, then the sentence p♦ϕq is true with regard to a possible world w if and only if ϕ is true with regard to some possible world that is accessible from the possible world w. We cast only a brief look at the quantified part of modal logic, by giving the criterion for the truth of an atomic sentence: For a constant α and a monadic predicate letter Φ, the atomic sentence pΦ(α)q is true with regard to a possible world w if and only if the object that is the extension of the constant α with regard to the possible world w is among the collection that is the extension of the predicate Φ with regard to the possible world w. (Here, the formal notation is more perspicuous: JpΦ(α)qKw = T if and only if JαKw is among JΦKw.)

The clauses for predicates with more than one place and for the quantifiers are relativized to possible worlds in a similar way.

4.6 Rigidity and direct reference in intensional semantics In the present framework which relativizes extensions to possible worlds, we can define a characteristic of some expressions that will turn out to be important when the framework is applied to natural language semantics. 469 (Rigidity) An expression is rigid if and only if it has the same extension with regard to every possible world. 470 469

470

The issues that are the topic of this section are connected to the names of Saul Kripke and David Kaplan, respectively. The seminal texts are Kripke 1980[1972] and Kaplan 1989. For the notion of rigidity, cf. Kripke 1980[1972], especially 84. Cf. Soames 2010, 78ff. In some uses of the framework of possible worlds, some expressions are allowed to have no extension with regard to some worlds. In that case it is advisable to adjust the above definition of rigidity by requiring of a rigid expression’s extension that it is the same with regard to every possible world with

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To give an example: The semantic values assigned to constants typically are constant functions; therefore constants typically are rigid. Predicates can also be rigid; e. g., given that we take the constant domain approach, the open formula that expresses self-identity (e. g., ‘x = x’), is satisfied by the same collection of objects (the collection of all objects) with regard to every possible world. An important application of the notion of rigidity to natural language will be the observation that while proper names are rigid (e. g., ‘Barack Obama’), most descriptions are not (e. g., ‘the president of the U.S. in the year 2016’). 471 In a semantic theory like the present one that fits into a Fregean framework because it conceptually distinguishes between semantic value (hitherto explicated as intension) and extension, it is possible to contrast the rigidity of some singular terms with another characteristic that some expressions are sometimes said to have: (Direct Reference) A singular term refers directly if and only if the object it refers to is its semantic value – or, more precisely, if and only if the object it refers to is construed as a direct constituent of its semantic value. 472 Why do we need the notion of direct reference besides that of rigidity? The need for this conceptual distinction can be made plausible by drawing attention to an intuitive difference in how – or why – an expression can be rigid. Let us contrast the term that is used traditionally to refer to the empty set, ‘∅’, with the description ‘the smallest set that is a subset of every set of concrete objects’ (or, ‘the smallest set of concrete objects’ for short). Given some standard modalized set theory, both the term and the description will refer to the empty set with regard to every possible world. But intuitively, there is a difference in how their rigidity comes about. Speaking figuratively, when we evaluate the description, we take a look at every possible world, and with regard to each one find that there the smallest set of concrete objects is the empty set (although in general, there are different sets of concrete objects in different possible worlds). But when we evaluate the term ‘∅’, we know that it refers to the empty set before we take a look at the extension it has with respect to some possible world. Whereas the description is only de facto rigid, the name is rigid de jure: Once we have understood a term like ‘∅’, we know that it must refer rigidly, while in the case of a description, we may need further information about the particular subject matter. 473 Now, rigidity de jure can be explained on the level of semantic theory with recourse to the notion of direct reference, by saying that an expression is rigid de

471 472 473

regard to which that expression has an extension. – It is often said that a rigid expression has the same extension with regard to every possible world with regard to which that expression’s extension exists (e. g., Kaplan 1989, 492), but, as Fitting and Mendelsohn argue convincingly, we have to distinguish between the question whether an expression has an extension relative to some parameter and the question whether the extension an expression has relative to some parameter exists relative to that parameter. E. g., the name ‘Socrates’ in the year 2016 refers to an individual that does not exist in 2016. Cf. Fitting /Mendelsohn 1998, 230ff. We will return to this matter in chapter 9. Cf. section 5.2. Cf., e. g., Kaplan 1989, 492ff. Here we anticipate part of what we will say about names and descriptions in section 5.2.

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jure if and only if it refers directly and hence is rigid, and that an expression is de facto rigid if and only if it is rigid but refers indirectly. However direct reference is explicated in a theory, it is clear that whereas direct referentiality always entails rigidity, the intensional semantics of some language can be such that there are rigid expressions that refer indirectly (i. e., which are de facto rigid). Typically, these are descriptions which (to speak metaphorically) only by accident turn out to refer to the same object with respect to every possible world, e. g., again, the description ‘the smallest set of concrete objects’ or, to give a clearer (but less perspicuous) case, ‘the object which is such that ((Barack Obama is president of the U.S. in the year 2016 and the object is the smallest prime number) or (Barack Obama is not president of the U.S. in the year 2016 and the object is the successor of one))’. 474 * How can the conceptual difference between the rigidity and the direct reference of singular terms be incorporated into a theory like intensional semantics? According to intensional semantics as we have described them in the preceding section, 475 every singular term is assigned a function from possible worlds to objects as its semantic value and therefore no singular term refers directly. But it is easy to formulate a variant of intensional semantics according to which the semantic values of some expressions are functions from possible worlds to extensions, while the semantic values of other expressions just are their extensions. 476 In this picture, the difference between rigidity and direct reference would be that between an expression’s semantic value being a constant function and such an expression’s semantic value being (or incorporating) the value of that constant function. But this picture is too crude. Although it is common to construe direct reference as the identification of sense with extension, 477 we could do with some more technical sophistication here. This can be shown as follows. If the semantic value of a directly referring expression was to be its extension (and nothing else) and the semantic value of an indirectly referring expression was to be its intension (and nothing else), then an indirectly referring expression and a second expression that refers directly to the intension of the first expression would have the same semantic value and would hence be construed as synonymous! E. g., let ‘Franny’ be a name of the function that maps each possible world to the human being who with regard to that world is president of the U.S. in the year 2016, and note that this name refers directly 478 to an intension. Then the semantic theory would entail that the name ‘Franny’ is synonymous with the description ‘the president of the U.S. in the year 2016’, because the description is also assigned that function as its semantic

474 475 476

477 478

For a similar example, cf. Kaplan 1989, 494f. Cf. section 4.5. This formulation of intensional semantics can be characterized as a hybrid between the first and second stage of the development of theoretical semantics as we present it here, because it entails that pure intensional semantics holds for some expressions, while other expressions are governed by extensional semantics. Cf., e. g., Kaplan 1989, 493; Soames 2010, 97. Here we anticipate the semantics of proper names; cf. section 5.2.

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value. 479 Furthermore, the semantic theory would entail that the sentence ‘Franny is the president of the U.S. in the year 2016’ is true, which it is not, because, being a function, Franny is not even eligible for presidency. This problem can be solved on the technical level by modeling a semantic value as a pair of, firstly, a (formal) object that encodes the information whether the reference is direct or indirect and secondly, either the extension or the intension of the expression. More specifically, the semantic value of a directly referential expression would be the pair of an object that stands for direct referentiality, D, and the extension of that expression; and the semantic value of an indirectly referential expression would be the pair of an object that stands for indirect referentiality, I, and the intension of that expression. Thus the semantic value of the name ‘Barack Obama’ would be the pair (D, Barack Obama), the semantic value of the description ‘the president of the U.S. in the year 2016’ would be the pair (I, Franny), and the semantic value of the name ‘Franny’ would be the pair (D, Franny). Because D 6= I, we also have (D, Franny) 6= (I, Franny). Thus, ‘Franny’ is no longer predicted to be synonymous with ‘the president of the U.S. in the year 2016’. 480 The need to solve the difficulty with Franny in this or a similar way is the reason why, strictly speaking, the sense of a directly referential expression should not be construed as identical to its extension, but merely as incorporating its extension as a direct constituent. 481 Here, the word ‘direct’ is important. As a function is commonly construed as a set theoretical construction from the objects of its domain and range (a set of ordered pairs), even an intension, the semantic value of an indirectly referential expression, strictly speaking incorporates the extension which that expression actually has (along with all its merely possible extensions). However, while the distinction between being a primary and being a non-primary constituent can serve to explicate the difference between directly and indirectly referential expressions, it does not capture our main intuition of just how different a directly referential expression is from an indirectly referential one. In terms of sense and extension, we can say that the important question concerns the direction of grounding: While the extension of an indirectly referential expression is grounded in its sense, it is the other way around in the case of a directly referential expression, where sense is grounded in extension. Or, to apply this observation about grounding to the role that direct reference plays in intensional semantics, we can paraphrase a statement of Kaplan’s and say: 479

480

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A similar difficulty is described by Kaplan in a footnote to “Demonstratives” (Kaplan 1989, 496, fn. 23; where Kaplan’s “Plexy” corresponds to our Franny). Kaplan’s technical solution differs from ours only on the surface level (he uses a set theoretical operation where we use pairing with additional objects), but not in spirit. An important difference between Kaplan’s treatment of this problem and ours is that Kaplan ignores it in his official theory, according to which the sense of a directly referential expression is identical to its extension. This variant is not any longer a hybrid between extensional semantics and pure intensional semantics, because the semantic value of an expression is neither its extension nor its intension. However, both extensional semantics and pure intensional semantics are aufgehoben in this technically sophisticated variant of intensional semantics, because the semantic values assigned by those theories occur as part of the pairs that are the semantic values here. Cf. section 4.2.

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What is characteristic of a directly referential singular term is that the extension determines the semantic value rather than the semantic value, along with the possible world, determining the extension. 482 Usually, a claim about grounding – about what is so in virtue of what – can be motivated or made plausible by telling a story of stages, even where the temporal dimension strictly speaking does not play any role. We already told a story of stages when we used the example of the contrast between the term ‘∅’ and the description ‘the smallest set of concrete objects’ to explain why we need to distinguish between de jure rigidity and de facto rigidity. Now we can tell that story in a more general way and say: While for an indirectly referential expression sense precedes extension, for a directly referential expression sense is subsequent to extension. 483

4.7 Two-dimensional semantics When intensional semantics was turned on natural language, it was found to be unable to adequately describe all important phenomena that can occur when extensions are relative to a parameter. One such phenomenon is what we will call Kaplan’s puzzle: 484 What a sentence expresses corresponds to a non-contingent state of affairs if and only if the sentence has the same truth value with regard to every possible world, i. e., if and only if it has a rigid extension. In that case it is also the case that every possible utterance of that sentence has the same truth value. E. g., the sentence ‘Two is the smallest prime number’ is not only true with regard to every possible world, it is also the case that every particular utterance of that sentence is true. But – and here comes the puzzle – there are sentences that have the second characteristic without having the first. For example, the sentence ‘I am here now’ is true whenever it is uttered, but what accounts for the truth of each of its utterances is a contingent fact. E. g., the sentence ‘I am here now’ is true when Ann utters it while sitting at her desk at 7 p. m., but its truth then is due to a contingent matter, because Ann at 7 p. m. also might have been in the garden. 485 That is Kaplan’s puzzle, and intensional semantics has nothing illuminating to say about it. 486

482

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485 486

We give a paraphrase because Kaplan’s terminology is quite different: “What is characteristic of directly referential terms is that the designatum (referent) determines the propositional component rather than the propositional component, along with the circumstance, determining the designatum.” (Kaplan 1989, 497). It is important to be clear about the fact that here and in similar contexts, stories of stages and the corresponding claims of precedence and subsequence have the status of mere metaphors – especially as I will argue later that in some contexts, facts of grounding do entail facts of subsequence in a literal sense. Cf. chapter 11. Kaplan appears to be prolific with puzzles, because at least two other problems have been called “Kaplan’s puzzle”, one about the cardinality of possible worlds and the other within cognitive dynamics. Cf. Kaplan 1989, 508f. Manuel García-Carpintero and Josep Macià more generally speak of “modal illusions” which arise in the framework of intensional semantics; García-Carpintero /Macià 2006b, 1f.

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4.7.1 Two dimensions: Contexts of use and circumstances of evaluation Kaplan’s puzzle can be solved by two-dimensional semantics, more specifically by what David Chalmers calls the “contextual understanding” 487 of the two-dimensional framework, which, like the puzzle, is due to Kaplan. 488 Understood in this way, two-dimensional semantics is based on the idea that an expression’s extension is parameter-relative in a two-dimensional way (hence the name), where one dimension consists of circumstances of evaluation and the other dimension of contexts of use. 489 Circumstances of evaluation are already known to us from the role played by possible worlds in intensional semantics. Contexts of use can be said to explicate what is specific to the particular utterances of an expression that were mentioned above, 490 and each utterance of an expression can be explicated as an occurrence, i. e., as a pair that consists of the expression (type) and the context of its use. 491 These notions already allow to anticipate how the conflicting intuitions that are elicited by Kaplan’s puzzle can be resolved, namely by observing that the sentence ‘I am here now’, though each of its occurrences is true, is not true relative to every pair of a context of use and a circumstance of evaluation. Circumstances of evaluation and contexts of use usually are modeled in terms of possible worlds and centered possible worlds, respectively. A centered possible world is constituted by a possible world and a position of use within that world, which might for instance be made up of a speaker, a moment in time, and a place. The idea behind the notion of a position of use is that of including a parameter that can be varied when all contingent states of affairs are held fixed. When the framework is used to do two-dimensional semantics in the contextual understanding, a position of use models what is specific about a particular utterance of an expression, for instance who made it when and where. We should note some differences between the two dimensions. Obviously, a context differs from a circumstance by having a center. But there are applications of two-dimensional semantics where positions of use play no role at all, so that a context can be modeled in the same way as a circumstance, as a possible world. 492 Perhaps less obviously, there are in general more 493 circumstances than contexts 487

488

489 490 491 492 493

Chalmers 2006, 64f.; 129. For Chalmers it is important that the same two-dimensional formal framework can be employed not only for the purpose of doing semantics (the “contextual understanding” that is of interest to us; Chalmers 2006, 65ff.), but also to model the differences and relations between metaphysical and epistemic modality (the “epistemic understanding”; Chalmers 2006, 75ff.). The seminal text on two-dimensional semantics is Kaplan’s “Demonstratives” (Kaplan 1989, which is based on a lecture in 1977; cf. Kaplan 1989, 481). As always, there were forerunners, notably Kamp 1971. Further literature on two-dimensional semantics and its multiple applications: Soames 2005; Predelli 2005; Chalmers 2006 and other contributions to the anthology García-Carpintero/Macià 2006a; Soames 2010, 93–105; and Schroeter 2010. Cf. Kaplan 1989, 494 & 500ff. Cf. Kaplan 1989, 494. Cf. Kaplan 1989, 522 & 546. E. g., Davies 2006, 145. Here, ‘more’ is meant in the sense of the possible worlds that constitute contexts being a proper subcollection of all possible worlds, which need not entail that there is a difference in cardinality.

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because in general some possible worlds will contain no (possible) utterances of expressions of the language we are concerned with and thus do not constitute contexts of use – but we may for a particular application of two-dimensional semantics restrict the collection of possible worlds to those that constitute contexts of use. Neither of these two observations, however, is relevant to what for us will be the main difference between contexts of use and circumstances of evaluation. It is often described as that between considering a possible world as counterfactual, which yields a circumstance of evaluation, and considering a possible world as actual, which yields a context of use. 494 We will say more about this at the end of the present section (in subsection 4.7.6), after we have seen two variants of two-dimensional semantics as well as some examples for the two-dimensional semantics of natural language expressions.

4.7.2 Two-dimensional profiles: The matrix variant of two-dimensional semantics Let us see how two-dimensional semantics assigns semantic values and extensions to some important kinds of expressions. According to a basic variant of two-dimensional semantics, which will already provide us with a valuable tool 495 and which we will call the matrix variant, the semantic value of an expression is a function that maps a pair of a context of use and a circumstance of evaluation – modeled as a pair of a centered possible world and a possible world – to the extension which that expression has at that context and with respect to that circumstance. 496 Let us call that function the two-dimensional profile of the expression. 497 As in intensional semantics, the two-dimensional profile of a singular term is a function on objects, the two-dimensional intension of a predicate is a function on collections of objects, and the two-dimensional intension of a sentence is a function on truth values. To refer to the extension an expression ϕ has at the context of use (w, p) and with respect to the circumstance of evaluation v, we can also say more briefly pthe extension ϕ has relative to the point ((w, p), v)q; and we write pJϕK((w, p), v)q to abbreviate this descriptive functor. In these terms, we can give a clause about how the truth value that a sentence has relative to such points depends compositionally 494 495

496

497

Cf., e. g., Chalmers 2006, 59f. To anticipate for those readers already familiar with Kaplan’s theory and terminology: We present first the basic matrix variant of two-dimensional semantics which does not include the distinction between character and content, because this already allows to define helpful notions like rigidity and context-freeness, and then we will sketch a more sophisticated variant of two-dimensional semantics which includes counterparts of characters and contents. Today, it is not uncommon to present twodimensional semantics only with the matrix variant, and thus to ignore Kaplan’s distinction between character and content. Cf., e. g., Chalmers 2006, where Kaplan’s notion of character is discussed only under the heading of “other varieties of two-dimensionalism” (Chalmers 2006, 112 and Chalmers 2006, 115ff.). We adopt a terminological convention which binds different prepositions to the different dimensions: We use ‘at’ only in connection with contexts of use and ‘with respect to’ only in connection with circumstances of evaluation – in contrast to ‘with regard to’, which we use in the intensional semantics. We reserve the term ‘two-dimensional intension’ for another notion; cf. subsection 4.7.3.

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on the point-relative extensions of the semantic parts of that sentence. This clause is part of the two-dimensional theory of extensions, but it also tells us something about how the two-dimensional profile of a singular term and of a monadic predicate determine the two-dimensional profile of a sentence that is formed from them: For a constant α and a monadic predicate letter Φ, the sentence pΦ(α)q is true at a context of use (w, p) and with respect to a circumstance of evaluation v if and only if the object that is the extension of the constant α at (w, p) and with respect to v is among the collection that is the extension of the predicate Φ at (w, p) and with respect to v. More formally: JpΦ(α)qK((w, p), v) = T if and only if JαK((w, p), v) is among JΦK((w, p), v).

The semantics of more complex sentences, i. e., sentences involving non-monadic predicates and sentences built up from others with the help of the sentential connectives, are given in a similar way as in intensional semantics, only relativized two-dimensionally. The same goes for the clauses for the quantifiers – but there is some leeway for the two-dimensional treatment of the domain of discourse: We can either carry over the treatment of the domain of discourse from the possible worlds framework unchanged, i. e., either assign a single domain to the whole twodimensional framework or let the domain depend on the possible world, 498 and hence on circumstance and context alike (and thus be independent of the position of use). Or we can distinguish between a circumstantial domain and a contextual domain, thus opening up the possibility of there being a difference between the domain of discourse of a possible world taken as a circumstance of evaluation and the same possible world taken as a context of use (as well as the possibility that the position of use may be relevant for the contextual domain). 499 With a view to the two-dimensional semantics of non-extensional operators, suffice it to say that only the dimension of circumstances is relevant for the semantics of the operators of necessity and possibility. * One thing the matrix variant of two-dimensional semantics has in common with intensional semantics 500 is that it does not go so far beyond a mere theory of extensions. Each one of the two semantic theories does go beyond a theory of extensions because it assigns semantic values to expressions which are distinct from their extensions, but neither one goes much farther because the semantic value they assign to an expression just is the function from a collection of parameters to extensions that already figures in the respective theory of extensions. In the case of the matrix variant of two-dimensional semantics, this function is the two-dimensional profile which both governs how the expression’s extension is relative to the two dimensions and at the same time explicates its semantic value. We might quip that, despite relativizing extension to two dimensions, it construes meaning in a rather one-dimensional way. 498 499 500

Cf. section 4.5. This theoretical option will be important in our approach to the Liar paradox. Cf. chapter 10. Cf. section 4.5.

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4.7.3 Two-dimensional intensions: The two-stage variant of two-dimensional semantics When we set out to describe how two-dimensional semantics assigns two-dimensional profiles to expressions, we qualified the matrix variant as being basic. Now we turn to a more sophisticated variant of two-dimensional semantics that allows to distinguish between two kinds of meaning, character and content. Let us return to the sentence ‘I am now here’. It is true when Ann utters it at her desk at 7 p. m., and likewise it is true when Beth utters it at the same time in the garden. There is a sense in which these two utterances of ‘I am now here’ mean the same, i. e., roughly, 501 that, at the time of the utterance, the speaker of the utterance is at the place of the utterance. That is the kind of meaning that Kaplan calls “character”. But in another sense, the two utterances differ in meaning, the first one meaning that Ann is at her desk at 7 p. m. and the second one meaning that Beth is in the garden at 7 p. m. That is the kind of meaning that Kaplan calls “content”. 502 This distinction between character and content allows to advance our understanding of Kaplan’s puzzle. We see now that the fact that every occurrence of ‘I am now here’ is true concerns the level of character – how could the speaker of the utterance at the time of the utterance be somewhere else than at the place of the utterance? –, and that the question of contingency enters on the level of content – both Ann and Beth could have been somewhere else at the respective times. The apparent conflict between every occurrence of ‘I am now here’ being true and our clear intuition of contingency is resolved by distributing these two observations among the two levels of character and content. 503 While character attaches to the expression as a type, content attaches to occurrences of the expression, or, in other words, the content is determined by the expression as a type and the context of use. It is in fact the character that does the determining: To get from the sentence type ‘I am now here’ to the content it has when uttered at a particular context, we have to ask: Who is the speaker of the utterance? When is it uttered? And where? We reach the content by evaluating the character at a particular context. And when we look at one of the two particular 501

502

503

The paraphrase we (implicitly) use above, ‘At the time of this utterance, the speaker of this utterance is at the place of this utterance’, is by no means a translation of the example sentence ‘I am now here’. The clearest difference is one of subject matter: While the paraphrase sentence is about an utterance, the example sentence clearly is not. What is more, the dimension of indexicality brought into the paraphrase sentence by the phrase ‘this utterance’ is quite untypical (cf. section 5.3.), in contrast to the personal, temporal, and spatial indexicality of the paradigmatic indexicals ‘I’, ‘now’, and ‘here’. Cf. Kaplan 1989, 518ff. It is customary to follow Kaplan in explaining the notion of content by describing it as “what is said” by uttering an expression at a context (cf., e. g., Kaplan 1989, 500). But this characterization can be misleading, because there is an intuitive sense of ‘what is said’ that goes beyond the state of affairs that is associated with an utterance because it includes the perspective of the speaker. Think of Ann uttering ‘I am now here’ while sitting at her desk at 7 p. m., but having lost track of the time, and Carla recording ‘Ann is at her desk at 7 p. m.’ Ann might well object ‘that is not what I said’, because she believes that she was at her desk at 6 p. m. By thus incorporating perspective into what is said, we anticipate arguments for the irreducibility of indexicality; cf. subsection 5.2.8. Cf. Kaplan 1989, 508f. and 538f.

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contents determined in our example, that Ann is at her desk at 7 p. m., we see that when content is determined some important questions are still open, because now we can observe that while the state of affairs that correlates to this content does in fact obtain, this need not have been the case, because Ann could have been somewhere else at that time (e. g., in the garden). We reach the observation of contingency by evaluating the content with respect to different circumstances. We can tell a story of stages: An expression starts out with its character, which, in a first step, when it is evaluated at a context of use, determines a content. In a second step, content determines extension when it is evaluated with respect to a particular circumstance. To quote Kaplan: “Character: Contexts → Contents Content: Circumstances → Extensions” 504

In this story of stages, contexts precede circumstances; the two dimensions of twodimensional semantics have been associated with the two stages of assigning a semantic value to an expression. 505 Having introduced the distinction of character and content on an intuitive level, we will now look at a sketch of the corresponding two-stage variant of twodimensional semantics. Even more obviously than the matrix variant, the two-stage variant of two-dimensional semantics incorporates intensional semantics. Note that intensional semantics is enough to explicate the meaning of the sentences ‘Ann is at her desk at 7 p. m.’ and ‘Beth is in the garden at 7 p. m.’, because each semantic part of these sentences depends for its extension at most on the dimension of circumstances (while neither contains a semantic part that depends for its extension on the dimension of contexts). Intensional semantics thus provides us with the second stage, because the semantic values assigned by it are enough to explicate content. 506 As long as we ignore the issues of direct reference, a content can in fact be identified with an intension, 507 i. e., a function from circumstances to extensions. This is reflected in Kaplan’s second formula “Content: Circumstances → Extensions”. His first formula, “Character: Contexts → Contents”, provides us with the first stage. A character can be explicated by a function from contexts of use to the semantic values assigned by intensional semantics. Ignoring the issue of direct reference, a character thus can be explicated as a function from contexts to intensions, and thus as a function on functions.

504 505 506

507

Kaplan 1989, 506; notation altered. The talk of stages and precedence is, again, meant only in a metaphorical way; cf. section 4.6. We might want to say that intensional semantics is aufgehoben in the two-stage variant of twodimensional semantics in the same way extensional semantics was aufgehoben in intensional semantics. To anticipate, when direct reference is incorporated into two-dimensional semantics, content can be explicated by whatever semantic value the sophisticated variant of intensional semantics of section 4.6 assigns, which will be either of the form (D, extension) or of the form (I, intension). Cf. subsection 5.2.6.

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4.7.4 Defined notions A two-dimensional intension being a function on functions allows to define the notion of a two-dimensional profile, which was a primitive notion of the matrix variant of two-dimensional semantics, in the terms of two key notions of the twostage variant of two-dimensional semantics. 508 Thus the two-stage variant of twodimensional semantics also assigns two-dimensional profiles. 509 Often our interest will concern only the truth of an occurrence (or token) of a sentence, or more generally, the extension of an occurrence (or token) of an expression. As an occurrence (or token) for our purposes can be modeled as a pair of the expression as such or as a type and the context of use where it is located, we can say that what we are interested in is whether a sentence is true relative to 510 the context it actually occurs at, disregarding the question whether it would have been true had the facts been different; or, more generally, our interest will concern only the extension an expression has relative to its context of use. In the terms of the two-dimensional framework this means that we are interested in the truth of a sentence at a context and with respect to the circumstance that is the possible world of that context; for a context (v, p) and a circumstance w we can set v = w and say: (Truth of an Occurrence) If ϕ is a sentence, then ϕ is true relative to a context of use (w, p) if and only if ϕ is true at the context (w, p) and with respect to the circumstance w. Or, more generally: (Extension of an Occurrence) If ϕ is an expression, then ϕ has the extension x relative to a context of use (w, p) if and only if ϕ has the extension x at the context (w, p) and with respect to the circumstance w. These definitions of being true relative to a context and having a particular extension relative to a context amount to viewing a two-dimensional profile as a twodimensional matrix (a table-like array of the extensions) and restricting our attention to the diagonal of that matrix. 511 For this reason the resulting function from contexts to extensions is sometimes called the “diagonal intension” of the expres508

509

510

511

For a function f with domain A and range B and a function g with domain B and range C, the resultant function g ◦ f with domain A and range C is defined by g ◦ f (x) = def g(f(x)). Cf., e. g., Enderton 2001, 4f. In contrast to our description of the matrix variant, we say nothing here about how the two-stage variant of two-dimensional semantics is applied to various kinds of expressions (its application to singular terms will be described in section 5.2) and how the character and content of a composite expression is determined by the characters and contents of its semantic parts. This happens in a similar way as in intensional semantics and in the matrix variant of two-dimensional semantics. Note how we thus add another locution, ‘relative to’, to the locutions ‘at’ and ‘with respect to’ which we use for the two dimensions of contexts and circumstances, and ‘with regard to’ which we use in the intensional semantics. Cf. subsection 4.7.2. For our talk of a diagonal, we need to ignore the position of use (and disregard geometry).

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sion in question. 512 By moving from a particular two-dimensional intension to the corresponding diagonal intension we lose information; in fact we go back from two dimensions to a single parameter – but (to speak figuratively) not by cutting off one of the two dimensions, but by merging them. 513 In some theoretical contexts we may be interested in a further reduction of information, sticking to how things actually stand and wondering only how the extension of an expression depends on its position of use. For this purpose, we can define being true relative to a position of use and having a particular extension relative to a position of use, or, in other words, we can define local truth and local extension. The way things actually are is encoded in the possible worlds framework by designating one possible world as the actual world, which we denote as ‘wactual’. Now we need only employ the two-dimensional notions of truth and more generally of extension which are given by two-dimensional profiles and restrict the contexts and circumstances to the actual world by setting w = v = wactual. (Local Truth) If ϕ is a sentence, then ϕ is true relative to a position of use p if and only if ϕ is true at the context (wactual, p) and with respect to the circumstance wactual. (Local Extension) If ϕ is an expression, then ϕ has the extension x relative to a position of use p if and only if ϕ has the extension x at the context (wactual, p) and with respect to the circumstance wactual. The notion of local truth will play a role for sentential indexicality and when we analyze the sententially indexical Liar sentence ‘This sentence is false’. 514

4.7.5 Rigidity and context-freeness Both the matrix variant and the two-stage variant of two-dimensional semantics assign two-dimensional profiles to expressions, and two-dimensional profiles are all that matters for the theory of extensions that is entailed by what any variant of twodimensional semantics says about the semantic values of expressions. Therefore, there will be no differences between any two-stage variant and the corresponding matrix variant as far as the two-dimensional theory of extensions is concerned. This is welcome because it allows us to put the more complex aspects of the two-stage variant to the side for the time being 515 and concentrate on the two-dimensional theory of extensions. On the basis of the theory of extensions that is entailed both by the matrix variant and the two-stage variant of two-dimensional semantics, we can 512

513

514 515

Cf. Schroeter 2010, section 1.1.3; Laura Schroeter suggests that this terminology goes back to Robert Stalnaker. When working with the two-stage variant of two-dimensional semantics, it is of course possible to define diagonal intensions directly, skipping the intermediate step of defining two-dimensional profiles. Cf. Kaplan 1989, 522. Cf. sections 5.3, 11.11, and 12.6. The two-stage variant of two-dimensional semantics and its notion of a two-dimensional intension will be taken up again in section 5.2.

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define two important pairs of notions that will allow to characterize some kinds of expression, namely rigidity as opposed to flaccidness and context-freeness as opposed to context-dependence. As circumstances of evaluation here play the role possible worlds have in intensional semantics, we can just repeat the definition of rigidity 516 in slightly different terms: (Rigidity) An expression is rigid if and only if its two-dimensional profile is such that its extension does not depend on the circumstance of evaluation; otherwise it is called non-rigid or flaccid. The additional dimension of the two-dimensional framework allows to define a corresponding, but different notion: (Context-Freeness, on the Level of Two-Dimensional Profiles) An expression is context-free if and only if its two-dimensional profile is such that its extension does not depend on the context of use; otherwise it is called context-dependent or context-sensitive. 517 In the two-stage variant of two-dimensional semantics, we can define an equivalent pair of notions by saying that an expression is context-free on the level of content if and only if its character is a constant function and context-dependent on the level of content otherwise, i. e., when its character is a variable function. Thus an expression is context-dependent on the level of content if and only if its content depends on the context of its use. In view of how two-dimensional profiles are defined in the two-stage variant, there will be no difference between the two notions of context-freeness and context-dependence on the level of two-dimensional profiles. Let us, for contrasting purposes, introduce a third pair of notions of contextfreeness and context-dependence for applications of two-dimensional semantics where attention is restricted to occurrences: (Context-Freeness, on the Level of Occurrences) An expression is context-free on the level of occurrences if and only if its diagonal intension is a constant function from contexts to extensions, and context-dependent on the level of occurrences otherwise, i. e., when its diagonal intension is a variable function. 516 517

Cf. section 4.6. As we want to put the two-stage variant of two-dimensional semantics to the side for the time being, we will postpone contrasting these two notions of constancy, i. e., rigidity and context-freeness, with the notion of direct reference (cf. section 4.6) until we will discuss some concrete example of singular terms in section 5.2. At that point we will also develop a sophisticated variant of two-dimensional semantics which allows to incorporate the issue of direct reference into the semantic theory (similar to the sophisticated variant of intensional semantics developed in section 4.6). It will turn out then that the incorporation of the issue of direct reference into two-dimensional semantics is possible in a satisfactory way only when we work with its two-stage variant, which provides us with another reason for going beyond the matrix variant.

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It is interesting to note that being context-free on the level of two-dimensional profiles is compatible with being context-sensitive on the level of occurrences, and perhaps even more interesting that being context-sensitive on the level of twodimensional profiles is compatible with being context-free on the level of occurrences. This is best illustrated by some examples. In our first example, we look at the two-dimensional profile and diagonal intension of a sentence that concerns a contingent matter. For the moment we decide to ignore positions of use, and model both circumstances and contexts with (uncentered) possible worlds. Let us stipulate that the possible worlds w1, w2, and w3 are such that with respect to w1 Barack Obama is president of the U.S. in the year 2016, with respect to w2 Mitt Romney is president of the U.S. in the year 2016, and with respect to w3 Hillary Clinton is president of the U.S. in the year 2016. 518 We display the relevant part of the two-dimensional profile of the sentence ‘Barack Obama is president of the U.S. in the year 2016’ in the form of a table, where circumstances are noted along the horizontal axis and contexts noted along the vertical axis, and the diagonal intension is marked out by the use of boldface:

w1 w2 w3 ...

w1

w2

w3

...

true true true ...

false false false ...

false false false ...

... ... ... ...

(table 1)

We see here that the example sentence is flaccid (because there is variation in the rows of the table), and that while it is context-free on the level of its two-dimensional profile (because there is no variation in any column of the table), it is context-dependent on the level of its occurrences (because there is variation along the diagonal of the table). This goes to show that even a context-free expression can be context-dependent on the level of occurrences, and it provides an illustration of the importance of the above admonishment to be careful to distinguish between being true as occurring at a context and being true at a context when the latter is elliptical for being true at a context and with respect to a circumstance. We find our second example in the scenario of Kaplan’s puzzle. The sentence ‘I am now here’ has only true occurrences; its diagonal intension is constant. Its two-dimensional profile, however, is not constant, and it in fact shows contextdependence. To see this, we hold a circumstance of evaluation fixed (say, in the above scenario, by focusing on the possible world with regard to which Ann is at her desk and Beth is in the garden at 7 p. m.) and vary the context of use (say, from a context where Ann is the speaker to one where Beth is the speaker). 519 518 519

The example is adapted from Schroeter 2010, section 1.2.1. In order to show context-dependence, it is crucial that what we vary is the context. We have already seen that Kaplan’s sentence is flaccid, but we know from the first example that flaccidness does not entail context-dependence (on the level of two-dimensional profiles).

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Speaking figuratively, we need only leave the diagonal of its two-dimensional profile and move vertically along the axis of contexts, 520 and we will reach a point where Kaplan’s sentence is false. Kaplan’s sentence is a rare case of a context-dependent (and flaccid) expression that is context-free on the level of occurrences. Its being context-free on the level of occurrences is due to which pairings of a context and a position of use are possible. In terms of centered possible worlds we can say that the collection of admissible centers of a possible world is restricted by how matters are with respect to that possible world (e. g., if a possible world is such that with respect to it Ann is at her desk at 7 p. m., it has an admissible center where Ann is the speaker, 7 p. m. is the time, and Ann’s desk is the place, but none where Ann is the speaker, 7 p. m. is the time, and the garden is the place). 521 More specifically, the place of an occurrence is determined by the context, speaker, and time of that occurrence, and therefore the extensions of the context-dependent expressions ‘I’, ‘now’, and ‘here’ are coordinated with the context of use in such a way that every occurrence of ‘here’ is co-extensional with an occurrence of ‘where I am now’ that is located at the same context. 522 In the case of Kaplan’s sentence this leads to the variability of the extensions of its subsentential expressions cancelling out each other for each occurrence of the sentence. With our third example we want to show that a similar effect of a contextdependent expression being context-free on the level of occurrences can be achieved without recourse to restrictions on admissible positions of use. It concerns the word ‘actual’ and is set in the same scenario of the contingent matter of the U.S. presidency as the first example, where the possible worlds w1, w2, and w3 are such that Obama, Romney, or Clinton are president respectively in the year 2016. We are interested in the two-dimensional profile of the sentence ‘The actual president of the U.S. in the year 2016 = the president of the U.S. in the year 2016’, which we will compute from the two-dimensional profiles of the two singular terms it contains. 523

520

521 522

523

When we move horizontally, along the axis of circumstances, we will of course also reach a point relative to which the sentence is false. But such variation along the dimension of circumstances of itself does not show context-dependence, but only flaccidness. Kaplan 1989, 509. It has been argued that ‘here’ can be translated as ‘where I am now’, thus reducing spatial indexicality to a combination of personal and temporal indexicality (e. g., Müller 2002, 205f.). This claim seems to me to be exaggerated, because although co-extensionality is given along the whole diagonal intensions of the two phrases, it is not given across the board, or rather, not across the whole field of their respective two-dimensional intensions. We will always be able to find a pair of a context and a circumstance relative to which ‘here’ refers to a different place than ‘where I am now’. Otherwise we would have to say that the current speaker is here of necessity. In this, we will use a two-dimensional criterion for the truth of identity statements: For singular terms α and β, Jpα = βqK((w, p), v) = T if and only if JαK((w, p), v) = JβK((w, p), v).

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Here is the two-dimensional profile of ‘the president of the U.S. in the year 2016’:

w1 w2 w3 ...

w1

w2

w3

...

Barack Obama Barack Obama Barack Obama ...

Mitt Romney Mitt Romney Mitt Romney ...

Hillary Clinton Hillary Clinton Hillary Clinton ...

... ... ... ...

(table 2)

And here is the profile of ‘the actual president of the U.S. in the year 2016’:

w1 w2 w3 ...

w1

w2

w3

...

Barack Obama Mitt Romney Hillary Clinton ...

Barack Obama Mitt Romney Hillary Clinton ...

Barack Obama Mitt Romney Hillary Clinton ...

... ... ... ...

(table 3)

Comparing the two tables shows that, speaking figuratively, adding the word ‘actual’ can amount to reflecting the two-dimensional profile across its diagonal. In the case at hand, the addition of ‘actual’ has turned a flaccid and context-free description into one that is context-sensitive and rigid. And while the two profiles are identical along the diagonal, they differ everywhere else. Therefore the identity statement formed from them has a profile which exhibits a high degree of symmetry. This symmetry becomes evident when we display the table for part of the two-dimensional profile of the sentence ‘The actual president of the U.S. in the year 2016 = the president of the U.S. in the year 2016’:

w1 w2 w3 ...

w1

w2

w3

...

true false false ...

false true false ...

false false true ...

... ... ... ...

(table 4)

This example shows that sentences of the form ‘The actual F = the F’ are similar to Kaplan’s sentence ‘I am now here’ in being context-dependent on the level of twodimensional profiles but context-free on the level of occurrences. They provide even more clear-cut examples because in their case what accounts for context-freeness on the level of occurrences is not some complicated interdependence between contexts and positions of use, but the effect of adding ‘actually’ alone.

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4.7.6 The main difference between the two dimensions With the last example we have reached a good stepping-stone from which to come back to what we have singled out as the main difference between the two dimensions of the two-dimensional framework, which is the difference between considering a possible world as counterfactual and considering a possible world as actual. The example provides us with a valuable illustration because those differences between contexts and circumstances which are of lesser importance are absent from its scenario: It involves no positions of use, and every possible world that figures as a circumstance also figures as a context. 524 There is nothing to distract us from what distinguishes incorporating a possible world into the two-dimensional framework as a circumstance from incorporating the same possible world into the framework as a context. The fact that the president of the U.S. in the year 2016 is Barack Obama is contingent, for somebody other than Obama could have been president. We have modeled the contingency of this fact by incorporating the different possible worlds w1, w2, and w3 as circumstances into the two-dimensional framework, and it is evident specifically from the variability of the first rows of the above tables 1 and 2. 525 Now it is also the case that the actual president of the U.S. in the year 2016 is Obama, but this fact is not contingent; and this is reflected in the constancy of the first row of table 3. 526 However, when we contrast the fact that the actual president is Obama with the fact that Obama is Obama we may well have the intuition that in some sense there must still be room for a variation of possibilities. This room is provided in the two-dimensional framework by the dimension of contexts. By giving the above table 3 we have already modeled the intuition that the reference of the description ‘the actual president of the U.S. in the year 2016’ refers rigidly to whatever person is president at the context it occurs at, in contrast to its counterpart description which lacks the word ‘actual’. That is, although it is necessarily the case that the actual president is Obama, it is only contingently the case that the description ‘the actual president of the U.S. in the year 2016’ refers to Obama. To generalize: What the contrast between a context-free and flaccid description of the form ‘the F’ to its context-sensitive and rigid counterpart ‘the actual F’ illustrates in an untypically neat way is that a variation of possibilities can become relevant to the extension of an expression in two entirely different ways. In terms of the schematic dichotomy between a language and the world it is about, we can say that for the question how the sense of an expression determines an extension both certain facts about the world and certain facts about language can be relevant. By these facts about the world we mean those facts (and more generally, those possible states of affairs) that correspond to what the sentences of the language are about (e. g., and schematically, the fact that a is the actual F), and by these facts about 524 525 526

Cf. the end of the subsection 4.7.1. For tables 1, 2, and 3, cf. subsection 4.7.5. Note that up to this point a collection of different possible worlds is employed only as circumstances, not yet as contexts, because we are holding the possible world w1 fixed as the actual world.

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language we mean facts about the expressions of the language that help determine their content (e. g., that Ann is the speaker) and which in some cases are correlated to those facts about the world (e. g., that Obama is the actual president). Perhaps surprisingly, even the question of who is president of the U.S. in the year 2016 can be pertinent to how matters stand with language, because the description ‘the actual president of the U.S. in the year 2016’ will pick out whoever is president with respect to the possible world at which it is uttered and refer to him or her rigidly. Thus we can characterize the main difference between the two dimensions of two-dimensional semantics according to the conceptualist understanding in the following way: The dimension of circumstances of evaluation models facts about the world and the dimension of contexts of use models facts about language. But let us be clear that the dichotomy of language and world occurs here within the input of two-dimensional semantics, because its output – in particular, the truth of a sentence relative to a context of use 527 – again models what is the case with language, but as determined by both linguistic and worldly facts. 528

4.8 Two reductionist programs The three semantic theories described in the preceding sections – extensional semantics, intensional semantics, and two-dimensional semantics – can be seen as three important stages in a simplified history of modern semantics, the younger theories adding new resources to the older ones. (In fact, hindsight allows to construe the older theories as special cases of the younger ones. 529) To conclude the present chapter on theoretical semantics, we will now have a look at some features shared by the three theories and contrast them with the Fregean framework sketched before. 530 This will help to make explicit a reductionist tendency that is present in all of them.

Truth relativized Each one of the three semantic theories – via its theory of extensions – delivers a theory of truth. This happens in the most direct way in extensional semantics, because there the semantic theory just is the theory of extensions, and the extension of a sentence is its truth value. We saw that, when we put all clauses for the truth of sentences together, extensional semantics provides a modernized variant of

527 528

529

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Cf. the definition (Truth of Occurrence) in subsection 4.7.4. In chapter 10, we will expand on this observation about how the language/world-dichotomy occurs in two-dimensional semantics. Extensional semantics can be construed as that special case of intensional semantics where there is exactly one possible world or as that special case of two-dimensional semantics where there is exactly one context of use and exactly one circumstance of evaluation. Intensional semantics can be construed as that special case of two-dimensional semantics where there is exactly one context of use. – In a similar vein, Schroeter calls extensional semantics “0-dimensional” (Schroeter 2010, section 1.1.1). Cf. section 4.2.

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Tarski’s recursive definition of truth, 531 which of course satisfies the Tarskian truth schema. 532 But what about the theories of truth that are part of intensional semantics and of two-dimensional semantics? Here, the Tarskian recursive definition of truth is relativized to a parameter – to possible worlds in intensional semantics and to pairs of a context and a circumstance in two-dimensional semantics. The sentences that correspond closest to T-sentences and that are entailed by this relativized definition of truth are of the form ‘Sentence s is true relative to x if and only if . . . ’. Sentences of the form ‘Sentence s is true if and only if . . .’, however, are not even grammatical in the language which this relativized definition of truth is formulated in, because its truth predicate is dyadic. Thus we lose the Tarskian truth schema; at least it no longer holds in an unrelativized form on the level of the semantic theory. The language that is the object of our semantic theory, however, can contain a truth predicate that applies to (some of) its sentences, and this allows to form T-sentences within the object language. But there is still a difference, despite the syntactic similarity, because the truth predicate of the object language will be context-sensitive in the sense that it has an extension only in relation to the parameter in question. 533 The observation that in two of the three theories the notion of truth is relativized to a parameter brings us to a more general fact.

Extension relativized For each one of the three semantic theories, the Fregean principle (Sense Determines Extension) holds, but in intensional and two-dimensional semantics only in a form that is relativized to a parameter. In each theory, an expression’s semantic value determines its extension. In extensional semantics this determination occurs in the simplest way possible, by the extension being identical to the semantic value. But for the other two theories we can also give a straightforward account of this determination, according to which the extension, though not identical to the semantic value, is identical to the respective value of the function which is the semantic value. In intensional and two-dimensional semantics we do not ask for the extension an expression has in an absolute way, but for its extension relative to some value c of the respective parameter. As the expression’s semantic value is identified with the appropriate function f from the parameter to a collection of extensions, the expression’s extension relative to c is the value f(c) which that function delivers for c. 534

531 532 533 534

Cf. section 4.4. Cf. section 3.2. We will say more about this in chapter 10. Strictly speaking, the considerations of this paragraph apply only to indirectly referential expressions, because in the case of directly referential expressions, the determination takes the converse direction.

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Reductionist tendencies It is interesting to note that a semantic value, besides doing its duty of determining the corresponding extension, does nothing else. In fact, the determination of extensions (in an absolute sense or relative to a parameter) is all there is to the semantic values which are posited by the three semantic theories. I see this as the result of an influential reductionist program, the program of reducing sense to extension. Throughout the modern tradition of theoretical semantics, from the invention of model theory by Tarski and others 535 via the beginnings of intensional semantics in Carnap’s Meaning and Necessity 536 to today’s technically sophisticated applications of two-dimensional semantics to natural language, 537 it seems to be an important guiding idea that a theory of extensions, if it only is fully specified, will give us the complete theory of sense. An early expression of this idea is the well-known claim that the meaning of a sentence is given by its truth conditions. 538 And many a contemporary semantician 539 seems to be led by the conviction that once we have given a complete description of how the extension of some expression depends on those facts about language and about the world that can be varied, we will already have characterized its sense. In this tendency to reduce all of semantics to a theory of extensions, by the way, we find an explanation for the Tarski-Quine tradition of using the term ‘semantic’ in a sense narrowed down to notions from the theory of extensions, so that for those who followed the reductionist agenda there was a systematic reason for using ‘semantics’ narrowly. 540 Maybe this reductionist spirit is unproblematic or even commendable for those semanticians who had their intuitions schooled in extensional semantics, but it should make any friend of a Fregean notion of sense feel uncomfortable. For can a function that delivers the correct extension for each value of the respective parameter be sufficient to explicate the cognitive value of a meaningful expression and the mode of presentation that goes with it? We have to acknowledge that in each one of the three semantic theories sense is reduced to extension, because each one posits semantic values that are nothing but constructions built up from extensions. Related to the semantic program of reducing sense to extension is the metaphysical program of reducing sense to objects. (Under the material understanding 541 of the semantic theories, we can even say that the semantic reduction entails (or presupposes) the metaphysical reduction, because on the material understanding of a semantic theory we have to take its metaphysical implications literally.) The metaphysical reduction of sense to objects is most evident when the three semantic theories are given in their usual formulation, which is set theoretical (and 535 536 537 538 539

540 541

Tarski 1956[1935]; 1956[1936]; and 1944. Carnap 1956[1947]; cf. already Carnap 1961[1942]. Cf., e. g., the contributions in García-Carpintero /Macià 2006a. E. g., Wittgenstein Tractatus, 4.024 & 4.431. A terminological remark: In view of our distinction between syntacticism and semanticism (cf. sections 3.1 and 4.8), we should refrain from calling people who do semantics ‘semanticists’. Cf. section 4.1. Cf. section 4.3.

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syntacticist). Then every item that the semantic theory is about is modeled as an object from a set theoretical universe: A single object, if not itself a set, is an urelement of that universe; a collection is a set (with the objects that are among the collection being the elements of the set); a function is a set of ordered pairs; and an ordered pair is a set. The basic items of semantics – (syntacticist) expressions, the objects that form the domain of discourse, the truth values, the possible worlds, the circumstances of evaluation, and the contexts of use – are all set theoretical objects (urelements or sets), and all further items of semantics – collections of objects, intensions, two-dimensional intensions – are made up from them by set theoretical construction, and therefore will also be sets. So, an expression’s sense is explicated as a set. And being a set entails being an object. But the metaphysical reduction of sense to objects does not depend on set theory. The metaphysical picture behind each one of the three semantic theories has objects at the bottom level; and the operations of forming a collection and of constructing a function according to all standard analyses (not only to the set theoretical ones) lead from objects to objects. The latter may be more complex or even of a higher order than the former, but objects they are nonetheless. The metaphysical reduction should be troubling for any friend of our Fregean framework, because it brings into conflict two of its central tenets about concepts. 542 As a concept is the sense of a predicate, the reduction of sense to objects entails that concepts are reduced to objects as well. But for the Fregean a concept is an unsaturated item and thus not an object! So each one of the three semantic theories departs from the Fregean framework at least in rejecting the claim that concepts are not objects. In order to illustrate the reduction of concepts to objects and why it is problematic from a Fregean perspective, let us look at a standard example. Imagine that we are interested in the material semantics of the English expressions ‘. . . is a cordate’ and ‘. . . is a renate’. This pair of predicates is standardly used to illustrate the motivation for moving from extensional to intensional semantics. 543 For we have the clear intuition that being an animal with a heart is not conceptually equivalent to being an animal with kidneys so that the two predicates express different concepts, but actually, the collection of animals with a heart is identical to the collection of animals with kidneys, so that the two predicates are assigned the same semantic value (the said collection of animals) by extensional semantics. 544 Intensional semantics, by contrast, will assign different semantic values, because the intension of a predicate is the function from possible worlds to collections of objects such that each possible world is mapped onto the collection of the objects that satisfy the predicate with respect to that world, 545 and there is a possible world with respect to which there is an animal that has a heart but no kidneys (or kidneys but no heart). 546 542 543 544 545 546

Cf. section 4.2. Cf. Lycan 2000, 142f. & 154. Cf. section 4.4. Cf. section 4.5. That state of affairs may well be biologically impossible, but it surely is metaphysically possible, and that is all we need to ascertain that distinct intensions are assigned to the two predicates.

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So intensional semantics respects the intuitive non-synonymy of the predicates, but only at the price of reducing the concepts expressed by them to objects. For, as the semantic value of ‘. . . is a cordate’ is a function from possible worlds to collections of possible objects, the concept of being an animal with a heart is reduced to a construction out of (possible) objects and thus itself is an object (that extends across the space of possibilia). Besides contradicting the Fregean tenet that no concept is an object, this arguably is not adequate as an explication of that concept understood in the Fregean way, i. e., as a mode of presentation or as a way that the objects falling under it are given to us. The move from extensions to intensions, metaphorically speaking, has brought us nothing but an increase in the number of pixels, but what we really want to grasp is the gestalt of the picture. When presenting the Fregean framework for semantics, we said that while the extension of a predicate, a collection of objects, is on the world side, its sense, a concept, is more on the side of language and its users. 547 One thing that is meant by that is that our epistemic access to a concept is more direct and hence usually better than our epistemic access to the collection of objects that fall under it. We can have a good grasp of a concept but know for only a few objects whether they fall under it. E. g., our grasp of the concept of being an animal with a heart is much better than our grasp of the collection of all cordates – let alone our grasp of all possible collections of co-possible cordates! And to be able to distinguish the notion of being a cordate from that of being a renate, we do not need to be made aware that it is remotely possible that a creature exists that has only one of these organs; it rather is the other way around: We know of that remote possibility already in virtue of our anterior grasp of the two concepts. 548 We will take these worries up again later, and I will give an argument against reducing concepts to objects and thus against reducing sense to extension. 549 For now it is enough to be aware of the reductionist tendency inherent in each one of the three important semantic theories we have presented in this chapter: Extensional semantics, intensional semantics, and two-dimensional semantics pursue the semantic reductionist program of reducing sense to extension as well as the 547 548

549

Cf. section 4.2; see also the diagram in section 4.3. In the above worries, we focus on the conceptual (or explanatory) side of the treatment of concepts in intensional semantics, and for the sake of argument accept its extensional adequacy, i. e., the correctness of its predictions about which predicates are synonymous. There are, however, examples of intuitively non-synonymous predicates that do share an intension, e. g., the pair ‘. . . is a cordate’ and ‘. . . is an element of the set of cordates’. Let us say that we are given some natural extension of set theory within the framework of possible worlds (cf., e. g., Fine 1981), according to which matters of elementhood are non-contingent, but the description pthe set of Φsq is flaccid if the predicate Φ is, because the description pthe set of Φsq with regard to a possible world w refers to the set of those objects from the domain of w that satisfy Φ with regard to w. Then intensional semantics will assign the same semantic value to the predicates ‘. . . is a cordate’ and ‘. . . is an element of the set of cordates’. But intuitively they are not synonymous, for there surely is a conceptual difference between being an animal of a certain kind and being an element of a certain set, which entails standing in a non-contingent relation to some abstract object! (The present use of the contrast between a collection of cordates with a set of cordates is inspired by Kit Fine’s use of the contrast between Socrates and the set that has Socrates as its sole element, singleton Socrates, in his argument against reducing essence to modality; cf. Fine 1994, 4f.) Cf. section 9.7.

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metaphysical reductionist program of reducing concepts to objects. But let us also note in conclusion that, despite any worries about reductionism, the three semantic theories together form a terrific toolkit, and we will put both intensional semantics and two-dimensional semantics to good use. Two-dimensional semantics will be helpful when we are doing the semantics of singular terms in the next chapter; and together with intensional semantics, we will later use it to model the interaction of language and world. 550 We will have to explain along the way why we are warranted to make copious use of these tools although we reject the reductionist programs that they are connected to. 551

550 551

Cf. subsection 5.2.5 and chapter 10. Cf. section 9.8.

Chapter 5

Singular Terms and the Spectrum of Liar Sentences

Raymond Smullyan: “Self-Reference in All Its Glory!” 552

After looking at semantics in general we are now slowly moving back towards the Liar paradox. The reunion needs to be slow because we should base our study of the semantics of Liar sentences and other self-referential sentences on a detailed description of the semantics of important kinds of singular terms as they are used in general. We start with a remark about reference and self-reference (in section 5.1), and then we will describe the semantics of names, descriptions, and indexicals in detail and as a single package (in section 5.2). After a word about the particular indexical ‘this sentence’ (in section 5.3), we will come to the central piece, which is an overview of the spectrum of (possible or purported) Liar sentences – with a name, an indexical, a description, or a quantificational phrase (in section 5.4). We will conclude the chapter by describing the semantics of quotation (in section 5.5) and explaining why quotation cannot be used to construct self-referential sentences (in section 5.6 and 5.7), which is why it does not occur in the spectrum of Liar sentences.

5.1 Reference and self-reference Reference by people and reference by expressions A remark on the relata of reference. Until now, we have without much comment often talked as if it was a singular term that refers to an object and sometimes also as if it was a sentence that refers to an object, saying for instance that a Liar sentence refers to itself. But there is a tradition of asking: Is it not people, rather than expressions, who refer to something? 553 And, is it not a singular term, rather than a whole sentence, that refers or is used to refer to an object? The first question, however, belongs to the foundational theory of meaning, and we do not need to take a stand on it here. We will for the most part continue to talk as if reference was a relation between an expression and an object; and this talk can be taken as abbreviatory or derivative by those who think that it is ultimately people who refer. At the rare points where the notion of reference by people and the notion of reference by expressions both play a role, we will speak of personal reference and 552 553

Smullyan 2006, 151. Cf. Strawson 1950, against Russell 1905. Cf. Künne 2003, 179.

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linguistic reference, respectively. With regard to the second question we should lay down definitionally that in the primary sense of ‘reference’, what refers to an object is a singular term, 554 but that we can say in a derivative sense of a sentence that it refers to an object if and only if a singular term is used 555 in that sentence to refer to that object. 556

Self-reference One point where we should be explicit about the distinction between personal and linguistic reference is when we introduce the notion of self-reference, because the two different kinds of reference lead to two different kinds of self-reference. Personal self-reference consists in a person referring (personally) to herself, and linguistic self-reference consists in an expression referring (linguistically) to itself. A person can refer to herself by making an appropriate pointing gesture, but usually she will use an expression like her name, the indexical ‘I’, or a description like (sadly) ‘the only female philosopher in the department’. Personal self-reference will play a role in this study only for contrasting purposes. Note, e. g., that for persons, selfreference is an accidental and temporary feature (in contrast to breathing, referring is not something a person would do all the time, much less referring to herself). Linguistic self-reference, however, if it occurs at all 557 will typically have a less fleeting character because many expressions that purportedly refer to themselves cannot help doing so: A sentence that refers to itself with a proper name or with the indexical singular term ‘this sentence’ is (or would be) self-referential to the core. The same goes for many sentences that refer to themselves with a description, at least on the level of particular tokens and in typical cases (‘the sentence printed in red on page 65 of the June 1969 issue of Scientific American’, ‘the last sentence uttered by Socrates’). But there are also cases where the self-referentiality of a sentence with a description arguably is contingent even on the level of concrete tokens. E. g., a token of the sentence ‘The longest sentence on the whiteboard in room 101 consists of thirteen words’ that is written on the whiteboard in room 101 will cease to be self-referential (and to be true) when a longer sentence is written beneath it. 558 Every self-referential sentence refers to itself in a derivative way, usually with the help of a singular term that is used in the sentence and that refers to the whole sentence. The most important kinds of natural language singular terms are proper

554

555

556

557

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This is in keeping with section 4.2, where we talk about the extension of singular terms, predicates, and sentences and reserve the term ‘reference’ for the extension of singular terms. It is of course not enough to mention a singular term in a sentence for that sentence to inherit that singular term’s reference. While ‘Socrates is wise’ (derivatively) refers to Socrates, ‘‘Socrates’ is a name’ does not. This definition should help to dispel the worries voiced by Hannes Leitgeb about whether a sentence can refer to anything; cf. Leitgeb 2002. We will argue in part III that the kind of self-reference needed in the Liar paradox despite appearances does not occur, but for the time being we will follow common practice and talk as if it does. In chapter 12, we will look in detail at examples of this kind.

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names, descriptions, and indexicals 559; and we will ignore other kinds (like demonstratives and the personal pronouns ‘he’, ‘she’, and ‘it’ in their anaphoric use) as incorporating them would not add any interesting further details to the study of self-referential sentences. A fourth kind of singular terms is important because it has its use exclusively in referring to expressions, namely quotation expressions. In the following sections we will treat the semantics of proper names, descriptions, and indexicals and give examples for how they can occur in Liar sentences (in sections 5.2 through 5.4). Only then will we turn to quotation expressions, because although they allow to refer to expressions, they cannot occur in self-referential sentences (in sections 5.5 through 5.7). Before we start with that, however, a brief remark about a distinction that is often made with regard to self-referential expressions. The following definitions are meant to capture the common notions of syntactic, semantic, and pragmatic selfreference: 560 A sentence is syntactically self-referential if and only if it ascribes a syntactic concept to itself. A sentence is semantically self-referential if and only if it ascribes a semantic concept to itself. A sentence is pragmatically self-referential if and only if it ascribes a pragmatic concept to itself. Examples for the three kinds of self-reference would be, respectively: ‘This sentence consists of six words.’ ‘This sentence is true.’ ‘Obey this order!’ The focus of this study is on what is called 561 semantic self-reference, because, as truth is a semantic concept, that is of course the notion under which Liar sentences fall.

5.2 Names, descriptions, and indexicals We will describe the semantics of proper names, descriptions, and indexicals within a single long section because this will allow us to compare characteristics of the three kinds of expressions. In giving this description, we will employ notions and theories from the preceding chapter. After giving some examples, we will in a first step use the distinction between sense and extension to compare names and descriptions, and to describe indexicals. In a second step, we will say something about logical 559

560 561

A terminological remark: We use the noun ‘indexical’ only to refer to indexical singular terms, and not to refer to indexical expressions in general. Cf., e. g., Blau 2008, 404. We are speaking cautiously here because later on we will partly reject the conceptual distinction between syntactic, semantic, and pragmatic self-reference. Cf. section 11.2.

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form, which will allow us in particular to distinguish the Fregean and the Russellian approach to descriptions and to draw attention to the phenomenon of pseudoreference with a quantificational phrase. In a third step, we will compare the twodimensional profiles of names, descriptions, and indexicals. In a fourth step, we will say something about the issue of direct reference and how it can be incorporated into the two-dimensional semantics of the kinds of singular terms we are concerned with. Although we will sometimes speak as if propounding a full-fledged semantic theory of the three most important kinds of singular terms, i. e., names, descriptions, and indexicals, our aim is actually less ambitious than the tone may suggest. What we say here is not meant to cover each and every case natural language has to offer, but the paradigmatic and typical ones; its purpose is primarily that of a preparation for an investigation into the semantics and later the metaphysics of linguistic selfreference.

5.2.1 Some examples Names 562 are singular terms like ‘Barack Obama’, ‘Venus’, and ‘three’ in its substantival use. 563 Descriptions are singular terms like ‘the U.S. president in the year 2016’, ‘the second planet of the Solar system’, and ‘the smallest odd prime’. These descriptions are context-free, but there are also descriptions of a context-sensitive variety, as is witnessed by the following counterparts of the three examples: ‘the current U.S. president’, ‘the heavenly body that is brightest in the morning sky of this planet’ (henceforth abbreviated as ‘the morning star’), and ‘my favorite number’. Simple indexicals 564 are singular terms like ‘I’ and ‘you’ in their typical uses, and like ‘here’, ‘now’, ‘yesterday’, ‘today’, and ‘tomorrow’ in their substantival uses; complex indexicals are terms like ‘the temperature here and now’ and, again, ‘my favorite number’. 565

5.2.2 A first characterization of names and descriptions The first thing to note when contrasting names and descriptions is that while a name is semantically atomic, a description is derived from a predicate. E. g., while the name ‘Barack Obama’ does not have any semantic parts, the description ‘the winner’ is formed (with the help of the definite article) from the predicate ‘. . . wins’, or, more generally and in Russell’s words, the description ‘the so-and-so’ is derived from the predicate ‘. . . is so-and-so’. 566 562 563

564 565

566

In our terminology, every name is a proper name, and we will often drop the by-word ‘proper’. The word ‘three’ is used as a singular term in the sentence ‘three is the smallest odd prime’, but in a quantificational way in the sentence ‘Only three students are in the library’. Cf. Recanati 2006, 249. The words ‘here’, ‘now’, and ‘yesterday’ are used as singular terms in the sentences ‘The place for putting up our tent is here and the time for doing so is now’, and ‘Yesterday was a fine day’, but they are used in an adjectival way (i. e., functioning as sentential operators) in the sentences ‘Here, the river merges with the sea’, ‘Let’s have a rest now’, and ‘Yesterday, we started our march to the sea’. Russell 1920, 167f. Cf., e. g., Ludlow 2011.

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In terms of the dichotomy of sense and extension, names are more on the side of extension and descriptions are more on the side of sense. In its typical use, the sole purpose of a name is to pick an object out, leaving it to a predicate or description to present that object in a particular way. E. g., in the sentence ‘Barack Obama is president of the U.S. in the year 2016’, the name ‘Barack Obama’ does nothing but refer to that person, while the predicate is used to express a concept that the sentence ascribes to that person. This is not to say that a name has no sense, but that (speaking figuratively) the extension of a name makes up an uncommonly large part of its sense, or that (in the form of a pun) the sole sense of a name is to refer to a particular object. This is sometimes expressed by saying that a name is nothing but a tag or a label. 567 A description, by contrast, relies heavily on the concept that is expressed by the predicate it is derived from. E. g., the description ‘the president of the U.S. in the year 2016’ manages to refer to Barack Obama only in virtue of the concept of being president of the U.S. in the year 2016, which is expressed by the predicate ‘. . . is president of the U.S. in the year 2016’. While a name refers to an object straight away, a description needs to take a detour via a concept. This is of course no shortcoming of descriptions: While like a name a description has the purpose of picking out an object, it is also essential to its purpose that it presents this object in a certain way.

5.2.3 A first characterization of indexicality and indexicals Before we come to indexical singular terms (indexicals, for short), let us characterize the notion of indexicality in general. I take the following to be the basic criterion of indexicality: An expression is indexical if and only if it has a stable sense but its extension can nevertheless vary with the context of its use. Because of the stability of sense it involves, indexicality precludes ambiguity: The extension of the word ‘bank’ also varies with use, but it does so only in a secondary or derivative manner, namely in virtue of its sense varying with use in the first place. There is no such variation of sense that would explain the variation in the reference of indexical singular terms like ‘I’. When I tell you ‘I am going to the bank’, you have every right to ask for a disambiguation of the word ‘bank’, but it would betray a lack of understanding if you went on: ‘And in which sense of ‘I’ do you mean that?’ Because of the variability of extension it involves, indexicality is a species of context-sensitivity. However, indexical expressions should be distinguished from other kinds of context-sensitive expressions, and in particular from co-text-sensitive and from demonstrative expressions. An expression is co-text-sensitive if and only if its extension depends on information given in the larger piece of text (be it an inscription or an utterance) it is a part of. A clear case of co-text-sensitive ex-

567

This doctrine is (or in some cases, should be) connected to the names of John Stuart Mill, Edmund Husserl, Ruth Barcan Marcus, Saul Kripke, Dagfinn Føllesdal, and David Kaplan.

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pressions are pronouns – ‘he’, ‘she’, ‘it’ – in their anaphoric use. Demonstratives are expressions like ‘this’ and ‘that’ in those uses where they must be accompanied by an appropriate pointing gesture (or referential intention) to determine their extension. The context-sensitivity of co-text-sensitive expressions lies more on the linguistic side and the context-sensitivity of demonstratives lies more on the worldly side, because the context relevant to fixing the extension of a co-text-sensitive expression (the co-text) is entirely internal to the language it belongs to, while the context relevant to fixing the extension of a demonstrative (the gesture or intention) is entirely external to the language it belongs to. In a way, the context-dependence of indexicals occupies a middle position in this spectrum, because the parameter needed to fix the extension of an indexical expression is external to the language (e. g., who utters the expression when), but the sense of the indexical expression determines which parameters are relevant (e. g., speaker and moment in time, but neither place nor addressee). Observe that while incorporating demonstrations and co-text into communication requires spontaneity and a considerable amount of inventiveness of both the speaker and the addressee, an indexical expression determines its extension in a nearly automatic way. This is so at least in the case of simple indexicals, where the semantic rule that is part of the indexical’s sense is so straightforward that it allows for a high degree of systematicity in the semantics of the term. E. g., the semantics of the simple indexical ‘I’ is very systematic because it is exhausted by the straightforward rule that an occurrence of ‘I’ refers to whoever uses it in an utterance. 568 Let us conclude this first characterization of indexicals by drawing attention to an interesting feature (which they have in common with demonstratives). According to Peirce, who gave an early analysis of indexicality, each occurrence of an indexical expression stands in some “existential connection” to the object it refers to. 569 This connection can be of different forms: An occurrence of ‘I’ is uttered by the speaker it refers to; an occurrence of ‘you’ is uttered to the addressee it refers to; an occurrence of ‘now’ is simultaneous with the moment in time it refers to; and an occurrence of ‘here’ is located at the place it refers to. (Something similar can be said about demonstratives, although here there is far less systematicity, because the pointing gesture (or the accompanying intention) that connects a demonstrative to the object it refers to will vary considerably from occurrence to occurrence.)

5.2.4 Logical form, descriptions, and pseudo-reference What is the logical form of sentences containing names, descriptions, and indexicals? That is, what are the counterparts (if any) of the three kinds of singular terms in a language of classical logic? These questions are highly relevant to the semantics of singular terms because, in view of the compositionality of language, logical 568

569

This automaticity makes indexicals amenable to formalization (in a way that demonstratives are not); cf. Kaplan 1989, 541ff. Burks 1949, 674.

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form determines in which way the semantic value and extension of a singular term contribute to the semantic value and extension of a sentence it occurs in. 570 Using classical logic to give the logical form of singular terms will also allow to contrast two approaches to the semantics of descriptions, both of which will be important later on. In typical languages of classical logic there are two important kinds of terms that can be made use of to give the logical form of singular terms of natural language, constants and iota-terms. While a constant is a semantic (and often also a syntactic) atom, 571 an iota-term is semantically complex because it is a term that results from prefixing a description-forming operator to an open formula, namely to an open formula that corresponds to the predicate the description is to be derived from. For a formula open in one variable like pΦ(x)q, the iota-term is p x Φ(x)q. 572 E. g., the description ‘the positive root of four’ can be formalized in a formal language of arithmetic as ‘ x (x2 = 4 ∧ x > 0)’. In a moment we will come to the question of whether an iota-term can be further analyzed. Constants are standardly used as the formal counterparts of natural language names, e. g., when the logical form of the (paradigmatic) sentence ‘Socrates is wise’ is given as (paradigmatic) atomic sentence ‘F(a)’. But given the right semantics (e. g., two-dimensional semantics, to which we will come in a moment), constants can also be used as formal counterparts of simple indexicals, so that the logical form of ‘I am wise’ could be given as ‘F(i)’. 573 When we are interested in the logical form of a fragment of natural language that includes both names and indexicals, it can be advisable to distinguish two kinds of constants (e. g., ‘a1’, ‘a2’, ‘a3’, etc. as formal counterparts of names and ‘i1’, ‘i2’, ‘i3’, etc. as formal counterparts of simple indexicals). Including indexical constants allows us to give the logical form of complex indexicals when we also have the iota-operator at our disposal (e. g.,

ι

ι

571 572

Cf. section 4.1. Cf. section 3.3. It is possible to assimilate descriptive functors (which will play a role in sections 6.1 and 6.2) into this picture. Unsaturated natural language expressions like ‘the successor of x’ and ‘the mother of x’ can be formalized by functors, e. g., as ‘s(x)’ and ‘m(x)’. Now complex singular terms of natural language like ‘the successor of zero’ and ‘the mother of Socrates’ can be formalized in a direct way as ‘s(0)’ and ‘m(a)’. But we can also construe the complex singular terms as descriptions because we can formalize them as iota-terms by first deriving a predicate from the respective functor and constant and then applying the iota-operator to that predicate; in the example we get the predicates ‘x = s(0)’ and ‘x = m(a)’ and thus the iota-terms ‘ x x = s(0)’ and ‘ x x = m(a)’. Often we can also go another more elegant way because there will be dyadic predicates, e. g., ‘x = y + 1’ and ‘M(x, y)’ with the meaning ‘x is (a) mother of y’, which when applied to constants allow to form the appropriate descriptive terms, e. g., ‘ x x = 0 + 1’ and ‘ x M(x, a)’. Sometimes constants are also given a context-free but non-rigid semantics, and it is said that their (two-dimensional) intensions then explicate singular concepts. But why should we countenance flaccid constants? For not only is there nothing constant about them – it also difficult (at least to my mind) to come up with any natural language counterpart for a flaccid term that is not a description and thus a singular term derived from a predicate, and thus better represented by an iota-term because in that way the connection to the relevant predicate is made explicit.

ι

ι

ι

573

ι

570

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the logical form of ‘my mother’ can be given as ‘ x M(x, i1)’, 574 where the predicate ‘M(x, y)’ is the formal counterpart of ‘x is a mother of y’ and the indexical constant ‘i1’ is the formal counterpart of the simple indexical ‘I’), which goes to show that a complex indexical is a description. While there is really not much more to say about names and indexicals as far as their logical form is concerned, matters get more interesting when we turn to the logical form of descriptions. Here there are two competing approaches, connected to the names of Frege and Russell. According to the Fregean approach to descriptions, we have already reached an endpoint of logical analysis when we represent a description as an iota-term. I. e., a sentence of the form ‘The F is G’ is actually (i. e., on the level of logical form) of simple subject-predicate structure, and an iota-term does actually (i. e., on the level of logical form) behave like a term, which is evident when we do the semantics of sentences which contain iota-terms, because on the Fregean approach an iota-term contributes its semantic value and extension in a straightforward compositional way to the sentence it occurs in. E. g., according to a typical semantics of Fregean descriptions, the sentence ‘G( x F(x))’ is true if and only if the extension of the iotaterm ‘ x F(x)’ is among the extension of the predicate ‘G(x)’, where truth and having an extension are perhaps relativized to some parameter. The fact that an iota-term, though syntactically complex, is semantically simple on the Fregean approach is reflected in its semantics: The semantic value of an iota-term is determined by the semantic value of the predicate the iota-term is derived from by a straightforward rule like the following: If the extension of the predicate consists of a single object, then the description refers to that object; otherwise the description has no extension, or as we can say in set theoretical parlance, it has an empty extension. The need to allow for terms to be empty may be seen as a cost for the advantage of the theoretical simplicity of the Fregean construal of descriptions. On the Russellian analysis of descriptions, by contrast, an iota-term is a term only on a syntactic surface level, but on a deeper, semantic level it is an incomplete symbol which can be explained only by a contextual definition. Famously, Russell analyzed a natural language sentence of the form ‘The F is G’ along the lines of ‘There is at least one F and there is at most one F and every F is G’. 575 I. e., concerning a formal language and for monadic predicates Φ and Ψ, Russell analyzed pΨ( x Φ(x))q as p∃x Φ(x) ∧ ∀x ∀y (Φ(x) ∧ Φ(y) → x = y) ∧ ∀x (Φ(x) → Ψ(x))q. This analysis is a contextual definition because it does not give us a term that would be equivalent to the iota-term in the sense that it would be possible to substitute it salva extensione for every occurrence of the iota-term in some larger expression. As a consequence, there is no straightforward rule which leads from the semantic value and extension of an iota-term to the semantic value and extension of a larger expression it occurs in. In fact, it is not advisable on the Russellian analysis to attribute any semantic value or extension to an isolated iota-term because it would not contribute them

ι

ι

ι

ι

574

575

Or, more directly, as ‘m(i1)’, where ‘m(x)’ is the formal counterpart of the descriptive functor ‘the mother of x’. Russell 1905 & 1920, 167–180.

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in a compositional way to the semantic value and extension of a sentence it occurs in. Rather, the semantic value and extension of the sentence pΨ( x Φ(x))q depends on the semantic values and extensions of the predicates Φ and Ψ in a way that is determined by the three conjuncts of Russell’s formula, and only in this sense can they be said to be determined compositionally. We do not need to take sides in the dispute about which is the right approach to descriptions and therefore will simply distinguish between Fregean descriptions and Russellian descriptions, where a description is Fregean if and only if the Fregean approach is an adequate characterization of its semantics and a description is Russellian if and only if the Russellian analysis is an adequate characterization of its semantics. Often we can leave open the question of whether the descriptions that occur in some language we are concerned with are of one or of the other kind. It should be noted that a Russellian description, strictly speaking, is not a singular term and thus not a device of reference. If a Russellian description of the form ‘the F’ occurs in a sentence of the form ‘The F is G’, then the sentence is of simple subject-predicate-form only on the syntactic level, while its logical form is something like ‘There is a unique F and every F is G’. Thus ‘The F is G’ really is a conjunction of two general statements, the first a claim of unique existence and the second a universally quantified conditional, and neither one refers to any particular object. But, speaking less strictly, in view of the unique existence of an object that satisfies ‘F’, which we can think of as being referred to by a name – say, ‘Felix’ –, the universally quantified conditional ‘Every F is G’ does a good enough job of emulating the subject-predicate sentence ‘Felix is G’. After all, in that situation both ‘Every F is G’ and ‘Felix is G’ are true if and only if the unique object that satisfies ‘F’ is among the extension of ‘G’. As this phenomenon plays an important role in the construction of self-referential sentences, let us introduce a name for it, pseudo-reference. We will say that a sentence of the form ‘Every F is G’ pseudo-refers to an object if and only if that object uniquely satisfies the predicate ‘F’. And for a formal language we can say that the sentence p∀x (Φ(x) → Ψ(x))q pseudo-refers to an object if and only if that object is in the domain of discourse and uniquely satisfies the open formula pΦ(x)q. 576 As a quantificational phrase (like ‘every F’) is used to achieve this effect, we will also sometimes speak of quantificational pseudo-reference. (Later on, we will sometimes allow ourselves to drop the cautionary ‘pseudo’.) In these terms we can say that a Russellian description, though never a device of reference in the strict sense, is a device of pseudo-reference in those cases where the predicate it is derived from is uniquely satisfied. And it is interesting to note that in those cases a Russellian description does not differ from the corresponding Fregean description as far as the theory of extensions is concerned: E. g., if there is a single rose in the vase, the sentence of the form ‘The rose in the vase is red’ is true if and

ι

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Note that while the primary linguistic relatum of reference in the strict sense is a singular term, and a sentence can refer only in a derivative sense, pseudo-reference from the start applies to a sentence – the reason being that there is not a singular term to be found in the logical form of a typical pseudoreferring sentence.

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only if that rose is red – regardless of whether we understand ‘the rose in the vase’ as a Fregean or as a Russellian description. But there are extensional differences between Fregean and Russellian descriptions if unique existence is not given, i. e., with regard to a description that is derived from a predicate that is not uniquely satisfied. E. g., if there are thirteen roses in the vase, the sentence ‘The rose in the vase is red’ will differ in truth value, depending on whether we understand ‘the rose in the vase’ as a Fregean or Russellian description. If it is a Fregean description, then it is a singular term with an empty extension, and, given a plausible criterion for how the truth value of a sentence depends on the extensions of its parts, the sentence will be neither true nor false. 577 However, if the phrase is a Russellian description, the whole sentence comes out as false.

5.2.5 Two-dimensional profiles We have seen that the representation of different kinds of singular terms in logical form takes its full effect only when a semantic theory is added. Now we want to characterize names, descriptions, and indexicals with the help of two-dimensional semantics. We thus move from the question of how an expression of one of the three kinds contributes its extension and semantic value to a sentence to the question of what it contributes, i. e., to the question of how its extension depends on the context of use and the circumstance of evaluation, and thus to the two-dimensional profile that according to the matrix variant of two-dimensional semantics is the semantic value of the expression. 578 We will also look at how the question of direct reference can be incorporated into the two-stage variant of two-dimensional semantics, in order to explicate an important difference between names, indexicals, and descriptions. Recall that the two-dimensional profile of an expression is a function that maps a pair of a context of use and a circumstance of evaluation onto the extension that expression has at that context and with respect to that circumstance. An expression is rigid if there is no variation of extension along the dimension of circumstances, otherwise it is flaccid. An expression is context-free if there is no variation of extension along the dimension of contexts, otherwise it is context-sensitive. 579 As our intuitions about how the extension of an expression depends on context and circumstance are much clearer with regard to natural language (and as many a run-of-themill formal language will need nothing more than extensional semantics, anyway), we will look at a bunch of examples for names, descriptions, and indexicals that are taken from the English language:

577

578 579

Thus the Fregean approach to descriptions leads naturally to a three-valued or a partial logic. Frege himself took another way, though, and assigned an arbitrary extension to every description derived from a predicate that is not uniquely satisfied. Cf., e. g., Fitting /Mendelsohn 1998, 250ff. Cf. section 4.6. Cf. section 4.6.

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example:

context of use:

circumstance of evaluation:

name:

‘Barack Obama’ ‘Venus’ ‘3’ ‘∅’

context-free

rigid

description:

‘the U.S. president in the year 2016’ ‘the second planet of the Solar system’ ‘the smallest odd prime’ ‘the smallest set of concrete objects’ ‘the mother of Barack Obama’ ‘the actual president of the U.S. in the year 2016’ ‘the local specialty’

context-free

flaccid

context-free

rigid

context-sensitive

rigid

context-sensitive

flaccid

description as well as complex indexical:

simple indexical:

‘the current U.S. president’ ‘the morning star’ ‘the temperature here and now’ ‘the favorite number of my brother’ ‘my mother’

context-sensitive

flaccid

context-sensitive

rigid

‘I’, ‘you’, ‘here’, ‘now’, ‘yesterday’, ‘today’, ‘tomorrow’ 580

context-sensitive (in a systematic way)

rigid

The examples for singular terms that are collected in this table have first been grouped together as names, descriptions, and indexicals, and then been classified according to our intuitions about their meaning as context-free or context-sensitive, rigid or flaccid. The idea behind this procedure is to formulate criteria within twodimensional semantics that will distinguish the three kinds of singular terms. But we find clear results only at the two extremes of the spectrum. Here we can generalize and say: A name is rigid and context-free; and a simple indexical is rigid and contextsensitive in a systematic way. In between, there is a wide spectrum of descriptions (some of which are complex indexicals), and we find all four combinations of the two pairs of characteristics. The only admissible generalization here would be that a description, in general, is flaccid and context-sensitive, but in special cases may also show constancy in one or both of the two dimensions. These observations, however, do not allow to formulate a criterion, because there are descriptions that behave like names, and there are descriptions that behave like simple indexicals – at least as far as their two-dimensional profiles are concerned. But more can be said about the two-dimensional semantics of singular terms. First of all, two-dimensional semantics does better than the other two semantic theories we presented in the preceding chapter. Extensional semantics cannot even distinguish between names and (Fregean) descriptions, let alone indexicals, because it assigns to every singular term whatsoever the object it refers to as its semantic value (e. g., ‘Barack Obama’, ‘the president of the U.S. in the year 2016’, 580

For a similar list, cf. Kaplan 1989, 489.

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and a token of ‘I’ that is uttered by Barack Obama would get the same semantic value). Intensional semantics does assign different semantic values to names and descriptions insofar as names are rigid and descriptions in general are flaccid, 581 but it can do nothing to distinguish names from simple indexicals as both kinds are rigid (e. g., ‘Barack Obama’ and an occurrence of ‘I’ that is uttered by Obama would still get the same semantic value). – So, what does two-dimensional semantics have to say about particular kinds of singular terms, descriptions and indexicals? Why do some descriptions 582 fail to meet the general characteristic of being context-sensitive and flaccid? A description can have a fully constant two-dimensional profile (i. e., be both rigid and context-free) in virtue of its subject matter – when it concerns abstract objects like numbers or sets (e. g., 3 or ∅), or when it is derived from a rigid and context-free predicate (e. g., ‘x is a mother of Barack Obama’). A description can be rigid but context-sensitive, again, in virtue of its subject matter (when it is a complex indexical derived from a rigid predicate like ‘x is a mother of y’ and a simple indexical like ‘I’). But a description can also be rigid in virtue of a rigidifying phrase like ‘actual’ or ‘actually’ that is added in the appropriate way (e. g., ‘the actual president of the U.S. in the year 2016’). What happens to the basic criterion of indexicality in a two-dimensional setting? We said that an expression is indexical if and only if its extension can vary with use despite its sense being stable. 583 Now, in the two-dimensional framework (as we use it here), stability of sense is guaranteed trivially, because the semantic value of an expression just is whatever function from contexts and circumstances to extensions is assigned to that expression by the two-dimensional theory of extensions. 584 Thus, indexicality here falls together with context-sensitivity. As this is too unspecific for a characterization of the semantics of simple indexicals, which are the paradigmatic indexical expressions, we should speak of indexicality in a wide sense, and contrast it with a narrow sense. (Wide Indexicality) An expression is widely indexical if and only if its extension depends on the context of use. (Narrow Indexicality) An expression is narrowly indexical if and only if its extension depends in a systematic way on some feature of the position of use which it has an existential connection to, but neither on any other feature of the context of use nor on the circumstance of evaluation.

581

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583 584

Intensional semantics already runs into the same problem: As modeled by the theory, some descriptions behave exactly like names, because they are rigid. In the present discussion of the two-dimensional profiles of descriptions, we can allow ourselves to ignore the distinction between the Russellian and the Fregean kind, because for now we are only interested in cases where the description succeeds in referring to an object. Cf. subsection 5.2.3. Cf. section 4.6.

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Recall that in the possible worlds framework of two-dimensional semantics, a context of use is modeled as a centered possible world, i. e., as a possible world taken together with a position of use. 585 A narrowly indexical expression (e. g., ‘now’ in its substantival sense) depends for its extension only on the center of the possible world that is its context of use (e. g., on the time of utterance), and each occurrence of it is existentially connected to that center (e. g., by being simultaneous with the time of utterance). Generalizing from the examples in the above table, we can observe that every simple indexical is narrowly indexical (which connects the study of logical form to two-dimensional semantics). As an indexical is a singular term that is indexical in the wide sense, there is considerable overlap between indexicals and descriptions: Any context-sensitive description is an indexical. In this region, a further sub-distinction can be introduced by appealing to logical form. Some indexical descriptions are complex indexicals, because they are derived from a predicate that includes a simple indexical (e. g., ‘the favorite number of my brother’ on the level of logical form involves a simple indexical much like ‘I’). But there are also indexical descriptions that are not complex indexicals, because they are derived from a context-sensitive predicate that does not involve any simple indexicals (e. g. ‘the local specialty’, ‘the actual U.S. president in the year 2016’).

5.2.6 The issue of direct reference Despite the descriptive detail that can be achieved by comparing the two-dimensional profiles that are assigned by the matrix variant of two-dimensional semantics, we saw that it does not in general allow to keep descriptions apart from names and from simple indexicals. This motivates an addition to our semantic theory, which we will find in the issue of direct reference. Here is the picture in broad strokes: Descriptions refer indirectly; names and simple indexicals refer directly. In a description, the sense (together with context and circumstance) determines an object that is referred to. For names and simple indexicals, it is the other way around because the sense (in a way) incorporates the object referred to. But here the strokes are already too broad, because saying only that these two kinds of singular terms refer directly would assimilate simple indexicals to names and thus obscure the important difference that whereas simple indexicals depend on the context for the object they refer to directly, names do not. And intuitively it is clear that the contextfreeness of names is not only given de facto, but de jure: 586 Once we understand that a given expression is a name, we know that its reference is independent not only of the circumstance but also of the context. In contrast, when we understand that a given expression is a simple indexical, we need the context to determine its reference before we know which object that occurrence of the simple indexical refers to in a direct way. Intuitively, direct reference occurs at different stages –

585 586

Cf. section 4.6. Cf. the comparison of de facto rigidity and de jure rigidity in section 4.6.

