New Frontiers in Truth 144386806X, 9781443868068

Table of contents :
Table of Contents
Introduction • Fabio Bacchini, Stefano Caputo and Massimo Dell’Utri
1 The Norm of Truth: a Dialogue • Pascal Engel
2 An Explanatory Role for the Concept of Truth • Boris Rähme
3 Faultless Disagreement and the Equal Validity Paradox • Annalisa Coliva and Sebastiano Moruzzi
4 Weak Indexical Relativism • Carlo Filotico
5 Radical Relativism, Retraction and “Being at Fault” • Filippo Ferrari and Dan Zeman
6 Contradictions, Disagreement and Normative Error • Samuele Iaquinto
7 Do we Have a Determinate Concept of Truth? • Fredrik Stjernberg
8 Alethic Pluralism and Logical Pardoxes • Michele Lubrano
9 Deflation and Reflection. On Tennant’s Criticism of the Conservativeness Argument • Ciro De Florio
10 How Simple Is the Simplicity of Truth? Reconciling the Mathematics and the Metaphysics of Truth • Andrea Strollo
11 The General Missing from the Hierarchy • Elia Zardini
Abstracts
Contributors
Index of Names

Citation preview

New Frontiers in Truth

New Frontiers in Truth Edited by

Fabio Bacchini, Stefano Caputo and Massimo Dell'Utri

New Frontiers in Truth Edited by Fabio Bacchini, Stefano Caputo and Massimo Dell'Utri This book first published 2014 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2014 by Fabio Bacchini, Stefano Caputo, Massimo Dell'Utri and contributors All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-6806-X ISBN (13): 978-1-4438-6806-8

TABLE OF CONTENTS

Introduction ............................................................................................... vii Fabio Bacchini, Stefano Caputo and Massimo Dell’Utri Chapter One ................................................................................................. 1 The Norm of Truth: a Dialogue Pascal Engel Chapter Two .............................................................................................. 15 An Explanatory Role for the Concept of Truth Boris Rähme Chapter Three ............................................................................................ 38 Faultless Disagreement and the Equal Validity Paradox Annalisa Coliva and Sebastiano Moruzzi Chapter Four .............................................................................................. 63 Weak Indexical Relativism Carlo Filotico Chapter Five .............................................................................................. 80 Radical Relativism, Retraction and “Being at Fault” Filippo Ferrari and Dan Zeman Chapter Six .............................................................................................. 103 Contradictions, Disagreement and Normative Error Samuele Iaquinto Chapter Seven.......................................................................................... 115 Do we Have a Determinate Concept of Truth? Fredrik Stjernberg Chapter Eight ........................................................................................... 132 Alethic Pluralism and Logical Pardoxes Michele Lubrano

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Chapter Nine............................................................................................ 144 Deflation and Reflection. On Tennant’s Criticism of the Conservativeness Argument Ciro De Florio Chapter Ten ............................................................................................. 161 How Simple Is the Simplicity of Truth? Reconciling the Mathematics and the Metaphysics of Truth Andrea Strollo Chapter Eleven ........................................................................................ 176 The General Missing from the Hierarchy Elia Zardini Abstracts .................................................................................................. 201 Contributors ............................................................................................. 206 Index of Names........................................................................................ 209

INTRODUCTION

Although philosophers have been concerned with truth since at least the age of Plato, the last thirty years have witnessed a veritable explosion of the philosophical debate on this topic. The touchpaper which lit the fuse for this was undoubtedly the Deflationist Renaissance (half a century after the seminal work of Ramsey) due, in the Seventies, to the development of the prosentential theory of truth by D. Grover, J. Camp and N. Belnap (1975) and the Quinean disquotational interpretation of the Tarskian truth definitions (Quine 1970) and, from the second half of the Eighties onwards, to the forceful defences of deflationary conceptions provided by H. Field (1986, 1994) and P. Horwich (1990). The philosophical struggle on deflationism has been thought-provoking: by arguing on the merits and shortcomings of such a conception, philosophers have come to broaden and deepen the discussion on truth beyond the boundaries of deflationism. An outstanding example thereof is C. Wright’s Truth and Objectivity (1992) which, starting from a critical discussion of deflationism, opened the door to two central focuses of the current debate on Truth: truth-pluralism (advocated for the first time by Wright himself) and truth-relativism. One of the central issues of Wright’s book was in fact how to better account for the truth of sentences belonging to so-called non-factual discourses, e.g. discourses (such as those concerning moral, aesthetic or personal taste matters) where truth, if any, seems not to depend on the existence in the world of objective facts whose discovery would allow for the settlement of disputes between people holding opposite views. Two different responses to this question are on the one hand the pluralist stance, according to which truth in these areas of discourse is of a different stuff than truth for sentences concerning, for instance, the physical features of reality, so that we must recognize the existence of different truth-properties in different domains of discourse, and, on the other hand, the relativist approach, revived by the works of M. Kölbel (2002) and J. MacFarlane (2005), according to which truth is one but it must be relativized to standards of evaluation, so that one and the same truth-evaluable content may be true relative to one of such standards and not true relative to a different one. Another example of how the debate on deflationism stimulated and deepened the investigations on truth is provided by the fruitful intertwining

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of metaphysical approaches, concerned with the nature, if any, of truth, and formal theories of truth, mainly interested in the construction of definitions, or axiomatic treatments, of non-paradoxes-engendering truthpredicates. This development was due mainly to two problems deflationists had to face. Firstly, given that a theory that embraces the general validity of T-biconditionals (sentences of the form “‘p’ is true if and only if p”) and classical logic is inconsistent, as Tarski showed, and that the deflationism’s core idea is that such biconditionals exhaust the theory of truth, how can a deflationary theory escape inconsistency? Secondly, S. Shapiro (1998) claimed that a sensible way to give a determinate content to the deflationist slogan that truth is an insubstantial property, is to resort to the concept of a theory being a conservative extension of another one (where, roughly, a theory A is a conservative extension of a theory B when there is nothing expressible in the language of B which is provable in A and not in B; that is to say when A does not provide us with any new knowledge concerning the things B is about). Truth would be, in this sense, an insubstantial property if and only if the theory of truth for a language in which a given theory B is expressed is a conservative extension of B. Shapiro noticed, in the meantime, that Tarski’s and Gödel’s works provided a blatant proof that a theory of truth for the language of arithmetic is not a conservative extension of arithmetic: in fact, as Gödel showed, arithmetic cannot prove its own consistency (although it can, in some sense, express it) and, as Tarski showed, a theory of truth for the language of arithmetic can prove such a consistency. Shapiro stressed moreover that a theory of truth made only by an infinite list of Tarskian biconditionals, though being a conservative extension of its base theory, is not able to prove the consistency of arithmetic. So, to Shapiro’s lights, the deflationist is forced to give up either the claim that truth is insubstantial or the claim that an adequate theory of truth should be able to prove sentences like “All the theorems of arithmetic are true”. The non-conservativity argument, besides burning the discussion between friends and enemies of deflationism, opened, in very recent times, an interesting new field of investigation concerning the best way to understand the presumed insubstantiality of truth. The problem of the normative status of truth provides another case where the philosophical discussion has profited from the debate on deflationism. Dummett (1959) had pointed out that the role of truth as a goal of our cognitive life poses a potential problem for any conception of truth that sticks uniquely to the T-schema: the instances of the latter seem in fact to be silent on the role that truth has as a guidance of our action and a central norm of our cognitive behaviour. Are deflationists able to give a

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satisfactory account of the normative role of truth? And what this normative role actually amounts to? The essays collected in this book provide new and enlightening ideas on each of the aforementioned topics. To begin with the last topic, Pascal Engel’s “The Norm of Truth: a Dialogue” is an exciting dialogue between Pilate and the Epicurean philosopher Aelius Lamia concerning the role of truth as the main norm of belief. After having pointed out the shortcomings of truth and justification in playing such a role and having gone through the question of the kind of normativity that is involved in our epistemic attitudes, the two interlocutors converge on the claim that knowledge should better be taken as the main norm to which our beliefs are subjected. Boris Rähme claims, in “An Explanatory Role for the Concept of Truth”, that the role of truth as a norm of belief poses a threat to deflationary theories that is more serious than advocates of such conceptions have thought. In particular such a role offers counterexamples to the claim, central to deflationism, that truth has no genuine explanatory role. Deflationists in fact typically explain away the normative role of truth using a truth-free schema such as “One ought to believe that p only if p”. But, according to Rähme, the concept of truth is inescapable when one comes to explain why the instances of such a schema hold. A second group of essays deals with relativism about truth. Annalisa Coliva and Sebastiano Moruzzi focus, in “Faultless Disagreement and the Equal Validity Paradox”, on what they call the equal validity paradox engendered by the putative phenomenon of faultless disagreement of which contemporary truth-relativists aim at giving an account. When two people are engaged in so-called disputes of inclination (those concerning what is morally right, beautiful, tasty) one can in fact have the feeling that, although they believe contradictory propositions, none of them is at fault, that is to say none of them is to blame for believing something false. The paradox engendered by the phenomenon is that it seems to force us into admitting that both a proposition and its negation are true, contra the principle of non-contradiction. Coliva and Moruzzi deploy eight alternative solutions to the paradox, some of which correspond to the main opposite camps in the contemporary discussion on truth-relativism, and argue (contrary to what advocates of such solutions usually think) that they all amount to taking a revisionary stance toward the concept of faultless disagreement, a stance according to which this concept is internally incoherent and needs, for this reason, either to be abandoned or to be revised in some respect.

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Carlo Filotico’s “Weak Indexical Relativism” is a defense of a moderate form of so-called indexical relativism (better known as contextualism) which has been one of the main competitors to truth-relativism in the recent debate. According to indexical relativism, speakers’ assertions within non-factual areas of discourse implicitly refer to a feature of the speaker, so their content depends on the context of utterance. For instance, utterances of a sentence such as “Matisse is better than Picasso” must be read as assertions of “Matisse is better than Picasso according to my own aesthetic standards” or “I prefer Matisse to Picasso”. So two different assertions of this sentence can get different truth-values just because they can express different propositions in different contexts. Therefore indexical relativists are ready to accept that, in a dispute of inclination, nobody is at fault; they think however that in such disputes there isn’t a real disagreement: in fact people involved in them aren’t really holding mutually incompatible beliefs. The weak version of this view advocated by Filotico, which consists of the claim that indexical relativism properly describes just some cases of disputes of inclination, aims at answering the standard objections that have been raised by truth-relativists against stronger versions of indexical relativism. Filippo Ferrari and Dan Zeman’s focus, in “Radical Relativism, Retraction and Being at Fault”, is the phenomenon of retraction, the speech act an agent performs when she takes back a previously made assertion whose content she currently considers false, although she does not blame herself for having being at fault in making the retracted assertion. After providing an analysis of this kind of speech act, Ferrari and Zeman claim that MacFarlane’s (2011, 2014) explanation of the faultless dimension of it fails to account for an asymmetry between retraction of assertions with moral content and retraction of assertions on matters of personal taste. In order to account for this asymmetry they put forward a new dimension of evaluation of assertions, called “circumstance accuracy”. Samule Iaquinto’s “Contradictions, Disagreement and Normative Error” compares two different ways of allowing for the possibility of a faultless disagreement, namely the relativist approach and the approach consisting in both the negation of the principle of bivalence and the subsequent adoption of a three-valued logic. His main point is that the first approach is preferable to the second on two grounds: firstly, it most easily accounts for the possibility of true semantic, and not just syntactic, contradictions; secondly, it is less metaphysically demanding than the three-valued approach.

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Both the essays of Frederik Stjernberg (“Do We Have a Determinate Concept of Truth?”) and Michele Lubrano (“Alethic Pluralism and Logical Paradoxes”) centre on alethic-pluralism. Stjernberg offers a powerful defence of alethic pluralism against what many have considered a knockdown objection to it. The objection (called “Instability Challenge”) is that once a plurality of truth predicates (T1 … Tn) is admitted one can readily define a new predicate by the disjunction of T1 …. Tn. Since this disjunctive predicate applies only to all of the sentences to which the former predicates apply, the result is that it is the universal truth predicate whose existence pluralists deny. The strategy Stjernberg opposes to such an objection is twofold. On the one side, he mounts a diagonalizing argument showing that for any given truth-predicate Ti one can introduce a further truth-predicate which applies to the sentence saying of itself that it is not Ti; but, if this is true, then there is no finite list of truth-predicates whose disjunction one can identify with the unique, universal truth-predicate. On the other side, he claims that neither formal axiomatic treatments of truth nor commonsense intuitions concerning it provide us with a determinate concept of truth. Stjernberg’s conclusion is that the pluralists who want to escape the instability challenge should apply to the concept of truth Waismann’s (1945) views about the “open texture” of concepts. Lubrano’s paper tackles the problem of how a pluralist about truth can deal with the Liar paradox. After having criticised Cotnoir’s (2013) solution to this problem, he puts forward his own proposal that amounts to a Tarskian, hierarchical version of pluralism. Lubrano firstly provides a formal treatment of this approach enlightening some constraints a pluralist hierarchical account of truth must satisfy in order not to fall prey to the Liar and, finally, shows how his treatment can escape not only the classical Tarskian liar-paradox, but also Kripke’s (1975) and Yablo’s (1993) paradoxes which are not based on the mechanism of selfreferentiality. Lubrano’s paper, with its intertwining of formal treatments of truth with metaphysical questions concerning its nature, sets the stage for the last group of essays which are devoted to topics to which this intertwining of formal and metaphysical matters is central. Ciro De Florio’s “Deflation and Reflection. On Tennant’s Criticism of the Conservativity Argument” is a critical examination of the debate between N. Tennant and J. Ketland on the conservativity argument mounted against deflationism by Shapiro (1998) and Ketland (1999) himself. De Florio’s main point is that Tennant’s way out from the Conservativity argument, based on the use of schematic principles such as

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“if ‘p’ is a theorem of T then p”, is after all committed to substantial notions that a deflationist should not be happy to buy. In fact, according to De Florio, the epistemic justification of the aforementioned schematic principle requires the exclusion of the non-standard models of arithmetic; the characterization of the standard model however requires in turn second order arithmetic; but the notion of logical consequence for second order logic may turn out to be a robust notion. So Tennant is chargeable for the same trick for which Shapiro blamed the deflationist who embraces the second-order manoeuvre: he is “hiding the robustness of truth in the second-order consequence relation” (Shapiro 1998, p. 510). The proper way of giving a precise content to the deflationist idea that truth is an insubstantial property is the focus of Andrea Strollo’s “How Simple Is the Simplicity of Truth? Reconciling the Mathematics and Metaphysics of Truth”. Strollo argues, in the first place, that the insubstantiality-claim (the idea that truth is an insubstantial property) is the real mark of deflationism, and that it is not entailed neither by the intersubstitutibility-claim (that “it is true that p” and “p” are conceptually equivalent) nor by the logical role-claim (that the truth-predicate serves only an expressive-logical role), also typically endorsed by deflationists. Then he tackles the endeavor of giving a precise content to the insubstantiality-claim, starting from Edwards’ (2013) and Asay’s (2014) idea that the substantial/insubstantial divide among properties may be understood in terms of the distinction between sparse and abundant properties famously put forward by Lewis (1983). Strollo’s central point is that the best way to exploit the sparse/abundant divide in giving a content to the notions of substantiality/insubstantiality of a property is to explain these notions via a model-theoretic concept which is closely related to that of conservativity: this is the concept of expandibility of models. According to Strollo this notion provides an invaluable tool for ordering in a precise way different theories of truth on a scale from a higher to a lower degree of substantiality. The last paper of this group, and the concluding one of the book, Elia Zardini’s “The General Missing of the Hierarchy”, centres on the locus classicus around which formal theories of truth have developed, notably the problem of semantic paradoxes. Making use of the notion of a schematic assertion, Zardini provides a way by which hierarchical approaches to paradoxes can account for some of the generalising uses of the truthpredicate (exemplified by the famous Dixon-Dean example put forward by Kripke (1975)) that pose a problem to such approaches. He also argues, however, that hierarchical approaches cannot account for another kind of generalising use of truth, which consists in using truth to characterise all

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instances of a certain kind of sentences, as happens when we are willing to express logical principles (such as the law of non-contradiction or the law of excluded middle) or to explain logical notions such as negation or conjunction. We believe that the variety of problems tackled by the essays in this book and their thought-provoking insights highlight how the land of Truth is still far from having been totally explored and how, in this intellectual endeavour, real progresses can be achieved. Sassari (Italy), September 2014 Fabio Bacchini Stefano Caputo Massimo Dell’Utri

References Asay, J. 2014, “Against Truth”, Erkenntnis 79 (1), 147 Cotnoir, A. 2013, “Pluralism and Paradox”, in Pedersen, N.J.L.L. and C.D. Wright (eds.), Truth and Pluralism: Current Debates, New York: Oxford University Press, 339 Dummett, M. 1959, “Truth”, Proceedings of the Aristotelian Society 59 (1), 141 Edwards, D. 2013, “Truth as a Substantive Property”, Australasian Journal of Philosophy 91 (2), 279 Field, H. 1986, “The Deflationary Conception of Truth”, in. MacDonald, G. and C. Wright (eds.), Facts, Science and Morality: Essays on A.G. Ayer’s “Language, Truth and Logic”, Oxford: Blackwell, 55 —. 1994, “Deflationist Views of Meaning and Content”, Mind 103 (411), 249 Grover, D.L., J.L. Camp and N.D. Belnap 1975, “A Prosentential Theory of Truth”, Philosophical Studies 27 (2), 73 Horwich, P. 1990, Truth, Oxford: Basil Blackwell Ketland, J. 1999, “Deflationism and Tarski’s Paradise”, Mind 108 (429), 69 Kölbel, M. 2002, Truth without Objectivity, London: Routledge Kripke, S. 1975, “Outline of a Theory of Truth”, The Journal of Philosophy 72 (19): 690 Lewis, D. 1983, “New Work for a Theory of Universals”, Australasian Journal of Philosophy 61 (4), 343 MacFarlane, J. 2005, “Making Sense of Relative Truth”, Proceedings of the Aristotelian Society 105 (1): 321

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—. 2011, “Epistemic Modals Are Assessment-Sensitive”, in Egan, A. and B. Weatherson (eds.), Epistemic Modality, New York: Oxford University Press, 144 —. 2014, Assessment-Sensitivity: Relative Truth and Its Applications, Oxford: Oxford University Press Quine, W.V.O. 1970, Philosophy of Logic, Cambridge (MA): Harvard University Press Shapiro, S. 1998, “Proof and Truth: Through Thick and Thin”, The Journal of Philosophy 95 (10), 493 Yablo, S. 1993, “Paradox without Self-Reference”, Analysis 53 (4), 251 Waismann, F. 1945, “Verifiability”, Proceedings of the Aristotelian Society, Suppl. Vol. 19, 119 Wright, C. 1992, Truth and Objectivity, Cambridge (MA): Harvard University Press

CHAPTER ONE THE NORM OF TRUTH: A DIALOGUE PASCAL ENGEL

1. Introduction In his short story The Procurator of Judea (1893)1 Anatole France imagines a dialogue between Pontius Pilate, retired in Naples, and the Epicurean philosopher Aelius Lamia. Their exchange happens in the year 50 of our era, near Pompeii. At that time the volcano had not given any sign of the activity which would destroy the city thirty years later. They talk about Pilate’s office in Judea, and about how difficult it was then to rule Syria, especially with the Jews. Pilate complains that he had a hard time coping with their laws while trying to tear their unfortunate victims away from death. At the end of their exchange, Lamia asks Pilate whether he remembers a young “thaumaturge from Galilea”, whose name was Jesus of Nazareth. Pilate answers: — “Jesus? Jesus, from Nazareth? I don’t remember.” The tale suggests that the most important episode of the history of humanity slipped out of the memory of its main protagonist. I took up the story at this point, and try to give it a sequel. It may be a surprise to some readers that Pilate had philosophical ideas, but he lived then among a community of trained philosophers, and probably knew Philodemus.2

2. The lost dialogue The next day Lamia meets again Pilate in his villa on the beautiful slopes of the Vesuvius. Aelius - So, Pontius, you still do not remember this Jesus from Nazareth? Pontius - Now, the episode came back to my mind. Those Jews wanted him to be crucified because he called himself “King of the Jews”. He