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and we cannot represent that fact as long as we stick to the matrix variant of twodimensional semantics, which assigns two-dimensional profiles to expressions in a single stage. 587 So let us turn to the two-stage variant of two-dimensional semantics. With its distinction between character and content, it allows to express the interesting difference between names and simple indexicals: Names refer directly on the level of character; simple indexicals refer directly on the level of content. To see how this fact can be explicated on the level of theoretical semantics, let us briefly recapitulate the two-stage variant of two-dimensional semantics as well as the sophisticated variant of intensional semantics which allowed to model the direct referentiality of some expressions. According to the two-stage variant of two-dimensional semantics, 588 the character of an expression is explicated as a two-dimensional intension, which is a function from contexts to the semantic values assigned by intensional semantics, and these semantic values explicate content. In a first stage, the character of an expression together with a context determines a content, which, in a second stage, together with a circumstance determines an extension. According to the sophisticated variant of intensional semantics, 589 an indirectly referential expression is assigned a semantic value of the form (I, intension) and a directly referential expression is assigned a semantic value of the form (D, extension), where the (formal) objects I and D respectively encode the fact that an expression is indirectly or directly referential (and where an intension is a function from circumstances to extensions). 590 Let us now combine these elements to form a sophisticated variant of twodimensional semantics which allows to distinguish between direct reference on the level of character and direct reference on the level of content. It is best to look at the second stage first, because to model the two kinds of content – that of an indirectly referential expression and that of an expression that is directly referential on the level of content – we need only employ the two kinds of semantic values assigned by the sophisticated variant of intensional semantics, (I, intension) and (D, extension). In the first stage, we can reuse the objects I and D to encode indirect and direct referentiality. To an expression that is indirectly referential on the level of character we assign a semantic value of the form (I, two-dimensional intension), where the two-dimensional intension is a function from contexts to contents, which can be either of the form (I, intension) or of the form (D, extension). To an expression that is directly referential on the level of character we assign a semantic value of the form

587 588 589 590

For the matrix variant of two-dimensional semantics, cf. subsection 4.7.2. For the two-stage variant of two-dimensional semantics, cf. subsection 4.7.3. For the sophisticated variant of intensional semantics, cf. section 4.6. Here we put readability first and are being a bit sloppy in our notation: When we augment our language of use with locutions like ‘(I, intension)’ and ‘(D, extension)’, we use italics where we strictly speaking would need a schematic formulation or a treatment with variables to express the intended generality in a precise way.

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(D, content), where the single content that is assigned again can be either of the form (I, intension) or (D, extension). 591 Now we can employ the sophisticated two-stage variant of two-dimensional semantics to characterize names, simple indexicals, and descriptions. For names, we can characterize the semantic value already on the level of expressions as such (or, the type level), but for simple indexicals and in general also for descriptions, we also need to take the level of occurrences (or tokens) into account. The semantic value of a name is of the form (D, (D, object)), where the object is of course the one referred to by the name, and the two occurrences of the object D encode the fact that a name is directly referential with regard to both dimensions. 592 E. g., the semantic value of the name ‘Barack Obama’ is the pair (D, (D, Barack Obama)). The semantic value of a simple indexical is of the form (I, two-dimensional intension), where the two-dimensional intension of a simple indexical is a systematic function from contexts to contents that are of the form (D, object). E. g., the semantic value of the personal indexical ‘I’ is the pair (I, g), where g is the function from contexts to contents which maps each context onto the pair (D, the respective speaker of the context). With a view to the level of the expression as such (or, on the type level), no more can be said. But for a particular occurrence (or token) of an indexical, we can add the content (D, object) that is determined by the character. In the example of the personal indexical ‘I’, a token that is uttered by Obama would thus have a semantic value of the form ((I, being the speaker of the context), (D, Barack Obama)), which differs clearly from the semantic value of a corresponding token of the name ‘Obama’, (D, (D, Barack Obama)). The semantic value of a description is of the form (I, two-dimensional intension), where the two-dimensional intension of a description in general is a function from contexts to contents that are of the form (I, intension), and the intension is a function from circumstances to objects. An example: The context-free description ‘the president of the U.S. in the year 2016’ has a constant two-dimensional intension 591

592

In section 4.6 we saw how the two-dimensional profiles (which in the matrix variant of two-dimensional semantics are primitive) can be defined in the (plain) two-stage variant of two-dimensional semantics. With regard to the sophisticated variant of two-dimensional semantics developed here, we can do the same. We need only in each case where direct reference occurs (either on the level of character or on the level of content) assign the corresponding constant function (either a constant two-dimensional intension or a constant intension), and then go on to combine these functions in the way described in section 4.6. Thus it is possible to retrieve the notion of a two-dimensional profile from the sophisticated two-stage variant of two-dimensional semantics; and this is a welcome fact because it shows that even after we have incorporated the issue of direct reference, we still can use the notions of rigidity / flaccidity and context-freeness / context-dependence. In “Demonstratives”, Kaplan is not officially concerned with the semantics of names (cf. Kaplan 1989, especially 489). We can however gather from some hints that the semantic value he would assign to a name, in present terms, would be of the form (I, constant character), with the single value of the constant character being of the form (D, object). That is, Kaplan does not appear to distinguish between the direct reference of a name and that of a simple indexical, insofar as he seems to situate both on the level of content. But in addition to the intuitive difference between de facto and je jure rigidity (cf. section 4.6), there surely is a similar intuitive difference between de facto and de jure context-freeness, both of which are modelled indiscriminately by constancy of content, but only the latter has its source in the character.

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which maps each context on the same intension; this intension maps each possible world on the person who with regard to it is president of the U.S. in the year 2016. As we have a name, ‘Franny’, for this particular intension, 593 we can present the semantic value of this description as (I, Franny). Note that the semantic value of a description does not incorporate the object referred to (which in the example would be Obama) – not even on the level of occurrences (or tokens) – because it is indirectly referential. In sum, the sophisticated variant of two-dimensional semantics captures the fact that whereas the direct reference of a name occurs already when it is assigned its character, the direct reference of a simple indexical occurs later because the direct reference of a simple indexical concerns only the level of content, which is determined by its non-constant character. Putting this together with the claim that all descriptions refer indirectly provides us with a simple criterion to distinguish names, simple indexicals, and descriptions: In the end the two-dimensional profile is not needed after all, because the question of direct reference is (almost) sufficient to distinguish the three kinds of singular terms. (Only almost, because we still have to incorporate one observation about the two-dimensional intension, that the character of a simple indexical is such that its content depends systematically on the position of use.) But the issue of direct reference allows to formulate the relevant distinctions only when it is incorporated into the two-stage variant of two-dimensional semantics; so we see that despite the importance the issue of direct reference has turned out to have, two-dimensional semantics provides more than just the distinctions between rigid and flaccid and between context-free and context-sensitive expressions. In concluding the investigation into direct reference and the two-dimensional semantics of singular terms, I would like to add a conjecture about how we are to interpret its result. We started out 594 by trying to distinguish between names, simple indexicals, and descriptions with a view solely to how context and circumstance determine extension, but we were unable to find a criterion formulated in the two-dimensional theory of extensions alone. What we did find might appear to be not much more than an ad hoc criterion that distinguishes between names, simple indexicals, and descriptions just in virtue of their being names, being simple indexicals, or being descriptions, because the tripartite distinction between direct reference on the level of character, direct reference on the level of content, and indirect reference might be just another formulation of that trichotomy. But this need not be a disadvantage of the present semantic theory – rather, the direct connection between the issue of direct reference and the distinction between names, simple indexicals, and descriptions might turn out to be entirely adequate to the semantics of these kinds of singular terms. After all, to start our story of stages at an earlier point, one of the first feats of someone who understands one of these singular terms is to discern whether it is a name, a description, or a simple indexical – this has to happen before any character can be apprehended. And the gist of this early feat is of course not the act of attaching the label ‘name’, ‘simple indexical’ or ‘description’ to

593 594

Cf. section 4.6. Cf. subsection 5.2.5.

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the singular term, but consists in reaching an understanding of the way in which it refers: whether it refers directly, and if so, on what level.

5.2.7 A brief summary Here is a brief summary of what we said in the preceding subsections in the form of an overview of the semantics of names, simple indexicals, and descriptions: logical form:

circumstance:

context:

mode of reference:

names:

(regular) constants

rigid

context-free

direct reference on the level of character

simple indexicals:

(indexical) constants

rigid

context-sensitive (in a systematic way)

direct reference on the level of content

descriptions:

iota-terms: Fregean or Russellian

flaccid (or rigid)

context-sensitive (or context-free)

indirect reference

By using classical logic to explicate logical form, applying some notions from two-dimensional semantics, and addressing the issue of direct reference, we have reached a nearly comprehensive account of what I take to be the contemporary common sense about the semantics of singular terms. This of course leaves many contentious questions open. But aside from my appeals to our intuitions about the examples, this is not the place to argue in detail for this account. With a view to the study of (purported) self-referential sentences that will form an essential part of our investigation of the Liar paradox, it is of less importance that the exact delineation of the different kinds is correct. There is one thing that is essential in this regard, that they indeed span the full spectrum of singular terms (besides quotation, which gets a separate treatment). But of that we can now be reasonably sure. What we should to do to make our account more plausible, however, is to deal briefly with possible objections that concern some rare uses of singular terms (in subsection 5.2.7) and to reject some claims that one of the various kinds is reducible to another (in subsection 5.2.8).

5.2.7 Atypical uses According to our account, a description by way of its sense – some would say, by way of an associated “descriptive condition” 595 – determines an object and introduces it into discourse. But this does not appear to be correct in all cases. Let us imagine that we are at a garden party, and I tell you ‘The woman with the cup of tea will soon get wet’ while pointing to a man with a cup of coffee who stands near the as yet inactive lawn sprinkler. Many will have the intuition that the content of my statement is true, even though the only woman on the party who has a cup of tea is sitting safely on the porch. Keith Donnellan has used examples like this to distinguish between an attributive use of descriptions, which conforms to the 595

E. g., Soames 2010, 98.

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semantics of descriptions as described above, and a referential use of descriptions, which is given in the example. 596 We need not follow Donnellan, though, and call for a substantial revision of the semantic theory of descriptions that is based solely on their attributive use. Rather, we have good reason to say the following: While the typical use of descriptions is attributive, they can be used in atypical ways, for instance by using a description as a demonstrative (or, more accurately, using a phrase that usually is a description in a demonstrative way). Atypical uses of descriptions are interesting in general, especially with a view to the interaction of semantics and pragmatics, but we do not need to take them into account in a study of the semantics of (purported) Liar sentences and self-referential sentences in general. The reason for this is that although it does not appear to be guaranteed that they do not occur, 597 they will add no new aspect to the phenomenon. According to our account, a name refers directly to an object and that is all it does; it does not have any descriptive content. But is this right for all cases? There are two kinds of candidates for counterexamples. Firstly, we have names like ‘Dartmouth’ and ‘The Holy Roman Empire’, which obviously allow to read off a descriptive content. Here, there is an easy way out because we have good reason to construe these phrases as former descriptions which were turned into names a long time ago, so that any descriptive content that is still perceptible is nothing more than a trace of the phrase’s ancient history, which is in keeping with the fact that in these cases, the descriptive condition need not be true of the object referred to. Secondly, there is an example due to Gareth Evans according to which someone lays down, for the discourse of a group of people who do not know who invented the zip, ‘Let us call the inventor of the zip ‘Julius’’. In the context of the ensuing conversation, the sentence ‘Julius invented the zip’ appears to be analytically true, in much the same way as ‘The inventor of the zip invented the zip’. 598 With regard to cases like the singular term ‘Julius’ as construed by Evans we can take a similar way out as with referentially used descriptions: We can say that here a name is used as a description (or, more accurately, an expression that usually is used as a name is used to abbreviate a description). Even when we accept for the sake of argument that such cases occur, this again need not deter us when we come to study the semantics of (purported) self-referential sentences. We will have to deal with both the issue of self-reference by name and of self-reference by description, anyway, and can put sentences that appear to refer to themselves with an Evans-style name into the same drawer as sentences that appear to refer themselves with a description.

596 597

598

Donnellan 1966, especially 287ff. Imagine a logic professor who intends to give an example of a Liar sentence and writes ‘The sentence on the whiteboard in room 101 is false’ on the whiteboard, not knowing that he is in room 102 instead. Arguably, a sentence of the form ‘The F is F’ is true (or analytically true) only if there is a unique object that satisfies ‘F’. But there are zips, and we have reason to believe that someone invented them.

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5.2.8 Claims of reducibility Historically, there have been several attempts to assimilate one kind of singular terms to another kind of singular terms. We will look briefly at two of these claims of reducibility, because their rebuttals are important roots of the present account of singular terms. They concern the reduction of names to descriptions and the reduction of indexicals to some context-free kind of referential device. The rebuttal of the claim that names can be reduced to descriptions is one of the rare clear cases where the historical development of philosophy resembles the progress regularly made in mathematics and the natural sciences. While in the first half of the 20th century, nearly every analytic philosopher thought that the semantic value of a name is like that of a description (or a descriptive bundle, i. e., roughly, a disjunction of conjunctions of descriptive conditions), since about the 1960s and ’70s nearly everyone in the field agrees that this cannot be the case. The initial plausibility of the description theory of names stems, famously, from its power to solve certain puzzles that were posed by Frege and Russell 599 and also from the phenomenon that someone who knows which object some name refers to, when asked to whom the name refers, will probably answer by describing that object. 600 However, the description theory of names makes the wrong predictions about the behavior of names in modal contexts. When we use a description to refer to a concrete object, we usually make use of some contingent fact which involves that object. But when we refer to an object by its name, we can make all kinds of counterfactual claims about it, among them the claim that matters could be different with regard to the contingent facts we used in our description. E. g., we might say ‘If Hillary Clinton had won the primary elections in 2008, then Barack Obama would not be president of the U.S. in 2016’. So names cannot have the same semantics as descriptions – they are directly referential and therefore rigid. – These claims and arguments are wellknown today because of the huge impact Kripke’s Naming and Necessity 601 made on the community of analytic philosophers. It is therefore worth mentioning that Ruth Barcan Marcus in her 1961 article “Modalities and Intensional Languages” had already formulated some of the theses and arguments which now are commonly attributed to Kripke. 602 While the claim that names can be reduced to descriptions (before its rebuttal) was made explicitly by Russell, Searle, and others, the claim that indexicals can be reduced to some context-free kind of referential device was little more than a tacit assumption shared by the many early analytic philosophers who were wont to put indexicality, together with ambiguity, into a shabby box labeled ‘the deficiencies of natural language’. 603 This changed at about the time modal arguments were used to show that names cannot be reduced to descriptions, and Arthur Prior, Hector599

600 601 602 603

These puzzles include the problematic use of names to refer to non-existent objects and the informativeness of identity statements with names. Cf. Lycan 2000, 37ff. Lycan 2000, 39f. Kripke 1980[1972]. Barcan Marcus 1961. Cf. Smith 1995. Cf., e. g., Quine’s rant against tense; Quine 1960, 170.

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Neri Castañeda, John Perry, and David Lewis formulated variants of a compelling argument for the ineliminability of indexicals from ordinary discourse. 604 The argument is based on the observation that, especially when we are concerned with the rationality of certain actions, the cognitive significance of certain sentences that contain indexicals (e. g., Prior’s “‘Thank goodness that’s over!’” 605 and Perry’s “‘I am making a mess.’” 606) is lost in each and every proposal for a non-indexical substitute (e. g., “‘Thank goodness the date of the conclusion of that thing is Friday, June 15, 1954’” 607, “‘[. . . ] John Perry is making a mess.’” 608). The ineliminability of indexicals is most obvious in the case of sentences that express self-localizing beliefs, which combine an indexical and a context-free device of reference (e. g., ‘Today is Friday, June 15, 1954’ and ‘I am John Perry’) and thus allow the person who knows them to be true to know his or her location on one of the dimensions of indexicality (e. g., the temporal or the personal dimension). The substitution of a coreferential context-free singular term for an indexical robs a sentence that expresses a localized belief of this cognitive value (as witnessed by the uninformative ‘Friday, June 15, 1954 is Friday, June 15, 1954’, ‘John Perry is John Perry’, and even by the informative but not self-localizing ‘The only bearded philosopher in a supermarket in San Diego is John Perry’). – Although Prior was first in formulating his “Thank Goodness”-variant of the argument against reducing indexicals to any kind of context-free device of reference, 609 its being generally accepted today is due largely to Perry’s better-known work on essential indexicality. 610

5.3 ‘This sentence’ is sententially indexical. We should stay for a moment with indexicals, turning from their semantics in general to the specific semantics of that particular indexical singular term that can be used to construct self-referential sentences, ‘this sentence’. This will not take long, 611 and we will shortly go on to give an overview of Liar sentences with a name, description, or with this indexical. 612 In contrast to the names and descriptions that might occur in self-referential sentences, but also elsewhere, the indexical ‘this sentence’ is tailor-made for this purpose (and not much good for anything else). It provides a clearer way of constructing a self-referential sentence than the complex indexical ‘what I am saying now’ – but only if it is regimented in the right way. In life outside logic, a token of the term ‘this sentence’ usually refers to another sentence, in a demonstrative or in

604 605 606 607 608 609 610 611 612

Prior 1959; Castañeda 1966 & 1967; Perry 1977 & 1979; and Lewis 1983[1979]. Prior 1959, 17. Perry 1979, 3f. Prior 1959, 17. Perry 1979, 4f. Müller lists Broad, Wittgenstein, and Geach as forerunners of Prior; Müller 2002, 200, fn. 265. Cf. Müller 2002, 199f. We will say more about the indexical ‘this sentence’ in sections 11.11 and 12.6. A preliminary version of the material in section 5.3 appears as part of Pleitz 2010b.

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179

a context-sensitive way. But used in an indexical Liar sentence the term is purely indexical, requiring no accompanying gesture and no background information to fix its reference. So we can say that by stipulation, 613 each occurrence of the term ‘this sentence’ that is used 614 as part of a sentence refers to that sentence. Recall that an expression is indexical if and only if there is a systematic way in which its reference depends on some aspect of its position of use, and that according to Peirce’s early analysis of indexicality, each token of an indexical expression stands in an “existential connection” to the object it refers to. 615 Any token of the term ‘I’ refers to the speaker who produced it, any token of the word ‘today’ refers to the day when it is uttered, etc. As the term ‘this sentence’ always refers to the sentence it is used in, it is a sententially indexical term in much the same way the term ‘I’ is personally indexical and the word ‘today’ is temporally indexical. The existential connection between each token of the term ‘this sentence’ and the sentence it refers to is given by the relation of parthood. And, just like ‘I’ and ‘today’, it is a simple indexical, because it has no semantic parts. We cannot split up the locution ‘this sentence’ that is regimented in a sententially indexical way into a demonstrative ‘this’ and a sortal predicate ‘sentence’, by analogy to ‘this table’ and ‘that woman’, because in the case of ‘this sentence’, there is no accompanying gesture needed to determine a collection of objects ‘this’ may refer to, one of which is then determined by the sortal predicate ‘table’ or ‘woman’. Rather, like the reference of ‘I’ and ‘today’, the reference of ‘this sentence’ is fixed in an entirely automatic way. We should note that by this construal of the semantics of ‘this sentence’, we add a peculiar new dimension of indexicality to the well-known temporal, spatial, and personal dimension. By stipulating that each occurrence of ‘this sentence’ refers to the sentence it is used in, we construe sentences as positions of use, on a par with moments, places, and speakers. Thus we incorporate a sentential dimension of indexicality into the more general semantic theory of context-sensitive expressions.

5.4 The full spectrum of Liar sentences After covering a lot of ground we are now in a position to give an exhaustive classification of Liar sentences. Or rather, of purported Liar sentences! Strictly speaking, we would need to add ‘purported’ each and every time we say ‘Liar sentence’, because it will be our aim in part III to argue that ultimately, i. e., once we have gained the correct understanding of the metaphysics of language and its consequences for semantics, we will see that there are no properly self-referential sentences and hence no Liar sentences. But there is of course no doubt that at 613

614

615

In their work on circular propositions, Jon Barwise and John Etchemendy make a similar regimentation, pointing out that “‘this proposition’ can also be used demonstratively, to refer to some other proposition immediately at hand” (Barwise /Etchemendy 1987, 16). If the term ‘this sentence’ is mentioned, it will in general not refer to the sentence it is a part of. An example is ‘‘This sentence’ consists of two words’, in contrast to ‘This sentence consists of six words’. Burks 1949, 674.

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least apparently there are Liar sentences and other self-referential sentences (most scholars think so). It is those apparent sentences that we will speak about in this section; and for the time being we will often drop the precaution of explicitly qualifying them as purported. Here is an overview of all the devices that can be used to refer to a sentence: names, indexicals, Fregean descriptions, Russellian descriptions, pseudo-reference with a quantificational phrase, and quotation. (We can ignore anaphoric uses of pronouns like ‘it’, as in the sentence ‘‘Snow is white’ was Tarski’s favorite natural language sentence, and he said that it is true.’ They depend for their extension on another referential device occurring nearby in the co-text (here, a quotation expression), and thus they do not allow to construct a self-referential sentence without the help of one of the other devices. 616) Referring to a sentence is what a self-referential sentence does, but it will turn out that not every device that allows to refer to a sentence also allows to form a self-referential sentence (even on the level of apparent self-reference): While names, indexicals, descriptions, and quantificational phrases can all be used to form self-referential sentences, quotation cannot – which is why we have already described the semantics of names, indexicals, and descriptions, but postpone our description of the semantics of quotation expressions. 617 We thus come to what is of central interest in this chapter, a list of ten exemplary Liar sentences, 618 seven of which are English while three belong to a formal language that we have used already when we presented variants of the formal Liar reasoning: 619 (1) ‘This sentence is not true’, where ‘this sentence’ is understood as regimented to refer to whatever sentence it is used in, (2) ‘¬True(iσ)’, where ‘iσ’ is a sententially indexical constant that refers to whatever well-formed formula it is used in, (3) ‘Larry is false’, under the assumption that it is named ‘Larry’, (4) ‘¬True(aL)’, which is an object in the domain of discourse of the formal language that is referred to by the constant ‘aL’, (5) ‘(5) is not true’, (6) ‘The only sentence written on the whiteboard in room 238 on February 28th, 2012, is not true’, under the assumption that it is the only sentence written on the whiteboard in room 238 on February 28th, 2012,

616

617

618 619

Anaphors can be used in the construction of self-referential sentences, but as they always depend on one of the other referential devices, this will add nothing essentially new to the spectrum of Liar sentences. E. g., ‘This sentence is problematic because it is not true.’, although ascribing untruth to itself by way of the pronoun ‘it’, achieves self-reference (mainly) in virtue of the indexical ‘this sentence’. We will give the semantics of quotation in section 5.5, and in sections 5.6 and 5.7 we will show that quotation cannot be used to construct self-referential sentences, at least not on its own. For another overview of Liar sentences, cf., e. g., Rheinwald 1988, 230f. Cf. section 3.5.

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181

(7) “The sentence printed in red on page 65 of the June 1969 issue of Scientific American is false”, which is the only sentence printed in red on page 65 of the June 1969 issue of Scientific American, 620 (8) ‘¬True( x F(x))’, which is the only object in the domain of discourse of the formal language that satisfies the predicate ‘F(x)’,

ι

(9) ‘Every natural language sentence that is mentioned on this page and starts with an English quantificational phrase is not true’, and (10) ‘∀x (F(x) → ¬True(x))’, which is the only object in the domain of discourse of the formal language that satisfies the predicate ‘F(x)’. From our considerations about singular terms and logical form we know that we can assimilate indexical constants to simple indexicals, (regular) constants to names, and iota-terms to descriptions. 621 Thus we can distinguish five groups of Liar sentences here, because self-reference can be achieved with a simple indexical, with a name, with a Fregean description, with a Russellian description, and with a quantificational phrase. 622 Accordingly, we can speak of the indexical variant, the name variant, the Fregean description variant, the Russellian description variant, and the quantificational variant of the Liar paradox. While some of the examples are clear cases, for others the classification will depend on how we understand the referential device they employ: (1) and (2) use a simple indexical, (3) and (4) use a name, (5) uses a name or a description (depending on how we construe the practice of numbering sentences that are displayed in a philosophy text), (6) through (8) use a description, and (9) and (10) use a quantificational phrase to achieve self-reference. For each of the Liar sentences with a description ((6) through (8), and possibly (5)) we have two readings, because we can understand each description (or iota-term) either as a Fregean or as a Russellian description. – Recall that a Liar sentence with a Russellian description or with a quantificational phrase strictly speaking stands to itself not in the relation of reference, but of pseudo-reference. 623 But although being the extension of a predicate, even its unique extension, is not the same thing as being referred to by a singular term, we will nonetheless respect common terminology and call these sentences ‘self-referential’. We should note that in most cases self-reference is secured not by the semantics of the Liar sentence alone. (3) through (10) are self-referential only because of an 620

621 622

623

Tarski 1969, 65; printed in red. – Rheinwald constructs a similar example: “Der kursiv gedruckte Satz, der in dem Buch ‘Semantische Paradoxien, Typentheorie und ideale Sprache’ von Rosemarie Rheinwald auf Seite 19 steht, ist nicht wahr”, which is the only italicized sentence on the nineteenth page of her book about semantic paradoxes (Rheinwald 1988, 19). Cf. also the fourth sentence that is mentioned in the list of five sentences in section 1.2. Cf. subsection 5.2.4. We refrain here from adding examples for variants of the Liar paradox with more than one sentence, i. e., for Liar cycles or Yablo sequences; but cf. section 1.3. To construct a Liar cycle or a Yablo sequence, again, all five kinds of referential device can be used (even in combination with each other) to construct the circular (or infinite) referential chains required for paradoxicality. Cf. subsection 5.2.4.

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extra assumption, which we have noted in most cases. E. g., if ‘Larry’ had been the name not of ‘Larry is false’, but of ‘Snow is white’, then ‘Larry is false’ would not be self-referential and not paradoxical but merely false. Similar considerations hold for those sentences that must be placed at the appropriate position (within a text, or – in examples that have become standard – on a whiteboard in a specific room) to refer to themselves. In the case of (5) and (9) the extra assumption is implicit in the use: To be self-referential, (5) must be numbered ‘(5)’; and (9) would fail to refer to itself if there was another sentence mentioned on the same page that starts with a quantificational phrase. 624 The sentences (1) and (2), in contrast to all other example sentences, achieve the feat of referring to themselves entirely on their own, at least when the semantics of ‘this sentence’ and of ‘iσ’ are regimented in the right way. 625 We thus have reason to conclude that it is only indexicals that allow to construct sentences that are self-referential in a semantically self-sufficient way. This is not to say that the indexical variant of the Liar paradox poses a more severe problem than the other variants – the need for an extra assumption in the non-indexical variants does nothing to alleviate the seriousness of the paradoxes engendered by them. When we considered Church’s paradox 626 we saw that even where the inference to a contradiction depends on some contingent, non-semantic fact we have a full-fledged paradox on our hands in any case because we can transform the contradictory reasoning into an inference to a contingent statement (here, the negation of the extra assumption that the contingent state of affairs obtains) that employs nothing but semantic premises and logic. More important than what might distinguish the variants of the Liar paradox on our list – at least for now 627 – is that the list is complete in the sense that it gives examples for all possible variants, so that it indeed spans the full spectrum of Liar sentences. To show that the list is exhaustive in this sense, we need to show further devices of referring to a sentence either do not allow to form selfreferential sentences or can be subsumed under one of the categories given above. Specifically, we will argue that no sentence can refer to itself by quotation alone (in sections 5.6 and 5.7) and that a certain construction which does involve quotation can be subsumed under the variant with a description (in section 6.1).

624

625 626 627

The extra assumption needed for example sentences (3) through (10) to be self-referential corresponds to what we called an external semantic information in section 3.5 and 3.6. In the case of those sentences which use a description to refer to themselves we may have to re-formulate the extra assumption to see what is semantic about it, e. g., by requiring that the predicate the description is derived from is such that it is satisfied by a unique object (in the given context). Cf. section 5.3. Cf. section 1.4. It will turn out that only one difference among Liar sentences is important for our one approach, which is the difference between direct and indirect reference. Cf. chapters 11 and 12.

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183

5.5 Quotation In the present chapter, we are interested in singular terms that allow to refer to expressions, because we want to understand linguistic self-reference. We have already studied names, descriptions, and indexicals, and illustrated how they might figure in Liar sentences. Names, descriptions, and indexicals are the three most important devices for referring to objects, where ‘object’ is understood in the most general sense of the word. Expressions are objects, too, and they can also be referred to by singular terms of these three kinds. There is another referential device, that is tailormade for referring to expressions but that (arguably) cannot be used to refer to any object that is not an expression: quotation. 628 We turn to quotation only in the present section, after our presentation of the spectrum of Liar sentences, because we will see in the next two sections that quotation cannot occur in this spectrum, and also because the three sections on quotation together prepare the way to chapter 6 nicely. Quotation – putting an expression between quote marks – has several uses. In direct quotation, it is used to mention the enclosed expression, i. e., to refer to it. In mixed quotation, it is used to mark the enclosed words as those of a certain speaker, and in shuddering, to distance oneself from them. And last but not least, quotation is used to mark citations in scientific texts. 629 Our main interest is in direct quotation, but we will first take a contrasting look at some of the other uses. We find many examples for mixed quotation in newspapers, where we will encounter sentences like the following: ‘The president declared that the government will do ‘everything in its power to stop the crisis’.’ What this sentence means is not that the government stands in some relationship to the expression ‘everything in its power to stop the crisis’, and accordingly the quotation expression ‘‘everything in its power to stop the crisis’’ does not function as a singular term here. Rather, quote marks are used in mixed quotations like the above example to transport, in addition to the meaning which the sentence would 628

629

The literature on quotation, although less extensive than that on the other kinds of singular terms, by now attests to a considerable interest of semanticians. Most of it exists in the form of single articles, e. g., Davidson 1979; Richard 1986; Bennett 1988; Washington 1992; García-Carpintero 1994 & 2004; Boolos 1995; Saka 1998; Recanati 2000 & 2001; and Cappelen /Lepore 2012. Monographs are fewer, but the authors of the concise Cappelen/Lepore 2007 are not correct when they assert “This is the first book ever written exclusively on the topic of quotation and metalinguistic representations more generally.” (Cappelen/Lepore 2007, 16), because they thus fail to mention two full studies of quotation and metalinguistic reference, Harth 2002 and Steinbrenner 2004. There are also several kinds of quote marks. This allows us to make some distinctions between uses of quotation visible; and we do so within the present study. Single apostrophes (‘‘. . .’’) will occur in this section as generic quote marks in our account of direct quotation, and we also use them within this study to mark when we mention an expression (as well as to mark when an expression is cited or mentioned within a citation). There are also, e. g., double apostrophes (‘“. . .”’), which we use to mark citations, and double angles (‘«. . . »’), which we use to form formal quotation expressions. Formal quotation was used already in section 3.5; and we will say more about it in section 5.7.

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have had without them, 630 the information that someone (here, the president) used the exact words between the quote marks. 631 (I conjecture that shudder quotes can be seen as an extreme case of mixed quotation, 632 because they allow to use an expression in the standard way whilst transporting the extra information that one distances oneself from a particular use of it, typically a use that is made by someone else. A certain German ‘newspaper’, e. g., notoriously used the abbreviation of the German name of the German Democratic Republic only within quote marks.) We now turn to direct quotation, but not yet to give an account of how it works, but to draw attention to a normative aspect that is special to it. The function of quote marks in their referential use is to make explicit that an expression is not used but mentioned. We have to be clear, however, that this use of quote marks – direct quotation – has the status of a regimentation of language; and that a characterization of it is meant in a prescriptive way – in contrast to, say, our characterization of the typical use of descriptions in the preceding section, which was descriptive in character. To which languages or language fragments does the prescription apply? It must be acknowledged that the use of quote marks in ordinary language, if not an utter mess, is hardly a systematic device of making explicit mention whenever it occurs. 633 But criticizing this is not among our tasks (or privileges) when doing philosophy; we can only try to keep calm and distinguish. We might, e. g., adopt a medieval distinction and say that each expression of ordinary language either has suppositio formalis, when it is used in the typical way, i. e., to refer to something beyond itself, or suppositio materialis, when it is used to refer to itself. 634 With regard to scientific texts, however, it is clearly commendable always to use quote marks when mentioning an expression. This is so especially with regard to texts which (like the present one) deal with logic, the philosophy of language, and the metaphysics of expressions, because these texts deal with expressions (and perhaps they even deal with dealing with expressions). We find advice to this effect in Frege’s Basic Laws of Arithmetic: “Someone may perhaps wonder about the frequent use of quotation marks. It is by this means that I distinguish cases in which I speak of the sign itself from cases in which I speak of its reference. Pedantic though it may seem, I nevertheless take this to be necessary. It is remarkable how an imprecise manner of speaking and writing, perhaps originally used for ease and brevity yet with full 630

631

632 633

634

Only roughly; and it is difficult to formulate a general rule. E. g., ‘‘The queen said that ‘I am not amused’’ of course does not mean that the queen said that I am not amused.’ Another nice example is “Quine said that quotation ‘has a certain anomalous feature’”, which is not about the expression ‘has a certain anomalous feature’ (Cappelen/Lepore 2007, 14; cf. Quine 1981[1940], 26). Cappelen and Lepore point out that in direct reports “mixed quotation is far more common than direct quotation” (Cappelen /Lepore 2007, 14f.), and they discuss it extensively (cf. the entry “mixed quotation” in the index of their monograph; Cappelen /Lepore 2007, 167). Cf. Cappelen /Lepore 2007, 16f., for a contrary opinion. It seems that contemporary ordinary language is in better shape when it comes to marking each citation by the use of quote marks. But the reasons for this have little to do with semantics, and more with the fact that intellectual ownership is highly priced (and legally protected) in contemporary societies. Cf. e. g., Gamut 1991, 12.

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awareness of its imprecision, can finally addle the thinking after this awareness had disappeared.” 635

Quine adds a plausible explanation of the error’s being committed so frequently: “In the literature on the logic of statements, and in other foundational studies of mathematics as well, confusion and controversy have resulted from failure to distinguish clearly between an object and its name [. . . ] The trouble comes [. . .] in forgetting that a statement about an object must contain a name of the object rather than the object itself. If the object is a man or a city, physical circumstances prevent the error of using it instead of its name; when the object is a name or other expression in turn, however, the error is easily committed.” 636

For some time, and presumably due to a degree to Quine’s influence, analytic philosophers and philosophical logicians did their best to mark each expression they mentioned by the appropriate number of quote marks. 637 But today, neglect of Frege’s and Quine’s good advice has become fashionable. 638 We, however, try to keep out of mischief within the present study and aim to always mark mention. The referential device of direct quotation is at once perspicuous and perplexing. On the perspicuous side, a minimal theory of quotation expressions 639 is easily stated: (Minimal Theory of Quotation) For any expression α, (i) the expression p‘α’q is the 640 quotation expression of α, and (ii) the quotation expression of α refers to α. When to quote an expression is to refer to that expression by means of a quotation expression, then this theory entails that we can quote any expression just by putting it between quote marks. That is in fact how Herman Cappelen and Ernest Lepore formulate the minimal theory of quotation when they give the principle “‘e’ quotes e”, 641 or in present terms and notation: 635 636 637

638

639

640

641

Frege 2013, 4 = Frege 2009[1883/1903], 27. Quine 1981[1940], 23. E. g., Davidson recalls: “When I was initiated into the mysteries of logic and semantics, quotation was usually introduced as a somewhat shady device, and the introduction was accompanied by a stern sermon on the sin of confusing the use and mention of expressions” (Davidson 1979, 79) Some authors follow this fashion trend with what Frege called “full awareness of its imprecision”. Michael Glanzberg: “Where clear and convenient, I am intentionally confusing use and mention. There are enough quotation marks in this paper already.” (Glanzberg 2001, 245) Ted Sider: “I have decided to be very sloppy about use and mention. When such issues matter I draw attention to them; but where they do not I do not.” (Sider 2010, vii) See also Smith 2007, 28f. & 129. Our formulation of the minimal theory is developed directly from Cappelen/Lepore 2007, 123ff. (see comments below for the differences). They note that something like it has been formulated before by a number of authors; cf. Cappelen /Lepore 2007, 124. An implicit idealizing assumption of this formulation of the minimal theory of quotation is that there is exactly one pair of quote marks available in the language. Under this assumption, there is a unique quotation expression for every expression, which justifies using the definite article ‘the’ in clause (i). Cappelen /Lepore 2007, 123.

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For every expression α, p‘α’q quotes α. But due to some qualms about circularity, we refrain from using the verb ‘quotes’ in our account of quotation, 642 and are content with noting the principle about quoting as one of its consequences. We can also note that the use of Quine corners and a meta-variable in this principle delivers instances of different kinds. We have not only the homophonic ‘‘YHWH’’ quotes ‘YHWH’, 643 but also variants with a name or a description of the expression in question: ‘‘YHWH’’ quotes the tetragrammaton and ‘‘YHWH’’ quotes the most important name of God in Judaism. What can we learn from the minimal theory about the range of objects that can be quoted? Two issues are connected to the notion of an expression, which plays a central role in the minimal theory. Firstly, as our talk about expressions primarily concerns the level of types, a particular quotation expression refers to the type expression an occurrence of which is enclosed between the quote marks. Secondly, the minimal theory does not limit the expressions that can be quoted in some language to expressions of the same language. It is a consequence of the minimal theory that we can, with the help of quote marks that belong to one language, quote expressions of another language. 644 And this is intuitive, as the following example shows: ‘Cantor chose the Hebrew letter ‘ℵ’ to form names of cardinalities.’ 645 The minimal theory entails a principle that Cappelen and Lepore call “Containment”, of which we formulate the following variant: (Containment) For any quotation expressions ř, the object that ř refers to is a proper part of ř. 646 In view of the minimal theory, we can be more specific: The object x is that proper part of the quotation expressions ř that results from dropping the outermost pair of quote marks from ř. The principle of (Containment) rests on the syntactic divisibility of quotation expressions. What about their semantic divisibility? In other words, what about the 642

643

644

645 646

In our slightly longer formulation, we also gain a nice separation of the syntax and the semantics of quotation into clause (i) and clause (ii). This is the variant Cappelen and Lepore focus on; they like examples like “‘‘Quine’’ quotes ‘Quine’.” (Cappelen /Lepore 2007, 123) Cappelen /Lepore 2007, 6f. – Beyond that, the line might be difficult to draw. Although we will probably want to quote expressions of possible languages, we should not go so far as to try to allow to quote all objects that possibly are linguistic expressions. But cf. Cappelen and Lepore, who explicitly reject the restriction of quotable objects to expressions (Cappelen /Lepore 2007, 22f.; 123f.; and 147ff.). E. g., Deiser 2002, 163f., who quotes Cantor’s definition from 1995, and Cantor 1967[1899], 113ff. Cf. Cappelen /Lepore 2007, 124.

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logical form of quotation expressions? 647 Taken as a whole, a quotation expression is a singular term and thus allows for the usual inferential moves like substitution by a co-referential singular term or existential generalization that are codified in quantified classical logic. E. g., from the fact that ‘YHWH’ has four letters we can infer the following: The tetragrammaton has four letters. Something has four letters. But from the fact that ‘YHWH’ has four letters, even though ‘YHWH’ does refer to the God of the Jewish religion and hence to something, we obviously cannot infer either of the following: ‘The God of the Jewish religion’ has four letters. 648 ‘Something’ has four letters. 649 These examples should be enough to illustrate that substituting co-extensional terms within quote marks is not in general truth-preserving, so that we cannot quantify into quote marks. In general, quote marks present an impenetrable barrier to the usual logical operations. Or, in our terminology, neither the quoted expression nor any one of its semantic parts is a semantic part of a quotation expression. E. g., ‘YHWH’ is not a semantic part of ‘‘YHWH’’, and ‘Socrates’ is not a semantic part of ‘‘Socrates is wise’’. Quotation expressions are semantically indivisible. From the fact that a quotation expression is syntactically but not semantically divisible, Tarski, Carnap, and Quine concluded that quotation expressions can be reduced to names or constants, which also have no semantic parts but in general do have syntactic parts. This reduction would explain why operating with our usual logical rules on ‘YHWH’ in ‘‘YHWH’’ is no more legitimate than operating with our logical rules on ‘cat’ in ‘cattle’, to give an example Quine was fond of. 650 But if a quotation expression were to have the same semantics as a name, it would be impossible to tell you my name by way of its quotation expression. Uttering the sentence ‘My name is ‘Martin’’, and thus following Frege’s and Quine’s advice to mark all mentioned expressions with quote marks, seems to be a perfectly good way to acquaint you with my name. According to the reductionist claim, however, it cannot be a way of telling you my name, because the claim entails that what I present to you is not my name, but a name of my name. And knowing a name is not yet knowing the named object (which in this higher-level scenario is itself a name). 651 We will return to the perplexing conundrum that quotation expressions, although 647

648 649 650 651

Recall from section 3.1 that what accounts for semantic parthood are inferential relations, and thus logical form. Künne uses the example of the tetragrammaton in a slightly different context; Künne 2007, 182f. For similar examples, cf. Cappelen /Lepore 2012, section 2. Quine 1980[1953], 140. This argument against the reduction of quotation expressions to names was given as early as 1938 by the Czech logician K. Reach, and the argument was later popularized by Elizabeth Anscombe (Reach 1938; Anscombe 1971[1959], 84; cf. Künne 2007[1983], 184).

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logically similar to names, cannot be reduced to names once we have learned more about the semantics of quotation. Let us take the semantic theory that we used in the preceding section to characterize names, descriptions, and indexicals, and apply it to quotation expressions. Thus, we can see quite easily that a quotation expression is rigid, context-free, and refers directly, and that these features are all entailed by the minimal theory. To illustrate the rigidity and context-freeness of quotation, we need only contrast, e. g., the description ‘the first sentence on page one of today’s edition of my favorite newspaper’ with a quotation of the sentence referred to by that description. While the reference of the description depends on the context (specifically, on the speaker and the day of utterance) and on contingent facts (what newspaper that speaker favors on that day, and what the first sentence on page one of that newspaper’s edition of that day is), the corresponding quotation expression will depend on none of these features. To show that the rigidity and context-freeness of quotation is entailed by the minimal theory we need only point out that clause (ii) of the minimal theory determines the reference of any given quotation expression without relativizing it to a circumstance of evaluation or a context of use (in fact, clause (ii) does not even mention circumstances or contexts). To see that a quotation expression is directly referential we need only make ourselves aware of how its rigidity and context-freeness comes about, which is not in an (to speak figuratively) accidental way, but de jure. E. g., ‘‘YHWH’’ refers to the tetragrammaton with regard to every circumstance of evaluation and at every context of use because it quotes the tetragrammaton. And this is also what the minimal theory says, where clause (i) and (ii) together entail that the quotation expression p‘α’q must refer to α. But there is more to be said about the direct referentiality of quotation expressions. When we contrasted names and indexicals, we observed that while both kinds of singular terms are directly referential, the direct reference occurs on different levels: While an indexical is directly referential only on the level of content, which first has to be determined by the character taken together with the context of use, a name is directly referential already on the level of character. In both cases the extension of the singular term is a direct constituent of its semantic value; for indexicals, this is so only on the level of occurrences (or tokens) of the expression, because the content will vary with the context of use, but for names, this is so already on the level of the expression as such (or, as a type), because that is what the character is connected to. 652 Now the surprising thing is that in the case of direct quotation, the direct reference occurs even earlier, or, on an even more basic level, because the extension of a quotation expression is a direct constituent of the quotation expression itself! This is in fact just what the principle of (Containment) says, which is a direct consequence of the minimal theory of quotation. So, whereas in the case of a typical directly referential expression the extension is a direct constituent of its semantic value (of the character of the expression as such, or of the content of an occurrence), in the case of a quotation expression the extension is a direct constituent of the expression itself. Let us use the label ‘super652

Cf. subsection 5.2.6.

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direct reference’ for this phenomenon. In order to illustrate and to bring out an interesting epistemic aspect, we will again tell a story of stages, which is meant in a figurative way. 653 What happens when a speaker encounters a singular term of her language that is new to her, and what is specific when that singular term is a quotation expression? This time, let us begin the story at a really early stage: 654 The first thing the speaker does is to grasp the syntactic shape of the expression she has encountered and classify it as an expression of her language. 655 Then she classifies it as a singular term; and thus she comes to a crossroads, because she will now want to know to which kind of singular term it belongs. As we are comparing quotation expressions to names, simple indexicals, and descriptions, there are four ways to choose from. Hence the story after this point has four parallel plotlines. What we want to show in our story of stages is that, among the four kinds of singular terms, a quotation expression provides the fastest route from a singular term to the object it refers to. From our earlier comparison of names, simple indexicals, and descriptions, we know that, in view of names being directly referential on the level of character, the route from a name to the object referred to is faster than the routes that start from a simple indexical or a description. 656 It will therefore suffice to contrast quotation expressions with names: When the speaker knows the syntactic shape of an expression and if she has classified it as a name, she therefore knows that it will refer directly to an object (she even knows that the direct reference occurs already on the level of character) – but at this stage she does not yet know to which particular object! However, when the speaker knows the syntactic shape of an expression and if she has classified it as a quotation expression, she already knows the object it refers to, because that object will invariably be the expression between the quote marks, and she is already acquainted with that expression because she already must have grasped the shape of the larger quotation expression it is contained in. Imagine, with regard to one of our main examples, two language users who are in a race to know the extension of the quotation expression ‘‘YHWH’’ and the name 653

654

655

656

Recall from section 4.6 that used like this a story of stages is no more than a metaphor to elicit intuitions about what determines what; the stages of the story are not to be understood literally as given in temporal of causal order, but as an illustration of the direction of grounding. We also used the device of telling a story of stages in section 5.2. We are interested in the lowest level of a grounding structure, and being low in a grounding structure corresponds to being early in the metaphor of a story of stages. It may not be possible to say in general what happens first, the grasping of syntax or the classification as belonging to a particular language. But in the case of written expressions of a large family of languages which both English and German belong to, the grasping of syntax precedes the classification as belonging to a particular language. This is witnessed by strings of letters like ‘bad’, ‘bank’, and ‘gift’, which are used to form words of more than one of those languages. The reason for this order is that, disregarding some diacritic signs and extra letters, the written languages of this family share the same alphabet (the Latin one) and rule of concatenation (of writing the letters beside each other). Cf. subsection 5.2.6.

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‘the tetragrammaton’, respectively. Clearly it is only the second one who might need to have a look at Wikipedia. Being aware of super-direct reference helps solve the conundrum that, because of the logical similarity of quotation expressions to names, it appears to be impossible to tell you my name by quoting it, although we know intuitively that this works very well. The point is that the logical similarity to names results from the semantic indivisibility of quotation expressions, but that is as far as the similarity between the two kinds of singular terms goes, because in the case of quotation expressions, their syntactic divisibility is directly connected to their semantics by the principle of (Containment). It is unproblematically possible to acquaint an addressee with a name by quoting it because the name is then contained in the quotation expression. We have described this mechanism when we illustrated the super-direct referentiality of quotation with a story: Getting acquainted with the quoted expression is part already of the first stage of getting to know a quotation expression, when we grasp it syntactically. This epistemic aspect of super-direct reference is basically the same phenomenon that Wolfgang Künne brings out by characterizing quotation expressions as “revealing designators”, where a singular term is a revealing designator if and only if one cannot understand the singular term without being able to identify the object it refers to. 657 Manfred Harth even goes so far as to call quotation a “non-lingual means of language” (“ein nicht-sprachliches Mittel der Sprache”) 658, because it is typical of all other linguistic devices that they stand for something in a mediated and arbitrary way. Thus it is super-direct reference that accounts for what we have called the perplexing side of quotation; it is a consequence of the semantics of quotation, which is easy to describe but which is also unlike the semantics of any other referential device. In fact, the super-direct referentiality of quotation can be used not only to argue against the claim that quoting is reducible to naming, but as an all-round argument against any proposal for reducing quotation to another referential device, at least under the presupposition that quotation is the only device of super-direct reference. 659 Of course, to be adequate an argument for a non-reductionist account of quotation would have to rest on detailed criticisms of its contemporary reductionist rivals, perhaps most importantly of Davidson’s demonstrative theory. 660 Here, I can

657

658 659

660

Künne writes: “A designator of an expression is [. . . ] revealing if and only if somebody who understands it can read off from it which (orthographically individuated) expression it designates. Thus the quotational designator ‘‘Das Ewigweibliche zieht uns hinan’’ is revealing, whereas the (co-designative) definite description ‘the last sentence in Goethe’s Faust’ is not.” (Künne 2003, 184) Harth 2002. The only viable candidate for another device of super-direct reference are true demonstratives. But even their reference is less direct than that of quotation expressions, because in the case of an occurrence of a true demonstrative, understanding the expression is not enough to know its referent (as it is in the case of a quotation expression), because the addressee must also understand the associated pointing gesture and perhaps even grasp the referential intention of the speaker. Davidson 1979; cf. Cappelen /Lepore 2007, 108–122; and Cappelen /Lepore 2012, section 3.3.

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only defer to the critical work 661 Cappelen and Lepore have done to support their claim that “[. . .] quotation – like every other legitimate linguistic category – is ineliminable from a language without a loss of expressive power. No other linguistic category in our language can serve the complete function of quotation.” 662

Thus a striking fact about the minimal theory of quotation is not that it is true – which most and perhaps all reductionists would grant –, but that it merits its name by being all that needs to be said concerning the semantics of quotation. 663

5.6 Quotation and the irreflexivity of proper parthood There is a mereological argument 664 that shows that no sentence can achieve selfreference with the help of direct quotation alone. When a sentence (derivatively) quotes some expression, then the quotation expression of that expression must occur in that sentence as a singular term, and must hence be a proper part of that sentence. Now, the minimal theory of quotation entails that any quoted expression is a proper part of its quotation expression (it is that part of it that occurs between the quote marks). 665 By the transitivity of proper parthood, a sentence in order to quote itself would thus have to be a proper part of itself. But it is mereologically impossible that something is a proper part of itself. Peter Simons expresses this mereological principle by noting that the relation of proper parthood is irreflexive: 666 “The most obvious properties of the part-relation [i. e., the relation of proper part] are its asymmetry and transitivity, from which follows its irreflexivity. [. . . ] These principles are partly constitutive of the meaning of ‘part’, which means that anyone who seriously disagrees with them has failed to understand the word.” 667

So, for mereological reasons, there can be no Liar sentence that refers to itself with quotation alone. 668 In order to illustrate the mereological argument, we can try and imagine what a Liar sentence that would achieve self-reference by way of quotation would look like. It would have to be infinitely long: 661 662 663 664

665 666

667 668

Cappelen /Lepore 2007, especially 81–122. Cappelen /Lepore 2007, 123. Cappelen /Lepore 2007, 124. By ‘mereological’ we mean ‘pertaining to the notion of parthood’. Mereology is the systematic study of the notion of parthood and related notions. Cf. Simons 1987; Ridder 2002; and Varzi 2016. Cf. the principle of (Containment) in section 5.5. A relation is called irreflexive (on a collection of objects) if and only if no object (that is among that collection) stands in that relation to itself. Simons 1987, 10f. and cf. Ridder 2002, 33 & 35. With regard to the question of sentential self-reference, direct quotation is thus a kind of mirror-image of the indexical ‘this sentence’ (cf. section 5.3): Where a quotation expression can be used to refer to all sorts of expressions, but never to refer to a sentence it occurs in, the phrase ‘this sentence’ can only be used for the latter purpose.

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‘The sentence ‘The sentence ‘The sentence ‘The sentence ‘The sentence . . . . . . is not true’ is not true’ is not true’ is not true’ is not true.’ But, even though there is no upper limit to the length of a sentence, there are no sentences of infinite length in any natural language – nor are there infinite sentences in any language of classical logic. 669

5.7 Formal quotation and the well-foundedness of syntax But what about formal languages that are non-classical? There are after all infinitary logics, the formal languages of which do contain infinite sentences, e. g., conjunctions and disjunctions of infinitely many sentences. 670 An infinite sentence is construed not as a concrete object of infinite length, but as an abstract object with infinitely many constituents. As it is the abstractness of formal expressions that makes it possible to construct formal languages that differ so strikingly from all possible natural languages with regard to the cardinality of the semantic parts of some of their expressions, we are well advised to have a look at what this abstractness entails for the properties of formal quotation. Is it governed by similar mereological principles as natural language quotation? Does formal quotation, in certain formal languages, allow for an expression to contain its own quotation expression? In order to transfer to a formal language the notion of a quotation expression which we described for natural language 671, we have to look at the relation between an expression and its parts, concatenation. In a written natural language, concatenation is achieved, roughly, by writing one expression more or less directly beside another (in the Latin script, on a horizontal line and progressing to the right). In a spoken natural language, although its syntactic elements are less crisp than letters, it is similarly easy to give a rough characterization; here concatenation is achieved by speaking one expression more or less immediately after another. With regard to a formal language the expressions of which are construed as abstract objects, we can only say that concatenation is an abstract relation between an expression and its parts that has certain properties. Therefore we need to be more explicit. First of all, we should distinguish between a full and a partial notion of concatenation. 672 A statement of full concatenation is of the form ‘expression α is the concatenation of expressions γ1, γ2, . . . to γn’, e. g., ‘The expression ‘Anna’ is the concatenation of the expressions ‘An’ and ‘na’’. 673 A statement of partial concatenation is of the form ‘expression α is concatenated from expression β’, e. g., ‘The 669

670 671 672

673

A similar mereological argument, this time presupposing the well-foundedness (cf. section 5.7) of the relation of proper part, will show that there cannot be Yablo sequences of sentences (cf. section 1.3) that employ only the referential device of direct quotation. Cf., e. g., Bell 2009. Cf. section 5.5. The contrast between our notions of full and partial concatenation is similar to the contrast Kit Fine makes between the notions of full and partial ground (Fine 2012, 50ff.). We will understand every statement of the form ‘expression α is the concatenation of expressions γ1, γ2, . . . to γn’ in such a way that the order of the γ1, γ2, . . . to γn matters. It is not the case, e. g., that ‘Anna’ is the concatenation of ‘na’ and ‘An’.

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expression ‘Anna’ is concatenated from the expression ‘na’’. The full notion of concatenation is basic, and the partial notion can be defined in terms of it: An expression α is concatenated from an expression β if and only if there are some expressions γ1 to γn such that alpha is the concatenation of γ1 to γn and β is identical to one of the γi. For the purpose of the present section, which is to study whether a formal language can be such that one of its expressions quotes itself, our account of concatenation need not be formal. 674 And it need not be comprehensive; it will suffice to list a selection of structural properties of full and partial concatenation. A principle of extensionality for full concatenation is implicit already in the use of ‘the’: If an expression α is the concatenation of some expressions γ1 through γn and an expression β is the concatenation of the same expressions γ1 through γn, then α = β. The converse does not hold; ‘Anna’, e. g., is the concatenation of ‘An’ and ‘na’, and also the concatenation of ‘A’, ‘n’, ‘n’, and ‘a’. For partial concatenation, we have a principle of transitivity: If an expression α is concatenated from an expression β which is concatenated from an expression γ, then α is concatenated from γ. 675 And so on. It is the partial notion of concatenation (being concatenated from) that corresponds to the mereological notion of parthood in the case of written or spoken language. (That is why it makes sense to extrapolate for the general case that partial concatenation is transitive.) However, while concatenation in written natural language is governed by the principles of mereology as applied to things and in spoken natural language is governed by the principles of mereology as applied to events, 676 we prima facie cannot say anything similar about concatenation in a formal language. This is due to the fact that formal expressions are neither things nor events, but are construed as abstract objects, 677 to which the principles of mereology do not in general apply. 678 We also have to adjust the minimal theory of quotation accordingly, because it implicitly involves the notion of concatenation: By saying that the quotation expression of an expression α is p‘α’q, 679 we lay down that the quotation expression of α is the concatenation of the left quote mark, the expression α, and the right quote mark. Thus transferring the notion of quotation to a formal language is not 674

675 676 677 678

679

In formal theories of syntax, concatenation is usually conceived to begin with as a triadic relation, which corresponds to construing the operation of concatenating as forming an expression α from a pair of given expressions β and γ. To express that α is the concatenation of β and γ we write pα = β*γq. Like addition and multiplication, concatenation is associative: α*(β*γ) = (α*β)*γ; cf., e. g., Corcoran et al. 1974, 628. Therefore we can speak, for any number of expressions, of the concatenation of these expressions – while the order of the expressions matters, the order in which they are concatenated in a pairwise fashion does not. Thus our informal notion of full concatenation, which is a one-many relation, can be reduced to the formal notion of concatenation, which is triadic. This will be entailed by a more complicated principle of transitivity for full concatenation. Cf. Pleitz 2011b & 2015b. Cf. Pleitz 2010a. On the abstract nature of the relation of concatenation that holds between the expressions of a formal language, cf. Shapiro 1997, 98ff. and Pleitz 2010a, 195 & 200f., especially fn. 32. Cf. clause (i) of the (Minimal Theory of Quotation) in section 5.5.

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merely a matter of exchanging one pair of quote marks (‘‘. . .’’) for another (‘«. . .»’), but strictly speaking calls for rephrasing the minimal theory of quotation in terms of the abstract relation of concatenation: (Formal Variant of the Minimal Theory of Quotation) For any expression α, (i) if an expression is the concatenation of ‘«’, α, and ‘»’, then it is a formal quotation expression of α, and (ii) a quotation expression of α refers to α. As the role that in natural language is played by the relation of being the mereological sum of in a formal language is played by the abstract relation of being the concatenation of, we are barred from using the mereological argument to show that a formal sentence cannot quote itself. But maybe we can find some requirement on the relation of concatenation that corresponds to the mereological principle that no object can be a proper part of itself. Let us first note a respect in which formal quotation is similar to its written counterpart, more specifically, a counterpart of the principle of (Containment): As the concatenation of ‘«’, α, and ‘»’ stands in the relation of partial concatenation to each one of the three expressions ‘«’, α, and ‘»’, 680 we can gather that a formal quotation expression (an expression p«α»q) stands in the relation of partial concatenation to the expression it quotes (the expression α). Therefore, if a sentence of some formal language were to refer to itself by formal quotation alone, then it would (in view of partial concatenation being transitive) stand in the relation of partial concatenation to itself. So we have to ask: Can an expression stand in the relation of partial concatenation to itself? And, more generally: Can the formal relation of partial concatenation be non-well-founded? * A relation is well-founded (with respect to a given collection of objects) if and only if every chain of an object (that is among that collection) standing in that relation to an object (that is among that collection) terminates. In other words, a relation R is well-founded (on a collection) if and only if there are no infinitely descending R-chains (within that collection). Otherwise it is non-well-founded. 681 E. g., the relation expressed by the two-place predicate ‘x rests on y’ is well-founded with respect to the bricks of a house because each chain of one brick resting on 680 681

That is a consequence of how partial concatenation can be defined in terms of full concatenation. Peter Aczel, e. g., defines: “A relation R is well-founded if there is no infinite sequence a0, a1, . . . such that an+1 R an for n = 0, 1, . . . .” (Aczel 1988, 116) Our definition differs from the customary mathematical definition of the notion of well-foundedness (cf., e. g., Smullyan/Fitting 2010, 99ff.), which is as follows: A relation R is well-founded on a class if and only if every set of objects that are members of this class contains an R-minimal element, i. e., an object that does not stand in the relation R to any element of that set. Given the axiom of choice, this definition is equivalent to the statement that there are no infinitely descending R-chains used in our definition. Note that in a system without any infinitely descending sequences, there may still be descending chains of infinite length, as is witnessed by the relation of being smaller than on sequences of ordinal numbers that extend into the transfinite, e. g., 1, 2, 3, . . . ř, ř+1, . . . .

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another will end in the basement; but the relation expressed by the predicate ‘x is the successor of y’ is non-well-founded with respect to the integers (i. e., the numbers . . . , –3, –2, –1, 0, 1, 2, 3 . . .); and there are some academics on whom the relation expressed by the predicate ‘x cites y’ is non-well-founded because there are citation cycles among their work. An object is well-founded with respect to a relation R (and a collection of objects) if and only if there is no infinitely descending R-chain starting from that object (within that collection); otherwise it is non-well-founded. Obviously, if a relation is well-founded, every object is well-founded with respect to that relation. If a relation is non-well-founded, there will be some objects which are non-well-founded with respect to that relation, but in general there may also be some other objects which are well-founded with respect to that relation. In our examples, and with regard to the respective relations of resting on, succession, and citing, we can say that every brick is well-founded, every integer is non-well-founded, and while some academics are non-well-founded, there are probably other, well-founded academics. On an historical note, one area where the question of well-foundedness is important is the development of set theory. Here, the relevant relation is elementhood: We call a theory of sets well-founded if and only if it entails that the relation of being an element of is well-founded on all sets, and non-well-founded otherwise. In the early 20th century, Dmitry Mirimanoff distinguished between well-founded and non-well-founded sets: “Let E be a set, E0 one of its elements, E00 any element of E0 , and so on. I call a descent the sequence of steps from E to E0 , E0 to E00 , etc. . . . . I say that a set is ordinary when it only gives rise to finite descents; I say that it is extraordinary when among its descents there are some which are infinite.” 682

Now, non-well-founded sets truly are extraordinary because prominent exemplars lead to paradox. E. g., Cantor’s paradox is centered on the set of all sets, which (crucially) is an element of itself, and the characterization of a set as the set of all sets that are not an element of themselves leads only to Russell’s paradox if elementhood is non-well-founded. 683 Accordingly, the axiomatic set theories which have become standard are well-founded. However, in the late 20th century, Peter Aczel worked on non-well-founded axiomatic set theories, and showed them to be consistent if the standard well-founded set theory is. 684 Aczel’s work has lent some respectability to non-well-foundedness in general, and, in some quarters, it has led to a tendency to try out non-well-foundedness in an area formerly thought to be well-founded, an activity which we might call Aczelism (without wanting to insinuate that it is exhibited by Aczel himself with regard to any issue besides set theory). 685 682 683

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Mirimanoff 1917; quoted from Aczel 1988, vii. In a well-founded set theory, Russell’s description is empty. (As in well-founded set theory every set is not an element of itself, one might at first think that Russell’s description here refers to Cantor’s set of all sets, but in view of well-foundedness the latter description is also empty.) Aczel 1988; especially 33ff. E. g., Barwise /Etchemendy 1987 (according to which the relation of being a constituent of a proposition is not well-founded); Barwise /Moss 1996 (according to which all kinds of relation usually thought

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An area almost anyone surely thinks of as well-founded is syntax; and we will see in a moment that Aczelism can lead to some astonishing results here. 686 The relevant relation is of course the one we are concerned with in this section, concatenation. We call a theory of syntax well-founded if and only if it entails that the relation of (partial) concatenation is well-founded on all expressions, and non-wellfounded otherwise. 687 And we call an expression ordinary or well-founded if and only if there is no infinitely descending chain of concatenation that starts from it; otherwise we call it extraordinary or non-well-founded. 688 The well-foundedness of syntax excludes all extraordinary expressions, among them every expression that is concatenated from itself. Thus well-founded syntax is governed by a counterpart of the mereological principle that no object can be a proper part of itself, and like the mereological principle it will lead to the impossibility of Liar sentences that refer to themselves by quotation alone, even when it is formal quotation. But do we have good reason to require that concatenation is well-founded? Let us have a look at what happens when we drop this requirement. In non-well-founded syntax, there are extraordinary expressions, and usually there will also be ordinary ones. Let us, in order to have a definite framework that allows to give some examples, work with the language of classical logic, for most of the time sticking to its propositional part, but of course without the requirement that concatenation is well-founded, which is implicit in the usual presentation of the (well- and ill-formed) expressions of that language as strings of symbols. 689 Now we can give examples for both ordinary and extraordinary expressions: (E1) the ill-formed ‘¬(’, which is the concatenation of ‘¬’ and ‘(’, (E2) the well-formed expression ‘p’, which is basic insofar as it is not concatenated from any expressions, (E3) the ill-formed ‘q’ which is the concatenation of ‘q’ and ‘r’, and (E4) the well-formed ‘s’ which is the concatenation of ‘s’, ‘∧’, and ‘t’. We need only follow the relation of concatenation to see that while everything is as usual in the first two examples, there are closed loops in the other two examples.

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to be well-founded are not well-founded); Cotnoir/Bacon 2011 (according to which the relation of being a proper part is not well-founded); and Priest 2014, 167ff. (according to which the relation of metaphysical grounding is not well-founded). Almost anyone. I would like to thank Dave Ripley for a stimulating and helpful conversation, where he took the role of a staunch advocate of the admissibility of non-well-founded syntax. From now on we will often drop the byword ‘partial’, taking it as understood that the question of well-foundedness is posed in terms of partial concatenation, but will have consequences for the corresponding notion of full concatenation. Note that the syntax of an infinitary language can be well-founded. This is due to the fact that the notion of well-foundedness does allow for a chain to be infinitely long, if it terminates. An infinite conjunction, e. g., will contain infinitely long chains of concatenation, but each one of them will terminate (e. g., in one of the atomic conjuncts). Cf. section 3.3. – If we wanted to carry out this process of de-well-founding the language of classical logic in a technically adequate way, we would also have to adjust the definition of well-formed expressions in a way that allows for some extraordinary expressions to be well-formed.

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Thus, while the expressions (E1) and (E2) are ordinary, (E3) and (E4) are extraordinary. Note that we can no longer specify the relations of concatenation an abstract expression stands in by displaying a single written token of it; this is witnessed by contrasting concrete inscriptions of the quotation expressions ‘‘p’’ and ‘‘q’’, which display single tokens, respectively, of the ordinary expression (E2) and of the extraordinary expression (E3). But in the cases at hand we can work with identity statements built from quotation expressions: 690 (E3) is determined by stating that what holds of it is (E3) = ‘q’ = ‘qr’ (and nothing else 691), and (E4) by stating that (E4) = ‘s’ = ‘s ∧ t’ is a complete specification. 692 The method of determining extraordinary expressions by giving identity statements with quotation expressions has the advantage that it makes their extraordinariness visible, because for all ordinary expressions, the schema ‘‘. . .’ = ‘. . .’’ delivers a true sentence if and only if both occurrences of the three dots are filled with occurrences of the same string of letters. 693 Another extraordinary expression that is not ruled out in our non-well-founded but otherwise classical language is a Liar sentence that quotes itself: (E6) = ‘u’ = ‘¬True(«u»)’ Here, what looks like simple propositional letter (‘u’) is internally structured because it is the concatenation of the negation symbol, the formal truth predicate, and a quotation of itself. 694

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Note that these identity statements augment our language and involve the usual quotation of written language – not the formal quotation which might behave strangely in the context of non-well-founded syntax. Nothing substantially else, that is – because the single identity statement that ‘q’ = ‘qr’ already entails an infinity of identity statements, i. e., that ‘q’ = ‘qr’ = ‘qrr’ = ‘qrrr’ = . . . . But we need not go into details, because shortly we will be arguing against non-well-founded syntax. We will take it as understood from now on that the given identity statements are a complete specification. For our method of specifying an extraordinary expression by way of an identity statement with distinct quotation expressions (i. e., with quotation expressions which in English would be seen as distinct) to work, two points are crucial. Firstly, what a quotation expression refers to is the expression as such (as a type) that that is contained between the quote marks, not to that particular occurrence (or token) of it. Secondly, we have adopted a convention (implicitly already when we motivated the formal variant of the minimal theory of quotation) according to which a quotation expression is meant to be understood just like the description ‘the concatenation of . . . , . . . , . . . and . . .’ that breaks down the expression between the quote marks into its smallest parts. E. g., we understand the quotation expression ‘‘Anna’’ just like the description ‘the concatenation of ‘A’, ‘n’, ‘n’, and ‘a’’. This is a natural extrapolation from spoken and written languages where concatenation is well-founded. I would like to thank Dave Ripley for forcing me to get clearer on this point. As ‘u’ = ‘¬True(«u»)’ = ‘¬True(«¬True(«u»)»)’ = ‘¬True(«¬True(«¬True(«u»)»)»)’ =. . ., we now do have a Liar sentence that can be illustrated as at the end of section 5.6: ‘¬True(«¬True(«¬True(«¬True(«¬True(«¬True(«¬True(«. . . »)»)»)»)»)»)»)’

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It gets worse, because we can also form the simpler extraordinary expression: 695 (E7) = ‘l’ = ‘¬l’ The sentence letter ‘l’ is the concatenation of the negation symbol and the sentence letter ‘l’. Using the clause for classical negation that a negation is true if and only if the negated sentence is not true, we can show immediately that (E7) is true if and only if not true. The paradoxical (E7) strictly speaking is not a Liar sentence because it does not ascribe untruth to itself; 696 in fact, it neither refers nor ascribes at all because below the sentence level it has no internal structure whatsoever – no predicate and no term is to be found in it. 697 However, it is Liar-like enough: Not only does the question for its truth value land us in a contradiction, but it also has a negative and reflexive nature, 698 for although it does not refer to itself, it surely (albeit in an unusual way) reflects on itself. 699 Non-well-founded syntax thus allows to construct a Liar sentence that quotes itself, and it even allows to construct a very Liar-like sentence without using a predicate of untruth or falsity and without employing any device of reference! But unwelcome though that may be, in the present context of an investigation into the Liar paradox any argument against non-well-founded syntax that is based on this fact alone is open to the charge of ad hocness. We will have to look elsewhere when we want to find reasons to require that the relation of concatenation is well-founded. * Our argument for the well-foundedness of concatenation will take the form of stamping the foot and complaining that the system of abstract objects characterized above as non-well-founded syntax is just too weird to be the syntax of something that is a language more than in name. This weirdness can be brought out by several observations; and in view of the topic of this section we might as well start with what happens to the referential device of quotation when applied to extraordinary expressions. If syntax is non-well-founded, quotation would no longer be revealing; i. e., it would no longer suffice to understand a particular written token of a quotation expression to know the expression it quotes. 700 E. g., neither the quotation expression ‘‘l’’ nor the quotation expression ‘‘¬l’’ suffices to identify expression (E7). In other words, quotation would no longer exhibit the super-direct referentiality which we saw to be characteristic of its semantics. 701 Note that this observation concerns much more than a possible device of formal quotation in some language with a non695

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The example is due to Graham Priest, who used it in his talk at the workshop “Paradoxes of Truth and Denotation” organized by the Logos group at the university of Barcelona in May 2011. Cf. section 1.2. The occurrence of the letter ‘l’ in the sentence ‘¬l’ is not a term – otherwise ‘¬l’ would be ill-formed. Cf. section 1.7. With equal ease, non-well-founded syntax allows to form counterparts of Curry sentences (cf. section 1.5), e. g., (E8) = ‘c’ = ‘c → 1 + 1 = 2’, which allows to show that ‘1 + 1 = 2’ is a theorem of propositional logic, without using a single arithmetical axiom. Cf. section 5.5. Cf. section 5.5.

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well-founded syntax, because it crucially concerns the device of quotation that is part of the very written language we use: Despite its well-founded syntax, English (and in fact every language) would loose the device of quotation the semantics of which we described in the preceding chapter if we gave up the requirement that extraordinary expressions do not exist in any possible language. 702 There are further unprecedented problems. We are accustomed to being able to identify any expression of a language by producing a token of it, be it when displaying the expression or when using it. We can, e. g., identify the expression ‘accustomed’ by pointing out that it is used as the third word in the preceding sentence. And this feature is shared by all better-known formal languages. E. g., we can identify a sentence that expresses that one plus one equals two by producing the inscription ‘1 + 1 = 2’. Arguably, this possibility of using a token to identify its type is an essential feature of any use of expressions. It would be lost, however, in non-well-founded syntax. Returning to the example of the extraordinary expression (E7), writing down ‘l’ does not give all of its internal structure, nor does writing down ‘¬l’ (and even writing down the identity statement ‘‘l’ = ‘¬l’’ does not suffice as long as we do not add that that is all). The problem that extraordinary expressions cannot be written down 703 does not only concern the production of tokens, its root is to be found already on the type level. This can be shown by subjecting extraordinary expressions to a requirement George Boolos formulates for formal languages: “[. . .] requirement of sequentiality: expressions must be codable as functions from a finite segment of the natural numbers into the set of symbols of the language, i. e., as finite sequences of symbols.” 704

An extraordinary expression would not even be sequential if the requirement of finiteness was dropped, because more than one sequence of symbols is needed to determine it. This explains why it cannot be tokened in written or spoken language, which require a (finite) linear order of the symbols that make up an expression. – Note that criticizing non-well-founded syntax by subjecting it to the requirement of sequentiality is not an ad hoc move. For Boolos originally formulates the requirement in a theoretical context where he is concerned with formal quotation, but not with whether syntax might be non-well-founded. 705 A further reason against non-well-founded syntax concerns a general picture we have according to which all the expressions of a language are constructed from a 702

703 704 705

Observe that the impossibility of picking out an extraordinary expression by way of a single quotation expression of our language goes way beyond certain difficulties that might occur when we quote expressions of another language that has a different principle of concatenation, like writing from right to left, or even in the two-dimensional fashion of Frege’s logical notation. Although a quotation expression in these cases would break the flow of writing from left to right in our language of use, we could still make use of a picture of the quoted expression. Trying to display an extraordinary expression, in contrast, is even more difficult than trying to give an adequate picture of a higher-dimensional object, like a hypercubus. Neither can they be uttered. Boolos 1998, 397. Boolos 1998, 392ff.

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collection of basic building blocks which are independent of each other. In written language, we have a finite number of letters which allow to construct all expressions simply by writing them beside each other in the appropriate manner, and spoken language is not so different. In a formal language, we standardly define all expressions recursively, by specifying an alphabet and a finite number of formation rules. As a consequence, we can analytically break down every expression into the concatenation of an ordered collection of some of the basic building blocks. If syntax was non-well-founded, this would no longer be possible. If there were nonwell-founded expressions in a language, it would no longer be possible to break up all its expressions into basic building blocks such that each one of them would be independent of all others, because there would be infinitely descending chains of concatenation. This would be bad, because the picture of language as being analyzable into basic building blocks is connected to an essential characteristic of all languages, which has to do with how every language is more than a mere system of objects, namely a tool that can at least in principle be used by intelligent beings. This characteristic should be shared even by the systems of abstract objects that we study as formal languages. * In sum, we see that what we have described as non-well-founded syntax is just too weird to characterize any language that merits the name. Hence, concatenation should be well-founded. 706 This allows to generalize the mereological argument against the existence of sentences which refer to themselves by quotation alone. As a sentence that quotes an expression stands in the relation of concatenation to that expression, and as concatenation is well-founded, no sentence can quote itself, and there can be no non-well-founded chains of sentences quoting each other. 707 Hence there are no Liar sentences with formal quotation, and no Yablo sequences of sentences with formal quotation, either. And of course the Liar-like sentence that is identical to its own negation is also ruled out. This ensures the completeness of the spectrum of Liar-sentences that was given in section 5.4. In addition to this being a welcome result, our detailed discussion of a purported formal language with a non-well-founded syntax in this section (apart from hopefully being interesting in its own right) has served the important function of introducing the notion of wellfoundedness, which will be important again in part III. With the present result about formal quotation we have reached a point where we can be sure that the device of direct quotation cannot ever be used on its own to construct self-referential sentences. 708 It thus turns out that for the purposes of 706

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Later on, in chapter 11, we will argue against the ontology comprised of expressions being non-wellfounded, i. e., against a metaphysical theory that entails that the relation of grounding is non-wellfounded. The well-foundedness of ontology arguably will provide a further reason against non-wellfounded syntax because concatenation plausibly entails grounding (an expression is grounded in every expression it is concatenated from). This is also noted by Tim Maudlin, who does not explicitly consider non-well-founded syntax, but observes that every (proper) semantic part of an expression contains fewer symbols than the whole. Cf. Maudlin 2004, 37f. Cf. sections 5.5 through 5.7.

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this study, the main interest of the referential device of direct quotation is that of providing a contrast for those referential devices that at least prima facie do allow to construct self-referential sentences. But although its own role in the construction of self-referential sentences cannot be that of a protagonist, we will see shortly that it can at least play the role of a supporting actor (cf. section 6.1). It is for that reason that quotation will provide a valuable contrast to the device of Gödel corners, which appears to be used in what is often seen as an impeccable form of self-reference, Gödel’s method of the arithmetization of syntax (cf. sections 6.2 through 6.4). With this we turn to the question whether there are Gödelian Liar sentences.

Chapter 6

Is There a Gödelian Liar Sentence?

Willard Van Orman Quine: “‘The result of putting the quotation of ‘The result of putting the quotation of w for S1 in w is not a theorem’ for S1 in ‘The result of putting the quotation of w for S1 in w is not a theorem’ is not a theorem.’ [. . .] This is more intelligible than it looks.” 709 Ulrich Blau: “Die Selbstreferenz ‘Dieser Satz’ ist auf nichtindexikalische Weise ausdrückbar nach Gödels Rezept”. 710

In the last chapter, we have characterized the semantics of the referential devices that might be used to construct self-referential sentences: Names, descriptions, indexicals, and their respective formal counterparts. We have seen that quotation, although the paradigmatic device of referring to expressions, on its own cannot be used to construct a self-referential sentence because thus the sentence would have to be a proper part of itself. In the present chapter we will look at certain constructions where quotation is combined with other devices in such a way that it appears to lead to self-reference (in section 6.1). These help complete the overview of the spectrum of potential devices of self-reference. They also provide a good starting point for the main topic of this chapter, which is the presentation and discussion of what is often seen as an impeccable method of achieving self-reference, or of achieving something very much like it: Gödelization. After outlining its technical side (in section 6.2), we will give a theory of meaning for Gödelian constructions (in section 6.3), on the basis of which we will come to see that we need to give a differentiated answer to the question whether there can be Gödelian self-referential sentences. It will include a surprisingly clear ‘no’ to the question that is the title of the present chapter (in section 6.4).

709 710

Quine 1981[1940], 307. “Following Gödel’s recipe, the self-reference of ‘this sentence’ is expressible in a non-indexical way.” (Blau 2008, 398; my translation)

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6.1 The Quinean and the Smullyanesque variant of the Liar paradox As long as quotation is on its own, it cannot be used to construct a self-referential sentence. 711 But it can be used as a device of sentential self-reference when it has some help from a referential device of another kind. This is witnessed by an ingenious (if initially unintelligible) construction that is due to Quine: “‘Yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation.” 712

According to its surface grammatical structure, Quine’s sentence is not about any sentence at all but about the expression ‘yields a falsehood when appended to its own quotation’, of which it says that it falls under the notion expressed by the complex predicate ‘. . . yields a falsehood when appended to its own quotation’. But intuitively, Quine’s sentence seems to be about itself ; and this intuitive understanding of the sentence can be backed up by looking at one of its consequences. On a natural understanding of the locutions ‘yields a falsehood’ and ‘when appended to its own quotation’, Quine’s sentence clearly entails what it does not say outright, namely that the following sentence is false: ‘‘Yields a falsehood when appended to its own quotation’ yields a falsehood when appended to its own quotation.’ As that is, again, Quine’s sentence, we have to acknowledge that it (albeit indirectly) ascribes falsity to itself and thus we should classify it as a Liar sentence. 713 Let us accordingly speak about the Quinean variant of the Liar paradox. And let us, to have a label for what goes on here, call the sense in which we can say that Quine’s sentence refers to itself Q-reference and the sense in which we can say that Quine’s sentence ascribes falsity to itself Q-ascription. The letter ‘Q’ is meant to be mnemonic of both ‘Quine’ and ‘quotation’. But why should we bother with the Quinean variant of the Liar paradox, in view of it being so obviously contrived? And why should we deal with Q-reference and Q-ascription? There are several reasons. First off, we saw in the first part of the study that, from a logical point of view, any Liar sentence presents us with the most serious of problems; 714 so we had better prepare the way for a solution to the Quinean variant as we did before by having a look at the semantics of the variants with a name, indexical, description or quantificational phrase. We also need to show that Quine’s Liar sentence does not present a counterexample to our claim that the feasible ways to achieve sentential self-reference are by name, by indexical, by description, or by a quantificational phrase; 715 i. e., we need to classify Q-reference 711 712

Cf. section 5.6. Quine 1976a, 7. – Douglas Hofstadter explains Quine’s construction at length (e. g., Hofstadter 2007, 139ff.). Following him, we could also use a slightly different phrasing: ‘‘Preceded by itself in quote marks yields a falsehood’ preceded by itself in quote marks yields a falsehood.’

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Cf. section 1.2. Cf. chapters 1 through 3 and especially sections 3.8 and 3.9. Cf. section 5.4.

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within the spectrum of referential devices described in the preceding chapter. But most importantly, understanding Q-reference will help to prepare the discussion of self-reference with the help of Gödel numbers in the next three sections (6.2 through 6.4). * What is the logical form of Quine’s sentence? How can we represent the selfreferentiality of Quine’s Liar sentence and Q-reference in general with the help of a formal language? We can go two ways, depending on whether we want to respect either the literal meaning indicated by the surface grammar of Quine’s sentence, or our intuitive understanding according to which it is like a Liar sentence in ascribing falsity to itself. When we choose the first way, the logical form of Quine’s sentence is easy to give. For according to its surface grammar, Quine’s sentence is about the expression ‘yields a falsehood when appended to its own quotation’, which is also used in the sentence as a predicate, so that its logical form should be something like pΦ(«Φ(x)»)q. (We obviously need formal quote marks, ‘«. . .»’, in the language we use to give the logical form.) Note, however, that thus all that is interesting about Quine’s sentence is now hidden in the semantics of the open formula pΦ(x)q! When we choose the second way, we get a chance to make explicit what is interesting about Quine’s sentence and to respect the intuition that it ascribes falsity to itself, but we will encounter some obstacles because we will have to disentangle the falsity predicate from the complex predicate ‘. . . yields a falsehood when appended to its own quotation’ and to attach what remains of the predicate to the term. Luckily, we need only follow Raymond Smullyan, who has already been there and done the formal work which shows how to parse Quine’s sentence. 716 Before we turn to Smullyan’s work on self-reference in formal languages, however, we should have a look at a specific problem that concerns every attempt to give the logical form of Quine’s sentence in a language of classical logic. This problem occurs already when we try to understand the result of our first way of giving the logical form of Quine’s sentence, the formal sentence pΦ(«Φ(x)»)q. When we try to formulate what the open formula pΦ(x)q has to mean for the sentence pΦ(«Φ(x)»)q to be a formalization of Quine’s Liar sentence, something curious happens. The open formula pΦ(x)q is meant as a formal counterpart of the complex natural language predicate ‘. . . yields a falsehood when appended to its own quotation’. But when we use this predicate in the usual direct translation from natural to formal language, i. e., when we understand the open formula pΦ(x)q as meaning something like ‘x yields a falsehood when appended to the formal quotation expression of x’, 717 we seem to be stuck. For what our formal sentence pΦ(«Φ(x)»)q thus appears to

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Cf. the end of the present section for a remark on the historical connection between Smullyan and Quine. In section 5.6 we defined the formal quotation expression of x as the concatenation of ‘«’, x, and ‘»’ (in that order).

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Q-refer to is the expression p«Φ(x)»Φ(x)q, which is ill-formed and thus far from being a falsehood. 718 The root of the present predicament is that what we can call semantic concatenation is realized by quite different syntactic operations in English and in the formal languages of quantified classical logic. By ‘concatenating semantically’ we mean forming a meaningful expression from meaningful expressions; and what we are interested in here is how a predicate and a (singular) term are semantically concatenated into a sentence. Now, while the semantic role of forming a meaningful sentence of subject-predicate structure is played in English by the syntactic operation of appending the predicate to the singular term, it is played in classical quantified formal languages by a different (and more complex) syntactic operation, namely by substituting the term for all occurrences of an unbound variable in the open formula which represents the predicate. Therefore, if we wanted to find a counterpart of Quine’s sentence in a language of quantified classical logic that is intuitively self-referential in a similar way, we would have to exchange the operation of appending for that of substituting. 719 But although it is possible to find a meaning for the open formula pΦ(x)q that respects this fact, 720 things would get just too messy for this to be of any more help in elucidating the intuitive selfreferentiality of Quine’s sentence and the phenomenon of Q-reference in general. In order to evade the said messiness, Smullyan in his early article on “[formal] languages in which self reference is possible” 721 transfers into a formal setting the welcome property of English so cleverly exploited by Quine, that the syntactic operation of appending a predicate to a singular term has the semantic role of fusing them semantically into a sentence of subject-predicate structure. He does this, simply, by specifying a formal language with a similar property (only with a reverse order of appendation). In it there is no need for substitution because functors and predicates are not represented by open formulas, but are primitive expressions which are unsaturated on the right hand side, so that the concatenation of a functor and a term is a term and the concatenation of a predicate and a term is a sentence. In a written version of Smullyan’s language, both predicates and functors are thus applied to a term simply by writing them to the left of the term. E. g., while the English sentence ‘The mother of Socrates is wise’ could be formalized as ‘W(m(s))’ in classical logic, in Smullyan’s language its logical form could be given as the simpler ‘Wms’.

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The same goes for its direct natural language counterpart ‘‘x yields a falsehood when appended to the quotation of x’ x yields a falsehood when appended to the quotation of x’. Thus we are in effect moving backwards to Gödel’s formal construction that Quine wanted to explain with the means of natural language. See the historical remark at the end of the present section. For the formal sentence pΦ(«Φ(x)»)q to be intuitively self-referential and ascribe falsity in the required way, the open formula pΦ(x)q would have to mean the same as ‘The result of substituting the quotation expression of the expression x for the variable ‘x’ in the expression x is false.’ Cf. Quine 1981[1940], 307. Smullyan 1957; cf. also Smullyan 1994.

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As Smullyan wants his language to allow self-reference, he adds formal quote marks, ‘«. . .»’, and a functor, ‘Norm’, the semantics of which is governed by the norm function, i. e., the function which maps each expression to the concatenation of itself and its quotation expression, i. e., it maps each expression ϕ to the expression pϕ«ϕ»q. 722 Smullyan’s ‘Norm’-functor can be described in the following way: (Syntax of ‘Norm’) If α is a term, so is pNormαq. (Semantics of ‘Norm’) If the term α refers to an expression, then the term pNormαq refers to the norm of the expression referred to by α. Or in formal terms (and here our notation grows clumsy): JpNormαqK = pJαK«JαK»q.

Given the role appendation has in Smullyan’s language, it is clear that for a predicate Φ and a term α, pΦαq is a well-formed sentence of that language. The semantics of such sentences of predicate-term-structure is easily transferred from the extensional semantics of classical logic: 723 For a predicate Φ and a term α, the sentence pΦαq is true if and only if the object JαK is among the collection JΦK.