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seemed to me to have committed no crime, and to be just one of those lunatic prophets of whom I had heard about from time to time, and he was not dangerous. But I had to accept their verdict. There had been several seditions among them, and I did not want to have another uprising. A. So, did you see this man Jesus at all? P. Yes, I did. I asked him: “So you are the King of the Jews”. To which he answered: “You just said it”. He added that his kingdom was not of this world, and that he had come to this world to testify for truth. So I jested: “What is truth?” A. And you wouldn’t stay for an answer, I suppose? P. No, I wouldn’t. Everyone knows what truth is. If what he said was true, then things were as he said they were, and if he said he was the King of the Jews, what he said was, by his own lights, true, for when one says something one says something that one represents, ipso facto, as true. If he said he was the King of the Jews, then he said that it was true that he was the King of the Jews. So if he said that he was the King of the Jews, and if what he said was true, then he had spoken the truth. I had nothing to add, and nothing to complain about this. Actually he could just as well have whistled it. The issue is not of what truth is, and this is why I asked this question—“What is truth?”—rhetorically. My question was not about the nature of truth, or about the definition of this too obvious notion. It was about the role of truth in our lives. My question was: what is truth good for? Is it good to believe the truth? Even if he had said the truth—and was the King of the Jews, or the one who brought them Truth, the problem was: was there for me any good motive to believe this? I found no reason to believe it. A. Nevertheless you sent him to death! And you did not believe that he was guilty. P. No, I thought he was innocent. I just took up the Sanhedrin’s decision. They deferred to me for their own decision. But I ask you to imagine the consequences of my taking the opposite decision. A. So you did not believe that it was true that he was the King of the Jews? That he was guilty? P. No, I did not believe that. But sometimes you do not act on your beliefs, and you even have to act against your beliefs. Sometimes also you want to believe that something is true. The Jews wanted to believe that this man Jesus was guilty. I did not accept that this Jesus was, as he said, the incarnation of Truth on earth. But I decided to believe that he was guilty nevertheless, in order to avoid greater evils for Rome. Why should we always believe the truth? Isn’t it good sometimes at least, to believe what is false, even when one knows that it is false? Don’t you, Lamia, like most

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people in this city, prefer to believe that the volcano will not erupt, in spite of the fact that it frequently emits smokes? The cost of believing otherwise would be great. And most likely, you are right. A. No, Pontius, I do not agree. As Epicurus said: “Do not spoil what you have by desiring what you have not”. You should always believe the truth, because truth is the aim of belief. The truth is what you ought to believe. The truth of a proposition is the best reason one has for believing it. For isn’t it a fatal objection to a belief to say that it is false? You may have had reasons to want to believe that Jesus was guilty. But these were not reasons to believe, these were reasons to want to believe. Even though it could be rational or useful in the prudential sense to believe that Jesus was guilty, it was not epistemically rational. It is not the right kind of reason to believe. Truth is the norm of belief. P. That is not so clear to me. What is this norm? I agree that to believe something is to believe that it is true. But that’s trivial. If I believe that the Capitol is between the Field of Mars and the Palatine, I believe that it is true. This may show that perhaps all our beliefs aim at truth because truth is the object of our beliefs (formaliter—forgive me for using a Latin word!—or as the formal object of belief).3 But why does this show that our beliefs aim at truth? Our imaginings also «aim at truth» in this sense. For if I imagine that I am Caesar Augustus, I imagine that it is true that I am Caesar Augustus. If I hope that I shall become a senator, then I hope that it is true that I shall be a senator. But that does not mean that imaginings or hopes “aim at truth”. Indeed I may imagine or hope something which is actually false. So to say that what one believes is what one believes true does not show that belief aims at truth or that truth is the norm of belief. It is not a good argument A. Concedo. That was not my claim when I said that truth is the aim of belief. What we want to say, when we say this, is that we have, in general, the goal of believing truths and of not believing falsehoods. In other words it is our epistemic goal to believe what is true and to disbelieve what is false. P. Come on! To believe everything that is true? Are we to believe anything whatsoever which is true? That there are 351 bricks in this wall? That the Vesuvius is a mountain? That Julius Caesar crossed the Rubicon? Must I believe that? There is no reason for me to believe these things either because they are useless or because I know them already. Why should I believe something which I already know? I suggest that the maxim that we ought to believe the truth has to be restricted. Why not say that our epistemic goal is to believe truths which are interesting or important for us? That would be better.

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A. I am not very happy with your suggestion, Pontius. What is interesting or not may vary according to circumstances, individuals, etc. Might not our epistemic maxim become empty or completely contextual: believe the truths which are important and interesting for you at the appropriate time or place? Or: believe what is important, unless it is unimportant, as the case may be? What kind of advice is that? I prefer to stick to my original formulation. Perhaps every truth, however trivial, is interesting and important. Our cognitive resources are limited, and better have few beliefs on important matters than a lot on unimportant ones. Nevertheless we never know when trivial or uninteresting beliefs are going to become important. It can be prudentially useful—but prudentially for our epistemic aims—to get as many beliefs as possible, interesting or not. So I suggest a maxim of Epistemic Consequentialism, which we might call the rule of truth maximisation:4 (EC) We must maximise true beliefs and minimise false ones.

P. Can you tell me how this gets us out of the previous difficulty? For instance, should we believe all the consequences of our beliefs (if we could do so)? Perhaps we ought to believe the truths from which we can infer a maximum of other beliefs, such as the most general truths (e.g., the laws of basic sciences or axioms). And avoid the too complex ones. But it would be absurd to say that, for instance, we ought only to believe propositions which express axioms or fundamental truths in a domain, and not those about simple matters of fact. What is worse, Epistemic Consequentialism prescribes that we believe everything that is true and avoid believing what is false. But it does not sort out those of our maximised true beliefs which we might have on the basis of false beliefs, and for bad reasons, from those which are true and based on true beliefs and for good reasons. We can think of two kinds of situations of this sort. The first one would be cases like the following. The mathematician Euclid is victim of a fatal illness and knows it. He has only a few weeks to live. He would need a year to be able to finish his Elements. But he knows that if he can manage to believe that he will live enough to finish his Elements, he will recover and live at least one more year. This belief, although false and based upon very little evidence, is epistemically useful, since publishing The Elements is going to maximise a lot of true beliefs for the future of humanity. But Euclid ought not to believe that he will live, since it is contrary to evidence. This is a case where a false belief, based on poor evidence—therefore a belief which one ought not to believe—is also to be had because it maximises true beliefs;

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hence it satisfies the norm of true belief maximisation. Here is another counterexample. One can infer a proposition which, by sheer luck, happens to be true from a false proposition: for instance, from the proposition that I have visited Carthage, which is false, you can infer that I visited a city in North Africa, but this proposition is true by sheer luck: as it happens, although you have no evidence for it, I have visited Alexandria. Clearly you ought not to believe that I have visited Carthage, since it is false, but since the proposition that can be inferred from it is true, you ought to believe the latter.5 Indeed these situations are in principle rare, but they show, like the previous case, that it can sometimes be good to believe truths which are not justified or which are based on weak or no evidence. The problem is that (EC) does not say anything about our reasons for believing, and it seems as important to have proper bases for our beliefs as to have true beliefs. A. I agree with you Pontius. But perhaps it shows also that there is something wrong with truth as a goal or aim of belief. Even in the cases of true beliefs held for bad reasons, nobody disputes what Epicurus said in this Canon, that our beliefs (hypolepseis) are correct or incorrect, and that a belief is correct if true, incorrect otherwise. That is, we have this Condition of Correctness: (CC) A belief is correct iff it is true.

Now this might seem like a definition of belief, as above: if one believes that P, then what one believes is supposed to be true. In other words, the condition of satisfaction of a belief is fulfilled when a belief is true, and not fulfilled when it is false. But saying that a belief is correct when it is true says more. “Correct” indicates a normative property of our attitude of believing: a belief, to be correct, ought to be true. If one identifies a belief as true, the correct attitude towards this belief is to adopt it. So the Condition of Correctness of a belief is that it is correct when it is true, and this condition can itself be translated as a prescription, the Norma Veritatis: (NV) One ought to believe that P iff P,

which itself can be decomposed into an if part and an only if part: (NVa) One ought to believe what is true (if P one ought to believe that P); (NVb) One ought not believe what is false (one ought to believe that P only if P).

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P. I do not see why this is better than (EC) and the idea that truth is our main epistemic goal. For (NVa) just says that we ought to believe any truth whatsoever, trivial or interesting, which is absurd, and (NVb) is vacuously satisfied when one does not believe that P or does not form any belief at all: for instance stones and people asleep satisfy it.6 You see Aelius, I know what a norma is: in our language it means a rule, a standard, as when we use a measuring instrument. Something is a norm not only if it has a condition of satisfaction and a condition of correctness, but also if it governs or guides our attitudes or our practices. Let us call this a condition of guidance. A norm is not a norm if it does not tell us how to comply with it. A norm, to be a norm, has to have motivational power, normative force. And neither (NVa) nor (NVb) have any motivational power, for they prescribe conditions which are either impossible to fulfill (believe all truths whatsoever) or conditions which are too easy to fulfill (if you fail to believe you obey the norm). Not only it is a law of the normative realm that ought implies can, but also that the norm ought not to be trivially satisfied. A. Don’t you agree, nevertheless, that (NVb) sounds to have some bite? When you entertain a certain proposition, and it appears to you true, you are disposed to assent to it, and when the proposition appears to you false, confused or unlikely to be true, you refrain from assenting to it. This picture should be familiar to you, Pontius, from the work of Chrysippus and other Stoics, when they talk about assent or katalepsis or what we call in our language assensio. So perhaps one can reformulate (NVa) and (NVb) thus: (NVC) If one considers P and asks oneself whether P, (i) If P is true, one ought to believe P; (ii) If P is false, one ought not to believe P.

We thus avoid having epistemic obligations with respect to truths that one has never considered. P. I am not sure that it can work, Aelius. (NVC) tells you that you ought to believe that P only if you consider P and P is true. But how can one consider P without assenting to it as true? As soon as you consider a winged horse, you believe that it exists. But if this is the case, then in order to apply the norm (NVC) you must already believe P. But if, in order to apply the norm for belief (NVC) and see whether you ought to believe P, you need to believe P, the norm is useless.7 Moreover, are you sure that (NVC) is without exceptions? For there are certain “blindspot”, propositions that one can consider, but which it is impossible to judge as true or false, such as

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(B) P is true and nobody believes it.

If you believe the first conjunct then the second conjunct falsifies the first.8 A. I agree that there are propositions which are hard to believe, hence hard to subject to a norm of truth. But even if that were correct, that would threaten only some of our believings.9 I am not impressed by the objection that the norm is useless. In the first place, it is not evident, pace Epicurus,10 that as soon as we consider a proposition we assent to it. There is such a thing as suspending judgment without assenting to a proposition, as the Stoics made clear, in my view. In such cases it makes sense to ask oneself whether we should follow an epistemic norm, be it that of truth or another one, for instance that one ought to believe what one finds highly probable. In the second place, it is not clear that when we are guided by epistemic norms, we are always guided by explicit prescriptions or by imperatives, either of an hypothetical form (“if condition C holds, believe P”) or of a categorical form, as if, when we consider a belief, we considered the positive prescription “You ought to believe the truth” or the negative one: “You ought not to believe the false”. Actually we almost never follow such prescriptions when we have to form a belief, although we may consider them when we have to maintain a belief that we already have. When such maintenance or conservation of belief is at stake, antecedent conditions can be taken into account.11 In the other cases it is far from clear that the norm of truth governs us in the manner of a deontic prescription or as an imperative, as if we had to perform some kind of mental action (judging, or accepting). It is far from clear that the Norma Veritatis, if it is the expression of a duty or an obligation, is the expression of an ought-to-do, or of some sort of moral prescription of the kind: “It is wrong, always, everywhere and for everyone not to believe what is true”. If the norm involves an ought, it seems that it involves an ought-to-be, rather than an ought to do. As when we say such things as “The world ought to be a better place”, or “Politicians ought to be honest”. Such oughts-to-be need not even imply that their realization is possible or possible in the short run. Similarly with beliefs. When we say that they ought to be true, we do not imply that we ought to believe any truth whatsoever, or than we can do so. We do not imply either that we know how and when we can believe the truth, and what kind of epistemic conditions have to be realized for us to believe the truth. I submit that when we say that belief is governed by a norm of truth, we express an ideal, the ideal condition that belief ought to satisfy. Ideals are such that we need not know how to realize them or how they

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can be reached. So I propose to interpret the Norma Veritatis thus, as an ideal of truth: (NVI) Truth is what one ought to believe, whether or not one knows how to go about it or whether one has reached it.12

The Norma Veritatis is an ideal of reason, in the sense that it tells you what you ought to ideally believe, namely the truth, and thus it belongs to the category of the ought-to-be rather than to the category of the ought to do (Kornblith 2001, p. 238; Millar 2005, p. 76; Chrisman 2008). The truth norm for belief does not give us any prescriptive—or even permissive— guidance. We can also say that it is a constitutive norm. An ideal is meant to describe an abstract situation which holds only “in principle” or a kind of conduct which only certain imaginary beings endowed with powers which are distinct from ours could follow (logical saints, believing all the consequences of their beliefs; perfectly rational agents). The status of the truth norm for belief is of this sort: it tells us what believing requires, but neither what kind of beliefs one must have before applying the norm, nor what kind of beliefs one must have once one has applied it. It is blind to the actual psychology of the agents. In this sense, it need not explain or guide our belief formation. P. This is all good to me, Aelius. We Romans, like Horatius Cocles or Regulus, know where duty is, whatever the consequences. Still, it does not tell me why and how we should apply the norm of truth. We need to be given a reason to follow the norm. And what reason do we have to believe all truths and avoid all falsehoods? Actually, do we have any reason at all to believe truths? Isn’t our primary reason to believe anything the evidence and the justification that we have for it? For truth, by definition, is inaccessible to us. Most of the time, we do not know it, and cannot know it. So how could there be for us a norm to follow the truth? Why should you believe the truth when you do not know it? In such cases, it is mandatory to suspend your judgement, because you do not have enough evidence. So it is much more plausible to say that our actual norm is the Norm of Evidence: (NE) For all P, one ought to believe that P if one has sufficient evidence for P.

One believes for good reasons, and these good reasons are most of the time the evidence that we have for our beliefs. But if it is a necessary condition, it can hardly be a sufficient one. For very often our evidence for believing is not sufficient, although we do not know that it is not. How

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many times do we take ourselves to be justified in believing something while we are not? We have evidence for our beliefs because we have evidence that these are true. Evidence is evidence for truth. Truth is our master norm. It dominates the evidential norm (NE). Moreover, how do you apply (NE)? It says that you ought to have sufficient evidence. But when do you know that you have sufficient evidence, or are justified in believing something? As soon as you epistemicize, so to say, your reasons for believing things become unmanageable. A. But can’t we say that there are actually two norms, one of truth (NV) and the other of evidence (NE)? The former tells us what we ought to believe, so to say, absolutely, independently of our justification or evidence, whereas the latter tells us what we ought to believe, given our evidence. P. I am not so sure. For actually the two norms can conflict. Suppose that a doctor believes, for good reasons and on the basis of what he takes to be sufficient evidence, that he must prescribe a certain drug to a patient. Unbeknownst to him, the drug may be fatal because the patient suffers from a certain illness which is presently unknown to medicine.13 Ought the doctor to believe that the drug will cure his patient? If he follows the norm of evidence, the answer is yes. But according to the norm of truth, he ought not to believe this, for the drug will actually kill the patient. So which norm are we to choose? If you say that it is the norm of evidence, the one that the doctor has from his epistemic perspective, so to say—call it the subjective norm—the doctor will certainly have done all that there is in his power to save his patient. But he will fail miserably, for on the objective side—let us call this the objective norm—he actually ought not to prescribe the drug. Once you realize this, Aelius, there are three possibilities, provided you accept the distinction between the objective and the subjective norm. Either you say that the objective norm (NV) is the only norm. But you then face such situations as the doctor’s one. Or you say that the subjective norm is primary, but you are equally at a loss, for in such cases as the doctor’s, one norm tells you to believe something, and the other tells you not to believe it. Indeed you could say that there is but one norm of truth, but that it is ambiguous, and has an objective side, which says what, in the absolute, one ought to believe (i.e., the truth), and a subjective side, saying what, relative to the information and the actual capacities of the believer, is to be believed. You could also say that the norm is contextual, telling you, in certain circumstances, to believe one thing, and in other circumstances to believe something else. But what kind of norm would this be? Does it make sense to say that there is a norm, which sometimes

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prescribes you something and at other times something else? Or that its instructions are systematically ambiguous or subject to exceptions? If (NV) or (NE) are to be norms, they cannot be systematically hypothetical. They have to be imperative in the categorical sense.14 Actually, if there is a conflict between the norm of truth and the norm of evidence, or if there are equal evidential reasons to believe that P and to believe that not P, the correct thing to do would be, as I said above, to suspend one’s judgment— i.e., not to believe the propositions in question. Or, as I did with Jesus, to decide to believe one of the propositions for practical purposes. A. It would be foolish and desperate, Pilate, to adopt such a rule. Beliefs are not like actions, which one can take indifferently when there is no reason to prefer an option rather than another. If you are indifferent to eating an apple or a pear, nothing serious ensues if you decide to choose one rather than the other. But in the case of belief, it is utterly unwise to adopt this policy. If it is all important for you to go to Rome and you have the choice between a road which will lead you there and another to Naples, but do not know which, it would be crazy to take one rather than another. From the previous discussion I conclude that the norm of truth is not a good candidate alone for being the norm of belief, and that the norm of evidence is not a good candidate either. A purely objective norm like (NV) is not enough because we need to appeal to the fact that our beliefs have to be epistemically constrained. If your belief is false, it does not show that there is something wrong with your belief. A purely subjective norm like (NE) would not do either because we need to appeal to the fact that our beliefs must not be only epistemically constrained, but also true. If your belief is justified but not true, it shows that there is something wrong with it. There is actually a simple solution, Pilate, to these problems. P. Which one? It seems that we have exhausted the possibilities. A. It consists in saying that the norm for belief is actually the norm of knowledge. When we discover that one of our beliefs is false we reject it. When we discover that one of our beliefs is unjustified, we try to revise it. But we would not revise it unless we hoped that in doing so we could get a true belief, and not simply one which would be better justified. We hope at one point to stop inquiring and get the truth. This speaks for the priority of the norm of truth over the norm of evidence. But the norm of evidence is not enough to rule our believings, for what we want is not simply evidence, but evidence for truth. It would be wrong, however, to say that we want truth alone, without justification, as when we guess. As Marcus Tullius Cicero says in his De Fato, what use would be divination if it aimed only at truth, without evidence? This is why astrologers are actually looking to evidence and signs of what they guess. The conclusion is that a

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good guess is not blind, it needs an epistemic justification. Hence I propose that the right norm is a norm of knowledge: (NK) One ought to believe that P iff one knows that P.

Knowledge is the perfect norm for belief. For if you do not know that P, but believe that P, aren’t you in a predicament comparable to the one in which you are in Moore’s paradox, when someone says: “P, but I do not believe that P”? For “P and I do not know that P” seems even more hard to assert. And given that knowledge is factive, that it entails truth, by definition what is known is true, hence the norm (NV) is a direct consequence of (NK). So the two norms are not incompatible: the one entails the other (Engel 2003, 2014; Smithies 2012). P. So, Aelius, you tell us that one must believe something because one knows it? But that’s stupid. If I know that P, why should I care about believing it? I believe it already, and moreover in the mode of knowledge. Ought I to believe that 2 + 2 = 4? That Rome is the capital of the Imperium Romanum? That Caesar was our first imperator? That Capri is an island? Moreover, how do we avoid the previous difficulties of the objective norm of truth? If I ought to know that P in order to believe it, our problem reproduces: for most of the time, I do not know that P, hence it is absurd to require of me that I know P in order to believe it. A. The answer, Pilate, is that the norm of knowledge does not require you to have actual knowledge of P. As I said before, a norm is first and foremost an ideal. It states what kind of condition you must be in, ideally, in order to believe. It does not say that you are actually in that condition, for it would be trivial or impossible, and it does not say how you have “to go about it”. In other words, the norm prescribes in general what it is to believe correctly, but it does not give you the details about “how to go”. How could it? There are so many different kinds of belief, so many different kinds of knowledge, that it would be useless to require of the norm that it told you how to go, even in rough outline. One thing is sure, however. The norm does not presuppose that it is possible to know that P. But it presupposes that, in order to believe that P, you must be in a position to know that P. P. Your are clever, Aelius. But aren’t you, like those Galenic doctors, someone who says that knowledge is epistemically constrained? Aren’t you a verificationist? A. I say that you must be so placed, cognitively and epistemically, that knowledge that P is at least possible. But that does not entail that everything that is true is known. There are indeed things that are true and that we do not actually know, and also perhaps things that we shall never

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know. For instance about the Gods, or about the limits of the universe. But if we are to believe anything about these, we must, even in an idealized sense, be in a position to know. Even if we would never know, the posture of a believer must be such that he takes himself to be in a position to know. P. But Aelius, with this guy Jesus, I wasn’t even in a “position to know”. I could not even figure out how to know whether he was the King of the Jews or some crazy impostor. A. I agree. But then you should have suspended judgment, and released him. P. I don’t know. But I agree with you that I should not have asked him “What is truth?”. I should have asked him simply: “What is knowledge?”