Given these semantics and a falsity predicate ‘False’, i. e., a predicate such that pFalseαq is true if and only if the sentence JαK is false, we can form a Smullyanesque simple Liar sentence: (σ) ‘FalseNorm«FalseNorm»’ Or, if we have a negation symbol ‘¬’, which as usual is applied to a sentence by prefixing, and a truth predicate ‘True’, i. e., a predicate such that pTrueαq is true if and only if the sentence JαK is true, we can form the Smullyanesque strengthened Liar sentence: (Σ) ‘¬TrueNorm«¬TrueNorm»’ Now let us have a look at the semantics of the sentence σ, working outwards (or rather, from right to left). In view of the semantics of (formal) quotation, the term ‘«FalseNorm»’ refers to the expression ‘FalseNorm’; and in view of the semantics of the ‘Norm’-functor, the term ‘Norm«FalseNorm»’ refers to the sentence ‘FalseNorm«FalseNorm»’. Thus, in view of the semantics of the falsity predi-

722

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Smullyan uses genuine quote marks only in his introductory characterization of the norm function that applies to English expressions. For the corresponding norm function that applies to expressions of his formal language he does not make use of quotation, not even of formal quotation. Smullyan defines for the formal case: “By the norm of an expression E [. . . ] we mean E followed by its own Gödel numeral (i. e., the numeral designating its Gödel number).” (Smullyan 1957, 56) But we are interested in a variant of the Smullyanesque construction of self-referential formal sentences that makes use of formal quotation, as a stepping stone on our way to explain the use of Gödelization. Therefore we will ignore Smullyan’s official practice in this matter. Cf. section 4.4.

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cate, the sentence ‘FalseNorm«FalseNorm»’ is true if and only if the sentence ‘FalseNorm«FalseNorm»’ is false, or shorter, σ is true if and only if σ is false, which, given the exhaustiveness and exclusiveness of truth and falsity, we know 724 to entail a contradiction in the meta-language that we use to give the semantics of Smullyan’s language. So, σ ascribes falsity to itself and gives rise to the usual basic Liar reasoning. And similarly for the strengthened Liar sentence Σ. What is specific and interesting about Smullyan’s construction of a self-referential sentence is the role of the norm function. It helps to circumvent the restrictions on the expressions a sentence can quote that are due to the well-foundedness of concatenation: Speaking metaphorically, a sentence by quotation alone can refer only to a smaller expression, but as the norm function takes each expression to a larger one, there is a chance that these effects of shrinking and enlarging cancel out each other, so that the ‘Norm’-functor when applied to a quotation expression can indeed refer to the sentence it occurs in. In fact, for an arbitrary predicate Φ of Smullyan’s language, the following sentence says of itself that it falls under the concept expressed by Φ: 725 pΦNorm«ΦNorm»q Thus in the Smullyanesque variant of the Liar paradox the norm function plays the role that in the Quinean variant is played by the operation of appending an expression to its own quotation, which there helps circumvent the irreflexivity of proper parthood. However, in our formal language a predicate is applied to a term not by simple concatenation, but by substitution of the term for each occurrence of a free variable in the open formula that represents the predicate. Therefore we cannot use the selfreferential sentence σ constructed by Smullyan to give the logical form of Quine’s sentence, at least not in a canonical way. But as long as it does not come to the arithmetization of syntax, which need not concern us at the moment, Smullyan’s method is easily transferred to classical languages – in fact, Smullyan describes the transferal himself. 726 We now need a different functor, ‘Diag(x)’, the semantics of which is governed by the diagonalization function. This function maps each formula open in one variable to the expression that results from substituting the quotation expression of the open formula for each occurrence of the variable in the open formula, i. e., it maps each open formula pΦ(x)q to the expression pΦ(«Φ(x)»)q. The ‘Diag’-functor can be described by adding the following rules to the syntax and semantics of a language of classical quantified logic:

724 725

726

Cf. section 2.4. As the semantics of Smullyan’s language is extensional, we strictly speaking would have to say that the sentence is true if and only if it is among the collection of objects which is the extension of Φ. Cf. Smullyan 1957, 56, fn. 5, where he writes: “In contrast with this construction, let us define the diagonalization of E as the result of substituting the quotation of E for all occurrences of the variable ‘x’ in E. [. . . ] This [. . . ] construction involves substitution (inherent in diagonalization), whereas the norm function involves concatenation (the norm of E being E followed by its quotation), which is far easier to formalize [. . .].”

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(Syntax of ‘Diag’) If α is a term, so is pDiag(α)q. (Semantics of ‘Diag’) If the term α refers to an open formula, then the term pDiag(α)q refers to the diagonalization of the open formula referred to by α, i. e., if α refers to pΦ(x)q, then pDiag(α)q refers to pΦ(«Φ(x)»)q. 727 Given the usual predicates for falsity ‘False(x)’ and for truth ‘True(x)’ (such that pFalse(α)q is true if and only if the sentence JαK is false and pTrue(α)q is true if and only if the sentence JαK is true), we can again form Liar sentences: (δ) ‘False(Diag(«False(Diag(x))»))’

(∆) ‘¬True(Diag(«¬True(Diag(x))»))’ With the sentence δ, we have found a way of giving the logical form of Quine’s Liar sentence that does justice both to its literal and to its intuitive meaning. While we can understand δ as referring to the expression ‘False(Diag(x))’ and ascribing to it the concept expressed by the complex predicate ‘False(Diag(x))’, we can also understand δ as referring to itself and ascribing falsity. The former understanding captures the literal meaning of Quine’s sentence, and latter captures its intuitive meaning. It all depends on whether, in parsing the sentence δ, we count the first occurrence of the ‘Diag’-functor as part of the term or as part of the predicate. What is the natural language counterpart of the formal ‘Diag’-functor? In natural language, a description that contains a singular term can be turned into a descriptive functor by the same method that turns a sentence into a predicate: 728 By exchanging some occurrences of a singular term for a variable. E. g., from the description ‘the mother of Socrates’ we get the descriptive functor ‘the mother of x’. As in the case of formal functors, the semantics of a natural language descriptive functor can be thought of as governed by a function from objects to objects. 729 E. g., the semantics of the descriptive functor ‘the mother of x’ is governed by the function which maps everyone to his or her mother. It is clear that the function that governs the semantics of the natural language counterpart of the ‘Diag’-functor maps each expression to the appendation of that expression to its quotation expression. Therefore, the descriptive functor that is the counterpart of ‘Diag’ would have to mean the same as ‘the appendation of x to the quotation expression of x’, which we may abbreviate as ‘the quotation-appendation of x’. In view of the curious phenomenon we encountered above, this paraphrase can be used only for a characterization of the intuitive semantics of Quine’s sentence, which under a literal reading contains no

727

728 729

In order to give the semantics of ‘Diag(x)’ in full, we would have to add an ‘otherwise’-clause for cases where α does not refer to an open formula; laying down, e. g., that in those cases the extension of pDiag(α)q is empty. Cf. section 4.2. In order to be able to describe a descriptive term in isolation, we presuppose a Fregean construal of descriptions here (cf. subsection 5.2.4) and we allow the function in question to be partial.

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descriptive functor but a complex predicate that is applied to a quotation expression. But for our purposes understanding the intuitive semantics of Quine’s sentence is enough, because it helps to understand the phenomenon of Q-reference, and more specifically 730 the phenomenon of Q-self-reference. When we had our first look at Quine’s sentence we saw already that it does not refer to itself by quotation. An important result of our investigation into its logical form is that we can supplement this negative point by the following positive observation: Quine’s sentence refers to itself by description. More specifically, it refers to itself by a description that is formed by applying a descriptive functor to a quotation expression: On the level of logical form, we apply the ‘Diag’-functor to a formal quotation expression. And according to the intuitive semantics of Quine’s sentence, it Q-refers to itself by a complex singular term formed from applying the descriptive functor ‘the quotation-appendation of x’ to a quotation expression. Thus the Q-self-reference achieved by Quine’s sentence can be classified within the spectrum of referential devices which allow to form self-referential sentences, namely in the category of descriptions. The Quinean variant and the Smullyanesque variant of the Liar paradox are special cases of the description variant of the Liar paradox. * On an historical note, and anticipating the topic of the next section, we should mention that there are two lines which lead from Quine’s natural language Liar sentence to counterparts that belong to formal languages, one back to Gödel and the other onwards to Smullyan. The chronology is like this: In 1931, Gödel developed a technique for constructing formal counterparts of self-referential sentences in the language of arithmetic, 731 and more specifically constructed a sentence of arithmetic that in a sense says of itself that it is not provable in the formal theory of arithmetic. This sentence, which by now is called a Gödel sentence of that theory, plays a pivotal role in the proof of Gödel’s Incompleteness Theorems. Its construction is difficult to understand for non-mathematicians. Quine did much to explain Gödel’s method to philosophers, and his Liar sentence is a result of that work. 732 In the culminating chapter of his 1940 introduction to mathematical logic, Quine had tried to explain Gödel’s construction of a self-referential formal sentence by an elaborate natural language paraphrase like the following:

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731 732

A sentence can also Q-refer to an expression that is distinct from it, i. e., Q-reference need not be Q-self-reference. E. g., the sentence ‘‘Is an expression’ yields a grammatical expression when appended to its own quotation’ Q-refers to the distinct sentence ‘‘Is an expression’ is an expression’. For similar examples, cf. Hofstadter 2007, 140. For a sketch of the method, cf. section 3.6. Quine’s construction and its relation to Gödel’s method of achieving self-reference are explained in great detail in Hofstadter 2007, 101–172; especially 138ff. While Quine has done a lot to explain Gödel’s work to philosophers, Douglas Hofstadter has done a lot to explain both Gödel’s work and Quine’s explanation of it to a more general public. Cf. also the classic Hofstadter 1985.

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‘The result of putting the quotation of ‘The result of putting the quotation of w for ‘w’ in w is not a theorem’ for ‘w’ in ‘The result of putting the quotation of w for ‘w’ in w is not a theorem’ is not a theorem.’ 733 And he had tried to reassure his readers: “This is more intelligible than it looks.” 734 But we can presume that Quine later (understandably) lost faith in its intelligibility, because he moved on to formulate the following cleared up variant: ‘‘Is not a theorem when appended to itself in quotation marks’ is not a theorem when appended to itself in quotation marks.’ This natural language paraphrase of a Gödel sentence is of course the archetype of the Liar sentence we have discussed in this section. 735 Quine’s sentence is quoted in Smullyan’s 1957 article on self-reference in formal languages in the following way: “‘Yields falsehood when appended to its quotation’ yields falsehood when appended to its quotation.” 736 So Smullyan evidently was inspired by the Quinean variant of the Liar paradox when he developed the method to achieve self-reference with the help of the ‘Norm’functor, which allowed us to give the following formal counterpart of Quine’s Liar sentence: ‘FalseNorm«FalseNorm»’ The historical line which runs from Gödel via Quine to Smullyan forms an interesting loop from formal language via natural language back to formal language, and from mathematical logic via philosophical logic back to mathematical logic. Observe how philosophy (or at least, philosophical logic) was thus able to provide concrete help to formal logic. We disregard the historical order in our presentation of Gödel’s, Quine’s, and Smullyan’s method of constructing self-referential sentences in this and the next two sections because in the present chapter we move from natural language to formal language, and also for a heuristic reason: The ‘Diag’-functor, which in the present section was introduced to allow to transfer Smullyan’s method to the classical language we work with, will be helpful when we take another look at Gödelian selfreference in the next two sections. 733

In order to have a self-contained sentence, we have altered Quine’s original sentence “The result of putting the quotation of ‘The result of putting the quotation of w for S1 in w is not a theorem’ for S1 in ‘The result of putting the quotation of w for S1 in w is not a theorem’ is not a theorem.” (Quine 1981[1940], 307)

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736

by putting the quotation expression ‘‘w’’ for each occurrence of ‘S1’, which Quine uses here as a name of the variable ‘w’. Quine 1981[1940], 307. Quine presents and discusses his sentence in the 1961 lecture “The Ways of Paradox”, which was published in 1966; cf. Quine 1976a, 7. Smullyan 1957, 56, fn. 6.

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6.2 The Gödelian construction of a self-referential sentence We have already encountered the Gödelian method of constructing a formal counterpart of a self-referential sentence when we investigated formal variants of the Liar reasoning. 737 Here is a good point to return to the matter of Gödelian selfreference; for several reasons. Firstly, as we said at the end of the preceding section, our discussion of the Quinean and Smullyanesque techniques for constructing self-referential sentences prepares the way for a presentation of Gödel’s original construction, which was the inspiration behind the work of Quine and Smullyan. Secondly, our (official) knowledge of semantics (both in the sense of a theory of meaning and the sense of a theory of extensions 738) has increased since we last had a look at Gödelian self-reference, 739 which puts us into a better position to judge whether we are really dealing with reference and ascription here, and if so, in what sense. Thirdly, and most importantly, there is a common impression that Gödelization delivers a counterpart of Liar sentences, which we should discuss critically from a semantic point of view here, before we will try to show in part III that we have good metaphysical reasons to doubt that there are Liar sentences of any kind. The material is split up into three sections. We will describe the technical side of Gödel’s construction and contrast it with the Quinean and Smullyanesque techniques in the present section, turn to the semantics of Gödelian self-reference in the next section (6.3), and answer the question whether there is a Gödelian variant of the Liar paradox in section 6.4. * Before we come to the interesting part, which is the indirect talk about expressions made possible by Gödelization, we should have a brief look at how the formal language of arithmetic allows to talk about what it is about first and foremost, namely natural numbers, and the way in which the formal theory of arithmetic says something about them. A rough sketch will suffice. Let us fix on a specific formal language of arithmetic, LA . It is a language of classical quantified logic with identity, which has terms to refer to natural numbers, among them a numeral for each number, 740 and symbols to express addition and multiplication (and perhaps exponentiation and the relation of being greater than). The formal theory of arithmetic, TA , allows to formalize proofs of important facts about natural numbers (Peano Arithmetic PA or the slightly weaker Robinson Arithmetic Q will provide all the tools needed for Gödelization).

738 739 740

Cf. section 3.6. Cf. section 4.1. Cf. chapters 4 and 5. We had a look at Gödelian self-reference in section 3.6. By a system of numerals we here mean a system of standard terms for the numbers. In formal languages of arithmetic, the numerals are often formed by prefixing a successor symbol to a constant which refers to zero (e. g., ‘o’, ‘so’, ‘sso’, . . .); in English and a wide range of further natural languages, the numerals are given by the Arabic numerals in decimal notation (‘0’, ‘1’, ‘2’, . . . ). Both in natural language and in the usual formal languages of arithmetic, there will also be terms that refer to numbers which are not numerals (e. g., ‘105 + 4’ and ‘ x x = so + sso’).

ι

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Now, in order to talk about numbers, it is not enough that we can refer to them (by way of numerals, for a start) – we will also want to ascribe certain numerical notions to them. E. g., we want to be able to say that the number 1010 + 8 is even, and we want our theory to assert that fact. At this point, in order to understand the formal ascription of a numerical notion to a number, it is important to distinguish between what we can express in our formal language and what we can capture with the help of our formal theory. 741 (Def. To Express) A language of arithmetic LA expresses a numerical concept c by an open formula pΦ(x)q if and only if for every number n, if n falls under c then pΦ(n)q is true, and if n does not fall under c then p¬Φ(n)q is true. 742 (Def. To Capture) A formal theory of arithmetic TA captures a numerical concept c by an open formula pΦ(x)q (of TA ’s language LA ) if and only if for every number n, if n falls under c then pΦ(n)q is a theorem of T , and if n does not fall under c then p¬Φ(n)q is a theorem of T . These two definitions can be extended in a natural way to predicates of more than one place and to functors. 743 As is common in mathematical contexts, we treat the matter of expressing and capturing in a purely extensional way and formulate it in the (informal) language of set theory. Therefore we construe falling under a concept as being an element of a certain set – the set that is the extension of a predicate which expresses that concept (to use the terms of our Fregean framework for semantics 744). In particular we will be content with representing a numerical notion as the set of those numbers which fall under it, 745 and later on with construing a linguistic notion as the corresponding set of expressions. Thus what really is expressed or captured by an open formula

741

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744 745

Recall that we use underlining to indicate that a numeral (e. g., ‘1’, and in general for any number n, pnq) or a descriptive term that refers to a number (e. g., ‘105 + 4’) belongs to the formal object language. Cf. section 3.6. Here we assume a semanticist understanding of the formal language of arithmetic. Under the usual syntacticist understanding, we would have to adjust the above definition by replacing both occurrences of ‘true’ by ‘true under the standard interpretation’. Cf. section 3.1. For the two definitions, cf. Smith 2007, 34f.; Boolos /Burgess/Jeffrey 2002, 199ff. & 207; and Enderton 2001, 90f. & 205f. The terminology differs significantly; for an overview cf. Smith 2007, 36, fn. 6. Cf. section 4.2. We can see this construal of numerical notions as sets of numbers as a consequence of giving an extensional semantics (cf. section 4.4) to the language of arithmetic and construing collections as sets (cf. section 1.7).

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in the technical sense defined here is a set of objects. Although this is usually taken to suffice in the context of characterizing a mathematical language, we should note that it is a far cry from what we mean when we say in the context of a theory of meaning that a predicate expresses a concept. 746 Returning to our example, we can now say that the open formula ‘∃y 2y = x’ of our language of arithmetic LA expresses the set of even numbers {2, 4, 6, . . . } (and thus the numerical notion of evenness) because each one of the formal sentences ‘¬∃y 2y = 1’, ‘∃y 2y = 2’, ‘¬∃y 2y = 3’, ‘∃y 2y = 4’, ‘¬∃y 2y = 5’, ‘∃y 2y = 6’, . . . is true, and that our formal theory TA captures this set with this formula because it proves each one of those sentences. Thus we can say in our language that the number 105 + 4 is even by using the formal sentence ‘∃y 2y = 105 + 4’, and because of the theoremhood of this sentence we also know our theory to assert this fact. As the same language allows to formulate theories of different strength it is clear that there will be cases where the two feats of expressing and capturing a given set come apart; we need only weaken the theory for a set that is expressed not to be captured. To anticipate, it is a consequence of (the proof of) Gödel’s First Incompleteness Theorem that in the case of a broad range of theories we cannot amend this even by strengthening the theory we are concerned with, because there will always be some set that can be expressed but not captured (e. g., crucially, the set of numbers which encode the true sentences of the language). 747 If a theory is correct in the sense that all its theorems are true, then it follows from the above definitions that every set it captures is expressed by its language. 748 As we can presuppose here that the formal theories of arithmetic we deal with are correct, we can therefore say that in the case of arithmetic, every set that is captured is also expressed. But we should also note that by far not every set of numbers is expressed in a language of arithmetic. This can be shown by a simple cardinality argument: Because of Cantor’s Theorem there are indenumerably many sets of natural numbers, 749 but a language of arithmetic has only enumerably many expressions 750 and thus only enumerably many open formulas; hence we cannot even associate each set of numbers with an open formula, let alone find an open formula that expresses it. What sets does a language of arithmetic express? For our present purposes it suffices to say: A language of arithmetic expresses at least all those sets of numbers for which we can specify their elements by a finite application of some particularly

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749 750

Again, cf. section 4.2. Cf., e. g., Smith 2007, 149f. For a theory that is not correct, however, there will in general be some set that is captured by the theory but not expressed by its language. This is another way in which expressing and capturing can come apart – but a way that is not important for our considerations here, because we presuppose that the formal theories of arithmetic we deal with are correct. Cf. section 1.7. We can list the expressions of a language with a finite alphabet by first listing the expressions that consist of one symbol in alphabetical order, then listing the expressions that consist of two symbols in alphabetical order, and so on.

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well-behaved numerical algorithm; 751 and this algorithmic grasp is of course given in the case of our example of the set of even numbers. * So, let us turn to Gödelization. We devise a particular Gödel numbering, i. e., a function that maps each expression of the language of arithmetic to a number (called its Gödel number) and which is effectively decidable in the sense that there is an algorithm finitely many applications of which decide it. 752 The effective decidability of a Gödel numbering, in other words, guarantees that both finding the Gödel number of a given expression and finding the expression (if any) that a given number is the Gödel number of can be described as a purely mechanical procedure that will deliver a result after finitely many steps. 753 The finitely algorithmic (effectively decidable) nature of a Gödel numbering is important for its role in the Gödelian method of encoding talk about expressions in talk about numbers. For the purpose of encoding reference to expressions in reference to numbers, it is enough that a Gödel numbering is a bijection between the expressions and a subset of the natural numbers, but for whole statements that concern expressions to be encoded in statements about numbers, it is necessary that we have an algorithm that brings us back and forth between them. 754 We will see shortly why this is so. A statement about expressions that is of a basic form will say of a particular expression that it falls under some linguistic concept. It will say, e. g., that the expression ‘1 + 2 = 3’ is well-formed. Encoding such a linguistic statement in a statement about numbers will amount to saying of the Gödel number of that expression that it is an element of the set of Gödel numbers of expressions that fall under that linguistic concept. In the example we will need a sentence of arithmetic that says of the Gödel number of the expression ‘1 + 2 = 3’ that it is an element of the set of Gödel numbers of well-formed expressions. Thus we see that to encode a linguistic statement in an arithmetical statement, we need to find an open formula of arithmetic that expresses the set of Gödel numbers of expressions falling under the respective linguistic concept. And here we find the reason for the requirement that a Gödel numbering be effectively decidable: As we said above, expressibility of a set is given when we have an algorithmic grasp 755 of its elements. As we are dealing 751

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In the terminology of mathematical logic, a language of arithmetic expresses the primitive recursive sets which are a proper subset of the effectively decidable sets. Cf. Smith 2007, 88f. On effective decidability cf. Enderton 2001, 61ff.; Boolos /Burgess/Jeffrey 2002, 23ff.,; and Smith 2007, 15f. To make this possible for infinitely many expressions usually each expression of the finite alphabet of the formal language is associated with its Gödel number, and then a function is algorithmically specified which maps any pair of Gödel numbers of two expressions to the Gödel number of the concatenation of those two expressions, which allows the treatment of the matter by a recursive definition. Cf. Enderton 2001, 184 & 225ff.; Boolos /Burgess /Jeffrey 2002, 187ff.; Smith 2007, 124ff. In saying ‘an algorithmic grasp’ and so on, we are being deliberately vague. A full treatment of these matters would have to be so detailed that it would not fit the purpose of the present chapter. It can be found, under the headings of “decidability”, “computability”, “recursion theory”, and “arithmetization of syntax”, in every textbook on mathematical logic. Cf., e. g., Smullyan 1992; Enderton 2001; Boolos / Burgess /Jeffrey 2002; and Smith 2007.

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with a formal language, we already have an algorithmic grasp of a lot of interesting concepts that apply to its expressions (like well-formedness, being an atomic sentence, and even being a derivation of ). Therefore, the effective decidability of our Gödel numbering guarantees that we will also have an algorithmic grasp of the Gödel numbers of the expressions that fall under those linguistic concepts. And in view of the indenumerable infinity of sets of numbers that are not arithmetically expressible it is evident that we need this guarantee if we want to encode linguistic statements in arithmetic statements. In order to have a perspicuous and definite example, 756 let us suppose our Gödel numbering is such that an expression will be well-formed just in case its Gödel number is even, and that the Gödel number of the expression ‘1 + 2 = 3’ is 105 + 4. 757 Then the formal sentence ‘∃y 2y = 105 + 4’, which under the usual reading makes the arithmetical statement that the number 105 + 4 is even, encodes the linguistic statement that the expression ‘1 + 2 = 3’ is well-formed. And we can even say that our formal theory of arithmetic indirectly asserts this linguistic statement because the formal sentence ‘∃y 2y = 105 + 4’ is one of its theorems. At this point it is tempting, and it is common practice, to introduce some abbreviations which make visible the meta-arithmetical statements that are encoded in the language of arithmetic. In general, we can use Gödel corners, ‘h . . . i’, to form for each expression of arithmetic its Gödel corner expression which abbreviates a term (e. g., a numeral) that refers to the Gödel number of the expression: For each expression of arithmetic ϕ, phϕiq is a term that refers to the Gödel number of ϕ; and in our example, the Gödel corner expression ‘h1 + 2 = 3i’ abbreviates the numerical term ‘105 + 4’. And given that we know the open formula ‘∃y 2y = x’ to express the set of Gödel numbers of well-formed expressions, we can abbreviate it as ‘NumWff(x)’. 758 With these abbreviations, we can write our formal sentence ‘∃y 2y = 105 + 4’ as ‘NumWff(h1 + 2 = 3i)’, which makes the encoded meta-arithmetical meaning visible. But we will have to be careful, especially with regard to the common device of Gödel corners, to remember that these abbreviatory devices are not expressions of the language of arithmetic. Thus, with respect to our example we must not forget that ‘NumWff(h1 + 2 = 3i)’ is the same sentence as ‘∃y 2y = 105 + 4’. So much for the idea of encoding; now let us turn to Gödelian self-reference. In his 1931 paper, Gödel constructed a sentence that encodes the statement that it is not provable. His construction can be generalized, which gives us a variant of the Diagonal Lemma (which we already stated in a general form in section 3.6) for the case where the standard name operator is realized by what we here call Gödel corners:

756 757 758

We already gave a briefer presentation of the following example in section 3.6. There is a Gödel numbering like that. I am not entirely comfortable with the proliferation of the prefix ‘Num’, but we will nonetheless use it each time we try to abbreviate an arithmetical formula that encodes a linguistic concept in a telling way, as a persistent reminder that the complex predicate thus abbreviated applies to numbers, not expressions. Besides ‘NumWff(x)’, there will be ‘NumProof(x, y)’, ‘NumProvable(x)’, and of course ‘NumTrue(x)’.

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(Diagonal Lemma, Special Variant) For every formula pΦ(x )q of the language LA that is open in one variable (here, ‘x’), there is a sentence ψ of LA such that pψ ↔ Φ(hψi)q is a theorem of the theory of arithmetic TA . 759 At this point we can already note that despite its considerable generality, the Diagonal Lemma does not guarantee that we will find, for every linguistic concept that is of interest to us, a sentence that ascribes it to itself. As the universal quantifier here ranges not over linguistic concepts, but over open formulas of arithmetic, we get self-referential sentences only for those linguistic concepts which can be encoded in a set of numbers that is expressed by some open formula of LA . 760 The Gödelian construction of a self-referential sentence, which provides the basic idea for the proof of the Diagonal Lemma, can be explained by recourse to what we saw in the preceding section – more specifically, by recourse to the second Smullyanesque construction, which we used to give the logical form of natural language sentences which are Q-self-referential, like Quine’s Liar sentence. From what we saw there we can glean a general recipe for constructing self-referential sentences in a language of classical logic which has been augmented by formal quote marks: For an arbitrary open formula pΦ(x)q which applies to expressions, the following sentence ascribes the concept expressed by pΦ(x)q to itself: pΦ(Diag(«Φ(Diag(x))»))q Here, self-reference is made possible by the semantics of the ‘Diag’-functor, which is governed by a diagonalization function which brings us from expressions to larger expressions in exactly the right way. To regain the Gödelian construction from this, we have to replace our Smullyanesque diagonalization function, which maps expressions to expressions, with a numerical diagonalization function, which maps numbers to numbers, more specifically, which maps the Gödel number of an expression to the Gödel number of the (Smullyanesque) diagonalization of that expression. It turns out that this function can be expressed and captured by a functor of arithmetic 761 which we will abbreviate as ‘NumDiag(x)’. Also we have to substitute Gödel corners (‘h . . . i’) for the occurrence of formal quote marks (‘«. . . »’). Thus we get the following recipe: (Gödelian Recipe) For an arbitrary open formula pΦ(x)q which applies to numbers, the following sentence exhibits Gödelian self-reference: pΦ(NumDiag(hΦ(NumDiag(x))i))q 759

760

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As we said above, we can insert here Peano Arithmetic PA or Robinson Arithmetic Q; it should however be pointed out that some theories of arithmetic are too weak to deliver the Diagonal Lemma. Presuming that there are at least as many linguistic concepts as there are different sets of expressions, we can use a variant of the above cardinality argument to show that not every linguistic concept can be expressed indirectly by an open formula of arithmetic. Showing this is the real work behind the proof of the Diagonal Lemma in the special variant with Gödel corners we discuss here. Cf., e. g., Smith 2007, 173.

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The direct contrast with the Smullyanesque self-referential sentence allows to point out some specific characteristics of the Gödelian self-referential sentence. Note first that, aside from the (mentioned) variable ‘x’ and the brackets, not one of the expressions displayed here belongs to the formal object language, because they all are meta-linguistic abbreviations: pΦ(x)q for an arbitrary open formula, ‘NumDiag(x)’ for a specific open formula that captures the numerical diagonalization function, 762 and the Gödel corner expression for a term of the formal language that refers to the Gödel number of the expression abbreviated here as ‘Φ(NumDiag(x))’. A second important characteristic of the Gödelian self-referential sentence is that it is not about expressions but about numbers. Only when the arithmetical formula happens to express a set of numbers which are the Gödel numbers of all and only those expressions which fall under some linguistic concept can we say sensibly that it encodes a self-ascription of that concept. In our example, where being a well-formed expression correlates in this way to being an even number, this does indeed work. We can take the open formula ‘∃y 2y = x’ which expresses evenness, form the sentence ‘∃y 2y = NumDiag(h∃y 2y = NumDiag(x)i)’, and will then be justified in saying that we have found a sentence which says of itself that it is well-formed; and this will warrant an abbreviation which makes this encoded meaning visible, namely ‘NumWff(NumDiag(hNumWff (NumDiag(x))i))’. But where we start with an open formula of arithmetic for which we cannot find a linguistic concept that is encoded in it, the sentence constructed from it by the Gödelian recipe can only be construed as speaking about a number, and we should not be overly impressed by the observation that it refers to its own Gödel number. 763 * A word on some similarities between Gödelian, Smullyanesque, and Quinean selfreference, and thus on a common thread that runs through sections 5.6, 5.7, 6.1, and the present one. In each one of the three cases, there is a serious obstacle to any attempt of constructing a sentence that is self-referential in a straightforward way: 764 Quine has to circumvent the irreflexivity of proper parthood, which prevents a sentence from quoting itself; 765 Smullyan has to circumvent the well-foundedness 762

763

764

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By contrast to the Smullyanesque ‘Diag’-functor, the open term abbreviated here as ‘NumDiag(x)’ is not primitive, but an open term of the formal language which might be quite complex. If we extend our example by stipulating that the Gödel number of the sentence ‘∃y 2y = NumDiag (h∃y 2y = NumDiag(x)i)’ is 1010 + 8, then (because of the semantics of ‘NumDiag(x)’) the term ‘NumDiag(h∃y 2y = NumDiag(x)i)’ will refer to 1010 + 8, and we can write our sentence also as ‘∃y 2y = 1010 + 8’. Now the fact that the number of which this sentence says that it is even is the Gödel number of the sentence itself can hardly be seen as more than a random and irrelevant extra fact – as long we do not add that the predicate part of the sentence encodes the linguistic concept of being well-formed. Here, ‘straightforward reference’ is an umbrella term for reference by a quotation expression and reference by a Gödel numeral. Besides the curious methods of referring described in the last and the present section, there are of course other respectable ways of non-straightforward reference, namely by a name that is not a numeral, by a description, or by an indexical, all of which have been used to construct self-referential sentences (cf. section 5.4). Cf. sections 8.2 and 8.4.

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of concatenation, which prohibits the construction of any formal sentence which stands in the relation of concatenation to a formal quotation expression of itself; 766 and what Gödel has to circumvent is a certain kind of monotonicity which precludes that a sentence contains the numeral of its own Gödel number. All standard 767 Gödel numberings are monotonous in the sense that (i) the Gödel number of an expression is bigger than the Gödel number of any one of its proper syntactic parts and (ii) the Gödel number of a numeral is bigger than the number it refers to. Therefore, a sentence cannot contain its own Gödel numeral under any standard Gödel numbering. However, a number can be referred to by a term much shorter than its numeral. 768 And when the Gödel numbering is monotonous, the ‘NumDiag’-functor will provide us which such a term. But it will do more, because it is not merely some arbitrary device to refer to a (very) large number, but its construction is connected in a direct (“canonical” 769) way to an explanation why that large number refers to the sentence the term occurs in. This is a further respect in which the Gödelian functor ‘NumDiag(x)’ is similar to the Quinean descriptive functor ‘the quotationappendation of x’ and to the Smullyanesque functors ‘Norm’ and ‘Diag(x)’: Once we understand them (or, are given their semantics), we need only put one and one together to know that and how they will allow to form self-referential sentences. In each case the semantics of the respective functor is governed by a function which does the trick of circumventing the obstacle that precludes straightforward self-reference. We can use a single metaphor of size to describe what happens in all three cases. If we liken the relation between an object and a sentence that refers to it straightforwardly to putting an object into a box, the impossibility of straightforward self-reference can be made evident by pointing out that no object fits into a box with an opening that is smaller than the object. Each one of the mentioned functions, however, leads from an object small enough to fit into the box to a larger object (in fact, to the object which is the box itself): Quine uses the function which maps an expression to its quotation-appendation (section 6.1) to circumvent the mereological obstacle of the irreflexivity of parthood (section 5.6); Smullyan uses the norm function and the diagonalization function to circumvent the obstacle of the well-foundedness of concatenation (section 5.7); and Gödel uses what we described as the numerical diagonalization function to circumvent the obstacle constituted by the monotonicity of standard Gödel numberings. But let us not forget that the numerical diagonalization function, in contrast to the functions that play a corresponding role in the Quinean and Smullyanesque method, does not bring us to the sentence we want to refer to itself, but merely to its Gödel number. As it needs a 766 767

768 769

Cf. sections 8.3 and 8.4. It is possible to characterize a Gödel numbering such that some sentence contains its own Gödel numeral (this point is alluded to by Kripke in Kripke 1975, 693, fn. 6), because we can make an exception to the usual recursive definition of Gödel numbers for some sentence that is antecedently given and still retain the effective decidability which is required of a Gödel numbering. But this will complicate the encoding of linguistic concepts in what the language of arithmetic expresses about numbers, and hence will make the construction of sentences which are self-referential in the Gödelian way no easier. Just compare ‘1030’ to ‘1 000 000 000 000 000 000 000 000 000 000’. Cf. Smith 2007, 144ff. & 176.

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scheme of encoding in addition to a function which does the enlarging that is needed to overcome the obstacle of monotonicity, the Gödelian method thus is less direct than its Quinean and Smullyanesque counterparts. 770

6.3 The semantics of Gödelian reference Now that we have seen an outline of the technical side of Gödel’s method of encoding statements about expressions in statements about numbers, we will turn to the semantics of Gödelian reference and in particular of Gödelian self-reference. Can talk about reference and self-reference be warranted here, in view of the indirectness we have seen the Gödelian method to have because of its reliance on encoding? In this section, we will give a differentiated answer to this question. This will prepare the way for a critique of the common misconception that the Gödelian method allows to construct formal counterparts of Liar sentences in the next section. We will start by describing briefly the prerequisites of and obstacles to reference that can be achieved by the Gödelian method. Let us suppose, at least for the sake of the argument of this and the next section, that each one of the obstacles can be overcome, so that we can go on to formulate a mini theory of G-meaning, which gives preconditions for a sentence to G-refer to an expression and to G-ascribe a linguistic concept to it. Then we will have a look at how those formal sentences constructed by the Gödelian method which are indeed G-self-referential can be translated into natural language. By criticizing some common natural language paraphrases of G-self-referential formal sentences we will complete the picture of what can and what cannot be achieved by the Gödelian method as far as semantics is concerned. 771

The usual prerequisites and obstacles Are we really justified in calling a formal sentence that is constructed according to the Gödelian method self-referential? As has been pointed out by cautious logicians and critically-minded philosophers, 772 there are a number of prerequisites for and potential obstacles to Gödelian self-reference: 770

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Hofstadter explains the Gödelian method by way of the analogy of fitting not an elephant, but its DNA into a matchbox (Hofstadter 2007, 139ff. and especially 143). This is a nice way of bringing together the aspect of enlargement, which is part of the Quinean, Smullyanesque, and Gödelian method, with the aspect of encoding, which is specific to the Gödelian method. The topic of self-reference via Gödelization pops up in several places in the present study. Each time we come at it from a different angle: In section 3.6 we wanted to know whether it allows a full formalization of the Liar reasoning; in section 6.2 we have recalled technical aspects that are relevant for what distinguishes Gödelian self-reference from Quine’s and Smullyan’s constructions; in the present section 6.3 we focus on a theory of meaning in the context of Gödelization; and we will turn to the question whether there can be Gödelian self-ascriptions of falsity or untruth in the following section 6.4. Cf., e. g., Smith 2007, 144–146 & 176f.

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(a) Gödelian self-reference requires that the language of arithmetic has its intended interpretation, 773 or, in present terms, is understood in a semanticist way. (b) Gödelian self-reference requires that the formal theory of arithmetic is correct. 774 (c) Gödelian self-reference must be indirect because it prima facie is reference to a number, and it requires a particular Gödel numbering for the indirect reference to be successful. 775 (d) Gödelian self-reference is indirect insofar as the Diagonal Lemma delivers only the theoremhood of a biconditional, not the truth of an identity statement. 776 (e) The Diagonal Lemma does not guarantee that the sentence it delivers has a structure which allows to recapitulate its construction as a self-referential sentence (in contrast to sentences that are self-referential in a Quinean or Smullyanesque way). 777 (f) In the setting of the Gödelian method, concepts are treated extensionally: The concept expressed by a predicate is represented by the extension of the predicate. This list allows and perhaps calls for an extensive discussion, but let us be brief. Doubts about the prerequisites (a) and (b) are highly relevant in a theoretical context that concerns the question of what can be achieved by formal methods alone, but given the semantic focus of the present section we can unproblematically assume that the language of arithmetic speaks of the natural numbers and that the formal theory of arithmetic describes them correctly. It is possible to circumvent obstacles (d) and (e) by working not with a sentence delivered by the Diagonal Lemma, about which we know only that it is provably equivalent to a (perhaps distinct) sentence that encodes a statement about it, but more directly with a sentence delivered by the (Gödelian Recipe) described in the last section. 778 Such a sentence does allow to read off its construction as self-referential (on the level of meta-linguistic abbreviation even in a way entirely similar to Quine’s and Smullyan’s sentences), and it is about itself, albeit indirectly. So, the only obstacles which remain to talk of Gödelian self-reference being warranted at least in some cases are (c) and (f). I think that the question whether they can be overcome will be difficult to settle. Should we adopt an algebraic understanding 779 of the formal theory of arithmetic? Does this understanding entail that we are justified in taking a theory of arithmetic to assert

773 774 775 776 777 778 779

Cf. Lacey /Joseph 1968; Milne 2007. Cf. Milne 2007. Lacey /Joseph 1968. We have emphasized the points (c) and (d) already in section 3.6. Cf. Smith 2007, 145f. Cf. section 6.2. Cf. section 3.1.

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facts about the expressions of its own language? 780 Should we defer to mathematical logicians and accept an extensional treatment of concepts in this particular case, contrary to the Fregean beliefs about concepts we have in general? 781 Let us here give affirmative answers to these questions, and accept at least for the sake of argument that is some cases the (Gödelian Recipe) does deliver sentences which in a sense are self-referential. As a reminder of obstacles (c) and (f) and to mark the particular sense in which we can speak of reference, ascription, and meaning here, we will speak of ‘G-reference’, ‘G-ascription’, and ‘G-meaning’ (where ‘G’ is of course mnemonic of ‘Gödelian’).

A mini theory of G-meaning In order to be able to answer the question of which formal sentences can properly be said to be G-self-referential – i. e., to be self-referential under the charitable assumptions made here for the sake of argument –, I here want to present a mini theory of G-meaning. It makes use of the technical points reported in the preceding section and puts them into a semantic setting. We presuppose a particular Gödel numbering, and we think of all indirectly self-referential sentences as being constructed in accordance with the (Gödelian Recipe). (Def. To G-Refer (on the level of terms)) For a term of arithmetic η that refers to a number n, we say that the term of arithmetic η G-refers to an expression eη if and only if the number n encodes the expression eη. (Def. To G-Express) For an open formula of arithmetic pΦ(x)q that expresses 782 a set of numbers NΦ which represents a numerical concept cΦ, we say that the open formula of arithmetic pΦ(x)q G-expresses a linguistic concept dΦ if and only if the set of numbers NΦ encodes a set of expressions EΦ which represents the linguistic concept dΦ. (Def. To G-Capture) For an open formula of arithmetic pΦ(x)q that captures 783 a set of numbers NΦ which represents a numerical concept cΦ, we say that the open formula of arithmetic pΦ(x)q G-captures a linguistic concept dΦ if and only if the set of numbers NΦ encodes a set of expressions EΦ which represents the linguistic concept dΦ. 784 780

781 782 783 784

This would amount to using the Gödel numbering not merely as a code, but as a means of re-interpreting the language of arithmetic. Cf. section 4.2. Cf. (Def. To Express) in section 6.2. Cf. (Def. To Capture) in section 6.2. Due to the solely semantic orientation of the present section, we are much more interested here in G-expressing than in G-capturing. However, we were already concerned with G-capturing in section 3.6 where we talked about what a theory can define, and in the next section we will again be

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The basic structure is the same in all three cases: Referring, expressing, and capturing are functions that bring us from expressions to numbers; encoding is a function that brings us from numbers to expressions; and thus (when everything goes well) there is a resultant function that brings us from expressions to expressions, which is what we call G-referring, G-expressing, or G-capturing, respectively. Up to this point, our mini theory of G-meaning is no more than a terminological variation on what we said about encoding in the preceding section. There we already noted the fact that not every open formula of arithmetic encodes, or in present terms, G-expresses, a linguistic concept. Now, even after we have decided to go the most charitable way in our interpretation of the results of the Gödelian method, we have to acknowledge that we are hardly warranted in saying of a sentence of arithmetic that it speaks about expressions when it does not contain any open formula that G-expresses a linguistic concept – for an arithmetical sentence to be G-meaningful it does not suffice when it merely contains a term that G-refers to an expression! This observation motivates the central claim of our mini theory, which is the following context principle: 785 (Context Principle for G-Meaning) A sentence of arithmetic pΦ(α)q G-means that an expression eα falls under a linguistic concept cΦ if and only if the term α G-refers to the expression eα and the open formula pΦ(x)q G-expresses the linguistic concept cΦ. Only in that case do we say that the sentence pΦ(α)q (derivatively) refers to the expression eα and ascribes to it the linguistic concept G-expressed by the open formula pΦ(x)q. Hence: (Corollary about G-Reference (on the level of sentences)) A sentence of arithmetic pΦ(α)q G-refers to the expression eα if and only if the term α G-refers to the expression eα and the open formula pΦ(x)q G-expresses a linguistic concept. And in particular: (Corollary about G-Self-Referentiality) A sentence of arithmetic pΦ(α)q G-refers to itself if and only if the term α G-refers to the sentence pΦ(α)q and the open formula pΦ(x)q G-expresses a linguistic concept. So, everything depends on the question whether the open formula of arithmetic in a sentence we are concerned with G-expresses a linguistic concept. There are two reasons why an open formula of arithmetic can fail to G-express a linguistic concept.

785

concerned with what a theory can G-capture when we contrast the proof of Tarski’s Theorem with the proof of Gödel’s First Incompleteness Theorem. This is of course inspired by Frege’s context principle (Frege 1988[1884], 10; cf. Frege 1988[1884] 69f.).

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Firstly, in a scenario where not every number in the set expressed by the open formula encodes an expression we cannot say that it encodes a set of expressions, not even in an indirect way. Secondly, in a scenario where every number in the set expressed by the open formula does encode an expression but the resulting set of expressions does not represent any sensible linguistic concept (i. e., no concept that we can recognize as genuinely applying to expressions, e. g., being well-formed, being a variable, being a theorem, and so on), we may have serious doubts about whether we should speak of G-expressing. The second reason is vague (because our list of genuine and sensible linguistic concepts is open-ended and incomplete) and weak (because in each case there will at least be the derivative concept of having a Gödel number that falls under the concept expressed by the open formula). But however that may be in a particular case, let us note the general point that as soon as we have reason to doubt that the open formula in a given sentence G-expresses a linguistic concept, the (Context Principle for G-Meaning) gives us as much reason to doubt that that sentence is G-self-referential. Furthermore, coming not from the side of open formulas of arithmetic but from the side of linguistic concepts, we should note again that although for some linguistic concepts there are G-self-referential sentences which G-ascribe them to themselves, we do not have the guarantee that we will find, for every linguistic concept that is of interest to us, a formal sentence which G-ascribes it to itself. For although every linguistic concept can be represented by a set of expressions and every set of expressions by way of the Gödel numbering is encoded by a set of numbers, 786 not every set of numbers is expressed by the language of arithmetic. 787 Thus we reach the preliminary result that even when we accept the Gödelian method as a way of constructing self-referential sentences, we do not yet know whether it allows to construct Liar sentences, because we (officially) do not know yet whether the linguistic concepts of falsity or of untruth can be G-expressed. But before we come to that (in the next section), let us discuss the semantics of Gödelian reference and self-reference as it is regimented by our mini theory of G-meaning, concentrating for now on successful cases. First off, let us retell the story of an example of the last section in the present terminology. Our Gödel numbering is such that (i) the Gödel number of the sentence of arithmetic ‘∃y 2y = 1010 + 8’ is 10 000 000 008 and (ii) an expression is well-formed if and only if its Gödel number is even. Because of (i) we can say that the arithmetical term ‘1010 + 8’ G-refers to the sentence ‘∃y 2y = 1010 + 8’; because of (ii) we can say that the open formula ‘∃y 2y = x’ G-expresses the linguistic notion of well-formedness, and hence we are justified to say that the sentence ‘∃y 2y = 1010 + 8’ is G-self-referential and G-ascribes wellformedness to itself. In order to move on from the example, note that when all goes well we do not need to make ourselves hostage to any concrete stipulation about the Gödel numbering we work with. Once we know that the notion of well-formedness can 786

787

The Gödel numbering is a bijection between the set of expressions and (a subset of) the set of numbers, which induces a bijection between the respective powersets, i. e., a bijection between the set of sets of expressions and the set of sets of numbers (which are elements of that subset). Cf. section 6.2.

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be G-expressed, we can make use of the (Gödelian Recipe) to construct the formal sentence ‘NumWff(NumDiag(hNumWff(NumDiag(x))i))’, and be sure that it is G-self-referential and G-ascribes well-formedness to itself. 788 And more generally, once we know that an open formula pΦ(x)q G-expresses some linguistic concept dΦ, 789 we can make use of the (Gödelian Recipe) to construct the formal sentence pΦ(NumDiag(hΦ(NumDiag(x))i))q, and be sure that it is G-self-referential and G-ascribes the linguistic concept dΦ to itself. Despite the existence of G-self-referential sentences, it would be wrong to count G-referential terms as a further kind of referential device, in addition to names, descriptions, and indexicals, 790 that can be used to construct self-referential sentences. The language of arithmetic has some referential devices that correspond to names: constants, and some that correspond to descriptions: complex terms. It is the latter which are employed in sentences of arithmetic which we are justified in classifying as G-referential (as witnessed by the complex term of arithmetic we tellingly abbreviate as ‘NumDiag(hNumWff(NumDiag(x))i)’). It would, however, be misleading to say that G-reference is a species of reference with a description, because of its indirect nature: What is described by the term of a G-referential sentence is not an expression but its Gödel number. And in view of our context principle, the indirect description reaches the expression it is meant to G-refer to only when the predicatepart of that sentence G-expresses a linguistic concept.

Trying to paraphrase G-meaningful sentences in natural language For a better understanding of what can be achieved by the Gödelian method, it is instructive to have a look at how G-meaningful sentences can be translated, or, perhaps more generally, how they can be adequately paraphrased in natural language. (Looking at these attempts at translation and paraphrase here is in keeping with the emphasis put on semantics in the present chapter and its way of moving back and forth between formal language and natural language.) There are two kinds of paraphrases of G-meaningful sentences in the literature, which are given frequently for heuristic reasons. The paraphrases of the first kind use an indexical like ‘this sentence’ or ‘I’, where ‘I’ is understood to refer to the sentence it occurs in, 791 and the paraphrases of the second kind use a complex description that is built from a term that refers to an expression and a descriptive functor meaning something like ‘the Gödel number of x’. The G-meaningful sentences so paraphrased usually are Gödel sentences, i. e., G-self-referential sentences that G-ascribe the linguistic 788

789 790 791

In contrast to the sentence ‘∃y 2y = 1010 + 8’ as it is displayed here, the same sentence (or another one) as abbreviated as ‘NumWff(NumDiag(hNumWff(NumDiag(x))i))’ is guaranteed to be canonically selfreferential (cf. Smith 2007, 145f.), because in view of what we know about the ‘NumDiag’-functor and Gödel corners we know that it contains a term (‘NumDiag(hNumWff(NumDiag(x))i)’) that G-refers to it. Cf. section 6.2. This question is often independent of the Gödel numbering. Cf., e. g., Smith 2007, 126f. Cf. section 5.4. Thus ‘I’ here is not understood in the usual way as personally indexical but as if the sentence it occurs in were itself the speaker. Thus it does not differ much in meaning from the locution ‘this sentence’ that is regimented as sententially indexical. Cf. section 5.3.

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concept of being unprovable to themselves. In the present context it is of less importance that these sentences G-refer to themselves, but it is crucial that they G-refer at all. This is indeed the case because the concept of being unprovable can be G-expressed. 792 Here are examples for both kinds of paraphrase of a Gödel sentence: (1) ‘I am unprovable.’ 793 (2) ‘The sentence with the Gödel number n is not a theorem.’ 794 The common impression seems to be that while (1) is valuable only for heuristic purposes, because the language of arithmetic clearly does not contain indexicals, (2) is much nearer the mark because it contains a term that refers to a number (even to the same number as that which is referred to by some arithmetical term in the formal sentence (2) is meant to paraphrase). In the light of what we have seen by now of the semantics of G-meaningful sentences, however, (2) turns out to be misleading and less adequate as a paraphrase than (1). The reason for this is that while (1) says of an expression that it falls under a linguistic concept and thus concerns only the linguistic level, (2) speaks both about an expression and about a number, and thus connects the linguistic and the arithmetical level. Let us spell out more of the logical form of (2), by employing the Russellian analysis of descriptions: (20 ) ‘There is a unique sentence s such that (n is the Gödel number of s and s is not a theorem).’ Now, while the notion expressed by the English predicate ‘x is not a theorem’ can be G-expressed by an open formula of arithmetic, the (relational) notion expressed by the English predicate ‘x is the Gödel number of y’ cannot be G-expressed. The first place of this two-place predicate must be filled by a singular term that refers to a number and its second place must be filled by a singular term that refers to an expression. And while arithmetic allows to talk directly about numbers and in some cases allows to talk indirectly about expressions, it does not allow to talk, neither directly nor indirectly, about how numbers relate to expressions. In the spirit of the above context principle, we can say that either all expressions which are the semantic parts of a sentence of arithmetic must be understood in a numerical way or they must all be understood in a linguistic way. As a consequence, we cannot express facts about the Gödel numbering we work with in the language of arithmetic. So, to generalize, we cannot adequately paraphrase a sentence of arithmetic that G-means that an expression e falls under a linguistic concept d as pThe sentence with the Gödel number η is ∆q or as pη is the Gödel number of an expression that is ∆q, even when the singular term η does refer to the Gödel number of e and the 792

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Showing that unprovability is G-expressible makes up a large part of the proof of Gödel’s First Theorem. Cf. Smullyan, 1984, 201ff. and Boolos /Burgess/Jeffrey, 2002, 228. Gödel in his 1931 paper does not give any outright paraphrase of the self-referential formula essential in his proof, but he writes “Wir haben also einen Satz vor uns, der seine eigene Unbeweisbarkeit behauptet.” (Gödel, 1931, 175), which can be translated as “Thus we have a sentence before us that asserts its own unprovability.” And that is just what one would say to report the meaning of a sentence like (1). Cf. Rheinwald 1988, 231.

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predicate ∆ does express the concept d. What we can use here, at least in general, is pE is ∆q, where the singular term E refers to the expression e, or, more fully, pThe expression E falls under the linguistic concept of being ∆q. But this way of paraphrasing will not help much in the case of G-self-referential sentences, because it is difficult to find a natural language singular term that mirrors the particular way a sentence constructed according to the (Gödelian Recipe) achieves reference to itself. It is my impression that in this special case the best thing we can do to make as much of the semantics of such a sentence explicit as possible is to use telling abbreviations instead of a natural language paraphrase. For a Gödel sentence this would give us, where ‘NumProvable(y)’ abbreviates an open formula of arithmetic that G-expresses the linguistic notion of being a theorem of the formal theory of arithmetic: (3) ‘¬NumProvable(NumDiag(h¬NumProvable(NumDiag(x))i))’ I think that (3) does better than (1) and (2), because, although it may be more intimidating, it makes explicit both the G-meaning of the arithmetical sentence it abbreviates and allows to recapitulate how that sentence achieves indirect reference to itself. But even on this way some caution is needed, with regard to at least two points. Firstly, in accordance with the (Context Principle about G-Meaning), we should be suspicious of any abbreviation which suggests a mixed meaning, containing both ordinary arithmetical expressions and abbreviations telling of some intended G-meaning (e. g., we should be suspicious of the abbreviation ‘NumWff(h1 + 2 = 3i) ∧ 1 + 2 = 3’). Secondly, we have to be careful in how we understand Gödel corners.

Gödel corners and quote marks A Gödel corner expression may look seductively similar to the corresponding quotation expression, but it is entirely unlike it. First of all, while a quotation expression either (in most cases) belongs to the same language as the quoted expression, or is part of a meta-language used to talk about the language the quoted expression belongs to, a Gödel corner expression belongs neither to the object language (arithmetic) nor to the meta-language used to talk about it (here, logically augmented English), because it exists only within the limbo of meta-linguistically inspired abbreviations of object language expressions. And secondly, while a quotation expression is a revealing designator, 795 a Gödel corner expression abbreviates some complex arithmetical term that is not revealing, because the language of arithmetic arguably does not contain any revealing designators. At least it has no revealing designators that refer to expressions, 796 the reason being that G-reference to expressions is essentially indirect because (to say it again) the language of arithmetic cannot express any facts about the Gödel numbering, not even indirectly. We can illustrate the essential indirectness of G-reference by an analogy. Imagine someone who speaks only English and someone else who understands only the 795 796

Cf. section 5.5. Some notations for numerals look pretty revealing, e. g., when each number n is referred to by the concatenation of n strokes (prefixed to the constant that refers to zero). But what these numerals revealingly designate are numbers, not expressions.

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formal language of arithmetic – or, weaker, who does speak English in addition to understanding arithmetic but is ignorant of the particular Gödel numbering we use to encode linguistic facts in the language of arithmetic. The English speaker will know from a quotation expression which expression it refers to. But the speaker of the language of arithmetic, although having a full grasp of that formal language, does not know which expression a given term of arithmetic G-refers to and thus has no way no way of using Gödel corners. This goes to show that even in a theory of G-meaning, we cannot coherently speak of ‘the semantics of Gödel corners’.

6.4 Is there a Gödelian variant of the Liar paradox? Thanks to the preparatory work of the preceding sections, everything will now fall into place nicely when we turn to the important question whether there is a Gödelian variant of the Liar paradox. Let us for contrasting purposes take a last detour and have a look at the role a Gödel sentence plays in the proof of Gödel’s First Incompleteness Theorem.

Gödel’s First Incompleteness Theorem The version of the proof of Gödel’s First Incompleteness Theorem that we will see here in very broad strokes is what has been called the semantic argument for incompleteness. 797 It is semantic insofar as it presupposes that the language of arithmetic LA we work with has the intended interpretation and that the formal theory of arithmetic TA we work with is correct. 798 (The central argument in Gödel’s 1931 paper was syntactic because this allowed him to derive a stronger result and was in line with the theoretical stance adopted by most mathematical logicians at the time.) First we show that the notion of being a theorem of TA (or, of being provable in TA ) can be G-expressed. This is so because all relevant operations on formal expressions of LA are mirrored in relations among numbers that LA expresses and some of which TA captures. Thus we know that there is an open formula of LA that G-expresses being a theorem, which we therefore abbreviate as ‘NumProvable(y)’. Now we use the (Gödelian Recipe) to construct a G-selfreferential sentence which G-ascribes being unprovable to itself: 799 (Γ) ‘¬NumProvable(NumDiag(h¬NumProvable(NumDiag(x))i))’ Because of our semantic presuppositions and the G-meaning of Γ we know: (*) Γ is true if and only if Γ is unprovable in TA .

797 798

799

Cf. Smith 2007, 139f. Thus we took this semantic orientation already when we accepted the prerequisites (a) and (b) in section 6.3. It is the same sentence we already discussed as sentence (3) in section 6.3.

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And we can argue semantically for the fact that Γ is an undecidable sentence, i. e., that neither Γ nor its negation is provable in TA : If Γ were provable in TA , it would (because of (*)) be false, and TA would prove a false theorem; but that cannot be the case because TA is correct. So Γ is not provable in TA . Hence, because of (*), Γ is true. Thus the negation of Γ is false. But then, again because TA is correct, the negation of Γ cannot be provable in TA , and so TA does not refute Γ, either. Thus we have shown semantically that TA is semantically incomplete, because there is a true sentence that it does not prove, and that TA is syntactically incomplete, because there is a sentence of its language that it neither proves nor refutes. 800 This result is clearly of deep significance. But we need not discuss the exact nature of its import 801 because here the sole function of presenting its proof is to provide a contrast to the proof that truth cannot be G-expressed.

The second part of Tarski’s Theorem: Truth cannot be G-expressed Tarski’s Theorem can be presented as having two parts. 802 With respect to the present scenario and in present terms, its first part is the claim that the notion of truth of sentences of LA is not G-captured by the theory TA . 803 The second part of Tarski’s theorem is the stronger claim that the notion of truth of sentences of LA is not even G-expressed. Its proof is highly relevant to our question of whether the (Gödelian Recipe) delivers formal Liar sentences, and it can be presented in the following way: 804 Assume for reductio that there is an open formula of LA which G-expresses the notion of truth of sentences of LA , which we therefore abbreviate as ‘NumTrue(y)’. Then we could use the (Gödelian Recipe) to construct a G-self-referential sentence which G-ascribes untruth to itself, which would look like this: (Λ) ‘¬NumTrue(NumDiag(h¬NumTrue(NumDiag(x))i))’ Because of our semantic presuppositions and the G-meaning Λ would have, we know that the following would hold of it: (*) Λ is true if and only if Λ is not true. As this entails a contradiction we have shown that the assumption we made for reductio is false. Hence there is no open formula of LA which G-expresses the notion of truth of sentences of LA . 800

801 802 803

804

The presentation here is dense, but for our present purpose we do not need a longer report of the mathematical matters. Cf. Smith 2007, especially 124ff., and Boolos /Burgess /Jeffrey 2002, 187ff. The usual presupposition that the theory TA is consistent is not needed here because we have taken a semantic approach and assumed that TA is correct. And in a classical setting, truth entails consistency. A good guide to what is not the import of Gödel’s theorems is Franzén 2005. Smith 2007, 180f. A proof of this was presented already in section 3.6, albeit in slightly different terms: Where we said ‘to define’ then, we now say ‘to G-capture’. Smith 2007, 181; Enderton 2001, 236.

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Similar reasoning allows to show that the notion of falsity of sentences of LA is not G-expressible in LA , either.

There are no Gödelian Liar sentences Given this result of the G-inexpressibility of untruth and falsity, our (Context Principle for G-Meaning) 805 entails that there are no G-self-referential sentences that G-ascribe untruth or falsity to themselves. The Gödelian method of encoding meta-linguistic meaning in a language of arithmetic does not allow to form Liar sentences, and not even indirect counterparts of Liar sentences. However, before we take some time to appreciate this result, we should deal with a couple of possible objections. First off, there is a daunting question: Why is this reasoning not ad hoc? For the proof of the second part of Tarski’s theorem certainly appears to concern a Liar sentence that was constructed according to the Gödelian method! Additionally, there are some doubts: Are we really justified in using the rule of reductio ad absurdum in what amounts to a solution to the (purported) Gödelian variant of the Liar paradox? As all assumptions can be called into doubt in the vicinity of a paradox, the rule of reductio needs to be applied very cautiously there. 806 To deal with these objections, we need only contrast the proof of Tarski’s theorem to the proof of Gödel’s theorem. In both proofs, a G-self-referential sentence plays a crucial role (a Gödel sentence and a Liar sentence, respectively). But while in the proof of Gödel’s theorem the existence of a Gödel sentence is shown constructively, in the proof of Tarski’s theorem the existence of a Liar sentence that would also be constructed according to the (Gödelian Recipe) is talked about only in a hypothetical mode, and it depends on an assumption that is later shown to be wrong. This is so not only in the present case of an assumption for reductio, but also in the proof of the first part of Tarski’s Theorem we gave earlier, where the existence of a Gödelian Liar sentence depends on an assumption for conditional proof. 807 We find an illustration of this hypothetical mode in a passage by Herbert Enderton, who in his presentation of the proof for Tarski’s Theorem writes: “Consider any formula β (which you suspect might [express the set of the Gödel numbers of true sentences]). By the [Diagonal Lemma] (applied to ¬β) we have a sentence σ [. . . ]. If β did [express the set of the Gödel numbers of true sentences], then σ would indirectly say ‘I am false’.” 808

Furthermore, the dangers of using reductio in the vicinity of a paradox concern theoretical contexts in which all assumptions which together entail a contradiction 805 806 807 808

Cf. section 6.3. We will say more about reductio in the vicinity of paradox in section 7.3. Cf. section 3.6. Enderton 2001, 236. To make what Enderton writes readable for the readers of the present chapter, we have adjusted his terminology (within square brackets). Specifically, we have substituted ‘expresses the set of the Gödel numbers of true sentences’ for his “defines # Th N” and ‘Diagonal Lemma’ for his “fixed-point lemma” (Enderton 2001, 236).

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are of more or less equal plausibility, so that it would be improper to disprove one of them by a reductio argument with the other assumptions as premises. 809 The present case, by contrast, is like the use contemporary set theory makes of the reasoning of Russell’s paradox to show that there is no Russell set. In the case at hand, the only dubitable assumption is that truth is G-expressible, so using the Liar reasoning to refute the G-expressibility of truth is not ad hoc at all, even given that it allows to refute in a further step that appeals to the (Context Principle for G-meaning) that there is a Liar sentence. For we have seen already when we introduced the Gödelian method that we have no guarantee that every linguistic concept that is of interest to us will be among the G-expressible concepts. 810 And now we have seen that one interesting concept that is not among them is truth – in contrast to provability. 811 * Looking back, we can see how it was good to interpret the results of the Gödelian method charitably in order to bring out this difference between truth and provability clearly. 812 For the contrast between a (constructible) Gödel sentence and a (hypothetically assumed but non-existent) Gödelian Liar sentence has enabled us to show the following: Even according to those charitable standards according to which there are arithmetical sentences which indirectly state their own unprovability, there are no arithmetical sentences which indirectly state their own falsity or untruth! It is important to state this fact clearly because the conviction is widespread that the method of the arithmetization of syntax devised by Gödel provides an impeccable way of constructing formal counterparts of self-referential sentences. And quite often this method is seen as delivering Gödelian Liar sentences. 813 We find an example for this particular conviction in Rheinwald’s study about the semantic paradoxes, where she gives the following paraphrase of a formal Liar sentence:

809 810 811

812 813

Cf. section 7.3. Cf. section 6.2 and 6.3. A further objection might come from philosophers working on axiomatic theories of truth (Halbach 2011 and cf. Halbach 2014 for an overview; see also Leitgeb 2007). As they take a primitive predicate that is meant to explicate truth and add it to a formal base theory that is typically as strong as arithmetic, they may well think that it is the case almost by definition that there is a sentence that ascribes untruth to itself. From our perspective, however, they have changed the subject, and moved on from truth to a different notion that is as truth-like as possible, given that the assumption is held fixed that there are means to construct self-referential sentences. That is not to deny that the technical sophistication will bring philosophically relevant results (I do not mean to dig in my feet and antagonize people who prefer to do philosophy in a formal way; cf. sections 1.9 and 1.10), but only to call attention to the fact that the technical results will be more difficult to interpret than talk of a “truth predicate” might suggest. Cf. section 6.2. Cf., e. g., Kripke 1975, 692f.; Field 2008, 30. Note also the cavalier attitude adopted by Blau in the second motto of the present chapter (Blau 2008, 398).

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“The sentence which according to the Gödel numbering . . . has the Gödel number . . . is false.” 814

In this section we have shown that even when we accept the widespread conviction to be correct in general, it is wrong in the particular case: Although the Gödelian method is powerful, as is witnessed by its role in the proofs of Gödel’s and Tarski’s Theorems, it is misleading at best to say that it can be used to construct Liar sentences. This result is not only interesting in its own right; it is also highly relevant for our attempt to solve the Liar paradox. In the following third part of this study, I will try to show that neither the natural language devices of names, descriptions, and indexicals nor their formal counterparts allow to form self-referential sentences of the sort needed for the Liar paradox. Therefore the present result is very welcome: It allows to put to the side what arguably is the only serious candidate for by-passing those more standard referential devices. But obviously, the bulk of the work remains to be done.

814

The original text is: “Der Satz, der bei der Gödelisierung . . . die Gödelnummer . . . hat, ist falsch.” (Rheinwald 1988, 231, ‘. . . ’ in the original). We should note that when Rheinwald gives this particular paraphrase, she aims at a broad overview and is not concerned with Liar sentences in formal languages or with Gödelization. When she is dealing specifically with the Gödelian method of constructing selfreferential sentences, her paraphrases are more cautious (e. g., Rheinwald 2012[1991], 257).

PART III

THE METAPHYSICS OF EXPRESSIONS AND THE LIAR PARADOX

William of Ockham: “As for insolubles, you should know that it is not because they can in no way be solved that some sophisms are called insolubles, but because they are solved with difficulty.”

Chapter 7

How Can the Liar Paradox be Solved?

Bertrand Russell: “the solution should, on reflection, appeal to what may be called ‘logical common sense’ [. . .] it should seem, in the end, just what one ought to have expected all along.” 815

Our aim in this third part of our study is to solve the Liar paradox. The present chapter will be no more than a short intermezzo, meant to give us a brief break – an opportunity to collect ourselves for the final part of the journey and to welcome those readers who have only skimmed over the previous two parts or skipped them entirely (hopefully with the intention of referring back as needed). 816 To provide a new starting point, we begin with a brief summary (in section 7.1). Then we turn to the main question of the chapter: 817 What does it mean to solve the Liar paradox? In answer to this question, we will list several desiderata for a solution (in section 7.2) and reflect on the use of reductio ad absurdum in the vicinity of a paradox (in section 7.3). Anyone trying to solve a paradox should of course be aware of attempts that have already been made. But instead of trying to give a detailed overview of current work on the Liar paradox, we will present a collection of quotes that attest to the sheer plenitude of proposals, which we will count as evidence for the historical claim that the Liar paradox is not solved yet (in section 7.4) and then give only a very concise overview of the spectrum of modern approaches to the Liar paradox (in section 7.5). We will conclude the chapter with a brief discussion of the approach that prima facie is the most promising, contextualism, also mentioning earlier work of mine (in section 7.6). Before we start, a disclaimer. Most of the points we make in this brief chapter will not have to carry a great argumentative weight later. We will therefore just state them here without much argument, apart from the support some of them should already have from parts I and II.

7.1 What we have learned Before we say what we expect from a solution, we should take stock and give a brief summary of what we have learned about the problem so far. The previous two parts of this study hopefully contain elements that are interesting in their own right, part I about logic and part II about semantics. Here, however, we will highlight those 815 816

817

Russell 1985[1959], 61. The motto for part III is taken from Ockham 1974, II-3, 46. It should be possible to assess our argument in part III adequately for any reader who is willing to follow back the references to parts I and II where necessary, which will be given in the footnotes. Answering this question is not the same as outlining the specific problem posed by the Liar paradox, which is what we did in part I and partly in part II.

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bits that will be especially relevant for our own proposal to solve the Liar paradox that we will give in the following chapters. In part I we have learned that although it is mostly those of us who do philosophy in a formal way who tend to take it seriously, 818 the Liar paradox should be seen as a severe theoretical threat by any philosopher interested in logic or language. We have shown this by giving a dual presentation of the Liar reasoning, both in informal logic and in formal logic. 819 In order to answer the question of formality – roughly, whether the Liar paradox is really more disruptive in a formal setting –, 820 we have distinguished between a syntacticist understanding and a semanticist understanding of a language. According to the syntacticist understanding, the expressions of a language are no more than strings of marks (or sequences of sounds) that play certain syntactic roles, like forming a singular term, a predicate, or a sentence of a natural language, or forming a constant, a variable, a predicate letter, or a sentence of a formal language. According to the semanticist understanding, in contrast, we cannot separate the expressions of a language from what we intuitively construe as their meaning. In these terms, one of our important observations was that we have to adopt a semanticist understanding of language when we study the Liar paradox. 821 This observation interestingly runs counter to the initial impression that the Liar paradox is a bigger problem for formal logic, and in fact we have seen that because the key notions of truth and falsity are not separable from meaning (perhaps explicated as translation or most simply as disquotation in one of the variants of the Tarskian truth schema), even the most formal variants of the Liar reasoning have an irreducible non-formal residue. 822 Of course, all this does not in any way speak against the use of formal logic! But it certainly points our attention to those aspects of expressions that we can learn about only when we construe them as meaningful, and this will be important when we develop our account of the metaphysics of expressions. Continuing the theme of the severe threat posed by the Liar paradox, we should also recount another thing we learned in part I, which is that the inconsistency of the Liar paradox is contagious in the sense that any language is inconsistent that allows to reason about a Liar sentence, even one belonging to another language, 823 and that any Liar sentence must be seen as an inconsistent object, in the sense of being an object that both falls and does not fall under a certain concept (to wit, the concept of truth). 824 Apart form lending even more drama to the situation of those who try to solve the Liar paradox, the last of these observations in particular provides a motivation for taking a metaphysical turn in our own approach: The Liar paradox, as long as it is unsolved, entails not only that some theory is defective (as in the case of Frege’s inconsistent Basic Law V), but it entails that there is a contradiction out 818 819 820 821 822 823 824

Cf. chapter 1 and in particular section 1.9. Cf. chapters 2 and 3, respectively. Cf. section 1.10. Cf. section 3.1. We will take up this point again in section 8.1. Cf. sections 3.2, 3.5, and 3.6. Cf. section 3.8. Cf. section 3.9.

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there in the world. So it might be a good idea to look at expressions as objects that are out there in the world, too, when we try to solve it. In part II we have learned about semantics, and in particular about the semantics of singular terms, that are of course relevant for a study of Liar sentences, and about semantic aspects of what has been called Gödelian self-reference. We understand semantics as a study that encompasses both the theory of extensions (the reference of singular terms, the extension of predicates, and the truth values of sentences) and an account of what we pre-theoretically call meaning, including the Fregean notion of sense. 825 In parallel to our distinction between syntacticism and semanticism in chapter 3, we have distinguished two parts of the enterprise of doing theoretical semantics, which are the modeling as understood in a formal way (which can concern both the expressions of a formal language and the abstract ontology that provides objects as their extensions) and our material understanding that complements it (again concerning both the language and the world it is about). 826 These two understandings will recur in the present part III, and in particular when we ourselves use semantic theories to model language and world in chapter 10. Another important take-away can be summarized in terms of a Fregean thesis we have endorsed, according to which objects and concepts form disjoint and irreducible categories. As objects are what singular terms refer to and concepts (for us, and here we depart from Frege) are the senses of predicates, 827 this distinction carries over to semantics: For some expressions, what we pre-theoretically would call their meaning has an irreducibly objectual element; in particular, the meaning of directly referential singular terms like names and simple indexicals is grounded in their extension. For other expressions, their meaning has an irreducibly conceptual element; in particular, the meaning of primitive predicates and of the descriptions derived therefrom is given by the concepts they express, which notably cannot be reduced to their respective extensions (or their intensions, i. e., functions from contexts as circumstances of evaluation to extensions). 828 The existence of both classes of expression will be important for our approach to the Liar paradox. 829 We have described in detail some standard tools of current theoretical semantics, and in particular, two-dimensional semantics. Despite the fact that sense ultimately is irreducible, these tools retain their value for us, especially in the theory of extensions. We have used two-dimensional semantics to characterize the behavior of different kinds of singular terms (names, indexicals, descriptions, quotation expressions), and have complemented it with a pair notation that enables us to make explicit 825 826 827

828 829

Cf. section 4.2. Cf. section 4.3. We are here only indicating the kind of item that singular terms and concepts are. It would be more cautious to say that objects are what singular terms can refer to and concepts are the senses of possible predicates. Cf. chapter 4 and section 5.2. To anticipate, the observation that the meaning of some expressions is irreducibly objectual and the meaning of other expressions is irreducibly conceptual will give us reason to distinguish two broad categories of (purported) self-referential sentences, treated separately in chapters 11 and 12, and the existence of expressions with irreducibly conceptual meaning will be crucial for our solution of the problem of expressibility; cf. section 13.5.

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that some singular terms are directly referential and others indirectly referential (e. g., by distinguishing between the semantic values (D, object) and (I, singular sense)). 830 Two-dimensional semantics will play a key role in chapter 10, and of course the characterization of different kinds of singular terms will also be highly relevant, especially in chapter 11 which is about directly referential candidates for self-referential sentences and chapter 12 which is about indirectly referential candidates for self-referential sentences. In connection with quotation expressions, we have had a first encounter with the structural property of well-foundedness and how it can make things difficult for the Liar paradox: For as the relation of concatenation is well-founded (it always bottoms out in expressions that have no proper semantic parts), no sentence can refer to itself by quotation alone, and hence there are no candidates for Liar sentences which use (only) a quotation expression. 831 Last but not least, we have discussed the method of Gödelization from a semantic angle, and come to the (perhaps surprising) conclusion that even under a charitable understanding which allows for some formal sentence to claim (in some sense) its own unprovability, Gödelization does not allow to form Liar sentences. 832 As people who are convinced that there is self-reference of the kind that leads to the Liar paradox tend to cite the Gödelian method as evidence, this is an important preliminary result for us.

7.2 Desiderata for a solution to the Liar paradox We turn now to the main question of this chapter: What does it take for an approach to the Liar paradox to really amount to a solution? Let us hear Bertrand Russell about solving the set theoretic and semantic paradoxes: “[. . .] there were three requisites if the solution was to be wholly satisfying. The first of these, which was absolutely imperative, was that the contradictions should disappear. The second, which was highly desirable, though not logically compulsive, was that the solution should leave intact as much of mathematics as possible. The third, which is difficult to state precisely, was that the solution should, on reflection, appeal to what may be called ‘logical common sense’ – i. e., that it should seem, in the end, just what one ought to have expected all along. [. . . ] The third condition is not regarded as essential by those who are content with logical dexterity. Professor Quine, for example, has produced systems which I admire greatly on account of their skill, but which I cannot feel to be satisfactory because they seem to be created ad hoc and not to be such as even the cleverest logician would have thought of if he had not known of the contradictions.” 833

As we are not dealing with a paradox that threatens the foundations of mathematics or the logicist project in particular, but with the Liar paradox, we can safely ignore Russell’s second requisite, which is about mathematics. But the other two 830 831 832 833

Cf. sections 4.7. and 5.2. Cf. sections 5.5 through 5.7. Cf. chapter 6. Russell 1985[1959], 61.

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observations made by Russell apply here, too. Thus we get the first two entries on a list of desiderata for a solution to the Liar paradox, which arguably are also the most important desiderata for the solution to any paradox: (Consistency) Prima facie, inconsistency is absolutely unacceptable. Therefore “the contradictions should disappear”; a solution to the Liar paradox should restore consistency. 834 In the terms developed in chapter 2, a solution to the Liar paradox should respect the exclusiveness of truth and falsity. 835 This is all the more important (if possible) because we found that the inconsistency of the Liar paradox is contagious and any Liar sentence is metaphysically inconsistent: 836 If there is a Liar sentence in any language, then every language that allows to talk about this is inconsistent. And turning from semantics to metaphysics, any Liar sentence would be an inconsistent object – an object, that is, that falls and does not fall under a certain concept. (Explanatoriness) The second-most important desideratum, I contend, is non-ad hocness, or to give it a positive name, explanatoriness. 837 The solution to a paradox is explanatory if and only if the modification it uses to dispel the apparent conflict that makes up the paradox is justified by reasons that are independent of the aim of solving the paradox. In Russell’s words, “the solution should [. . .] appeal to what may be called ‘logical common sense’” 838 and thus be such that even a hypothetical not-so-clever logician who never heard about the paradox might have thought of it. 839 But we want more. For the sake of brevity, we will continue to state desiderata as unconnected points with neat labels. Now we are no longer following Russell, but adding further desiderata, which are more or less specific to the Liar paradox:

834

835 836 837 838 839

Nowadays we need to characterize the notion of a solution to a paradox in a way that does not exclude paraconsistent approaches by definition; cf. section 1.1. But by giving the desideratum of (Consistency) top priority on the list of desiderata we do not take back this earlier move of admitting paraconsistent approaches among the candidates for a solution. This priority means no more than that paraconsistent approaches will have to do very good with regard to the other desiderata in order to compete with consistent approaches. Cf. section 2.3. Cf. sections 3.8 and 3.9. Cf. Haack 1978, 140. Russell 1985[1959], 61. The desideratum of (Explanatoriness) is also required (in spirit, if not in name) in Susan Haack’s distinction between a “philosophical solution” and a “formal solution” (Haack 1978, 138f.) and in Rheinwald’s distinction between explanation (Erklärung) and prevention of a paradox (Vermeidung; Rheinwald 1988, 10f.). Also cf. Bromand 2001, 16f.

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(Generality) A solution to the Liar paradox should be general. First of all, it should allow to deal with Liar sentences of all kinds, 840 including revenge sentences. 841 Also, a solution should solve not only the Liar paradox itself, but also all of its closer relatives: Liar cycles, Yablo’s paradox, Church’s paradox, Curry’s paradox, and perhaps other semantic paradoxes. 842 Ideally, a solution to the Liar paradox will also include at least some information about the relation of the semantic paradoxes to the set theoretic paradoxes. 843 (Simplicity of Truth) Any proposal for a solution to the Liar paradox is likely to meddle with one of its essential ingredients: the concept of truth. 844 Therefore anyone who tries to solve the Liar paradox should take care to capture as much as possible of our ordinary intuitions about truth. I. e., they should not tarry too far from the intuition of correspondence that is explicated by the naïve truth principle or the Tarskian truth schema, 845 and we should be able to justify any departure. Among our ordinary intuitions are also, I contend, that truth is a simple concept that for its explication should not need a gerrymandered theory, 846 and that any truth predicate is univocal, i. e., that, as the surface grammar of our languages suggests, there is only one truth predicate in each language, and that these are synonymous: ‘. . . is true’ is not elliptical for ‘. . . is true in language L’. (Unity of Language) A solution to the Liar paradox should work for any language, and in particular both for natural languages and for formal languages. We have seen in part I that the Liar paradox occurs both in an informal and in a formal setting; and we have found reason to believe that (at least with regard to the Liar paradox) there is less of a gap to be bridged between the informal and the formal stance than is commonly thought. 847 It is therefore desirable that a solution to the Liar paradox be based on a single idea that works in the same way for all languages. (Therefore a solution to the Liar paradox must also include a solution to those variants of the problem of Expressibility where 840 841 842

843 844

845 846

847

Cf. section 5.4. Cf. section 2.6. For Liar cycles and Yablo’s paradox, cf. section 1.3; for Church’s paradox, cf. section 1.4; for Curry’s paradox, cf. section 1.5; and for other semantic paradoxes than the Liar paradox, cf. section 1.8. Cf. section 1.8. Cf. Leitgeb 2007, with the telling title “What Theories of Truth Should be Like (but Cannot be)”, for a list of desiderata that a theory of truth ideally should meet, which as a topic is not far away from our list of desiderata for a solution to the Liar paradox. Cf. sections 2.2 and 3.2. We require simplicity of the notion of truth pace John Burgess, who wrote an article on the Kripkean approach to the Liar paradox with the telling title “The Truth is Never Simple”; cf. Burgess 1986. Cf. section 1.9 as well as chapters 2 and 3.

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the paradox-free formal language and the paradox-solving fragment of natural language come apart. 848) (Semantic Closure) There is a strong intuition that natural language is semantically universal, i. e., that it is flexible enough to allow to talk about anything whatsoever, and in particular that it is semantically closed, i. e., that it allows to talk about itself and in particular about semantic features of its expressions (notably including the truth or falsity of sentences). As semantic closure is usually seen as entailing the existence of self-referential sentences, someone trying to solve the Liar paradox might well feel forced to compromise with regard to the semantic closure of his or her official language. But because of the strong intuition it is desirable that he or she does not concede too much with regard to semantic closure. In particular, a solution to the Liar paradox should be able to address the Reflective problem, and, more generally, it should be able to address the Expressibility problem. 849 And the final points on our list of desiderata again concern the solution to any paradox: (Independence of Logic) Logic should not be hostage to paradox. As logic explicates our ordinary reasoning, it is desirable that we are able to read off the correct logic 850 from ordinary cases of our reasoning – without having to tailor it to fit the extraordinary problem presented by a paradox. This does not mean to say that classical logic is sacrosanct, but that the legitimate reasons for logical revision do not include the paradoxes. Even though there might be good reasons to call classical logic into question, the Liar paradox hopefully is not among them. (Accommodation of the Previous Debate) When all of the above has been achieved, there still remains something to be done. To solve a paradox it is not enough to explain away the apparent conflict between its premises and its conclusion, but ideally we will want to explain the initial appearance, too. In the case of the Liar paradox, there by now is a vast plurality of approaches, and a variety of incompatible stances have been adopted. 851 This is itself a datum that calls for an explanation. The desideratum to accommodate the previous debate about the paradox in question stands in an odd tension to the desideratum of (Explanatoriness), which (one is tempted to say, paradoxically) calls for an attitude that is unspoilt 848 849 850 851

Cf. section 2.8. Cf. sections 2.7 through 2.10. . . . or the correct logics, if we are logical pluralists. Cf., e. g., Beall /Restall 2006. Cf. sections 7.4 and 7.5.

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by too much interest in solving the paradox. I think that the two desiderata are complementary in the sense that when we try to solve the paradox, we ideally should in a first step learn about the debate about it so that we know as much as possible about the problem, then in a second step somehow imagine to forget again all that we have learned to enable us (with luck and skill) to find an explanatory solution, and in a final third step turn back to the debate in order to understand it better in light of our solution. We see the desiderata for a solution to the Liar paradox here in a tentative order of decreasing importance. It is good to be clear about our priorities, because trying to satisfy all these desiderata is a tall order. We have to acknowledge that the majority of people working on the Liar paradox today think that it is impossible to satisfy all desiderata at once. One who shares this pessimist near-consensus is Michael Sheard: “As everyone knows, this is an area in which one simply cannot have all that one might reasonably ask for. The selection of any particular approach necessarily involves a trade-off, and in the face of criticism that a given system has some flaw (usually of omission), the most appropriate response is typically concession, accompanied by an argument that the advantages outweigh the failings. The resulting colloquy may be informative or even inspiring, but it can go on forever.” 852

We will have to see whether this pessimism is warranted.

7.3 Reductio ad absurdum and paradox We should supplement the list of desiderata of the previous section with a brief remark about the use of reductio ad absurdum 853 in the vicinity of a paradox. This means to elaborate on the desideratum of (Explanatoriness), in particular. Imagine someone who wants to solve the Liar paradox by rejecting the claim that there are Liar sentences and who argues by assuming for reductio that there is a Liar sentence, reasoning in the usual way until a contradiction has been inferred, and concluding by the rule of reductio that there are no Liar sentences. A Concise Solution to the Liar paradox (1) There is a Liar sentence. assumption for reductio ad absurdum (2) That Liar sentence is true and not true. (1), (part of) the usual Liar reasoning 854 (3) There are no Liar sentences. (1) through (2), reductio ad absurdum

Although it surely looks valid, this argument is thoroughly unsatisfying as a solution to the Liar paradox. Why? It is not that the logical rule of reductio in the vicinity of a paradox miraculously looses its force. Rather, its application as in this Concise Solution pushes ad hocness to its limit – the proposed solution is not at all 852 853 854

Sheard 1994, 1033. Cf. section 2.1. E. g., line (2) through (7) of Variant 2 of the basic Liar reasoning with a strengthened Liar sentence in section 2.5. The place of the premise in line (1) of Variant 2 is here taken by a hypothetical assumption.

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explanatory. It also is quite arbitrary, because we could use reductio ad absurdum in an entirely analogous way to argue against any one of the apparently true premises of an antinomic paradox. 855 The last observation about arbitrariness calls for a clarifactory contrast: In contemporary set theory it is common to use the reasoning behind Russell’s paradox 856 in a reductio argument that shows that it is not the case that for every open formula of set theory there is a unique set of just those objects that satisfy that formula. That is, a claim that Frege endorsed as an axiom of his formal theory 857 by contemporary set theorists is assumed only hypothetically and then proven false. 858 What is it that makes this way of arguing more respectable than the above Concise Solution to the Liar paradox? The justification can be put like this: In the present situation of the historical development of set theory and of mathematics in general, it is no longer arbitrary which of the premises that led to Russell’s paradox is to be put to the test of hypothetical assumption, eventually to be discarded after a brief reductio argument. Thus the contrast shows this: When we have good reason to hold on to a collection of principles, but are unsure about a further principle, we are allowed to try to use reductio to show that the precarious principle does not hold. But in the vicinity of the Liar paradox (or any other proper paradox, for that matter), we cannot be sure of any one of the principles that are involved. This difference is really not of a systematic, but of an historical nature: A hundred years of mathematical practice have led to rather firm convictions about foundational principles of set theory within the scientific community. 859 But as of now, no such consensus has been reached with regard to the principles involved in the Liar reasoning – as we will see in the next section. The moral is clear: When trying to solve a paradox, keep your hands off reductio arguments – at least as long as no proposal for a solution has won the stable approval of a majority of the experts! (And then the need for solution has become less pressing, anyway.) In effect, reductio is just unsuited as a tool for dealing with any paradox that merits the name; but it may offer a concise way of dealing with a solved paradox, that is, with a former paradox. 860 Although the present point about the ad hocness of reductio ad absurdum has been noted sometimes, 861 it is worth emphasizing again, because there are quite

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Another case in point is Church’s paradox, which is a reductio argument about Epimenides’s utterance that all Cretan utterances are false which shows that some other Cretan utterance is true. Cf. section 1.4. Cf. section 1.8. What Frege’s infamous Basic Law V says is, roughly, that for every open formula of set theory there is a unique set of just those objects that satisfy that formula. Cf. sections 1.8 and 3.9. Cf., e. g., Deiser 2002, 187; Potter 2004, 25ff. A similar contrast is provided by the pseudo-paradox of the barber who is said to shave just those people who do not shave themselves. Here we are usually thought to have every right to use a reductio argument to show that there just is no one who fits that description (cf., e. g., Rheinwald 1988, 12). Cf. the remark about the element of historicity in the notion of a paradox at the end of section 1.1. E. g., cf. Martin 1978[1970]c, 91.