References Berker, S. 2013, “The Rejection of Epistemic Consequentialism”, Philosophical Issues 23 (1), 363 Boghossian, P. A. 2003, “The Normativity of Content”, in Sosa, E. and E. Villanueva (eds.), Philosophical Issues, 13: The Metaphysics of Epistemology, 31, reprinted in his Content and Justification, Oxford: Oxford University Press Bykvist, K. and A. Hattiangadi 2007, “Does Thought Imply Ought?” Analysis 67 (296), 277 Bykvist, K. and A. Hattiangadi 2013, “Belief, Truth and Blindspots” in Chan, T. 2013, 100 Chan, T. (ed.) 2013, The Aim of Belief, Oxford: Oxford University Press Chrisman, M. 2008, “Ought to Believe”, The Journal of Philosophy 105 (7), 346 Engel, P. 2013, “Belief and Correctness”, Proceedings of the Aristotelian Society, Sup vol. 37, 199 —. 2013a, “In Defence of Normativism about the Aim of Belief”, in Chan, T. 2013 —. 2014, Volontà, credenza e Verità, Milano: Jouvence Fairweather, A. and L. Zagzebski (eds.) 2001, Virtue Epistemology: Essays on Epistemic Virtue and Responsibility, Oxford: Oxford University Press Gibbard, A. 2003, “Thoughts and Norms”, Philosophical Issues 13 (1), 83 Gibbons, J. 2013, The norm of Belief, Oxford: Oxford University Press Glüer, K. and Å. Wikforss 2009, “Against Content Normativity”, Mind 118 (469), 31

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Hattiangadi, A. 2010, “The Love of Truth”, Studies in History and Philosophy of Science 41 (4), 422 Kornblith, H. 2001, “Epistemic Obligation and the Possibility of Internalism”, in Fairweather A. and L. Zagzebski (eds.) 2001, 231 Millar, A. 2004, Understanding People: Normativity and Rationalizing Explanation, Oxford: Oxford University Press Smithies, D. 2012, “The Normative Role of Knowledge”, Noûs 46 (2), 265 Sosa, E. 2001, “For the Love of Truth?” in Steup, M. (ed.) Knowledge, Truth and Duty, Oxford: Oxford University Press Steglich-Petersen, A. 2010, “The Truth Norm and Guidance: A Reply to Glüer and Wikforss”, Mind 119 (475), 749 Teroni, F. 2007, “Emotions and Formal Objects”, Dialectica 61(3), 395

Notes 1

Anatole France, Le procurateur de Judée, in L'Etui de nacre, Œuvres, Pléiade, Gallimard, Paris, 1984, tome I, pp. 877 ff. 2 See M. Gigante, Philodemus in Italy. The books from Herculaneum, Ann Arbor: University of Michigan Press, 1995; and Pierre Vesperini, “«Que faisaient dans la baie de Naples Pison, Philodème, Virgile et autres Epicuriens?» Pour une approche hétérotopique des pratiques philosophiques romaines”, Melanges de l’Ecole française de Rome, 121, 2, 2009, pp. 515-543. 3 Cf. Teroni (2007), Engel (2013).  4 See Berker (2013) for a statement, and a rejection of (EC). 5 The informed reader will notice that Pontius Pilate anticipates here, remarkably, Edmund Gettier. 6 Bykvist and Hattiangadi (2007). 7 Sosa (2001, p. 54), Glüer and Wikforss (2009, p. 44). 8 If you believe a conjunction then you believe each of its conjuncts, so if you believe (B) then (B) is false. Now, suppose (B) is true and you have a doxastic attitude to it (this would be an attitude of withholding or of disbelief). By (NVCi) you ought to believe (B). But if you believe (B) then it is false, and so by (NVCii) you ought not believe it. So, in the case where (B) is true and you form a doxastic attitude to it, (NVCi) entails that you are subject to a requirement that you cannot satisfy, in the sense that you cannot do what the requirement requires you to do while the requirement is in place. Indeed, if you did what the requirement requires you to do, you would thereby do something the requirement requires you not to do. This violates the plausible principle that you cannot be subject to oughts that are impossible to satisfy (Bykvist and Hattiangadi 2007). 9 About the blindspot objections, see Bykvist and Hattiangadi (2013). 10 The view would become later associated with the names of Spinoza and Hume. 11 Steglich-Petersen (2010). 12 Boghossian (2003).

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13 The example comes from Gibbard (2003), and is taken up by Hattiangadi (2010). 14 The options are well discussed in Gibbons (2013).

CHAPTER TWO AN EXPLANATORY ROLE FOR THE CONCEPT OF TRUTH BORIS RÄHME

1. Introduction Deflationism about truth (henceforth, deflationism) comes in a variety of versions 1 Variety notwithstanding, there is widespread consensus among advocates of different stripes of deflationism (disquotationalism, minimalism, prosententialism, etc.) with respect to the following noexplanatory-role claim concerning the concept of truth: (NER) The concept of truth has no explanatory role to play in philosophical explanations (nor, for that matter, in non-philosophical explanations).

Versions of (NER) can be found in Armour-Garb (2012), Brandom (2002), Dodd (2013), Field (2001), Grover (2002), Horwich (1998, 2010), Soames (1999) and Williams (2002, 2007), to give just a few examples. In one way or another, all of these authors seem to hold that (NER) follows from their respective deflationary accounts of truth. To be sure, they do not intend to deny that truth talk is sometimes useful (or even indispensable) for the purposes of formulating and expressing explanations of, say, our epistemic practices, meaning and propositional content, practical success or the success of scientific theories. But, they insist, truth talk does not and cannot contribute any genuinely explanatory content to the explanations which we formulate with its help. Its contribution to them is, as Michael Williams puts it, “wholly expressive, thus never explanatory” (Williams 1999, p. 547). In what follows I argue that (NER) is false. My argument begins with the question of why the following conditional holds:

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Chapter Two An assertion of is correct only if some dogs are vicious.2

My contention is that the best available answer to this question—the best available explanation of why the conditional holds—is in terms of an explanatory “because”-statement whose explanans-clause contains truth talk that is both inaccessible to standard deflationary treatment and explanatory.3 I take this to amount to a counterexample against (NER). Of course, nothing in what follows hinges on the peculiarities of assertions of . Michael Williams (2002, p. 157) suggests that an example of a genuinely explanatory use of the concept of truth, i.e. a counterexample to (NER), would amount to a refutation of deflationism—not just of this or that specific deflationary account of truth but of the deflationary outlook on truth quite generally. I will be cautious with regard to the question of whether Williams is right. Maybe a counterexample to (NER) should not per se be taken to amount to a refutation of deflationism because, maybe, there is no good reason for deflationists qua deflationists to commit themselves to (NER) in the first place—contrary to what many of them seem to think. In fact, the claim that deflationism entails or in some other, logically weaker, way requires acceptance of (NER) has recently come under criticism. Nic Damnjanovic (2005, 2010), for instance, points to a way in which, arguably, one can be a deflationist about truth without committing oneself to (NER), and Leon Horsten, in outlining his inferentialist version of deflationism, suggests that “perhaps we should divorce deflationism from the claim that the concept of truth has no explanatory function in specific philosophical disciplines” (Horsten 2011, p. 92; see also 2009, 2010). The next two sections prepare the ground by rehearsing the standard deflationary account of the role and function of truth talk. In the fourth section I present a counterexample to (NER). The fifth section discusses various objections that my line argument is likely to provoke. The final one contains a very brief discussion concerning the question of whether my counterexample to (NER) should be taken to show the deflationary outlook on truth to be mistaken. In what follows the focus is on deflationary accounts of truth for propositions. However, what I have to say carries over, mutatis mutandis, to deflationary accounts of truth for (utterances of) sentences.

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2. Deflationism and the no-explanatory-role claim Consider the distinction between revealing and unrevealing contexts or environments of truth talk (see Soames 1999, p. 230). A context, C, of truth talk, is revealing if and only if the proposition(s) which truth is predicated of in C is (are) immediately recoverable from C alone. By way of example: is true. If NASA’s press releases are to be believed, then is true.

Notice that, in the sense intended here, predicating truth of a proposition does not always amount to asserting or endorsing that proposition. In the antecedent of “if is true, then some dogs are dangerous” truth is predicated of , but is not asserted or endorsed. The same holds for the consequent in the second example of revealing truth talk given above. A context C, of truth talk, is unrevealing if and only if C is such that the propositions which are at issue are not immediately recoverable from it—either because we have no information as to what they are or because there are too many of them: Some of what Khrushchev asserted in 1960 is true. Every proposition of the form “p o p” is true.

Deflationists sometimes say that unrevealing environments of truth talk give us a hint as to the raison d’être of truth locutions in our languages (see, for instance, Horwich 2010, pp. 4-5). If it were not for the utility of the truth predicate in formulating general statements of the kind just displayed, use of that predicate would be dispensable quite generally (see Soames 1999, p. 230; Horwich 1998, p. 39). Assuming, plausibly, that all contexts of truth talk are either revealing or unrevealing, the deflationist can be interpreted as offering the conjunction of the following two claims as an account of the role of the truth predicate in our discourses: (a) When the truth predicate is used in a revealing environment, use of that predicate is usually dispensable in the following sense: what is said by predicating truth of some given proposition can be said without loss of, or other changes in, relevant content by directly expressing that proposition.

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(b) Whenever the truth predicate is used in unrevealing contexts, its contribution to the expression of what is said in those contexts can be exhaustively accounted for in terms of its role as a device for facilitating the formulation of a specific kind of generalisation over propositions.

There are two qualifiers in (a), “usually” and “relevant”. The qualifier “relevant” is needed to make room for denying the simple redundancy claim according to which the meaning of a revealing truth ascription (“ is true”, say) just is the proposition that truth is ascribed to ()—a claim that most contemporary deflationists want to deny (see, for instance, Horwich 1998, p. 124; Damnjanovic 2010, p. 45). The qualifier “usually”, on the other hand, is required once the restriction to relevant content is in place. In fact, once it is admitted that the content of a revealing truth ascription is not exhausted by the proposition that truth is ascribed to, it cannot be ruled out that competent speakers sometimes utter revealing truth ascriptions precisely because the content they want to express could not be expressed without employing propositionally revealing truth talk. In other words, it cannot be excluded that sometimes the extra content of revealing truth ascriptions is (part of) the relevant content.4 Deflationists maintain that in order to see why truth talk is (usually) dispensable in revealing contexts and how the truth predicate performs its generalising role in unrevealing environments it is sufficient to appeal to the (non-paradoxical) instances of the equivalence schema (ES)

is true l p.

I will come back to this point in the next section. How do these deflationary theses concerning the role of truth talk relate to the no-explanatory-role claim (NER), which concerns the concept of truth? (NER) states that the notion of truth is explanatorily inert. Deflationists who subscribe to (NER) provide this sweeping claim with a tangible interpretation. One aspect of this interpretation—the one that will be particularly relevant to the following discussion—is this claim: any explanatory work that can be done by some revealing truth ascription, R, can equally well be done by the proposition(s) that truth is ascribed to in R (see Horwich 1998, p. 49). As mentioned above, deflationists do not maintain that utilising the word “true” in philosophical explanations is a somehow defective practice. In subscribing to (NER) they do not intend to prohibit the use of truth talk in such explanations. 5 Rather, they distinguish between truth-talk’s involvement or occurrence in formulations of explanations on the one

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hand, and truth-talk’s contributing explanatory content to what is expressed by those formulations on the other—in order to then insist that whenever “is true” occurs in the formulation of explanations its use is either dispensable in the sense of (a) or can be entirely accounted for in terms of its role as a device for expressing a certain kind of generalisation in the sense of (b). The deflationary account of the role of truth talk relates to (NER) via a line of reasoning that deflationists who advocate (NER) often leave implicit but clearly rely upon. In large outlines, the reasoning goes as follows: (DeflatioNER) 1. For any explanation, E, involving truth talk, if the truth talk in E can be adequately accounted for along the lines of either (a) or (b), then it does not contribute explanatory content to E. 2. For any explanation, E, involving truth talk, the truth talk in E can be adequately accounted for along the lines of either (a) or (b). 3. Therefore, for any explanation, E, involving truth talk, the truth talk in E does not contribute explanatory content to E.

I accept the first premise of (DeflatioNER) for the sake of argument, and I take issue with the second. More precisely, I take the argument overall to suggest a promising strategy for identifying candidate counterexamples to (NER). The strategy suggested by (DeflatioNER) is to try to find counterexamples to its second premise. One way to pursue this strategy is to try to identify an occurrence of the predicate “is true” in the explanans-clause of a true explanatory “because”-statement, where the occurrence of “is true” is such that its function (or role) cannot be accounted for by either (a) or (b). Of course, not all explanations can be couched in terms of “because”-statements since, for instance, some explanations answer to “what is”-questions rather than to “why”-questions (cf. Jenkins 2008). In order for the following to go through, however, it is sufficient to make the uncontroversial assumption that one paradigmatic way of asking for an explanation is by means of “why”-questions and that one paradigmatic way of giving explanations is by means of “because”statements. Before stating what I take to be a counterexample to the second premise of (DeflatioNER)—and arguing that it is also a counterexample to (NER)—it will be useful to rehearse the deflationary treatment of a norm of assertion whose normative content prima facie involves the notion of truth.

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3. A norm of assertion and its deflationary treatment Assertions are open to normative assessment along many different lines. They can be objectionable or praiseworthy for moral reasons, they can be epistemically justified or unjustified, boring or interesting, relevant or irrelevant to their conversational context etc. Speech acts of assertion can exemplify all of these qualities and defects (and many more) quite independently of whether their propositional contents are true or not. However, consider the following semantic constraint on the correctness of assertoric speech acts: (SCA) It is correct to assert a proposition only if it is true.

It would seem that the truth predicate makes an important contribution to the normative content of this constraint. Not so, says the deflationist. Granting the use which is here being made of “is true” as entirely proper and unobjectionable, she insists that, in (SCA), “is true” is only being used to achieve the right kind of generality: it makes an important contribution to the formulation or expression of the general constraint, but it does not contribute anything to (SCA)’s normative content proper. In fact, what deflationists have to say about the role of the truth predicate in formulations of norms for assertion echoes what they have to say about the role of that predicate in explanations. They do not intend to prohibit the use of “is true” for the purpose of formulating norms of assertion. Rather, they appeal to the distinction between the mere occurrence of truth talk within formulations of such norms and truth-talk’s contributing genuinely normative content to what is expressed by those formulations—in order to then insist that whenever “true” is being used in the formulation of norms of assertion its use is either dispensable in the sense of (a) or can be entirely accounted for in terms of its role as a device for the expression of generalisations in the sense of (b). The deflationist’s argument for the claim that truth talk does not contribute genuinely normative content to norms of assertion like (SCA) is simply (DeflatioNER) with “norm” in place of “explanation” and “normative” in place of “explanatory”.6 In order to establish the more specific claim that the truth predicate makes no contribution to the normative content expressed by (SCA) deflationists offer the following line of argument. With the variable x taking propositions as values and singular terms denoting propositions as substituents, (SCA) can be stated more formally in this way: (SCA) x (It is correct to assert x o x is true).

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If we express the semantic constraint by means of a first-order universal quantification, the last occurrence of the bound referential variable x requires a predicate—and the predicate best suited for the job is, of course, “is true”. However, it would be hasty to take the fact that truth talk facilitates the expression of the normative content of (SCA) as evidence for the thesis that the concept of truth is involved in the normative content expressed by (SCA). This becomes clear, says the deflationist, when we consider an arbitrarily chosen instance of (SCA), say: (T+) It is correct to assert o is true.

There is an obvious way of expressing the normative content of (T+) without using the truth predicate. Since (T+) is a context of truth talk that is both propositionally revealing and transparent, we can immediately apply the left-to-right direction of the relevant instance of (ES), i.e. of is true l some dogs are vicious,

to the consequent of (T+) and write (T-) instead of (T+): (T-) It is correct to assert o some dogs are vicious.

At this point a deflationist will point out that the normative contents of (T-) and (T+) are identical even though, contrary to (T+), (T-) does not involve truth talk at all. The claim that (T+) and (T-) have identical normative contents can be spelled out along the following lines: every assertion of that satisfies the necessary condition of assertoric correctness specified by (T+) also satisfies the necessary condition specified by (T-), and vice versa. Assuming (T+) and (T-) to be declarative equivalents of the imperatives “[assert ] only if is true!” and “[assert ] only if some dogs are vicious!”, respectively, another way to make the same point is this: there is nothing a person could rationally do in order to comply with (T+) that she could not also rationally do in order to comply with (T-), and vice versa. Notice, in passing, that to say that the normative contents expressed by (T+) and (T-) are identical is not to say that (T+) and (T-) express identical contents tout court. So the normative point of (T+) can be expressed without employing the concept of truth and, importantly, in doing so, there is no need to replace use of that concept with use of any other.7 All that needs to be appealed to

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is the relevant instance of (ES). The treatment just applied to (T+) works for every instance of (SCA); and this, according to the deflationist, should be taken to show that the only reason why the predicate “is true” appears in (SCA) is that it—together with the apparatus of referential quantification—conveniently enables the formulation of a universal generalisation which entails every instance of (T+schematic) It is correct to assert

o

is true

and, together with the relevant instances of (ES), every instance of (T-schematic) It is correct to assert

o p.

For what follows it is important to emphasise that I do not take issue with the deflationary treatment of (SCA). Again for the sake of argument, I accept the claim that for each instance, I+, of (T+schematic), there is at least one instance, I-, of (T-schematic), such that the normative contents of I+ and Iare identical (and vice versa for each instance, I-, of (T-schematic)). Deflationists treat this point as a satisfactory end to the debate on the role of the truth predicate in (SCA). As far as the normative content of (SCA) is concerned, they may very well be right in doing so. However, there is a sequel to that debate. The sequel concerns the question of how we are to understand the instances of (T-schematic) and, or so I will argue in the remainder, it is bound to involve truth talk in a way that cannot be accounted for along the standard deflationary lines, (a) and (b), introduced in the last section.

4. A counterexample to the no-explanatory-role claim The instances of (T-schematic) stand in need of explanation. With regard to every single one it is legitimate to ask: why does it hold? In what follows I use (T-) as an arbitrarily chosen example-instance of (T-schematic). Consider the following question, then: (Q) Why is an assertion of correct only if some dogs are vicious?

Alternatively: (Q) Why does (T-) hold?

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Here it might be objected that (T-) does not hold and that, consequently, (Q) should be dismissed. I will come back to this worry in the next section. For the time being I assume that (Q) cannot be dismissed so easily. So, given that (Q) is a good question, a good answer to (Q) would provide us with an explanation of why (T-) holds—to put it in fewer words, with an explanation of (T-). What, then, should one expect from an explanation of (T-)? At the very least it should help us understand why the existence (or non-existence) of vicious dogs is relevant to the correctness (or incorrectness) of assertions of . It should break an explanatory path from the existence or non-existence of vicious dogs to the normative status (correctness or incorrectness) of assertoric speech acts which have as their propositional content. In asserting we do not assert dogs or any of their dispositions. Therefore the detour via the proposition that some dogs are vicious, i.e. via the propositional content that is shared by all assertions of , seems inevitable in answering (Q). It is hard to see how the relevance of whether or not there are vicious dogs to the normative status of assertions of could be thought of as not mediated by some property (substantial or not) that can (fail to) exemplify—by something that appertains to only if some dogs are vicious. This suggests that what we have to do in order to explain (T-) is identify a predicate G of propositions that satisfies (at least) the following two conditions: (i) G applies to only if some dogs are vicious, and (ii) ’s being G or not being G partly determines the normative status of assertions of , such that an assertion of is correct only if is G.

Unsurprisingly, I think that the predicate that best meets conditions (i) and (ii) is the predicate “is true”. So my claim is that the following “because”-statement is (part of) the best available explanation of (T-), i.e. (part of) the best available answer to (Q): (A+) [It is correct to assert only if some dogs are vicious] because [ is true only if some dogs are vicious].

In (A+) “is true” occurs in the explanans-clause of a “because”statement whose explanandum-clause is (T-). It is used in a (partial) explanation of the relevance of the existence or non-existence of vicious dogs for the normative status of assertions of . Given the deflationist’s advice not to confuse the mere occurrence of the truth predicate in the explanans of some explanation, E, with evidence for the

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claim that the notion of truth contributes genuinely explanatory content to E, it is fair to ask: if its role in (A+) is not explanatory, then what role does the truth predicate play in (A+)? Can the deflationist account for the role of the truth talk involved in (A+)? More precisely, can she account for it in one of the standard deflationary ways, (a) and (b), introduced above? It is obvious that in (A+) “is true” is not being used to formulate a generalisation of the kind that is dear to deflationists (and non-deflationists as well). There is only one proposition at play here and, importantly, we know exactly which one it is. The explanans-clause of (A+) is a propositionally revealing context of truth talk. So the truth talk involved in it cannot be accounted for along the lines of (b). But it is not accessible to deflationary treatment in terms of (a) either. Since the deflationist claims that any explanatory work that can be done by a revealing truth ascription, R, can equally well be done by the proposition that truth is ascribed to in R, one might expect her to hold that the truth talk in (A+) is dispensable in precisely this sense: the explanatory work that can be done with it can equally well be done without it—by using a sentence that expresses instead of one that expresses in the antecedent of (A+)’s explanans-clause. However, the deflationist would be ill-advised to put forward this claim. Unlike “ is true only if some dogs are vicious”, the tautology “some dogs are vicious only if some dogs are vicious” on its own cannot be used to state even a partial explanation of (T-). The proposition it expresses has no explanatory value at all with regard to the question of why the normative status of assertions of (partly) depends on whether there are vicious dogs. The following “because”-sentence expresses a false proposition, despite the truth of the propositions expressed by the two clauses that it connects by “because”: (A-) [It is correct to assert only if some dogs are vicious] because [some dogs are vicious only if some dogs are vicious].