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a few arguments with the aim of solving the Liar paradox where it has not been properly heeded. 862

7.4 The Liar paradox is as yet unsolved. By now, very many proposals have been made to solve the Liar paradox. Rather than trying to list them all, I will resort to citing a number of remarks on that plenitude. It is more than forty years since Charles Parsons asked: “Why is it that today, more than sixty years after Principia Mathematica and nearly forty years after the first publication of Tarski’s Wahrheitsbegriff, the Liar paradox is still discussed as if it were an open problem?” 863

And about fifteen years ago, Michael Glanzberg took up Parsons’s question: “About twenty-five years ago, Charles Parsons published a paper that began by asking why we still discuss the Liar Paradox. Today, the question seems all the more apt. In the ensuing years we have seen not only Parson’s work (1974), but seminal work by Saul Kripke (1975), and a huge number of other important papers. Surely, one of them must have solved it! In a way, most of them have. [. . .] What we lack is not solutions, but a way to compare and evaluate the many ones we have.” 864

Glanzberg is by no means alone in this assessment. Here are some further quotes which make a similar point. Anil Gupta and Nuel Belnap wrote in 1993: “Paul of Venice, a fourteenth-century logician, discusses fourteen proposals for solving the problem before offering one of his own. A modern-day Paul of Venice would need to discuss as many as fourteen times fourteen solutions.” 865

Graham Priest wrote in 1993: “Trying to find a solution to the semantic paradoxes has been a perennial theme in logic this century. A comprehensive review of all the solutions suggested would fill many volumes. Nor has the stream of proposed solutions dried up of late: the last twenty years have seen a number of solutions of elegance and technical virtuosity that few previous proposals can match.” 866

Greg Restall wrote in 1993: “There is almost universal agreement that no obviously correct account of the selfreferential paradoxes has yet surfaced.” 867 862

863 864 865 866 867

Cf., e. g., van Fraassen 1978[1970], 13ff. and especially 16, where Bas van Fraassen states that a simple Liar sentence “presupposes a contradiction, and hence cannot have a truth value”; Parsons 1974a, 387; and Glanzberg 2004, 33f. Parsons 1974a, 381. Glanzberg 2001, 217. Gupta /Belnap 1993, 10. Priest 1993, 60. Restall 1993, 282.

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Albert Visser wrote in 2004: “The semantical paradoxes are not a scientific subject like Inductive Definitions, Algebraic Geometry or Plasma Physics. At least not yet. On the other hand the paradoxes exert a strong fascination and many a philosopher or logician has spent some thought on them, mostly in relative isolation. The literature on the paradoxes is vast but scattered, repetitive and disconnected.” 868

In some texts, the overwhelming vastness of the literature on the Liar paradox is observed mainly to justify the author’s decision not to give a full treatment of every important proposal. 869 And I admit that I am giving the above collection of quotes partly with a similar intention of explaining myself, for we will have only the briefest of looks at past proposals for solving the Liar paradox. 870 But that is not the main function of the quotes here, which is rather to provide textual evidence for a prima facie striking historical claim: As of 2017, the Liar paradox has not yet been solved. Our reasoning is as follows. Recall that a paradox is an apparently valid argument from apparently true premises to an apparently unacceptable conclusion; and a solution to a paradox is an explanation that dispels some of these appearances as illusory. We saw when we introduced these notions 871 that the status of being a paradox involves an intersubjective (or even democratic) element: To constitute a true paradox, the conflicting appearances that make it up must be widely shared. It is only natural to require a similar intersubjective element of a true solution: The explanation it consists in must be good enough (or more precisely, convincing enough) to sway the intuitions of a majority (or at least a comparatively large number) of the experts in a way that dissolves the conflict of appearances. Thus a paradox is only truly solved when there is a solution which the majority of experts can agree upon. 872 In the case of the Liar paradox, it may well be that today a majority of experts think that it is solved in some way. But their reasons for thinking so differ; there is no majority agreeing on any one of the contending and often incompatible proposed explanations meant to dispel the conflicting appearances. A true solution to the Liar paradox would induce revised intuitions, which (at least in the long run) would be as widely shared as the intuitions about language, truth, and logic, which currently make it so easy to lead almost anyone through the basic Liar reasoning. So the contemporary abundance of different proposals for solving the Liar paradox must, with all due respect for the good work of their proponents, be seen as an indication 868 869 870 871 872

Visser 2004, 149. Cf. Gupta /Belnap 1993, 10; Visser 2004, 149. Cf. section 7.5. Cf. section 1.1. Arguably this is so in the contrasting cases of the Achilles paradox and Russell’s paradox. Here there is a majority of people who think that the paradoxes are solved, and they will more or less all be convinced by the same solution, gesturing towards the advances in the mathematics of infinity and in axiomatic set theory, respectively. Cf. sections 1.1, 1.6 and 1.7.

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that, as of now, no true solution has been reached. 873 And we should construe Glanzberg’s and Priest’s talk of there being several solutions as nothing more than a polite way of saying that several proposals have been made, so that there really is no solution yet. 874

7.5 The spectrum of modern approaches to the Liar paradox Instead of presenting a select few important proposals in full, we will now look at a coarse overview of the spectrum of modern approaches to the Liar paradox. 875 In the preceding section we have seen that, in the strict sense of the word, no solution to the Liar paradox has yet been reached. Because of the intersubjective element inherent in the notion of a solution to a paradox, it was possible to argue for this claim without discussing each important proposal in turn, and in fact without looking at any particular approach at all. But an overview of the full range of approaches will be helpful to orient us on our way into the investigation that we will undertake in the rest of this study, and for contrasting purposes. 876 * For an approach to a paradox to amount to a solution it must involve a rejection of at least some of the essential elements of that paradox. It is fair to say that in the case of the Liar paradox, everything has been tried by now, from denying the existence of Liar sentences via modifying the Tarskian truth schema or rejecting those rules of classical logic that are used in the Liar reasoning (often in combination with each other) to accepting the contradictory conclusion (and thus also departing from classical logic). It turns out, however, that distinguishing these four options is not the best route to a description of the full spectrum of approaches to the Liar paradox. A taxonomy based only upon this fourfold distinction would be just too coarse-grained, and it would be misleading, especially in view of the considerable overlap between modifications of the truth schema and of the logical system. In order to describe the wide spectrum of approaches taken to the Liar paradox in the last one hundred years in a more telling way, I suggest that we extend (and maybe overstretch) an old medical metaphor that likens a paradox to a symptom of a disease. Tarski wrote in 1969:

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A similar point is made by Willem Groeneveld. From the “vast number” of proposed solutions he tentatively draws a similar conclusion: “The large number of purported solutions might even lead to the contention that there is no real solution.” (Groeneveld 1994, 267) This is obviously so in the case of Priest, who at one point adopts an history of science perspective and counts the very proliferation of proposals as evidence for his claim that the quest for a solution must fail (Priest 2006a, 24f.). The ongoing modern phase of the discussion about the Liar paradox starts in the late 19th century (cf. section 1.6), with Cantor’s Diagonal argument (cf. section 1.7). For more detailed overviews of (some larger part) of the spectrum of approaches to the Liar paradox, cf. Brendel 1992, 55–215; Simmons 1993, 45–98; Koons 1992, 85ff.; Bromand 2001, 81ff.; Dutilh Novaes 2008; and Beall /Glanzberg 2014, section 4.

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“I am not the least inclined to treat antinomies lightly. The appearance of an antinomy is for me a symptom of disease.” 877

Ten years later, Charles Chihara took up the medical metaphor when he distinguished two problems raised by the Liar paradox, the “preventative problem” of devising a paradox-free language that to a certain extent is semantically closed and the “diagnostic problem” of “explaining how and why” the conflict of appearances that makes up the paradox arose in the first place. 878 The medical metaphor can fulfill different functions. Tarski used it to reconcile the common first impression according to which the Liar paradox is no more serious than a joke with his own perception of it as a sign of grave trouble. 879 By likening the paradox to a symptom, which in itself is often by no means as uncomfortable as the disease it is a sign of, Tarski claimed the role of the physician, of an expert capable of perceiving the danger more clearly than the layperson, who might feel no more than an irritating itch. Chihara used the same metaphor to make explicit his demand that an approach to the Liar paradox should meet what we have formulated as the desideratum of (Explanatoriness) – prevention alone is not enough, it must be complemented by insightful diagnosis. 880 For us the metaphor will be valuable for a third reason. Likening a paradox to a symptom of a disease opens up a wide repertoire of remedies which provide analogies for the different theoretical stances that have been adopted towards the Liar paradox. By a ‘theoretical stance’ we mean, besides the broad strokes of the propositional part of a theory, the guiding background conviction of what that theory is for, of what the theorist tries to achieve with it. * In a rough chronological order, the modern approaches to the Liar paradox can be grouped systematically under the following labels: prohibitive prevention, defeatist diagnosis, psychotherapy of hypochondriasis, contextualist cure, medieval medicine, sophisticated surgery, and palliative care by paraconsistency. The story goes like this: The approaches of prohibitive prevention denied the existence of Liar sentences by restrictions on syntax that make sentential self-reference impossible – Bertrand Russell by appealing to an ontological hierarchy of all entities 881 and Alfred Tarski by forming a semantic hierarchy of an object language

877 878 879 880

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Tarski 1969, 66. Chihara 1979, 590f. Cf. section 1.9. Cf. section 7.2. – The distinction between prevention and diagnosis has become widespread and is used often to remind us of the desideratum of explanatoriness. Barwise and Etchemendy, e. g., write: “Logicians, it is said, abhor ambiguity but love paradox. Perhaps that is why they are so inclined to give formal prescriptions for avoiding the famous Liar paradox, but so loathe to diagnose the underlying problem that gives rise to it.” (Barwise /Etchemendy 1987, 4) It is not obvious, though, that Barwise and Etchemendy escape their own criticism. Their formal approach is sophisticated and elegant, but they do not say much to motivate it philosophically. Cf., e. g., Russell 1967[1908] and Rheinwald 1988, 57ff.

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and the associated meta-languages. 882 Russell thus intended to solve all set theoretic and semantic paradoxes in one fell swoop, including the Liar paradox. Tarski, by contrast, aimed only at the Liar paradox (and by extension at the other semantic paradoxes). The importance of Tarski’s hierarchical approach as a source of the current model theoretic way of understanding truth and logic should not be underestimated (and the same goes for its status as the orthodoxy among those philosophers who do not themselves work on the paradoxes). As decreeing restrictions on syntax works only for formal languages, Charles Chihara gave the defeatist diagnosis (which in a sense had already been formulated by Tarski) that whereas formal languages can be kept paradox-free, natural languages are inconsistent, a resigned echo of Tarski’s pessimism about natural language. 883 Ordinary language philosophers came to the rescue of natural language by denying the existence of Liar sentences in a fashion quite different from that of the earlier logicians. Gilbert Ryle, Yehoshua Bar-Hillel, William Kneale, John Leslie Mackie, and more recently Dorothy Grover, argued that any (purported) Liar sentence is in some sense meaningless, often by pointing to the pragmatic presuppositions for a sentence to express a proposition. 884 As these approaches try to convince us that the paradox is an artifact of our wrong explication of the workings of language and that there was no real problem to begin with, they are like the psychotherapy of hypochondriasis. Note that an irrational fear of a certain ailment can itself be a serious condition, it just concerns a different part of the patient than what the hypochondriac thinks. Thus the meaninglessness approaches should not be mistaken as underestimating the problem. In the 1970s, the general attitude to the Liar paradox changed in two important ways. Especially since Saul Kripke’s important 1975 paper, most people have been convinced that neither syntactic nor semantic reasons suffice to show that there are no Liar sentences 885 – neither the existence nor the meaningfulness of selfreferential sentences stands in question, they think, and many of them feel free to poke some fun at trying to “ban” self-reference. Also, both philosophical logicians and ordinary language philosophers now see the difference between formal and natural language as less pronounced than before. The first development led to questioning the Tarskian truth schema and classical logic, both of which had been held sacrosanct by all proponents of the three earlier groups of approaches. The second development led (among other things) to admitting an element of contextdependence into formal work. Approaches of contextualist cure that have been proposed by Charles Parsons, Tyler Burge, Haim Gaifman, and more recently Michael Glanzberg 886 try to over-

882 883 884 885

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Cf. Tarski 1956[1935]; Tarski 1944. Chihara 1969, 608ff. and Tarski 1956[1935], 154–165. Ryle 1952; Bar-Hillel 1957; Kneale 1972; Mackie 1973, 237ff.; Grover 2005. Kripke 1975, 691ff. – Cf. Gupta 1982, 1; Priest 2006a, 11f.; Field 2008, 23ff.; Beall/Glanzberg 2014, section 2.3.1. Parsons 1974a; Burge 1979; Gaifman 1988 & 1992; Glanzberg 2001 & 2004.

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come the proliferation of truth predicates of the earlier hierarchical approaches by construing truth as context-dependent. 887 Contextualization of truth also appears to promise to account for the reflective step 888 that is often seen as being forced upon us in reasoning about the Liar paradox: After a strengthened Liar sentence has been diagnosed as paradoxical and hence as not true it seems to come out as true, after all, because it says that it is not true. 889 We saw already in our brief overview of its pre-modern history that there was a multitude of medieval approaches to the Liar paradox and its relatives. 890 Although most of these can be treated as precursors of one of the modern approaches (and therefore do not occur in our spectrum), some stand alone and are of systematic interest. In particular, we should mention the medieval medicine prescribed by Stephen Read: Inspired by Bradwardine, he has developed a theory that revises our notion of truth, laying down that a sentence is only true if everything it signifies is the case. Crucially, a sentence can signify things other than what it intuitively says, and thus the Liar sentence comes out as false only. 891 Approaches of sophisticated surgery have been given by Saul Kripke, Anil Gupta (and Nuel Belnap), Jon Barwise and John Etchemendy, Vann McGee, Keith Simmons, and more recently by Hartry Field. 892 In this connection we should also mention philosophers working on axiomatic theories of truth. 893 All these theoretical surgeons have, in accounts of a high degree of technical sophistication, proposed what amounts to restrictions of the Tarskian truth schema, often supplemented by a non-classical logic. They allow for a differentiated treatment of self-referential sentences, sometimes ascribing an unproblematic status to all but the paradoxical ones. 894 Finally, since Graham Priest called in 1979 to “stop banging our heads against a brick wall trying to find a solution, and accept the paradoxes as brute facts” 895, there is a growing (and by now sizeable and rightfully respected) group of paraconsistent logicians who propose to solve the Liar paradox by giving up the (Exclusiveness) of truth and falsity. They think that some contradictions are true, but keep these from doing the greatest possible harm, trivialization, by giving up the classical ex falso quodlibet principle according to which a contradiction entails any sentence. As they

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Parsons 1974a, 381ff.; Burge 1979, 195ff. Cf. section 2.7. Cf. Parsons 1974a, 388f.; Burge 1979, 178f.; Gupta 1982; Gupta /Belnap 1993; Gaifman 1992, 223ff.; Glanzberg 2001, 230 & 2004, 32–35. The reflective step sometimes goes by the name of ‘the Son of the Liar’ (e. g., Rheinwald 1988, 53ff.) or ‘the Strengthened Liar’ (e. g., Burge 1979, 178f.). Cf. section 2.7. Cf. section 1.6. Read 2002, 2006, 2008, and 2009; and cf. the other contributions in Rahman/Tulenheimo/Genot 2008. Kripke 1975; Gupta 1982; Gupta/Belnap 1983; Barwise /Etchemendy 1987; McGee 1991; Simmons 1993; Field 2008. Halbach 2011 and cf. Halbach 2014 for an overview; see also Leitgeb 2007. E. g., Gupta 1982, 48ff. Priest 1979, 220. Cf. Priest 2006a.

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try to make the unacceptable acceptable by alleviating the pain, their approach can plausibly (if perhaps polemically) be characterized as one of palliative care. 896 * Much more would have to be said to enable us to even start evaluating the proposals made so far to solve the Liar paradox in a way that would do any of them justice. But I think that we should not aim to, as Priest said, “fill many volumes” 897 with a discussion of past and present proposals, and we will therefore refrain from going beyond our schematic characterization of the spectrum of modern approaches. Instead of taking us through a substantiated critique, I will do no more than give a personal report of what has been my view since I have started reading the current literature on the Liar paradox. It is my opinion that in the current situation, contextualism presents our best hope of finding a safe and sane middle way between the desperation of diagnostic and palliative approaches and the naïve heroism of prohibitive and therapeutic approaches, and of bypassing the unnaturalness of approaches of sophisticated surgery (which in a way are both desperate and naïvely heroic) as well as avoiding the strong side effects of the medieval medicine. This opinion of mine can be presented as the result of confronting the spectrum of approaches with the desiderata listed earlier in the chapter. 898 The approaches of depressing diagnosis and of palliative care by paraconsistency would appear to be desperate because they wholly give up meeting the most important desideratum of (Consistency), and at least in the case of the diagnostic approaches fail also with regard to (Explanatoriness). 899 The approaches of prohibitive prevention and the psychotherapy of hypochondriasis appear naïvely heroic insofar as their proponents ignore bravely the evidence (and the word of their colleagues) that there really is a dangerous problem to be solved, thereby failing utterly with regard to the desideratum of the (Accommodation of the Previous Debate), and each group failing in its own way also with regard to the (Unity of Language) because the prohibitive approaches work only for formal languages and the therapeutic approaches usually are formulated only for natural languages. The approaches of sophisticated surgery appear unnatural because they fail with regard to the desiderata of the (Simplicity of Truth), of the (Unity of Language), and in some cases also of (Explanatoriness). To me they even appear to be motivated by a combination of desperation and naïve heroism insofar as we can understand their proponents as starting out by accepting the unsolvability of the problem they are concerned with and reacting by plunging into the heroic work of creating very detailed technical systems which are meant to make the best of the desperate situation in which they believe to find themselves. 896

897 898 899

Although the six categories developed above are meant to be exhaustive of the field of approaches to the Liar paradox, not all of them are mutually exclusive. Contextualist elements often get incorporated into approaches of sophisticated surgery (e. g., in Simmons 1993 and Barwise /Etchemendy 1987), and the contextualist approach of Glanzberg has a high degree of technical sophistication. The monumental monograph by Ulrich Blau is a highly technical approach which includes an element of paraconsistency that is contextualized, and so brings together elements from these three approaches (cf. Blau 2008). Priest 1993, 60; quoted in full in section 7.4. Cf. section 7.2. Priest, in contrast, has done a lot to explain why there are true contradictions; cf. especially Priest 2002.

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The approaches that prescribe medieval medicine again remove us far from our intuitions about truth and might run into some trouble with regard to (Explanatoriness), because to show that the Liar is only false they use part of the Liar reasoning. Contextualist approaches, by contrast, will probably perform admirably with regard to the desiderata of (Simplicity of Truth) and (Unity of Language) and thus promise a high degree of naturalness, and at least prima facie there is no reason to suppose that they will fail with regard to any of the other desiderata. Thus what presents our best hope for fulfilling all the desiderata on the list to a reasonable degree is contextualism. 900

7.6 The limits of semantic contextualism We will conclude the chapter with a cautioning remark about what we have singled out as the most promising cure for the Liar paradox, i. e., about taking a contextualist approach. Our focus will (again) not be on the current literature, but rather on a simple initial idea about how contextualism promises to solve the Liar paradox, followed by the immediate realization that it is altogether too simple. This will point the way for developing our own brand of contextualist approach. Contemporary contextualists typically employ the Reflective step to show that the truth value of a Liar sentence is relative to context. I. e., they argue that the basic Liar reasoning with its contradictory result shows that a strengthened Liar sentence is defective and hence neither true nor false, but not being true is just what a strengthened Liar sentence ascribes to itself – so it must be true, after all! 901 The specific twist given by most contextualists is that this situation can be consistent if a change of context occurs within this reasoning, so that the Reflective step appears to provide evidence for such a shift of context. This way of arguing has been criticized in a convincing way by Christopher Gauker. 902 In our own terms, there is bound to be a problem with the desideratum of (Explanatoriness) here because part of the Liar reasoning itself is used to support a claim that is meant to solve it, which is awfully similar to how one gets entangled by using a reductio arguments in the vicinity of paradox. 903

Contextualist hope Thus we lose one possible reason for the claim that the truth of a Liar sentence is relative to context, but we might still retain our hope that context-relativity can help solve the Liar paradox. How can we support this hope, especially if it is not 900

901 902 903

These considerations about how all current approaches to the Liar paradox except for contextualism fail to meet our desiderata, I should like to repeat, are little more than a report about my own initial impression of the literature. It might even be the case that this impression was itself the result of an initial sympathy for contextualism, likely stemming from my prior work in the philosophy of time. Cf. section 2.7. Gauker 2006. Cf. section 7.3.

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indicated by the Reflective step? Here is a simple idea: The Tarskian truth schema, which plays a crucial role in the Liar reasoning, governs the behavior of the truth predicate only when it expresses an absolute notion of truth. But where truth is relative to context, the Tarskian truth schema does not apply. This observation is made, e. g., by Hartley Slater: “[. . .] consider for a start the sentence that is said to lead to the Truth Teller paradox: ‘This sentence is true’. The most obvious feature of this sentence is that it contains a demonstrative ‘this sentence’, so in particular Tarski’s unamended Truth Scheme does not apply, i. e. one cannot say: ‘This sentence is true’ is true if and only if this sentence is true, any more than one can say ‘he is happy’ is true if and only if he is happy. If truth is still to be attached to such sentences then the Tarskian Truth Scheme must be modified in some way: ‘he is happy’ said of John is true if and only if John is happy, ‘this sentence is true’ said of s is true if and only if s is true.” 904

But as the Tarskian truth schema does not apply – might not then the Liar reasoning be blocked? We can try to follow Slater’s way of modifying the Tarskian schema and apply it to an indexical Liar sentence: 905 ‘This sentence is not true’ as said of s is true if and only if s is not true. Abbreviating the quotation expression ‘‘This sentence is not true’’ as ‘Λ’ and instantiating the variable ‘s’ as Λ we get the following instance:

Λ as said of Λ is true if and only if Λ is not true. And this is not inconsistent! Or more cautiously: It is not evident that this is equivalent to the biconditional ‘Λ is true if and only if Λ is not true’, the likes of which we have used in the Liar reasoning to derive a contradiction. For ‘Λ as said of Λ is true’ need not be equivalent to ‘Λ is true’. This consideration might lead us to entertain the hope that we have found a way in which context-relativity blocks the Liar reasoning. 906

Contextualist revenge Alas, this hope is soon dashed. 907 For note that in our reasoning that was inspired by Slater’s modification of the Truth schema, we have only taken into account the context-relativity of the singular term (‘this sentence’). If we also take into account 904 905

906 907

Slater 2014, 253. We should note that the need for a modification of the Tarskian truth schema from this point onwards plays a different role in our argument than in Slater’s argument, where it is used to discuss self-reference. This was indeed my own first idea for solving the Liar paradox, as reported in Pleitz 2010. The following is loosely based on Pleitz 2011a, where I have reported my realization that my first idea for solving the Liar paradox is flawed.

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the context-relativity of the truth predicate itself, we will arrive at a principle like the following: 908 (Local Truth of L) ‘L is not true’ is true relative to a context c if and only if the extension that ‘L’ has relative to the context c is not true relative to the context c. Here, ‘L’ as a schematic singular term. It might be instantiated as ‘this sentence’, as a name, or as a description, but let us keep our considerations general. By giving this criterion for the truth of ‘L is not true’, we are making explicit that the truth predicate in that sentence behaves in a context-relative way. Now, what about a scenario where there is indeed sentential self-reference? I. e., let us assume that for a certain context cL, the following holds: (Assumption of Self-Reference) ‘L is not true’ = the extension that ‘L’ has relative to the context cL. On the basis of this assumption we get the following instance of (Local Truth of L): ‘L is not true’ is true relative to the context cL if and only if ‘L is not true’ is not true relative to the context cL. But this is again a biconditional of the form ‘p if and only if not p’, which makes it easy to derive a contradiction: ‘L is not true’ is true relative to the context cL and ‘L is not true’ is not true relative to the context cL. Thus fails our simple idea for how contextualism solves the Liar paradox. One way to understand what has happened here is to see ‘L is not true’, where ‘L’ is stipulated to provide sentential self-reference relative to the context cL, as a revenge sentence tailored to make trouble for contextualism, because it is the fact that the truth predicate is construed as context-sensitive, too, that leads to the contradiction. 909 In a similar vein, Gauker has confronted the contextualist approach with slightly more complex revenge sentences of the form ‘This sentence is not true in the current context’ and ‘This sentence is not true in any context’, which again lead to a contradiction if we assume that they are self-referential in the intended sense. 910 908

909

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This principle concerns the same notion of local truth that we talked about in section 4.7.3, but it does not follow from what we said there. For a more substantiated and more detailed variant of the present argument, cf. section 13.2. In Pleitz 2011a, 186–193, I have distinguished an “atemporal” and an “omnitemporal” from the present “indexical” reading of the predicate ‘. . . is not true’ in the prospective Liar sentence, and shown that it is only on the third reading that the present problem arises. Gauker 2006, 412f.; notation changed. Cf. Pleitz 2011a, 196f.

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Beyond semantic contextualism The simple idea of how contextualism might solve the Liar paradox fails because of our (Assumption of Self-Reference); the possibility of sentential self-reference is the crux of the matter. 911 What can be said to call it into doubt? What reason could there be to reject the assumption that there can in principle be a singular term such that for some context, the singular term relative to the context refers to the sentence it is part of? We have by now seen enough of the semantics of singular terms to be highly doubtful that a reason can be found in some purely semantic consideration. Sure, neither quotation alone nor Gödelization allows to construct a self-referential sentence of the kind that leads to the Liar paradox. 912 But no aspect of the semantics of names, indexicals, and descriptions holds great promise of preventing a scenario as described in the (Assumption of Self-Reference) from occurring. The upshot of the considerations of the present section is that contextualist approaches to the Liar paradox are likely to fail as long as context-dependence is understood as a merely semantic phenomenon. I submit, however, that this failure can be overcome once context-dependence is understood as a metaphysical phenomenon (and we heed the semantic consequences of this). In the present third part of this study we will accordingly develop a metaphysical approach to the Liar paradox that in a sense is contextualist. Our metaphysical approach to the Liar paradox will be based on two core ideas: Firstly, linguistic expressions and truth bearers in particular form a contextualist ontology insofar as their very existence is relative to context. Secondly, in a contextualist metaphysics, reference to an object is possible only relative to some contexts. In the setting based upon these two ideas, close attention to problems that might arise from the interrelations between the metaphysics and semantics of a self-referential language will show that we have good reasons to adopt a principle that restricts the contexts relative to which some expression may be referred to in such a way as to preclude the formation of any self-referential sentence. In particular there will be no Liar sentences relative to any context, so that this approach will solve the Liar paradox. This will be a contextualist solution in the sense that according to our account, even in the absence of the usual context-sensitive linguistic devices, the extension of ‘true’ will be relative to context – but primarily in a derivative way, in virtue of the domain of truth bearers itself being context-dependent. And yet the fact that our account is contextualist will be of great help when we address the problem of expressibility.

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This strengthens the result of Gauker, who – while unproblematically presupposing the possibility of sentential self-reference – concludes his detailed critique of contextualist approaches thus: “So to avoid these negative results, it would suffice to deny the possibility of reflexive reference to contexts.” (Gauker 2006, 416) The reason behind this is that he constructs his revenge sentences from complex predicates of the form ‘. . . is not true in the current context’ and ‘. . . is not true in any context’, making use of a relational truth predicate and means to refer to the current context. For our own revenge sentence for contextualism, in contrast, it suffices to understand the truth predicate in a context-relative (“indexical”) way. And that is hardly something that the contextualist can object to. Cf. sections 5.6 and 5.7 and cf. chapter 6, respectively.

Chapter 8

The Changing World of Expressions

Ludwig Wittgenstein: “Die Philosophie der Logik redet in keinem anderen Sinn von Sätzen und Wörtern, als wir es im gewöhnlichen Leben tun, wenn wir etwa sagen, ‘hier steht ein chinesischer Satz aufgeschrieben’, oder ‘nein, das sieht nur aus wie ein Schriftzeichen, ist aber ein Ornament’, etc. Wir reden von dem räumlichen und zeitlichen Phänomen der Sprache; nicht von einem unräumlichen und unzeitlichen Unding.” 913 Michael Dummett: “We perceive the meaning in the words.” 914

The six chapters from the present chapter 8 to chapter 13 constitute an extended argument with the aim of solving the Liar paradox. Finally, we take the plunge! We start in the present chapter by outlining the metaphysics of expressions, which will be the basis of our account. Thus we change our methodology once more (or rather, we broaden it). The three parts of this study correspond to three philosophical subdisciplines that are interested in language: logic, semantics, and the metaphysics of expressions. Each one of them is concerned with the very same expressions, but in three different, specific ways. Logic focuses on the inferential relations among expressions, and semantics is concerned with the relation of synonymy which holds among expressions as well as with other relations like reference and satisfaction which hold between an expression and another object, and of course with the concepts of truth and falsity. It is not that these relations and concepts play no role in the metaphysics of expressions – far from it –, but our gaze on expressions and on how they fall under these concepts and relations changes when we look at them from a metaphysical point of view. In a way, our view normalizes, because in the metaphysics of expressions we treat expressions like any other kind of object. Now there is no longer a difference in principle between Socrates being wise and the cat being on the mat on the one side and the sentence ‘snow is white’ being true and the name ‘Hesperus’ referring to Venus on the other side. While logic and

913

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“The philosophy of logic speaks of sentences and words in exactly the sense in which we speak of them in ordinary life when we say e. g. ‘Here is a Chinese sentence’, or ‘No, that only looks like writing; it is actually just an ornament’ and so on. We are talking about the spatial and temporal phenomenon of language, not about some non-spatial, non-temporal phantasm.” (Wittgenstein 2003, 80 (= Philosophical Investigations, note between 108 and 109); Wittgenstein 1958[1953], 46f.) Dummett 2004, 11; his emphasis.

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semantics establish a clear boundary that separates expressions from the objects they are about (as well as from other non-linguistic objects), the metaphysics of expressions is more open and encourages expressions to mix with (other) ordinary objects. There are of course several connections between the subdisciplines of logic, semantics, and the metaphysics of expressions, and one recurring theme in our own approach to the Liar paradox will be that the bonds are even closer than is commonly thought. In fact, we have already strayed from the realm of logic into that of the metaphysics of expressions. This has happened to us when we uncovered the metaphysical inconsistency of the Liar paradox, i. e., when we brought to light that any Liar sentence would be an inconsistent object. 915 Then, we noted that the Tibetan Liar sentence of our story – a concrete inscription on a shard of pottery that ascribes falsity to itself in the language of Classical Tibetan – is as inconsistent an object as Sylvan’s box of Priest’s story. Seen like this, the Tibetan Liar sentence and Sylvan’s box are alike: One is true and not true, the other contains and does not contain a figurine, but both are equally out there in the world, and apart from their inconsistency, both are equally ordinary objects. 916 We noted at the time that this metaphysical inconsistency adds to the severity of the problem posed by the Liar paradox, and that ironically we will have to adopt the metaphysical perspective again when we want to solve it. 917 Now we have reached this point: We cross over again, from logic and semantics to the metaphysics of expressions. Our bridge will be the question of the primary truth bearers, which belongs to logic and semantics, broadly construed, but connects them to the metaphysics of expressions. We will distinguish semantic expressions from syntactic expressions and show that it is semantic sentences that are the primary truth bearers and hence what is relevant for the Liar paradox (in section 8.1). Then we will explore the metaphysics of expressions in more detail, drawing in several places on work by Kit Fine (in sections 8.2 through 8.6). We will conclude the chapter by highlighting the contextualist nature of the ontology of expressions and drawing an important parallel to the tensed theory of temporal reality that will prepare us for the next chapter (in section 8.7).

915 916

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Cf. sections 3.8 and 3.9. Or rather, they would be out in the world if they were real. Sylvan’s box, however, is merely fictional, and the existence of Liar sentences is exactly what is at issue in the present third part of this study. Cf. the comments towards the end of section 3.9.

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8.1 Semantic sentences as primary truth bearers An object is called a truth bearer if and only if it is true or false (or is of a kind of objects that can at least in principle be true or false). 918 Traditionally, several kinds of objects have been seen in the role of truth bearers: sentences, the propositions they express, the statements made with their help, or the judgements that warrant their assertion. Often, several kinds of truth bearers are countenanced, but one of them is singled out as primary. Frege, for example, thought that propositions (which he called Thoughts) are what is true or false in the first place, and that a sentence is true or false only derivatively, insofar as it expresses a true proposition or a false proposition, respectively. In the present section, we will also ask for the primary truth bearers, but we will not give the same answer as Frege. Observe first that for a Liar sentence to be a possible source of paradox, the object it refers to clearly must be a truth bearer. After all, the question whether that object is true or false plays an essential role in the Liar reasoning. 919 The situation is more complex with regard to the relation of the Liar paradox to primary truth bearers. Even when it is given that one kind of truth bearers is primary, there is no reason that what a Liar sentence refers to must be a primary truth bearer. For suppose – to have a specific example – that Frege’s answer is correct, and propositions are primary. Then both ‘This sentence does not express a true proposition’ and ‘The proposition expressed by this sentence is not true’ would be Liar sentences (at least in a wide sense of the word), 920 but only the second one refers to a proposition. And yet, it makes good sense, both for the advocate of the paradox and for the philosopher who tries to solve it, to concentrate on cases that involve a primary truth bearer. These will provide the hardest problems, and if all goes well, attention to them will also provide the most thorough solutions. In the example of an adherent of the Fregean answer that propositions are the primary truth bearers, it would make more sense to focus on the two Liar sentences that we just mentioned than on the more standard ‘This sentence is not true’. 921 Here, it might even be profitable to focus on a Liar proposition (e. g., the proposition that this proposition is not true). 922 918

919 920

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The proviso is needed here to keep open the theoretical possibility that some truth bearer is gappy, i. e., neither true nor false. E. g., somebody who endorses the view that sentences about contingent future events are neither true nor false should not be committed to denying that ‘There will be a sea-battle tomorrow’ is a truth bearer. Cf. sections 2.4 and 3.5. We have characterized a Liar sentence as a sentence that ascribes falsity or untruth to itself (section 1.2). Under this notion, it is a conceptual truth that a Liar sentence refers to a sentence, regardless of the answer to the question of primary truth bearers. For the present consideration we would therefore need a weaker conception of the object that stands in the center of the Liar paradox. We could lay down that a Liar-sentence-like truth bearer is a truth bearer that ascribes falsity or untruth to itself; but our argument about Liar sentences and primary truth bearers should be clear enough without such an additional notion. Under the assumption that propositions are the primary truth bearers, the truth of sentences would be derivative, and it is likely that ‘This sentence is not true’ must be analyzed as ‘This sentence does not express a true proposition’, anyway. That is the reason Barwise and Etchemendy start their investigation of the Liar paradox by giving an account of circular propositions (Barwise /Etchemendy 1987, 26–74).

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All in all, we would do well to answer the venerable old question that lately has suffered some neglect: What are the primary truth bearers? But before we can do that, we should distinguish between semantic expressions and syntactic expressions, and in particular between semantic sentences and syntactic sentences, because thus we get a more comprehensive list of candidates for an answer and can pinpoint our own answer more precisely. * It is crucial for the following to distinguish semantic expressions from syntactic expressions. A syntactic expression typically is a string of letters that fulfills some syntactic function (like belonging to a certain grammatical category); a semantic expression can be characterized in a figurative way as a syntactic expression taken together with its meaning. 923 When we observe, e. g., how the noun ‘gift’ differs in the English and the German language, we can say with the help of this distinction that there is one and the same syntactic expression (a noun made up of the letters ‘g’, ‘i’, ‘f’, and ‘t’, in that order) that constitutes two distinct semantic expressions in the distinct languages (one of them meaning present, the other meaning poison). The figurative characterization of a semantic expression as a syntactic expression taken together with its meaning is only preliminary for us because it suggests that meanings are objects that are prior to semantic expressions, and we do not want to commit to either the objecthood or to the priority of meaning. The account can be made more precise by comparing the criteria of identity that characterize syntactic expressions and semantic expressions: For syntactic expressions α and β, α = β if and only if α is of the same syntactic shape as β. 924 For semantic expressions α and β, α = β if and only if α is of the same syntactic shape as β and α is synonymous to β. Here, the equivalence relation of being synonymous does all the work that in the figurative characterization was done by meanings understood as objects that are prior to semantic expressions. But we need not go into any more details, because for 923

924

My attention to the distinction between semantic sentences and syntactic sentences owes a lot to Rheinwald’s threefold distinction between expressions as physical objects, expressions as mentioned syntactic objects and expressions as used syntactic objects (Rheinwald 1988, 95–98). Cf. also Wittgenstein about signs and symbols in Tractatus 3.32. In section 3.1 we defined the syntactic shape of an expression as its alphabetical shape together with its syntactic (or, grammatical) structure. This will need to be spelled out in different ways for formal languages, written languages, and spoken languages. But the details are not important here; all that matters for present purposes is that at least in principle, we have a specifically syntactic equivalence relation at our disposal that will allow us to characterize syntactic expressions. For more detailed discussions of syntactic equivalence, cf., e. g., Goodman 1976, 127ff. and Scholz 2004, 113.

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our present purposes it suffices to give the distinction between syntactic expressions and semantic expressions in broad strokes. As a semantic sentence incorporates the syntactic sentence that constitutes it, it differs from what is usually conceived as a proposition, i. e., the abstract object that encodes the meaning of the semantic sentence in question. A proposition is often explicated as the set of possible worlds with respect to which the semantic sentence is true, or, somewhat more fine-grained, as a structured proposition which also incorporates the logical form of the semantic sentence and in some cases the extensions of some of its parts. 925 But both construals abstract away from the specific sentence that expresses the proposition in question. Thus it is evident from the criteria of identity that even structured propositions are more coarse-grained than semantic sentences (e. g., the same structured proposition is expressed by the distinct semantic sentences ‘The cat is on the mat’ and ‘Die Katze ist auf der Matte’). Using the dichotomy of abstractness and concreteness to formulate the debate about metaphysical realism, we can say that, between the platonist extreme of construing a truth bearer in a purely abstract way as a proposition, and the nominalist extreme of construing a truth bearer in a purely concrete way as a mere sentence, i. e. either as a token of a syntactic sentence or as a physical object of a certain kind, the notion of a semantic sentence occupies an Aristotelian hylomorphist middleposition: 926 The physical matter of a particular semantic sentence is concrete, but its semantic form is abstract. 927 This middle-position might seem unstable; and I am aware that it is non-standard. The usual way of setting up things recognizes only mere sentences – construed either as physical objects of a certain kind or as what we call syntactic sentences – and (perhaps) the propositions expressed by them on certain occasions. In these terms, what we call a semantic sentence would be an amalgam of a mere sentence and the proposition expressed by it, and thus a derivative entity. But that would be putting the cart before the horse. Really, semantic expressions are more basic than propositions, and probably, a proposition is nothing more than an abstraction over synonymous semantic sentences. I take this to be a common sense position. For what we encounter in real life are neither propositions and their parts nor syntactic expressions as such, but semantic expressions. In the words of Michael Dummett: “It is an essential feature of linguistic exchange that we do not merely apprehend the meaning of what is said but are aware of the words used to say it.” 928

925 926

927

928

Cf. section 4.5 and 4.6. Note that a hylomorphist construal of expressions does not entail global hylomorphism, i. e., the theory according to which every object is constituted by its matter and its form. The situation is actually more complicated because there are three levels involved; as we will see shortly, the matter of a semantic expression is a certain syntactic expression and the matter of a syntactic expression is a certain physical object (cf. section 8.3). And in some contexts, the type /tokendistinction must also be heeded. But the point remains that according to our construal an expression incorporates both matter and form, and that at least in a relational sense, the matter is concrete and the form is abstract. Dummett 2004, 16.

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“in conversing or reading in our languages, or any others that we know well, we do not normally carry out two successive activities: first that of listening to a sequence of sounds or looking at a string of marks on paper, then that of interpreting them. Rather, as Edmund Husserl insisted, we hear or read the words as saying whatever it is that they say; only by a heroic effort can we hear them as mere sounds or see them as mere shapes. We perceive the meaning in the words.” 929

I am convinced that in this particular case things are indeed as they appear to be: The meaning is in the words, and in particular, the proposition is in the sentence. Not only should we distinguish semantic expressions from syntactic expressions; we need to understand that it is the former that a language is comprised of first and foremost. Observe that the present view of the primacy of semantic expressions is the metaphysical flip side of a position in the theory of meaning that we have already expressed sympathies for: semanticism. According to semanticism, even a formal language should be understood as meaningful, and thus as more than the system of de-semantified expressions countenanced by syntacticism. 930 And when those who adopt a semanticist understanding of language get into a metaphysical mindset, they can express their view very well by saying that language is comprised of semantic expressions, and that according to their syntacticist opponents, language is comprised of syntactic expressions. * The primacy of semantic expressions is already a strong indication that semantic sentences are the primary truth bearers: When language is composed first and foremost of semantic sentences, from which syntactic sentences can be derived (by desemantification) and perhaps also propositions (by abstraction), then it is plausible that truth and falsity apply first and foremost to semantic sentences, and only derivatively to syntactic sentences or to propositions. We will add to this justification by formulating reasons for preferring semantic sentences to syntactic sentences as primary truth bearers and reasons for preferring semantic sentences to propositions as primary truth bearers. Why should we prefer semantic sentences to syntactic sentences as primary truth bearers? Firstly, there is the strong intuition, described at length by Dummett, that truth and meaning are inseparable. 931 It would simply be ungrammatical to ascribe truth simpliciter to a syntactic sentence, a mere sting of letters of the appropriate grammatical category. And although it would (technically speaking) be grammatical to ascribe truth relative to an interpretation to a syntactic sentence, this would separate the truth of a sentence from its meaning, which under this construal would not show itself in the true sentence, but be hidden in the interpretation. Speaking figuratively, as truth cannot be separated from meaning and meaning cannot be separated from semantic expressions (due to their primacy), truth cannot be separated from semantic sentences. Secondly there is a related intuition, formulated 929 930 931

Dummett 2004, 11; his emphasis. Cf. section 3.1. E. g., Dummett 2004, 39.

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by Davidson, that truth simpliciter is prior to truth under an interpretation. At least some ascriptions of truth should be absolute in this sense. 932 Now, if truth were a property of syntactic sentences, then every ascription of truth would be relative to an interpretation and the truth predicate would always be elliptical for a predicate expressing truth under some interpretation. 933 Thus, to save the intuition of absoluteness, truth should rather be conceived as a monadic concept that applies to a semantic sentence, than as a dyadic concept that relates a syntactic sentence to a language. E. g., instead of saying of the string of Latin letters ‘Snow is white’ that it is true in English (and meaningless in German), we rather should say of the English sentence ‘Snow is white’ that it is true simpliciter (and that there is no German sentence of the same syntactic shape). 934 And why should we prefer semantic sentences to propositions as primary truth bearers? Admittedly, to report what is said by a given sentence and hence what is true if that sentence is true, it is quite often irrelevant which words were used to say it. But sometimes the words do matter. Let us look at an example. Intuitively, the following inference is valid: 935 (P1) He says: ‘I am Pavel Tichý.’ (P2) What he says is true. (K) His name is ‘Pavel Tichý’. If (P2) ascribed truth to the proposition expressed by the sentence mentioned in (P1), then the inference to (K) would not be justified, because the name is no constituent of the proposition. But if (P2) ascribes truth to the semantic sentence mentioned in 932

933

Davidson claims that, at least in the philosophy of language, we should be interested only in absolute ascriptions of truth (Davidson 1984, 68f.). He notices that this does not fit well with his call for local truth conditions of indexical sentences (Davidson 1984, 75), but does not resolve the tension (as far as I know). On the background of the present distinction and of what we learned about semantic theory in chapters 4 and 5, it can be resolved like this: What the intuition in question requires is that truth is not relative to an interpretation, where ‘interpretation’ is meant in the technical sense of adding semantic values to a language that is understood in a syntacticist way. Absoluteness in this sense is compatible with the truth of (semantic) sentences being relative to certain other parameters, like positions of use for indexicals, and more generally, like contexts of use and circumstances of evaluation. We will later (in particular in chapter 10) adopt a framework where all ascriptions of truth (and falsity) are made only in relation to such parameters; but at the same time we will stick to the absoluteness of truth (and falsity) insofar as we will not vary the interpretation that assigns semantic values to syntactic expressions. For this reason, the Tarskian truth schema is often given not as a principle about the predicate ‘is true’, but as a principle about the relativized predicate ‘is true in L’, like this: s is true-in-L if and only if p.

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Cf., e. g., Soames 1999, 68 and Künne 2003, 83. On the present account, quotation is systematically ambiguous because a given quotation expression might refer either to the syntactic expression or to the semantic expression between the quote marks. But any quotation expression can be disambiguated by prefixing it with an appropriate phrase, e. g., ‘the syntactic expression ‘. . . ’’ or ‘the semantic expression ‘. . .’’. Note how attention to the metaphysics of expressions provides a reason to augment our semantic account of quotation (e. g., that of section 5.5). Cf. Tichý 1984, 233f. Although the present argument is inspired by an example of Tichý’s, it is not likely that he would have drawn the same conclusion from it.

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(P1), then the inference to (K) is justified, because the name is part of the semantic sentence. The present point can be further substantiated by alluding to the problem of hyperintensionality, which really is a sharpened variant of Frege’s puzzle of the informativeness of identity statements like ‘a = b’. 936 E. g., the knowledge that Mark Twain is Mark Twain need not amount to the knowledge that Samuel Clemens is Mark Twain. Also, an assertion that Mark Twain wrote The Adventures of Tom Sawyer need not amount to an assertion that Samuel Clemens wrote The Adventures of Tom Sawyer. However, according to the usual construals, the proposition that Mark Twain is Mark Twain just is the proposition that Mark Twain is Samuel Clemens, and the proposition that Mark Twain wrote Tom Sawyer just is the proposition that Samuel Clemens wrote Tom Sawyer. Not so, of course, for semantic sentences – the distinctness on the level of semantic expressions (which here is due to syntactic differences) can account for the distinctness on the level of knowledge and on the level of assertability. Now, because of the intimate connection of both knowledge and assertability to truth, it is highly plausible that the objects of these attitudes are none other than the primary truth bearers. Thus the phenomenon of hyperintensionality provides us with an indirect argument to prefer semantic sentences to propositions as the primary truth bearers. 937 To sum up. We should reject the view that syntactic sentences are the primary truth bearers because it would lead to an unacceptable separation of truth and meaning, and an equally undesirable global relativism about truth. We should reject the view that propositions are the primary truth bearers because propositions, both construed as sets of possible worlds and as structured propositions, are too coarsegrained for that role. Hence we are justified in reserving the role of primary truth bearers for semantic sentences, which is already very plausible on the background of the common sense view that a language is first and foremost comprised of semantic expressions. 938 936 937

938

Frege 2002c[1892]. It is likely that the primary truth bearers are also the primary objects of cognitive attitudes like belief, hope, and so on. Thus our preference of semantic sentences over propositions for this role would carry over. This holds promise for solving Kripke’s puzzle about belief – which is about the rational believer Pierre, who has come to believe that he is warranted to assert both ‘Londres est jolie’ and ‘London is not pretty’ (Kripke 1979, 257). If propositions are the objects of belief, there is a blatant contradiction within Pierre’s beliefs, but if semantic sentences are the objects of belief, that is much less clear. For note that in contrast to quotation expressions, the names ‘Londres’ and ‘London’ are not revealing designators (cf. section 5.5), so that a speaker can grasp them without thereby being acquainted with the object they refer to. In the context of similar considerations about hyperintensionality and cognitive attitudes, Mark Richard has proposed that what a sentence expresses is not a proposition as standardly construed, but an object he calls a “Russellian annotated matrix”, which is, roughly, a structured pair of the sentence and the corresponding proposition (Richard 1990, 137ff.). As far as I understand, Richard’s Russellian annotated matrices are very much like our semantic sentences. But although both accounts should include similar criteria of identity, there are differences in the aspects they highlight. We can understand this as a case of coming from different directions to the same thing, and therefore seeing different sides of it. In terms of the different genesis of the accounts, we can say that Richard arrives at a Russellian annotated matrix by enriching a proposition as traditionally construed with information about which

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8.2 The distinctness of an expression and its basis Let us look more closely at the metaphysics of expressions. What kind of objects are expressions and how do they relate to objects of other kinds? As we have distinguished semantic expressions from syntactic expressions, and as there certainly are some non-linguistic objects – objects that are not expressions –, we really need to answer a larger number of questions: What kind of objects are semantic expressions? What kind of objects are syntactic expressions? What is their relation, and how do they relate to non-linguistic objects? The first thing to note is that the expressions of a language are always realized in non-linguistic objects – there is no such thing as a disembodied word. In the paradigmatic cases of spoken and written language, the non-linguistic objects that are used to realize language will be physical objects, either events or things. 939 In spoken language, typically a certain sound event will be used as the basis of a syntactic expression, which will in turn be used as the basis of a semantic expression. And in written language, typically a certain mark on paper 940 will be used as the basis of a syntactic expression, which will in turn be used as the basis of a semantic expression. E. g., a particular inscription token of the semantic name ‘Venus’ is realized by a particular string of letters ‘Venus’ which in turn is realized by a particular series of marks on paper. Clearly, the general picture is the following: A non-linguistic object 941 helps to realize a syntactic expression, which in turn helps to realize a semantic expression. So there are always three levels. Given this characterization of the basic situation, it is easy to contrast the account we will endorse with a view that is held by many philosophers, and that may well be the standard view. What is distinctive of our account is that it crucially entails that an expression is distinct from its basis. It entails, more particularly, that a semantic expression is distinct from the syntactic expression that realizes it and that a syntactic expression is distinct from the non-linguistic object that realizes it. On the standard view, by contrast, an expression is identical to its basis – there are only non-linguistic objects, and an expression just is the non-linguistic object that realizes it. On this identity theory, when we talk about an expression, what we really do is talk about a non-linguistic object under a certain description, and although that

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syntactic expressions express which of its parts, whereas we find ourselves with a semantic sentence because we have declined to follow the usual practice of dividing it up into a syntactic sentence and a proposition. Cf. Pleitz 2011b and Pleitz 2015b. Given the aims of the present investigation, we can safely ignore that today, an expression is of course often realized by a particular configuration of pixels on an LED screen. Arguably, the non-linguistic object that realizes an expression need not always be a physical object (although that is the case both for spoken and for written language). In the context of Gödelization, we can develop an understanding of the expressions of the formal language in question according to which every expression is realized by its Gödel number, and thus by an abstract object. Cf. Pleitz 2010a, 211ff.

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description entails that the non-linguistic object in question is used as an expression, this is meant in a way that is not metaphysically committing. 942 Now for an argument for the three level account of the ontology of expressions, and hence against the identity theory. When a certain semantic expression is realized by a certain syntactic expression that is in turn realized by a certain nonlinguistic object, their pairwise distinctness is entailed by the principle of the indiscernibility of identicals – or, contraposing, by the principle of the distinctness of discernibles – together with the fact that there are concepts that allow to discern the semantic expression, syntactic expression, and non-linguistic object in question. For one thing, some concepts are such that only objects of one or two of the three kinds can fall under them. E. g., only semantic sentences can fall under the monadic concept of being true; only semantic singular terms can fall under the concept of referring to some object. Only syntactic expressions (and perhaps semantic expressions in a derivative way, but certainly not physical objects) can fall under the concept of being grammatical (or of being well-formed, to use logician’s parlance). And although many concepts that a physical object falls under will be inherited by the syntactic and the semantic expression that it realizes, e. g., having a certain shape, there are concepts that only physical objects fall under. E. g., of the three kinds of objects under discussion only physical objects fall under the concept of being ontologically independent of the role they play within a linguistic community. As a result of this way of arguing, it is clear that physical objects, syntactic expressions, and semantic expressions are not only distinct in each particular case of realization, but that they also form three disjoint categories of objects. For there is always a categorical difference between two kinds of objects when only one of the two kinds is the range of applicability of some concept. In order to supplement (and illustrate) the above very general way of arguing, let us also note that in each particular case, there will also be many differences. In the example of a particular inscription token of the semantic name ‘Venus’, we can argue thus: The semantic name refers to Venus, but the particular string of letters ‘Venus’ that realizes it does not refer to anything. Therefore the semantic name ‘Venus’ is distinct from the syntactic name ‘Venus’ that realizes it. And as the series of marks that realizes this particular string of letters also does not refer to anything, the semantic name ‘Venus’ is also distinct from this series of marks that (indirectly) realizes it. Further, the syntactic name ‘Venus’ is a noun but the series of marks that realizes it does not belong to any grammatical category, therefore the syntactic name ‘Venus’ is distinct from the physical object that realizes it. And so it goes in any particular case. We can supplement this argument for the distinctness of semantic expressions, syntactic expressions, and non-linguistic objects which makes use of the distinctness of discernibles with another argument, which makes use of the criteria of identity for expressions we gave in section 8.1. Succinctly put, syntactic expressions are identical if and only if they are syntactically equivalent and semantic expressions 942

Instead of an overview of the proponents of the standard view, suffice it here to quote what Donald Davidson wrote in a different context: “there is no such thing as a language apart from the sounds and marks people make, and the habits and expectations that go with them.” (Davidson 1997; 18)

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are identical if and only if they are both syntactically equivalent and semantically equivalent. By themselves, the two criteria of identity do not show that syntactic and semantic expressions are disjoint – it all depends on what the two equivalence relations hold of. 943 But given that syntactic equivalence is having the same syntactic shape (being same-spelled and belonging to the same grammatical category) and that semantic equivalence is synonymy, we can appeal to some basic facts about what is of the same syntactic shape with what and about what is synonymous with what. In our stock example of the English noun ‘gift’ and the German noun ‘gift’, we have the same shape but no synonymy. On the basis of these facts, the criterion of identity for semantic expressions entails that these are not the same semantic expression – very plausibly they are semantic expressions, but then they have to be distinct. But if they are distinct semantic expressions, they cannot also be syntactic expressions (which the identity theory would entail), because the criterion of identity for syntactic expressions would entail that they are identical, contradicting their distinctness. It is very natural to dissolve the tension by countenancing three distinct linguistic objects, the semantic expression that is the English noun ‘gift’, the semantic expression that is the German noun ‘gift’, and the syntactic expression that is the string of letters ‘gift’ in the role of a noun. All three will of course be distinct from the pattern of marks on paper that realizes the string of letters ‘gift’, because marks on paper do not stand in the relation of having the same syntactic shape or of synonymy to any object. Applying this kind of reasoning to a broader range of objects will again allow to show that semantic expressions, syntactic expressions, and non-linguistic objects form three disjoint categories, so that every semantic expression is distinct from its syntactic basis and its non-linguistic basis and every syntactic expression is distinct from its non-linguistic basis.

8.3 Grounding, ontological dependence, and coincidence Despite their distinctness, there will of course be close connections between an expression and its basis. Of these, the most important are metaphysical grounding, ontological dependence, and coincidence. We will explain these notions as we go along, starting with the notion of metaphysical grounding. We say that one object grounds another if and only if the former helps explain the latter in a metaphysical way, so that we are inclined to say that the second is what it is (at least partly) in virtue of what the first one is. 944 The paradigm example is that Socrates grounds {Socrates}, i. e., Socrates grounds the set that has him as its 943

944

Imagine – contra what you know about the plurality of actual languages – syntactic equivalence and semantic equivalence to be co-extensional. For the ensuing scenario, the criteria of identity would entail that the syntactic expressions just are the semantic expressions. For introductions to the notion of metaphysical grounding, cf. Correia/Schnieder 2012a, and in particular Fine 2012 and Correia/Schnieder 2012b, as well as Bliss/Trogdon 2014 and Schaffer 2010. We are talking here about grounding between objects, i. e., about “predicational grounding” in the terminology of Correia /Schnieder 2012b, 10ff. But there can also be grounding between states of affairs, that is expressed by a two-place sentential operator (Correia’s and Schnieder’s “operational grounding”, see also Fine 2012, 46). Besides the predicational and operational variant of grounding,

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sole element, but not vice versa. Now we can observe that, despite their distinctness, every expression is grounded in its basis. E. g., when a particular inscription token of the semantic name ‘Venus’ is realized by a particular string of letters ‘Venus’ which in turn is realized by a particular series of marks on paper, then that series of marks grounds that token of the syntactic name ‘Venus’, which in turn grounds that token of the semantic name ‘Venus’. Grounding is transitive, asymmetrical, and (hence) irreflexive. (Irreflexivity of Grounding) No object is grounded in itself. (Asymmetry of Grounding) If an object b is grounded in an object a, then the object a is not grounded in the object b. (Transitivity of Grounding) If an object b is grounded in an object a and an object c is grounded in the object b, then the object c is grounded in the object a. These properties are natural to assume because of the connection of grounding to explanation. 945 To give some examples from the metaphysics of expressions, no semantic or syntactic expression will be grounded in itself, no syntactic expression will be grounded in a semantic expression that it realizes, and every semantic expression will be grounded in the non-linguistic object that realizes the syntactic expression it is realized by. A kindred notion of grounding is that of ontological dependence. It is modal: An Object b depends ontologically on an object a if and only if, necessarily, if the object b exists then the object a exists. In the terminology of possible worlds, this is equivalent to the condition that every possible world where the object b exists is a possible world where the object a exists. The present framework, however, is more general than possible worlds semantics, because contexts play the role of possible worlds. So we can say that an object b depends ontologically on an object a if and only if every context relative to which the object b exists is a context relative to which the object a exists. In a tense logical framework (like the one that we will be using later on 946) it is possible to contrast ontological dependence with a weaker notion, remote ontological dependence: An Object b depends ontologically in a remote fashion on an object a if and only if, necessarily and omnitemporally, if the object b exists then the object a exists or it was case that the object a exists. In the terminology of moments in time, this is equivalent to the condition that for every

945 946

which are often recognized, there can in our metaphysical framework also be grounding between a concept and an object, or between concepts. That is due to the fact that we countenance not only objects (and states of affairs), but also concepts as a separate category of items. Cf. Schaffer 2010, 36 and Fine 2012. Cf. chapter 9 and chapter 10, and in particular section 9.1.

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moment where the object b exists, the object a exists relative to that moment or there is an earlier moment relative to which the object a exists. Ontological dependence entails remote ontological dependence, but not vice versa. How do these notions of ontological dependence relate to the notion of grounding? It might seem natural to assume that grounding entails (the converse of) ontological dependence: If an object a grounds an object b, then the object b depends ontologically on the object a. For grounding is clearly accompanied by ontological dependence in many cases, e. g., when a set is grounded in its elements or a grin in the grinning person. But we should not commit to this entailment, because grounding without ontological dependence is at least conceivable. One may, e. g., find reason to assume that some living beings are related to their ancestors not only in a causal way, but by grounding (this might be a way of spelling out the idea that ancestral origin is essential to some living beings). However, living beings are in most cases outlived by their descendants, so that there is no ontological dependence here (it is not the case that whenever I exist, my parents exist, to say nothing of my great-grandparents). 947 Note that in the scenario of this example, there is still remote ontological dependence: For any living being that exists at present, it was the case that its ancestors existed. We have no reason to doubt the more cautious general claim that grounding entails (the converse of) remote ontological dependence. And in fact, we will later see reason to adopt an equivalent principle of non-priority for grounded objects, 948 according to which no object can exist prior to any object it is grounded in. But however that may be, grounding is not reducible to remote ontological dependence, nor to any other purely modal notion for that matter. This can be seen from the example of Socrates and {Socrates}, because plausibly, these two objects exist in the very same possible worlds and relative to the very same moments in time, but the relation of grounding holds between them only in one direction. 949 Now, when we apply the notion of ontological dependence to the ontology of expressions, we get a counterpart of what we said about expressions and grounding: An expression depends ontologically on its basis – it exists only relative to contexts relative to which its basis exists –, but not vice versa. And in our example it would indeed be weird to think that the particular token of the semantic name ‘Venus’ or of the syntactic name ‘Venus’ might exist relative to a context where the corresponding series of marks did not exist – it would be no less weird than a set that exists without its elements, or the grin of the Cheshire cat, which lingers on for a bit when the cat has already disappeared. So we do not need the notion of merely remote ontological dependence to describe the ontology of expressions – but it will have a role to play later on. 950 Another notion that is relevant here is that of coincidence. An object a coincides with an object b if and only if the object a and the object b occupy the same region 947

948 949 950

On a methodological note, it suffices for the present argument for the conceptual possibility of grounding without ontological dependence that a metaphysical account of ancestry is conceivable according to which some living beings are grounded in ancestors whom they can outlive. Cf. section 9.10. Cf. Fine 1994, 4f. Cf. section 9.10.

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in space. So for instance even those philosophers (like me) who take a statue to be distinct from the piece of matter it is made of will hold that the statue coincides with the piece of matter as long as the statue exists – but not vice versa, because the piece of matter was there before the statue was created. Likewise, any expression that is realized in a physical way 951 coincides with its basis, but not vice versa. The token of the name ‘Venus’, both as a semantic expressions and as a syntactic expression, is located in the very same spatial region as the corresponding marks. Therefore an expression of written language will inherit the geometrical shape of its physical basis. I take it that, of the metaphysical claims of the last section and the present section, only the claim of distinctness is controversial. For the claims of grounding and of coincidence are really what remains of the rival account of the metaphysics of expressions, the identity theory: There is a close connection and a close resemblance between an expression and its basis. The existence of this remainder can also account for the intuitiveness of the rival account: As an expression is grounded in its basis and coincides with it, one might well take them to be identical. But that would be a mistake, as has been argued. By way of a summary, the present account of the metaphysics of expressions can be formulated as a list of five principles: (Realization) Any particular semantic expression is realized by a syntactic expression which in turn is realized by a non-linguistic object (typically a sound or a mark on paper). (Distinctness) And yet, the semantic expression, the syntactic expression and the physical object are pairwise distinct. (Grounding) The semantic expression is grounded in the syntactic expression which in turn is grounded in the physical object. (Ontological Dependence) The semantic expression depends ontologically on the syntactic expression which in turn depends ontologically on the physical object. (Coincidence) The semantic expression, the syntactic expression and the physical object are coincident, i. e., they occupy the same spatiotemporal position when they exist. 951

If it is correct that we can construe the expressions of some formal languages as realized by their Gödel numbers (as in Pleitz 2010a, 211ff.), then we have a case at our hands where there is no clear counterpart of coincidence, because abstract objects are not located in space.

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These claims about the metaphysics of expressions can of course be spelled out in more detail, and we will see one explication shortly, in terms of qua objects. 952 But before we come to that, let us have a brief look at the social aspect of expressions, introducing the dimension of contexts into the metaphysics of expressions, which will play a crucial role in its qua objects explication and its application to selfreferential languages.

8.4 Social ontology Now, how does a physical object manage to realize a syntactic expression without being identical to it? And how is it possible for a syntactic expression to realize a distinct semantic expression? The full answer is, in general: They achieve these feats not entirely on their own, but by being used in an appropriate way by the members of some linguistic community. A physical object realizes a syntactic expression by playing a syntactic role; a syntactic expression realizes a semantic expression by playing a semantic role. And they play these roles only within a certain linguistic community and under certain circumstances. Thus, syntactic and semantic expressions form a social ontology, and both are social objects in the sense of John Searle. 953 Their constitution from lower-level objects can be described by Searle’s concise formula: “x counts as y in context c.” 954 For example, the string of letters ‘Venus’ counts as a name of the second planet of our solar system in the context of the English language. And an upright stroke, ‘I’, counts as the numeral that refers to the number one in the context of the Latin numeral system. Describing it in the terms of social ontology highlights that a pragmatist element is essential to the metaphysics of expressions – for expressions exist only (and can only exist) when some objects that help constitute them are used in a certain way. But in a description of the metaphysics of expressions that follows Searle’s slogan, this pragmatist element can to a large extent be left implicit. In the present metaphysical framework, it is hidden behind the theoretical role played by contexts. When we say ‘in the context of the Latin numeral system’ or ‘in the context of the English language’, we implicitly allude to an important background of the members of some linguistic community following certain practices. In order to theorize about the metaphysics of expressions, however, it is not necessary to go into the details of a description of those practices. Most of the time we can adopt the viewpoint of the members of the respective linguistic community, for whom it does not look as if the existence of the expressions of their language was tied to what they do. But at one point in the train of thought of the following chapters, it will be crucial to remember that the expressions of a language are social objects. 955 952 953 954 955

Cf. section 8.5. Searle 1995, 4ff. Searle 1995, 27ff. Cf. sections 11.14 and 12.9.

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8.5 Expressions as qua objects The ontology of physical objects, syntactic expressions and semantic expressions can likely be explicated in many different ways. We will spell it out with the help of a modified variant of Kit Fine’s theory of qua objects, which strikes me as a good contemporary explication of hylomorphism. 956 Here is the theory in Fine’s original formulation: A qua object x qua F is a pair of an object x, called the basis of the qua object, and a property F, called the gloss of the qua object. 957 “Existence. The qua object x qua ϕ exists at a given time (world-time) if and only if x exists and has ϕ at the given time (world-time). Identity. (i) Two qua objects are the same only if their bases and glosses are the same. (ii) A qua object is distinct from its basis (or from the basis of its basis, should that be a qua object, and so on). Inheritance. At any time (world-time) at which a qua object exists, it has those normal properties possessed by its basis.” 958

Fine goes on to explain: “According to the first of these principles, a qua object exists just when its basis does and satisfies the gloss. According to the first part of the second principle, the identity of qua objects requires both identity of basis and gloss. Thus Socrates qua philosopher is distinct from Socrates qua Greek, since the properties of being a philosopher and of being Greek are distinct; while Socrates qua philosopher is distinct from Plato qua philosopher, since Socrates and Plato are distinct. According to the last principle, a qua object inherits its properties from its basis. However, not all properties are inherited. Mrs Thatcher existed ten years ago, Mrs Thatcher qua Prime Minister did not. Again, Mrs Thatcher is not a qua object with Mrs Thatcher as basis, but Mrs Thatcher qua Prime Minister is. We exclude these cases by requiring that the property not be formal and that its application only concern the time (or world-time) in question. That is the force of the qualification ‘normal’.” 959

We will work with the following modified variant of the theory of qua objects: A qua object b-qua-g is constituted by an object b, called its basis, and a concept g, its gloss.

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Fine outlines another explication of hylomorphism in his “Things and Their Parts” (Fine 1999), in a different terminology and with a higher degree of generality. For the present purpose of explicating the metaphysics of expressions, however, we do not need to go beyond his older and more specialized theory of qua objects (Fine 1982). Fine 1982, 100. Fine 1982, 100. Fine 1982, 100f.

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(Qua-Existence) The qua object b-qua-g exists relative to a context c if and only if the object b exists relative to the context c and the object b falls under the concept g relative to the context c. (Qua-Identity) Qua objects are identical only if their bases are identical and their glosses are equivalent. (Qua-Distinctness) Each qua object is distinct from its basis (and from the basis of its basis, should that be a qua object, and so on). (Qua-Inheritance) For any context relative to which a qua object exists and for any normal concept, if the qua object’s basis falls under that concept relative to that context, then so does the qua object. We will follow Fine in his delineation of the notion of normalcy, so that we will require of a normal concept that it is not formal and concerns only the context in question. With Fine we can say that a qua object is like an “amalgam” of the given object and the concept, much like the given object but wearing the concept “on its face” 960. The modifications of Fine’s original theory are that here contexts replace possible worlds and times, and that concepts replace properties. The first change is no more than a move to achieve greater generality. The second change is more significant because it concerns the basic metaphysical framework in the background of our variant of the theory of qua objects. Whereas Fine’s background theory is a platonist metaphysics of concrete objects and their properties, we decided already when concerned with the theory of meaning on (what might be called) a Fregean metaphysics of objects and concepts. 961 The most important difference between Fine’s platonist and our Fregean background metaphysics is that properties are usually construed as abstract objects. Now, the concepts of our framework arguably are also abstract insofar as it would be ungrammatical to ask where or when they are, 962 but they are not objects. 963 (For that reason we talk of the equivalence and not of the identity 960

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The original quote is: “the qua object should be regarded as some sort of amalgam of the given object and the property, like the given object but wearing the property on its face” (Fine 1982, 100). Cf. section 4.2. The common figurative talk of abstract entities being neither in time nor in space is misleading because it suggests that they are located in some (non-regional) region outside time and space. This feature will become important at some points in the following arguments (and in the larger project in which I intend to embed the current study in the future), notably when we are concerned with what a primitive predicate or a typical description are grounded in (and in the context of the set theoretic paradoxes).

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of glosses in our condition of (Qua-Identity), for identity arguably applies only to objectual items.) It is worth noting that, despite concepts being abstract in a sense, the present framework of objects and concepts is to a certain extent independent of the question whether the realm of the abstract should be construed in a platonist way or in an idealist way. This independence is due to the fact that the question of objecthood is orthogonal to the question of mind-dependence. So the door is open to construe concepts (and other senses) as belonging to a realm which is non-physical but dependent in some way on subjectivity. 964 But given both the Fregean provenance of the framework of objects and concepts, and its Finean application here, it is certainly more natural to construe concepts as no less mind-independent than physical objects, and go the platonist way of Frege’s article “Der Gedanke” 965 and Fine’s monograph The Limits of Abstraction. 966 Let us apply our variant of the theory of qua objects to the ontology of expressions. In order to explicate the ontology of expressions, we will represent syntactic roles and semantic roles with the help of context-sensitive concepts. They need to be context-sensitive because a particular physical object will fall under a given syntactic role concept only relative to some contexts, and because, similarly, a particular syntactic expression will fall under a given semantic role concept only relative to some contexts. Now, a syntactic expression can be modeled as a qua object that has a physical object as its basis and a syntactic role concept as its gloss. And a semantic expression can be modeled as a (second level) qua object that has a syntactic expression as its basis and a semantic role concept as its gloss. Because of the context-sensitivity of the glosses, the condition (Qua-Existence) entails that these qua objects – the syntactic and semantic expressions – exist only relative to some contexts. A few examples: It is likely that the syntactic expression ‘Duschschlauch’ exists only relative to the German language (in contrast to the syntactic expressions ‘Gift’ and ‘Angst’). Outside the context of a (rather broad) range of languages, the string of letters ‘Venus’ does not fall under the semantic role concept of realizing a name of Venus, and hence the semantic name ‘Venus’ does not exist relative to those contexts. And it is plausible to say that when in the distant future some extraterrestrials examine the remains of human civilization (without of course having a clue about human languages), there will still be some marks left, but none of the expressions they realize today. 967 The context-dependent existence of expressions that is entailed by the qua objects account is entirely appropriate, because expressions are social objects 968 and therefore have a certain fragility.

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In this regard the present approach is compatible with a constructivist (or idealist) approach to the paradoxes (cf., e. g., Rohs 2001 and von Kutschera 2009, chapters 2 and 3). But that is not where our commitments lie. Although mind-dependence would go well with some of the following arguments against the existence of self-referential expressions, it is not needed and we decline to presuppose it. Frege 2010[1918]. Fine 2008[2002]. They might conjecture something in the past tense, e. g., that once upon a time, these marks must have been signs, in much the same sense as we would say of a dead body that is must have been a living being. Cf. section 8.4.

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The condition of (Qua-Identity) entails that for expressions a and b to be identical, the basis of the expression a needs to be identical to the basis of the expression b and the gloss of the basis of the expression a needs to be equivalent to the gloss of the expression b. So, be it on the level of semantic expressions or on the level of syntactic expressions, there are two different possible sources for the distinctness of expressions: the distinctness of basis or the non-equivalence of gloss. To give an example on the level of semantic expressions, the English word ‘gift’ is distinct from the German word ‘Gift’, because although they (arguably) are based in the same syntactic expression, 969 it is glossed by a semantic role concept that corresponds to the meaning present in the first case and with a semantic role concept that corresponds to the meaning poison in the second case, so that the semantic glosses are not equivalent. And the English word ‘snow’ is distinct from the German word ‘Schnee’, because, although they arguably have the same meaning so that the relevant glosses are equivalent, their bases are distinct, because ‘snow’ is not of the same syntactic shape as ‘Schnee’. 970 The condition of (Qua-Distinctness) entails a controversial claim that we saw reason to endorse, 971 namely, that every expression is distinct from its basis. It entails, in particular, that every syntactic expression is distinct from its physical basis, and every semantic expression is distinct both from its syntactic basis and its physical basis. A particular token of the semantic name ‘Venus’, e. g., thus is distinct from the particular token of the syntactic expression ‘Venus’ that realizes it, as well as from the marks on paper or whatever else that realizes this token of the syntactic expression. Finally, what the condition (Qua-Inheritance) means in the context of our qua objects account of expressions is that an expression will share many, though not all characteristics of its basis. E. g., the semantic name ‘Venus’ is five-lettered, just like its syntactic basis. And a particular inscription of the string of letters ‘Venus’ shares the geometrical shape and location of its physical basis. What the expression does not share with its basis are such characteristics as being the basis of an expression, being a non-linguistic object, as well as characteristics that concern existence relative to certain contexts. E. g., the particular physical object that is used to realize a particular token of the name ‘Venus’ is not itself an expression, and it is possible that it will still exist when it is no longer used as a basis of that name (when the extraterrestrials arrive after our civilization has died). To conclude, a disclaimer. We have seen that our variant of the theory of qua objects provides us with a theory that entails our claims about the ontology of expressions. It is valuable to have found this way of spelling out our intuitions, but we should also note here that we do not need to commit ourselves wholesale to 969

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For the sake of the example, let us ignore the difference that the first starts with a lowercase letter and the second with an uppercase letter; and to account for the multiple uses of the Latin alphabet, let us allow that the relevant syntactic expressions can, in principle, belong to more than one language. These examples concern expressions on the type level. The condition (Qua-Identity) will probably allow to give a similar account on the level of tokens, and to say something interesting about the relation of a type to its tokens. But these details need not concern us here. Cf. sections 8.2 and 8.3.

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Fine’s theory of qua objects in order to endorse those claims about expressions. Aside from the possibility of finding an alternative explication, we should be aware that we need not accept Fine’s theory of qua objects tout court, with its proliferation of objects. E. g., we need not countenance such (distinct) objects as Socrates qua philosopher and Socrates qua Greek, let alone such (distinct) objects as a certain rock qua rolling down a certain hill and the rolling rock itself, if we restrict the principle of (Qua-Existence) to the special sort of role concepts that are used in the constitution of expressions and of other social objects. A restriction of the qua objects theory along these lines would not only take some argumentative weight off our shoulders, it might also improve on the qua objects theory in general, which has its most compelling examples in the realm of artifacts and other social objects, but delivers counter-intuitive results when applied to the merely physical. 972

8.6 An example by Kit Fine In our use of his theory of qua objects, and less explicitly in the above argument for the distinctness of an expression and its basis and the considerations of grounding and coincidence 973, the current theory about the metaphysics of expressions owes a lot to work done by Kit Fine in the metaphysics of ordinary objects, in particular on the distinctness of an object and its matter. 974 For instance, Fine has argued forcefully against the general metaphysical thesis that coincidence entails identity. 975 But even though one of his prime examples is that of a letter, Fine has not yet shown any particular interest in the metaphysics of expressions. 976 In order to make up for this lacuna here, let us put his example of a letter into the context of the metaphysics of expressions as explicated with his own tools. 972

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974 975 976

The option of restricting our use of Fine’s theory of qua objects to social ontology harmonizes with a particularity of our application of Fine’s theory which concerns the glosses. While Fine in his paradigmatic example of the statue “made of bronze and in the image of Goliath” (Fine 1982, 97) uses ordinary physical characteristics like being shaped in a certain way, we always use characteristics like being used as the first numeral or being used as a name of the second planet, which have an essentially social aspect (‘used’) and include the assignment of a certain function (‘as’). Transferred to the case of the statue, this means that we see being used as an image of Goliath as the gloss of the bronze, which is quite different from having the appropriate shape, being the appropriate configuration of molecules, and similar purely physical characteristics. Thus we evade two problems raised by Simon Evnine for Fine’s qua objects account of ordinary objects, which are that it does not capture the element of intentionality and that it gives no clue which of many distinct qua objects is the one that is identical to the ordinary object in question (Evnine 2009, 206f.). Cf. section 8.3. Our argument for the distinctness of an expression and its basis is inspired by wellknown arguments for the distinctness of a statue, say Michelangelo’s David, and the stone that constitutes it: The stone was there before the statue was and the stone, but not the statue, could have had a different shape, so they must be distinct. Cf., e. g., Fine 1982, 101. I take it that the current approach is also in the spirit as Fine’s work on grounding (e. g., Fine 2012). Cf. Fine 2000 & 2003. In “Things and Their Parts” which generalizes his theory of qua objects, Fine does mention words, but only in passing: “A token of a given word will be the corresponding expression token under the description of being an instance of that word (thus the token must be understood in terms of the type, not the type in terms of the token).” (Fine 1999, 68).

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Fine gives several counterexamples to Locke’s thesis that for objects of the same sort, coincidence entails identity. 977 Then he comes up with an example designed to make plausible that there might even be distinct objects that coincide necessarily, and that is what we will focus on: “Fluent wishes to write a letter to his elder daughter, who only understands Prittle, and a letter to his younger daughter, who only understands Prattle. The peculiarities of Prittle and Prattle are such that the same inscriptions will often serve to convey different contents in the two languages and, being adept at exploiting these peculiarities, Fluent is able to write the letter to his elder daughter in Prittle while simultaneously writing the letter to his younger daughter in Prattle. The first letter (under a rough translation from Prittle) begins as follows: ‘I write this letter, dear first-born, in order to demonstrate my skill at parallel penmanship – for as I write this letter to you in Prittle, I am simultaneously writing a letter to your sister in Prattle . . .’, and the second letter begins in a similar vein, though [it] is much more florid in its style and extravagant in its claims.” 978

There clearly is only a single physical object at the center of this scenario – a piece of paper with the marks Fluent made on it. It is perhaps unclear whether the number of syntactic expressions is one or two, depending on how fine-grained we construe the syntactic role concepts involved. But on the level of semantic expressions there are clearly two distinct objects – the two letters –, and each one of these two semantic expressions is realized by the same physical object, the single piece of paper marked by Fluent. Because of their distinctness, at least one of the two letters must be distinct from that physical object. And for reasons of similarity, we should take both semantic expressions to be distinct from that physical object. Generalizing, every semantic expression should be construed as distinct from its physical basis. Thus reflecting on Fine’s example provides further evidence for our claim that an expression is distinct from its basis. 979 Another feature of Fine’s example concerns the role of contexts in the ontology of expressions. Let us accept what Fine wants to show, that the two letters coincide necessarily 980 – relative to every possible world and to every moment in time (or, relative to every possible world-state 981), either both exist and they occupy the same region or neither one of them exists. But what about the contexts that are relevant to the existence of expressions? From the perspective of Fluent it might indeed seem as if the two distinct letters existed relative to a single context. Things start to look different once we adopt the perspective of one of his daughters, who might well 977 978 979 980

981

Fine 2000. Fine 2000, 359f.; I added quote marks around the displayed passage. Cf. section 8.2. This is plausible if the criterion for the individuation of letters is strict enough to entail that, had Fluent written only one letter, it would have been distinct from each one of the two actual letters. We can call the points of an abstract space that has moments in time as one dimension and possible worlds as another dimension possible world-states. Then a possible world-state is the state of a certain possible world at a certain moment in time.

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recognize only the existence of her own letter, and think of her sister’s letter as an illusion. It has been observed that, depending on context, the same object can realize different expressions or signs. 982 E. g., an upright stroke can realize not only the Latin numeral ‘I’ but also the English indexical ‘I’. Now, typically it will not be the very same upright stroke that is used both in the context of the Latin numbering system and the English language. 983 And in general, it will typically not be the same physical object that is used at two contexts. But I think that this is just how we need to understand what is happening in Fine’s example. There we have a single physical object, the marked piece of paper, framed by a single situation of Fluent writing to his two daughters. And yet there are at least two different contexts, namely the two languages Prittle and Prattle! I submit that in the scenario of the example, we should construe the languages Prittle and Prattle as two distinct contexts of the sort relative to which semantic expressions can exist (or not). This explains Fluent’s point of view, relative to which there are two distinct letters, as well as the point of view of each daughter, who more likely will think only her letter is real, because Fluent understands both Prittle and Prattle and thus occupies the two contexts at the same time, whereas each daughter understands only one of the two languages and thus occupies only one of the two contexts. In the way attention to contexts has enabled us to disentangle what is happening in Fine’s example, there is an important lesson: The contexts that are relevant to the existence of expression – which we might want to call semiotic contexts – are neither moments in time, nor possible worlds, nor possible world-states! Our notion of a context is meant to be strictly more general than these, so that it encompasses all of them as well as semiotic contexts. Thus it is meant as an extension of what we are accustomed to from the model theory of modal logic or tense logic.

8.7 The contextualist ontology of expressions and the internal stance There is more to be said about the ontology of linguistic expressions and the metaphysics of signs in general. 984 But for our present purpose of preparing an investigation of self-referential languages that will help us deal with the Liar paradox, we have learned enough. We have learned that there are at least three levels to ontology: At the bottom level, there are non-linguistic objects (typically physical objects like sounds or marks on paper); some of these help constitute syntactic expressions which form the second 982 983

Cf. Scholz 2004, 103; Steinbrenner 2004, 26ff.; and Cappelen /Lepore 2007, 149ff. That untypical cases are possible is witnessed by typographical arrangements like the following: I think therefore I am. II III IV

984

Cf. Goodman 1976, Kaplan 1990, Scholz 2004, Steinbrenner 2004, 15ff., Hawthorne/Lepore 2011, as well as Pleitz 2007 and 2015b.

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level; and these in turn help constitute semantics expressions which form the third level. We have learned that despite being grounded in it and sharing many of its characteristics, an expression is distinct from the object that helps constitute it. We have learned that in addition to the object that is its matter, a syntactic or semantic role concept is needed to complement the constitution of an expression. We have seen that there is a good way of explicating this broadly hylomorphist account of the metaphysics of expressions, namely by means of a variant of Fine’s theory of qua objects, but we remain open to other possible explications. There is another important thing that we have learned and which we will dwell on here. Whether or not we spell out the metaphysics of expressions in terms of the qua objects analysis of the preceding sections, 985 syntactic and semantic expressions exist only relative to some contexts of use. Thus they form a contextualist ontology, where a theory about ontology is contextualist if and only if in its scope the question ‘What is there?’ needs to be answered with regard to or from the viewpoint of a context; i. e., when the question ‘What is there?’ is always elliptical for the question ‘What is there with regard to the context c?’. The notion of a contextualist ontology can be embedded in the broader notion of a contextualist metaphysics, where not only the question ‘What is there?’ but also the more general question ‘What is the case?’ can be answered only with regard to or from the viewpoint of a context. The contextualist ontology and metaphysics of expressions is a distant cousin of the temporal ontology and metaphysics of those ordinary things which exist only relative to some moments in time. 986 It is distant insofar as the contexts of the present framework are not moments in time (or possible world-states), 987 but the result of a generalization from our physical notion of time to something more abstract. And yet the contextualist ontology of expressions is still akin to the temporal ontology of transient things, because some important aspects of temporal ontology will be preserved in this generalization. Therefore we will be well-advised to look for inspiration in the philosophy of time when we try to tackle the main task of the following chapter, which is to describe how we must construe reference in a language that is about a contextualist ontology, and more generally how we must construe the semantics and the metaphysics of a language that is about a contextualist ontology. One aspect, however, is best addressed already here, because it allows to complete our account of the contextualist ontology of expressions. With regard to the metaphysics of time, there are two importantly different approaches. According to the tenseless theory, objects fall (tenselessly) 988 under concepts relative to moments 985 986

987

988

Cf. sections 8.5 and 8.6. As temporary existence typically goes hand in hand with contingent existence, it might be better to say that ordinary things exist only relative to some possible world-states, i. e., relative to some pair of a moment in time and a possible world. In some cases, there will be a close connection between the contexts relative to which expressions exist and the moments of time (or the possible world-states), e. g., in a story about the historical development of a language. But even these cases are best construed as embedding the abstract, semiotic contexts of the metaphysics of expressions in a temporal structure, not as an outright overlap. Cf. Rheinwald 1988, 241–259. The falling under a concept is tenseless in the sense that the verb is used in a tenseless way in the requisite locutions. E. g., saying that Obama tenselessly falls under the concept of being president of

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(and if existence is not atemporal, objects exist (tenselessly) relative to moments), and that is all there is to temporal reality. The tenseless theory stands in opposition to the tensed theory, according to which past, present, and future – beyond their grammatical role – are integral aspects of temporal reality: Objects both exist and fall under concepts in a present tensed way, and moments are constructions that are to be understood in terms of the temporal perspectives of the past and the future. 989 Generalizing, there are two importantly different approaches to ontology and metaphysics: According to the external stance on ontology and metaphysics, both existence and more generally the falling under a concept is really a relation between objects and contexts, and that is all there is to reality. Therefore we will also speak of relational metaphysics for the generalization of the tenseless theory of temporal reality. The external stance of relational metaphysics stands in opposition to the internal stance of contextualist metaphysics, according to which context-sensitivity is not merely a grammatical phenomenon, but an integral aspect of reality: Objects both exist and fall under concepts in a context-sensitive way, and contexts ultimately are constructions that are to be understood in terms of contextual perspectives. An overview: external stance: internal stance:

temporal reality:

reality in general:

tenseless theory tensed theory

relational metaphysics contextualist metaphysics

Just as the tensed theorist denies that ultimately we can leave the temporal perspective of the present moment behind in a legitimate description of temporal reality, the theorist who takes the internal stance denies that ultimately we can leave the perspective of the current context behind in a legitimate description of contextualist metaphysics. Or in other words, the tenseless and context-free God’s eye view on reality is no more than a fiction. This is the orientation that we take in this study – we take the internal stance on reality, and in particular on the metaphysics of expressions. (At least ultimately, because in some places we will speak as if our metaphysics was relational, to follow established custom or to keep the picture simple. 990) As a theoretical decision, this reaches too deep to be fully supported by arguments, and I will not try. 991 But let us have a look at an illustration that might be helpful to elicit intuitions in favor of adopting the internal stance in the case at hand. What does taking the internal stance mean for the metaphysics of expressions? Let us take up again the example by Fine from the previous section. There we said that from the viewpoint of Fluent who speaks both Prittle and Prattle, there are

989

990 991

the U.S. relative to the year 2016 can be cashed out by saying that Obama (tenselessly) is president of the U.S. relative to the year 2016. The literature on the metaphysics of time is vast. For an overview, cf. Markosian 2016, and for a spectrum of exemplary positions, cf. Oaklander/Smith 1994. Our own approach owes a lot to the work of Arthur Prior; cf. Prior 1968[1957], 1967, and 2003. Cf. in particular chapter 10. With regard to an important special case, the metaphysics of time, we can gesture to the main part of the works of Arthur Prior, which in my opinion constitutes a powerful argument for the tensed theory and thus for taking the internal stance on temporal reality. Cf. Prior 1968[1957]; 1959; 1960; 1967; and 2003.

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two distinct letters, addressed to his two daughters, but from the viewpoint of each daughter who understands only one of these languages, there is only one letter. Now, taking the internal stance means that these viewpoints ultimately cannot be transcended – although one of the daughters can change her viewpoint (arduously, by forgetting one language and learning another), in the end of the day there just is no legitimate viewpoint-free representation of the matter, which would construe the existence of the letters as a relation to the respective contexts. And so it is in general. In lieu of a decisive argument for this rejection of the external stance on expressions, just imagine how crowded the viewpoint-free universe of expressions would have to be! Turning back to the example: Not only would both daughters have to recognize two letters as originating from Fluent’s writing. But because of the infinity of possible languages and the correlated infinity of viewpoints relevant to the existence of expressions, Fluent’s scribbles alone would give rise to infinitely many distinct letters. 992 Relational metaphysics would be just too much. Contextualist metaphysics, in contrast, admits only those objects into reality that we intuitively countenance, without denying the possibility of others – in fact, we can always admit more objects if only we change the context in the appropriate way. 993

992

993

In a formal setting, this thought experiment can be made precise, for instance along the following lines. Let us construe a syntactic expression as a qua object with the expression’s Gödel number as a basis. Now consider a language of arithmetic with infinitely many numerals (e. g., ‘o’, ‘so’, ‘sso’, . . .). Even when we hold the standard interpretation fixed, the requirements on Gödel numberings give us so much leeway that the number 17, say, can be the Gödel number of any numeral whatsoever. According to our construal, it would thus be the basis of infinitely many distinct expressions. Rather than countenance the redundantly abundant ontology that contains all of them, we are well advised to understand each Gödel numbering as a context relative to which only one of the many possible systems of syntactic expressions exists. Cf. Pleitz 2010a, 212. In terms borrowed from the philosophy of mathematics, we can say that the universe of expressions is indefinitely extensible and thus never given as an absolute totality – just like the universe of sets and the universe of ordinal numbers (Dummett 1991, 316ff. and 1993, 441; cf. also Fine 2006 and Shapiro/ Wright 2006).

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Charles Sanders Peirce: “The (merely) possible [. . .] is necessarily general [. . .] it is only actuality, the force of existence, which bursts the fluidity of the general and produces a discrete unit.” 994 Gottlob Frege: “Freilich auf den ersten Blick scheint der Satz ‘alle Walfische sind Säugetiere’ von Tieren, nicht von Begriffen zu handeln; aber, wenn man fragt, von welchem Tiere denn die Rede sei, so kann man kein einziges aufweisen.” 995

We have seen that linguistic expressions form a contextualist ontology. In terms of a temporal metaphor, expressions constitute a world that is constantly changing, because again and again new expressions come into existence. As our quest in this part of the study is to understand self-reference, we are interested in a language that allows to speak about expressions. So we need to ask: What does it mean for a language to be about a changing world of expressions? What, in general, is characteristic of a language that allows to talk about a changing world of objects? How can we describe a contextualist metaphysics from within? Let us take the temporal metaphor of change as a cue and look again at the paradigm of time. We will start with a sketch of tense logic (in section 9.1). From there on, there will be three main topics: We will see that a contextualist metaphysics puts certain restrictions on which objects can be referred to (in sections 9.2 through 9.4), that in some cases where reference to objects is impossible, an irreducibly conceptual element can take over (in sections 9.5 through 9.9), and how particularities of the semantics of a language that is about a contextualist metaphysics lead in turn to particularities in the metaphysics of its expressions (in sections 9.10 and 9.11). In conclusion, we will remark briefly on the interconnectedness of metaphysics and semantics that surfaces in this chapter (in section 9.12).

994 995

Peirce, Collected Papers, 4.172; quoted in Prior 1960, 690. “It is true that at first sight the proposition ‘All whales are mammals’ seems to be not about concepts but about animals; but if we ask which animal then are we speaking of, we are unable to point to anyone in particular.” (Frege 1988[1884], 61 (= Foundations of Arithmetic, § 47); Frege 1980, 60)

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9.1 Tense logic and temporal context-sensitivity The standard method of modeling temporal context-sensitivity is by way of the formal semantics of tense logic, which is a species of modal logic. 996 The semantics of tense logic accordingly is a species of the possible worlds semantics of modal logic, with moments in time playing the role of possible worlds and being later than playing the role of the relation of accessibility. The key idea is that a sentence of the object language does not have its truth value in an absolute way, but in relation to a moment. E. g., the sentence ‘Obama is president of the U.S.’ is false with regard to the year 2006 but true with regard to the year 2016 – it is tensed in a non-trivial way and therefore temporally context-sensitive. Similarly, the singular terms and predicates of the object language will have their extension only in relation to a moment (as witnessed, e. g., by the description ‘The president of the U.S.’ and the complex predicate ‘. . . is giving a press conference’). 997 A formal language of tense logic has, in addition to the usual logical vocabulary, sentential operators that are meant to express the past or future tense; sentences without such operators are understood as present tensed. This can be spelled out differently in different tense logical systems. Traditionally, e. g., ‘P’ and ‘F’ are used with the intuitive reading ‘It was (sometimes) the case that . . .’ and ‘It will (sometimes) be the case that . . .’, respectively. It is standard to complement them with their respective duals ‘H’ and ‘G’ for ‘It was always the case that . . .’ and ‘It will always be the case that . . .’, because both the pair of ‘P’ and ‘H’ and the pair of ‘F’ and ‘G’ bears a structural similarity to the pair of the operators ‘’ and ‘♦’ of modal logic. 998 The behavior of these operators in different systems can be characterized either by a deductive system (e. g., axiomatically) or model theoretically (via requirements on the relation of accessibility). 999 The details need not concern us here. 1000 We are interested only in the key ideas because in the following we will not aim at delineating a specific system of tense logic, but will be content with giving hints for how this can be done. Some further ideas about tense operators should be mentioned, though: These can also be metric, and they can also be modal – in fact they can be both metric and modal, and that is how we will prefer them to be.

996 997

998

999 1000

Cf. section 4.5. The feature that every extension is had only in relation to a moment does not exclude the possibility that an expression’s extension is constant over all moments. Tense logics are often called bi-modal because they have two pairs of operators like ‘’ and ‘♦’. Note that the similarity is only structural, much like the similarity between the pair of locutions ‘always’ and ‘sometimes’ and the pair of locutions ‘necessarily’ and ‘possibly’. The box-like temporal operators do not express metaphysical necessity, but (a restricted variant of) omnitemporality. The difference becomes clearer once tense and metaphysical modality are combined. Then we can distinguish between ‘possibly always’ and ‘necessarily sometimes’, and so on. Cf. section 3.9. First on the list of the primary literature about tense logic are of course the relevant works by Prior; cf., again, Prior 1968[1957]; 1967; and 2003. A textbook with an exemplary place on the list of the secondary literature is Øhrstrøm /Hasle 1995.

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In contrast to their purely qualitative cousins ‘P’ and ‘F’, metric tense operators allow to express a temporal perspective that is quantitatively specified. E. g., we can use ‘P100 years’ to formalize the locution ‘One hundred years ago it was the case that . . .’; and we can use ‘F100 years’ for ‘One hundred years hence it will be the case that . . .’ – at least in a deterministic setting. 1001 This caveat brings us to the issue of combining tense with modality, or to the question of the open future. If the past is settled but the future open, then the metric past operator ‘P100 years’ makes good sense, but not so its counterpart, the metric operator ‘F100 years’. Rather, we will then need to work with ‘〈F100 years〉’ for ‘One hundred years hence it can be the case that . . .’ and its dual ‘[F100 years]’ for ‘One hundred years hence it must be the case that . . . ’. 1002 (Putting the metric future tense operator between angle brackets, ‘〈. . .〉’, or between square brackets, ‘[. . .]’, is meant to be reminiscent of the box and the diamond operator, ‘’ and ‘♦’, respectively.) A word on what these options mean in terms of the relation of accessibility. As long a tense is not combined with modality, it makes good sense to construe accessibility – being later than – as a linear order. To account for the metric tense operators, we need to add metric variants of this relation (e. g., being 100 years later than) to the framework. When tense is combined with modality in a way that the past is settled but the future open, we need to relax the requirements on accessibility in such a way that we liberalize from a linear order of moments to a branching tree structure of possible world-states, with the branches opening only towards the later possible world-states. (Accessibility is thus no longer the relation of being later than, but that of being later than and historically possible from.) One theoretical option that will be particularly interesting in the following comes into focus only when we look at predicate logical variants of tense logic: We will be able to work with a variable domain, i. e., with a domain that varies from context to context. If we do this, then what is meant by quantificational phrases of the form ‘all Fs’ and ‘some Fs’ will in general also depend on the moment, and so will the extension predicates – even the extension of stable predicates, where a predicate is stable if and only if some object satisfies it with regard to some moment just in case the object satisfies it with regard to every moment where the object is in the domain. (E. g., if the domain is variable, then the extension of sortal predicates like ‘. . . is human’ and even of the predicate of self-identity, ‘x = x’, can vary from moment to moment, although these predicates are stable.) The present remarks are of course a mere sketch of tense logic, and they indicate only in rough strokes the tense logical system (or rather, family of systems) that will deliver what we need for our purposes. But that is alright, because within the broad scope of the present study it is enough to make plausible that a rigorous account can be given – and it is prudent to resist the temptation to carry out the full technical 1001

1002

On metric tense logic, cf. Prior 1968[1957], 11ff.; 1967, 95ff.; and 2003, 159ff.; Øhrstrøm/Hasle 1995, 210ff., as well as Pleitz 2015c. For those who are familiar with the tense logic of indeterminism, this decision against making room between the possible future and the necessary future for an actual but contingent future amounts, in terms coined by Prior, to preferring a Peircean over an Ockhamist tense logic (Prior 1967, 113ff.; cf. Belnap /Green 1994 and Strobach 2007, 58ff., as well as Pleitz 2009).

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work, with its dangers of getting lost in formal details, of settling on a system that some formal-minded readers might dislike for extraneous reasons, and of losing some not-so-formally-minded readers, which would mean to risk gambling away the chance to present the basic ideas of our argument in a convincing way. Anyway, we are not interested in tense logic for its own sake, but in what we can learn from it for an adequate representation of contextualist metaphysics in general. Thus we will in due course generalize from moments in time to contexts of a more general sort, which will also be ordered by a suitable relation of accessibility. Taking the internal stance, we can say that we will generalize from the present moment to the current context, as the only legitimate vantage point for a representation of reality.

9.2 Future objects: Contextual restrictions on reference Getting back to the paradigm of a language that represents temporal reality, we will now look at restrictions on reference that result from taking the tenses of past, present, and future seriously. It has often been observed that statements about future objects and about merely possible objects must be general, where a future object is an object that will exist but has not yet come into existence and a merely possible object is one that could exist but does not exist. In the words of Richard Gale and Irving Thalberg: “Any prediction referring to a future individual must be general” 1003. In a temporal ontology, reference is restricted in a way that precludes referring to an object that has not yet started to exist, because there is nothing there yet to refer to. We cannot refer to future objects! 1004 While Charles Sanders Peirce, Gilbert Ryle, and Nelson Goodman do little more than state this fact, 1005 Arthur Prior gives an argument: “Suppose there is some person living before the existence of Caesar or Antony who prophesies that there will begin to be a person who will be called ‘Caesar’ who will be murdered, etc., and another person who will be called ‘Antony’, who will dally with Cleopatra, etc. And then suppose this prophet to say, ‘No, I’m not sure now that it will be like that – perhaps it is the second of the people I mentioned who will be called ‘Caesar’ and will be murdered, etc., and the first who will be born later and be called ‘Antony’, etc.’ This, it seems to me, really would be a spurious switch; and after Caesar and Antony had actually come into being and acted and 1003

1004

1005

Cf. Gale/Thalberg 1965, 202. – Note that it is not the case that all statements about the future must be general, as there are future tense statements about present individuals (cf. Gale/Thalberg 1965, 195ff.), witness my present utterance of ‘Next year, I will visit New York’, which is about me and New York. Some phrases need to be taken with a grain of salt here. According to our official theory, we can refer to no future object. But strictly speaking in saying so, we appear to do just that. What we really mean is that relative to any moment, it is not possible to refer to an object that starts to exist relative to a later moment. Please think of the phrase ‘future objects’ as cashed out thus in terms of later objects – at least where necessary. For note that some uses of the phrase are unproblematic, e. g., when we said, long enough after the fact, that at the time of the prophet, Caesar and Antony were future objects. Cf. Peirce, Collected Papers, 4.172; quoted in Prior 1960, 690.; Ryle 1954, 25ff.; and Goodman 1973, 55.

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suffered as prophesied, it would be quite senseless to ask, ‘Are these, I wonder, really the two people he meant?’ and if possible more senseless to ask, ‘Is it – if either of them – our man’s first prophecy, or his second suggested alternative, that has come to pass?’” 1006

From the temporal perspective of the prophet envisioned by Prior, Caesar and Antony would still have been no more than future objects. What Prior’s thought experiment shows is that although the prophet would have been able to describe some individuals in a way that is qualitatively identical to how we, from our present perspective, would characterize Caesar and Antony, the prophet would not yet have been able to describe them. As they still were future objects, he would not have been able to refer to them in the sense of picking out these unique individuals. 1007 It is worth noting the contrapositive of the observation that we never refer to objects that start to exist in the future: When we do refer to an object, then it has started to exist in the past or it starts to exist right now. We will later find arguments to strengthen this principle by excluding the possibility that we can refer to an object at the very moment it starts to exist. When we generalize from moments in time to contexts of a more general sort, the observation that we never refer to future objects is transformed into a principle of non-priority for reference: Reference is in no case prior to the existence of the object that is referred to. 1008 In these general terms, what we will later argue for is a principle of subsequence for reference: Reference is in all cases subsequent to the existence of the object that is referred to. 1009 But that is still far off. At this point, we can already draw a general moral for the case of a language that is about a contextualist ontology: When objects exist only relative to some contexts, reference to these objects will in general also be restricted to some contexts. But we need to be more specific than that.

1006 1007

1008 1009

Prior 1960, 690. Note that the above argument to the conclusion that we cannot refer to future objects does not presuppose indeterminism. Our intuitions about the prophet in the thought experiment have nothing to do with the course of history being fixed; they concern only how the identity of an object is tied to its past or present existence. However, the conclusion that we cannot refer to future objects would certainly get additional support from indeterminism. If we were to endorse it, we could echo Quine’s questions about the identity of possible men in the doorway (Quine 1980[1953], 4) and ask: Which first child born a hundred years hence do you mean? The heavy one or the bald one? Or are they the same? How do we decide? And we could imagine asking the prophet: Which possible human called ‘Caesar’ do you mean? The one you describe, or the possible one who will never meet Cleopatra and die in his sleep? Or are they the same? How do we decide? All these questions can only be answered in hindsight, because only then will it be settled which of the many branches of the treelike tempo-modal structure has become actual. Note that for this way of arguing from indeterminism, it is relevant that we have chosen to take a Peircean rather than an Ockhamist approach within modalized tense logic; cf. the last footnote to section 9.1. By ‘reference’ we here mean the concrete act or event of referring, i. e., reference relative to a context. Cf. chapters 11 and 12.

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9.3 Past objects: Identifiability without existence How can the general moral that reference to objects that form a contextualist ontology is itself restricted to certain contexts be made more specific? A plausible first idea is the following candidate for a restrictive condition that is meant to apply to all languages that are about a contextualist ontology: Relative to every context, a singular term can refer only to those objects that exist relative to that context. 1010 However, another look at the temporal case shows that this idea is too simple. For note that when Prior described a (partly fictional) past situation, he himself could without any problem refer to Caesar, Antony, and Cleopatra, and so can we, even after they have been dead for about two thousand years. But on the highly plausible assumption that for humans to exist is the same as to be alive, Caesar, Antony, and Cleopatra do no longer exist relative to the moments of Prior’s and our acts of referring to them. So we can refer to past objects, where a past object is an object that has existed, but exists no longer. There is thus an interesting asymmetry in the temporal case, for whereas we cannot refer to future objects, we can refer to past objects. Generalizing we can draw the conclusion that the direction of the relation of accessibility (in the case of temporal reality, the relation of being later than) matters for what we can refer to. And we see that we have to reject the candidate for a restrictive condition according to which what we currently can refer to falls together with what currently exists. For now we have observed that in a contextualist ontology, there can be two kinds of currently non-existent objects: Those that we currently cannot refer to (like future objects) and those that we currently can refer to (like past objects). This observation motivates the following definition: An object is identifiable relative to a context if and only if it can be referred to relative to that context, where by ‘can be referred to’ we mean ‘can be referred to in any language whatsoever’. 1011 In these terms, we can sum up what we have gleaned from Prior’s thought experiment about the temporal case (in this section and the preceding one) by saying that while past objects stay identifiable even after they have stopped to exist, future objects are not yet identifiable because no object is identifiable before it starts to exist. It is again worth noting the contrapositive of the second observation: Every object that is identifiable at present exists at present or has existed in the past. Thus, while identifiability can succeed existence, identifiability cannot precede existence, so that we might want to say that identifiability always starts from existence. This is again a principle of non-priority: The identifiability of an object can never be prior to its existence. In contrast to the corresponding principle of non-priority for reference that we formulated at the end of the preceding section, the principle of non-pri1010

1011

For all we know (officially) at this point, this restrictive condition might be compatible with what we saw to hold in the case of temporal reality in section 9.2: Relative to every moment, a singular term can refer only to those objects that start to exist relative to that moment or relative to an earlier moment. Cf. Fitting/Mendelsohn 1998, 230ff. – In another context, we might want to work with a notion of identifiability that is even more general, according to which an object is identifiable relative to a context if and only if it can be referred to, or thought about, or otherwise represented or picked out relative to that context. I would like to thank Christian Nimtz for raising this point. In the scope of the present study of self-referential expressions, though, recourse to reference is all we need.

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ority for identifiability is modally strengthened and no longer bound to a specific language. The general lesson we can draw from the observation that identifiability can succeed existence is that identifiability does not in general entail existence, or, to be more precise, that identifiability relative to a certain context does not in general entail existence relative to that context. What about the converse question: Does existence entail identifiability? In the temporal case, it prima facie looks that way. For we appear to be able without any problem to refer to every present object (where a present object of course is an object that exists at present). But is this possible relative to every moment of the existence of a present object, even relative to the very first moment of its existence? This question – or rather its generalized variant, concerning reference in a contextualist ontology – is crucial to the approach to the Liar paradox that we will outline in the following chapters. In chapters 11 and 12, I will give a negative answer, arguing that no object is identifiable at the very first context of its existence (or to put it more generally, for every object, there is a context relative to which it exists, but that is prior to every context relative to which it is identifiable), so that identifiability is always subsequent to existence. But that is still far off; here is not the place to give a preview of the argument for this claim of subsequence. But to lessen the astonishment, let us briefly consider an analogy: the role of causality in the temporal case. Plausibly, we can refer (in a nongeneral way) only to what we are already in causal contact with. In the case of an object of our acquaintance, i. e., an object that is currently present to us, we typically establish contact through our senses to pick out the object before we say something about it. In the case of a past object, there is what Kripke has described as a causalhistorical chain of reference that goes back to the object we talk about. 1012 Hence, as a causal connection needs to be established to enable reference, existence will precede identifiability in all these cases, if only by the briefest of durations. In view of the limit set to causal interaction by the speed of light according to the theory of special relativity, this effect gets more pronounced when it comes to reference to objects that start to exist at present, but are located far away from the place where we talk. 1013

9.4 The modal logic of existence and identifiability We have seen that in a language that concerns a contextualist ontology, existence and identifiability can come apart. How can this observation about semantics be explicated on the level of logical form, and with the help of formal semantics? Just as in our sketch of tense logic in section 9.1, our aim here is not to give a full description of a logical system, but to present a basic idea. In fact, the idea will shortly prove to be too basic, insofar as we will run into a problem that we will be able to solve

1012 1013

Kripke 1980[1972], 91ff. We will give an example for a scenario where there is existence without identifiability because of special relativity in more detail in section 10.10.

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only later when we will have developed further technical resources that allow for a crucial modification of the basic idea. 1014 In modal logic and also in tense logic in particular, there are two standard ways of explicating existence that is relative to context. In the constant domain approach, there is a single domain of possibilia and a primitive existence predicate with an extension that can vary from possible world to possible world, often writing ‘E!(x)’ for ‘x exists’. In the alternative variable domain approach, it is the objects in the domain of discourse that vary from possible world to possible world. In this approach, there is no need for a primitive existence predicate, because an existence predicate can be defined using the tools of quantification, with the open sentence ‘∃y x = y’ expressing what we mean by the colloquial ‘x exists’. 1015 Neither of the two standard approaches allows to deal with the distinction between existence and identifiability in a satisfactory way. The variable domain approach does not allow to distinguish between existence and identifiability. The constant domain approach does allow to draw a distinction by using the extension of the existence predicate to model existence and the constant domain to model identifiability. But this entails that identifiability cannot differ from possible world to possible world, and will thus not help model our observation. For in view of the context-relativity of existence, our observation that identifiability is never prior to existence entails that in general, identifiability will be relative to context, too. What I propose to do is to stay within the paradigm of quantified modal logic, and more specifically, within the paradigm of quantified modalized tense logic, and use tools from both standard approaches. Let us work with a logic that has both a variable domain and a primitive existence predicate, ‘E!(x)’! Then a natural idea would be to use the primitive existence predicate to model the context-relative notion of existence and the variable domain to explicate the context-relative notion of identifiability. It is, after all, the domain of discourse, the collection of all objects we can talk about. In this, we will construe the points of the frame of this modal (tense) logic in the most general way as contexts of a general sort, which can be possible worlds, moments in time, possible world-states, or the contexts relevant to the existence of expressions. 1016 In the tense logical framework it makes sense to construe the domain of identifiabilia as growing, to model the persistence of identifiability. 1017 And as we have seen that identifiability always starts from existence, a restriction should be added so that it does not happen that an object enters the 1014 1015 1016 1017

Cf. chapter 10. Cf., e. g., Fitting /Mendelsohn 1998, 163ff. Cf. the observation about contexts at the end of section 8.6. We say that a property is persistent if and only if, if an object has that property relative to some context (moment), then it has it relative to every subsequent context (later moment). Requiring the persistence of identifiability models the assumption about language that no object is ever forgotten. This is a useful idealization, but strictly speaking it would not be needed for the following investigation of self-referential languages. – Note that if a property is stable, then it is persistent, but the converse does not hold, because a persistent property is not stable if its negation is not persistent, too. E. g., being older than 18 years is persistent but not stable because being 18 years old or younger is not persistent. Persistence and stability are both compatible with indefinite extensibility (cf. the last footnote to section 8.7).

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domain before it satisfies the existence predicate. (We will need to look at this in more detail, 1018 but that will be worth the effort only after we have enriched our framework for the treatment of existence and identifiability in the next chapter.) In the language of this logic with a variable domain and a primitive existence predicate, we can say truthfully of some object that it is a merely possible object, i. e., that it does not exist but could exist, because the formal sentence ‘¬E!(a) ∧ ♦E!(a)’ can be true with regard to a certain context c, if with regard to c the constant ‘a’ refers to an object that is in the domain of c but does not satisfy the existence predicate with regard to c (meaning that relative to c, the extension of ‘a’ is identifiable but does not exist). And if our logic has metric tense operators, we can use it to account for talk about past objects. E. g., we can express that although Socrates does not exist, it was the case 2420 years ago that he existed, because the formal sentence ‘¬E!(s) ∧ P2420 years E!(s)’ 1019 is true with regard to the year 2016 if we use the constant ‘s’ to refer to Socrates. Some things can be true of Socrates when he does not exist – not only that he does not exist any longer, as in our example, but also more positive features like being famous. But certainly, many other things cannot be true of Socrates when he does not exist, e. g., that he sits or that he feels pain. This motivates a distinction between two kinds of predicates: A predicate is existence-entailing if and only if it can only be satisfied by an object with regard to a context with regard to which that object satisfies the existence predicate. Accordingly, a predicate is not existenceentailing if an object satisfies it with regard to a context where that object is in the domain but does not satisfy the existence predicate. Thus ‘. . . sits’ and ‘. . . feels pain’ are existence-entailing, but ‘. . . is famous’ is not existence-entailing. 1020 The distinction carries over in a natural way from predicates to concepts: A concept is existence-entailing if and only if falling under it entails existence, for every object and relative to every context. The present construal of existence and identifiability in terms of a variable domain and a primitive existence predicate, however, will work only as a first approximation. It will work for all scenarios where existence entails identifiability. Here we can say that the objects that are identifiable relative to a context are just the objects in its domain of discourse, and the objects that exist relative to a context are those objects from its domain that are in the extension of the existence predicate relative to that context. Things get more complicated if we also want to be able to account for scenarios where there can be existence without identifiability. And we will want to account for such scenarios! 1021 But in a modal logic or a tense logic with a variable domain and an existence predicate that is characterized by a semantics along the lines of the above construal, it is not possible to account for existence without identifiability. For only objects from the domain of discourse can fall in the extension of the existence predicate, and according to the above construal the objects in the domain are the identifiabilia. 1018 1019 1020 1021

Cf. chapter 12. Cf. section 9.1. So is the predicate ‘. . . is dead’, which is even non-existence-entailing. Cf. chapter 12 and in particular sections 12.7 through 12.9.

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Here is a clue for a solution: Would it help to construe the variable domain as the union of the objects that are identifiable and the objects that exist, and add a primitive identifiability predicate, ‘I!(x)’, to the language? Thus we would be able to talk not only about identifiability without existence (as before), but also about existence without identifiability. 1022 But then we would run into a new problem. In predicate logic, the domain is what the quantifiers range over. And the domain is meant to explicate what we can talk about. So the quantifiers should range over identifiabilia and nothing else. But according to this idea for a solution, the quantifiers would also range over objects that exist but are not identifiable. This would allow us to identify an object and say truthfully about it that it is not identifiable. But we should not be allowed to do that! What is to be done? The present problem of how to represent existence without identifiability is more than a technical difficulty. We will only be able to solve it when we will have gained a deeper understanding of what it means to model both a contextualist metaphysics and a context-sensitive language that is about it with the tools of formal semantics, and when we will have adapted these tools for this specific, dual purpose. 1023 For now, we leave the problem unsolved, and turn to other matters in the vicinity.

9.5 Future objects: Irreducibly conceptual talk In order to make a further observation about the semantics of a language that is about a contextualist ontology, let us return to the paradigm of time. In the last sections, we have studied some intricacies concerning the interplay of existence and identifiability. For the purposes of the present section, though, it is enough to keep in mind one observation about the temporal case, that we cannot refer to future objects. 1024 As we saw, there are no future objects among present identifiabilia. But what, then, are we to make of our present talk that does seem to concern future objects? And more generally, what are we to make of current talk that does seem to concern currently non-identifiable objects? For it appears possible to make meaningful and even true statements that certainly look as though they are about future (and hence currently non-identifiable) objects. As Russell demonstrates: “We do not know who will be the inhabitants of London in a hundred years, and we don’t know that there will be any. But we do know that any two of them and any other two of them (granted that they exist) will make four.” 1025 1022

1023 1024 1025

Both ‘I!(x) ∧ ¬E!(x)’ and ‘¬I!(x) ∧ E!(x)’ would in principle be satisfiable by some objects relative to some contexts (but of course not by the same object relative to the same context), as well as ‘I!(x) ∧ E!(x)’, which could therefore also be used in an informative way. It is only ‘¬I!(x) ∧ ¬E!(x)’ that would not be satisfiable relative to any context, because no object in the variable domain would be both non-identifiable and non-existent relative to the same context. We will say more about how this restriction of the technical apparatus can be motivated in section 10.7. We will give a detailed account in chapter 10. Cf. section 9.2. Quoted from Edwards 1949, 155.

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Russell’s example shows that mathematics applies even to future objects; or, to phrase it more cautiously, that mathematics applies with regard to future tense statements that look as though they are about future objects. So does logic. As a variation of Russell’s example shows: We know that a hundred years hence it will be the case that (there are more than four inhabitants of London, or it is not the case that there are more than four inhabitants of London). 1026 In both cases, the part of the statement that purports to be about future objects is in fact irreducibly general. A third way, besides applying mathematics or logic, to arrive at truths that look like truths about future objects (or, more generally, about non-identifiabilia) but that are in fact irreducibly general is to rely on conceptual connections. For example: We know that it will be the case one hundred years hence that (if London is still inhabited by humans, then London is inhabited by rational animals). What we know here because of conceptual connections is that the following sentence is true at present: ‘It will be the case one hundred years hence that (if London is still inhabited by humans, then London is inhabited by rational animals).’ Note that the content of this sentence is not purely conceptual. It refers to London, and so has an objectual element, too. But alongside this objectual element, there is also an irreducibly conceptual element in the sentence. For our justification in believing it to be true at present lies in a purely conceptual truth, namely the truth that being human entails being a rational animal. 1027 This truth is about concepts in a way that cannot be cashed out in terms of objects; it concerns much more than the objects that are currently identifiable (and, speaking metaphorically, it even goes beyond the objects that could yet become identifiable). 1028 To summarize, future tense sentences that seem to be about future objects are really statements that have an irreducibly general element. In certain cases we have access to their truth via logic, mathematics, or conceptual connections. Only the latter will concern us in the following. And to generalize this observation from the paradigm of time to any language that is about a contextualist ontology: In certain 1026

1027

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The brackets and regimented grammar are meant to convey that the whole disjunction is in the scope of the future tense operator. This is more perspicuous in the contrast of the corresponding sentences of metric tense logic, ‘[F100 years] (p ∨ ¬p)’ and, e. g., ‘[F100 years] p ∨ ¬[F100 years] p’. (The metric tense operator ‘[F100 years]’ formalizes the locution ‘One hundred years hence it will (necessarily) be the case that . . . ’; cf. section 9.1.) Please be charitable with our examples for conceptual truths, or think up your own more intuitive ones. The present conception of statement with an irreducibly conceptual element presupposes a philosophical rehabilitation of the notion of a purely conceptual statement, which calls for defending against its modern critics (e. g., Williamson 2006; Quine 1976b; Quine 1980[1953], 20ff.) what has been called “analyticity” (Frege, Kant), “relations of ideas” (Hume) and “truths of reasoning” (Leibniz). Cf., e. g., G. Russell 2008.

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cases we do have access to truths that appear to be about currently not-identifiable objects – it is given to us by purely conceptual truths.

9.6 The logic of objectual and conceptual quantification Let us again, as in the case of existence and identifiability, have a look at how our semantic observations – about the irreducibly conceptual elements of some statements and about purely conceptual truths – can be explicated on the level of logical form. As before, the sole aim is to fix the basic idea of how this can be done. To account for irreducibly conceptual elements of some statements, we have drawn attention to purely conceptual truths. But how are we to construe purely conceptual statements in our framework, with its domain of discourse restricted to what is identifiable (or at most, restricted to what is identifiable or exists)? 1029 And more generally, how can the distinction between the objectual elements and the conceptual elements of a sentence be made visible on the level of logical form? Let us look at purely conceptual statements of the form ‘Every F is G’ first. To see what is characteristic of them, we should contrast them with various kinds of objectual statements that are of the same form. So, compare the following four sentences: (1) ‘Every human knows about Coca Cola.’ (2) ‘Every human is a featherless biped.’ (3) ‘Every human is descended from Eve.’ (4) ‘Every human is a rational animal.’ We can easily tell a background story, making some factual (and metaphysical) stipulations and idealizations, so that it becomes plausible that all four sentences are true, but the quantificational phrase ‘every’ means something different in each of them. (A) Every human alive today knows about Coca Cola. But clearly most humans in the past did not know about Coca Cola, and probably many humans in the future will not know about Coca Cola. (B) Whereas every past human was a featherless biped and every present human is a featherless biped, a human may have feathers or a number of legs other than two. (C) A long time ago, there was a female hominid in Africa, whom we will call ‘Eve’. Eve was the first human being, and every other human is descended from her. As ancestry is transitive, it is also true of every (possible) future human that he or she is descended from Eve. Under the metaphysical stipulation that its ancestral origin is essential to a species (in a certain way), this will even be true of every possible human. (D) Being human conceptually entails being a rational animal. 1029

Cf. section 9.4.

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Given these stipulations, it is natural to understand the phrase ‘every human’ to have an increasingly wider range or meaning in the four sentences. 1030 Roughly, 1031 sentence (1) is about all present humans and thus about all existent humans, sentence (2) is about all past and present humans and thus about all identifiable humans, and sentence (3) is about all possible humans in a sense that even goes beyond the currently identifiable humans and thus encompasses not only all present and future identifiabilia, but all possible identifiabilia. Sentence (4), finally, is about the very concept of being a human being. So, as each one of the four sentences on the surface is a universal statement of the form ‘Every F is G’, we will need to look at their logical form if we want to make these differences in meaning explicit. The most important line is to be drawn between sentences (1), (2), and (3) on the one side and sentence (4) on the other side. In a way, the meaning of ‘every’ in the first three sentences can be cashed out by wider and wider ranges of possible objects, but not so in the fourth sentence. In terms of ranges of possibilia, we would need to widen the circle from the metaphysically possible to the conceptually possible to account for this conceptual truth. But what would it even mean to talk about an object that is conceptually possible but metaphysically impossible? On the level of logical form, this contrast between what can be cashed out in terms of possible objects on the one side and conceptual truths on the other side is reflected in the fact that we can formalize the full meaning of sentences (1), (2), and (3) by the logical resources already at our disposal – the universal quantifier that ranges over a growing domain, the primitive existence predicate, and the intensional operators of our tense logic –, but not so with sentence (4). Here something new is called for. What I propose is to incorporate a second pair of quantifiers (‘Π’ and ‘Σ’), which I call conceptual quantifiers, in addition to the standard objectual quantifiers (‘∀’ and ‘∃’), together with a second style of variables (‘ξ’, ‘υ’, . . ., in addition to the standard ‘x’, ‘y’, . . . ). This enables us to distinguish the conceptual truth that everything is F, ‘Πξ F(ξ)’, from the universal statement that all current objects are F, ‘∀x F(x)’. Note that even though the resulting logic has two pairs of quantifiers, it is still a predicate logic of first order, because all quantification is into singular term position. To give the briefest of sketches of the system, the rules for the formation of well-formed formulas will be extended in the natural way, we can think of each pair of quantifiers as governed by the usual quantifier rules, 1032 and there will be 1030

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We need only imagine that the four sentences are uttered by someone who knows the facts of this background story, who intends to say something true, and who furthermore intends to follow a Gricean maxim of being as informative as possible. Many phrases in the following explanation again need to be taken with a grain of salt (cf. section 9.2). According to our official theory, we cannot talk about non-identifiable objects. Thus strictly speaking locutions like ‘possible identifiabilia’ cannot mean what they are supposed to mean here, and the same goes for many uses of ‘future objects’. But these phrases do their job of indicating the different intended readings of the four sentences, and that is all we need to motivate a formal language that will allow to talk about these matters in a way that needs less additional salt. There is some leeway for variation with regard to the exact quantifier rules. For the objectual quantifiers, we might want to allow that the domain can be empty. This would mean to adopt what Quine dubbed “inclusive logic” for the objectual pair of quantifiers, which can be seen as a variant of free logic (Quine 1954; cf. Uzquiano 2016, section 2.1, Nolt 2014, sections 1.3 and section 6). In fact, our

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one principle about their interaction, according to which, for every open formula pΦ(. . .)q from among a broad collection of open formulas, 1033 pΠξ Φ(ξ)q logically entails p∀x Φ(x)q (as well as stronger forms like p ∀x Φ(x)q, should the logic have resources needed to form them), but not vice versa. This principle captures the fact that conceptual truths hold for all objects. 1034 Before we take a look at the semantics of this logic, let us see how it allows us to make explicit the logical form of our four example sentences. This can be done in the following way: 1035 (1)

‘Every human knows about Coca Cola.’

‘∀x (E!(x) ∧ H(x) → C(x))’

(2)

‘Every human is a featherless biped.’

‘∀x (H(x) → B(x))’

(3)

‘Every human is descended from Eve.’

‘ ∀x (H(x) → B(x))’

(4)

‘Every human is a rational animal.’

‘Πξ (H(ξ) → R(ξ))’

There is some leeway for the formalization of the first three sentences, 1036 but given what we said about the semantics of the objectual part of our logic, these will do well enough to capture the different meaning of the predicate logical part of the sentences. Given the growing domain semantics, the objectual quantifier ‘∀x’ in the formalizations of (1) and (2) ranges over all currently identifiable objects; and in the formalization of (1) the existence predicate ‘E!(x)’ restricts this range to all currently existing objects. In the formalization of (3), the objectual quantifier ‘∀x’ is in the scope of a modal operator, and will thus range over all objects that are identifiable from the point of view of some context – in the temporal case, it will thus range over all possible past, present, and future identifiabilia. Sentence (4), in contrast, is formalized with the help of a conceptual quantifier, ‘Π’, that binds a variable of the corresponding style, ‘ξ’. Thus the resources of our formal language allow to make explicit a fourfold distinction that is implicit in the use of the quantificational phrase ‘every’ of natural

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different understanding of the two pairs of quantifiers would provide a good motivation for adopting inclusive logic for the objectual quantifiers but sticking to classical logic for the conceptual quantifiers. For consider ‘∃x x = x’ and ‘Σξ ξ = ξ’. Thus only the second would keep its theoremhood, reflecting that it is not a logical truth that there currently is an object but it is a logical truth that self-identity is conceptually coherent. But none of the issues we are concerned with in the present study is connected to the question of whether the domain of objects can be empty. Therefore we leave such details for another occasion. Maybe we will want to require of the open formula that it does not contain intensional operators, or that it does not contain objectual quantifiers, or that is does not contain conceptual quantifiers. Because of the duality of the conceptual quantifiers (pΠξ ϕq is equivalent to p¬Σξ ¬ϕq), and the duality of the objectual quantifiers (p∀x ϕq is equivalent to p¬∃x ¬ϕq), this interaction principle entails a dual principle according to which p∃x Φ(x)q logically entails pΣξ Φ(ξ)q. This captures the fact that existence entails conceptual possibility. We use the formal predicates ‘H(x)’ for ‘x is a human’, ‘C(x)’ for ‘x knows about Coca Cola’, ‘B(x)’ for ‘x is a featherless biped’, and ‘R(x)’ for ‘x is a rational animal’. Given the stipulations (B) and (C), we might want to assign a more complicated logical form to sentences (2) and (3), e. g., with the tense operators ‘H’ for ‘It always was the case that . . .’ and ‘[G]’ for ‘It will always necessarily be the case that . . . ’, ‘H ∀x (H(x) → B(x)) ∧ ∀x (H(x) → B(x))’ for sentence (2) and ‘ (H ∀x (H(x) → B(x)) ∧ ∀x (H(x) → B(x)) ∧ [G] ∀x (H(x) → B(x)))’ for sentence (3).

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language: Whether we want to talk about all existing objects as in the case of (1), or to talk about all identifiable objects as in the case of (2), or about all possible objects as in the case of (3), or – in an entirely different vein – to make a purely conceptual statement as in the case of (4). It would be misleading to say that sentence (4) is ‘about’ the concept of being human in the same way that the sentence ‘Socrates is human’ is about Socrates – rather, it connects the concepts of being human and being a rational animal without reifying them. This is reflected in the logical form we assign to (4), where the concept is not referred to by a singular term, but expressed by a predicate. 1037 The conceptually quantified formal sentence ‘Πξ (H(ξ) → R(ξ))’ is meant to be strictly stronger than its counterpart with an objectual quantifier, ‘ ∀x (H(x) → R(x))’ (which it entails by the principle about the interaction of conceptual and objectual quantifiers). How is this reflected in its formal semantics? The challenge here is to give the semantics in a way that does not lapse back into objectual quantification (as for example when we were to take the conceptual quantifiers to range over an even wider realm of objects, the purported conceptually possible objects). This can be achieved by counting a conceptually quantified statement true if and only if it is derivable from a collection of meaning postulates in Carnap’s sense. 1038 Because of the interaction principle, the true purely conceptual statements that are explicated by what is derivable from meaning postulates will usually entail some objectual truths. They will thus put restrictions on admissible (material) re-interpretations of the language with regard to the variable domain of discourse that is relevant for the semantics of the objectual quantifiers. Going this way amounts to giving a standard model theoretic or representationalist semantics for the objectual quantifiers, but an inferentialist semantics for the conceptual quantifiers. 1039 This dualist semantics reflects the fact that the truth of a sentence can derive from two distinct sources – from the world, which is explicated by the growing domain of objects and what is the case with them relative to the contexts, and from the conceptual connections that are expressed by purely conceptual truths, which are explicated by Carnapian meaning postulates and what can be derived from them. Now we can state more precisely what we meant when we said in the last section that statements that appear to be about future objects have an irreducibly conceptual element. When a statement has an irreducibly conceptual element, this need not be reflected in its logical form by the occurrence of conceptual quantifiers. Rather, its counterpart in the formal semantics is that its truth is explained not by the representationalist semantics of the objectual quantifiers alone, but only in combination with the inferentialist semantics of the conceptual quantifiers, based on the truth of some purely conceptual statement. This means that for a true statement with 1037

1038 1039

This is relevant to Frege’s puzzle of the concept horse; cf. Frege 2002d [1892]. However, the new possibilities of formalization do not constitute a full solution, because not all reification of concepts can be circumvented by conceptual quantification. But that is a matter for another occasion. Carnap 1952. The need for an inferentialist account of the conceptual quantifiers pertains to the material level of theoretical semantics (cf. section 4.3). Nothing speaks against giving a standard model theoretic account as long as it is clear that this can only be a formal semantics.

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an irreducibly conceptual element, one source of its truth is not in the world of objects, but in the conceptual realm. E. g., the sentence ‘It will be the case one hundred years hence that (if London is still inhabited by humans, then London is inhabited by rational animals)’ can be formalized, roughly, as ‘[F100 years] ∀x (L(x) ∧ H(x) → R(x)))’. 1040 This formal sentence itself contains no conceptual quantifier, but it can be inferred from the purely conceptual statement ‘Πξ (H(ξ) → R(ξ))’, which expresses that being human conceptually entails being a rational animal. Thus ‘[F100 years] ∀x (L(x) ∧ H(x) → R(x)))’ is true relative to every moment in time, but not because of how matters are at the moment one hundred years later than that, but because ‘Πξ (H(ξ) → R(ξ))’ is an axiom (or a theorem) with the role of a meaning postulate that describes how certain of the concepts involved relate to each other.

9.7 The irreducibility of sense and of concepts Let us take a step back to see what our considerations about reference, logical form, and formal semantics mean for the material semantics of a language that is about a contextualist ontology. 1041 We have seen that we cannot refer to future objects, 1042 but are nevertheless sometimes able to make true future tense statements (that look as though they are about future objects) because of our grasp of purely conceptual truths, 1043 and we have proposed a logic that makes the difference between statements about objects and conceptual statements visible. 1044 Now it will turn out that some of these considerations provide us with an argument against the two reductionist programs we discussed when we introduced theoretical semantics, the program of a semantic reduction of sense to extension and the program of a metaphysical reduction of concepts to objects. Recall that we understand extensional semantics, intensional semantics, and twodimensional semantics as reducing the sense of an expression to its extension (perhaps in a complex way, as a function from contexts to extensions). And as we take a concept to be the sense of a predicate, this semantic reduction of sense to extension goes hand in hand with a metaphysical reduction of concepts to objects. 1045 As we noted, both reductions go against the Fregean spirit of our semantico-metaphysical framework. Specifically, the semantic reduction jars with Frege’s clear distinction between sense and extension (“Bedeutung”), and the metaphysical reduction jars with his clear distinction between concepts and objects. This already casts some doubt on the reductionist programs. 1046 Now, however, we are in a position to add

1040

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In addition to the formal predicates ‘H(x)’ for ‘x is a human being’ and ‘R(x)’ for ‘x is a rational animal’ that we introduced in section 9.5, we use ‘L(x)’ for ‘x inhabits London’. For the distinction between the material and the formal level of theoretical semantics, cf. section 4.3. Cf. section 9.2. Cf. section 9.5. Cf. section 9.6. Cf. section 4.8. Cf. sections 4.2 and 4.8.

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an argument that would have surprised (and probably not convinced) Frege because it takes the paradigm of time seriously. The present argument is based on the observation that we give the semantics of all our statements, even those that seem to concern future objects and other statements with an irreducibly conceptual element, not from a God-like non-perspectival perspective outside our temporally structured world, but from the perspective of the present moment. To generalize, when concerned with a contextualist ontology that we are ourselves situated in, 1047 we need to take the internal stance: Ultimately, we can only try to understand our ontology from within, and all seemingly contextfree descriptions we give of it have the status of metaphors only. 1048 When we adopt the internal stance with respect to the contextualist ontology of our temporal world and try to do semantics according to the material understanding, we will soon realize that there are just not enough identifiable objects available at present to reduce sense to extension or to reduce concepts to objects in a satisfactory way! As there are no future objects in our material ontology, it is impossible to construct semantic values that are guaranteed to get future extensions right. And in the general case, two expressions which differ in sense may be co-extensional relative to the current context (or even, co-extensional up to the current context) because just the same currently identifiable objects are among their extensions, and two different concepts might be objectually equivalent insofar as just the same currently identifiable objects fall under them. 1049 Thus both a semantic reduction of sense to extension and a metaphysical reduction of concepts to objects will deliver too coarse-grained results because currently there are not enough identifiable objects available to model all intuitive distinctions. The shortage of current identifiabilia that we have to acknowledge once we take the internal stance turns out to be beneficial. What initially might appear as a drawback for semantic theory harmonizes in fact with those Fregean intuitions about semantics and its background metaphysics that arose in a more general context, without a view to the semantics and metaphysics of time.

9.8 Irreducible sense in the semantics of singular terms What does the irreducibility of sense mean for the semantics of singular terms? This question is interesting not only in general, that is, from the viewpoint of the semantic theory of a language that is about a contextualist ontology, but also more specifically, because the answer is relevant for our project in the following chapters of giving the semantics of a self-referential language.

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That we are situated in a contextualist ontology does not entail that we ourselves are objects of that ontology, in the sense of belonging to the domain; it means merely that this ontology is given to us from the perspective of its contexts. Cf. section 8.7. Recall that we use the indexical expression ‘current’ to generalize the temporally indexical ‘at present’ to an arbitrary dimension of indexicality.

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I take the above argument for the irreducibility of sense to justify not an outright rejection but a modification, or more precisely, a restriction of the standard theories of intensional and two-dimensional semantics. To see why and how, let us start with a methodological observation. David Lewis explains the reductionist program of intensional semantics (and by analogy that of two-dimensional semantics) in the following way: “In order to say what a meaning is, we may first ask what a meaning does, and then find something that does that. A meaning for a sentence is something that determines the conditions under which the sentence is true or false. It determines the truth-value of the sentence in various possible states of affairs, at various places, for various speakers, and so on.” 1050

More generally, and in present terms, what Lewis would have to say is that the sense of an expression determines a function from contexts and circumstances to extensions. 1051 And given what he says in the first sentence, he would have to take this as justifying us in saying that sense is such a two-dimensional function on extensions. However, given our argument to the effect that sense is irreducible, we will have to deviate. Even when we follow Lewis’s methodology, our argument shows that meaning does more than determine the truth values of sentences and other extensions. At least from the point of view of the present moment (or more generally, from the point of view of the current context), it turns out that meaning, in addition to determining a present (or current) extension also guarantees to do this relative all other moments (or to all non-current contexts) – but in a way that at present (or currently) is inscrutable because the relevant objects are not yet identifiable. Thus, taking the internal stance forces us, if not to give up, to restrict the reduction of sense to two-dimensional intension: In general, we will have to take sense to be irreducible, but with respect to those contexts that have only identifiabilia that are among the present (or current) identifiabilia, we can still employ the standard reductive methods of intensional and two-dimensional semantics. So, relative to each context and circumstance the sense of an expression does determine a function on extensions, but sense cannot be fully reduced to any such function because the sense determines the function only with recourse to the objects identifiable relative to that context and circumstance. Thus we need to restrict intensional and two-dimensional semantics in the following way: Where we construed a function from a collection of parameters as a semantic value or as a constituent of a semantic value, we work with the irreducible sense instead; but as far as the theory of extensions is concerned, we can stick to the functions, at least when it suffices for our theoretical purposes to work with objects that are currently identifiable. (We need not change anything concerning the semantics of expressions that are directly referential, though. Here we can still say that the object referred to is a constituent of the semantic value of the expression. But as a future object is not yet there to be the constituent of the semantic value of a directly referential expression, there are 1050 1051

Lewis 1970, 22. Cf. Lycan 2000, 153. For the moment we are bracketing the issue of direct reference.

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consequences for the metaphysics of directly referential expressions. More on that later. 1052) This modification of semantic theory forces us to modify our accounts of the semantics of singular terms and predicates. 1053 Nothing changes for names: As a name is directly referential on the level of character, the named object is a direct constituent of its semantic value. But for predicates and descriptions the modification is drastic. We need to take the semantic value of a primitive predicate to be its sense, that is, as a concept understood as irreducible. This concept will determine a twodimensional intension relative to each context and circumstance, depending on the domain of identifiabilia of that context and circumstance. As descriptions are derived in a certain way from predicates, a similar change is needed in their semantics. So the semantic value of a typical description will not be a two-dimensional intension, but an irreducible sense derived from a singular concept that determines a corresponding two-dimensional intension, depending on which objects are currently identifiable. To give a schematic example: When we derive a description ‘the F’ from a primitive predicate ‘. . . is F’, the sense of ‘. . . is F’ will be a singular concept, 1054 i. e., a concept that is such that (at most) one object falls under it with regard to any context, and thus the sense and hence the semantic value of ‘the F’ will be a corresponding mode of presentation of that single object (where the object itself can vary from context to context). 1055 For a simple indexical, there will be a similar modification, because we have to acknowledge that its character is not reducible to a function from contexts to contents that incorporate its extension directly, but merely determines such a function with a range among currently identifiable objects. When we want to describe the behavior of certain particular expressions, we will in many cases be allowed to idealize, and assume that all relevant objects are identifiable, so that we can resort to the standard practice of doing intensional or two-dimensional semantics. But let us note two points that connect the present observation about irreducible sense to things we said before about semantics. When we used intensional and two-dimensional semantics to describe the semantics of singular terms, we introduced a pair notation to be able to model the difference between direct and indirect reference in the formal semantics. We used the abstract objects D for direct reference and I for indirect reference to mark the difference, assigning a semantic value (D, object) to a name and a semantic value (I, intension) to

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Cf. sections 9.10 and 9.11. Cf. section 5.2. Note that not all predicates and hence not all descriptions are entirely free of an objectual element. E. g., the complex predicate ‘. . . is president of the U.S. in 2016’ and the corresponding description ‘the president of the U.S. in 2016’ refer to the U.S., and their semantic values thus are not purely conceptual, but have an objectual constituent, too. (In fairly untypical cases, a description derived from a complex predicate can even have its extension as a constituent of its semantic value, e. g., ‘the person who is identical to Obama’ has Obama as a constituent of its semantic value – although not in the same immediate way as in the case of the name ‘Obama’.) We need to resist the temptation to actually identify the singular sense of ‘the F’ with the singular concept expressed by ‘. . . is F’, at least as long as we want to adhere to a Fregean theory of sense, because the latter sense is unsaturated, but the former is not.

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a description. 1056 While at the time the motivation of this pair notation might have seemed to be only technical, it now finds a more material warrant in our observation that the sense of an indirectly referring expression in the end of the day cannot be reduced to objects. When we depict the semantic value of a name in the form ‘(D, object)’ and the semantic value of a description in the form ‘(I, singular sense)’, we now make visible that there is a categorical difference between the meaning of a directly referential expression and the meaning of an indirectly referential expression. The second point concerns our use of senses in our formal ontology (e. g., when assigning the semantic value (I, singular sense) to a description). It is likely that this will strike many who have learned their trade in standard intensional and twodimensional semantics as an unprincipled way of doing formal semantics. For is not the realm of objects so much more exact and open to our grasp than the realm of senses? Well, it is not – our grasp of senses is just of a different kind than our grasp of objects. In our formal semantics, this difference is reflected in the interplay between a model theoretic semantics for objectual truths and an inferentialist semantics for conceptual truths. 1057 So, we need not worry that the modification of the semantics of singular terms that results from acknowledging the irreducibility of sense will complicate the actual semantic practice overmuch. But viewed from a theoretical viewpoint, the modification remains drastic. And we will see that it has important consequences for the metaphysics of the expressions involved, 1058 that will also bear on its application to Liar sentences.

9.9 ‘The first child born a hundred years hence’ There is a possible objection to the present account of the semantics of a language that is about a contextualist ontology, and it appears to be very intuitive (I have heard it from many people by now). It concerns our apparent ability in certain cases to talk about future objects. For consider a present utterance of the following sentence: (1) ‘Probably, the first human child who will be born a hundred years hence will live outside of Europe.’ Intuitively, this sentence is meaningful and its present utterances may well be true. But that seems to be precluded by the principle that we cannot refer to future objects. For whenever the sentence is uttered, the child in question is surely no more than a future object. So how can we refer to it? Are we confronted with a counterexample?

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Cf. section 4.6. In section 5.2, we extended this method to also distinguish between indexicals and names. But in the context of the present methodological remark we can ignore this more complicated account. Cf. section 9.6. Cf. section 9.10.

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We are not. To see why, we need only observe what happens when we construe such a future-oriented description (and more generally, any occurrence of a description with an orientation towards subsequent contexts) not as a singular term, but as an incomplete symbol which has no direct counterpart on the level of logical form. Then it becomes clear that the example sentence does not really say of some child that it falls under the concept of being born in the year 2115 and the concept of probably being born outside of Europe, for there is no future child among the current identifiabilia. Rather, its logical form will be something along the lines of the following regimented sentence: (2) ‘It will probably be the case one hundred years hence that (there is an x such that x is a human child born in the present year and for every y that is born in the present year but not later than x, x = y, and x is born outside of Europe).’ This paraphrase amounts to giving what might be called a Prior-Russellian analysis of the example sentence, i. e., something along the lines of a Russellian analysis of a sentence with a description in subject position, but one which is Priorean in so far as the objectual quantifiers of the Russellian analysis are in the scope of future tense operators. The Russellian part already shows that by using a definite description we are not talking about a specific object, and the Priorean part implies in the present framework that we are not even talking in an unspecific way about an identifiable object. This becomes clear when we contrast the paraphrase we just gave of the example sentence with the following: (3) ‘There is an x such that it will probably be the case one hundred years hence that (x is a human child born in the current year and for every y that is born in the present year but not later than x, x = y, and x is born outside of Europe).’ This statement, if true, would indeed be about a currently identifiable object. But it is clearly false, and it does not give the logical form of our example sentence (1). More generally, in a tense logical framework with a growing domain, a statement like (3) is not equivalent to a statement like (2), because a formal sentence like p[Fn] ∃x Φ(x)q does not entail the corresponding formal sentence p∃x [Fn] Φ(x)q. 1059 In sentence (2), which has the logical form p[F100 years] ∃x Φ(x)q, 1060 there is no objectual quantifier that ranges solely over currently identifiable objects. In fact the sentence makes an irreducibly general statement that despite some of its elements being objectual (e. g., the geographical region of Europe) also has some irreducibly conceptual elements (e. g., the notion of being a human child which cannot be cashed out in terms of the identifiable objects currently available). 1061

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In technical terms, this is connected to the fact that neither the Barcan formula nor its converse hold in our modalized tense logic, which we sketched in section 9.1. We can safely ignore here that ‘probably’ does not mean the same as ‘necessarily’. Cf. the remarks towards the end of section 9.6.

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The role of conceptual connections can be made more obvious by a change of our example that does away with the objectual element of sentence (1): (4) ‘The first human child born a hundred years hence will be a rational animal.’ The truth of (4) is all but entailed by the truth of the purely conceptual statement that every human is a rational animal. 1062 It is contingent only on the continued existence of the human species, and if we want to get rid of this remnant of uncertainty, we can conditionalize: (5) ‘If there are still humans born a hundred years hence, then the first human child born a hundred years hence will be a rational animal.’ Thus we have arrived a statement that, although it might still look as though it was about a particular future object, is fully general, and the truth of which is guaranteed by conceptual connections alone. 1063 Even though it is about a future to which we have scant epistemic access, we can know today that it is true, because meaning postulates are accessible at any time whatsoever. In the corresponding tense logic, the truth (relative to any context) of a formal sentence that gives the logical form of sentence (5) is derivable from the axiom that explicates the meaning postulate about humans being rational animals, together with the interaction principle that pΠξ Φ(ξ)q logically entails p[F100 years] ∀x Φ(x)q. 1064 For our rejection of the objection about future oriented descriptions like ‘the first human child born a hundred years hence’, it is important that we can explain our epistemic access to the truth of some sentences that contain such descriptions. Given the Prior-Russellian analysis we gave for the example sentence, our present access to purely conceptual truths explains why we have the impression that we can know a lot about future objects. We see that it is the addition of conceptual quantifiers to objectual quantifiers that range over a variable domain that allows us to justify certain inferential connections and thus to accommodate the intuitions behind such examples into our theory, which does not countenance any reference to future objects. But despite this advantage for the current discussion, in the end of the day we do not need to commit to analyze future-oriented descriptions along Russellian lines. We are not precluded from construing a future-oriented description in a Fregean way, as a singular term that cannot be further analyzed. 1065 But in that case it is crucial to remind ourselves that a Fregean description can be empty. And in the case at hand, we see from the theoretical possibility of giving a PriorRussellian analysis why ‘the first child born a hundred years hence’, construed as Fregean description, would have no extension. In cases like our example, even a Fregean description does not allow to refer to a future object. 1062 1063

1064 1065

Cf. section 9.6. Note that not all future-oriented descriptions are entirely free of an objectual element. E. g., ‘the first child to be born in London one hundred years hence’ contains a word that refers to London. Cf. section 9.6. Cf. section 5.2.

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9.10 Reference, grounding, dependence, and accessibility Let us now turn our attention from the objects that language is about, back again 1066 to the linguistic objects that are used to talk about them: the semantic expressions. In other words, let us return from the semantics of a language that is about a contextualist ontology to its metaphysics. Recall our distinction between semantic and syntactic expressions: 1067 In the terminology of the present chapter, we can roughly identify a semantic expression with the syntactic expression that is its basis taken together with its semantic value. Because of this role of semantic values in the constitution of semantic expressions, certain semantic restrictions will in some cases go hand in hand with corresponding metaphysical restrictions. What does the restriction of reference to contexts mean for a referring expression? We have seen in the temporal example that Caesar could not have been referred to before he existed. Hence, the semantic name ‘Caesar’ – the name understood as a semantic expression, and thus as an expression which does refer to Caesar – could not have existed before Caesar did. In this way, a fact about semantics (when some object can be referred to) is interconnected with a fact about metaphysics (when some other object can exist). Let us look at this in more detail. In some way, a semantic name seems to be grounded in the named object. For without recourse to the named object, we cannot fully explain what a certain name, understood as a semantic expression, is. 1068 But in this case, grounding is not accompanied by ontological dependence proper, for names usually continue to exist even when the named object has gone out of existence. The continued existence of the semantic name ‘Caesar’, up to the present day, is a case in point. More generally, the failure of ontological dependence between a name and its bearer is just the metaphysical counterpart of our semantic observation that we can talk about past objects. 1069 But there surely is remote ontological dependence: Whenever a name exists, it is or was the case that the named object exists. This is the metaphysical counterpart of the semantic observation that identifiability always starts from existence – that present identifiability entails present or past existence. The phenomenon of remote ontological dependence is easily overlooked because the syntactic basis of a name does not depend on the named object. Thus, the semantic name ‘Caesar’ remotely depends ontologically on Caesar, but the mere string of letters ‘Caesar’ has existed long before Caesar himself. The remote ontological dependence of a name on the named object can be captured by the following principle: 1070

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Cf. chapter 8. Cf. sections 8.1 and 8.2. The notions of grounding, ontological dependence, and remote ontological dependence have been explained in section 8.3. Cf. section 9.3. The (Principle of Non-Priority for Names) is actually the contrapositive of the claim that a name depends ontologically in a remote way on its bearer.

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(Principle of Non-Priority for Names) No name exists prior to its extension. I. e., if an object b is a name of an object a, then the object b does not exist prior to the object a. However, the matter is more general. This is best 1071 seen by a detour via grounding. When we introduced the notions of ontological dependence and grounding, 1072 we already considered endorsing a principle that we can now formulate in the following way: (Principle of Non-Priority for Grounded Objects) No object that is grounded in another exists prior to it. I. e., if an object a grounds object b, then the object b does not exist prior to the object a. Plausibly, this principle captures what is true of our original intuition that grounding entails ontological dependence proper, which we recognized as too strong. Therefore we shall endorse the (Principle of Non-Priority for Grounded Objects). And it can help us generalize the (Principle of Non-Priority for Names). For the observation behind that principle, according to which a name is grounded in the object it refers to, generalizes to all directly referential singular terms. The reason is that, as a semantic expression, a singular term incorporates its semantic value, and the semantic value of a singular term that is directly referential in turn incorporates its extension. (In contrast to what we said about names, we now need to be clear that this observation concerns the token level, because indexical singular terms, although directly referential like names, have a context-dependent extension, so that the extension of an indexical may vary from token to token. 1073) So, every particular token of a directly referential singular term is grounded in its extension. Together with the (Principle of Non-Priority for Grounded Objects), this result entails the following: (Principle of Non-Priority for Directly Referential Expressions) No token of a directly referential expression exists prior to its extension. I. e., if an expression e refers directly to an object a, then the expression e does not exist prior to the object a.

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We should note that the (Principle of Non-Priority for Directly Referential Expressions) follows already from our observation that we cannot refer to subsequent objects, or the contrapositive of that observation, the principle of non-priority for reference (cf. section 9.2), together with the fact that a directly referential expression has a constant reference and the fact that the dyadic predicate ‘x refers to y’ is existence-entailing in the first argument place. Cf. section 8.3. Cf. section 11.11. There we will see that to study the sentential indexical ‘this sentence’ we actually need to generalize, and talk about a particularized expression, as an umbrella term for both tokens and occurrences of expressions. But at this point this is not yet important.

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This principle covers both names and indexicals. 1074 It entails the corresponding principle for names because every token of a name has the same extension. 1075 As names are the paradigm example of directly referential expressions, it is no surprise that the principle of non-priority for names generalizes to all directly referential expressions. It is important to note, because it will bear on our investigation of self-referential languages and of Liar sentences in particular, that the situation is entirely different for expressions that are indirectly referential. This is a consequence of the irreducibility of sense and the ensuing modified semantics. 1076 For a primitive predicate or for a typical description, it is not the case that the extension is a constituent of the semantic value – not even in a mediated way (as it would be if the semantic value was an intension, which is a function from contexts to extensions and thus a set theoretical construction that involves the actual extension, among other objects). This can be illustrated in terms of the pair notation that models the semantic value of a name as a pair (D, object) and the semantic value of a typical description as a pair (I, singular sense): A name understood as a semantic expression will thus be constituted from its syntactic basis taken together with the pair (D, the named object), whereas a description understood as a semantic expression will be constituted by its syntactic basis taken together with the pair (I, a certain singular sense that describes the object). Thus neither a primitive predicate nor a typical description is grounded in its extension, neither a primitive predicate nor a typical description is remotely ontologically dependent on its extension (not even on the level of concrete tokens), and finally, we do not have any reason to endorse a principle of non-priority that restricts the existence of primitive predicates or typical descriptions. 1077 An example: Obama is a constituent of the semantic name ‘Obama’, but not of the description ‘the president of the U.S. in 2016’. Therefore there is no reason to think that the said description is grounded in Obama, or that it depends remotely on him. In fact, there is no problem about taking the description to predate its extension – but we would have to take it as having been empty at the time. Mark Twain, e. g., may well have uttered a meaningful sentence containing the description ‘the president of the U.S. in 2016’ in 1889, but it would not have referred to anyone. Which brings us to an important semantic consequence of the metaphysical differences between names and descriptions: On the present account, names need to have an extension but descriptions can be empty. 1078 As a description is grounded only in its sense (besides its syntactic basis), it can exist without having an extension. But as a name is grounded in the named object (in addition to its basis), there can be no name that 1074

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It also covers quotation expressions, which are directly referential, too (cf. section 5.5). But these are not in our focus here because they cannot be a device of self-reference (cf. section 5.6). Let us bracket the issue of ambiguous names. Cf. section 9.8. To see why these claims need to be restricted to primitive predicates and typical descriptions, consider the non-primitive predicate ‘. . . is identical to Obama’ and the corresponding untypical description ‘the person who is identical to Obama’. Because of the occurrence of the name ‘Obama’, they are both grounded in Obama, depend ontologically on him, and could not have existed before 1961. This fact will be highly relevant to our investigation of Liar sentences in chapters 11 and 12; in fact it is an important reason for splitting up that investigation into two chapters.

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has no extension. (Of course there can be a syntactic expression that looks like a name, but it will not succeed in realizing a semantic name as long as the object in question does not exist. 1079) * Returning to directly referential expressions and the principle of non-priority that holds for them, let us now look at what this principle means for contexts in general. So far we have described the metaphysics of directly referential expressions within the paradigm of time, but we want to generalize what we have found to an abstract notion of contexts that includes those contexts that are relevant to the existence of expressions. What is clear from the temporal case is that, in a setting where existence is relative to contexts, the order of contexts plays a crucial role. The semantic name ‘Caesar’ could not have existed before Caesar did. What does this mean for the ordering of contexts in general? In the case of time, the ordering relation among the contexts (in this case, among the moments) is, roughly, that of being later than. 1080 The relation of being later than (or more accurately, the relation of accessibility of our tense logic) is irreflexive, asymmetrical, and transitive: 1081 (Irreflexivity of Being Later) No moment is later than itself. (Asymmetry of Being Later) If a moment m2 is later than a moment m1, then it is not the case that the moment m1 is later than the moment m2. (Transitivity of Being Later) If a moment m2 is later than a moment m1 and a moment m3 is later than the moment m2, then the moment m3 is later than the moment m1. And the same goes, mutatis mutandis, for the converse relation of being earlier than. Do the second level properties of irreflexivity, asymmetry, and transitivity carry over to the accessibility relation that orders the contexts relevant to the existence of expressions? The answer is ‘yes’ – the reason being that we want the above principles of non-priority to carry over to the more general setting, because we do not want to lose again what we have just learned from the temporal paradigm about grounding and direct reference. So let us generalize the notion of being a later (earlier) moment in time to the notion of being a subsequent (prior) context. We stipulate that the relation of subsequence is irreflexive, asymmetric, and transitive:

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There is of course the problem of how to account for fictional names and apparent names of mythical creatures. But we need not get into that here, because if there actually were self-referential sentences, then they would refer to themselves in a non-fictional, non-mythical way. I say ‘roughly’ because we are really looking at the relation of accessibility of a tempo-modal structure of branching time. Irreflexivity is entailed by asymmetry – but we list it here, anyway.

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(Irreflexivity of Subsequence) No context is subsequent to itself. (Asymmetry of Subsequence) If a context y is subsequent to a context x, then the context x is not subsequent to the context y. (Transitivity of Subsequence) If a context y is subsequent to a context x and a context z is subsequent to the context y, then the context z is subsequent to the context x. And the same goes, mutatis mutandis, for the converse relation of priority. As the relation of subsequence is meant to generalize the temporal relation of being later than, we need to think about more structural properties than just irreflexivity, asymmetry, and transitivity. We will assume (without going into the matter now) that like time, subsequence exhibits the structural property of connectedness, but we will make no requirements about branching. 1082 This will allow to capture the fact that a language can be extended in different ways; e. g., different new names can be added to it in different ways. * Note that the relation of being later than (being earlier than) holds among moments in time and the relation of subsequence (priority) holds among contexts in general – but not among the objects of the temporal ontology or our contextualist ontology. However, derived notions can be defined in a natural way. In the temporal case, we typically will want to say that an object b is later than (earlier than) an object a if and only if the object b starts to exist later than (earlier than) the object a starts to exist. According to this simple criterion, Cleopatra is later 1083 than Caesar, simply because she was born later. Let us give a graphic illustration of one object being later than another: 1084

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Cf. section 9.1. Note that we thus we allow branching not only towards subsequent contexts, which corresponds to the openness of the future, but also towards prior contexts. This would correspond to something like the openness of the past, which makes no sense for time, but does under certain circumstances for our generalized contexts: When contexts are individuated mainly by what exists relative to them, there may be advantages in allowing alternative prior histories of one and the same context, corresponding to different procedures for introducing some of the objects that exist relative to that context. It would sound less harsh to say that Cleopatra is younger than Caesar, but the tenseless ‘later’ is more appropriate than the phrase ‘younger’, which derives from the tensed notion of being of a certain age, because we here describe a temporal or contextualist ontology from an external perspective, taking the tenseless (or context-free) God’s eye view. The structure given by the relation of being later on moments (or the relation of subsequence on contexts) need not be linear; it can be tree-shaped. The diagram, however, depicts either a linear structure or only one branch of a tree structure. We will sometimes talk as if being later (or subsequence) was a linear order and ignore branching. For most of our purposes, this is a harmless idealization.

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object a exists:

*

*

*

*

object b exists: moments:

t1

t2

t3

t4

t5

*

*

*

*

*

*

*

*

t6

t7

t8

t9

*

... ...

t10

...

direction of being later →

We see here that one object being later than another does not rule out that they co-exist relative to many moments, as witnessed by Cleopatra and Caesar. 1085 We also see that an object can be later than another and yet cease to exist prior to it, as witnessed by Wittgenstein and Russell. (Another thing that is exemplified here is that we do not expect any existence gaps for objects that exist relative to moments in time. We will come back to that shortly.) But note that the simple criterion according to which an object is later than another just in case it starts to exist later covers only those scenarios where for each object there is a first moment of its existence, and that is not something we can presuppose in full generality. Before we differentiate, it will be helpful to define some notions, both for moments in time and for contexts in general: (Def. Beginning, First Moment, First Context) A moment t is a beginning of an object a if and only if the moment t is a minimal moment of the existence of the object a in terms of being later than, i. e., if and only if the object a exists relative to the moment t and the moment t is later than no other moment relative to which the object a exists. A context c is called a beginning of an object a if and only if the context c is a minimal context of the existence of the object a in terms of subsequence, i. e., if and only if the object a exists relative to the context c and the context c is subsequent to no other context relative to which the object a exists. A moment t is the first moment of an object a if and only if the moment t is the earliest moment (i. e., a unique minimal moment) of the existence of the object a, i. e., if and only if the object a exists relative to the moment t and every moment relative to which the object a exists is identical to or later than the moment t. A context c is the first context of an object a if and only if the context c is the least context in terms of subsequence (i. e., a unique minimal context) of the existence of the object a, i. e., if and only if the object a exists relative to the context c and every context relative to which the object a exists is identical or subsequent to the context c. Thus the first moment or context of an object is its unique beginning. 1086 A word on when and why we need these notions. We need to differentiate between starting in a beginningless way and having a beginning as long as we do not presuppose that the moments or contexts form a well-founded order, where every 1085 1086

This will also be the case in our own application of this notion to expressions and their extension. Talk of the first moment or context of the existence of an object does not entail that there is a second moment or context of its existence in the normal sense of the word, both because the ordering of moments or contexts need not be linear and because the ordering of contexts need not be discrete.

9.10 Reference, grounding, dependence, and accessibility

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collection of moments or contexts has a beginning. We need to differentiate between a beginning and a first moment or first context as long as we do not presuppose that moments or contexts form a linear order, where these notions fall together. And as long we do not make these presuppositions (and even later we will make only one of them), we also need to work with a more general definition of what it means for one object to be later than another. So let us define more generally that an object b is later than an object a if and only if every moment mb relative to which the object b exists is later than some moment ma relative to which the object a exists but the object b does not exist. And we will say that an object a is earlier than an object b just in case the object b is later than the object a. 1087 In a scenario where for each object there is a unique first moment of its existence, this is equivalent to our original simple criterion according to which an object is later (or earlier) than another just in case their respective first moments of existence are. But in a scenario where not every object has a beginning, we can still work with the more general notion. This matter will recur after we have generalized from moments in time to contexts in general. In the case of contexts in general, we will work with the following definition: (Definition of Subsequence Among Objects) An object b is subsequent to an object a if and only if every context cb relative to which the object b exists is subsequent to some context ca relative to which the object a exists but the object b does not exist. An object a is prior to an object b just in case the object b is subsequent to the object a. Some remarks to justify this way of transferring the notion of subsequence from contexts to objects. The definition entails that no object is subsequent to itself, because otherwise there would have to be a context relative to which it exists and does not exist. Hence subsequence among objects is irreflexive. Further structural features carry over from the relation of subsequence among contexts, most notably asymmetry and transitivity – at least in the presence of the highly plausible further assumption that there are no existence gaps: (No Existence Gap Principle) An object that exists relative to two contexts exists relative to every context in between. I. e., if an object exists relative to a context c1 and relative to a distinct context c3 and there is a context c2 such that the context c3 is subsequent to the context c2 and the context c2 is subsequent to the context c1, then the object exists relative to the context c2. 1087

It need not be the case in a tensed ontology that for each object there is a first moment or even a beginning of its existence. To see what this means for our criterion for the relation of being later among objects, let us assume that time has the same structure as the real numbers, so that we can use certain sets of real numbers to model temporal intervals. Then we might have an object a that exists in the temporal interval {t | 0 ≤ t ≤ 1}, or [0, 1], and an object b that exists in the temporal interval {t | 0 < t ≤ 1}, or ]0, 1]. In this scenario, the object b is subsequent to the object a according to the more general definition, but the more specific criterion does not apply, because there is no first moment of the existence of the object a.

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We will endorse this assumption. It is already highly plausible because an object taking a brief break from existing is not something that has ever been reported (or could be with any credibility), and even more so because we should understand this assumption not (or not primarily) as making any demand on what exists, but (primarily) on how objects are individuated. For note that a state of affairs that might be described as one object exhibiting an existence gap is equally well – or more credibly – described as a state of affairs where two objects that are similar in some respect exist relative to disjoint consecutive temporal intervals. Now, because there are no existence gaps, the asymmetry of subsequence among contexts is transferred via our definition to the relation of subsequence among objects, and similarly for transitivity and other structural features. 1088 And because of the shared structural features of this derived relation, we are warranted in again talking of subsequence. Our (Definition of Subsequence Among Objects) is phrased in a way that is general enough to cover cases, too, where an object comes into existence without there being a first context or even a beginning of its existence. This degree of generality is desirable; but let us also have a look at an important special case, which is given in scenarios where the following principle holds:

1088

To see how we need to reason here, let us have a detailed look at how, when subsequence among contexts is asymmetric, the (No Existence Gap Principle) entails asymmetry among objects. We show the contrapositive, that without the asymmetry among objects the (No Existence Gap Principle) does not hold: Assume that there are objects a and b such that (i) the object a is subsequent to the object b and (ii) vice versa. Now let c3 be a context relative to which the object a exists. (Although our framework strictly speaking allows for objects that exist relative to no context at all, we will not consider these here.) Because of (i) and the (Definition of Subsequence Among Objects), the context c3 is subsequent to a context c2 relative to which the object b exists but the object a does not exist. Now similarly, because of (ii) and the definition, the context c2 is subsequent to a context c1 relative to which the object a exists but the object b does not exist. In sum, the object a exists relative to the context c1, does not exist relative to the context c2, and exists relative to the context c3. As subsequence among contexts is asymmetric, c1 6= c3. Thus we would have two distinct contexts relative to which the object a exists with another context in between relative to which the object a does not exist, contradicting the (No Existence Gap Principle). The case of transitivity is similar. To see that the (No Existence Gap Principle) is indeed needed to ensure that asymmetry and transitivity are transmitted from contexts to objects, consider the following scenario where it does not hold: There are two objects which exist alternately for all of (beginningless) time; call them Chicken and Egg. According to the (Definition of Subsequence Among Objects), Chicken is subsequent to Egg and vice versa, which is a counterexample to the asymmetry of subsequence among objects, and because of the irreflexivity of subsequence among objects also a counterexample to the transitivity of subsequence among objects. And note how the (No Existence Gap Principle) precludes the existence both of Chicken and of Egg – plausibly dividing them up into infinitely many chickens and infinitely many eggs, each existing in a gapless way, and all of them arranged in an infinite alternating sequence.

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(First Context Principle) For every object, there is a first context of its existence. I. e., for every object a there is a context c1 such that the object a exists relative to the context c1 and every other context relative to which the object a exists is subsequent to the context c1. 1089 In contrast to the (No Existence Gap Principle), the (First Context Principle) is not something which we endorse in general. For who knows whether there was a first moment of the existence of the cosmos? And many other physical objects might similarly sneak into existence in a beginningless way rather than start abruptly. But for some of the objects we will be concerned with in the following – most notably, for syntactic and semantic expressions – we will in some places assume the (First Context Principle). But then we will assume it not because we want it to carry any great argumentative weight, but rather because it will enable us to complement important principles with simpler formulations. 1090 One such a simplification becomes already evident when we assume the (First Context Principle) for the scenarios to which we apply our (Definition of Subsequence Among Objects). For in a scenario where for each object there is a first context of its existence (‘least’ in terms of subsequence), 1091 our definition entails that the following simple criterion holds, which should look familiar because it brings us back to our first idea about transferring the relation of being later than from moments to objects that exist in time: (Criterion of Subsequence Among Objects) An object b is subsequent (prior) to an object a if and only if the first context relative to which the object b exists is subsequent (prior) to the first context relative to which the object a exists. When we later apply the notions that we have introduced here, conditions that are formulated in terms of first contexts will sometimes suffice. * 1089

1090

1091

By requiring a unique beginning of existence, we are actually conflating two features which on another occasion might also be treated separately: The feature that existence always starts from a definite context (‘beginning’) and the feature that there is only one possible way in which each object can start to exist (‘unique’) – a variant of the necessity of origin. While the first feature is more on the side of topology (think open versus closed temporal intervals), the second feature first and foremost concerns the individuation of objects. But we need not go into this here as we will not in general endorse the (First Context Principle). With a view to Yablo’s paradox, we will in the end argue for the well-foundedness of the relation of subsequence among those contexts that are relevant for the existence of semantic expressions (cf. section 11.13). This well-foundedness of subsequence among semiotic contexts will entail a weak variant of the (First Context Principle), namely that the collection of contexts relative to which a given object exists always contains a minimal context, although perhaps not a unique one (well-foundedness alone does not entail the necessity of origin). What we said in a previous footnote about the first moment of the existence of an object applies here (mutatis mutandis) to the first context of the existence of an object.

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In terms of the derived notion of being later than for objects and of the derived notion of subsequence for objects, we can rephrase the (Principle of Non-Priority for Directly Referential Expressions) thus: No directly referential expression is earlier than its extension, and more generally, no directly referential expression is prior to its extension. 1092 It may be helpful for the following to contrast this principle of non-priority, which we have already found reason to endorse, with a corresponding principle that we will later argue for, 1093 the (Principle of Subsequence for Directly Referential Expressions): 1094 (Principle of Non-Priority for Directly Referential Expressions) No directly referential expression is prior to its extension. I. e., if an expression e refers directly to an object a, then the expression e is not prior to the object a. (Principle of Subsequence for Directly Referential Expressions) Every directly referential expression is subsequent to its extension. I. e., if an expression e refers directly to an object a, then the expression e is subsequent to the object a. Because of the asymmetry of subsequence among objects, being subsequent precludes being prior. Therefore the (Principle of Subsequence for Directly Referential Expressions) entails the (Principle of Non-Priority for Directly Referential Expressions). But it is properly stronger. For consider a scenario where every directly referential expression exists relative to the very same contexts as its extension. There, the (Principle of Non-Priority for Directly Referential Expressions) would hold, but not the (Principle of Subsequence for Directly Referential Expressions). And we should be clear that the reasons we gave for the (Principle of Non-Priority for Directly Referential Expressions) – a generalization of the observation that we cannot refer to future objects – do not in themselves constitute sufficient reasons to adopt the (Principle of Subsequence for Directly Referential Expressions). 1095

1092

1093 1094

1095

And we can rephrase the (Principle of Non-Priority for Grounded Objects) thus: No object that grounds another can be prior to it. Cf. chapter 11. For directly referential indexicals, both principles apply not to the respective semantic expression as such, but to the corresponding particularized expressions, i. e., to its particular tokens or occurrences. But this detail need not concern us yet (cf. section 11.11). We have already encountered a related difference when we were concerned with reference to past and present objects in sections 9.3 through 9.12. Then we saw that identifiability presupposes simultaneous or prior existence – but that it is a further question whether existence entails identifiability.

9.11 The flexibility of language

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9.11 The flexibility of language We come to a final feature that, if not indispensable, is highly desirable for any language that is about a contextualist ontology. What we have described so far in this chapter can be subsumed as the requirement that a language about a contextualist ontology needs to be semantically flexible: The extension of some of its expressions will be context-relative so that the truth values of its sentences will in general also be context-relative, and even the objects that are candidates for being in the extension of its expressions will vary from context to context. 1096 What we now require is that a language that is about a contextualist ontology also needs to be metaphysically flexible. The point is this. In view of the fact that a semantic name cannot exist prior to the named object, a language that is about a contextualist ontology should be able to survive the addition of (at least some) new names. For otherwise only those objects would be nameable that exist relative to all contexts where the language exists! Natural languages are of course flexible in this way. It would be ludicrous to hold that every time a baby is baptized or some other thing is given a name, we leave one language behind and move on to the next. Rather, we conceive of our language as remaining the same throughout its growth. By contrast, formal languages as usually conceived do not have this flexibility (even formal languages of tense logic, which exhibit semantic flexibility, are not flexible in this metaphysical way). They are brittle in the sense that the addition of a single new expression (e. g., a new constant) leads from one language to another, distinct language. In a logic textbook, we would read something like ‘let us extend our language L1 to a language L2 by adding the constant ‘a0’’. 1097 In temporal terms, a language that is about a changing world must itself be capable not only of semantic change, but also of metaphysical change, for it needs to be able to grow in the sense of acquiring new parts or constituents. The requirement is perfectly general, at least as far as names are concerned: 1098 A language that is about a contextualist ontology should itself constitute a contextualist ontology.

1096 1097

1098

Cf. sections 9.1 through 9.12. How is the requirement of metaphysical flexibility for a formal language to be spelled out in its formal meta-theory? On one possible construal, what we intuitively understand as a growing formal language is explicated as a sequence of distinct languages, each one a formal language in the standard sense, and each extending its predecessor in the sequence. It is likely that there will be technical challenges. Note for instance that we will not be able to separate as clearly between syntax and semantics as we are accustomed to, for under this construal not only the facts about semantics, but also the facts about syntax will vary from context to context. And anyway this construal moves us too far from our original intuition that it is one and the same language that goes through changes. Another way to go would be to allow that certain terms have no extension in relation to some contexts, with the intention of modeling not that the corresponding semantic expression does not refer to anything, but of modeling that the corresponding semantic expression does not exist. This is the route we will explore in chapter 10. In contrast, it is much more plausible that a language that is about a contextualist ontology has a fixed stock of primitive predicates.

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9.12 Metaphysico-Semantics Let us rest briefly and look back at the path that our arguments have taken in the present chapter. We have seen that a language that is about a contextualist ontology needs to be context-sensitive, and that there are semantic restrictions on which objects can be referred to relative to a context, which in turn lead to metaphysical restrictions on which semantic expressions can exist relative to a context. We have looked at purely conceptual truths and we have found an irreducible conceptual element in some statements, like those about future objects. On the basis of a contextualist metaphysics with its growing universe of identifiable objects, we have argued for a semantic conclusion, the irreducibility of sense, which entails a metaphysical conclusion, the irreducibility of concepts, which again has a semantic consequence, that descriptions can be empty. All in all, our path has circled back and forth several times between metaphysics and semantics. There is a methodological moral here: Far from being clearly separated, 1099 metaphysics and semantics are inextricably interconnected. Within our own Fregean framework with its categorical distinction between objects and concepts, we can correspondingly make out two main connections. The main connection regarding objects stems from the observation that it is not possible to refer to subsequent objects. 1100 This can either be seen as a restriction on semantics (about which objects are in the variable domain of which context) or as a restriction on metaphysics (about which objects of reference and which semantic expressions can exist in which contextual order). The main connection regarding concepts stems from the fact that a concept is the sense of a predicate, but on a realist picture, a concept is also what an object falls under relative to a context when the corresponding state of affairs obtains. Thus semantics and metaphysics overlap (to speak figuratively) in the notion of concepts. The inseparability of metaphysics and semantics that we have noticed here will remain with us throughout the rest of the study.

1099

1100

Kit Fine likes to say that “metaphysics is metaphysics, and semantics is semantics” (June 2016, in conversation), presumably meaning that there is no noticeable overlap between these two subdisciplines of philosophy, and that we can do our work in one without bothering about the other. This conviction is probably quite common, but it stands in stark contrast to what we have found to be the case, that metaphysics and semantics are inextricably linked. I conjecture that we can reconcile this apparent tension when we apply the distinction between the formal and the semantic level of section 4.3 in the following way: Fine speaks about metaphysics in the material understanding and about semantics in the formal understanding, and these are indeed located in two disjoint realms, but not because of any great gulf that separates metaphysics and semantics, but because of the crucial difference between the material level and the formal level of our understanding of reality. When we, however, observe a close link between metaphysics and semantics, both are understood in the material way. The same will of course hold to the slight strengthening of this requirement that we will argue for in chapters 11 and 12.