This point reinforces the claim that in explaining (T-) we cannot contend ourselves with talk about dogs and their properties. The detour via some property that can (fail) to exemplify is indispensable in answering (Q). Not any old property that can reasonably be claimed (not) to exemplify will do. The only one that has explanatory import with respect to (T-) is the property of truth. Therefore, given her commitment to (NER), the deflationist would seem to be in a rather difficult dialectical position when it comes to the question of why the existence or non-existence of vicious dogs is relevant to the normative status of assertions of .

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At this stage, if not before, the deflationist might complain that the line of argument just presented is wrong-headed. After all, the explanansclause in (A+) is just the left-to-right direction of an instance of (ES), and all (non-paradoxical) instances of (ES), together with the propositions expressed by them, are available to the deflationist by default, as it were. The proposition expressed by the explanans-clause of (A+) is, for example, entailed by an axiom of Horwich’s Minimalist Theory of truth (see Horwich 1998, pp. 17-20). So, the thought would have to go, the deflationist can dismiss the demand for an account, in terms of (a) or (b), of the truth-predicate’s role in (A+) since there is nothing in her deflationism which prevents her from simply accepting (A+) as a (partial) explanation of (T-). These points are important. But they do not lend support to the complaint that the line of argument presented above is on the wrong track. Rather, they speak in favour of the claim that deflationists should be more cautious in answering the question of whether their respective version of deflationism commits them to (NER). Accepting (A+) as a partial explanation of (T-) amounts to accepting that the left-to-right direction of the relevant instance of (ES) is capable of doing explanatory work. Since the left-to-right direction of that instance, when employed in the context of explaining (T-), contains truth talk which is not accessible to deflationary treatment in terms of (a) or (b), a deflationist who accepts (A+) would seem to be forced to grant that the explanans-clause of (A+) is capable of doing its work precisely in virtue of the conceptual contribution that the truth predicate makes to the explanatory content which is expressed by it. If this is correct, then there is a rather blatant tension between (NER) and the claim that instances of (ES) can be used explanatorily. If the latter claim holds good, then (NER) is false, and if (NER) is true, then instances of (ES) cannot be used explanatorily. The deflationist will have to make a choice here. She can either maintain the thesis that the concept of truth is explanatorily inert, i.e. she can stick to (NER), or she can hold that instances of the equivalence schema can have explanatory import. But she cannot consistently hold both. The tension just described is quite evident in Horwich, for example. On the one hand, he holds that “all of the facts whose expression involves the truth predicate may be explained […] by assuming no more about truth than instances of the equivalence schema” (Horwich 1998, p. 23). On the other hand he claims that “whenever we deploy the concept of truth nontrivially – whether in logic, ordinary language, science or philosophy – it is playing this role: a device of generalization” (Horwich 2002, p. 138). Of

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course, the description of the (normative) fact registered by (T-) does not (have to) involve the truth predicate. But still, that fact is one whose explanation should at least not be rendered impossible by one’s account of truth—and deflationism would seem to have precisely this undesirable effect. I do not want to put too much weight on these points, however. For the sake of argument, let us assume that they do not establish that accepting (A+) precludes one from accepting (NER), and vice versa. I have already alluded to the fact that the explanation of (T-) in terms of (A+) is incomplete or partial. By itself this observation does not disqualify (A+) as an answer to (Q). A “because”-statement can be true— and thus have explanatory value—even if what it expresses is only a partial or incomplete explanation of what is described by its explanandumclause (Schnieder 2006, p. 32). No doubt, there are many different possible approaches to rendering (A+) more complete as an explanation of (T-). However, there is one explanatory lacuna in (A+) which any such approach will have to fill in order to be acceptable. What is clearly missing in (A+) is an explicit statement of the explanatory link between the truth or untruth of and the correctness or incorrectness of assertions of . As far as the possible approaches to filling this explanatory gap in (A+) are concerned there is not much of a choice. Eventually appeal will have to be made to (T+). In fact, the most immediate way to render (A+) more complete consists in adding (T+) to its explanans-clause. This gives us: (B) [It is correct to assert only if some dogs are vicious] because [ is true only if some dogs are vicious, and it is correct to assert only if is true].

While the first conjunct of the explanans-clause in (B) might—for the reasons discussed above—be available to the deflationist, its second conjunct, i.e. (T+), is not available to her in the context of explaining (T-). She has already used (T-) in her account of (T+) (see section 2, above), and having done so precludes her from now using (T+) in explaining (T-). Moreover, and more importantly, the role of the truth predicate in the second conjunct cannot be accounted for along the lines of either (a) or (b). While the deflationist might be able to accommodate this point with respect to the first conjunct, it is hard to see how she could do so with regard to the second. The truth talk involved in the second conjunct of (B)’s explanans-clause does not serve the expression of a generalisation. That excludes (b). But neither is it dispensable in the sense that the explanatory work that is done by (T+) in the explanans of (B) can equally

An Explanatory Role for the Concept of Truth

27

well be done by (T-). (T-) is our explanandum, and it cannot be used to explain itself. That excludes (a). Due to her commitment to (NER), which is based on the argument (DeflatioNER), it would seem that the deflationist is left with no choice but to reject (B) as an explanation of (T-). And given that (Q) is a legitimate request for explanation, the rejection of (B) commits her to offering an explanation of (T-) which, while being at least as good as (B), either does not involve truth talk at all or uses it exclusively as a device for expressing generalisations. But such an explanation does not seem to be forthcoming. I will come back to this shortly. (B) is a counterexample to the second premise of the argument (DeflatioNER)—a counterexample, that is, to the claim that for any explanation, E, involving truth talk, the truth talk in E can be adequately accounted for along the lines of either (a) or (b). In order to see that (B) is also a counterexample to (NER) it is sufficient to appreciate that, in (B), it is the truth predicate which forges the explanatory link between the existence or non-existence of vicious dogs and the normative status of assertions of . Contrary to what many deflationists maintain, there is a genuinely explanatory role for the concept of truth in philosophical theorizing. It can be found in what is expressed by the explanans-clause of (B)—and, arguably, already in what is expressed by the explanans-clause of (A+). Moreover, since the assertion of is an arbitrarily chosen example there would seem to be a plethora of genuinely explanatory roles for the concept of truth.

5. Discussion of objections It might be objected (1) that (T-) does not hold (and therefore doesn’t need explaining), (2) that (T-) can, after all, be explained without making use of the concept of truth, (3) that (T-) registers a basic or brute constraint on the correctness of assertoric speech acts which have as their content, (4) that the failure of the standard deflationary ways in accounting for the role of truth talk in the explanans-clause of (B) is simply due to the fact that “because” creates non-extensional contexts, (5) that the explanations (A+) and (B) of (T-) do not pose a problem for the deflationist advocate of (NER) because (NER), as she intends it, concerns the role of the concept of truth in causal explanations only. I discuss these rejoinders in turn. (1) In asking “why”-questions speakers presuppose that what they respectively request an explanation for is indeed the case. If (T-) does not hold, then (Q)—“why does (T-) hold?”—should be dismissed as ill-posed.

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Consequently, one strategy to block the line of argument presented above would be to reject (T-). Is this a viable strategy to pursue for the deflationist? I do not think so. If there are no vicious dogs then any speaker who asserts commits a mistake. Taking the expression “correct” in (T-) to mean “not mistaken” in the sense just hinted at, it seems rather difficult to find fault with the claim that (T-) expresses a valid constraint on the correctness of assertions of . The denial of (T-) would amount to the claim that the existence or non-existence of vicious dogs is irrelevant to the normative status of (some) assertions of . That seems plainly wrong, however, and it flies in the face of what, arguably by default, is the central point of asserting . Moreover, it seems to be something that no deflationist would want to be committed to—at least not in virtue of her deflationism alone. This objection, then, can be set to one side. (2) I have argued for the claims that (B) is the best available explanation of (T-) and that the deflationist’s rejection of (B) commits her to explaining (T-) in a way—at least on a par with (B) as regards explanatory force—that either steers completely clear of the concept of truth or employs it exclusively in its generalising function. Moreover, I have claimed that such an explanation does not seem to be forthcoming. Is this last claim warranted? As mentioned in section 3, deflationists offer an explanation of the general semantic constraint (SCA)—it is correct to assert a proposition only if it is true—which they, rather convincingly, take to establish that the truth talk involved in (SCA) can be exhaustively accounted for in terms of its role as a device for formulating a useful kind of generalisation, i.e. in terms of (b). However, as far as explaining (T-), or rather, as far as explaining particular instances of the schema “it is correct to assert

, only if p” is concerned, the deflationary literature has little to offer. To date both deflationists and their opponents have tended to ignore that particular norms like (T-) will appear to be unproblematic only as long as a prior understanding of the speech act of assertion is simply presupposed. Both sides, that is, have tended to ignore that, once the concept of assertion is put on the agenda, questions like (Q) will become legitimate requests for explanation. This tendency can probably be construed as a result of the prevalent focus of current debates concerning the explanatory role and the normative import of the concept of truth. Understandably, the focus of these debates has been squarely on statements of general epistemic norms and on general theses that purport to explain, for instance, meaning in terms of truth conditions or practical success in terms of desire and true

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belief. Deflationists have made a remarkably strong case for the claim that the truth talk involved in such theses figures “merely [as] a device of generalisation” (Horwich 2002, p. 140). However, that claim rests on treating the observation that particular instances of the norms and theses in question can be reformulated sans truth talk and without loss of (relevant) content as the natural end of the debate. What about the “true”-free reformulations of the particular instances themselves? Sticking to my example, what about (T-)? The closest that one gets to reading an attempt at offering a “true”-free explanation of particular norms like (T-) in the works of advocates of deflationism is Horwich’s account of the “normative significance of truth” (Horwich 1998, p. 139) for belief. Horwich generally translates issues concerning normativity into issues concerning desirability. In the present context, this general approach leads him to turn the question of why truth has normative significance for belief into the question of why the following principle (see Horwich 2002, p. 135), let’s call it (DT), holds good: (DT) It is desirable to believe only what is true.8

Horwich, of course, accepts both the claim that truth is normatively significant for belief and the claim that (DT) holds good. His answer to the question of why (DT) holds, i.e. why it is desirable to believe only what is true, begins with a “true”-free reformulation of a particular instance of (DT)—an instance which concerns what Horwich calls “a directly actionguiding proposition”: [It] is easily seen why I should want it to be the case, for example, that I believe that if I run I will escape, only if I will escape if I run. I want this because, given a desire to escape, that belief would lead to a certain action (running), and that action would satisfy my desire if indeed it implies escape. This is why I would like it to be that I believe that I will escape if I run, only if I will indeed escape if I run (Horwich 1998, p. 139).