Chapter 10

Changeable Truth in Language and World

Alfred Tarski: “‘It is snowing’ is a true sentence if and only if it is snowing.” 1101

We have learned a lot in the preceding chapter about what is characteristic of a language that is a about a changing world – i. e., we have characterized a contextsensitive language that describes a contextualist metaphysics. But we encountered one problem that we were unable to solve with the tools at hand: We found no way of representing a scenario where there is existence without identifiability. In a nutshell, the difficulty was that when we say of something that is exists but is not identifiable, we thereby identify it. 1102 But if we cannot say this, how can we represent the corresponding state of affairs? In the present chapter, we will use the tools of formal semantics to deepen our understanding of what it means to model both a contextualist metaphysics and a context-sensitive language that is about it. Once we have adapted the tools of intensional semantics and two-dimensional semantics for this specific, dual purpose, we will be able to solve the problem of existence without identifiability. But that is not our only aim here. Along the way, we will describe the behavior of truth and falsity in a way that allows to transfer Tarski’s truth schema into a contextualist setting and to ask whether it holds. In the theory of truth and falsity that we will endorse, we will take into account both that the metaphysics of expressions is itself contextualist and that there are certain restrictions on reference in a context-sensitive language that is about a contextualist metaphysics. The generalized principles about truth and falsity that fall out of this will not in all cases make the same predictions as the contextualist counterpart of Tarski’s truth schema. They will of course be highly relevant for our study of the Liar paradox in the following chapters. Chapter 10 will be more technical in character than the other chapters of this third part of the study. (For the more impatient reader who is reluctant to work through all the details we have included summaries of the most important points in later chapters; it is therefore possible to treat the present chapter as one would a thematic digression or a technical appendix.) What we will do here is to take one or two of the several threads from the preceding chapter and put them under a microscope (which is provided by theories of formal semantics that were sketched in chapter 4 and section 5.2). In chapter 9, we noted the following characteristics of

1101

1102

Tarski 1956[1935], 156. Observe that charity requires to read both ‘is snowing’ and ‘is a true sentence’ in a tensed way here. This may have been an oversight on Tarski’s part. Later, he switched to more clearly tenseless sentences like the famous ‘Snow is white’ when giving instances of his truth schema (Tarski 1944, 343). Cf. section 9.4.

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a context-sensitive language that is about a contextualist metaphysics: contextualist restrictions on reference, a distinction between objectual and conceptual quantification, the irreducibility of sense, and the flexibility of language. Here we will put the conceptual aspects to the side and concentrate on the objectual aspects, which are the contextualist restrictions on reference to the objects of a contextualist ontology and the desirable flexibility of a language construed as a system of objects that belongs to a contextualist ontology. And although we will stop short of presenting a full formal theory that is rigorous in all details, we will hopefully say enough to make plausible that not only these but all the threads from chapter 9 can be explicated adequately, and within a single theory. We will proceed as follows. As a start, we will develop a quite general variant of a formal framework for representing both language and world (in sections 10.1 through 10.5). The general variant of the framework will already be sensitive to the context-relative existence of expressions, but will still be silent about the difference between existence and identifiability. Then, we will incorporate this difference into the framework by characterizing the connections of existence and identifiability to truth and falsity (in section 10.6 through 10.10). In particular, we will derive generalized principles about truth and falsity and discuss them, especially in contrast to the contextualist counterpart of the Tarskian truth schema (in section 10.9) and with a view to the possibility of truth value gaps (in section 10.10). We will conclude the chapter with our solution to the problem of existence and identifiability (in sections 10.11 and 10.12).

10.1 A dual framework When we tried to solve the problem of existence without identifiability in the previous chapter, this is how far we got: We had the idea to introduce both a predicate for existence, ‘E!(. . . )’, and a predicate for identifiability, ‘I!(. . .)’, into our formal language. Thus we would be able to formalize the sentence ‘Object a exists but is not identifiable’, as ‘E!(a) ∧ ¬I!(a)’. But we also had reason to lay down that in the intensional semantics of this formal language, the variable domain of a context is made up exactly by those objects that are identifiable relative to that context. Thus the aforementioned formal sentence could not be true with regard to any context, and we would be barred from representing any state of affairs where an object exists but is not identifiable relative to a certain context. 1103 Fortunately, our idea for a solution of the problem of accounting for existence without identifiability can be made to work if we add two-dimensional semantics to our toolbox. More precisely, we will be able to solve it eventually, when we know how to work with both intensional semantics and two-dimensional semantics. To see why we should do this – and not only in order to solve this specific problem, but in general –, note that what is characteristic of both semantic theories, namely that an expression has its extension only in relation to a contextual parameter,

1103

Cf. section 9.4.

10.1 A dual framework

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and in particular that a sentence is true or false only in relation to that contextual parameter, can actually serve two entirely different functions when it comes to a material understanding 1104 of the formal semantics: Firstly, it allows to describe a changing world from within, or more generally, it can model how the features of objects vary from context to context in a contextualist metaphysics. Secondly, it allows to describe a changing language, or more generally, it can model how the features of expressions vary from context to context when these belong to a contextsensitive language. E. g., when our formal semantics says that ‘Obama is president of the U.S.’ is true in relation to the year 2016, we can understand this either as modeling that the tensed state of affairs that Obama is president of the U.S. obtains relative to the year 2016, or that the tensed sentence ‘Obama is president of the U.S.’ is true relative to the year 2016. It is not only in this special case that we need to distinguish clearly between tensed facts and tensed states of affairs on the one side and tensed sentences on the other side. Generalizing from moments in time to all kinds of contexts, what we see here is the distinction between contextual states of affairs and context-sensitive sentences, i. e., between a contextualist metaphysics and a context-sensitive language. There are several factors that can explain why the corresponding important distinction between the two possible functions of context-relative formal semantics is often overlooked. One factor is that not everyone agrees that metaphysics is contextualist and that we accordingly should take the internal stance on reality (think of adherents of the tenseless theory of time). And if your metaphysics is relational and thus context-free, you will not want (or need) a context-relative semantics to model it from within. Another factor is that on the basis of a contextualist metaphysics it is at least prima facie highly plausible that what we will call the Tarskian equivalence holds globally, i. e., that a context-sensitive sentence is true relative to just the same contexts relative to which the corresponding contextual state of affairs obtains. 1105 And as it is initially so plausible that what is true and what obtains is extensionally equivalent, it is understandable if that distinction is sometimes blurred. For us, however, there are compelling reasons to be very clear about the distinction between the function of modeling a contextualist metaphysics and the function of modeling a context-sensitive language. To start with, the question does arise for us, because we both endorse a contextualist metaphysics of linguistic expressions and want to describe a context-sensitive language that allows to talk about these expressions. 1106 Then, we are interested in the possibility of self-reference and that means that for us the context-sensitive language may itself be part of the

1104 1105

Cf. the distinction between the material and the formal level in doing semantics made in section 4.3. In our example we want to say something like the following: The sentence ‘Obama is president of the U.S.’ is true relative to a moment in time if and only if Obama is president of the U.S. relative to that moment.

1106

Actually, once we will have developed the technical apparatus of this chapter, we will phrase the Tarskian equivalence in a different way. But that is not important here. Cf. section 10.4 and 10.9. Cf. chapter 8 and in particular section 8.7.

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contextualist metaphysics – so that we have all the more reason never to lose sight of the conceptual distinction between being the case and being true. Most importantly, we will want to allow for failures of the Tarskian equivalence – situations where something is the case without the corresponding sentence being true – so that being the case and being true would not even be guaranteed to be extensionally equivalent. For these reasons let us decide upon the following: We will use intensional semantics to model our contextualist metaphysics and we will use two-dimensional semantics to model the behavior of the context-sensitive language we are interested in. Here is an overview: formal level:

material level:

intensional semantics

two-dimensional semantics

models

models

contextualist metaphysics

context-sensitive language

But before we look at this dual framework in detail, a brief recapitulation is in order. Recall from our survey of semantic theories 1107 that while in intensional semantics, i. e., the standard possible worlds semantics of modal logic, an expression has its extension in relation to (the points of) one dimension, namely with regard to a circumstance of evaluation, in two-dimensional semantics an expression has its extension in relation to two dimensions, namely with respect to a circumstance of evaluation and at a context of use. 1108 In the standard possible worlds scenarios, a circumstance is one of the possible worlds known from (one-dimensional) modal logic, and a context is a pair of a possible world and a position of use in that world (usually a speaker, a moment in time, and a spatial location). But positions of use will not matter for our purposes. Therefore what we can use for both dimensions are contexts in our general sense that encompasses not only possible worlds and moments in time, but also the semiotic contexts relevant to the existence of expressions. 1109 But we will of course have to distinguish clearly between a context understood as a circumstance of evaluation and a context understood as a context of use. Recall also that we have adopted a terminological convention that has two different prepositions for the two dimensions of two-dimensional semantics, ‘with respect to’ in connection with circumstances of evaluation and ‘at’ in connection with contexts of use. (They will never occur separately, but always together in a locution of the form ‘with respect to the context c and at the context d’.) Note that in connection with the circumstances of evaluation of the intensional semantics, we use a third preposition, ‘with regard to’. The similarity of the prepositions ‘with respect to’ and ‘with regard to’ is meant to convey that despite the different functions of 1107 1108

1109

Cf. sections 4.6 and 4.7. When we introduced two-dimensional semantics, we distinguished the matrix variant, which assigns an extension in relation to a pair of a context of use and a circumstance of evaluation, and a two-stage variant, where first a context determines the content of an expression which then in conjunction with a circumstance determines the extension of that expression (subsections 4.7.2 and 4.7.3). For most purposes of the present chapter, the less complex matrix variant will suffice. Cf. section 8.7.

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the two semantic theories, there is a close connection between the single dimension of intensional semantics and the first dimension of two-dimensional semantics. This will get clearer once we give explicit criteria for truth 1110 in the two semantic theories. On a methodological note, we should say something to justify the copious use we will make of intensional semantics and two-dimensional semantics, in the face of the anti-reductionist critique we have subjected these theories of formal semantics to earlier. 1111 There are two points to be made. Firstly, even though one of our main objectives in this chapter will be to model the internal stance, we will take the external stance in the formal meta-theory that we use to do this. But our antireductionist critique of the semantic theories was formulated from the internal stance that we want to take in the end of the day. Secondly, all we need in this chapter is the theory of extensions for the language in question, which describes the behavior of its expressions in terms of reference, satisfaction, truth, and falsity. But our antireductionist critique of the semantic theories concerned the question whether the theoretical entities that can be postulated within these theories – intensions and two-dimensional profiles – are adequate to explicate the notion of the sense of an expression and in particular to explicate the concepts that are the senses of predicates. Within the scope of this chapter, we can bracket the question of how sense and concepts are to be represented in the formal theory, because we are focusing on the objectual aspects of both the context-sensitive language and the contextualist metaphysics that we want to model.

10.2 The role of the intensional semantics Both the intensional semantics and the two-dimensional semantics are formulated in a context-free meta-language. In a way, this context-free meta-language already allows to describe what is the case with the world that the context-sensitive object language is about. With regard to the paradigmatic special case of temporal reality, where we can use a tenseless meta-language to give the semantics of a tensed object language, the corresponding fact is that the tenseless meta-language already allows to describe temporal reality. Where we say in the intensional semantics that the tensed sentence ‘Obama is president of the U.S.’ is true with regard to the year 2016, we can already say in the tenseless fragment of English that Obama is president of the U.S. relative to the year 2016. Why is the tenseless representation not enough? Why do we need a tensed representation, and more generally, why do we need a contextsensitive representation, when we can already represent every state of affairs in a context-free way? These questions grow even more pressing because in the case at hand, the question of the coherence of existence without identifiability, the problem 1110

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We will also give criteria for falsity. In a fuller and more systematic account, we would start with criteria for reference and satisfaction and derive the criteria for truth and falsity from them, in accordance with the principles that a monadic sentence is true (false) if and only if the object referred to by the singular term satisfies (does not satisfy) the predicate. Cf. sections 9.7 and 9.8.

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does not appear to arise as long as we we look at the matter in a purely contextfree way. For we can coherently represent existence without identifiability; in fact, we have done so all along when we outlined the problem. We need only say of an object a and a context c that a exists relative to c but a is not identifiable relative to c. There you have it: existence without identifiability. When the apparent problems of representing it coherently resulted only from the specifics of the context-sensitive object language, 1112 why not just drop context-sensitivity? In order to remind ourselves why we need a context-sensitive object language to describe the world it is about, recall what we said about the difference between taking the external stance and taking the internal stance on reality, that is, the difference between relational metaphysics and contextualist metaphysics. We have decided to endorse a contextualist metaphysics of expressions because the universe of expressions (and of everything else) would be just too crowded if metaphysics was relational. 1113 Now, on the level of linguistic representation, contextualist metaphysics is given by a context-sensitive language and relational metaphysics is given by a context-free language. So, when we say of a certain object that it exists but is not identifiable relative to a certain context, we are talking in a context-free way and representing existence without identifiability within the relational metaphysics. The problem arises only when we try to take the internal stance, when we try to represent existence without identifiability within the contextualist metaphysics. This is no accident; it results from the notion of identifiability, which is tied essentially to the internal stance. For what would identification be without a viewpoint? Recall that we introduced the notion of identifiability in the context of a representation of reality from a temporal viewpoint, when we observed that while past objects are identifiable even when they no longer exist, no future object is identifiable. 1114 As being identifiable means being representable from a certain viewpoint, it becomes evident why the problem arises: How can not being representable from a certain viewpoint ever be represented from that very viewpoint? Now that we have gained a better understanding of the problem we eventually want to solve within our dual framework of intensional semantics and two-dimensional semantics, let us look at the role of the intensional semantics of the contextsensitive language in more detail. When we formulate the intensional semantics of a context-sensitive language in a context-free language, the former takes the role of object language and the latter the role of meta-language, because the latter is used to say something about the former. 1115 But to give a full account of what is the case with an object language (which of its sentences are true and so on), its metalanguage needs to do more; it needs to enable us to give a full description of the world that the object language is about (which of its states of affairs obtain and so on). Thus we can distinguish a linguistic fragment and a worldly fragment of the

1112 1113 1114 1115

Cf. section 9.4. Cf. section 8.7. Cf. sections 9.2 and 9.3. Cf. section 2.2.

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context-free meta-language. Its linguistic fragment is used to give the intensional semantics of the context-sensitive object language (and its two-dimensional semantics); 1116 its worldly fragment is used to describe the world that the context-sensitive object language is about. It is important to be clear whose world this is. 1117 For the world that the meta-language is about must of course include the expressions of the object language (and facts that involve these expressions), but these will in general not be part of the world that the object language is about – at least as long as the object language is not self-referential. In the paradigmatic case of a description of temporal reality that does not involve self-reference, we will have a tensed object language that is about a world of physical objects and persons (and what is the case with them), and a tenseless meta-language the worldly fragment of which can also describe these physical objects and persons, albeit in a tenseless and relational way, but that also has a linguistic fragment that allows to formulate the intensional semantics of the tensed object language. To take up the above example again, the intensional semantics of a tensed fragment of English could very well include the following statement: ‘Obama is president of the U.S.’ is true with regard to the year 2016 if and only if Obama is president of the U.S. relative to the year 2016. Here, the left hand side of the biconditional belongs to the linguistic fragment of the meta-language, but the right hand side of the statement belongs to its worldly fragment. It is easy to see what the right hand side is about, namely Obama and the year 2016, as well as the (dyadic) concept of being president of the U.S. in relation to. To see what the left hand side is about, we need to keep in mind that we have decided to use the intensional semantics not to model the behavior of the object language, but rather to model the contextualist metaphysics. Therefore the left hand side is not about the tensed sentence ‘Obama is president of the U.S.’, nor is it about a peculiar species of truth, truth-with-regard-to. Rather, the left hand side is about the tensed state of affairs that Obama is president of the U.S. (which is modeled by the tensed sentence of the object language) to which it ascribes the obtaining of this state of affairs in the year 2016 (which is modeled by the technical notion of truth with regard to the year 2016). So we can see in the example how the worldly fragment of the context-free meta-language models the world understood as exhibiting a relational metaphysics, while the intensional semantics of the context-sensitive object language models the world understood as exhibiting a contextualist metaphysics. Here is an overview of this relation between two kinds of metaphysics:

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To anticipate, we will shortly use the linguistic fragment of the meta-language to formulate a third semantic theory, the relational semantics of the object language. Our use of the word ‘world’ is technical here; it does not mean the world in an absolute sense, because it is always relative to a language – much like the common phrase ‘universe of discourse’.

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metaphysics: internal stance:

our example: intensional semantics of the object language models contextualist metaphysics

‘Obama is president of the U.S.’ is true with regard to the year 2016

worldly fragment of the context-free meta-language

Obama is president of the U.S. relative to the year 2016.

if and only if external stance:

models relational metaphysics

How does the relation of the context-sensitive object language to the worldly fragment of the context-free meta-language look in general? In the particular case of Obama being president, it is very plausible that the biconditional that has provided our example does indeed hold. And there is nothing to be said against generalizing. So we can say, roughly, that a sentence is true with regard to a context if and only if what that sentence says obtains relative to that context. In order to make this rough formulation more precise, we need to compare how things are phrased in the context-sensitive object language and in the context-free meta-language. Note that in our example, there is a difference between the two predicates: ‘. . . is president of the U.S.’ is monadic and context-sensitive, but ‘. . . is president of the U.S. in relation to . . .’ is dyadic and context-free. In the meta-language predicate, the contextsensitivity of the object language predicate is traded in for an additional argument place. And something similar goes for singular terms. Although in our example we use the same singular term, ‘Obama’, in both the object language and the metalanguage, this can be otherwise in other examples. Witness, e. g., the following statement of the intensional semantics: ‘The president of the U.S. is eloquent’ is true with regard to the year 2016 if and only if the president of the U.S. in the year 2016 is eloquent relative to 2016. Here we have two different singular terms, again trading in context-sensitivity (of ‘the president of the U.S.’ and ‘is eloquent’) for an additional argument place (of ‘the president of the U.S. in . . .’ and ‘is eloquent relative to . . . ’). In order to generalize, we will require that there is a standard translation 1118 between the context-sensitive object language and the worldly fragment of the context-free meta-language, so that an object language predicate ‘F(. . . )’ is correlated with a meta-language predicate ‘. . . is F0 relative to c’ and an object language singular term ‘a’ is correlated with a metalanguage singular term ‘a0 -of-context-c’, where ‘c’ is a schematic context variable. 1119 Note that in concrete examples this singular term need not be of this exact form; like 1118

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The notion of standard translation belongs to the correspondence theory of modal logic. Cf. Blackburn 2006, 335f. For a locus classicus, cf. van Benthem 1991. We use a prime symbol to distinguish between the object language expression and the corresponding meta-language expression, and we attach it to the latter because we ultimately will want to privilege the object language.

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the object language term ‘a’, its meta-language correlate ‘a0 -of-context-c’ is meant in a schematic way. It can either be instantiated by a complex context-free description that is roughly similar (like the tenseless description ‘the president of the U.S. in the year 2016’ of our current example) or by a name, maybe even the same name as the one used in the object language sentence (like the name ‘Obama’ 1120 in our other stock example). More generally, we can drop the relativization to context in many natural language examples. 1121 Note also that the meta-language term ‘a0 -ofcontext-c’ can lack an extension for certain values of the variable ‘c’. In terms of the standard translation, we can give a schematic statement of the general principle that characterizes the interaction between the intensional semantics and the worldly fragment of the meta-language. We do this here only for monadic sentences (we will allude with less rigor to more general cases where necessary): 1122 (Int+ ) prelim ‘F(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (a) a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c is in the domain of the context c. It may seem redundant that context c occurs in two places in clause (a). But with a view to some cases, this is necessary. Both the singular term and the predicate can contribute to a monadic sentence in a context-sensitive way, and in some cases both will do so in the same sentence (e. g., in ‘The president of the U.S. lives in Washington’). Still with a view to clause (a), observe that the schematic criterion for 1120

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When we give the standard translation of ‘Obama is the president of the U.S.’, we do not want to end up using the phrase ‘Obama-of-the-year-. . . ’. This is also the case with stable predicates like ‘. . . is human’. Cf. section 9.1. In the scope of the present chapter, we express generality in our statements about the object language not with the use of meta-variables and Quine corners, but with schematic letters. E. g., we therefore write ‘‘F(a)’ is true’ here, and not ‘pΦ(α)q is true’ as we would have in chapter 4. Our main reason to express generality in a schematic way here is that we will often, within the same claim, both mention an expression and use an expression that is closely associated to it (via the standard translation). The apparatus of meta-variables and Quine corners, however, would only enable us to mention expressions in a general way; there is no natural way to extend it so that it would also enable us to use expressions in a general way. Take for example our claim: ‘‘F(a)’ is true with regard to a context c only if a0 -of-context-c is F0 relative to the context c.’ A variant with meta-variables and Quine corners would have to look like this: ‘pΦ(α)q is true with regard to a context c only if α0 -of-context-c is Φ0 relative to the context c.’ However, while ‘α’ is a meta-variable that ranges over singular terms, ‘α0 ’ (or ‘α0 -of-context-c’) would need to be a variable that ranges over the objects referred to by these singular terms, or at least a singular term that refers to one of these objects, but these objects will in general not be expressions. There is a related problem for the predicate. While ‘Φ’ is a meta-variable that ranges over predicates, and thus a singular term, ‘Φ0 ’ would need to have the grammatical category of a predicate because it is used. An additional reason for a schematic formulation is that we will in due course be mentioning quotation expressions, as in ‘‘True(«F(a)»)’ is true with regard to a context c’, where using the apparatus of meta-variables and Quine corners might lead to the false impression that we are quantifying into quote marks.

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truth does not make any detour through the extension of the singular term and of the predicate, 1123 but connects the truth of an object language sentence that is mentioned (‘F(a)’) directly to a meta-language sentence that is used (‘a0 -of-context-c is F0 relative to the context c’). In our natural language example, we mention the object language sentence ‘The president of the U.S. is eloquent’ and we use the meta-language sentence ‘The president of the U.S. in the year 2016 is eloquent relative to 2016’. With a view to clause (b), we come upon another apparent redundancy in the criterion. Saying that there is a certain object of the context which is in the domain of that context may seem like making the same requirement twice over, that there be the object in relation to the context. But these are actually two distinct requirements. The first one is due to the fact that some singular terms of the object language will not refer to an object with regard to every context (witness ‘the president of the U.S.’ with regard to the year 1788), and might refer to different objects with regard to different contexts (witness ‘the president of the U.S.’ with regard to the year 1789 in contrast to the year 2016). The second requirement is due to the fact that even when a singular term of the object language does refer to a certain object with regard to a context, this object cannot in all cases figure in a state of affairs that obtains relative to that context. This is witnessed by the description ‘the president of the U.S. in the year 2016’ with regard to the year 1889: From the external stance of the intensional semantics, it seems plausible that Obama is the referent of this tenseless description already with regard to 1889; but relative to that year, Obama could not yet figure in what was the case. 1124 With a view to certain applications we should add a corresponding criterion for falsity. (Int− ) prelim ‘F(a)’ is false with regard to a context c if and only if there is an object a0 -of-context-c such that (a) it is not the case that a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c is in the domain of the context c. Both in the criterion for truth and in the criterion for falsity we require that there is an object a0 -of-context-c and that it is in the domain of the context c. The rationale behind this requirement is that when we evaluate a context-sensitive singular term, we will in general not have a guarantee that with regard to every context there is an object referred to and that it can figure in what is the case relative to that context. Just think of ‘the king of France’, which was empty with regard to the year 1905! 1125 It bears repeating that, because of the way we use the intensional semantics in the 1123

Contrast the following more standard principle (again formulated with schematic letters): ‘F(a)’ is true with regard to a context c if and only if J‘a’Kc is among J‘F’Kc (i. e., if and only if the object that is the extension of ‘a’ with regard to the context c is among the collection that is the extension of ‘F’ with regard to the context c).

1124 1125

We will say more about this reason for not being in the domain in section 10.7. Similarly, ‘the president of the U.S.’ does not refer to anything with regard to any year before 1789.

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present framework, in our material understanding this does not model anything about a singular term and its reference, but about what is the case – it models that there was no (unique) king of France in the year 1905. As a consequence of the clause requiring that there be an object that is in the domain both for truth and for falsity, it can easily happen that there are truth value gaps in the intensional semantics. To use the obvious example, ‘The king of France is bald’ is neither true with regard to the year 1905 nor false with regard to that year. This lack of a truth value in the formal semantics models that, correspondingly, there is no fact of the matter in our material understanding of the world this sentence is a about – there is a metaphysical gap: Neither did the tensed state of affairs that the king of France is bald obtain relative to the year 1905, nor did the tensed state of affairs that the king of France is not bald obtain relative to the year 1905. Note that we treat truth and falsity separately in the intensional semantics, and we will again treat them separately in the two-dimensional semantics and in further, related semantic theories. This has several advantages. Besides construing truth and falsity not as objects but as the concepts that they are, it enables us to tap into common sense intuitions about truth and falsity while at the same time keeping open not only the classical two, but four options for the extension of a sentence, which otherwise would have to be represented by four distinct truth values: A sentence can be true only, a sentence can be false only, a sentence can be neither true nor false (gappy), and a sentence can at least in principle be both true and false (a dialetheia). Thus the question of bivalence – the double question of whether the principles of the (Exhaustivity) and of the (Exclusivity) of truth and falsity hold – is not settled by definition; it is in principle open. 1126 We will eventually arrive at different answers, depending on whether we use the present framework as the background for a diagnosis of the Liar paradox or for our proposal for a solution. 1127 * With regard to the metaphysical understanding of the principle (Int+ ) that charprelim acterizes the interaction between the intensional semantics and the worldly fragment of the meta-language, we should note that ‘if and only if’ is symmetrical – it privileges neither of the two sides of the statement in its scope. Therefore, just as the tensed theoretician and the tenseless theoretician can agree on the criterion for truth with regard to a moment that we used in our example about Obama being president, it is the case more generally that an adherent of a contextualist metaphysics and an adherent of the corresponding relational metaphysics can agree on our schematic criterion for truth with regard to a context. The crucial difference between the two 1126

1127

These matters have already been discussed in section 4.4. On the technical level, our treatment of truth and falsity can be understood as a conceptual variant of the relational semantics of FDE (Dunn 1976; cf. Priest 2008, 142ff. and Priest/Tanaka/Weber 2015, section 3.6). Instead of complex relational predicates like ‘R(x, T)’ and ‘R(x, F)’ for ‘the extension of x is the True’ and ‘the extension of x is the False’, our meta-language has simple predicates like ‘x is true’ and ‘x is false’ (of course relativized to a parameter, e. g., a circumstance of evaluation). In the diagnostic application of the framework, neither (Exhaustivity) nor (Exclusivity) will hold so that there will be both gappy sentences and dialetheias; in its therapeutic application, only (Exhaustivity) will fail so that there will be gappy sentences but no dialetheias. Cf. section 13.2.

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sides of the metaphysical debate is which side of ‘if and only if’ they take to be metaphysically committing. The arguments must be found elsewhere, because laying out the intensional semantics cannot settle this dispute. At least not in general! For we should mention that in the dispute about the metaphysics of time, the situation has often been seen as asymmetrical. Often, the criterion for truth with regard to a moment has been formulated with a language of purely qualitative tense logic as object language, which can make only the rough distinction between past, present, and future, but a richer meta-language, which can access each moment in time. In this situation, the meta-language can express strictly more than the object language. With the aim of counter-balancing, Arthur Prior set out to strengthen the tensed object language, to regain a situation where it can express everything that the worldly fragment of the tenseless meta-language can. We will abstain from going into the technicalities. But let me mention that this point was the reason for us to incorporate metric tense operators 1128 (or some generalized counterpart) into the object language. Under certain conditions this enhancement will compensate the deficiency of standard languages of tense logic, as compared to their standard tenseless meta-languages. 1129 In this situation, where we can think of the context-sensitive object language and the worldly fragment of the context-free meta-language as being of equal expressive strength, so that the criterion for truth with regard to a context gives us no way to adjudicate between the two contrary metaphysical understandings, we need to be clear about our endorsement of a contextualist metaphysics. This means that with a view to the picture in the formal semantics, where a context-free meta-language is used to describe the behavior of a context-sensitive object language, we take the object language to be what is metaphysically committing. The representation given by the context-free meta-language is helpful for several reasons – not least for the reason that this is currently the standard way of doing formal semantics –, but ultimately it is no more than a ladder to be kicked away in the future. 1130 In other words, although for the time being there is some methodological value in the fiction of taking the external stance on reality as it is represented by the meta-language, we are ultimately committed to taking the internal stance of what in the framework of the formal semantics is the object language.

10.3 The role of the two-dimensional semantics We have seen enough for the moment of how the intensional semantics models the contextualist metaphysics. Let us now turn to the second half of our dual framework, to how the two-dimensional semantics models the behavior of the context-sensitive language. Here it is important to keep in mind the difference between a language 1128 1129 1130

Cf. section 9.1. For a fuller account of this debate, cf. Prior 2003, Blackburn 2006, and Pleitz 2015c. In the scope of the present study, we are well-advised to hold on to the ladder. We will only be in a secure place where we can kick it away comfortably when we have shown in a rigorous way how a particular formal language of tense logic can indeed express all we need it to express. But we leave this technical task for another occasion.

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occurring in one of the formal semantic theories and a language occurring in our material understanding of the formal semantics. At least according to how we make use of it in our dual framework, a language that occurs in the formal semantics need not model a language in the material understanding. That is exactly the situation when we use the intensional semantics of a context-sensitive formal language to model a contextualist metaphysics: Here, the formal language does not model a language, but a world of context-sensitive states of affairs. But when we use the two-dimensional semantics of a context-sensitive formal language to model the behavior of a context-sensitive formal language, we are indeed using a language to model the behavior of a language. These languages need not be identical, but they should be very similar; it is natural, for example, to require that the relation between the two is formalization, as when we model a fragment of a natural language with a formal language that gives its logical form adequately. Although the object language of the intensional semantics and the object language of the two-dimensional semantics also need not be identical, we will require here that they are. After all, what we want to model with our dual framework is a situation where, figuratively speaking, a changing world is described adequately from within by a changing language. Now, this changing world (the contextualist metaphysics) is what is modeled by the context-sensitive object language of the intensional semantics, and this changing language (a context-sensitive language that is part of our material understanding) is what is modeled by the context-sensitive object language of the two-dimensional semantics. Therefore, requiring the object languages of the two semantic theories to be identical makes good sense. It amounts to making the assumption that the language that we model is fully adequate to describe the world it is about, which we also model. 1131 (This corresponds to a similar requirement we made implicitly when we described the relation between the context-sensitive object language of the intensional semantics and the worldly fragment of the context-free meta-language: Modulo the standard translation, these are equivalent: The worldly fragment of the meta-language will not outstrip the object language, and vice versa.) Especially as we work with the same object language in both the intensional semantics and the two-dimensional semantics, we need to answer the following questions: Why does it take one dimension more to model the behavior of a changing language than to model the changing world it is about? How is modeling a context-sensitive language different from modeling the corresponding contextualist metaphysics? The general point is, to put it figuratively, that we need to think of the expressions of the language as being situated in the world they represent, so that both how an expression is situated in the world and what is the case in the part of the world represented by the expression will be relevant to the expression’s behavior – its behavior in terms of truth, reference, or satisfaction. Of these two 1131

Although this assumption might be contentious in other settings, it is harmless in the context of the present study. We are interested mainly in phenomena that occur when a language becomes selfreferential. Issues that in other scenarios might call into question that a language can be fully adequate to describe the world it is about – the issues of vagueness, of representing the individual behavior of indenumerably many objects, and of whether there might be ineffable experiences – do not arise.

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factors relevant for modeling the behavior of the language only one is relevant for modeling the world it is about, namely what is the case. When we use a contextsensitive object language to model a contextualist metaphysics, we do not need to think of its expressions as situated; even when we use tensed or context-sensitive sentences to model tensed or context-sensitive states of affairs, we do not need to think of these sentences as objects that are part of the world. This explains, on the figurative level, not only why we need one more dimension to model the behavior of a language than to model the world it is about, but also why one of the dimensions of the two-dimensional semantics coincides with the single dimension of the intensional semantics: The two semantic theories share the dimension of circumstances of evaluation, which models what is the case in the world, and this is complemented in the two-dimensional semantics by the dimension of contexts of use, which models how an expression is situated in the world. In order to fill in this picture with more details, we can of course draw on everything we said in the second part of this study about semantics, and in particular about the semantics of singular terms. 1132 Recall that among the expressions that provided Kaplan with a rationale for going two-dimensional are indexicals. Now, the indexical ‘I’ refers to whoever utters it. This already illustrates that how an expression is situated in the world can matter for determining its extension. Moving on to the complex indexical singular term ‘my favorite color’, observe that its extension depends both on the speaker who utters it and on what is the case with this person’s preferences among the spectrum of colors. Here, both aspects matter: how the expression is situated in the world, and how the world is. But although the paradigmatic phenomenon of indexicality provides good illustrations, let us put it to the side again. Thus we are spared the additional complexity of incorporating the positions of use into our framework that are needed to deal with indexicals, 1133 and can use the same contexts, sometimes in the role of contexts of use and sometimes in the role of circumstances of evaluation. Also, we would like to have a first look at something that will be characteristic of our own application of the dual framework to model the metaphysics of expressions and our talk about it. As you will recall, expressions form a contextualist ontology. The contexts that are relevant to whether an expression exists are the same as the contexts that are relevant to what (if anything) the extension of an expression is. Now, unlike a person being famous, which can be the case even when the person does not exist any longer, an expression having an extension just cannot be the case relative to a context where the expression does not exist. 1134 Therefore we will require the following: Relative to every context, the existence of an expression is a precondition for its having an extension. In particular, to be true (or false) relative to some context, a sentence has to exist relative to that context. Similarly, a singular

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Cf. chapters 4 and 5, and in particular sections 4.7 and 5.2. We will of course make an exception in sections 11.11 and 12.6 that concern the specific indexical ‘this sentence’. To bring the Cheshire cat into the fray again: The having of an extension of a non-existent expression would be no less incoherent than the grin of an absent cat (perhaps more so).

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term always has to exist to refer to an object, 1135 and a predicate always has to exist to be satisfied by some object. (Note that the question of existence we are dealing with here has nothing to do for the time being 1136 with the question of existence that figures in the problem of existence without identifiability, which we want to solve later in this section, for the former concerns the expression in question and the latter concerns the object in question.) In the two-dimensional semantics, our new requirement that the expression in question exists concerns the context of use, i. e., the second dimension. To see how it can be incorporated, let us look at a preliminary formulation of a criterion for truth in the two-dimensional semantics (again for monadic sentences): (2D+ ) prelim A sentence ‘F(a)’ is true with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i) ‘F(a)’ is true with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is in the domain of the context d. The criterion for truth in the two-dimensional semantics that we will later officially endorse will look a bit different, especially with regard to the variable domain. 1137 But ignoring the domain for the time being, we can already read off some important points from this preliminary criterion for two-dimensional truth. Both dimensions do some work. The first dimension, which is the dimension of circumstances of evaluation, coincides with the single dimension of intensional semantics (that is why we chose the similar sounding prepositions ‘with respect to’ and ‘with regard to’, respectively); it models what is the case in the part of the world represented by the sentence, insofar as clause (i) models that the corresponding state of affairs obtains relative to the relevant context, which here has the role of a circumstance of evaluation. The second dimension models how the sentence is situated in the world, insofar as clause (ii) models that the sentence exists relative to the relevant context of use. In other applications of the our dual framework, the second dimension of the two-dimensional semantics will in addition (or instead) model other aspects of how the expression in question is situated in the world, like the position of use in the case of an indexical expression. (This is also the place where we will add a further requirement on our way to solve the problem of existence without identifiability.) In contrast, no requirement like clause (ii) is needed in the intensional semantics: For a context-sensitive sentence of the formal object language to model a contextsensitive state of affairs that obtains relative to a certain context, the corresponding sentence of the language that is part of our material understanding need not exist relative to that context. In our contextualist metaphysics, there will usually be context-sensitive states of affairs where there are no context-sensitive sentences. 1135

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As we have seen in the case of past objects (cf. section 9.3), the converse does not hold: An object does not have to exist to be referred to. This will change in the case of self-referential expressions. Cf. section 10.8.

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And in the temporal reality as it is described by the tensed theory of time (that we also endorse), it is likewise: There were tensed states of affairs before there were tensed sentences. Let us look at an example that concerns dinosaurs. 1138 As there were dinosaurs one hundred million years BCE, we are justified to assert: ‘There are dinosaurs’ is true with regard to the year one hundred million BCE. 1139 That is a statement of the intensional semantics. In the two-dimensional semantics, things are more complicated: ‘There are dinosaurs’ is not true with respect to the year one hundred million BCE and at the year one hundred million BCE, because no (English) sentence existed then so that (ii) does not hold. Nor is ‘There are dinosaurs’ false with respect to and at the year one hundred million BCE. At least if we put the analogous criterion for falsity beside the preliminary criterion for truth we are discussing here: 1140 (2D− ) prelim A sentence ‘F(a)’ is false with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i) ‘F(a)’ is false with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is in the domain of the context d. However, ‘There are dinosaurs’ is true with respect to the year one hundred million BCE and at the year 2016. This is because (ii) holds of the year 2016 – we can say ‘There are dinosaurs’, and although we say it at 1141 the present year, we can still say it with respect to the year one hundred million BCE. We can even say something 1138 1139

I would like to thank Niko Strobach for getting me interested in this example. Strictly speaking, we are not yet fully justified because both in the intensional semantics and in the two-dimensional semantics, we have criteria for truth as of now only for monadic sentences. But these are easily complemented by the relevant criteria for the truth and falsity of quantified sentences: (∃-Int+ ) ‘Something is F’ is true with regard to a context c if and only if some object that prelim belongs to the domain of the context c is F0 relative to the context c. (∃-Int− ) ‘Something is F’ is false with regard to a context c if and only if no object that prelim belongs to the domain of the context c is F0 relative to the context c. (∃-2D+ ) ‘Something is F’ is true with respect to a context c and at a context d if and only if prelim some object that belongs to the domain of the context d belongs to the domain of the context c and is F0 relative to the context c and ‘Something is F’ exists relative to the context d. (∃-2D− ) ‘Something is F’ is false with respect to a context c and at a context d if and only if prelim no object that belongs to the domain of the context d belongs to the domain of the context c and is F0 relative to the context c and ‘Something is F’ exists relative to the context d.

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In addition, we only need the unproblematic assumption that the dinosaurs were in the domain of the year one hundred million years BCE. Again, there are strictly speaking some more steps to go, because we can show that ‘There are dinosaurs’ is false in the two-dimensional semantics only on the basis of a criterion for the falsity of quantified sentences. But that is easily supplied; it will differ from the respective criterion for truth by having the condition on the right hand side negated. My apologies for the harsh use of language, but in view of the complexity of the dual framework it is vital that we stick to our terminological convention regarding prepositions.

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about the matter that is true both with respect to and at the present year if we use a metric tense operator: ‘One hundred million and 2016 years ago it was the case that there are dinosaurs.’ Although it could not have been said that there are dinosaurs when that tensed state of affairs obtained, we can say it now, and we can even express it with the corresponding present tensed sentence, if that sentence is embedded in a more complex construction. Within the two-dimensional semantics, an expression has its extension always in relation to two parameters; accordingly, the locution ‘with respect to c and at d’ should never be taken apart. But as we use the two-dimensional semantics to model the behavior of a language, we will often want to know whether a certain sentence is true relative to some context, whether a certain singular term refers to some object relative to some context, or whether some object satisfies a certain predicate relative to some context! How do we get back from two parameters to a single parameter? The answer is, in the form of a slogan: Relative truth is truth on the diagonal, and similarly for relative falsity, relative reference, and relative satisfaction, i. e., in general for having an extension in a relative way. More precisely, truth relative to a certain context is two-dimensional truth with respect to and at that context, and similarly for falsity: 1142 (Diag+) A sentence s is true relative to a context c if and only if the sentence s is true with respect to the context c and at the context c. (Diag–) A sentence s is false relative to a context c if and only if the sentence s is false with respect to the context c and at the context c. 1143 In effect, these bridge principles merge the two dimensions again into a single parameter. As long as no positions of use are involved so that we can use the same context both in the role of a circumstance of evaluation and in the role of a context of use, they hold quite generally in all applications where two-dimensional semantics is used to model the behavior of a context-sensitive language. But when we combine the bridge principle (Diag+) with our preliminary criterion for two-dimensional truth (2D+ ), we can infer that relative truth presupposes existence: 1144 If the prelim sentence s is true relative to the context c, then the sentence s exists relative to the context c. To illustrate, we can now complement the dinosaur example by saying that it is not the case that the sentence ‘There are dinosaurs’ is true relative to the 1142

1143 1144

In contrast to the criteria for truth and falsity that we have seen so far, the bridge principles (Diag+) and (Diag–) are not preliminary; also, they apply to all sentences, not only the monadic ones. The corresponding principles for reference and satisfaction are entirely similar. We could in fact infer a preliminary criterion for relative truth, but we forgo doing this in detail at this point because we will shortly be able to formulate our official criteria, including one for relative truth. Cf. section 10.8. The same goes for relative falsity.

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year one hundred million BCE, and that this is so solely because the sentence did not exist relative to that year. Also, it is not the case that ‘There are dinosaurs’ is false relative to the year one hundred million BCE. This follows when we combine the bridge principle (Diag–) with our preliminary criterion for two-dimensional falsity (2D− ). Hence ‘There are dinosaurs’ is gappy (neither true nor false) relative to prelim the year one hundred million BCE. We will say more about this kind of truth value gap in due course. 1145

10.4 The Tarskian equivalence The principles about relative truth and relative falsity connect us back from the twodimensional semantics of the context-sensitive object language to the context-free meta-language. The sentence ‘‘Obama is president of the U.S.’ is true relative to the year 2016’, e. g., is as tenseless and context-free as they come, despite the embedded sentence being context-sensitive, and the same goes for the sentence ‘‘There are dinosaurs’ is false relative to the year 2016’. The relational semantics 1146 that these sentences belong to takes the external stance on reality – specifically, it takes the external stance on that part of reality that is comprised of the object language. In this regard it is just like the relational metaphysics, which takes the external stance on the world the object language is about. In order to take the internal stance on the object language, seen as a part of reality, we need to repeat what we did when we connected the relational metaphysics to the contextualist metaphysics. Let us translate the relational truth predicate of the context-free meta-language into the context-sensitive object language, connecting it to a context-sensitive truth predicate! In fact, we can again follow the same recipe as when we translate monadic sentences of the worldly fragment of the meta-language into the (worldly fragment) of the object language. For that we use the preliminary criterion for truth in the intensional semantics. Recall: (Int+ ) prelim ‘F(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (a) a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c is in the domain of the context c.

1145 1146

Cf. in particular section 10.10. A terminological remark: By ‘relational semantics’ we do not mean what is usually called the “relational semantics of FDE” of Dunn 1976 etc., which we mentioned in a footnote to section 4.4 and a footnote to section 10.2. Our relational semantics is relational insofar as it treats a sentence as true or false only relative to a context, whereas Dunn’s semantics is relational insofar as it treats a sentence as standing in a relation to its truth value, as opposed to determining it functionally. (Strictly speaking, things are a bit more complicated, because – like our intensional semantics and our two-dimensional semantics – our relational semantics can be understood as a conceptual variant of Dunn’s semantics; cf. the mentioned footnote to section 10.2.)

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It is easy to extend this criterion so that it also delivers a translation of the truth predicate. First, we need to extend the object language by adding a context-sensitive truth predicate, ‘True(. . .)’, that is meant to apply to sentences of the object language, as well as singular terms that refer to sentences of the object language, for example by supplementing it with formal quote marks, ‘«. . . »’. 1147 (Thus we are complementing the worldly fragment of the object language with a linguistic fragment.) Now we need only instantiate the schematic predicate letter ‘F’ in the criterion (Int+ ) with the object language truth predicate ‘True(. . .)’ and conprelim strue the schematic singular term ‘a’ as referring to a sentence (with regard to some context). Here is what we get: (True-Int+ ) prelim ‘True(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is true relative to the context c and (ii) a0 -of-context-c is in the domain of the context c. Given the plausible precondition that only sentences can be true, clause (i) will entail that the object in question is a sentence relative to the context in question (and therefore whenever it exists, because being a sentence is a stable concept). We have arrived at the intensional semantics of ascriptions of truth to sentences of the object language. 1148 Just as the intensional semantics of more worldly sentences models an internal stance on what is the case in the world the object language is about, the intensional semantics of ascriptions of truth models an internal stance on the truth of sentences of the object language. When we compare the two, we will come to a point where we can state something very much like the homophonic variant of the Tarskian truth schema – its contextualist cousin, so to say. But before we turn to the details of this application, some general remarks. The bridge principle (True-Int+ ) 1149 enables us to move on from what we can state prelim about truth in the relational semantics to the intensional semantics of sentences that ascribe truth to sentences of the object language, in parallel to our earlier move from the relational metaphysics to the intensional semantics of sentences that characterize the world the object language is about. As this earlier move brought us to the contextualist metaphysics, we might want to describe what we have here as the kernel of the contextualist semantics of the object language, which allows to take the internal stance on the theory of extensions. 1150 Besides being of general interest, 1147 1148 1149

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Cf. section 5.7. The intensional semantics of ascriptions of falsity will look similar. It is preliminary only insofar as we will later be more specific about the variable domain; a definitive principle will be given in section 10.9. We are concerned here only with the particular case of context-sensitive truth, and as yet we can describe its behavior only from the external stance. Even when we give an internal characterization of its behavior (in section 10.9), this will be only for monadic sentences, and we will still be many steps away from a comprehensive theory. For that, we would additionally need principles about reference and satisfaction as the basis of a recursive definition of truth and falsity, and we would need

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it is of course highly relevant in the scope of a study about the Liar paradox that by adopting (True-Int+ ) we extend the object language to include its own truth prelim predicate. We thus take a great stride towards the object language becoming selfreferential. Given that in the intensional semantics we use the formal object language to model not the behavior of a language, but of the world, however, there is still another step to go until what we model is really a language talking about itself. What the intensional semantics of a self-referential formal object language model is rather taking the internal stance on what is the case with the language we model, i. e., the contextualist semantics. The principle (True-Int+ ) has a high level of generality because it is schematic prelim in the singular term ‘a’. We get a more restricted schema that covers an important range of cases when we instantiate the mentioned term ‘a’ with a quotation expression of the object language and all occurrences of the used term ‘a0 -of-context-c’ with a quotation expression of the meta-language: ‘True(«. . .»)’ is true with regard to a context c if and only if (i) ‘. . .’ is true relative to the context c and (ii) ‘. . .’ is in the domain of the context c. This homophonic schema for translating the truth predicate can be applied to the case of monadic sentences: ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true relative to the context c and (ii) ‘F(a)’ is in the domain of the context c. This still has some generality because we can think of both the predicate letter ‘F’ and the term ‘a’ as schematic, and it is all we need for the following discussion. The principle (True-Int+ ) relates the internal stance on ascriptions of truth prelim to the external stance on ascriptions of truth; it relates the contextualist semantics to the relational semantics. Given what we have already learned about truth in the relational semantics and in the two-dimensional semantics in the previous section, we can now go on to derive a principle that connects the contextualist semantics back to the contextualist metaphysics – that relates the internal stance on what is true to the internal stance on what is the case. When we apply (Diag+) to clause (i), we get: ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with respect to the context c and at context c and (ii) ‘F(a)’ is in the domain of the context c.

to address the question of how the information of the contextualist semantics that is lost in the step from the intensional semantics to the two-dimensional semantics could be retrieved. We leave the task of developing such a theory for another occasion.

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When we apply (2D+ ) to clause (i), we get: prelim ‘True(«F(a)»)’ is true with regard to a context c if and only if (i-i) ‘F(a)’ is true with regard to the context c, (i-ii) ‘F(a)’ exists relative to the context c, (i-iii) a0 -of-context-c is in the domain of the context c, and (ii) ‘F(a)’ is in the domain of the context c. For the particular case of expressions, and given the role of the domain in the intensional semantics, 1151 we can assume that clause (i-ii) entails clause (ii), which allows to simplify and re-number the clauses: ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with regard to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is in the domain of the context c. This principle tells us how the truth on a monadic sentence is correlated to the obtaining of the corresponding state of affairs, whilst taking the internal stance on both sides of the correlation. But is this the correlation we would have expected before we started to develop the dual framework? * To see what we originally would have expected for the correlation between what is true and what is the case, we should turn back to the Tarskian truth schema. When we apply its homophonic variant to the case of monadic sentences, we get this: 1152 ‘F(a)’ is true if and only if F(a). In order to be able to compare this with what we have derived in our dual framework, we need to transfer it into a contextualist setting. Consider the following principle: (Tarskian Equivalence) ‘True(«F(a)»)’ is true with regard to a context c if and only if ‘F(a)’ is true with regard to the context c. This differs in at least three points from the homophonic Tarskian truth schema for monadic sentences: The truth predicate in the sentence mentioned on the left

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We will focus on the role of the domain in sections 10.6 and 10.7. As of now, we are mainly interested in how truth interacts with the existence of expressions and we do not pay much attention to the domain. Think of the meta-language as augmented so that it can contain formal sentences like ‘F(a)’ as parts of semi-formal larger expressions.

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hand side of the equivalence is context-sensitive, the sentence mentioned on the right hand side must also be understood in a tensed way, and the two sentences are not connected by a biconditional, but by the statement that they are true with regard to the same context. I submit that the (Tarskian Equivalence) merits its name because it is the most natural candidate for a contextualist cousin of the (homophonic) Tarskian truth schema (as applied to monadic sentences). 1153 But does it hold? Initially, we appear to have little reason to doubt it. In our stock example, the following will be the case: ‘‘Obama is president of the U.S.’ is true’ is true with regard to a moment m if and only if ‘Obama is president of the U.S.’ is true with regard to the moment m. This is so because the two mentioned sentences are true with regard to the same moments (roughly, the years 2009 through 2016). 1154 Many situations are like this, and we are under no great pressure to distinguish between the truth of a sentence and the obtaining of the corresponding state of affairs, because usually they will be materially equivalent across all contexts. But we have already observed that even then, there is a conceptual difference because the truth of a sentence and the obtaining of a state of affairs concern different realms, 1155 as the truth of a sentence concerns what is the case with language and the obtaining of a state of affairs concerns what is the case with the world (whether it is talked about or not). In our example, the truth of ‘‘Obama is president of the U.S.’ is true’ with regard to some year models a fact about the sentence ‘Obama is president of the U.S.’, whereas the truth of ‘Obama is president of the U.S.’ with regard to that year models a fact about

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An alternative candidate for transferring the Tarskian truth schema into a contextualist setting is this: ‘True(«F(a)») ↔ F(a)’ is true with regard to every context c.

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Under the most natural intensional semantics of the biconditional, (↔-Int), a biconditional statement is true if and only if both of its sides are true or both of its sides are false, and it is false if and only if one of its sides is true and the other is false. Hence a biconditional is gappy if and only if at least one of its sides is gappy – gappiness spreads. This criterion for the biconditional is inspired by the criteria that are given in Dunn’s semantics for FDE for the connectives of conjunction, disjunction, and negation; cf. Priest 2008, 143. More precisely, the criterion belongs to the logic FDE↔ that is characterized in Petrukhin 2017, section 2. Given (↔-Int), the alternative candidate makes a stronger requirement than the (Tarskian Equivalence). The alternative candidate requires that the two mentioned sentences are true with regard to the same contexts, false with regard to the same contexts, and gappy with regard to no context. The (Tarskian Equivalence) requires only that the two sentences are true with regard to the same contexts. In a scenario where (Exhaustivity) and (Exclusivity) hold, the alternative candidate and the (Tarskian Equivalence) are materially equivalent. If someone does not feel a difference, this might therefore be due to their classical, bivalent intuitions. Note that by formulating the context-sensitive cousin of the Tarskian truth schema not with a biconditional but as an equivalence, we are able to ignore what does not yet interest at this point, namely that arguably both sentences are gappy with regard to the years before 1961. Well, they concern different realms until the object language gets self-referential. Then (part of) the language will be (part of) the world it is about! More on that in chapters 11 and 12, and in particular in section 11.1.

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Obama and the U.S. Because of the material equivalence that appears to hold in cases like this one, we may be lulled to think that despite the conceptual difference, we need not fear any failures of the Tarskian equivalence. But alas, according to our account there can easily be failures of the Tarskian equivalence. This is a consequence of taking seriously the fact that the expressions of a language form a contextualist ontology, and we see it already on the basis of the preliminary presentation of the dual framework that we are giving here. For compare the (Tarskian Equivalence) to the principle about truth ascriptions that we derived from criteria for truth we endorse in our account: (Tarskian Equivalence) ‘True(«F(a)»)’ is true with regard to a context c if and only if ‘F(a)’ is true with regard to the context c. Our principle about truth ascriptions: ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with regard to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is in the domain of the context c. Clearly, if the existence of the sentence in question is not guaranteed for every context with regard to which we evaluate it in the intensional semantics, there can be contexts with regard to which the (Tarskian Equivalence) fails. This is readily shown in the Dinosaur example (for which we move on from monadic sentences to quantified sentences). We have already seen that the existential claim ‘There are dinosaurs’, although true with regard to the year one hundred million BCE, is not true relative to that year. When we translate the latter claim into the object language, we find that the truth ascription ‘‘There are dinosaurs’ is true’ is not true with regard to the year one hundred million BCE. (So is the falsity ascription ‘‘There are dinosaurs’ is false’.) But there should be nothing deeply troubling about the fact that the Tarskian equivalence fails locally in this case. It merely reflects the historical fact that there were no English sentences when there were dinosaurs. We will say more about our acceptance of failures of the Tarskian equivalence when we come to this point for a second time, after developing the dual framework in more detail. 1156

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Cf. section 10.9.

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10.5 Overview of the dual framework We have seen enough details of our dual framework to get a good impression of how the apparatus works in general. But since it has so many moving parts, it will be helpful to round off the general presentation with an overview.

internal stance:

external stance:

metaphysics:

semantics:

intensional semantics of the object language

two-dimensional semantics of the object language

models

models

contextualist metaphysics

context-sensitive language

worldly fragment of the context-free meta-language

linguistic fragment of the context-free meta-language

models

models

relational metaphysics

relational semantics

To be clear about what is going on here, we need to keep in mind three distinctions, between the formal level and the material level, between metaphysics and semantics, and between the external stance and the internal stance. On the formal level, we are working with two interacting languages that serve multiple functions for what this framework models in our material understanding. The same contextsensitive object language of the formal level in our material understanding models both a contextualist metaphysics and a context-sensitive language. The first function is served by its intensional semantics, the second function by its two-dimensional semantics. The context-free meta-language of the formal level has a fourfold purpose; or more precisely, we can think of it as having four fragments, each with its own purpose. The worldly fragment of the context-free meta-language models the relational semantics, a part of its linguistic fragment is used to formulate the intensional semantics, another part of its linguistic fragment is used to formulate the two-dimensional semantics, and a final part of its linguistic fragment is used to formulate the relational semantics. The relational semantics gives an external stance on the semantics of the language. The four theories form a square, and they are related in such a way that it makes sense to move around the four corners in a clockwise manner: From the relational metaphysics we get to the intensional semantics by way of the standard translation that maps the worldly fragment of the meta-language to the object language. From the intensional semantics we get to the two-dimensional semantics by way of the criterion for two-dimensional truth because one clause of the criterion will usually connect this back to truth in the intensional semantics (in the preliminary framework we discussed this is witnessed by clause (i) of (2D+ )). From the two-dimensional prelim semantics we get to the relational semantics because relational truth is truth on the diagonal (as witnessed by the bridge principle (Diag+)). From the relational semantics, we can continue onwards, starting the next round. If the object language is not self-referential, then its expressions do not belong to the

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world it is about. In that case, we should think of the relational semantics as disjoint from the relational metaphysics: Both are given from the external stance, but they constitute separate realms. But if the object language is self-referential so that (some) of its expressions belong to the world it is about, the relational semantics will overlap the relational metaphysics. In both cases, however, we can get on from the relational semantics to the intensional semantics, now as applied to ascriptions of truth (and the like) that are formulated in the context-sensitive object language, when we complement the worldly fragment of the object language with a linguistic fragment that allows to represent facts about expressions of the object language and then apply the standard translation to the truth predicate (and the like) of the context-free meta-language. We thus arrive at the contextualist semantics, which gives the internal stance on the semantics of the object language. We are now in a position to evaluate claims that have the form of the Tarskian equivalence, i. e., we can ask whether a sentence of the form ‘‘. . . ’ is true’ is true with regard to the same contexts as the embedded sentence ‘. . .’. From there we could take yet another step by applying the criteria for two-dimensional truth and for truth in the relational semantics (each for the second time), and would thus arrive at the relational semantics of statements of the contextualist semantics, modeling what we can express internally about the internal stance on what is the case with our language. 1157 There is nothing to stop us going round the square of theories indefinitely, and in fact I am confident that this will under certain favorable conditions be a way to give a recursive definition of truth and falsity (and of other semantic notions) on the basis of worldly facts alone. We will be able to specify these favorable conditions, however, only when we have solved the Liar paradox, which as we will see threatens to block the progress around the square of theories. But for now we have seen enough, and I leave the rigorous implementation of this recipe for a recursively defined theory of extensions for another occasion. To conclude the overview, let us highlight how the schematic dichotomy between language and world occurs in our dual framework. The language and the world of the dichotomy are the two parts of the reality that we want to model, i. e., they are the two parts of the material understanding of our formal framework with its two interacting formal languages. The duality of language and world in fact occurs in two places, both in the input and in the output of the framework. It occurs in the input of the framework, and more specifically, in the input of the two-dimensional semantics, insofar as the first dimension of circumstances of evaluation models certain facts about the world and the second dimension of contexts of use models certain facts about the language. It occurs in the output of the framework insofar as the intensional semantics models the world construed as a contextualist metaphysics and the two-dimensional semantics models the contextsensitive language that allows to talk about it. How does this fit together? To see that this double duality does not lead to any tension within the two-dimensional semantics, 1158 let us be more specific. What a context of use models are certain quite 1157 1158

We will touch upon this towards the end of section 10.9. It is more easy to see that no tension arises within the intensional semantics, because the worldly output of the intensional semantics just is the worldly input of the two-dimensional semantics.

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specific facts about language – paradigmatically (in the centered contexts that play no great role in the present investigation) the speaker, time, and place of an utterance of an expression. More generally, the context of use models the situation in which a linguistic expression is used; it encodes not only who used it when and where, but also against which background of worldly facts it was used (which is important for the semantics of ‘the actual F’). That is the linguistic input into the two-dimensional semantics, modeled as its second dimension. The output of the two-dimensional semantics (and the resulting relational semantics) is also linguistic, but it models distinct facts about the language. It models the semantic properties (truth or falsity, reference, and satisfaction) of the expression in question. To fully determine the linguistic output, both the worldly input and the linguistic input is needed: What is the case with the world and the worldly background against which a sentence is used determine only together whether that sentence is true or false, and similarly for the reference of a singular term and the satisfaction of a predicate. 1159

10.6 Criteria for truth and falsity in terms of two variable domains We return now to the problem of modeling the internal stance on existence without identifiability. When we last looked at this problem, we found ourselves to be torn in two directions concerning the variable domain. Are we to include non-identifiable existents in the domain or not? That was the question – but both answers seem unsatisfactory. If we do not include non-identifiable existents, then we cannot represent the internal stance on existence without identifiability. If we do include them, then we appear to contradict ourselves by what we say, because to be in the domain understood as the domain of discourse is to be the possible object of talk and hence to be identifiable. 1160 We can see now, however, that underlying this dilemma about the extent of the domain was indecision about our material understanding of the domain. But the need to accommodate two different understandings no longer forces a dilemma upon us because now we are working within a dual framework. Figuratively speaking, it gives us enough flexibility to move in two directions without having to feel torn apart. In order to solve the problem at hand, let us develop our dual framework one step further. What we will need is the following: The distinction between a context understood as a circumstance of evaluation and a context understood as a context of use opens up the possibility of working with two variable domains for each 1159

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It is important not to confuse our characterization of how truth etc. are determined by both the linguistic input and the worldly input into the two-dimensional semantics with a common insight according to which truth depends on both meaning and fact. The common insight presupposes a syntacticist understanding of language, where the truth of a sentence construed as a string of letters does indeed vary both with how it is interpreted and with what is the case. We, however, make our characterization on the basis of a semanticist understanding of language, where the truth of a sentence that is already construed as meaningful may vary with both the situation it is used in and with what is the case. For the distinction between syntacticism and semanticism, cf. section 3.1. Cf. section 9.4.

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context, one of them relevant to the worldly side of the framework, and the other relevant to its linguistic side. Let us speak briefly of the circumstantial domain of the context c for the domain of the context c understood as a circumstance of evaluation. And let us speak of the contextual domain of the context c for the domain of the context c understood as a context of use. Later we will characterize the objects that make up the circumstantial domain and the contextual domain in our application of the dual framework. But we can see already on the basis of the function that they have in general that the contextual domain will always be a subcollection of the circumstantial domain. For the circumstantial domain is comprised by what can figure in states of affairs and the contextual domain is comprised by what can figure in sentences. And surely nothing can figure in a (declarative) sentence relative to a context when it cannot figure in a corresponding state of affairs relative to that context! To illustrate the respective contribution that the circumstantial domain and the contextual domain make to the dual framework, let us look at the respective roles they play in the criteria for the truth of a monadic sentence in the intensional semantics and in the two-dimensional semantics: (Int+ ) dual ‘F(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (a) a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c is in the circumstantial domain of the context c. (2D+ ) dual A sentence ‘F(a)’ is true with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i) ‘F(a)’ is true with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is in the contextual domain of the context d. These criteria differ from the preliminary ones only insofar as the variable domain has been specified: The criterion for truth in the intensional semantics (Int+ ) 1161 prelim + is turned into (Intdual ) by specifying the domain referred to in clause (b) as the circumstantial domain. The criterion for truth in the two-dimensional semantics (2D+ ) 1162 is turned into (2D+ ) by specifying the domain referred to in clause prelim dual (iii) as the contextual domain. These different specifications of the variable domains are what enables us to move on, from the two preliminary variants which

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Cf. section 10.2. Cf. section 10.3.

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(strictly speaking) could be used only separately, 1163 to two criteria that are two complementary parts of the dual framework (hence the subscript ‘dual’). When we combine the two criteria (Int+ ) and (2D+ ), we get: dual dual A sentence ‘F(a)’ is true with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i-a) a0 -of-context-c is F0 relative to the context c, (i-b) a0 -of-context-c is in the circumstantial domain of the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is in the contextual domain of the context d. 1164 This principle about truth in the two-dimensional semantics, when combined with the bridge principle (Diag+), entails the following principle about relative truth: A sentence ‘F(a)’ is true relative to a context c if and only if there is an object a0 -of-context-c such that (i-a) a0 -of-context-c is F0 relative to the context c, (i-b) a0 -of-context-c is in the circumstantial domain of the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is in the contextual domain of the context c. As the contextual domain of a context c is a subcollection of the circumstantial domain of the context c, (iii) makes a stronger requirement than (i-b), and we can drop the latter: (Rel+ ) dual A sentence ‘F(a)’ is true relative to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is F0 relative to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is in the contextual domain of the context c. This criterion for relative truth makes no reference anymore either to the intensional semantics or the two-dimensional semantics. Rather, it connects a context-free description of how things are with the world directly to a context-free description of how things are with language. * 1163

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We have already used the intensional semantics and the two-dimensional semantics in a complementary way in our considerations about the Tarskian equivalence, but that was due only to our decision to ignore the role of the variable domain within our preliminary presentation. The clauses (i-b) and (iii) taken together require the object in question to be in the intersection of the circumstantial domain of the context c and the contextual domain of the context d. We could use this fact to define this intersection as the two-dimensional domain with respect to c and at d, but we will not bother. Our main interest is in what happens on the diagonal, where things will again be simpler insofar as they can be described in terms of the contextual domain alone.

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In the same way that led us to the criteria for truth (Int+ ), (2D+ ), and (Rel+ ), dual dual dual we get the corresponding criteria for falsity in our dual framework by specifying the respective domain in the preliminary criteria for falsity: 1165 (Int− ) dual ‘F(a)’ is false with regard to a context c if and only if there is an object a0 -of-context-c such that (a) it is not the case that a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c is in the circumstantial domain of the context c. (2D− ) dual A sentence ‘F(a)’ is false with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i) ‘F(a)’ is false with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is in the contextual domain of the context d. (Rel− ) dual A sentence ‘F(a)’ is false relative to a context c if and only if there is an object a0 -of-context-c such that (i) it is not the case that a0 -of-context-c is F0 relative to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is in the contextual domain of the context c. Putting these criteria for falsity beside the respective criteria for truth shows that now various ways open up in which truth value gaps can occur. We will comment on this later, 1166 because with this overview we want to conclude the general presentation of our dual framework for modeling the interaction of a language and the world it is about, and turn back to the specific problem that motivated us to develop it in the first place.

10.7 The two domains in terms of existence and identifiability The distinction between the circumstantial domain and the contextual domain has enabled us to formulate criteria for truth and falsity in the intensional semantics and in the two-dimensional semantics that interact in a way that is appropriate to our dual framework. Now we will see that the ensuing possibility of the distinctness of the circumstantial domain and the contextual domain of a given context will allow us to solve the problem of accounting for existence without identifiability. 1165 1166

Cf. sections 10.2 and 10.3. Cf. section 10.10.

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Given our understanding of what the two semantic theories model, the circumstantial domain of a context will encompass the objects that can figure in the states of affairs that obtain relative to that context and the contextual domain will encompass the objects that can figure in sentences that are used at that context. How can this general characterization of the two variable domains be spelled out in terms of the two notions that pertain to the problem we want to solve, existence and identifiability? First off, the circumstantial domain of a context should include all objects that exist relative to that context, be they identifiable or not, because whatever exists can figure in a state of affairs that obtains (e. g., in the state of affairs that it exists). In addition, the circumstantial domain of a context should also include all objects that are identifiable relative to that context, even if they do not exist relative to it. In case this second point needs some evidential support, let us return to the paradigm of time and in particular to Prior’s thought experiment. Reflection about what Prior said about Caesar, Antony, and Cleopatra allowed us to show that we can refer to past individuals despite their present non-existence. 1167 Accordingly, it is unproblematic to say that the tensed sentence ‘A long time ago, Caesar dallied with Cleopatra’ is true today. But there is also no problem in saying that the tensed state of affairs that a long time ago, Caesar dallied with Cleopatra, obtains today. To give further examples for present facts that involve past objects, note that Socrates is still famous many centuries after his death and that Frege became famous only after his death. So past objects can figure in present states of affairs. Thirdly, we need to say something about which objects not to include in the circumstantial domain. Staying for the moment with the temporal paradigm, it should be clear that future objects cannot figure in present states of affairs. There is just nothing there yet for anything to be the case with it! When we first used Prior’s thought experiment, we understood it with him as showing that the imagined prophet could not refer to Caesar, Antony, and Cleopatra, so that they could not have figured in any of the prophet’s sentences. 1168 By now we are more aware of the distinction between a context-sensitive language and the contextualist metaphysics it is about, and in particular more aware of the distinction between the objects that could figure in the one and the objects that could figure in the other relative to a given moment. 1169 On this basis, we can ask whether the thought experiment (or a variation thereon) can also show that Caesar, Antony, and Cleopatra could not even have figured in any tensed state of affairs that might have obtained at the prophet’s time. So suppose that with his hindsight Prior wants to evaluate the prophet’s claims, and in his attempt to do so, hypothesizes that at the time of the prophecy, it was already the case with some future objects that one of them was going to be a person called ‘Caesar’ who would be murdered, etc., and another of these future objects was going to be a person called ‘Antony’, who

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Cf. section 9.3. Cf. section 9.2. Cf. section 10.1.

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would dally with Cleopatra (a third future object), etc. Then suppose Prior to hesitate and say: ‘No, I’m not sure now that it was like that – perhaps it was the second of the future objects I mentioned that was going to be a person called ‘Caesar’ and be murdered, etc., and the first future object that was going to be a person born later and called ‘Antony’, etc.’ This switch, it seems to me, would have been as spurious as the one Prior imagined in his original thought experiment. And it would be senseless to ask which of the two distinct future tensed states of affairs imagined by Prior really did obtain at the time of the prophet. Caesar, Antony, and Cleopatra were simply not there yet for anything to be the case with them, not even in the future tense. 1170 We will take this to show that future objects are not only not in the contextual domain – they are not even in the circumstantial domain. 1171 What distinguishes future objects on the one side from present objects and past objects on the other side is that they are neither identifiable nor existent. Generalizing, and putting things together, we see that the circumstantial domain of a context should encompass just those objects that are identifiable relative to that context or exist relative to it (or both); thus the circumstantial domain of a context is the union of what is identifiable relative to that context and what exists relative to that context. With regard to the contextual domain, our initial intuition about the formal semantics of a logic of existence and identifiability 1172 is still valid: The contextual domain of a context should encompass just those objects that are identifiable relative to it, for they are the objects that can figure in sentences that are true (or false) relative to that context. This is really what we meant with the notion of identifiability all along: To be a candidate for being talked about – or in other words, to belong to the universe of discourse. Let us sum up what we argued for here in the following two principles: (Circumstantial Domain) An object x is in the circumstantial domain of a context c if and only if the object x exists relative to the context c or the object x is identifiable relative to the context c.

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This variation of Prior’s thought experiment is written in close analogy to the original passage (Prior 1960, 690) that is quoted in section 9.2. ‘There just are no future objects!’, we might want to exclaim, and not as an expression of a pessimistic outlook, but of our awareness of the current limits of the changing world we live in. And we would be right – in our technical terms, the sentence is true with regard to every moment (and relative to every moment). Strictly speaking, the work done in our deliberations by the term ‘future objects’ would always need to be cashed out in terms either of later hindsight on the moment in question (as in our variation of Prior’s thought experiment) or of a (fictional) tenseless external representation, in which case the relevant distinction would really be between the earlier objects, simultaneous objects, and later objects of a moment. Cf. section 9.4.

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(Contextual Domain) An object x is in the contextual domain of a context c if and only if the object x is identifiable relative to the context c. According to this construal of the circumstantial domain as encompassing both what exists and what is identifiable (as well as what exists and is identifiable) and the contextual domain as encompassing what is identifiable (but nothing else), the two domains relate in the following way: In all scenarios, the contextual domain is a subcollection of the circumstantial domain. (We have already argued for this on more general grounds.) What is more, in scenarios where existence entails identifiability, the circumstantial domain and the contextual domain are equivalent in the sense of being identical for every context. Thus our first approximation for a formal semantics of existence and identifiability, with its single domain comprised of the identifiabilia, 1173 turns out to be a limit case of the present dual framework.

10.8 Criteria for truth and falsity in terms of existence and identifiability On the basis of the two principles (Circumstantial Domain) and (Contextual Domain), 1174 as well as the general formulation of criteria for truth in the dual framework, 1175 we finally arrive at the specific formulation of the criteria for truth and falsity that is relevant for our application of the dual framework to the problem of existence without identifiability. 1176 We present the criteria in the intensional semantics and in the two-dimensional semantics together with the resulting criterion in the relational semantics: (Int+) ‘F(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (a) a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c exists or is identifiable relative to the context c. (2D+) A sentence ‘F(a)’ is true with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that

1173 1174 1175 1176

Cf. section 9.4. Cf. section 10.7. Cf. section 10.6. We drop the subscript ‘dual’ because these are the variants of the criteria that we will refer back to.

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(i) ‘F(a)’ is true with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is identifiable relative to the context d. (Rel+) A sentence ‘F(a)’ is true relative to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is F0 relative to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is identifiable relative to the context c. The corresponding criteria for falsity differ again only in the respective first clause: (Int–) ‘F(a)’ is false with regard to a context c if and only if there is an object a0 -of-context-c such that (a) it is not the case that a0 -of-context-c is F0 relative to the context c and (b) a0 -of-context-c exists or is identifiable relative to the context c. (2D–) A sentence ‘F(a)’ is false with respect to a context c and at a context d if and only if there is an object a0 -of-context-c such that (i) ‘F(a)’ is false with regard to the context c, (ii) ‘F(a)’ exists relative to the context d, and (iii) a0 -of-context-c is identifiable relative to the context d. (Rel–) A sentence ‘F(a)’ is false relative to a context c if and only if there is an object a0 -of-context-c such that (i) it is not the case that a0 -of-context-c is F0 relative to the context c, (ii) ‘F(a)’ exists relative to the context c, and (iii) a0 -of-context-c is identifiable relative to the context c. It is characteristic of the criteria for truth and falsity in this formulation that all talk of the two variable domains has been dropped in favor of talk of existence and identifiability. That is a precondition for transferring the representation of truth and falsity into the context-sensitive object language, which we will need to do (in sections 10.9 and 10.10) in order to solve the problem of representing the internal stance on existence without identifiability (in section 10.11).

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10.9 The Generalized Truth Equivalence Now, to internalize the representation of truth and falsity, we need to start on another round through the square of semantic theories, 1177 which begins with the step from the relational semantics which include the criteria (Rel+) and (Rel–) to the contextual semantics which include the intensional semantics of sentences that ascribe truth or falsity. To take this step, we need the standard translation of the predicates of existence and identifiability from the context-free meta-language (where they express a relation between an object and a context) to the context-sensitive object language (where they express monadic concepts that apply to objects). As instances of our general criterion for truth in the intensional semantics (Int+), 1178 we get the following two principles governing the existence predicate ‘E!(. . . )’ and the identifiability predicate ‘I!(. . .)’ of the object language: 1179 ‘E!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c exists relative to the context c and (ii) a0 -of-context-c exists or is identifiable relative to the context c. ‘I!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is identifiable relative to the context c and (ii) a0 -of-context-c exists or is identifiable relative to the context c. These are easily simplified: (E!-Int+) ‘E!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c that exists relative to the context c. (I!-Int+) ‘I!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c that is identifiable relative to the context c. Of course we also need the standard translation of the truth predicate here, and later we will need the standard translation of the falsity predicate. We phrase them here in a way that is warranted by the principle (Circumstantial Domain): 1180 1177 1178 1179 1180

Cf. section 10.5. Cf. section 10.8. We have introduced these predicates of the object language in an informal way in section 9.4. We gave preliminary variants of the standard translation for truth and falsity, (True-Int+ ) and prelim (True-Int− ), in section 10.4. prelim

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(True-Int+) ‘True(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is true relative to the context c and (ii) a0 -of-context-c exists or is identifiable relative to the context c. (False-Int+) ‘False(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c such that (i) a0 -of-context-c is false relative to the context c and (ii) a0 -of-context-c exists or is identifiable relative to the context c. Putting (E!-Int+), (I!-Int+), and (True-Int+) together allows us to translate the principle (Rel+) which is formulated in the linguistic fragment of the context-free meta-language into the linguistic fragment of the context-sensitive object language. We give a formulation with formal quote marks, ‘«. . . »’ 1181 that concerns a monadic sentence of a formal object language: (Generalized Truth Equivalence) ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with regard to the context c, (ii) ‘E!(«F(a)»)’ is true with regard to the context c, and (iii) ‘I!(a)’ is true with regard to the context c. Given the natural intensional semantics for conjunction, 1182 we could also write: ‘True(«F(a)»)’ is true with regard to a context c if and only if ‘F(a) ∧ E!(«F(a)») ∧ I!(a)’ is true with regard to the context c. Or, if we transpose the first formulation into a scenario with a natural language: The object language sentence ‘‘F(a)’ is true’ is true with regard to a context c if and only if (i) the object language sentence ‘F(a)’ is true with regard to the context c, (ii) the object language sentence ‘‘F(a)’ exists’ is true with regard to the context c, and 1181 1182

Cf. section 5.7. Under the most natural intensional semantics of conjunction, (∧-Int), a conjunctive statement is true if and only if both of its sides are true, and it is false if and only if one of its sides is true and the other is false or both of its sides are false. Hence a conjunctive statement is gappy if and only if at least one of its sides is gappy – gappiness spreads. This criterion for conjunction is inspired by the criteria that are given in Dunn’s semantics for FDE for the connectives of conjunction, disjunction, and negation; cf. Priest 2008, 143. More precisely, the criterion belongs to the logic FDE↔ that is characterized in Petrukhin 2017, section 2.

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(iii) the object language sentence ‘a is identifiable’ is true with regard to the context c. To illustrate how the (Generalized Truth Equivalence) works, and to make plausible that it does what it is meant to do, let us apply its natural language variant to our stock example. There it entails the following: ‘‘Obama is president of the U.S.’ is true’ is true with regard to a moment m if and only if (i) ‘Obama is president of the U.S.’ is true with regard to the moment m, (ii) ‘‘Obama is president of the U.S.’ exists’ is true with regard to the moment m, and (iii) ‘Obama is identifiable’ is true with regard to the moment m. This particular prognosis of the (Generalized Truth Equivalence) is clearly correct. ‘‘Obama is president of the U.S.’ is true’ and the sentence mentioned in clause (i), ‘Obama is president of the U.S.’, are true with regard to the same moments (the days from January, 20 in 2009 until January, 19 in 2017). The sentences mentioned in clause (ii) and in clause (iii), respectively, are true with regard to the moments of a longer interval (starting from some time in 1961, and going on into the foreseeable future). Therefore clause (ii) and (iii) do not make a difference in this case, and the entire equivalence holds indeed omnitemporally. In the formulation of the (Generalized Truth Equivalence) that we give here, we use a specific device, quotation, to refer to the monadic sentence that is the object of the ascription of truth in question. There is a peculiarity of the referential device of quotation that calls for a remark about the sentence mentioned in clause (ii) the (Generalized Truth Equivalence), ‘E!(«F(a)»)’ (or, ‘‘F(a)’ exists’). As a quotation expression (in contrast to a name or a typical description) presents the expression that it refers to, 1183 mentioning this sentence in clause (ii) may appear to be selfverifying. For how can we present something without thereby ensuring its existence? But in the case at hand, this would be a misunderstanding. Recall that we need to distinguish between the object language that is part of our formal framework and the language that it models in our material understanding, and recall also that we are still working within the intensional semantics here. The expressions of the formal object language exist, anyway; what the variable extension of the existence predicates tracks is the existence of the expressions of the language that is modeled by it. As we are in the intensional semantics, an ascription of existence to an expression could actually fail to be true with regard to a context. 1184 It is only on the diagonal of the two1183 1184

Cf. section 5.5. Let us complement the criterion for the truth of existence-ascriptions that we argued for at the beginning this section with a criterion for their falsity, which is an instance of (Int–) that has also been simplified: (E!-Int+) ‘E!(a)’ is true with regard to a context c if and only if the object ‘a’ is meant to refer to exists relative to the context c.

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dimensional semantics and in the relational semantics, which model the behavior of the language in question, that the second conjunct must always be true. The reason for this is that in the two-dimensional semantics and the ensuing relational semantics, the existence of a sentence is a precondition for its being true. Here we can therefore indeed argue in accordance with our intuition. 1185 There is an important difference between the criterion for relative truth (Rel+) of the previous section and the (Generalized Truth Equivalence) that is entailed by it: Where the former is entirely context-free on its right hand side, the latter employs the truth of a context-sensitive sentence with regard to the context in question. Both principles describe how truth correlates not only with what is the case, but with what is the case taken together with both the existence of a sentence and the identifiability of an object. But of the two, only the (Generalized Truth Equivalence) takes the internal stance on both sides of this correlation. 1186 This will later be important in our approach to the Liar paradox; 1187 here it moves us close to a solution of the problem of existence without identifiability. But before we turn to that (in section 10.11), we should deal with some matters that are of more general interest: a comparison with the Tarskian equivalence (in the rest of this section) and the matter of truth value gaps (in the following section 10.10).

(E!-Int–) ‘E!(a)’ is true with regard to a context c if and only if the object ‘a’ is meant to refer to does not exist but is identifiable relative to the context c.

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A brief look at (E!-Int+) shows that ‘E!(«F(a)»)’ is not true with regard to a context relative to which ‘F(a)’ does not exist. As a consequence of (E!-Int–), it is also the case that ‘E!(«F(a)»)’ is not false with regard to such a context, either. In fact, an ascription of existence to an expression cannot plausibly be false with regard to any context. For as identifiability cannot be prior to existence and given that expressions are persistent, an expression exists whenever it is identifiable (in contrast to anything that ceases to exist, like humans). Thus the condition that is the left hand side of (E!-Int–) cannot ever be met by an expression (in contrast, again, to dead people and so on; e. g., the sentence ‘Socrates exists’ is false with regard to the year 2016 because Socrates is identifiable but non-existent relative to that year). We need not go back to the relational semantics to represent the non-existence of an expression, though, because the untruth of a sentence like ‘E!(«F(a)»)’ in the intensional semantics is enough to carry the information that ‘F(a)’ does not exist relative to the context in question. There is a clear correlation: In the intensional semantics, an existence-ascription to an expression is true where the expression exists and gappy where the expression does not exist. However, although this artifice does the work, it is far from elegant. In principle, a way of talking about expressions would be preferable that is more neutral with regard to the question of their existence. But as the task of devising such a way of talking is not trivial from a technical point of view, we will leave it for another occasion. An extra principle that we should make explicit is that when an expression exists relative to a context, so does each of its parts. Hence the existence of the sentence ‘E!(«F(a)»)’ relative to a context c entails the existence of the quotation expression ‘«F(a)»’ relative to c, which because of the particularities of quotation entails the existence of the quoted sentence ‘F(a)’ that it contains. Cf. the principle (Containment) in section 5.5. Showing that the (Generalized Truth Equivalence) holds is a big second step on the way to turning the contextualist semantics (cf. section 10.4) into a full theory. Cf. section 13.5.

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We call the principle we have derived here the generalized equivalence about truth because the Tarskian equivalence can be seen as its limit case. This becomes evident when we compare the two statements for the same monadic sentence: 1188 (Generalized Truth Equivalence) ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with regard to the context c, (ii) ‘E!(«F(a)»)’ is true with regard to the context c, and (iii) ‘I!(a)’ is true with regard to the context c. (Tarskian Equivalence) ‘True(«F(a)»)’ is true with regard to a context c if and only if (i) ‘F(a)’ is true with regard to the context c. It is evident that in every scenario where the existence of expressions and the identifiability of objects is not an issue – where the sentence in question exists and the object in question is identifiable relative to every relevant context –, the (Generalized Truth Equivalence) entails the (Tarskian Equivalence). In these scenarios clause (ii) and (iii) on the right hand side of the more general principle do not make any difference and could be dropped, so that its right hand side is reduced to clause (i), which also is the right hand side of the (Tarskian Equivalence). In the usual formulation, the Tarskian equivalence has a wider range of application than the formulation that we displayed above for contrasting purposes. It applies not only to monadic sentences, but to all object language sentences whatsoever: ‘True(«. . .»)’ is true with regard to a context c if and only if ‘. . .’ is true with regard to the context c. The principle for which we needed it as a contrast, the (Generalized Truth Equivalence), is formulated only for monadic sentences of the form ‘F(a)’ because clause (iii) on its right hand side, which requires the identifiability of the object a, is not as easily extended to other kinds of sentences as the other conjuncts. Very tentatively, we might want to try something like the following, using a non-formal object language: ‘‘. . .’ is true’ is true with regard to a context c if and only if (i) ‘. . .’ is true with regard to the context c, (ii) ‘‘. . .’ exists’ is true with regard to the context c, and (iii) ‘what ‘. . .’ is about is identifiable’ is true with regard to the context c. 1188

The (Tarskian Equivalence) has been introduced in section 10.4.

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But care is needed in concrete cases with a view to the sentence mentioned in clause (iii), ‘what ‘. . .’ is about is identifiable’. It is phrased in a deliberately open way here because this clause will have to work differently for atomic sentences and for quantified sentences. The reason behind this is best seen in an example. Relative to a context where neither Hugh the cyborg nor any other cyborg is identifiable yet, but all the relevant sentences exist already, ‘There are cyborgs’ is false, 1189 but ‘Hugh is a cyborg’ is neither true nor false. Hence both ‘‘There are cyborgs’ is false’ and ‘‘Hugh is a cyborg’ is neither true nor false’ will be true with regard to that context. Thus the same lack of identifiability can lead to different truth values for a quantified and for a monadic sentence. Nevertheless, we need not expect any obstacles on the way onward from our (Generalized Truth Equivalence) for monadic sentences to a corresponding principle that applies to all sentences of the object language and that is formulated with equal precision. But in the semi-technical spirit of this study, we can allow ourselves to leave this task for another occasion, especially as our main interest is directed at monadic sentences. In the case of monadic sentences, we can already learn something important when we contrast the Generalized truth equivalence with the Tarskian equivalence: (Generalized Truth Equivalence) ‘True(«F(a)»)’ is true w.r.t. c if and only if (i) ‘F(a)’ is true w.r.t. c, (ii) ‘E!(«F(a)»)’ is true w.r.t. c, and (iii) ‘I!(a)’ is true w.r.t. c.

(Tarskian Equivalence) ‘True(«F(a)»)’ is true w.r.t. c if and only if (i) ‘F(a)’ is true w.r.t. c.

From this contrast, it is clear that in our account, there can in general be failures of the Tarskian equivalence. Whenever clause (i) of the (Generalized Truth Equivalence) holds, but clause (ii) or clause (iii) does not, the (Tarskian Equivalence) will fail locally. We can discern two kinds of reason for a failure: firstly, the non-existence of the monadic sentence in question, and secondly, the non-identifiability of the object it is meant to refer to. In effect, we have already discussed failures that are due to the fact that the sentence in question does not exist in the dinosaur example, although for the case of a quantified sentence. Then we contrasted the sentence ‘There are dinosaurs’ being true with regard to the year one hundred million BCE with its being neither true nor false relative to that year. 1190 As we can now see, the corresponding truth ascription ‘‘There are dinosaurs’ is true’ is also neither true nor false with regard to the year one hundred million BCE, and the Tarskian equivalence for the sentence ‘There are dinosaurs’ fails. Such failures are due to the fact that, even though there is a fact of the matter, the sentence that would record it does not exist to be true or false. In addition, there can in principle be failures of the Tarskian equivalence that are due to the fact that clause (iii) does not hold while clause (i) does hold – due, in other words, to the object in question being not identifiable although something is the matter with it. In fact, this kind of failure brings us squarely to our original problem, 1189 1190

Here we employ the intensional semantics of the existential quantifier, (∃-Int); cf. section 10.3. Cf. section 10.3.

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to the scenario of existence without identifiability. For whenever there is such a failure, clause (i) must hold, and as the object in question is not identifiable because clause (iii) does not hold, it must exist because of the principle (Circumstantial Domain). E. g., the (Tarskian Equivalence) for ‘a is not identifiable’ fails with regard to a context relative to which object a exists but is not identifiable. When we stick to this kind of failure, but move from monadic sentences to quantified sentences, it becomes clear that the Tarskian equivalence can fail even when there is no truth value gap. For consider the sentence ‘Some object is not identifiable’. With regard to a context where some object from the circumstantial domain exists but is not identifiable, it is true. But relative to that context, it is false, 1191 because every object in the contextual domain is identifiable because of the principle (Contextual Domain). Therefore the corresponding truth ascription ‘‘Some object is not identifiable’ is true’ will be false with regard to that context. 1192 Thus the Tarskian equivalence fails here not because truth meets a gap, but because truth meets falsity. 1193 Let us be clear that the fact that we endorse the (Generalized Truth Equivalence) means that we have to accept that there can be failures of the (Tarskian Equivalence). The (Generalized Truth Equivalence), however, follows from principles we have found good reason to endorse. Hence we have equally good reason to accept the ensuing possibility of local failures of the (Tarskian Equivalence). But there is more to be said. In the context of an investigation about the Liar paradox it is important to be clear that our deviation from a global endorsement of the contextualist cousin of the Tarskian truth schema is not an ad hoc move with the aim of sidestepping contradiction, but motivated by the following independent factors: our understanding of the intensional semantics as modeling facts about the world and the two-dimensional semantics as modeling facts about a language that is about it (i. e., our use of the dual framework developed in this section), our conviction that the semantic expressions of a language form a contextualist metaphysics (argued for in chapter 8), and our suspicion, gleaned from the paradigm of tensed reality, that in a contextualist metaphysics cases cannot be ruled out where there is existence without identifiability (which arose in chapter 9). * And yet, we are accustomed to thinking that something like the Tarskian equivalence holds globally. Therefore it is welcome that there is a phenomenon which 1191 1192

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We presuppose here that the sentence exists relative to the context in question. This is in accordance with the fact that the corresponding falsity ascription ‘‘Some object is not identifiable’ is false’ will be true with regard to that context. For this reason, the failure of the Tarskian equivalence in our account cannot be dealt with by a simple restriction of the Tarskian truth schema, as in the conditionalized variants: If . . . or it is not the case that . . . , then ‘. . . ’ is true if and only if . . . . If ‘. . . ’ is true or ‘. . . ’ is false, then ‘. . . ’ is true if and only if . . . . Cf., e. g., Slater 2004a.

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promises to make its local failures more palatable. It will become visible when we move on yet again and extend our second round through the square of semantic theories, stepping from the intensional semantics of truth ascriptions (a part of the contextualist semantics) to the two-dimensional semantics of truth ascriptions and then to their relational semantics. 1194 Then we can compare the Tarskian equivalence, which is a principle of the intensional semantics, to its counterpart in the relational semantics: (Tarskian Equivalence in the Relational Semantics) ‘True(«F(a)»)’ is true relative to a context c if and only if ‘F(a)’ is true relative to the context c. On the face of it, this looks very similar to the (Tarskian Equivalence); it is only that both occurrences of the preposition ‘with regard to’ have been changed to ‘relative to’. But the slightly changed statement models something entirely different. The (Tarskian Equivalence) is a statement of the intensional semantics and therefore it gives an internal representation of a specific correlation that might hold between what is the case with the language and what is the case with the world (a correlation that on our account does not hold in general). In contrast, the (Tarskian Equivalence in the Relational Semantics) models externally what we can express in the language about its correlation with the world. And for the very reason that there are failures of the (Tarskian Equivalence), we cannot expect that what is the case and what we can express about what is the case are in general the same! This means that the (Tarskian Equivalence) need not pass on its failures to the (Tarskian Equivalence in the Relational Semantics). In fact, we can show that although the (Tarskian Equivalence in the Relational Semantics) will develop some minor failures of its own, it does not inherit any of the failures of the (Tarskian Equivalence). We can study the behavior of the (Tarskian Equivalence in the Relational Semantics) by applying the criterion for relative truth (Rel+) 1195 to its left hand side: ‘True(«F(a)»)’ is true relative to the context c if and only if (i) ‘F(a)’ is true relative to the context c, (ii) ‘True(«F(a)»)’ exists relative to the context c, and (iii) ‘F(a)’ is identifiable relative to the context c. This application of our criterion for relative truth is warranted because ‘True(«F(a)»)’ is a monadic sentence, so that we can instantiate the schematic predicate letter ‘F’ in (Rel+) as the truth predicate ‘True(. . .)’ and the schematic term ‘a’ as the quotation expression ‘«F(a)»’. 1196 1194 1195 1196

Fear not, for that is the farthest we will go around the square of semantic theories on this occasion. Cf. section 10.8 For this, we do not even have to think of the range of (Rel+) as extended from the worldly part of the object language to a separate linguistic part of that language (as we did when we extended the range of (Int+) in order to give the intensional semantics of truth ascriptions), because now, at last,

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From a comparison with this result, it is evident that the (Tarskian Equivalence in the Relational Semantics) fails if and only if clause (i) holds but clause (ii) or (iii) does not hold. Now mind you, this is not impossible on our account! We are talking about a context relative to which ‘F(a)’ exists but either ‘True(«F(a)»)’ does not exist or ‘F(a)’ is not identifiable (or both). As we endorse a contextualist metaphysics of expressions, nothing speaks against expressions coming into existence one by one, so that it is conceivable that ‘True(«F(a)»)’ comes to exist only after the quoted sentence ‘F(a)’ exists already. And as we try to open the door in this chapter for scenarios where an object exists but is not identifiable, nothing speaks against identifiability being subsequent to existence in the case of expressions, so that it is conceivable that the sentence ‘F(a)’ comes to be identifiable only after it exists already. Actually, we will in due course argue for a principle that requires just that, that a directly referential expression exists only later than its extension and that any object is only identifiable after it exists. 1197 Thus the present account does not preclude contexts where the (Tarskian Equivalence in the Relational Semantics) fails, and our own theory will require that there are such contexts. But even so, from the present perspective these failures are of much less consequence than those of the (Tarskian Equivalence). Plausibly, we are talking about only one or two contexts relative to which ‘True(«F(a)»)’ is not true but relative to which ‘F(a)’ is true, 1198 in contrast to millions of years with regard to which ‘‘There are dinosaurs’ is true’ fails to be true (due to the non-existence of sentences), but with regard to which ‘There are dinosaurs’ is true (due to the existence of dinosaurs). Or to look at the matter from the positive side: Relative to every one of all those contexts where it is possible to say the things that the two sides of the (Tarskian Equivalence in the Relational Semantics) model as being said, it does indeed hold! For it is only possible to say those things when all the mentioned expressions exist, and when ‘True(«F(a)»)’ exists, ‘F(a)’ must also exist, 1199 and ‘F(a)’ must arguably be identifiable then because of the peculiarities of quotation. 1200

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the object language has indeed become self-referential: By giving the two-dimensional semantics and the relational semantics of a sentence like ‘True(«F(a)»)’, we are modeling that the language of our material understanding that is represented by the formal object language is used to talk about its own expressions. Cf. chapters 11 and 12, respectively. It is plausible that if the principle that requires identifiability to be subsequent to existence is true, then there will nonetheless be no delay of any length. Rather, at least in the case of the successive introduction of expressions we can think of it like this: One moment, an object exists, and immediately afterwards, it is identifiable (and it stays identifiable at least as long as it exists). We have already made a background principle explicit according which when an expression exists relative to a context, so does each of its parts. Hence the existence of the sentence ‘True(«F(a)»)’ relative to a context c entails the existence of the quotation expression ‘«F(a)»’ relative to the context c, which because of the particularities of quotation entails the existence of the quoted sentence ‘F(a)’ that it contains. As reference by quotation is super-direct, a quotation expression presents the quoted expression (cf. section 5.5). Therefore the quoted expression must not only exist relative to all contexts relative to which the quotation expression exists, as entailed by the principle of (Containment), but it must also be identifiable there. For how can we present something without identifying it? – Note that this argument will not work for non-homophonic variants of the Tarskian equivalence, i. e., variants that

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We see that the (Tarskian Equivalence in the Relational Semantics) has its own failures, and that these are minor from our present perspective. But why does it not inherit the not so minor failures of the (Tarskian Equivalence)? These failures occur when the sentence ‘F(a)’ is true with regard to the context in question, but either the sentence ‘F(a)’ does not exist or the object ‘a’ is to refer to is not identifiable relative to that context. But as (Rel+) requires both the existence of the sentence ‘F(a)’ and the identifiability of the object ‘a’ is to refer to, the sentence ‘F(a)’ cannot be true relative to such contexts. Inspecting our above result again we see that thus neither side of the (Tarskian Equivalence in the Relational Semantics) can hold for such contexts. In sum, the contexts that are counterexamples to the (Tarskian Equivalence) do not constitute counterexamples to the (Tarskian Equivalence in the Relational Semantics). 1201 And this should not be too surprising, either. The failures of the (Tarskian Equivalence) result from a mismatch in the correlation between what is the case and what can be represented. Moving on to the relational semantics means moving on to what can be represented of this correlation, and the mismatch gets lost on the way because it can itself not be represented. In a sense we thus recapture the Tarskian equivalence in the relational semantics. Focusing on the main reason for this – the non-existence of the mentioned sentences – also sheds a different light on the failures of the Tarskian equivalence in the intensional semantics, allowing to see them in a more favorable way. To recapitulate: When a local failure of the Tarskian equivalence is brought about by the non-existence of the sentence in question, then a sentence that ascribes truth to the sentence that is mentioned on the left hand side of the equivalence does not exist, either. (We can argue for this from the peculiarities of quotation for the homophonic variant of the Tarskian equivalence, or in the many cases that are like the dinosaur example from the non-existence of any expression.) This promises to soften the blow considerably. In our example, ‘‘There are dinosaurs’ is true’ fails to be true with regard to the year one hundred million BCE, but like ‘There are dinosaurs’ (which is true with regard to that year) it also fails to exist relative to that year. In cases like this, the failure of the Tarskian equivalence is no more than an artifact of the formal framework we use to model the behavior of language and world: The formal theory makes a prognosis about the truth values of sentences even with regard to contexts where these sentences do not exist. At least according to our material understanding of the theory, there is nothing there to fail to be true (or false), and we are warranted to count such failures of the Tarskian equivalence as immaterial. 1202 This explains

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employ a device other than quotation to refer to the sentence in question. But even then there will be no major problem, again because for an expression, the state of being existent but not identifiable is so fleeting that for the vast majority of contexts where an expression exists it is identifiable, too. E. g., the year one hundred million BCE provides a counterexample to the (Tarskian Equivalence) for ‘There are dinosaurs’ because there were dinosaurs then but not the corresponding sentence to record it. The same year does not constitute a counterexample to the (Tarskian Equivalence in the Relational Semantics), because relative to it, the sentence ‘‘There are dinosaurs’ is true’ was just as non-existent as the sentence ‘There are dinosaurs’. They are immaterial in the double sense that, firstly, the relevant sentence does not exist, and secondly, they (therefore) do not correspond to anything on the material level.

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why we recapture the Tarskian equivalence in the relational semantics: As long as we restrict our attention to its failures (in the intensional semantics) that are due to the non-existence of expressions, every failure of the Tarskian equivalence to be true is immaterial 1203 – in other words, it is true whenever the relevant sentences exist! So it is small wonder when some people initially did not expect it to fail, ever. To make the point in a different way: Whenever we can talk freely about the weather, Tarski’s biconditional “‘It is snowing’ is a true sentence if and only if it is snowing” 1204 is a true sentence. Even though the biconditional is not true with regard to every moment in time (it has failed to be true with regard to every moment before the existence of the English language), it is true relative to every moment when it exists. This follows from the (Generalized Truth Equivalence) together with the intensional semantics of the biconditional, (↔-Int+). 1205 As a Tarski biconditional is true whenever we are free to pronounce it, it is small wonder when many people did not expect the Tarskian truth schema to fail, ever.

10.10 Different kinds of truth value gaps We have already had several brief encounters with truth value gaps – sentences that are neither true nor false in relation to a certain context. Now is the time to have a more principled look at them. As we are dealing with truth and falsity separately in all the semantic theories, truth value gaps can occur in each of these theories. In the intensional semantics, we can say that a sentence is gappy with regard to a certain context if and only if it is neither true nor false with regard to that context. And we can give similar characterizations, mutatis mutandis, for gappiness in the two-dimensional semantics and gappiness in the relational semantics. Because of the interconnections between the formal theories, gaps from all the theories come together in the contextualist semantics, 1206 which will be our main focus here.

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There can also be cases where the Tarskian equivalence holds immaterially. E. g., the following biconditional statement of the meta-language is indeed true (because both of its sides are false): ‘‘The king of France is bald’ is true’ is true with regard to the year one hundred million BCE if and only if ‘The king of France is bald’ is true with regard to the year one hundred million BCE.

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When for a certain state of affairs there is neither a fact of the matter nor an existent sentence to express it (or to anticipate the terms of section 10.10, when a metaphysical gap meets a semiotic gap), the Tarskian equivalence will hold locally. But it holds only because the two kinds of absence cancel each other out, and we can count such cases as immaterial. Tarski 1956[1935], 156. We are again presupposing that whenever the quotation expression of an expression exists, the quoted expression is identifiable. To see that gaps are always passed along from the intensional semantics via the diagonal of the twodimensional semantics into the relational semantics, we need only note how truth and falsity in the intensional semantics (the respective left hand side of (Int+) and (Int–)) recurs as the first clause in

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The first thing to do is to complement the (Generalized Truth Equivalence) with a corresponding principle about falsity. Just as the former arose from asking for the intensional semantics of truth ascriptions, the latter will arise from asking for the intensional semantics of falsity-ascriptions. We again focus on monadic sentences. Let us be quicker this time! We already have a criterion for the ascription of falsity to a monadic sentence ‘F(a)’ in the relational semantics, (Rel–), 1207 and a principle that allows to translate it into the intensional semantics, (False-Int+). 1208 When we take these together with the intensional semantics of existence and identifiability, (E!-Int+) and (I!-Int+), 1209 we get: (Generalized Falsity Equivalence) ‘False(«F(a)»)’ is true with regard to a context c if and only if (i) ‘¬F(a)’ is true with regard to the context c, (ii) ‘E!(«F(a)»)’ is true with regard to the context c, and (iii) ‘I!(a)’ is true with regard to the context c. To show this, we also presuppose a clause from the intensional semantics of negation, (¬-Int–), according to which ‘¬F(a)’ is true with regard to a context c if and only if ‘F(a)’ is false with regard to the context c. 1210 The condition in the ensuing criterion for falsity is similar to the one in the criterion for truth except for the first clause, where a negation sign has been added to the mentioned sentence. This sets the standard for falsity high – in fact as high as the standard for truth (albeit with the opposite sign). We should be clear about the status, again: Just like the (Generalized Truth Equivalence), the (Generalized Falsity Equivalence) follows from principles we have found good reasons to endorse. 1211

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the criteria for truth and falsity in the two-dimensional semantics, (2D+) and (2D–), from where it is passed on into the criteria for truth and falsity in the relational semantics, (Rel+) and (Rel–). Cf. section 10.8. Cf. section 10.9. Cf. section 10.9. Under the most natural intensional semantics of negation, (¬-Int), the negation of a sentence s is true with regard to a context c if and only if the sentence s is false with regard to the context c and the negation of the sentence s is false with regard to the context c if and only if the sentence s is true with regard to the context c. Hence a negation is gappy if and only if the negated sentence is gappy – gappiness spreads. This criterion for negation is the same as the one given in Dunn’s semantics for FDE; cf. Priest 2008, 143. The criterion also belongs to the logic FDE↔ that is characterized in Petrukhin 2017, section 2. Although it might appear otherwise, we are still true here to what we endorsed when we discussed Aristotle’s original formulation of the naïve truth principle and the naïve falsity principle in section 2.2, that falsity is at root a mismatch with what is the case, rather than the truth of negation, as Priest for instance holds (Priest 2006a, 64). It might appear otherwise because we require in clause (i) of the (Generalized Falsity Equivalence) that ‘¬F(a)’ is true with regard to the relevant context, which does look like requiring the truth of negation. But as our dual framework consists of several semantic theories, we are working with several notions of truth and falsity. The notion of truth at play in clause (i) belongs to the intensional semantics and in our material understanding does not model truth, but

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When we juxtapose the (Generalized Truth Principle) with the (Generalized Falsity Principle) for the same monadic sentence, we can see that there are three distinct ways in which it can come about that a monadic sentence is neither true nor false. In other words, we can distinguish three kinds of truth value gaps. (Generalized Truth Equivalence) ‘True(«F(a)»)’ is true w.r.t. c if and only if (i+) ‘F(a)’ is true w.r.t. c, (ii) ‘E!(«F(a)»)’ is true w.r.t. c, and (iii) ‘I!(a)’ is true w.r.t. c.

(Generalized Falsity Equivalence) ‘False(«F(a)»)’ is true w.r.t. c if and only if (i–) ‘¬F(a)’ is true w.r.t. c, (ii) ‘E!(«F(a)»)’ is true w.r.t. c, and (iii) ‘I!(a)’ is true w.r.t. c.

From this juxtaposition, it is evident that the respective conditions of the two criteria differ only in their first clauses, (i+) and (i–). Hence the sentence ‘F(a)’ is true or false if and only if clause (i+) or clause (i–) holds and clause (ii) holds and clause (iii) holds. In slogan form, and mirroring to the three clauses: There is no truth or falsity . . . without there being a fact of the matter about what the sentence says, without existence of the sentence, or without identifiability of the object in question. Correspondingly, there are three kinds of truth value gaps. To characterize the three ways in which a sentence can be gappy more precisely, let us define: A sentence of the form ‘s is gappy’ is true with regard to a context c if and only if neither ‘s is true’ is true with regard to the context c nor ‘s is false’ is true with regard to the context c. (In the relational semantics, this corresponds to laying down that a sentence is gappy relative to a context c if and only if it is neither true relative to the context c nor false relative to the context c. But as we have progressed to the intensional semantics of ascriptions of truth and falsity by now, we prefer to take the internal stance on gappiness.) We will say that a gappy sentence is gappy in a metaphysical way if and only if its gappiness is due to a failure of both clause (i+) and clause (i–), that it is gappy in a semiotic way if and only if its gappiness is due to a failure of clause (ii), and that it is gappy in a semantic way if and only if its gappiness is due to a failure of clause (iii). Although we thus make the threefold distinction by defining adverbs that specify the predicate ‘. . .

fact. What clause (i) models therefore is indeed a mismatch with fact. If we wanted to follow Priest and others and characterize falsity as truth of negation in our dual framework, we would have to correlate the falsity predicate of the object language with the truth predicate of the object language and use a definition like the following: ‘False(«F(a)»)’ is true with regard to a context c if and only if ‘True(«¬F(a)»)’ is true with regard to the context c. But although this will be a consequence of a natural extension of our framework (we still need the two-dimensional semantics of negation, (¬-2D+), to derive it), it is not how we define falsity.

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is gappy’, we can of course nominalize and distinguish between a semantic gap, a semiotic gap, and a metaphysical gap. In both manners of speaking, the terminology commends itself because it entails the following criteria: A gap is metaphysical if and only if there is a gap already in the intensional semantics, modeling that there is no corresponding fact of matter (either way); 1212 a gap is semiotic if and only if it is due to the failure of the expression in question to exist; and a gap is semantic if and only if it is due to lack of identifiability of the object in question. Note that a gap thus can have more than one of the three characteristics, and for example be both semiotic and semantic. Whereas a metaphysical gap is already noticeable in the intensional semantics, semiotic and semantic gaps show themselves only on the diagonal of the two-dimensional semantics. All three kinds of gaps are passed on into the relational semantics. Here are some examples. The sentence ‘The king of France is bald’ is gappy both with regard to and relative to the year 1905 when we construe the description in a Fregean way, and arguably, the sentence ‘There will be a sea-battle tomorrow’ is gappy with regard to and relative to any moment if indeterminism is true (as long as the possibility of a sea-battle is still imminent). If this is so, then the gaps in question are metaphysical because they already occur in the intensional semantics which models what is the case. As we have already seen, the sentence ‘There are dinosaurs’ is gappy relative to the year one hundred million BCE. This gap is semiotic because it is due to the non-existence of the sentence at the time. 1213 To give an example for the third kind of gap, we need to anticipate what we will show officially only in the next section: That existence without identifiability is coherent. To flesh this out with a scenario that does not presuppose the theory we will later develop, 1214 let us turn again to the theory of special relativity, and in particular to its claim that nothing can move faster than light. 1215 It is at least plausible that there can be no identifiability without the possibility of a causal connection, and it could also be argued that there is no similar restriction on what is the case, in particular with a view to the existence of objects. Now suppose that an alien species lives in the Sirius system, and the first infant born there in the year 2000 has blue tentacles (which is a rare sign of great joy), and is therefore given the name ‘Bt’. Then the sentence ‘Bt has blue tentacles’, although true with regard to the year 2000, will be gappy relative to that year, at least in relation to our own location, because given the speed of light and the distance of Sirius, Bt became identifiable on Earth only late in 2008. 1216

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Cf. section 10.2. There is no corresponding metaphysical gap as the sentence is not gappy with regard to the year one hundred million BCE. Cf. chapter 12. At the end of section 9.3, we have already argued that tying identifiability to causality will lead to cases of existence without identifiability, and mentioned special relativity. This gap will be not only semantic, but also semiotic. As a name (construed as a semantic expression) cannot exist without its referent being identifiable, the sentence ‘Bt has blue tentacles’ (construed as a semantic expression) did not exist in 2000 on Earth. For an example for a semantic gap that is not also a semiotic gap, we can use the sentence ‘The first alien infant born in the Sirius system in the year

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Note that each of these three ways of how a monadic sentence can be gappy is due to the singular term being empty. In fact, we can and should make an entirely similar threefold distinction for a singular term being empty. Let us say that our usual (schematic) singular term ‘a’ is empty relative to some context c. Then we will say that it is empty in a metaphysical way if and only if it is already empty in the intensional semantics, i. e., if and only if there is no object a0 -of-context-c that is in the circumstantial domain of the context c. It is empty in a semiotic way if and only if the singular term ‘a’ does not exist relative to the context c. And it is empty in a semantic way if and only if the object a0 -of-context-c is not identifiable relative to the context c. To give examples: The description ‘the king of France’, construed in a Fregean way, is metaphysically empty relative to the year 1905; it is empty because there was no king of France then. The name ‘Obama’ is semiotically empty relative to the year 1889; it is empty because the name ‘Obama’ did not exist then. And the name ‘Bt’ is semantically empty relative to the year 2000 (on Earth); it is empty because Bt was not identifiable then (and there). 1217 For predicates, there is no phenomenon that corresponds to the emptiness of a singular term or to the gappiness of a sentence, because a predicate always has an extension, even when it is not satisfied by any object. 1218 On those accounts, however, that assign both an extension and an anti-extension to a predicate, 1219 there is a similar phenomenon for the question whether a given object satisfies a predicate, which will then lead to a monadic sentence that contains the predicate being gappy even if the singular term is not empty. For if we assign not only an extension but also an anti-extension to a predicate ‘F’, then it can in general be the case that an object that is referred to by ‘a’ falls into neither, so that ‘F(a)’ is neither true nor false. 1220 Note that although we could easily incorporate it into our framework, we

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2000 has bluish tentacles’. It existed relative to the year 2000 (on Earth), but presupposing a Fregean construal of descriptions, it was gappy then (and there). We should note that these science fiction examples are more than a playful diversion, because they serve a serious function in our overall argument: They help make plausible that there can be existence without identifiability, and in a way that is independent of our own motivation for requiring this as part of our proposal to solve the Liar paradox (cf. chapter 12). For similar considerations, cf. Strobach 2007, 111f. and 240f. Like the gappiness of the corresponding example sentence, the emptiness of ‘Bt’ is both semantic and semiotic. For a singular term that is empty in a semantic but not in a semiotic way, witness the Fregean description ‘the first alien infant born in the Sirius system in the year 2000’ relative to the year 2000 (on Earth). This is particularly clear on the standard set theoretic construal of the theory of extensions (cf. section 4.2), because then we can say that a predicate that is not satisfied by any object has the empty set as its extension. C.f., e. g., Kripke 1975, 700ff. Here we also presuppose the natural adaptation of the criteria for the truth and for the falsity of a monadic sentence, according to which ‘F(a)’ is true if and only if there is an object referred to by ‘a’ that is among the extension of ‘F’ and ‘F(a)’ is false if and only if there is an object referred to by ‘a’ that is among the anti-extension of ‘F’.

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do not need the apparatus of extensions and anti-extensions to account for gaps, because we can explain that a monadic sentence is gappy by the fact that the singular term is empty alone. 1221 * Getting back from singular terms and predicates to sentences again, we should have a look at what the different kinds of gaps will mean for our material understanding. Despite the speaking labels, the threefold distinction between metaphysical, semiotic, and semantic gaps belongs to the formal semantic theories that make up our dual framework. What does it mean for our material understanding? We work with an understanding according to which the intensional semantics models facts about the world and the two-dimensional and relational semantics model facts about language. We also work with an account of language according to which the objects that are candidates for having an extension are semantic expressions, which are distinct from syntactic expressions, and exist only relative to some contexts. 1222 Correspondingly, there are several questions that concern our material understanding of gaps. Firstly, does the absence occur already in the world, or does it occur first in the language of our material understanding? Secondly, does the relevant sentence understood as a semantic expression exist? And thirdly, if it does, does it have an extension, i. e., is it true or false? With a view to the first question, we can say that a metaphysical gap, i. e., a gap that occurs in the intensional semantics, in our material understanding models an absence in the world – what we might call the worldly absence case: Relative to a context, neither a certain state of affairs nor its negative counterpart obtains (there is no fact of the matter). 1223 If a gap is metaphysical but neither semantic nor semiotic, i. e., if there is a gap in the intensional semantics that is not reinforced in the relational semantics by a failure of clause (ii) or (iii), then there will be a corresponding gap in language (we can express something about which there is no fact of the matter). With a view to the second and third question, let us note that there can be two quite different facts about language (construed as part of the material ontology) that can correspond to the failure of sentence of our formal language to have a truth value relative to a certain context. Either the relevant sentence fails to exist, or the relevant sentence exists but has no extension. Thus we should distinguish between a non-existence case and a proper gap case. A nonexistence case occurs if and only if there is a semiotic gap in the formal framework. For a proper gap case, it is more complicated. It can occur only if there is no semiotic gap in the formal framework. But on that basis, there are two possible sources for the lack of a truth value. If a metaphysical gap is the source, then we model an existent sentence that is neither true nor false because there is no corresponding fact of the matter. If a semantic gap is the source, then we are modeling an existent 1221

1222 1223

In fact, all gaps that will show up in the following investigation of (purported) self-referential sentences will be due to the singular term being empty in some way, and not to a predicate having both an extension and an anti-extension. Cf. chapter 8. With regard to this absence we should no longer use the word ‘gap’ because we are now talking about our material understanding, and what we find there – the failure of a state of affairs and its negative counterpart to obtain – in general has nothing to do with a sentence being neither true nor false.

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sentence that is neither true nor false because it cannot represent what is (or might be) the case. With a view to a monadic sentence ‘F(a)’ that is gappy relative to a context c because the term ‘a’ is empty relative to the context c, we can be more specific and distinguish the following two different scenarios on the level of our material understanding: 1224 (Non-Existence Case) The sentence ‘F(a)’ does not exist relative to the context c because the term ‘a’ does not exist relative to the context c. (Proper Gap Case) The term ‘a’ and the sentence ‘F(a)’ exist relative to the context c, but the sentence ‘F(a)’ is neither true nor false relative to the context c because the term ‘a’ is empty relative to the context c. There is a metaphysical variant and a semantic variant of the (Proper Gap Case), depending on whether ‘a’ is metaphysically empty so that ‘F(a)’ is metaphysically gappy or ‘a’ is semantically empty so that ‘F(a)’ is semantically gappy. It is important to distinguish these kinds of absence that we can discern on the material level, because the (Non-Existence Case) and the semantic variant of the (Proper Gap Case) will correspond to the two main branches of the approach to the Liar paradox I will propose. 1225 Of the three kinds of absence that gaps in the formal framework can correspond to according to our material understanding – the worldly absence case, the nonexistence case, and the proper gap case – only the third does actually involve an existent sentence that is neither true nor false. A worldly absence case and a nonexistence case can be real enough (albeit each in a negative way), but neither models a sentence being neither true nor false in our material understanding. Thus we should be clear that we have three kinds of truth value gap on the formal level, but only one kind of truth value gap, properly so called, on the material level (hence the label). For a correct understanding of how our formal framework models language and world on the material level, we should also note that both the worldly absence cases that are due to the singular term being metaphysically empty 1226 and the nonexistence cases can be seen as equally spurious as those failures of the Tarskian 1224

1225 1226

Strictly speaking, we would have to distinguish expressions on the formal and on the material level. For there is no question about the existence of the formal sentence ‘F(a)’ or of the formal term ‘a’. What stands in question is the existence of their material counterparts. Another simplification is that we restrict our attention to scenarios where it is only the existence of singular terms and not that of predicates that stands in question. This is plausible as long as we deal only with primitive predicates, which express pure concepts. Cf. chapters 11 and 12, respectively, and section 13.1. Things would be different if we were to extend our framework by adding the apparatus of extensions and anti-extensions. Then we would be able to model that an object neither falls under a concept nor under the negative counterpart of that concept, as in the case of some metaphysical accounts of vagueness (e. g., Merricks 2001 and Barnes /Williams 2011). This metaphysical absence would not be spurious, at all, because the object would be there to be incomplete with regard to the concept.

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equivalence that we have characterized as immaterial. It is not only that neither of them constitutes a deviation from the principle of the (Exhaustivity) as materially understood, because in general, neither concerns truth or falsity in our material understanding. Beyond that, neither of them constitutes a deviation from a worldly counterpart of the principle of (Exhaustivity) which requires that reality is complete. For formal gaps that are due to a singular term being metaphysically or semiotically empty and that do not correspond to a proper gap on the material level can be characterized as artifacts of the formal theories. In the meta-language, we can always talk about expressions although these do not always exist in the world, and we can always characterize states of affairs although there is not always a corresponding fact of the matter in the world. 1227 Thus the spurious gaps are due to the exorbitant expressive resources that the formal meta-language needs because its purpose is to take the external stance on language and world. 1228

10.11 Solution to the problem of existence without identifiability The problem that initiated our development of the dual framework for modeling language and world was the following: When we are laying out the relational metaphysics, we can easily say of an object that relative to some context, it exists but is not identifiable. As we endorse a contextualist metaphysics, we would like to take the internal stance on such a scenario. But how can we do that without thereby identifying what we claim to be not identifiable? The basic idea for a solution that has guided us in the present chapter is to raise our awareness of the distinction between language and world. In a nutshell: It is one thing that it can be the case with an existent object that it is not identifiable, and quite another to say of an object that it exists but is not identifiable (which we can never do, on pain of thereby identifying it). To see how these two things can be compatible, we employ our technical apparatus where the intensional semantics model what can be the case and the two-dimensional semantics as well as the relational semantics and the contextualist semantics model what we can say. Specifically, we look at the behavior of the object language predicates of existence and of identifiability.

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With the apparatus of extensions and anti-extensions, we would also be able to model that an object falls both under a concept and under the negative counterpart of that concept, as in the case of other metaphysical accounts of vagueness (Priest 2010; Ripley 2012), or in the fictional example of Sylvan’s box (Priest 2006c; cf. section 3.9). This would be a way of explaining that there is metaphysical inconsistency (cf. section 3.9) that does not depend on self-reference. These considerations are compatible with the world being the totality of things, and not incomplete in any real way, because we ultimately take the internal stance on it. E. g., if the future is unreal as we think according to our tensed theory of time, then there being no fact of the matter corresponding to the sentence ‘There will be a sea-battle tomorrow’ does not mean that there is a hole in the fabric of reality; rather, our changing world is complete without extending into tomorrow at all. Something similar can be said about a singular term that is empty in a semiotic way. On the formal level, there is an expression without an extension, but on the material level, it does not correspond to an empty container, but to no container at all.

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To recapitulate: The context-sensitive object language has a predicate ‘E!(x)’ with the intended meaning ‘x exists’ and a predicate ‘I!(x)’ with the intended meaning ‘x is identifiable’. 1229 These are correlated via the standard translation with their relational counterparts in the context-free meta-language, ‘x exists relative to the context c’ and ‘x is identifiable relative to the context c’: 1230 (E!-Int+) ‘E!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c that exists relative to the context c. (I!-Int+) ‘I!(a)’ is true with regard to a context c if and only if there is an object a0 -of-context-c that is identifiable relative to the context c. The predicates of existence and identifiability are special insofar as they play an important role in determining the two variable domains: 1231 (Circumstantial Domain) An object x is in the circumstantial domain of a context c if and only if the object x exists relative to the context c or the object x is identifiable relative to the context c. (Contextual Domain) An object x is in the contextual domain of a context c if and only if the object x is identifiable relative to the context c. On the basis of these four principles we can see that, firstly, every object in the circumstantial domain satisfies ‘E!(x) ∨ I!(x)’, 1232 and, secondly, that every object in the contextual domain satisfies ‘I!(x)’. When we call a predicate trivial on the domain if and only if it is satisfied by every object in the domain, 1233 we can say that ‘E!(x) ∨ I!(x)’ is always trivial on the circumstantial domain and ‘I!(x)’ 1229 1230 1231 1232

1233

Cf. section 9.4. Cf. section 10.9. Cf. section 10.7. Here we presuppose the intensional semantics of disjunction. Under the most natural construal, (∨-Int), a disjunctive statement is true if and only if both of its sides are true or one of its sides is true and the other false, and it is false if and only if both of its sides are false. Hence a disjunction is gappy if and only if at least one of its sides is gappy – gappiness spreads. This criterion for disjunction is inspired by the criteria that are given in Dunn’s semantics for FDE for the connectives of conjunction, disjunction, and negation; cf. Priest 2008, 143. More precisely, the criterion belongs to the logic FDE↔ that is characterized in Petrukhin 2017, section 2. E. g., the predicate of self-identity, ‘x = x’, is always trivial in this sense.

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is always trivial on the contextual domain. Given our material understanding of the dual framework, this is as it should be. To see what the first fact models, note first what it excludes: ‘¬E!(x) ∧ ¬I!(x)’ can never be satisfied by an object in the circumstantial domain. What this means is that we cannot take the internal stance on non-existence-cum-non-identifiability. This is a generalized variant of a metaphysical insight we have argued for with a view to temporal reality: Not only can future objects not be referred to, they cannot even figure in any state of affairs that might obtain at present. 1234 The second fact – that the predicate of identifiability is trivial on the contextual domain – models an intuition that we brought up more than once in our description of the dilemma of expressing that there is existence without identifiability: That we can never truthfully say of an object that it is not identifiable. Let us now turn to cases where some of these predicates are not trivial on the respective domain, at least not according to the basic principles. The fact that the existence predicate is not trivial on the contextual domain models that we can in some cases say truthfully of an object that it does not exist, e. g., when talking about people who are dead (like Socrates and Frege). In scenarios where existence entails identifiability, the identifiability predicate will be trivial not only on the contextual domain, but also on the circumstantial domain, modeling that only identifiabilia can figure in what is the case. But if we do not make this assumption, then the identifiability predicate need not be trivial on the circumstantial domain. There might be an object in the circumstantial domain of a certain context that does not satisfy it with regard to that context – given that this object satisfies the existence predicate with regard to it. I. e., a scenario is possible where for a particular substitution instance of the term ‘a’ and a particular context c0 the following is the case: ‘E!(a) ∧ ¬I!(a)’ is true with regard to the context c0. Thus we are finally there: Now we know how to represent taking the internal stance on an object existing without being identifiable, and we know that nothing speaks against this being so in some cases. Additionally we can show that our formal framework reflects our original intuition that we cannot express this, at least not from the viewpoint of the very same context, because we also know that the following is entailed by the triviality of the identifiability predicate on the contextual domain: 1235 ‘E!(a) ∧ ¬I!(a)’ is not true relative to any context. The compatibility of the possibility of taking the internal stance on existence without identifiability with the impossibility of expressing it right away is ensured by working with distinct semantic theories: We have truth in the intensional semantics 1234 1235

Cf. section 10.7. To show this rigorously, we need to employ (∧-Int+) and (¬-Int+) to analyze the sentence ‘E!(a) ∧ ¬I!(a)’ into its parts (or we construe ‘E!(x) ∧ ¬I!(x)’ as one complex predicate that we can use to instantiate the schematic predicate in the relevant principles), and to combine the principle (Contextual Domain) either with the principles (2D+) and (Diag+), or directly with the principle (Rel+).

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and untruth in the relational semantics. This discrepancy in truth value will of course lead to a local failure of the Tarskian equivalence for the sentence ‘E!(a) ∧ ¬I!(a)’ with regard to the context c0. But this should no longer trouble us. 1236 In order to take the internal stance on the matter, let us employ the (Generalized Truth Equivalence). 1237 To make it applicable to the formal sentence that we are currently focusing on, ‘E!(a) ∧ ¬I!(a)’, we can think of the complex predicate ‘E!(x) ∧ ¬I!(x)’ as an instance of the schematic predicate ‘F(x)’. 1238 Then we get: ‘True(«E!(a) ∧ ¬I!(a)»)’ is true with regard to a context c if and only if (i) ‘E!(a) ∧ ¬I!(a)’ is true with regard to the context c, (ii) ‘E!(«E!(a) ∧ ¬I!(a)»)’ is true with regard to the context c, and (iii) ‘I!(a)’ is true with regard to the context c. Note that because of (∧-Int+), the right hand side requires that both ‘I!(a)’ and ‘¬I!(a)’ is true with regard to the context c, which because of (¬-Int) entails that both ‘I!(a)’ and ‘¬I!(a)’ is false with regard to the context c. Therefore, if ‘True(«E!(a) ∧ ¬I!(a)»)’ were indeed true with regard to a particular context c0, ‘I!(a)’ would be both true and false with regard to the context c0. This is incompatible with the principle of (Exclusivity) for truth and falsehood in the intensional semantics, which in our material understanding models the requirement that the world does not contain inconsistent states of affairs, and in particular, no inconsistent objects. 1239 That is bad enough, but things are even worse. For ‘I!(x)’ is no ordinary predicate, because it is tied directly to one of the main parts of our framework, the contextual domain. According to (I!-Int), if ‘I!(a)’ were indeed both true and false with regard to the context c0, then there would be an object that both is and is not in the contextual domain of the context c0. Thus inconsistency would not only spill over from the language we model into the world we model, but even into our formal meta-theory itself. 1240 This price is certainly too high to pay (as long as there are no compelling independent arguments for dialetheism), 1241 and we conclude: 1236 1237 1238 1239 1240

1241

Cf. the end of section 10.9. Cf. section 10.9. Alternatively, we can use (∧-Int+) and (¬-Int+) to analyze the sentence into its parts. Cf. the notion of metaphysical consistency of section 3.9. Note that, in contrast, our formal framework is not in general prejudiced towards the principle of (Exclusivity), neither in the intensional semantics nor in the two-dimensional and relational semantics. If we implement the apparatus of extensions and anti-extensions for ordinary predicates, then it can happen for an ordinary predicate ‘F’ and a term ‘a’ that ‘F(a)’ is both true and false with regard to the same context, modeling that there is an inconsistent state of affairs in the world, and that ‘F(a)’ is both true and false relative to the same context, modeling that there is a dialetheia in the language of our material understanding. In fact, we do not even need the apparatus of extensions and anti-extensions to account for dialetheias. For the Liar paradox of course provides a way to bring about a breach of the principle of (Exclusivity) if we combine the principles that we already endorse with the possibility of sentential self-reference. Cf. section 13.2. For those who are already convinced of dialetheism, however, there is a road open to a dialetheist account of our identification of the non-identifiable, and more generally, to a dialetheist account of

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‘True(«E!(a) ∧ ¬I!(a)»)’ is true with regard to no context. Formulating this as a principle of the contextualist semantics means that we take the internal stance on the inexpressibility of non-identifiability. By similar reasoning from principles about falsity and in particular from the (Generalized Falsity Equivalence), 1242 we can also show: ‘E!(a) ∧ ¬I!(a)’ is not false relative to any context. ‘False(«E!(a) ∧ ¬I!(a)»)’ is not true with regard to any context. In sum, the sentence ‘E!(a) ∧ ¬I!(a)’ is gappy for every context both in the relational semantics and in the contextualist semantics. As this is due to non-identifiability, the sentence is gappy in a semantic way. It may even be gappy in a purely semantic way and thus a (Proper Gap Case), because if ‘a’ is a Fregean description, nothing speaks against the existence of the sentence even when ‘a’ is empty, so that the sentence need not be gappy in a semiotic way. In that case, the failure of the Tarskian equivalence for a context c0 relative to which there is an existent but non-identifiable object will be material. 1243 Thus we do deviate from classical logic even in the material understanding. But there is no inconsistency; and anyway, our account is well motivated from attention to the difference between language and world and the need to take the internal stance. In a nutshell, our formal result is that although the truth ascription ‘True(«E!(a)∧ ¬I!(a)»)’ is gappy with regard to any context, the embedded sentence ‘E!(a) ∧ ¬I!(a)’ can be true with regard to a particular context. Thus we model that although nonidentifiability is necessarily inexpressible, non-identifiability is nevertheless possible. We take the internal stance on each one of these two compatible phenomena by using the intensional semantics of the linguistic and the worldly fragment of the object language. In this way our dual framework solves the problem of existence without identifiability in an adequate way. * And yet, we are accustomed to thinking that existence entails identifiability. Therefore it is welcome that there is a phenomenon which promises to make cases of existence without identifiability like the one above more palatable. It will become visible when we compare the behavior of the sentence ‘Everything that exists is identifiable’ in the intensional semantics to its behavior in the relational semantics. (We can echo our remarks about recapturing the Tarskian equivalence in the relational semantics 1244 here because the situation is structurally similar.) In a scenario where there is existence without identifiability, ‘Everything that exists is identifiable’ will be false with regard to some contexts, because it will be false with regard to

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our expression of the inexpressible. The latter is indeed endorsed by Priest; cf., e. g., Priest 2002, 179ff. and Priest 2014, 194ff. We use (∧-Int–) and (¬-Int–) to analyze the sentence ‘E!(a) ∧ ¬I!(a)’, and combine the principle (Contextual Domain) either with the principles (2D–) and (Diag–), or directly with the principle (Rel–). Cf. section 10.9. Cf. section 10.9.

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every context relative to which there is an existent but non-identifiable object. This models on the material level that non-identifiable existence is possible. But in the relational semantics, the picture is different. The main point is that here, the contextual domain is relevant, which is comprised of identifiabilia. Therefore, ‘Everything that exists is identifiable’ will not be false relative to any context. It might be gappy relative to some contexts, because it will be gappy relative to contexts where the sentence itself does not exist or is not identifiable. But with a view to what we want to model with the relational semantics on the material level, i. e., with a view to the use of the context-sensitive language we are interested in, we can neglect these cases. It is plausible that where we are free to talk about expressions at all, we are free to talk about the sentence ‘Everything that exists is identifiable’, because it is natural to think of the primitive predicates of existence and identifiability as being among the very first expressions to come into existence and identifiability, and of the object language truth predicate as being among the first expressions that are introduced once the language gets self-referential. 1245 In sum, even when we do not in general have the principle that existence entails identifiability in the intensional semantics, we can recapture it in the relational semantics. The same point can be made by comparing the two formal sentences ‘∀x (E!(x) → I!(x))’ and ‘True(«∀x (E!(x) → I!(x))»)’, which formalize the above sentence ‘Everything that exists is identifiable’ and the corresponding truth ascription, but now both within the intensional semantics. In a scenario where there is existence without identifiability, ‘∀x (E!(x) → I!(x))’ will be false with regard to every context relative to which there is an existent but non-identifiable object. The truth ascription ‘True(«∀x (E!(x) → I!(x))»)’, however, will not be false with regard to any context, because within the scope of a truth ascription, the quantifiers range only over the contextual domain of identifiabilia. What is more, ‘True(«∀x (E!(x) → I!(x))»)’ will be true with regard to every context where we are free to talk about the expression ‘∀x (E!(x) → I!(x))’, i. e., with regard to every context where that sentence of the formal object language exists and is identifiable and the formal truth predicate exists, which plausibly is about every context where we are free to talk about expressions at all, because it is natural to think of the primitive predicates ‘E!(. . .)’, ‘I!(. . .)’, and ‘True(. . .)’ as being available once the language gets self-referential. In sum, the different behavior of the two formal sentences in the intensional semantics models, firstly, that it is possible that the state of affairs that existence entails identifiability fails to obtain and secondly, that it is virtually impossible that the corresponding sentence fails to be true. Thus we have another case where what is the case and what is true come apart – a case that also helps us accommodate our original (but perhaps misguided) intuition that existence entails identifiability.

1245

An additional assumption is again that even if identifiability is subsequent to existence (as we will argue in chapter 12), it will follow it closely in the examples we are mainly interested in, which are the development of languages construed as a contextualist ontology of expressions (and not the existence of aliens in a far off star system, where the lag between existence and identifiability would be more pronounced).

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10.12 Kaplan’s puzzle and the status of our solution Continuing the theme that local failures of the Tarskian equivalence are less surprising and more palatable than we might have thought, let us compare our solution of the problem of existence without identifiability in our dual framework (that has two-dimensional semantics as an integral part) to the usual two-dimensional solution of puzzles like the one posed by Kaplan’s sentence ‘I am here now’ and of certain puzzles concerning utterances about utterances. To bring out the analogy, compare the following sentences: (1) (2) (3) (4) (5)

‘Object a is not identifiable.’ ‘Something is not identifiable.’ ‘I am not here now.’ ‘There is no utterance.’ ‘The actual president of the U.S. is not the president of the U.S.’

Sentences (1) and (2) are similar to the sentences that came up in our discussion of the problem of existence without identifiability; sentence (3) is a negative counterpart of Kaplan’s sentence; 1246 sentence (4) is a negative example from a range of puzzling sentences about utterances that also include the positive examples ‘There are utterances’ and ‘This is an utterance’, 1247 and sentence (5) is an instance of a negative counterpart of a form of sentence that is of interest in many discussions of two-dimensional semantics, ‘The actual F is F’. One thing that the sentences (1), (2), (3), (4), and (5) have in common is that they cannot have true utterances (in a normal scenario, and given the normal use of the words), 1248 which is puzzling because what they express does not appear to be impossible at all. Two-dimensional semantics solves the conundrum in the same way for all five cases: The respective sentence is never true on the diagonal (i. e., never true with respect to and at a context c), modeling that it has no true utterances, but sometimes true beyond the diagonal (i. e., true with respect to a context c and at a context d where c = / d), modeling that it corresponds to a possible state of affairs. In our dual framework, we can say the same thing because of the way we use the twodimensional semantics to model the behavior of the language. But we can do more than that, because our dual framework is comprised of several semantic theories (including the two-dimensional semantics) that are interconnected. Like the twodimensional semantics in its usual application we can model both facts about the world and facts about language, but in our dual framework we can keep them apart more clearly, and take both the external and the internal stance on them. Taking the external stance, we can solve the puzzle in each case by distinguishing between the relational metaphysics (formulated in the worldly part of the meta-language) and the 1246

1247 1248

Sentence (3) might also be called a Dylan sentence, in view of “I’m not there” being the title of a wellknown song by Bob Dylan and of a film about him by Todd Haynes. I would like to thank Niko Strobach for reminding me of examples of this kind. Sentences (2), (3), (4), and (5) are always false when they are uttered, sentence (1) is false if the object in question is identifiable and otherwise gappy.

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relational semantics (formulated in the linguistic part of the meta-language), saying for example with a view to sentence (4) that there is a context relative to which no utterance exists, but there is no context relative to which sentence (4) is true. Taking the internal stance, we can solve the puzzle in each case by distinguishing between the contextualist metaphysics (given by the intensional semantics of the worldly part of the object language) and the contextualist semantics (given by the intensional semantics of the linguistic part of the object language), saying for example with a view to sentence (4) that there is a context with regard to which sentence (4) is true, but there is no context with regard to which an object language sentence that ascribes truth to sentence (4) is true. Thus we do not correct, but we complement the usual two-dimensional solution, and we also extend it beyond the usual example sentences to sentences that have been bothering us. Given the understanding of the two-dimensional semantics, relational semantics, and contextualist semantics as modeling facts about language and the intensional semantics as modeling facts about the world, this solution amounts to saying that it is no more than a fact about language that there cannot be true utterances of (1), (2), (3), (4), or (5), a fact about language that is not associated with a corresponding fact about the world – in contrast, e. g., to the sentence ‘Socrates is identical to Plato’, which cannot have a true utterance because it concerns an impossible state of affairs. In this, our solution to the problem of existence without identifiability is similar to the standard two-dimensional solution of the puzzle posed by Kaplan’s sentence ‘I am here now’ and the puzzle of certain utterances about utterances. I take this similarity to show that there is nothing too mysterious about the phenomenon of existence without identifiability, because it removes the appearance of inconsistency between the fact about language that we cannot truthfully say of some object that it is not currently identifiable (because thus we would identify it) and the fact about the world that it is nevertheless possible that some object is not currently identifiable. * In concluding this section and chapter, we should remind ourselves, however, that – aside from brief considerations of plausibility about causal contact with past and present objects and some thoughts inspired by the theory of relativity – we have not yet argued for the possibility of existence without identifiability. What we have done here is to show that, should we find reason to demand that there can be existence without identifiability (and we will find reason to demand it when we look at self-referential languages), we will not run into the technical problem that existence without identifiability cannot even be represented on the level of the formal semantics. But let us note also that the construal in our dual framework, together with our brief consideration of plausibility, already invites us to be openminded about the possibility of existence without identifiability, independently of the issue of self-reference. We have of course achieved more in this chapter than a solution to the problem of existence without identifiability. One of the more general benefits of having the dual framework for modeling a context-sensitive language and the corresponding contextualist metaphysics at our disposal is that it provides a principled account of the dichotomy of a language and the world it is about. This will be valuable when we

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turn to self-referential languages, where the danger of blurring the distinction between language and world is especially great. Along the way, we have accounted for the flexibility of language and outlined a theory of changeable truth and changeable falsity. That is, we have described within a contextualist setting how truth and falsity depend on what is the case in the world, on which expressions exist, and on what is identifiable. We have done this in a rigorous way for the case of monadic sentences (which are our main interest because most Liar sentences are monadic sentences). But it should have become plausible that we need not expect any greater obstacles on the way to extending the rigorous account to all sentences of the language. With regard to truth, it is again important to be clear about the distinction between the formal and the material level. We have pointed out that not every truth value gap in the formal theories corresponds to a truth value gap on the material level. 1249 The deeper reason behind this is that not every notion of truth or falsity that is used in the formal theories corresponds to a notion of truth or falsity on the material level. Truth in the intensional semantics models the obtaining of a state of affairs; falsity in the intensional semantics models the obtaining of the negative counterpart of a state of affairs. It is only truth or falsity in the two-dimensional semantics and in the relational semantics that models the truth or falsity of a sentence on the material level. Thus we can say that what we did in this chapter was to use several notions of changeable truth on the formal level to model both changeable facts and changeable truth on the material level.

1249

Cf. section 10.10.

Chapter 11

A Self-Referential Language as Its Own Changing World I: Subsequentist Metaphysics

Raymond Smullyan: “Oh yes, let me get back to the statement ‘The Tao is nameless’. I find this statement highly suggestive, mysterious, poetic and beautiful. But what does it mean? [. . . ] Is it completely out of the question that there may be objects in the universe which are so sensitive that the very act of naming them throws them out of existence? Now I am not suggesting that the Tao behaves like that; I hardly think the Tao goes out of existence if one so much as names it. But it might well be that the Tao is so remarkably sensitive that when named, it changes ever so slightly – it is not quite the same Tao it was before it was named. Indeed, if we identify the Tao with the universe as a whole, this must be the case, for the act of naming the universe is itself an event in the universe, hence the universe is not quite the same after as before the event.” 1250

Finally, we are in a position to approach the issue of self-reference in a principled way. In the present chapter, we will combine the core results from two of the preceding chapters. In chapter 8 we saw that the expressions of a language form a contextualist ontology; and in chapter 9 we saw that a language that is about a contextualist ontology needs to have certain features. As we now turn to languages that allow to talk about their own expressions, we will have to heed both results at once: As the expressions of any language belong to a contextualist ontology, a selfreferential language must be about a contextualist ontology. What happens when the circle closes – when a language is its own changing world? The results of the preceding chapter 10 have a more indirect role for the investigation in the present chapter. We saw there how the interplay between a contextsensitive language and the contextualist metaphysics it is about can be represented in a rigorous way, should the need arise. The dual framework that we developed there for this purpose is an assurance in the background, on the basis of which we can again take a less technical approach in the present chapter. 1251 We will start with some general remarks about self-referential languages and self-referential expressions (in sections 11.1 and 11.2). Then we turn to the main topic of the chapter, which are self-referential names, give some examples, and 1250 1251

Smullyan 1977, 25f. In section 11.1, we will refer back to the Generalized equivalence about truth that we developed as an alternative to the contextualist counterpart of the Tarskian truth schema in sections 10.4 and 10.9, but this will not carry much argumentative weight.

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study how they could be introduced into a language (in sections 11.3 through 11.5). After that, we describe four undesirable phenomena that would arise if there were self-referential names (in sections 11.6 through 11.10) or sentential indexicals like ‘this sentence’ (in section 11.11). On this basis, we will argue for the non-existence of self-referential names and indexicals by outlining a systematic measure for the prevention of the four undesirable phenomena (in sections 11.12 and 11.13). We will conclude with a methodological reflection on this particular way of establishing a non-existence claim (in section 11.14).

11.1 Overlap between language and world Let us say that a language is self-referential if and only if it allows to talk about some of its expressions (i. e., to refer to some of its expressions and to express that they fall under some concepts) and recall that we say that an expression is self-referential if and only if it refers to itself (or to a larger expression it is a part of). 1252 In this section, we will look at self-referential languages in general, turning to the issue of self-referential expressions in the following sections. Note that a self-referential language need not contain self-referential expressions. Take, for instance, a language that is about some non-linguistic items and add, besides some predicates that can be satisfied by expressions, quote marks as the only device of referring to expressions. This language allows to talk about every one of its expressions – by means of a quotation expression that quotes it. However, as a quotation expression is strictly larger (in terms of proper parthood) than the expression it quotes, no expression can achieve self-reference with the help of quotation alone. 1253 So this language would be self-referential without containing a single selfreferential expression. The possibility of a self-referential language that contains no self-referential expressions is important for the argument in this chapter and the following two. What we ultimately want to show is not only that there are no (properly) selfreferential expressions in any language, but also that nevertheless every (non-selfreferential) language can be extended to a self-referential language that achieves semantic closure to a large extent. In fact, arguments against the existence and for the dispensability of self-referential expressions are only interesting against the 1252

1253

Cf. section 5.1. Recall also from there that we have adopted a terminology according to which it is not people, but expressions that refer to objects. This is not meant to suggest a substantial thesis about meaning, and it can be taken as mere shorthand that is compatible with the claim that, ultimately, it is people who use expressions to refer to objects. And although, primarily, the job of referring to an object is done by a singular term, we have also chosen to say that a sentence (derivatively) refers to some object if and only if it contains a singular term that refers to that object, e. g., when saying that a Liar sentence refers to itself. Cf. sections 5.6 and 5.7. Strictly speaking, we will need to exclude devices that would allow to construct Quinean or Smullyanesque self-reference here, like the functor ‘the self-appendation of . . . ’ that takes quotation expressions as an argument. But that can be done – imagine, e. g., a language that (apart from some resources to talk about physical objects) has quote marks and a truth predicate as its sole devices that allow to talk about expressions.

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backdrop of the goal of constructing a language that is self-referential and that achieves semantic closure. For without this goal, we have no reason to introduce expressions that refer to expressions to begin with, and might as well just ban any such devices in order to prevent the paradoxes. However, even though it is only the possibility of self-referential expressions and not the possibility of a self-referential language that we will want to call into question in the following, we will often talk as if both were real. The reason is methodological. In the scope of this and the next chapter, we will often assume hypothetically that a self-referential language does contain self-referential expressions, to study the consequences of this assumption, and ultimately to provide reasons against the possibility of self-referential expressions. Strictly speaking, we would have to add ‘purported’ in very many places; and say, e. g., ‘a purported Liar sentence would refer to itself’ instead of ‘a Liar sentence refers to itself’. But in the interest of readability, we will often refrain from doing this, and take the hypothetical mode as understood. To start applying our previous results to the issue of self-reference, we should note that a self-referential language needs to be context-sensitive 1254 and metaphysically flexible. 1255 Being about expressions, it is about a contextualist ontology. 1256 So, expressions like ‘all expressions’ will have a context-dependent extension and, as new expressions may come into existence, it would be very inconvenient if we were unable to add new names to the language. Note that adding a name to a self-referential language will not only enrich the language but will also change the context, 1257 because with the name a new object enters the domain of discourse of the language. This observation leads us to an interesting double-role played by some objects when language gets self-referential. Even in the case of a language that is not meant to be self-referential, some physical objects may play a similar double role. Typically, some physical objects are used to realize language, but physical objects also make up the world we usually use language to talk about. We use certain sounds or marks on paper to talk or write, but we may also, if the occasion arises, talk or write about sounds or marks on paper. Usually, this can be ignored – we can continue to picture the relation between language and the world as if there were a neat division between the linguistic expressions here and the things they are about over there. But in the case of a self-referential language, (part of) the language will be (part of) the world it is about, so that there is an overlap between the language and the very world it is about that can no longer be ignored: Some of the expressions of the language will be in its domain of discourse, at least relative to some contexts. The overlap between language and world that we have just observed occurs with regard to the background ontology, i. e., the realm to which the expressions of the 1254 1255 1256 1257

Cf. section 9.1. Cf. section 9.11. Cf. sections 8.5 through 8.7. In this framework we can distinguish two kinds of change of context. A change of context is nonfactual if and only if it implies only a variation of the position of use but not of the possible world; a change of context is factual if and only if it implies a variation of the possible world. And a change of context that implies a change in some fact about existence obviously is a factual change of context.

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language belong and from which the extensions of singular terms and of predicates are taken. But is that all? Let us also have a look at the extension of sentences. That is, let us turn to truth and falsity and their relation to states of affairs. Here, we can again use the question of the distinctness of language and world to characterize what sets a self-referential language apart from a language that is not self-referential. As long as a language is not self-referential, there is a neat division between truth and fact. But there is of course a close connection between the two sides, which is usually characterized by the naïve truth principle 1258 and standardly explicated by the Tarskian truth schema. 1259 As we have seen in chapter 10, in a contextualist setting the Tarskian truth schema needs to be reformulated as the Tarskian equivalence, 1260 and once we take into account that expressions form a contextualist ontology and that a context-sensitive language can speak about objects only when they are identifiable, we must supplant this by a Generalized truth equivalence. 1261 We have also seen that under certain fairly common circumstances, the Tarskian equivalence is the limit case of the Generalized truth equivalence – namely, with regard to all contexts where all relevant expressions exist and all relevant objects are identifiable. Under these circumstances, we can even take one further step back towards the standard picture, because with regard to such contexts, the usual Tarski biconditionals will be true. 1262 Let us look at an example from this vicinity, so that we do not unnecessarily complicate the matter when we study how the close connection between truth and fact changes once a language becomes self-referential. ‘‘It is snowing’ is true if and only if it is snowing’ is true with regard to every moment when all expressions involved exist. As this claim belongs to the intensional semantics, the mentioned biconditional models a (complex) tensed state of affairs, which is said to obtain for all the relevant moments. Its left hand side makes a context-sensitive statement about language (a certain sentence is true), and its right hand side makes a context-sensitive statement about the physical world (the weather is a certain way). Their material equivalence (expressed by the locution ‘if and only if’) connects the two sides closely. But the truth of sentences belongs to a different realm than the weather; nothing calls into question yet the neat division between the language and the world it is about. This changes once we turn to a self-referential language. Now statements about sentences can occur on both sides of a Tarski biconditional. E. g., to move one level up from the above sentence: ‘‘‘It is snowing’ is true’ is true if and only if ‘It is snowing’ is true’ is true with regard to every moment when all expressions involved exist. Contrasting the second mentioned biconditional with the first mentioned biconditional, we see how the realms of truth and fact have started to overlap. The very 1258 1259 1260 1261 1262

Cf. section 2.2. Cf. section 3.2. Cf. section 10.4 Cf. section 10.9. Cf. section 10.9.

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same sentence, ‘‘It is snowing’ is true’, occurs both on the left hand side of the first mentioned biconditional and on the right hand side of the second mentioned biconditional. (And in each case, it is not mentioned but used within the biconditional.) Speaking figuratively, the sentence moves over from the truth side to the fact side of the Tarski schema due to the semantic ascent that is involved in the step from the first to the second claim. And more generally, we now ascribe truth to sentences not only when we want to say what is true, but also when we want to say what is the case. The realms of truth and fact are no longer disjoint. 1263 To sum up, when a language is self-referential, there will be overlap between the language and the world it is about, and this overlap will concern not only the background ontology, but also the realm of what is true and of what is the case. This overlap between truth and fact will lead to interesting effects, and it may lead to rather weird effects. As we will see, some of these effects would be so weird that the danger they would constitute provides us with a conclusive argument against the existence of expressions that exhibit semantic self-reference. But we have yet some way to go before we reach that goal. Our path will take us through the metaphysics of a self-referential language in the present chapter, and through its semantics in the next chapter.

11.2 Self-referential expressions In connection with the Liar paradox, the most important self-referential expressions are of course self-referential sentences. We have already had a detailed look at the semantics of self-referential sentences in part II. 1264 In the present section, we will apply what we have learned about the metaphysics of expressions to the case of self-referential expressions. Or, to phrase it more cautiously: We have already investigated how the semantics of purported self-referential sentences would look like, if there were any, and here we will investigate their metaphysics in a similar hypothetical mode. According to our account of the metaphysics of expressions, syntactic and semantic sentences are distinct and belong to different categories of objects. This thesis 1263

The same point can be made in greater generality by comparing two instances of the (Generalized Truth Equivalence) involving the monadic sentence ‘Fred runs’: ‘‘Fred runs’ is true’ is true w. r. t. t0 if and only if (i) ‘Fred runs’ is true with regard to t0, (ii) ‘Fred runs’ exists relative to t0, and (iii) Fred is identifiable relative to t0.

1264

‘‘‘Fred runs’ is true’ is true’ is true w. r. t. t0 if and only if (i0 ) ‘‘Fred runs’ is true’ is true with regard to t0, (ii0 ) ‘‘Fred runs’ is true’ exists relative to t0, and (iii0 ) ‘Fred runs’ is identifiable relative to t0.

As in an instance of the Tarskian equivalence, truth is modeled by the respective left hand side of the equivalence. But now fact is modeled not by the whole right hand side, but only by its first conjunct, clause (i) or (i0 ). Although the matter is more complicated due to the presence of the respective second and third clause, the main point remains: The truth ascription ‘‘Fred runs’ is true’ occurs on the left hand side of the left equivalence as well as on the right hand side of the right equivalence. Thus it models what is true in the first instance and it models what is the case in the second instance. Cf. sections 5.4, 5.6, 5.7, and chapter 6.

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of distinctness has crucial consequences for the characterization of self-referential expressions. Let us observe first that the candidates for falling under the notion of a self-referential expression are always semantic expressions (and not syntactic expressions, to say nothing of their physical bases), and in particular, the candidates for falling under the notion of a self-referential sentence are always semantic sentences (and not syntactic sentences). As it is semantic sentences that refer to something, a self-referential sentence properly so called will be a semantic sentence that refers to itself, and thus a semantic expression that refers to a semantic expression. But a semantic sentence may of course refer to something other than a semantic sentence – to a syntactic expression, or to a physical object, for that matter. In particular, a semantic sentence may refer to the syntactic sentence that is its own basis, as is plausible, e. g., in the case of the sentence ‘This sentence consists of six words’. When we investigated the semantics of self-referential expressions, we have decided to follow common usage and call such sentences ‘syntactically self-referential’. 1265 But here we should note that properly speaking they are not self-referential at all because each of them refers not to itself but to a distinct object, its syntactic basis. Some (semantic) sentences can even be said to be physically self-referential, because they refer to the physical basis of their syntactic basis. This will be the case, e. g., with certain inscriptions of the sentence ‘This sentence is three inches long’. And this is how we should construe the example given by Stephen Read, of a sticker on the back of a car that reads: “If you can read this, you are too close.” 1266 So, given this construal of what is usually called syntactic self-reference and of what can be called physical self-reference as reference of a semantic expression to its basis, the only proper form of sentential self-reference that is conceivable is semantic sentential self-reference. It is given when a semantic sentence refers to itself, and not (only) to its syntactic or physical basis. This bears repeating, because it is of course this kind of self-reference that a Liar sentence must exhibit to be the root of paradox. For if we construed a given Liar sentence as saying of its syntactic basis (or of its physical basis) that it is false, or that it is not true, the paradoxical reasoning would be blocked at the very beginning. To see this, recall that in our formulation, the basic Liar reasoning starts from a premise like: 1267 (1) There is a sentence λ that means that λ is false. (2) There is a sentence Λ that means that Λ is not true. And in standard formulations, the basic Liar reasoning will start from a premise like: (3) λ = ‘λ is false.’ (4) Λ = ‘Λ is not true.’

1265 1266 1267

Cf. section 5.1. Read 1995, 154. Cf. sections 2.4 and 2.5.

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But if we construed the respective candidate for a Liar sentence as referring to its (syntactic or physical) basis, then we could not argue for any of these four claims. We would be left with the following decisively weakened variants: (10 ) There is a sentence λ0 that means that the basis of λ0 is false. (20 ) There is a sentence Λ0 that means that the basis of Λ0 is not true. (30 ) λ0 = ‘δ is false’, where δ is the basis of ‘δ is false’. (40 ) Λ0 = ‘∆ is not true’, where ∆ is the basis of ‘∆ is not true’. But on the basis of any of the weakened premises (10 ) through (40 ), the basic Liar reasoning would not go through. Unlike the simple Liar sentence λ that is characterized by (1) or (3), the sentence λ0 that is characterized by (10 ) or (30 ) would be just false. The sentence λ0 says that its basis is false, but as neither syntactic sentences nor their physical bases can be false, the sentence λ0 is itself false. And unlike the strengthened Liar sentence Λ that is characterized by (2) or (4), the sentence Λ0 that is characterized by (20 ) or (40 ) would be just true. The sentence Λ0 says that its basis is not true, and all syntactic sentences and all physical objects are not true, so the sentence Λ0 is itself true. In sum, we have a false sentence with an unfalse basis and a true sentence with an untrue basis – what in the case of a simple or strengthened Liar sentence would result in contradictory ascriptions of truth values to one and the same object is here neatly split up between two distinct objects. * So, syntactically self-referential expressions and physically self-referential expressions are harmless. The crucial question for our investigation can be narrowed down from the question whether a language can be self-referential, via the question whether an expression can be self-referential in the wide sense that encompasses also syntactic and physical self-reference, to the question whether an expression can exhibit semantic self-referentiality. However, although this is the right kind of question, we are well advised to expand our circle of inquiry again a bit. Semantic self-reference is an important special case of the phenomenon of semantic ungroundedness, which has played an important role in the study of the Liar paradox since the work of Hans Herzberger and of Saul Kripke in the 1970s. 1268 A sentence is semantically ungrounded if and only if there is a referential chain that starts from that sentence which does not terminate 1269 – i. e., there is semantic ungroundedness if and only if the relation of reference is not well-founded. 1270 The main types of semantically ungrounded sentences are semantically self-referential sentences (e. g., ‘This sentence is true’), sentences that form a referential cycle with other sentences (e. g., ‘The next sentence is true. The previous sentence is true.’), and sentences that are followed by an infinite

1268 1269

1270

Herzberger 1970; Kripke 1975. Cf. Yablo 1985 and Bromand 2001, 127ff. The present concept of semantic groundedness is meant to be co-extensional with Kripke’s concept of groundedness, which is defined in a different way. Cf. Kripke 1975, 693ff. and 701ff. Cf. section 5.7.

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chain of sentences, each of them referring to some of the subsequent ones (e. g., ‘The second sentence is true. The third sentence is true. The fourth sentence is true. . . . ’). Another example is the sequence of sentences, each one ascribing untruth to all of the following, that gives rise to Yablo’s paradox, 1271 which is of course the main reason why we should expand the circle of our attention from semantic self-reference to semantic ungroundedness. On the other side, the most clear-cut case of a semantically grounded sentence is one that refers only to non-semantic entities, like ‘Berlin is about four hours by train from Münster’. But a sentence can be grounded and yet refer to other semantic sentences, like ‘The sentence about the train connection from Münster to Berlin that is mentioned in this paragraph is true’. In the context of the present investigation, it is important to be very clear about the distinction between semantic ungroundedness, which occurs when the relation of reference is not well-founded, and what might be called metaphysical non-wellfoundedness, which occurs when the relation of metaphysical grounding is not wellfounded. 1272 The two notions belong to two different realms, those of semantics and of metaphysics. Even more importantly, as they give the same structural description (non-well-foundedness) of distinct relations (reference and metaphysical grounding), we can expect reasons for requiring the respective contrastive notions of semantic groundedness and metaphysical well-foundedness to hold of a certain collection of objects to be of different kinds. However, even though strictly speaking we should be concerned with the more general phenomenon of semantic ungroundedness, we will focus on semantic selfreference in the following investigation. Because of the paradigmatic status of the Liar paradox, it will be enough if the present study succeeds in making a convincing case against the existence of expressions that are semantically self-referential, as long as we indicate in some places how our arguments could be generalized to a case against all expressions that are semantically ungrounded. * So, in the following, we will investigate whether there really can be semantic sentences that are semantically self-referential – given what we have learned so far about the metaphysics and semantics of self-referential languages. Again: Please think of these considerations as formulated in a hypothetical mode throughout! The methodology is that of a thought experiment: What would happen, how would it be, if there were indeed referential devices that would allow to construct selfreferential expressions? In our investigation of the semantics of self-reference, we have seen that semantic self-reference cannot be achieved by means of a quotation expression alone, 1273 and that (contrary to a common conception) there are no Liar sentences that achieve

1271 1272

1273

Cf. section 1.3. The terminology may be a bit confusing here, because it has come about that “groundedness” refers to a (second-level) semantic notion, whereas “grounding” refers to a (first-level) metaphysical notion. Cf. sections 5.4, 5.6, and 5.7.

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self-reference by Gödelization. 1274 Thus we have learned that if it is possible at all that a sentence achieves semantic self-reference, then only by means of a simple indexical, or a name, or a description of the Fregean or the Russellian kind, or a quantificational phrase. Here, however, we do not need to go through the indexical variant, the name variant, the Fregean description variant, the Russellian description variant, and the quantificational variant of sentential self-reference case by case. Rather, we will in the remainder of this chapter study self-referential sentences with a name and generalize our results to all self-referential sentences that are directly referential, and in the following chapter turn to self-referential sentences that are indirectly referential. This way of structuring the material is preferable because it will turn out that there is a uniform way of arguing against all directly referential cases and another uniform way of arguing against all indirectly referential cases. Thus what will be crucial about our semantic results for the present metaphysical question about whether there are semantically self-referential sentences is not so much the specific means – names, indexicals, descriptions, and quantificational phrases – that would allow to achieve self-reference, but that once we have dealt with all directly and all indirectly referential cases, we will have excluded all options.

11.3 Some weird sentences Can sentential self-reference be achieved with the help of a name? Let us see how this could be the case. An example would be the Liar sentence ‘Larry is false’ – which itself bears the name ‘Larry’, so that the following would be the case: Larry = ‘Larry is false’. However, in our investigation of sentences that achieve semantic self-reference with a name, we should keep our focus away from Liar sentences. Our aim is to argue against the existence of such sentences in a way that is not ad hoc. Therefore, our argument needs to be based on some general feature shared by all sentences that achieve semantic self-reference with a name (or another means of direct reference). So let us put Larry together with some less paradoxical examples of sentences that look as though they refer to themselves with a name. We list them roughly in increasing order of apparent viciousness: (1) ‘Jack is short’, which is called ‘Jack’, 1275 (2) ‘Frodo consists of five words’, which is called ‘Frodo’, (3) ‘Marge is meaningful’, which is called ‘Marge’, (4) ‘Sam is a semantic sentence’, which is called ‘Sam’, (5) ‘Tony is true’, which is called ‘Tony’, and (6) ‘Larry is false’, which is called ‘Larry’.

1274 1275

Cf. chapter 6. Cf. Kripke 1975, 693.

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A remark on the choice of predicates in these examples. It is natural to work under the presumption that these sentences would (or could) be true, if they indeed were to refer to themselves 1276 – imagine that someone utters the sentences with the intention of saying something self-referential. Therefore sentences (1) and (2) stand out a bit, insofar as each of them would already be true if construed as referring to its syntactic base, which is possible because each of the predicates ‘. . . is short’ and ‘. . . consists of five words’ has a reading under which it is true of certain syntactic expressions. The predicates in sentences (3) through (6), however, are most naturally applied to semantic sentences (they can also be applied to syntactic expressions, but then the resulting sentence will be false no matter what). So we can stipulate that the names in the sentences (1) and (2) are meant to refer to the syntactic sentence that is the basis of the respective sentence, whereas the names in the sentences (3) through (6) are meant to refer to the respective semantic sentence itself. Under this stipulation, sentences (1) and (2) are harmless. They are syntactically self-referential, i. e., on the present account that construes semantic expressions as distinct from syntactic expressions, they are not properly self-referential at all. This can be brought out by noting that Jack is distinct from sentence (1) because (1) is a semantic sentence and Jack is the syntactic basis thereof, and that Frodo is similarly distinct from sentence (2). In the present metaphysico-semantic framework, their existence and their semantic properties will turn out to be unproblematic. And with a view to the aim of solving the Liar paradox, there is no need to argue against the existence of either sentence (1) or (2). On the contrary, it will turn out to be an advantage of the present approach that it admits all syntactically self-referential sentences. We need not bother with the syntactically self-referential sentences (1) and (2), which are the semantic sentences realized by Jack and Frodo. What we do need to be concerned about are the semantically self-referential sentences (3) through (6). They are properly self-referential. This can be brought out by noting that the following identities hold: (3) = Marge, (4) = Sam, (5) = Tony, and (6) = Larry. And we will find grave metaphysical and semantic problems with Marge, Sam, Tony, and Larry – problems that stem from the fact that they refer to themselves with a name. 1277 Of course, not everyone will agree that all four sentences are problematic (or would be problematic if they were to exist). Marge and Sam will appear harmless to a large group of people. Tony will appear harmless at least to some people. It is only Larry that will strike most people as not harmless at all – just about everyone except the dialetheists will see it as dangerous. However, these assessments do not stem (as ours will) from general considerations about the metaphysics and the semantic structure of such sentences, but from assuming their existence to begin with (perhaps hypothetically), and then evaluating the outcome of that thought experiment for each particular case. Then people will go on to observe that it is very natural to suppose that Marge is meaningful and Sam is a semantic sentence – and this is in fact no independent supposition, but equivalent to the assumption that Marge and Sam

1276 1277

With the exception, of course, of the Liar sentence Larry. Cf. sections 11.6 through 11.10.

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exist. 1278 And similarly in the case of Tony: Given that it exists, people will observe that it can have either truth value without contradicting the Tarskian truth schema, and quite a few will see this underdetermination of Tony and other Truth-teller sentences as less vicious than the disruptive overdetermination of Larry and other Liar sentences. This is not the place to dispute these evaluations of the hypothetical scenarios, 1279 but we need to say a word about the apparent meaningfulness of Marge, Sam, Tony, and Larry. For (apart maybe from their names, which we are accustomed to associate with human beings) they do strike us as very similar to ordinary, well-behaved sentences of our language – they appear to be meaningful. The appearance of meaningfulness can be explained by the following aspects: The predicates in all four sentences are unproblematic. In ordinary discourse, there is a presumption of well-behaved reference for all names that a speaker encounters but is not yet familiar with – a name is presumed referential until proven otherwise –, because in real life, we often do not initially know the names used by other speakers. 1280 All four sentences are grammatical – they are formed in the right way from the singular term and the predicate –, which leads us to presume that the compositionality of meaning applies to them. In addition there is probably a general attitude of charity. The aspect that we will call into question in the next sections is that semantically self-referential names are indeed as well-behaved as other names. Here it is enough to note that, independently of the question whether the presumption of meaningfulness for semantically self-referential names is really warranted, there is a good explanation why many people have the intuition that sentences that contain such names are meaningful. They presume that semantically self-referential names work just like other names of expressions. That is, they will construe our sentence Sam, ‘Sam is a semantic sentence’, along the same lines as the seemingly similar sentence Jon, ‘Snow is white’. As ‘Snow is white’ is a true semantic sentence, Jon is a true semantic sentence. By analogy, Sam seems to be a true semantic sentence, too. It is our job in the following to show that this analogy is illusory, and that the name ‘Sam’ is not as well-behaved as the name ‘Jon’. To get a first impression of the specific weirdness of semantic sentences that achieve self-reference with a name, observe that it is not only the case that Sam is called ‘Sam’, 1281 but that also, conversely, that ‘Sam’ is a part of Sam. The same goes for the names ‘Marge’, ‘Tony’, and ‘Larry’ – they, too, are part of what they name. It is not the case, however, that ‘Frodo’ is a part of Frodo, because ‘Frodo’ 1278 1279

1280 1281

But is there an air of begging the question here? But cf. section 13.3, where we will show that it is actually an advantage of our approach to the Liar paradox that it precludes the existence of sentences like Tony along with the existence of sentences like Larry. Probably, the use of names in fiction and in myths is a further reason for this presumption. The homophonic statements that Jack is called ‘Jack’, and so on for all six of our example sentences, are as natural as the more common-place statement that Nixon is called ‘Nixon’, and like them, they are trivially true at typical contexts of use.

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is a semantic expression and Frodo is a syntactic expression; and similarly for ‘Jack’ and Jack and ‘Jon’ and the sentence about snow that it names. 1282 So this weirdness concerns only semantically self-referential sentences. We will make more of this.

11.4 Names due to circular baptism? How can self-referential names be introduced into a language? Let us look at the general picture first. Paradigmatically (if not in the majority of cases), a name is introduced by a baptism. And usually, an object will already exist before it is baptized. Think of a ship or a baby. Here there is the object that is to be named, and there comes someone with the appropriate authority to conduct a baptism, names it, and lo, there will be a new name in the language. In terms of the present account of the metaphysics of expressions, with the addition of the new name to the language, there is a new semantic expression – an object has come into existence. (Recall that we construe a language as metaphysically flexible, so that it can accommodate the addition of new expressions. 1283) A similar story can be told without any problem about how the name of the syntactic basis of a syntactically self-referential expression is introduced into the language. Here are, e. g., the syntactic sentences ‘Jack is short’ and ‘Frodo consists of five words’. They are to become the bases of the semantic sentences ‘Jack is short’ and ‘Frodo consists of five words’, i. e., of sentences (1) and (2), 1284 which will then be syntactically self-referential sentences. As of now they are not fully meaningful, because the syntactic expressions in subject position do not yet realize semantic expressions. But there comes someone with the appropriate authority, names them ‘Jack’ and ‘Frodo’, respectively, and lo, there are two new names in the language, and with them there are the two new semantic sentences (1) and (2), amongst many other semantic expressions that contain them. For, with regard to the level of types, it is plausible that with the introduction of a new expression, all expressions that can be formed from it exist, too. 1285 Thus the syntactically self-referential name ‘Frodo’, e. g., can without any problem be introduced by a speech act of baptism: ‘Let the syntactic sentence ‘Frodo consists of five words’ be called ‘Frodo’!’

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1283 1284 1285

This does not contradict that, just as the syntactic name ‘Frodo’ is a part of the syntactic sentence ‘Frodo consists of five words’, the semantic name ‘Frodo’ is a part of the semantic sentence ‘Frodo consists of five words’ – but that is not what it refers to, so it is not a part of what it refers to. Cf. section 9.11 and chapter 10. Cf. section 11.3. More precisely, the plausible principle in the background is that relative to any context, for all expressions that exist, every expression that is their concatenation exists, too. In a slogan, concatenation is instantaneous. And even if concatenation were not instantaneous, the genesis of the semantic sentences (1) and (2) would still be unproblematic; their formation would just take a little bit longer.

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This speech act is perfectly ordinary: As in the baptism of a ship or a baby, the syntactic sentence pre-exists its baptism, 1286 and after the baptism, its name has come to exist, too. But can this story also be told about the introduction of a semantically selfreferential name? Just barely, if at all. For, in contrast to the names ‘Jack’ and ‘Frodo’, the names ‘Marge’, ‘Sam’, ‘Tony’, and ‘Larry’ can be introduced into the language only by a baptism that is circular (if they can be introduced at all). In a circular baptism, a name and the named object must be introduced in one fell swoop, because the name figures importantly in determining the named object. To illustrate this, let us look at the introduction of the semantically self-referential name ‘Sam’. Suppose that, like ‘Frodo’, ‘Sam’ can be introduced by a speech act of baptism: ‘Let the semantic sentence ‘Sam is a semantic sentence’ be called ‘Sam’!’ In surface form, there is no obvious difference between this and the speech act of baptism that we used to introduce ‘Frodo’. But note that speech acts of baptism are essentially temporal. And the difference between the introduction of ‘Sam’ by baptism and the introduction of ‘Frodo’ by baptism becomes evident once we pay attention to the dynamic aspects of the two imagined speech acts. We can easily conceive of the syntactic sentence ‘Frodo consists of five words’ as existing prior to its baptism, just like a ship or a baby. But when the name ‘Sam’ is introduced, this can only happen at the very same moment when the sentence ‘Sam is a semantic sentence’ is introduced, because not only can no name exist prior to the object it names, but also, no semantic sentence can exists prior to a name (or any other expression) that is a semantic part of it. ‘Sam’ and Sam can come into being only simultaneously, and the baptism that introduces the name ‘Sam’ must be circular. Note that not only is the name ‘Sam’ due to a circular baptism – so is (if this is to work at all) the baptized object itself, for the semantic sentence ‘Sam is a semantic sentence’ would not be what it is if the name that is in subject position did not refer to it. This need to bring about the named object by the very act of naming allows to cast doubt on the conceivability of the particular speech act of baptism that is involved. Of course we can try and proclaim: ‘Let the semantic sentence ‘Sam is a semantic sentence’ be called ‘Sam’!’ But it is doubtful that we will succeed in saying something meaningful, which is a precondition for our speech act to succeed. For we can easily imagine a bystander to interrupt our circular baptism before we have uttered the last words of the formula, and rightfully demand that we specify the object we want to christen before we go on. But in contrast to what is the case in ordinary baptisms of ships, babies, and syntactically self-referential sentences like Frodo, the circularity of the baptism we try to conduct here will prevent us from complying. So the first half of our formula would appear to be meaningless as long as it is not completed. 1287 It is hard to see how this can be compatible with the compositionality of meaning.

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1287

It is most natural (although not mandatory) to think that all syntactic expressions pre-exist all semantic expressions. These doubts are similar in spirit to the considerations about Liar sentences in Ryle 1952, 67f.

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11.5 Names due to circular definition? We have seen that the circular baptism that would be needed to introduce a semantically self-referential sentence into the language would be difficult to conduct, because it can only succeed if the name and the named object are introduced in perfect simultaneity. But these problems are due to the temporal aspects of this way of introducing a new expression – so we might want to investigate whether there are other ways to effect this introduction, which are in less danger of being thwarted by interruption because they are more on the atemporal side. 1288 In contrast to baptism, definition is a way of introducing expressions that is less obviously an act that happens in time. Let us again start by looking at the introduction of a syntactically self-referential name ‘Frodo’. It can be introduced by a definition in the usual way: ‘Frodo =def the syntactic sentence ‘Frodo consists of five words’.’ This definition is perfectly ordinary. In terms of the standard distinction between a nominal definition, which gives the meaning of the defined expression, and a real definition, which gives the essence of the defined entity, what we have here is a nominal definition that gives the meaning of the singular term ‘Frodo’. As this definiendum is a name, it is most natural to understand the description that is the definiens as doing no more than fixing its reference (i. e., we do not want to say that according to this definition, ‘Frodo’ is synonymous with the complex description ‘the syntactic sentence ‘Frodo consists of five words’’). There is no indication that this nominal definition of ‘Frodo’ goes hand in hand with a real definition of Frodo – it is most natural to think of the syntactic sentence ‘Frodo consists of five words’ as being what it is independently of this definition. Note also that although the string of letters ‘Frodo’ occurs on both sides of the definition, there is no real circularity here, because it is no more than a syntactic name on the right hand side, but the basis of a semantic name on the left hand side. All in all, the definition of ‘Frodo’ is similar in most respects to the following standard definition of the numeral ‘one’: ‘One =def the successor of zero.’ But can this story also be told about the introduction of a semantically selfreferential name? The answer is: Just barely, if at all. For, in contrast to the names ‘Jack’ and ‘Frodo’, the names ‘Marge’, ‘Sam’, ‘Tony’, and ‘Larry’ can be introduced into the language only by a definition that is circular (if they can be introduced at all). In a circular definition, the definiendum occurs in the definiens. More specifically, a nominal definition of a singular term is circular if and only if the singular term that is the definiendum is used in the singular term that is the definiens. 1289 1288

1289

In due course, we will see that the temporal aspect of baptisms is indeed paradigmatic for an adequate understanding of the whole problem (which will be solved by a principle of subsequence that is formulated in section 11.12). But this needs yet to be established by the arguments of the following sections. It is important to be clear that we are dealing with definitions that are meant to introduce a singular term, in contrast to definitions that are meant to introduce a predicate. The latter are of the form

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Suppose that, like ‘Frodo’, ‘Sam’ can be introduced by a definition: ‘Sam =def the semantic sentence ‘Sam is a semantic sentence’.’ In surface form, the definition of ‘Sam’ does not differ from that of ‘Frodo’. Thus it is not immediately clear that the definition of ‘Sam’ is circular. But we can paraphrase the definition: ‘Sam =def the semantic sentence that is the concatenation of (i) the semantic name ‘Sam’ and (ii) the predicate ‘. . . is a semantic sentence’.’ Now it has become evident that the semantic name ‘Sam’ occurs on both sides. There is still a difference, though, which might deter us from diagnosing outright circularity, because the name is used on the left hand side and mentioned on the right hand side. But let us unfold the definition further by making more of the constitutional structure of the semantic name ‘Sam’ explicit: 1290 ‘Sam =def the semantic sentence that is the concatenation of (i) the semantic name that has the string of letters ‘Sam’ as its basis and the pair (D, Sam) as its semantic value and (ii) the predicate ‘. . . is a semantic sentence’.’ 1291 Though cumbersome and less perspicuous than the previous formulations, this variant of the definition is explicit about the semantic name ‘Sam’ being used on both sides – as the definiendum and as part of the definiens. It is clearly circular. The important contrast to the definition of ‘Frodo’ is that in the definition of ‘Sam’, the name ‘Sam’ that is also part of the definiens is a semantic expression. As a semantic expression is constituted by both its syntactic basis and its semantic value, and as the semantic value of a name has the named object as a constituent, the semantic name ‘Sam’ has the named object Sam as a constituent. This feature is not only evidence for the circularity of the definition – it is also the source of further weirdness. For note that the definition can only succeed to introduce ‘Sam’

1290

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‘F(x) ↔def G(x)’, and a definition of this sort is circular if the predicate ‘F’ that is the definiendum occurs also in the definiens: ‘F(x) ↔def . . . F(x) . . .’. It is this kind of circular definition – a circular definition that is meant to introduce a predicate, not a singular term – that figures prominently in the work of Anil Gupta and Nuel Belnap (Gupta/Belnap 1993; cf. Gupta 2015, section 2.7). Doubts about the legitimacy of the circular definition of singular terms need not carry over to doubts about the legitimacy of the circular definition of predicates. According to our qua objects account of semantic expressions (cf. section 8.5), we would construe the semantic name ‘Sam’ as a qua object with the syntactic expression ‘Sam’ as basis and a semantic role concept that is associated with the semantic value (D, Sam) as gloss. Here, we simplify by omitting the intermediate step of associating a semantic role concept with a semantic value, and just take a semantic expression to have two constituents, its syntactic basis and its semantic value. On both construals, we borrow a theoretical entity from semantics, the semantic value of the expression in question, to do metaphysics. For the pair notation that we have adopted so that we can refer to semantic values in a telling way, cf. sections 4.6 and 5.2, and in particular subsection 5.2.6.

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if it introduces Sam, too. It is a nominal definition that is inextricably intertwined with the corresponding real definition! This is a stark contrast to the definition of ‘Frodo’, which was clearly only nominal, and even to the definition of one as the successor of zero, which might be read either as a nominal definition of the numeral ‘one’ or as a real definition of the number one, 1292 but which does not mingle the two kinds of definition on either reading. In this section and the one before it, we have looked at the linguistic means – baptism, definition – that we might try to use to introduce semantically self-referential names. We have seen that weirdness abounds. But maybe this is not decisive in casting doubt on the existence of such expressions. For if they were to exist, they would surely be untypical objects, so that their introduction would likely be untypical, too. Let us therefore leave it at that with regard to the question of introduction, and turn to the objects themselves.

11.6 The problem of ontological co-dependence We are now in a good position to enquire whether the circularity that we saw to characterize the introduction of Marge, Sam, Tony, and Larry leads to problems of a metaphysical nature. To do this, we will in the present section look at what is characteristic of these sentences and their names in terms of the notion of ontological dependence, turning to the kindred notion of metaphysical grounding in the next section. Each of the two metaphysical problems that will emerge will turn out to have a semantic flip side that is also problematic (sections 11.9 and 11.10). We have said that an object b depends ontologically on an object a if and only if every context relative to which the object b exists is a context relative to which the object a exists. And we can say that an object b depends ontologically in a remote fashion on an object a if and only if for every context relative to which the object b exists, the object a exists relative to that context or there is a prior context relative to which the object a exists. 1293 We have already observed that a name and more generally, any directly referential expression, depends ontologically in a remote fashion on the object it refers to. 1294 We will endorse a further principle, that an expression depends ontologically on its parts. Arguably, not every object depends ontologically on its proper parts, for it is plausible that living beings and many ordinary physical objects can survive the loss or exchange of some of their parts. 1295 In the case of expressions, however, things are surely different. The meaningful expressions that are used to form a larger meaningful expression are necessary parts of that expression. Just imagine deleting one of the semantic parts of a given 1292 1293

1294 1295

Fine 1994, 14. This generalizes the account given in section 8.3, where remote ontological dependence was defined only for the special case of existence relative to moments in time. Cf. section 9.10. Some will even endorse the more radical claim that it is the parts that depend ontologically on or are grounded in the whole. Cf. the monism of Schaffer 2010. Note that even if monism were true for all physical objects, ontological dependence and grounding could still go the other way for syntactic and semantic expressions, which are distinct from the physical objects that are their bases.

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sentence, e. g., turning ‘Snow is white’ into ‘Snow’, whilst denying that you have thus turned to a distinct expression! 1296 To see what this means for semantically self-referential names, let us pick out the name ‘Sam’ and the sentence it names. As the name ‘Sam’ is a proper part of the sentence ‘Sam is a semantic sentence’, the sentence depends ontologically on its name. That is a consequence of the general principle that a semantic expression depends ontologically on its proper parts. But as the name ‘Sam’ refers directly to the sentence Sam, it is also the case that the name depends ontologically in a remote fashion on the sentence. That is a consequence of the general principle that a semantic expression depends ontologically in a remote fashion on an object that it refers to directly. In sum, the sentence depends on the name and the name depends remotely on the sentence. In order to characterize this situation, let us introduce the two notions of ontological co-dependence and weak ontological co-dependence. They can be distinguished in the following way. Some objects are ontologically co-dependent if and only if they exist at just the same contexts. Some objects are weakly ontologically codependent if and only if they start to exist at just the same context. 1297 Ontological co-dependence 1298 entails weak ontological co-dependence, but not vice versa. In terms of these notions, the name ‘Sam’ and the sentence ‘Sam is a semantic sentence’ need not be ontologically co-dependent, but they are weakly ontologically codependent: Although it is possible that the name survives the sentence, the sentence cannot survive the name, and they have to start their existence relative to the same context. 1299 As we have not appealed to any feature of Sam other than that it refers to itself with a name, we can conclude that Marge, Tony, and, most importantly, the Liar sentence Larry are also weakly co-dependent with their respective name (or would be, if they existed). Why are ontological co-dependence and weak ontological co-dependence weird phenomena? Note first that although ordinary physical objects will often exhibit ontological dependence in one direction – like a grin that depends ontologically on the person who is grinning, without the person depending on the grin –, there are no natural scenarios where ontological dependence connects two distinct contingent 1296 1297

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These considerations apply both on the type level and on the token level. According to these definitions, the notion of ontological co-dependence presupposes the general framework of modal logic, with the truth of sentences relative to the points of modal space, and the notion of weak ontological co-dependence presupposes the framework of tense logic, where there is a structure of a certain kind on these points. This difference is similar to the difference between ontological dependence and remote ontological dependence. Given that the relation of accessibility has enough structure to talk about weak ontological codependence. The possibility of the name existing when the sentence has ceased to exist makes much more sense on the token level than on the type level. On the token level, we can easily conceive of the partial destruction of an inscription, that leaves of a sentence only the name that was in subject position. On the type level, however, expressions are most naturally understood as persistent objects, i. e., as objects that will continue to exist without end once they have started to exist. Given this understanding, the expression types ‘Sam’ and Sam will actually be ontologically co-dependent in the proper sense: Neither can be without the other.

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objects in both directions, and weak ontological co-dependence is quite rare. The only cases that come to mind are of objects that originate in fission, like the daughter cells of an amoeba that reproduces by cell division, and perhaps like monozygotic twins. But these cases differ crucially in their mereological structure from the case of the name ‘Sam’ and the sentence ‘Sam is a semantic sentence’. For in the case of cell division, the weakly co-dependent objects have developed from parts of a single whole, whereas in the case of the name and the sentence, one of the weakly co-dependent objects is part of the other. And for ordinary expressions, it is highly unusual for the part to depend in this way on the whole (although the whole may of course depend on its parts). To give a very specific example, nobody would think that the 14m high letter ‘H’ of the famous Hollywood Sign could exist only when the other letters do. So it is probable that the weak ontological co-dependence of ‘Sam’ and ‘Sam is a semantic sentence’ is metaphysically problematic, because it contradicts the usual ontological independence of the part from the whole. There is an additional reason, because the weak ontological co-dependence of ‘Sam’ and ‘Sam is a semantic sentence’ also contradicts the usual ontological independence of an object from an expression that refers to it. Given the metaphysical flexibility of language, different names will exist relative to different contexts, and more generally, different expressions will exist relative to different contexts. We might feel compelled to take concatenation (on the type level) to occur instantaneously, so that once a certain expression exists, all expressions that it is a part of (and that are otherwise formed from expressions already in existence) exist, too. It is entirely natural, however, and it fits the flexibility of language, to allow that an expression exists for several contexts before it is named (if it is named at all) – so that naming should not be required to be instantaneous. But as this matter is more of a semantic than of a metaphysical nature, we will return to it later. 1300

11.7 The problem of metaphysical ill-foundedness The (weak) ontological co-dependence of Marge, Sam, Tony, and Larry on their respective names is not the only metaphysically problematic feature they exhibit. To see this, we will in the present section look at what is characteristic of these sentences and their names in terms of the notion of metaphysical grounding. We say that one object grounds another if and only if the former helps explain the latter in a metaphysical way. 1301 And we have already observed that a name and more generally, any directly referential expression, is grounded in the object it refers to. 1302 We will endorse a further principle, that the relation of being a proper semantic part is of a grounding nature, too. Although, arguably, not every object is grounded in its proper parts, 1303 this is surely so in the case of semantic expressions. The meaningful expressions that are used to form a larger meaningful 1300 1301 1302 1303

Cf. section 11.9. Cf. section 8.3. Cf. Section 9.10. Cf., again, Schaffer 2010.

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expression are essential parts of it; they help explain what that expression is. Just imagine exchanging one of the semantic parts of a given sentence for a distinct one, e. g., turning ‘Snow is white’ into ‘Snow is green’, whilst denying that you have moved on to a distinct sentence! To see what this means for semantically self-referential names, let us again look at the name ‘Sam’ and the sentence it names. As the name ‘Sam’ is a proper part of the sentence ‘Sam is a semantic sentence’, the sentence is grounded in its name. That is a consequence of the general principle that a semantic expression is grounded in its proper parts. But as the name ‘Sam’ refers directly to the sentence Sam, it is also the case that the name is grounded in the sentence. That is a consequence of the general principle that a semantic expression is grounded in an object that it refers to directly. In sum, the sentence is grounded in the name and the name is grounded in the sentence. As grounding is transitive, this entails that both the name ‘Sam’ is grounded in itself and the sentence Sam is grounded in itself. Thus the name ‘Sam’ and the sentence Sam both are metaphysically circular objects, where an object is called metaphysically circular if and only if it is grounded in itself. 1304 (Of course, as grounding is irreflexive, there cannot be any metaphysically circular objects. 1305 But that is the point. For we are arguing in a hypothetical mode with the aim of finding reasons against the existence of semantically self-referential sentences that achieve self-reference with a name.) As we have not appealed to any feature of Sam other than that it refers to itself with a name, we can conclude that Marge, Tony, and, most importantly, the Liar sentence Larry are also metaphysically circular objects (or would be, if they existed). * Metaphysical circularity is a special case of metaphysical non-well-foundedness. An object is metaphysically non-well-founded if and only if there is a chain of metaphysical grounding which starts from that object and does not terminate. The paradigm case of an metaphysically well-founded object is a substance, i. e., an object that is not grounded in any other object – like, according to a standard conception, a physical object. 1306 But an object can be metaphysically well-founded and yet be grounded in another object – like a grin, which is grounded in a grinning person (or in a grinning cat). There are basically two ways in which a relation can fail to be well-founded. If we view the relation as a graph, they correspond to circular branches and to infinitely descending branches. Correspondingly, an object can be metaphysically non-wellfounded either because there is a grounding chain that circles back to that very 1304

The metaphysical circularity of Sam can also be brought out by a method similar to the one we employed in section 11.5 to show that the definition of ‘Sam’ needs to be circular. For if this circular definition were to succeed, then the following would be the case: Sam = the sentence that is the concatenation of the semantic name that has the syntactic basis ‘Sam’ and the semantic value (D, Sam) with the predicate ‘. . . is a semantic sentence’.

1305 1306

Thus Sam would contain itself (in a non-trivial manner) as a constituent. Self-referential names thus give rise to what Fine calls “puzzles of ground” (Fine 2010). But cf. Priest 2014, 167–193 and, again, Schaffer 2010, for opposing views.

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object or because there is a grounding chain from that object that goes downwards without end. We have seen that the first reason for non-well-foundedness is given in the case of semantically self-referential sentences like Sam and Larry, and it is also given for Liar cycles, whereas the second reason for non-well-foundedness is given in the case of the sequence of sentences that figures in Yablo’s paradox. It is this possibility of extending the present approach to Yablo’s paradox that warrants generalizing from metaphysical circularity to metaphysical non-well-foundedness. I submit that the tripartite ontology formed by physical objects, syntactic expressions, and semantic expressions needs to be metaphysically well-founded. In other words, where metaphysical foundationalism is the view that grounding is well-founded with regard to a certain category of objects, I endorse metaphysical foundationalism for our tripartite ontology. 1307 As we can presume every physical object to be well-founded, this amounts to the claim that every expression is an ontologically well-founded object. (Metaphysical foundationalism appears to me so evident and important that I will sometimes use the pejorative shorthand ‘illfoundedness’ for the metaphysical non-well-foundedness of an object.) But how does one go about showing something so basic, that every object of a certain category either is a substance or is grounded in substances, i. e., that metaphysical foundationalism holds of them? 1308 What I can do here is to allude to some praise of metaphysical well-foundedness that can be found in recent work on metaphysical grounding, and in some writings on the philosophy of mathematics. Jonathan Schaffer is adamant: “There must be a ground of being. If one thing exists only in virtue of another, then there must be something from which the reality of the derivative entities ultimately derives.” 1309

As the following remarks from the philosophy of mathematics are a bit older, the terminology differs of course from current grounding talk. Kurt Gödel praises metaphysical well-foundedness with regard to the construction of objects: “the construction of a thing can certainly not be based on a totality of things to which the thing to be constructed belongs.” 1310

Kit Fine praises metaphysical well-foundedness with regard to ontological definition: “[When] the point of a definition [. . . ] is to introduce a certain ontology of objects, we should not appeal to that ontology in explaining how the objects are to be introduced” 1311 1307 1308

1309 1310 1311

Cf. Schaffer 2010, 37f.; Bliss /Trogdon 2014, section 6.2. After all, not every relation is well-founded. Neither the relation of being to the left of nor the relation of being later than are well-founded, and the well-foundedness of the relation of being caused by is highly contentious. So the claim that the relation of grounding is well-founded does stand in need of an argument. Schaffer 2010, 37. Gödel 1944, 136. Fine 2008[2002], 87.

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And, given that the iterative conception of sets is naturally construed as entailing that a set is grounded in its elements, we can understand George Boolos as enthusiastically praising the metaphysical well-foundedness of sets when he praises the iterative conception: “the iterative conception of set [. . .] often strikes people as entirely natural, free from artificiality, not at all ad hoc, and one they might perhaps have formulated themselves” 1312

Boolos goes on: “For when one is told that a set is a collection into a whole of definite elements of our thought, one thinks: Here are some things. Now we bind them up into a whole. Now we have a set. We don’t suppose that what we come up with after combining some elements into a whole could have been one of the very things we combined (not, at least, if we are combining two or more elements).” 1313

Taken together, these quotes make a strong case against metaphysical non-wellfoundedness, and show in particular that, in an expression, metaphysical ill-foundedness is even more of an undesirable feature than weak ontological co-dependence.

11.8 From well-foundedness to subsequence Besides putting the ‘ill’ in ‘ill-foundedness’, the direction taken by Boolos in the second quote brings us back from the static picture that is standardly painted of mathematical objects to the dynamics of a contextualist ontology. It points to an important connection one might see between the notions of metaphysical wellfoundedness and of the subsequence of objects. Given the intuitions evoked by what Gödel, Fine, and Boolos say, it would appear to be natural to endorse the following principle: (Principle of Subsequence for Grounded Objects) The grounded object is subsequent to its grounds. That is, if an object b is grounded in an object a, then the object b is subsequent to the object a. To see how this principle can indeed be natural and intuitive in the present context, recall that we work with an abstract notion of context that is much more general than the notion of a moment in time. 1314 This is important because the objects talked about by Gödel, Fine, and Boolos are abstract objects, and so their existence is not relative to moments in time, nor to possible worlds in the standard sense. But even for mathematical objects as the sets of the iterative conception, it does make sense to construe their existence as relative to an abstract kind of contexts 1312 1313 1314

Boolos 1998, 16. Boolos 1998, 18. Cf. sections 8.6 and 8.7.

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(the levels of the set-theoretic hierarchy, say). 1315 And more importantly, we have already seen that the objects we are concerned with in the present study, the semantic expressions of a self-referential language, form a contextualist ontology. As being subsequent precludes being prior, the (Principle of Subsequence for Grounded Objects) entails the (Principle of Non-Priority for Grounded Objects) that we have already discussed. 1316 But subsequence is properly stronger than nonpriority: In a scenario where every object were to start its existence relative to the same context as every object that grounds it, the (Principle of Non-Priority for Grounded Objects) would hold, but not the (Principle of Subsequence for Grounded Objects). And we should be clear that whatever reasons we might have for the (Principle of Non-Priority for Grounded Objects) – they will not in themselves constitute sufficient reasons to adopt the (Principle of Subsequence for Grounded Objects), too. When we think of the objects of a given realm as being introduced in a process (or better, a quasi-process) that starts with a collection of substances, then the (Principle of Subsequence for Grounded Objects) guarantees the well-foundedness of the objects that are thus introduced. Most importantly, it excludes metaphysical circularity. For according to the principle of subsequence, a metaphysically circular object would be subsequent to itself, but that is impossible because the relation of subsequence is irreflexive. And if we require that the introduction of objects is a process with a start, then the principle of subsequence for grounded objects will also exclude non-terminating chains of grounding of the non-circular kind. 1317 It is important to note that, even though talk of ‘introduction’ and ‘construction’ might suggest otherwise, our dynamic conception of grounding differs significantly from constructivism. Roughly speaking, constructivism is the view that what we take to be abstract objects are really the products of the (maybe mental) constructive actions of humans or of ideal agents who are at least quasi-human, and it entails that we should be skeptical about actual infinity, thus shutting us out of Cantor’s paradise of the transfinite. 1318 But our official conception of dynamic grounding is given by the (Principle of Subsequence for Grounded Objects), and it is this principle that we need to look at when we want to assess what is characteristic of the conception. And the principle of subsequence says no more than that the grounded object is subsequent to its grounds – there is no mention of a (real or mythical) agent who performs the construction of objects, and there are no problems with actual infinity, either, for the principle of subsequence is compatible both with infinitely long grounding chains and with the simultaneous coming into existence of an infinity of objects in the step from one context to the next. 1319 1315

1316 1317

1318 1319

For references to the literature on set theory and its philosophy, cf. section 1.7. For a construal of the set theoretic universe as a contextualist ontology, cf., e. g., Linnebo 2010 and Studd 2013. Cf. sections 8.3 and 9.10. Another way of guaranteeing the metaphysical well-foundedness of all objects is to stick to the principle of subsequence for grounded objects, but require the well-foundedness of the relation of subsequence instead of requiring a start to the process of introduction. Cf., e. g., Tiles 1989, 95ff. As we said in section 8.5, the present approach is compatible with a constructivist approach to the paradoxes (cf., e. g., Rohs 2001 and von Kutschera 2009, chapters 2 and 3) because it does leave open

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For the record, I am inclined to endorse the (Principle of Subsequence for Grounded Objects) in full generality, and I conjecture that this would solve not only the Liar paradox, but give a uniform solution to the larger family of both the semantic paradoxes and the set theoretic paradoxes. However, there are many issues to be dealt with on the way to that goal. In particular, we will have to rethink the relation of grounding to time. For grounding gives an explanatory order of reality that typically (and importantly) is distinguished from the temporal order of reality, but the (Principle of Subsequence for Grounded Objects) re-connects grounding to a quasi-temporal order – it requires that we construe grounding in a quasi-dynamic way. As we are ultimately given reality from within, we need to take the internal stance and give a (quasi-)tensed representation of realms that are standardly thought of not only as tenseless, but as atemporal, from the expressions of a language (which we already think of in a contextualist way) to sets and certain other mathematical objects. But I believe that this extension of the tensed theory will enable us to characterize the relevant parts of the platonic realm in an adequate way, and that it will be beneficial because the tools of a tensed representation of time promise a way of finally solving the vexing problems of expressibility that beset so many approaches to paradox. However, although we have done a lot in chapters 9 and 10 to make reality conceived of as a (quasi-)tensed representable in a rigorous way, there remains much to spell out and to argue for with regard to grounding. Therefore I will leave the project of giving a full (quasi-)tensed theory centered around the (Principle of Subsequence for Grounded Objects) for another occasion. Thankfully, this is possible in the context of the present study, because there is a more specific principle that will already allow us to deal with the Liar paradox in a satisfactory way. It is the following, which we have already discussed (although not endorsed yet): 1320 (Principle of Subsequence for Directly Referential Expressions) Every directly referential expression is subsequent to its extension. I. e., if an expression e refers directly to an object a, then the expression e is subsequent to the object a. In a framework like the present one where direct reference is taken to be a relation of grounding character, the (Principle of Subsequence for Directly Referential Expressions) is entailed by but strictly weaker than the (Principle of Subsequence for Grounded Objects). It states that a specific relation of grounding character – direct reference – meets the requirement of subsequence, but it does not say this of every relation of grounding character. It helps establish the metaphysical well-foundedness of expressions, though, because it removes the plausibility of ill-foundedness in the ontology of expressions (as we will see in a moment). As we have found illfoundedness to be a bad thing, this already gives us a plausible reason to endorse

1320

the option of construing concepts in an idealist way. Here we should add that despite this compatibility with regard to the metaphysical status of concepts, there is a marked contrast to these constructivist approaches with regard to considerations of cardinality, because the (Principle of Subsequence for Grounded Objects) is compatible with the actual existence of transfinite infinities. Cf. section 9.10.

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the (Principle of Subsequence for Directly Referential Expressions). But we will say more about this later. 1321

11.9 The problem of referential stickiness So far, we have seen two weird and undesirable effects that the existence of semantically self-referential names would have, ontological co-dependence and metaphysical ill-foundedness. Let us now change the perspective from metaphysics to semantics and look at two problems that grow visible when we focus on the relation between a language and the world it is about. Because of the overlap between language and world that will occur in a scenario that includes self-referentiality, 1322 the relations between language and world are not fully extricable from the metaphysics of expressions here. Therefore it is not surprising that what will emerge are not two entirely distinct, additional problems, but the semantic flip sides of the problems of ontological co-dependence (in the present section) and of metaphysical ill-foundedness (in the next section). Because of the ontological dependence of a sentence on its parts, the self-referential sentence Sam could not exist if it were not named ‘Sam’ and the Liar sentence Larry could not exist if it were not named ‘Larry’. We have already mentioned that this contradicts the usual ontological independence of an object from an expression that refers to it. But that observation was made from the perspective of the metaphysics of expressions; now we will take the semantic perspective. What we have here are objects that must have a certain name. But although every name designates rigidly, i. e., refers to the same individual with respect to every possible world, 1323 the converse is not true of name-bearers. E. g., Obama could have had a name different from ‘Obama’ – and Romeo need not be called ‘Romeo’. To quote William Shakespeare: “Juliet:

Romeo:

What’s in a name? That which we call a rose by any other name would smell as sweet; So Romeo would, were he not Romeo call’d, Retain that dear perfection which he owes Without that title. Romeo, doff thy name; And for thy name, which is not part of thee, Take all myself. I take thee at thy word: Call me but love, and I’ll be new baptized; Henceforth I never will be Romeo.” 1324

While Romeo need not be called ‘Romeo’, Sam must be called ‘Sam’, and Larry must be called ‘Larry’ – their names are undoffable. 1321 1322 1323 1324

Cf. section 11.12. Cf. section 11.1. Cf. section 5.2. Shakespeare 1951, 912 (= Shakespeare Romeo and Juliet, Act II, Scene 2).

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Let us say that an object is reversely rigid if and only if there is a singular term such that relative to every context, the object is referred to by that singular term. And let us say that a singular term is referentially sticky if and only if there is an object such that relative to every context, the object is referred to by that singular term. A reversely rigid object clearly needs a referentially sticky singular term that refers to it, and vice versa. And as a pair, reverse rigidity and referential stickiness are the semantic flip side of the weak ontological co-dependence that characterizes the sentence Sam and its name ‘Sam’, the sentence Larry and its name ‘Larry’, and so on. For the name ‘Sam’ is referentially sticky and the sentence Sam is reversely rigid, the name ‘Larry’ is referentially sticky and the sentence Larry is reversely rigid, and so on. In contrast, the names ‘Obama’ and ‘Romeo’ do not exhibit referential stickiness, nor do any other ordinary names, and no ordinary object exhibits reverse rigidity, because every ordinary object could have had a different name, or no name at all. Describing the situation in these terms reveals how very far we have moved away from the ordinary functioning of language by assuming that there are semantically self-referential names. For note how strange it would be if there indeed were any referentially sticky names in some language! It would lead to this language having an extraordinary modal advantage over the world it is about, and to speakers wielding an uncanny modal power over the world they talk about in this language. Merely by introducing a certain name – they need not even use it! –, they would be able to create an object distinct from it. Or, to look at it from the side of the named object, naming would no longer be arbitrary. At least for some objects – those referred to by a referentially sticky name – there simply would be no option of having another name or of having no name at all. 1325

11.10 The problem of semantic magicality Things are even worse. This grows evident when we turn to the semantic flip side of metaphysical circularity. 1326 We saw that the sentence Sam is grounded in its name, and similarly for any semantic sentence that refers to itself with a name. We have already mentioned that this contradicts the irreflexivity of grounding (because every name is grounded in its extension). Let us now switch from the perspective of the metaphysics of expressions to the perspective of semantics. Then we see that what we have here are objects that are grounded in their names. But although every name is grounded in its bearer, it would be weird to hold the converse of some object. It certainly cannot be true of ordinary objects. E. g., Obama is not grounded in his name, and neither is Romeo. 1327 1325

1326 1327

We are here ignoring the possibility that an object has two names – focusing on what could be called its main name, as it were. Cf. section 11.7. Granted, maybe, that as a fictional character, Romeo is grounded in his (or its) name. But that means viewing him from a point external to Shakespeare’s play. When we say here that Romeo is not

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Let us call a singular term a magical designator if and only if the object it refers to exists only in virtue of being referred to by it, so that a magical designator grounds its extension. Then the name ‘Sam’ is a magical designator because the semantic sentence ‘Sam is a semantic sentence’ exists only in virtue of being referred to by ‘Sam’. The name ‘Larry’ is a magical designator because the semantic sentence ‘Larry is false’ exists only in virtue of being referred to by ‘Larry’. And so on. In contrast, the names ‘Obama’ and ‘Romeo’ are not magical, nor are any other ordinary names, and no ordinary object can be explained by this kind of magic. We see again how very far we have moved away from the ordinary functioning of language by assuming that there are semantically self-referential names. For note how strange it would be if there indeed were any magical designators in some language! It would lead to this language having an extraordinary metaphysical advantage over the world it is about, and to speakers wielding an uncanny metaphysical power over the world they talk about in this language. Merely by introducing a certain name – they need not even use it! –, they would be able to shape the very essence of an object distinct from it. Or, to look at it from the side of the named object, naming would no longer be irrelevant to the essence of the named object. At least for some objects – those referred to by a magical designator – their very nature would be explained by how they are referred to. Telling the story in this way shows why we are justified in using the word ‘magical’, which (in the present context) is pejorative. For there is an analogy here, based on the fact that both causation and grounding can be seen as species of a wider notion of explanation. 1328 Just as an appeal to witchcraft means to offer an extraordinary kind of causal explanation, an appeal to semantic magicality means to offer an extraordinary kind of metaphysical explanation. Just as a magical spell causes something that could in ordinary circumstances not be caused by words (at least not by these words), a magical designator grounds an object that would in ordinary circumstances not be grounded in words (at least not in these words 1329). Put this way, it emerges that creating the sentence ‘Marge is meaningful’ by proclaiming ‘Let ‘Marge is meaningful’ be called ‘Marge’!’ is no less uncanny than conjuring up a guardian creature by casting the spell “Expecto Patronum!”. 1330

1328 1329

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grounded in his name, we are taking the internal viewpoint of the fiction, where Romeo is not a fictional character but a young man – as we did in section 11.9, when we discussed the doffability of his name. Cf., e. g., Fine 2012, 37 and Schaffer 2015. Some objects are grounded in words, because linguistic expressions are objects and every longer expression is grounded in the words that are its semantic parts (cf. section 11.7). But that is no reason to characterize these words as magical designators, as long as they do not also refer to the whole they are parts of. Rowling 1999, 253ff. It might appear that for Anselm, the word ‘God’ is a magical designator, because he seems to presuppose in his ontological argument for the existence of God (Anselm 1965 [Proslogion], chapters 2 and 3) that the word ‘God’ by its meaning grounds that it has an extension. But Anselm’s argument is translatable, and so it is not the word itself, but merely its meaning that must be construed as magical. We might say that although he does not use ‘God’ as a magical designator, it does express a magical concept for Anselm. Thus the notion of magicality may shed further critical light on Anselm’s

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Rejecting magical designators does not mean taking issue with performative speech acts and their role in the creation of certain objects in general. There is, e. g., a certain utterance of ‘Let this ship be called ‘Titanic’!’ that grounds an object, namely the semantic name ‘Titanic’. But the baptism is not magical in the way that those of Sam and Larry are; for it does not contain a name of the object that it introduces. For that object is the semantic name ‘Titanic’, and although the baptism formula refers to the ship, it neither uses nor mentions the semantic name ‘Titanic’ – the only thing it mentions in this regard is the syntactic name (the string of letters ‘Titanic’) that is to serve as the basis of that semantic name. However, there certainly are performative speech acts that ground a certain item in an expression. Even the baptism of the Titanic leads to one expression – the semantic name ‘Titanic’ – being grounded in another expression that is not a part of it, namely in the whole formula ‘Let this ship be called ‘Titanic’!’. But let us study the matter further with a focus on another example, where the grounded item is not itself an expression. When in the right kind of circumstances a person with the appropriate authority says ‘I pronounce you married partners!’, this utterance will ground a particular social item, namely the marriage of those partners. And many social objects will be grounded in a similar way. So there might be material here for an objection to the justification of our rejection of semantic magicality. But it is all about the direction of fit! 1331 The sentences we are discussing (the examples (1) through (6) and other self-referential sentences) belong to that large group of sentences that are typically used in assertions, and the distinctive goal of an assertion – saying something true – is reached when the sentence does indeed match the world. I. e., an assertion has a language-to-world direction of fit, and so do the self-referential sentences we are discussing. However, when a performative speech act has reached its goal, then the world will match the sentence used in that speech act; a performative has a worldto-language direction of fit. 1332 Now, what makes a spell magical in the usual causal sense of magicality and what makes a semantically self-referential name magical in our metaphysical sense is that they introduce the direction of fit from world to language into a setting that is otherwise characterized by the direction from language to world. 1333 Thus containing a magical designator will turn an assertive sentence

1331 1332

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ontological argument. Conversely, Anselm’s use of a magical concept provides a welcome analogy and contrast, that can bring out more of the specific character of names that are magical designators. Cf., e. g., Anscombe 1963, 56; Searle 2010, 11ff. and 27ff.; and for an overview, Green 2015, section 3.1. In Searle 2010, 12f. the corresponding kind of speech acts are construed as having a double direction of fit, both language-to-world and world-to-language. This is likely correct in some variants of the speech acts of interest to us (‘I name this ship ‘Titanic’.’), and doubtful in others (‘Let this ship be called ‘Titanic’!’). But for our purposes it is enough to highlight that all such performative utterances have a world-to-language direction of fit, whether in addition to a language-to-world direction of fit or not, because they thus already differ crucially from assertions, which only have language-to-world direction of fit. ‘Read this sentence!’ and ‘I promise to read this sentence’ can be construed as self-referential sentences that are used in performative speech acts. This goes to show that self-reference can occur not only in assertions but also in performatives and thus in combination with both directions of fit (if it can occur at all).

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into a wrong-way driver – it alone moves with the performatives, and thus against the stream of all other assertions. 1334 Semantic magicality can be seen as a more severe problem than referential stickiness. The general reason is that grounding is a stronger notion than ontological dependence, and the notions in question entail that the extension of a magical singular term is grounded in that singular term and the extension of a referentially sticky singular term depends ontologically on that singular term. So it should be possible for there to be a particular case where an expression exhibits referential stickiness but not semantic magicality. It is difficult, however, to come up with a realistic example of a sticky name that is not magical. (In the self-referential cases that we have been discussing, the magicality explains the stickiness.) But we can conceive of a religious story that depicts such a situation. For imagine a faith according to which the highest deity has its name of necessity (religions have been known to make all sorts of claims about the names of their deities), but which also depicts this highest deity as the ground of all being, that is not grounded in anything else, and a fortiori not grounded in its name. Then the name of the deity will be referentially sticky, but not semantically magical (according to that faith). The mere conceivability of this situation shows how semantic magicality goes beyond referential stickiness.

11.11 ‘This sentence’ exhibits the four problems, too. Among singular terms that are directly referential, names are arguably paradigmatic. But in our overview of the semantics of singular terms we saw that indexicals also exhibit direct reference, although in a different way. 1335 So we can expect those indexicals that are used to construct semantically self-referential expressions to exhibit the same problems that we saw to beset semantically self-referential names – although perhaps in a different way. Here, we will specifically have to look at the sententially indexical phrase ‘this sentence’, as regimented to refer to the sentence it is used in. 1336 Continuing our list of examples for self-referential sentences in section 11.3, here are three sentences that achieve self-reference with an indexical singular term:

1334

1335 1336

I conjecture that this observation can be generalized by extending our definition of a magical designator from assertions to performatives and laying down that any magical designator reverses the direction of fit of the sentence it occurs in. This conjecture would make our considerations applicable to selfreferential singular terms that occur in imperatives or promises, for example to the indexicals in the Liar-like ‘Do not follow this order!’ and ‘I promise not to keep this promise!’, the Truth-teller-like ‘Follow this order!’ and ‘I promise to keep this promise!’, as well as the Sam-like ‘Treat this speech act as an order!’ and ‘I promise to treat this speech act as a promise!’. If this is correct, there would be a natural way of extending the present approach from the Liar paradox to related paradoxes and puzzles about performatives. Cf. section 5.2. Cf. sections 5.2, 5.3, and 5.4.

11.11 ‘This sentence’ exhibits the four problems, too.

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(7) ‘This sentence consists of six words.’ (8) ‘This sentence is meaningful.’ (9) ‘This sentence is false.’ Sentence (7) corresponds to sentence (2) of section 11.3, ‘Frodo consists of five words’, in so far it is most naturally construed as being self-referential in a merely syntactic way: Just as ‘Frodo’ is the name of the syntactic basis of sentence (2), the occurrence of ‘this sentence’ in sentence (7) then refers to the syntactic basis of sentence (7). Thus sentence (7) is harmless, and we should put it to the side. Sentence (8) corresponds to sentence (4), i. e., to the sentence Sam we focused on in our discussion of self-referential names, in so far it is most naturally construed as semantically self-referential. And sentence (9) under the most natural construal corresponds to sentence (6), i. e., to ‘Larry is false’, which is named ‘Larry’ – sentence (9) is an indexical Liar sentence. As before, we will concentrate on a sentence that is semantically self-referential but not paradoxical (or rather, not paradoxical in the sense of the Liar paradox), i. e., on sentence (8). How are we to understand the indexical ‘this sentence’? By putting the merely syntactically self-referential sentence (7) to the side, we are in effect taking the locution ‘this sentence’ to be regimented in such a way that it refers to the semantic sentence it is used in – we are taking ‘this sentence’ as short for ‘this semantic sentence’. Understood in this way, ‘this sentence’ is an indexical of a rather special sort. An important aspect that sets indexicals apart from other expressions that are directly referential – names and quotation expressions – is their context-sensitivity. In this regard, we now come upon a first important difference between ‘this sentence’ and other indexicals, and more generally, between the sentential dimension of indexicality of ‘this sentence’ on the one hand and the personal, temporal, and spatial dimensions of ‘I’, ‘now’, and ‘here’, respectively, on the other hand. Because of context-sensitivity, we can no longer study the extension of expressions solely on the level of expressions as such. (This was still possible for names and even for certain descriptions like ‘. . . is a prime number’, where all particular instances of an expression behave in the exact same way.) And once we leave the level of expressions as such, we need to make a twofold distinction because we are concerned with the particular, sentential dimension of indexicality and want to contrast it with other dimensions of indexicality. For there are two ways in which something can be a particular instance of an expression as such: firstly, as a token of the expression, i. e., as a concrete physical instantiation of it; and secondly, as an occurrence of the expression, i. e., as that expression as embedded (at a certain position) in a larger expression. 1337 In places where we want to talk indiscriminately about to-

1337

Linda Wetzel gives a precise account of what an occurrence of an expression is (Wetzel 2009, 125ff., and in particular 129). But apart from the point that the distinction between an expression and its occurrences is orthogonal to the distinction between a type and its tokens, the details need not concern us here.

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kens and occurrences, we will use the umbrella notion of being a particularized expression. 1338 Now, in general, the extension of an indexical is never given on the level of the expression as such, and we always have to look at the corresponding particularized expressions. In the case of normal, run-of-the-mill indexicals like ‘I’, ‘now’, and ‘here’, it usually is the respective concrete, physical token of the expression that determines the context of use – it is the token that will determine who uttered it, when it was uttered, or where it was uttered, as the case may be. (As of now, we need not worry about some apparent exceptions, as the example that Hegel gave of a slip of paper with the inscription ‘Now it is night’ on it, which appears to be a case where a single token sentence persists through several moments and therefore changes its truth value at sunset and sunrise. 1339) In the case of sentential indexicality, however, it is the occurrence that determines the context of use, because the crucial thing relevant for determining the extension of a sentential indexical is the expression it is embedded in (perhaps together with its position in the larger expression) – in many cases leaving open the question of whether it is the expression on the type level or the expression on the token level that we are concerned with. 1340 In these terms, we can spell out the difference between the other, more standard kinds of indexicality and sentential indexicality in the following way: Where we can say with regard to standard situations that every token of the personally indexical singular term ‘I’ refers to the speaker who uses it in an utterance, every token of the temporally indexical singular term ‘now’ refers to the moment when it is used in an utterance, every token of the spatially indexical ‘here’ refers to the place where it is used in an utterance, we should say that it is every occurrence (be it type or token) of the sententially indexical singular term ‘this sentence’ that refers to the sentence it is used in. 1341 There is a corresponding difference with regard to the contexts of use 1338

In the usual set theoretical fashion, a particularized expression can arguably be modeled as an ordered pair of an expression and a context. The most important aspect of this construal is the criterion of identity: Expression e1 in context c1 = expression e2 in context c2 if and only if e1 = e2 and c1 = c2.

1339

1340

1341

Depending on how the contexts are construed, this can then be used both to model tokens of expressions, where the contexts would incorporate temporal, spatial, and personal positions of use, and to model occurrences of expressions, where the contexts would themselves incorporate larger expressions that the expression can occur in (an idea that will shortly become clearer) as well as its position therein. As of now, we do not need to endorse this construal (but we will in section 12.8, where we also give a more careful formulation of a criterion of identity for tokens). Here its main purpose is to illustrate the parallel between the two most general species of particularized expressions, tokens and occurrences. Hegel 1988, 71. Such examples will become important in section 12.7 and will be discussed in section 12.8. We could distinguish between two readings, made explicit as ‘this sentencetype’ and ‘this sentencetoken’. But let us not get lost in details beyond necessity. Note that for a sentence to be the extension of the indexical ‘this sentence’, we need to require more than that the indexical occurs in that sentence, because merely mentioning the indexical is not enough. E. g., the indexical expressions that occur between quote marks in the (true) sentences ‘‘I’, ‘now’, and ‘here’ are indexicals’ and ‘‘This sentence’ consists of two words’ arguably do not refer to anything because they are mentioned, not used.

11.11 ‘This sentence’ exhibits the four problems, too.

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relevant to the respective kinds of indexicals. It is very similar to the difference with regard to their reference, because each one of the four indexicals ‘I’, ‘now’, ‘here’, and ‘this sentence’ is centric insofar as it refers to its respective context of use: 1342 In contrast to the speaker for ‘I’, the moment for ‘now’, and the place for ‘here’, the context of use relevant to the extension of ‘this sentence’ is the respective sentence that it is used in. (It might seem strange to construe sentences as contexts of use, as they are usually themselves evaluated relative to contexts. But then, ‘this sentence’ is a strange indexical. 1343) This much – that in the case of ‘this sentence’, we need to talk about occurrences instead of tokens and to construe sentences as contexts of use – is already entailed by the regimentation of ‘this sentence’ to refer to the (semantic) sentence that it is used in. * Now we are in a position to turn to our main question: What, if anything, is wrong with sentence (8), ‘This sentence is meaningful’? In order to investigate, it will be helpful to recall that, as we learned from Peirce’s early characterization of indexicality, each token of an indexical expression stands in an “existential connection” to the object it refers to. 1344 A token of the personal indexical ‘I’ must be produced by the person it refers to, a token of the temporal indexical ‘now’ must be simultaneous with the moment it refers to, a token of the spatial indexical ‘here’ must be located at the place it refers to, and so on. In the same vein, an occurrence of the sententially indexical ‘this sentence’ must be used in the semantic sentence it refers to. ‘This sentence’ as used in another sentence – in sentence (9), say – does not refer to sentence (8). The crucial difference between the sentential dimension of indexicality and the standard dimensions of indexicality is that while the sentence referred to by an occurrence of ‘this sentence’ also stands in a similar existential connection to that occurrence because it is a part of it, a speaker does not stand in a similar existential connection to any token of the term ‘I’ he or she produces, a moment does not stand in a similar existential connection to any token of the term ‘now’ that is simultaneous with it, and a place does not stand in a similar existential connection to any token of the term ‘here’ that is co-located with it. It is only in the case of sentential indexicality that the existential connection goes both ways! In our example, the occurrence of ‘this sentence’ in sentence (8) stands in an existential connection to sentence (8) because it is used in it, and sentence (8) stands in an existential connection to this occurrence of ‘this sentence’ because it is a semantic part of it. The metaphysical framework that we have already used when we studied selfreferential names allows us to give two distinct explications of the Peircean notion of an existential connection: in terms of (weak) ontological dependence, and in terms of metaphysical grounding. Let us do both!

1342

1343 1344

This is in contrast to excentric indexicals which refer to a context distinct from their context of use, like ‘you’, ‘yesterday’, ‘over there’, and ‘the previous sentence’. Cf. Pleitz 2010b and 2011a. Burks 1949, 674. Cf. section 5.3.

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To see that an occurrence of the indexical ‘this sentence’ and the semantic sentence it refers to exhibit ontological co-dependence, note first that the semantic sentence referred to by an occurrence of ‘this sentence’ is identical to the sentence that this particular occurrence of ‘this sentence’ is used in. Thus the sentence that ‘this sentence’ is used in is the relevant context of use. E. g., sentence (8) is the context of use relevant to determining the extension of the occurrence of the indexical ‘this sentence’ in sentence (8). Nothing more is needed to show that an occurrence of the indexical ‘this sentence’ and the semantic sentence it refers to exist relative to the same contexts of use, so that they are ontologically co-dependent. 1345 To see how strange this is (or would be), let us have a contrasting look at standard indexicals. It may well be the case that a token of the word ‘I’ depends ontologically on the speaker who produced it, that a token of the word ‘now’ depends ontologically on the moment it is simultaneous with, and that a token of the word ‘here’ depends ontologically on the place it is located at. But neither does a speaker depend ontologically on any token of the word ‘I’ he or she produces, nor a moment on any token of the word ‘now’ that is simultaneous with it, nor a place on any token of the word ‘here’ that is located at it. In fact, speakers, moments, and places can exist very well without being referred to by any token of the respective indexical. As before, the metaphysical phenomenon of ontological co-dependence has the equally problematic, connected semantic phenomena of referential stickiness and reverse rigidity as its flip side. In our example, it is not only the case that the occurrence of the indexical ‘this sentence’ in sentence (8) needs to be used in sentence (8) to refer to sentence (8) – it is also the case that it could not have been separate! For any possible occurrence of ‘this sentence’ that is not embedded in sentence (8) is distinct from the occurrence of ‘this sentence’ in sentence (8). 1346 And from the viewpoint of the semantic sentence referred to this means, of course, that sentence (8) could not exist without being referred to by that occurrence of ‘this sentence’, because it would not be sentence (8) any longer. Thus sentence (8) is a reversely rigid object – which speakers, moments, places, and all other ordinary objects are not. Before we can show that an occurrence of the indexical ‘this sentence’ and the semantic sentence it refers to exhibit metaphysical circularity, we need to take a closer look at the constituents of the relevant occurrence of ‘this sentence’ construed as a semantic expression. We know that we can model a semantic expression e 1345

1346

Arguably, an occurrence of ‘this sentence’ and the sentence it is used in are ontologically co-dependent in more than the weak way, in contrast to self-referential names. For how could the respective occurrence of ‘this sentence’ survive the destruction of the sentence that it is used in, given that it is construed as a semantic expression? In this, ‘this sentence’ differs from a token of a (purported) selfreferential name, which like a token of any other name can continue to exist after the named object has gone out of existence. We could make our arguments for referential stickiness and reverse rigidity more specific by being explicit about the identity conditions for an occurrence of an expression: The occurrence of expression a in expression b in place n = the occurrence of expression c in expression d in place m if and only if a = c and b = d and n = m. (Wetzel 2009, 132. The notion of being in place n is needed here because an expression can occur more than once in an expression, as does ‘Macavity’ in the line “Macavity, Macavity, there’s no one like Macavity” (Eliot 1952, 163). For details cf. Wetzel 2009, 125ff.)

11.11 ‘This sentence’ exhibits the four problems, too.

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roughly as an object constituted by the syntactic basis of e taken together with the semantic value of e. 1347 As indexicals are both context-sensitive and directly referential, what we need to model is a particularized expression, or, an expression in a particular context of use, 1348 not the expression as such. 1349 The reason for this is that, because of the direct referentiality of an indexical, its respective extension must be an essential constituent of the semantic value of the indexical and hence also of the indexical construed as a semantic expression; but because of the contextsensitivity of the indexicality, we are precluded from construing the extension as an essential constituent of the expression as such – for just like an essential part, an essential constituent cannot vary. So, what is the semantic value of an indexical in a context? Recall that when we characterized singular terms with the help of two-dimensional semantics, we modeled a simple indexical like ‘I’, ‘now’, or ‘here’ – and these are paradigmatic for our understanding of the sentential indexical ‘this sentence’ –, as having a twodimensional intension, i. e., a (systematic) function from contexts of use to contents, each of which is a constant function mapping each circumstance of evaluation to one and the same object, corresponding to the direct reference of the indexical in a context of use. 1350 Later we noted that, as concepts and sense in general are not reducible to objects in a contextualist ontology with a growing domain, we cannot in general model a sense as an intension, not even as a two-dimensional one. In the end of the day, we have to stick to irreducible concepts when doing semantics. 1351 For the example of the personal indexical ‘I’ this means that we can do no more to characterize the semantic value of this expression as such than to say that ‘I’ has the character of being the current speaker, 1352 which relative to any context of use then picks out an object – the respective speaker of that context – which is then referred to in a direct way. What does this mean for the semantic value of an indexical in a particular context of use? What does it mean for a particular token of the indexical ‘I’ that has been uttered by Obama, say? It would be mistaken to think that someone who understands this particular token apprehends only that it refers to Obama (in a direct way).

1347

1348

1349

1350 1351 1352

As we said in a footnote to section 11.5, we are giving a simplified account of the a semantic expression as an object with two constituents, a syntactic basis and a semantic value, where the latter is a theoretical entity imported from semantics to do metaphysics. Here we are identifying a particularized expression with the corresponding expression as such in the respective context. Strictly speaking this is rash, because we will only later establish that each particularized expression exists relative to at most one context (cf. sections 12.7 and 12.8). But these details do not matter in our present investigation of the indexical ‘this sentence’. As we said earlier in this section, in our terminology being an expression as such is opposed to being a particularized expression; this does not in general coincide with the opposition of an expression on the type level and on the token level. Cf. section 5.2. Cf. sections 4.8 and 9.8. Recall that we use the word ‘current’ to generalize the personally indexical ‘I’, the temporally indexical ‘now’ or ‘at present’, and so on to an arbitrary dimension of indexicality, so that it is our umbrella centric indexical.

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Rather, they would also learn that it refers to him as being the current speaker. 1353 To generalize, the semantic value of an indexical in a particular context of use is comprised by both the character of this indexical (as an expression as such) and the object referred to directly by the indexical (as an expression in that context). In the pair notation that we have adopted as a way of referring to a semantic value in a way that makes something of its structure visible, we can say that the semantic value of an indexical in a particular context is the pair of pairs ((I, the character of the indexical), (D, the object picked out by the character in the context)), where (D, the object picked out by the character in the context) is the content determined by the indexical’s character in the context. 1354 The semantic value of the indexical ‘I’ in a context where it is uttered by Obama would thus be constituted by both the character of being the current speaker and Obama as the constant content that is referred to in a direct way, ((I, being the speaker of the context), (D, Barack Obama)). In the case of sentential indexicality, it is not the token but the occurrence that determines the context of use. Therefore the general result that we are interested in is the following: The semantic value of an occurrence of the sentential indexical ‘this sentence’ is comprised by the character of being the current sentence and the sentence it occurs in as the constant content that is referred to in a direct way. In the example of the occurrence of ‘this sentence’ in sentence (8) that is of particular interest to us here, the semantic value will be comprised by being the current sentence as the character and sentence (8) as the content that this character determines relative to the relevant context (i. e., relative to sentence (8) itself) – it will be ((I, being the sentence of the context), (D, sentence (8))). Let us have a brief look at how these considerations allow to distinguish indexicals from names and descriptions. The distinction concerns the relation between an expression as such and that expression in a context of use, i. e., the corresponding particularized expression. We saw that the semantic value of an indexical as a particularized expression has two constituents, the character of the indexical as an expression as such and the content that this character determines in the context of use of the respective particularized expression, of a particular object as being referred to directly. In contrast, the semantic value of a name in a context is just the content of the name, i. e., the extension of the name that is referred to in a direct way, and the semantic value of a typical description in a context is just the singular concept that is the sense of the description, that will determine an extension with respect to each circumstance of evaluation. Thus the semantic values of a name and of a typical description are not context-sensitive, and therefore we can assign them to the respective expression as such, too. 1355 On the metaphysical level, this means that we are free to model a name or a description as an expression as such 1353

1354 1355

That is why the sentence ‘I am Barack Obama’ that is uttered by Obama is informative. Or more precisely, that is why the sentence ‘I am Mark Twain’ that is uttered by Mark Twain would be informative in an entirely different way than an utterance of ‘Samuel Clemens is Mark Twain’. Cf. section 8.1. Cf. sections 4.6 and 5.2, and in particular subsection 5.2.6. Although the extension of a typical description need not be constant at all and can even be empty, this variation concerns circumstances of evaluation, not contexts of use. Also, the extension does not figure in the semantic value because a typical description is indirectly referential.

11.11 ‘This sentence’ exhibits the four problems, too.

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(i. e., typically, on the type level) – and this is what we have done for names in the preceding sections and what we will do for descriptions in the next chapter. It is only in the case of an indexical that we have no alternative but to work on the level of particularized expressions, i. e., on the level of tokens for ordinary indexicals like ‘I’, ‘now’, and ‘here’, and on the level of occurrences for the sentential indexical ‘this sentence’. Now we can see why an occurrence of the indexical ‘this sentence’ and the semantic sentence it refers to exhibit metaphysical circularity. Briefly, as a semantic sentence is grounded in its semantic parts and an occurrence of the sentential indexical ‘this sentence’ is grounded in the sentence it refers to directly, a sentence that refers to itself by way of the indexical ‘this sentence’ is grounded in itself. We are able to tell this story in more detail with the help of what we have learned about the semantic value of an indexical in a context of use. In the case of sentence (8), ‘This sentence is meaningful’, what we have learned allows us to describe the metaphysics of the occurrence of ‘this sentence’ in sentence (8) in the following way: The occurrence of the semantic expression ‘this sentence’ in sentence (8) = the semantic expression that is constituted by the syntactic expression ‘this sentence’ and the semantic value that is constituted by the character of being the current sentence and the content of sentence (8) that is referred to directly, i. e., ((I, being the sentence of the context), (D, sentence (8))). The metaphysics of sentence (8), in turn, can be modeled in the following way: 1356 Sentence (8) = the semantic sentence that is the (semantic) concatenation of (i) an occurrence of the semantic expression ‘this sentence’, and (ii) the predicate ‘. . . is meaningful’. Putting these two identity statements together (e. g., starting with the second one and using the substitution of identicals as licensed by the first one) we get the following somewhat complicated identity statement: Sentence (8) = the semantic sentence that is the (semantic) concatenation of (i) an occurrence of the semantic expression that is constituted by the syntactic expression ‘this sentence’ and the semantic value that is constituted by the character of being the current sentence and the content of sentence (8) that is referred to directly, i. e., ((I, being the sentence of the context), (D, sentence (8))), and (ii) the predicate ‘. . . is meaningful’. 1357

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We have used a similar paraphrase already in section 11.5. But there our objective was to give evidence for definitional circularity, whereas what we want to show here is metaphysical circularity. We could have unfolded the structure of the second part of sentence (8) in a similar way, by making explicit that the predicate ‘. . . is meaningful’ is a semantic expression that is constituted by the syntactic expressions ‘. . . is meaningful’ and the semantic value that is the concept of being meaningful. But

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Sentence (8) occurs on both sides of this identity statement – it is its own essential constituent. Now, essential constitution goes hand in hand with grounding – an object is grounded in its essential constituents. Therefore sentence (8) and the occurrence of ‘this sentence’ in sentence (8) form a grounding circle. As with names, this metaphysical circularity has as its flip-side the semantic magicality of the occurrence of the indexical ‘this sentence’ in sentence (8) – this occurrence of ‘this sentence’ helps ground the object that it refers to, i. e., sentence (8). As we have not made use of any particularities of sentence (8) other than that it refers to itself with the sentential indexical ‘this sentence’, we can generalize and say that every occurrence of the indexical ‘this sentence’ that is used in some sentence is semantically magical, and exhibits metaphysical circularity with the sentence it is used in. We know that metaphysical circularity and semantic magicality are undesirable because they contradict quite general intuitions concerning metaphysics (about the well-foundedness of grounding) and semantics (about the direction of fit of descriptive sentences), respectively. 1358 In the case at hand, we can also illustrate how weird the metaphysical circularity and semantic magicality of every occurrence of the indexical ‘this sentence’ would be by contrasting its predicament with the situation of standard indexicals. Construed as a semantic expression, a token or occurrence of an indexical includes its extension as a constituent, just as a name does on both the token level and the type level. It is this relation of constitution that accounts for the metaphysical circularity and the ensuing semantic magicality of each occurrence of ‘this sentence’, just as in the case of self-referential names. Now, just as no ordinary, non-self-referential name is a constituent of the object that it names, no token of the indexical ‘I’ is a constituent of the speaker who produces it, no token of the indexical ‘now’ is a constituent of the moment it is simultaneous with, and no token of the indexical ‘here’ is a constituent of the place it is located at. With ordinary, non-sentential indexicals, grounding goes only in one direction, from the object that is referred to directly to the token of the respective indexical, but not also back from the token to the object. Therefore there is no metaphysical circle between me and my use of ‘I’, and my utterance of ‘I’ has no magical power over me.

11.12 The principle of subsequence for directly referential expressions We have seen that semantic self-reference that is achieved with a directly referential singular term – with a name or an indexical – goes hand in hand with the phenomena of metaphysical circularity, weak ontological co-dependence, referential stickiness,

1358

with a view to the question how sentence (8) is among its own constituents, this would have added no further information. The main point to note here is that as concepts are not reducible to objects, sentence (8) will not be a constituent of the concept of being meaningful, regardless of whether it falls under this concept or not. Cf. sections 9.7 and 9.8. Cf. sections 11.7. and 11.10.

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and semantic magicality. And we have found these phenomena to be undesirable. 1359 This gives us good reason to deny the existence of semantically self-referential names and indexicals. 1360 But how can they be rejected in a systematic way? The short answer is: By endorsing the (Principle of Subsequence for Directly Referential Expressions). 1361 Our task in the present section and the next is to spell out the details of this answer, and to justify it. But before we turn to that, let us recapitulate what we have found to go wrong when we assumed that directly referential singular terms can be used to construct semantically self-referential sentences. Under the hypothetical assumption that there are indeed names that refer to a semantic expression that they are themselves a part of, we saw two metaphysical problems to arise, each with a semantic problem as its flip side. Under the hypothetical assumption that there are self-referential indexicals, we saw these problems to recur on the level of occurrences. Here are the four problems as they occur in our main example of the name ‘Sam’ that is self-referential because it refers to the sentence ‘Sam is a semantic sentence’. The name and the sentence must start to exist relative to the same context, i. e., they must exhibit weak ontological co-dependence. Insofar as this contradicts the usual modal independence of the part from the whole, this is a metaphysical problem for the name ‘Sam’. Seen from a semantic point of view, it means that the sentence Sam bears its name ‘Sam’ of necessity, so that the sentence is reversely rigid and the name is referentially sticky, which contradicts the arbitrariness of naming. 1362 Things get worse, because in addition to this uncommonly strong modal connection between the name ‘Sam’ and the sentence ‘Sam is a semantic sentence’, there is an uncommonly strong connection by grounding. As direct reference is a relation of grounding character, the name is grounded in the sentence. But as an expression is grounded in its proper parts, the sentence is also grounded in the name. Because of the transitivity of grounding, both the name and the sentence exhibit metaphysical circularity (which is a special case of the phenomenon of metaphysical ill-foundedness). Insofar as this contradicts the irreflexivity of grounding, it is a metaphysical problem for both the name and the sentence. (In the more general case of metaphysical ill-foundedness, the problem is that it contradicts the wellfoundedness of grounding.) Seen from a semantic point of view, the fact that the sentence Sam is grounded in its name ‘Sam’ means that the name ‘Sam’ exhibits semantic magicality, which contradicts the usual facts about an assertion’s direction of fit. 1363 To show these consequences, we did not have to rely on anything specific to ‘Sam’ and Sam. So clearly, all self-referential names will exhibit similar problems. Here is an overview of the four problems and how they are related:

1359 1360

1361 1362 1363

Cf. sections 11.6 through 11.11. How can the undesirability of these consequences warrant the denial of an existence claim? This is an important methodological question, and we will address it in section 11.14. Cf. section 11.8. Cf. sections 11.6 and 11.9. Cf. sections 11.7 and 11.10.

406 Chapter as Its Own Changing World Chapter11 11 A ASelf-Referential Self-Referential Language Language as Own Changing World I III 406412 Chapter Chapter 11 A Self-Referential Language asIts ItsIts Own Changing World 406 Chapter as Its Own Changing World 184 11 A Self-Referential Language as Own Changing World III 184 Chapter 11 11 A ASelf-Referential Self-ReferentialLanguage Language as Its Own Changing World 184 Chapter 11 A Self-Referential Language as Its Own Changing World metaphysical metaphysical problems: problems: metaphysical problems: ontological ontological co-dependence co-dependence ontological co-dependence (section (section 11.6) 11.6) (section 11.6)

flip flip side side flip side

strengthening strengthening strengthening metaphysical metaphysical ill-foundedness ill-foundedness metaphysical ill-foundedness (section (section 11.7) 11.7) (section 11.7)

semantic semantic problems: problems: semantic problems: referential referential stickiness stickiness referential stickiness (section (section 11.9) 11.9) (section 11.9) strengthening strengthening strengthening

flip flip side side flip side

semantic semantic magicality magicality semantic magicality (section (section 11.10) 11.10) (section 11.10)

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With regard to the present approach, this particular 1367 move, and as too strong a measure that is unwarranted in the face of an apparent abundance of harmless cases. With regard to the present approach, this particular charge of of adharmless hocness iscases. not justified, fourpresent problems are move, and as too strong a measure that is the unwarranted in we theencountered facethis ofparticular an apparent abundance Withbecause regard to the approach, 1367 1367 With charge ad is justified, because the four problems we abundance harmless cases. regard to the present approach, this 1367 charge of adof hocness is not not justified, because the four problems we encountered encountered are abundance ofhocness harmless cases. With regard to thethe present approach, this particular particular moreof general than the contradiction that arises from assumption that there is aare charge of ad hocness is not justified, because the four problems we encountered are abundance of harmless cases. With regard to the present approach, this particular more general than the contradiction that arises from the assumption that there is charge of ad hocness is not justified, because the four problems we encountered Liar sentence, and differ from it in character. Likewise, our distinction between semore general than the contradiction that arises from the assumption that there is charge of ad hocness is not justified, because the four problems we encountered are more general than theiscontradiction arisesthe from theproblems assumption there is aaa are charge of ad hocness not justified,that because four we that encountered are Liar sentence, and differ from it in character. Likewise, our distinction between semantic expressions and syntactic expressions, and the ensuing observation that only more general than the contradiction that arises from the assumption that there Liar sentence, and differ from it in character. Likewise, our distinction between semore general than than the contradiction contradiction that arises arises fromour thedistinction assumption that there there is Liar sentence, and differ from it in character. Likewise, between se- is more general the that from the assumption that is mantic expressions and syntactic expressions, and the ensuing observation that only semantic self-reference is self-reference properly so-called, allow us to show that the aaamantic Liar sentence, and differ from it in character. Likewise, our distinction between mantic expressions and syntactic expressions, and the ensuing observation that only Liar sentence, and differ from it in character. Likewise, our distinction between expressions and syntactic expressions, and the ensuing observation that only Liar sentence, and differ from it in character. Likewise, our distinction between apparent abundance ofand self-referential expressions is illusory. It is neversemantic self-reference is self-reference properly us show that semantic self-reference isharmlessly self-reference properly so-called, so-called, allow us to toobservation show that the thethat semantic expressions syntactic expressions, and ensuing semantic self-reference is self-reference properly so-called, allow us to show that the semantic expressions and syntactic expressions, and the theallow ensuing observation that semantic expressions and syntactic expressions, and the ensuing observation that theless prudent to take extra care. Doing no more than banning each expression that apparent abundance of harmlessly self-referential expressions is illusory. It is neverapparent abundance of harmlessly self-referential expressions is illusory. It is neveronly semantic self-reference is self-reference properly so-called, allow us to show apparent abundance of harmlessly self-referential expressions is illusory. It is neveronly semantic self-reference is self-reference properly so-called, allow us to show only self-reference is self-reference properly so-called, allow us to show wesemantic recognize as of the four unwanted would still leave us theless prudent to take care. Doing no than banning each that theless prudent toexhibiting take extra extrasome care. Doing no more more than features banning each expression expression that It that the apparent abundance of harmlessly self-referential expressions is illusory. theless prudent to take extra care. Doing no more than banning each expression that that the apparent abundance of harmlessly self-referential expressions is illusory. It that the apparent abundance of harmlessly self-referential expressions is illusory. It open to a more general charge adthe hocness. It would therefore be goodstill to have a us we recognize as exhibiting some of four unwanted features would leave we recognize asprudent exhibiting someof of the four unwanted features would still leave us we recognize as exhibiting some of the four unwanted features would still leave us is nevertheless to take extra care. Doing no more than banning each expresis nevertheless prudent to take extra care. Doing no more than banning each expresisopen nevertheless to explains takeof extra Doing no than banning each expresuniform storyprudent to tell that the care. non-existence of more the problematic expressions. to more general charge It would therefore be good have openthat to aaawe more general as charge of ad ad hocness. hocness. It would therefore befeatures good to towould have aaastill open to more general charge of ad hocness. It would therefore be good to have sion recognize exhibiting some of the four unwanted sion that we recognize as exhibiting some of the four unwanted features would still sion that we recognize as exhibiting some of the four unwanted features would still uniform story that the of expressions. uniform story to to tell tell that explains explains the non-existence non-existence of the theItproblematic problematic expressions. *of ad hocness. uniform story to tell that explains the non-existence of the problematic expressions. leave us open a more general charge would therefore be good leave us open to a more general charge of ad hocness. It would therefore be good leave us open to a more general charge of ad hocness. It would therefore be good to have aa uniform story the * to 1364 have uniform story to to tell tell that that explains explains the non-existence non-existence of of the the problematic problematic to have uniform to tell that explains the non-existence of the problematic ** Cf.a section 11.11. story expressions. 1365 Cf. section 11.2. expressions. expressions. *** a central part of those approaches to the Liar 1366 1364 conviction that self-reference is problematic is often 1364 Cf.Asection 11.11.

Cf. section section 11.11. 11.11. Cf. paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. Cf. section 11.2. Cf. section 11.2. Cf.section section7.5). 11.2. The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) A that self-reference is is a central part of approaches to Liar A conviction conviction that self-reference is problematic problematic is often often central part of those those approaches to the the Liar A conviction self-reference is problematic is often central of those to the and traces11.11. ofthat it can be found in the modern discussion in aaRyle 1952,part Mackie 1973,approaches Goldstein 1985 andLiar 1364 1364 Cf. section paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. 1364 Cf. section 11.11. paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. Cf. section 11.11. paradox thatGrover we have above characterized as attempting the psychotherapy of hypochondriasis (cf. 2000, and 2005. 1365 1365 Cf. section 11.2. section 7.5). The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) 1365 Cf. section 11.2. 1367 section 7.5). The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) Cf. section 11.2. section 7.5). already The idea back medieval approaches of “restriction” (Spade/Read section 2.4) We noted ingoes section 7.5to that since Kripke’s 1975 article, many experts doubt that2009, self-reference 1366 1366 A and traces of can be the discussion in 1952, Mackie 1973, Goldstein 1985 and conviction that is problematic is often aa central of approaches to 1366 and traces of it itmeaninglessness can self-reference be found found in in the modern discussion in Ryle Ryle 1952, part Mackie 1973, Goldstein 1985 and Liar A conviction that self-reference ismodern problematic is sentences often central part ofe.those those approaches to the the Liar leads to the orthe non-existence of the involved. Cf., g., Kripke 1975, 691ff.; and traces of it can be found in modern discussion in Ryle 1952, Mackie 1973, Goldstein 1985 and A conviction that self-reference is problematic is often a central part of those approaches to the Liar 2000, and Grover 2005. paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. 2000, and Grover 2005. paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. Gupta 1982, 1; Priest 2006a, 11f.; Field 2008, 23ff.; and Beall /Glanzberg 2014, section 2.3.1. 2000, and Grover 2005. paradox that we have above characterized as attempting the psychotherapy of hypochondriasis (cf. 1367 1367 We noted already in section 7.5 that since Kripke’s 1975 article, many experts doubt that self-reference section 7.5). The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) 1367 section We noted already in section 7.5 that since Kripke’s 1975 article, many experts doubt that self-reference 7.5). The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) We noted already in section 7.5 that since Kripke’s 1975 article, many experts doubt that self-reference section 7.5). The idea goes back to medieval approaches of “restriction” (Spade/Read 2009, section 2.4) leads to of involved. Cf., Kripke 1975, and it in the in 1952, 1973, Goldstein 1985 leadstraces to the theof meaninglessness or non-existence of the the sentences sentences involved. Cf., e. e. g., g., Kripke 1975, 691ff.; 691ff.; and traces ofmeaninglessness it can can be be found foundor innon-existence the modern modern discussion discussion in Ryle Ryle 1952, Mackie Mackie 1973, Goldstein 1985 and and leads to the meaninglessness or non-existence of the sentences involved. Cf., e. g., Kripke 1975, 691ff.; and traces of it can be found in the modern discussion in Ryle 1952, Mackie 1973, Goldstein 1985 and Gupta 1982, 1; 2006a, 2000, Grover 2005. Guptaand 1982, 1; Priest Priest 2006a, 11f.; 11f.; Field Field 2008, 2008, 23ff.; 23ff.; and and Beall Beall /Glanzberg /Glanzberg 2014, 2014, section section 2.3.1. 2.3.1. 1364 1365 1365 1365 1366 1366 1366

11.12 The principle of subsequence for directly referential expressions

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When we discussed what is wrong with the metaphysical ill-foundedness of expressions, we encountered the (Principle of Subsequence for Directly Referential Expressions), which is a way of ensuring the well-foundedness of the relation of direct reference in a contextualist setting. 1368 And we said at the outset of the present section that endorsing this principle is what will provide us with a principled way of rejecting expressions that exhibit ill-foundedness and the other problems – it will give us the desired systematicity. On our way to showing why this is the case, let us start by contrasting the principle of subsequence with the corresponding principle of non-priority: (Principle of Non-Priority for Directly Referential Expressions) No directly referential expression is prior to its extension. That is, if an expression e refers directly to an object a, then the expression e is not prior to the object a. 1369 (Principle of Subsequence for Directly Referential Expressions) Every directly referential expression is subsequent to its extension. That is, if an expression e refers directly to an object a, then the expression e is subsequent to the object a. 1370 Although any principle of non-priority is strictly weaker than the corresponding principle of subsequence, it is already the smallest of steps that takes us from the former to the latter. In fact, the move is similar to strengthening a statement like ‘x ≤ y’ to the statement ‘x < y’. Therefore it is interesting to observe that it is this small step that brings us from a principle that cannot do what we expect of it to a principle that does just that. For the existence of semantically self-referential expressions that are directly referential is compatible with the (Principle of NonPriority for Directly Referential Expressions), but incompatible with the (Principle of Subsequence for Directly Referential Expressions). To see that the (Principle of Non-Priority for Directly Referential Expressions) does not preclude the relevant kind of self-reference, note first that the hypothesis that they indeed form a self-referential circle does not require the name ‘Sam’ to be prior to the sentence ‘Sam is a semantic sentence’, nor does it require the sentence to be prior to the name. The same goes for all self-referential names (and something similar goes for all occurrences of self-referential indexicals). In fact, we have seen that a self-referential expression and what it refers to directly need to start relative to the same context: 1371 A baptism or definition that introduces a self-referential name

1368 1369 1370 1371

Cf. section 11.8. Cf. section 9.10. Cf. sections 9.10 and 11.8. We present the situation here in a way that presupposes that there are first contexts of the existence of the objects we deal with, as would be required by the (First Context Principle) considered in section 9.10. However, although it is difficult to see how an object that is introduced by an act of baptism does not start to exist at a unique moment – the moment when this act occurs –, it is strictly speaking not something that we endorse at this point. We nevertheless argue without loss of generality.

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needs to be circular. 1372 (And construed as a semantic expression, no occurrence of a semantically self-referential indexical can start to exist before the expression that it is a part of and that it therefore refers to.) But simultaneity is obviously compatible with non-priority. This also goes to show that the (Principle of Non-Priority for Directly Referential Expressions) cannot preclude directly referential expressions from exhibiting weak ontological co-dependence or referential stickiness, because these phenomena presuppose no more than a simultaneous start of the two expressions in question. Neither can it preclude them from exhibiting metaphysical circularity or semantic magicality, at least not as long as the only requirement that governs the dynamics of grounding is the (Principle of Non-Priority for Grounded Objects). 1373 In contrast, the principle of subsequence is obviously incompatible with a simultaneous start of existence. From this it is already clear that the relevant kind of self-reference is precluded by the slightly stronger (Principle of Subsequence for Directly Referential Expressions)! Let us look at this in more detail, sticking to our favorite example sentence ‘Sam is a semantic sentence’ and its name ‘Sam’. The (Principle of Subsequence for Directly Referential Expressions) requires that the name ‘Sam’ is subsequent to the sentence Sam – it requires that the sentence starts to exist relative to a context that is prior to the context relative to which the name starts to exist. But as a semantic expression depends ontologically on its parts, 1374 a semantic expression cannot be prior to its parts, and so the sentence ‘Sam is a semantic sentence’ cannot be prior to the name ‘Sam’ – it cannot meet the requirement of subsequence. So the name ‘Sam’ cannot exist, after all; and hence the sentence ‘Sam is a semantic sentence’ cannot exist, either. In other words, the principle of subsequence precludes the circular baptism or the circular definition that would create the sentence Sam and its name ‘Sam’ in one fell swoop. 1375 (Recall that what is in question is the existence of semantic expressions; we are not casting any doubt on the existence of the syntactic expressions or the strings of letters ‘Sam’ and ‘Sam is a semantic sentence’.) Note that the way how the (Principle of Subsequence for Directly Referential Expressions) precludes the existence of the sentence ‘Sam is a semantic sentence’ and its name ‘Sam’ is directly connected to how it precludes their weak ontological coWere we to spell out the principles involved in a more detailed way, we would see that our arguments would even go through in cases where not all objects involved have a first context of existence. To show this properly, we would for example have to use the (Definition of Subsequence Among Objects) from section 9.10 to spell out the (Principle of Subsequence for Directly Referential Expressions) thus: If an expression e refers directly to an object a, then every context relative to which the expression e exists is subsequent to some context relative to which the object a exists but the expression e does not exist.

1372 1373 1374 1375

(We do not need to relativize the antecedent to a context, because directly referential expressions have their extension in a stable way, i. e., relative to every context when they exist.) Now, given this formulation of the principle of subsequence and allowing that expressions can sneak into existence without a definite beginning, the following arguments would still be valid (mutatis mutandis). Cf. sections 11.4 and 11.5. Cf. section 11.8. Cf. section 11.6. Cf. section 11.4 and 11.5.

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dependence, together with the semantic flip side of this problem, the reverse rigidity of the sentence and referential stickiness of its name. For the principle requires that the name is subsequent to the sentence, and thus they obviously cannot start to exist relative to the same context, and so they cannot be weakly co-dependent, nor can the sentence be reversely rigid, nor the name referentially sticky. The (Principle of Subsequence for Directly Referential Expressions) also precludes the metaphysical circularity, together with its flip side, the semantic magicality of the name ‘Sam’. To see why, we need to appeal to a fact about the dynamics of grounding. Although for the scope of the present study we have declined to adopt and defend the very general (Principle of Subsequence for Grounded Objects), 1376 we have found a good reason to endorse the weaker (Principle of Non-Priority for Grounded Objects), according to which no object can be prior to any object that grounds it. 1377 With the non-priority of grounding in the background, the (Principle of Subsequence for Directly Referential Expressions) entails that the name ‘Sam’ cannot be a metaphysically circular counterpart of the sentence ‘Sam is a semantic sentence’, and that the name ‘Sam’ cannot be semantically magical. For both phenomena involve that the sentence is grounded in its name, so that by the (Principle of Non-Priority for Grounded Objects), the sentence could not be prior to the name. But this contradicts what is required by the (Principle of Subsequence for Directly Referential Expressions). 1378 To apply the (Principle of Subsequence for Directly Referential Expressions) in our argument to the effect that the name ‘Sam’ and the sentence ‘Sam is a semantic sentence’ do not exists and could not exhibit the four problems, we did not need to appeal to any feature of the (hypothetical) sentence Sam other than that it refers to itself with a name. Therefore the principle will also preclude the existence of the sentences Marge, Tony, and, most importantly, of the sentence Larry and other Liar sentences that achieve semantic self-reference with a name. It is important to note that nothing specific to the Liar paradox plays a role in this argument against the existence of Liar sentences with a name. For the argument does have a reductio-like structure (we have shown that the hypothetical assumption that there are directly referential Liar sentences would lead to very unwelcome consequences, and from this we concluded that there are no such Liar sentences). And we have seen that we need to be very careful when we apply reductio ad absurdum in the vicinity of a paradox. 1379 But here we need not fear any charges of begging the question, because we have focused in our argument not on Larry but on its prima facie non-paradoxical cousin Sam.

1376 1377 1378

1379

Cf. section 11.8. Cf. section 9.10. There might appear to be a certain redundancy in showing both that some problematic object (like the self-referential name ‘Sam’) does not exist and that it cannot be the case that certain undesirable features (like magicality) are instantiated by it. But in the interest of showing that the (Principle of Subsequence for Directly Referential Expressions) precludes the troublesome phenomena in a systematic way, it is good to see how its preclusion of the existence of ‘Sam’ goes hand in hand with the preclusion of the problems we saw to beset that name. Cf. section 7.3.

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In fact, the result is even more general. In our argument to the effect that the (Principle of Subsequence for Directly Referential Expressions) precludes the existence of semantically self-referential names we have appealed to no feature of names other than that a name refers directly to its extension. Therefore we can establish in a similar way that the (Principle of Subsequence for Directly Referential Expressions) precludes the existence of any token or occurrence of a directly referential expression that is semantically self-referential, and in particular, it precludes the existence of any occurrence of a semantically self-referential indexical. The consequence that the (Principle of Subsequence for Directly Referential Expressions) has for indexicals can be seen as more drastic than its consequence for names. In effect, the principle enforces that the indexical expression ‘this sentence’ (regimented to refer directly to the semantic sentence it occurs in) is banned tout court from the language, whereas the principle leaves all names that are neither self-referential nor otherwise ungrounded untouched. But then, ‘this sentence’ is a weird expression – at least in the regimentation by logicians that ensures its self-referential use, as opposed to its natural deictic (or anaphoric) use as a means of referring to another sentence. It is unclear what good it has ever done, apart from providing those who believe in sentential self-reference with a perspicuous example! To my mind, this particular indexical is well worth losing.

11.13 Drawing the line in the right place So the (Principle of Subsequence for Directly Referential Expressions) does what we want it to do – it precludes the existence of self-referential names like ‘Sam’ and of the sententially indexical singular term ‘this (semantic) sentence’. It is equally important, however, that it does not do much more than that! Yes, we want it to exclude all semantically ungrounded expressions of the directly referential variety – even the non-cyclical ones like the sentences of a Yablo sequence. But before we come to that (at the end of the present section), we need to make sure that it does not preclude the existence of any unproblematic expression. Therefore we need to ask: Is the principle of subsequence compatible with the existence of semantically grounded expressions? Let us look first at a language that is not self-referential, and in particular at the paradigmatic case of a language that allows us to talk about physical objects. We need to make sure that the principle of subsequence does not interfere with this most basic kind of discourse. As physical objects exist in time, we can distinguish them into past physical objects (which exist only relative to past moments), present physical objects (which exist relative to the present moment), and future physical objects (which exist only relative to future moments). 1380 Just like the corresponding principle of non-priority, 1380

Cf. sections 9.2 and 9.3. – As we mentioned already in section 9.10, it is highly plausible to suppose that there are no existence gaps for objects that exist relative to moments in time – to suppose, that is, that any object that exists relative to two moments will exist relative to every moment in between. Under this plausible supposition, the distinction of past objects, present objects, and future objects is not only disjoint, but also comprehensive, and thus a threefold partition of all temporal objects.

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our principle of subsequence is compatible with direct reference to past physical objects and incompatible with direct reference to future physical objects. 1381 With regard to present objects, things are more complicated. Like the principle of nonpriority, the principle of subsequence is in almost all cases compatible with direct reference to present physical objects – namely in all those cases where the present physical object in question has started to exist in the past. But it is incompatible with direct reference to those present objects that start to exist relative to the present moment, and in this point, it differs from the principle of non-priority. We have already alluded to the Kripkean view that every name is connected to the object it refers to by a causal-historical chain, and seen how it can explain our dual claim that while we cannot name future objects, the names of past objects still allow to identify them after they have gone out of existence. 1382 With regard to present objects, the idea that direct reference is mediated by a causal-historical chain explains why we cannot refer directly to an object at the very moment when it comes into existence, for this would require the causal-historical connection to be instantaneous. And with a view to our talk about ordinary objects, this restriction is very natural. Something pops into existence, and only then can it figure in an act of baptism and thus acquire a name. (Or, in the case of indexicals: Only when something has already popped into existence can it figure as a constituent of the semantic value of a context-sensitive but directly referential singular term in a context of use. 1383) In the everyday world it is actually a quite rare event that a new object starts to exist (many an apparent case can be construed as an old object changing in some drastic way, like a caterpillar turning into a butterfly), and we always have enough time to wait until it is fully there, so that we can pick it out with a name or an indexical, and talk about it. (If we know that a new object will soon start to exist, we can of course already talk in a general way about what it will be like when such an object is 1381

1382 1383

This is a special case of our earlier more general insight that we can talk about past objects, but not about future objects. Cf. sections 9.2 and 9.3. Cf. section 9.3. This is a good point to address a possible objection, which goes like this: What about centric temporal indexicals like ‘now’ and ‘today’ in their substantival use (cf. subsection 5.2.1)? It would surely be counterintuitive to think that they do not refer to the present moment or to the present day – but the (Principle of Subsequence for Directly Referential Expressions) seems to preclude this! In response, we should make several observations. First of all, the contexts we are concerned with in the case of the contextualist ontology of expressions are not temporal; in contrast to moments in time these contexts form discrete sequences and we should not expect all intuitions we have about time to carry over to them. In particular, when we use temporal indexicals like ‘now’ and ‘today’, we arguably are always referring to temporal intervals – the present day in the case of ‘today’ and an interval of a length that can vary with situation and co-text in the case of ‘now’ (just compare ‘The moment the race starts is now’ to ‘Now is the time global warming starts to become noticeable’). Even if some moments in time are not extended intervals but properly pointlike, it is not those pointlike moments that we usually refer to with temporal indexicals. Thus the (Principle of Subsequence for Directly Referential Expressions) does not preclude the standard use of temporal indexicals like ‘now’ and ‘today’ – except perhaps at the very first (pointlike) moment of the interval in question, so that we cannot use ‘today’ exactly at midnight. Anyway, the picture behind the possible objection might very well involve a mistaken reification, because it presupposes that we can construe the objects referred to by temporal indexicals as being just as temporal as physical objects. But are times really things in time?

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there, 1384 and we can talk in this way at the exact moment relative to which it starts to exist – but that does not mean that we are then already able to talk about it.) So the (Principle of Subsequence for Directly Referential Expressions) does not hamper much our ordinary talk about physical objects. Now, once we are talking about physical objects, and in particular, once we have introduced names for physical objects, we will probably want to talk about our talk about physical objects, and in particular, to give names to the names of physical objects. And so on. (Within the scope of this scenario, we can safely construe the expressions as objects that are in time just like the physical objects that realize them and the physical objects they are about.) Again, the principle of subsequence does not stand in the way, as long as we introduce new directly referential expressions step by step. A schematic story of stages could go like this: 1385 At stage zero, all physical objects in question exist. 1386 At stage one, there is a name for every physical object in question. Call these the first-level names and expressions formed from them the first-level expressions. At stage two, there is a name for every first-level expression. Call these new names the second-level names and expressions formed from them the secondlevel expressions. And so on. This story concerns a language with names as the only means of referring to expressions. Similar stories could be told about other kinds of directly referential devices: indexicals and quotation expressions. Of course, in accordance with the principle of subsequence no occurrence of the semantic expression ‘this sentence’ can come into existence at any stage; but there can be indexicals like ‘the previous sentence’. 1387 For quotation expressions – here understood as referring to semantic expressions –, the story of stages will mirror the recursive definition that we would standardly give of a language with quotation expressions as its only means of referring to expressions. And there could also be mixed variants, incorporating names, certain indexicals, and quotation expressions. (If you miss descriptions here, recall that we will turn to indirect reference to expressions in the next chapter.) The process will of course never terminate: There is no stage relative to which all semantically grounded expressions – all that seem to be possible from a purported 1384 1385

1386

1387

Cf. section 9.9. This story of stages is schematic in so far as the (Principle of Subsequence for Directly Referential Expressions) does not require that the process occurs in this very orderly, stagewise fashion. As we are concerned only with the possibility of semantically ungrounded expressions, we can bracket the fact that very probably, there will always be new physical objects. Here is an example for an unproblematic piece of discourse with indexicals that is compatible with the principle of subsequence: At stage one: ‘Snow is white.’ At stage two: ‘The previous sentence is true.’ At stage three: ‘The previous sentence is true.’ ... ...

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God’s eye view – are in existence. But for every possible semantically grounded expression we can find a story about its introduction that does not violate the principle of subsequence – for every possible semantically grounded expression there can be a stage relative to which it starts to exist. 1388 This picture about the introduction of semantically grounded expressions that are directly referential generalizes easily from expressions and physical objects that exist in time to any contextualist ontology of expressions and some other objects that is in accordance with the principle of subsequence. So, the principle of subsequence does not exclude the existence of semantically grounded expressions that are directly referential. But it gives us a picture of a language as a complex object that is continuously growing 1389 – or quasi-growing, along the dimension of the non-temporal but timelike contexts that are relevant for the existence of semantic expressions. By now we have seen that with regard to directly referential expressions, the principle of subsequence precludes self-reference but is compatible with semantic groundedness. What about the remainder – what about those cases of semantic ungroundedness that are not circular? In other words, what about Yablo sequences of directly referential sentences, and their like? The first thing to notice is that the (Principle of Subsequence for Directly Referential Expressions) alone does not preclude such cases of non-circular ungroundedness. This can be shown by the example of an infinite sequence of sentences, each referring to a prior one, like the following: ... At stage minus two: At stage minus one: At stage zero:

... ‘The previous sentence is true.’ ‘The previous sentence is true.’ ‘The previous sentence is true.’

If we think of this scenario as temporal, then we have something like a sequence of utterances that has no beginning in the past and reaches up to a present utterance. As each utterance refers to the one immediately preceding it, there is no conflict with the principle of subsequence. 1390 But construed as constituted by semantic expressions, the sequence is just as metaphysically ill-founded as the usual Yablo sequence. So, an additional assumption is needed. One way to go would be to require that every story of stages about the introduction of expressions needs to have the form of the particular story of stages we told above, which started with the existence of the physical objects. The crucial thing to require would be that every process of introducing expressions needs to have a beginning. This motivates another, more general way to go, which we will take: We will require the relation of subsequence among the contexts to be well-founded. Then every collection of contexts will 1388

1389

1390

There is a structural similarity here to the indefinite extensibility of the set theoretic hierarchy, and of the ordinals, which I want to exploit on another occasion to generalize the present account of expressions to a uniform approach to both the semantic and the set theoretic paradoxes. Cf. section 9.11. Rheinwald also relies on the flexibility of language in her approach to the semantic paradoxes; Rheinwald 1988, 236 and 241ff. I would like to thank Leon Horsten for alerting me to this point.

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have a minimal context with regard to the ordering by subsequence, and therefore every story whatsoever will have a beginning. 1391 This additional requirement can be justified in the framework of the present proposal by recourse to the idea that the contexts in question are those contexts that are relevant to the existence of expressions – together with the picture of language as a growing thing. For in contrast to other growing things (like, e. g., the cosmos), a language cannot naturally be conceived to grow in a process that is without a beginning. And (again in contrast to other growing things) we must conceive of the growth of a language as a discrete process, i. e., as a process each stage of which has an immediate predecessor (if it has a predecessor at all). Hence the contexts relevant to the growth of a language need to be well-founded. 1392

11.14 A social contract about ontology All this taken together gives us good reason to endorse the (Principle of Subsequence for Directly Referential Expressions). We have seen above how the principle precludes that there are expressions that exhibit the problems of metaphysical circularity and so on, as well as systems of expressions that exhibit the more general problem of metaphysical ill-foundedness – without tampering with the existence of expressions that are not beset by these problems. We have seen before that these effects are bad, 1393 which gives us reason to reject the existence of expressions that exhibit them, at least if we can do so in a systematic way. And we have seen in the last two sections how the principle does provide us with a systematic way of rejecting the problematic expressions. With regard to this way of arguing, it is worthwhile to return to our above contrast of the two corresponding principles: (Principle of Non-Priority for Directly Referential Expressions) No directly referential expression is prior to its extension. (Principle of Subsequence for Directly Referential Expressions) Every directly referential expression is subsequent to its extension. Even though the principle of subsequence is only slightly stronger than the otherwise similar principle of non-priority (as ‘