In the context of the passage just quoted, Horwich’s aim is to explain why the general principle (DT) holds good and, moreover, to do this in a way that establishes that the truth talk involved in (DT) serves merely as a device for expressing the required generalisation over propositions. To this end, he argues that an explanation analogous to the one given in response to the question of why it is desirable for me to believe , only if I will escape if I run, can be given for all my beliefs that have directly action-guiding contents. However, “directly action-guiding

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beliefs are derived from other beliefs” and “any of our beliefs might be a premise in some such inference” (p. 140). Therefore, says Horwich, (DT) holds good. Note that this explanation does indeed steer clear of using the concept of truth explanatorily. The predicate “is true” is used only in the final generalising step that takes us from particular instances of the schema “if it is desirable to believe

, then p” to the explicit generalisation (DT). Arguably, at that point all explanatory work has already been done. For present concerns, the most relevant part of Horwich’s explanation of (DT) is the long quote above. Let us grant that it contains the ingredients of a good explanation of why it is desirable for me to believe , only if I will escape if I run, and more generally for why we want our beliefs to be true. Can the explanatory strategy exemplified in the quote from Horwich be adapted so as to provide a good explanation of, say, (T-)—which states that it is correct to assert only if some dogs are vicious? The answer, I think, has to be that it cannot be so adapted, for the following simple reason: while (T-) holds without exception and, therefore, independently of the various practical goals we might have, the claim that it is desirable to assert , only if there are vicious dogs, does not hold without exception or independently of our practical goals. What we desire the propositional contents of our assertions to be like is one thing, what the propositional contents of our assertions have to be like in order for our assertions to be correct is quite another. To interpret “why does (T-) hold?” as “why is it desirable to assert , only if some dogs are vicious?” amounts to changing the subject. Questions about assertoric correctness just don’t translate quite as smoothly as Horwich would have us believe into questions about what we want the propositional contents of our assertions to be like. Of course, these considerations cast doubt only on the prospects of explaining (T-) along the lines of Horwich’s account of (DT). They do next to nothing to establish that “true”-free explanations of (T-) are impossible. It merits emphasis, however, that even if an acceptable “true”-free explanation of (T-) were to be given, the line of argument presented in section 4 would be weakened only in part. What would remain standing is the point that the explanatory use of the truth predicate in (B) is both perfectly legitimate and not accessible to standard deflationary treatment. What, then, can the deflationist say about the role of truth talk in (B)? (3) Another rejoinder to the argument given above might be to concede that (T-) holds but then to go on and say that (T-) registers a basic constraint on the correctness of assertions of —a constraint that neither requires nor allows of explanation. In order to make good on this rejoinder the deflationist would have to make a case for the claim that (Q),

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despite being a prima facie legitimate request for explanation, can ultimately be dismissed as unanswerable. No one can answer it, and therefore the fact that the deflationist cannot answer it should not be held against her deflationism—or so the thought would have to go. But this kind of manoeuvre should really be the last resort. At any rate, to reply to (Q) by declaring (T-) to register a brute or basic normative fact would seem to be an inappropriate over-reaction. After all, (Q) would certainly not seem to be unanswerable. In fact, above I have offered an answer to it. If deflationism should make it impossible for its advocates to offer one as well, then this must be taken to cast doubt, not on the possibility of giving an answer to (Q), but on deflationism. In order to appreciate that (T-) does indeed stand in need of explanation it suffices to notice that norms analogous to (T-) do not hold with respect to all speech acts. The following, for example, does not express a valid norm for asking the question of whether some dogs are vicious: [It is correct to ask whether some dogs are vicious] only if some dogs are vicious.

Why, then, is it correct to assert only if there are vicious dogs, whereas the existence or non-existence of vicious dogs is irrelevant to the normative correctness of asking whether there are such dogs? If (Q) were unanswerable, then so would be this question. Claiming the latter, however, would seem to be outré, to say the least. A somewhat more subtle variant of the objection just dismissed would start from the concession that (Q) is answerable, but then go on to insist that all one can reasonably expect by way of an answer to (Q) is this: because (T-) captures an important aspect of what we mean by “correct assertion of ”. This rejoinder might go hand in hand with the consideration that (T-) can be construed as part of an implicit definition of the concept of correct assertion, i.e. as part of a definition that would comprise, among many others, all sentences of the form “it is correct to assert

only if p”. Appeal to the concept of truth would thus be avoided, and this can be expected to make the rejoinder under discussion attractive to deflationists. As far as they go, these points are fair. But in order to constitute an objection against my line of argument, they would have to be supplemented by a convincing case for the claim that something is wrong with the explanation (B) of (T-) that has been given above. Recall that the target of my argument is not deflationism per se, but (NER), i.e. a claim that many deflationists assume themselves, rightly or wrongly, to be committed to in virtue of their deflationary accounts of truth. Deflationists

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rest their commitment to (NER) on the claim that all occurrences of truth talk in explanations can be adequately accounted for in terms of the standard deflationary moves (a) and (b). I take the considerations in section 4 above to establish that this claim is false. As long as no further arguments are forthcoming—arguments, that is, to the conclusion that, despite all appearances to the contrary, the concept of truth does not play an explanatory role in (B)—(B)’s status as a counterexample to (NER) remains untouched. (4) In (A+) and in (B), the predicate “is true” is involved in sentences which are multiply embedded. In (B), for instance, its first occurrence is in the antecedent of a conditional, its second in the consequent of another conditional. To complicate matters, the two conditionals in which the truth predicate occurs are the conjuncts of a conjunction which, in turn, is embedded in a non-extensional compound: the conjunction is the explanans-clause of a “because”-sentence. The line of argument presented above repeatedly relies on the claim that the role of the truth talk involved in (A+) and (B) is not accessible to standard deflationary treatment in terms of (a)—the fact that it is not accessible to (b) can be ignored in the present context. Now, a deflationist might concede this claim but complain that it cannot be used in the way suggested above, since the failure of (a) in accounting for the role of the truth talk involved in (B) is simply due to the fact that the sentential connective “because” generates contexts which are not transparent. Indeed, many deflationists restrict the intersubstitutivity of a given revealing truth ascription, R, and the sentences expressing the propositions that truth is ascribed to in R, to extensional contexts (see, for instance, Field 2008, p. 210; Armour-Garb and Woodbridge 2010, p. 66). Therefore, or so the rejoinder would have to go, (B) cannot be used to show that there is anything wrong with the deflationist’s interpretation of and commitment to (NER). But this objection can be resisted for several reasons. In the first place, recall that (NER) is a thesis about the role of the concept of truth in explanations and that one paradigmatic way of expressing explanations is by means of “because”-statements. So it seems legitimate to expect (NER) to cover the claim that the conceptual contribution which the truth predicate can make to the contents expressed by explanatory “because”statements is itself never explanatory. If that expectation is warranted, then a restriction of (a) to extensional contexts has the effect of rendering useless the deflationist’s main argument for (NER), i.e. (DeflatioNER). Notice, moreover, that there is nothing wrong with logically complex explanantia per se. We use them all the time.

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More importantly, we can evacuate the sentences whose conjunction constitutes the explanans-clause of (B) from their opaque context and write them down as premises in a little argument that has (T-) as its conclusion: 1. is true only if some dogs are vicious, 2. (T+), therefore: 3. (T-). Next, we can say of this argument that it is (part of) the best available explanation of (T-) and run the reasoning from section 4 with respect to what is expressed by the sentences 1. and 2.—which, of course, are no longer embedded in a non-extensional context. The upshot of this line of reasoning is the same as that of section 4 above. The deflationist cannot offer the argument as an explanation of (T-) because, while maybe being entitled to the first premise, she is not entitled to the second, i.e. to (T+). More importantly, the role of the truth predicate in premise 2. cannot be accounted for along the lines of (a)—restricted to extensional contexts or not. (5) The last rejoinder that I want to anticipate would try to make good on the claim that deflationists should remain unmoved by all that has been said here concerning the explanation of (T-) in terms of truth—because what they are committed to is not (NER), but rather: (NERcausal) The concept of truth has no explanatory role to play in causal explanations.9

Now, the explanations (A+) and (B) of (T-) are certainly not causal explanations. Maybe they can best be described as offering conceptual explanations of (T-) (see Schnieder 2006, pp. 31-35; 2011). Commitment to (NERcausal) is, of course, compatible with granting the notion of truth any non-causal explanatory role whatsoever. But if a deflationist were to insist that all she wanted to deny is that the notion of truth has causalexplanatory import, it would be difficult to help the impression that she betrays her cause. After all, deflationists take the thesis that the notion of truth is explanatorily inert to be something that distinguishes their accounts from traditional, non-deflationary approaches to truth. (NERcausal), however, expresses a claim that many advocates of traditional correspondence or coherence theories, say, will find utterly unobjectionable. This suggests that deflationists, even when they explicitly endorse only (NERcausal), have a broader thesis in mind, one that covers but is not exhausted by the claim that the notion of truth has no explanatory role to play in causal explanations.

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6. Concluding remarks I have claimed that (B) can be used to show that the argument (DeflatioNER), the driving force behind the deflationist’s commitment to (NER), is unsound. (B) can be used to show this because it is a counterexample to (DeflatioNER)’s second premise. Furthermore, I have claimed that (B) should be accepted as a counterexample to (NER). It is hard to see how the conceptual contribution that the truth predicate makes to the explanatory content expressed by (B)’s explanans-clause could be understood as anything other than itself explanatory. In fact, the truth predicate is precisely what makes the content expressed by (B) an explanation of (T-). Assume, for the sake of argument, that these claims are correct. Should we then also take (B) to amount to a general refutation of the deflationist outlook on truth, along the lines suggested by Williams (2002, p. 157)? Is (NER) really that central to deflationism? The question cannot be dealt with at any useful length here. Notice, however, that while (NER) is central to many deflationary accounts there is another, equally central, deflationary thesis which would seem to be perfectly compatible with accepting (B) as a counterexample to both (DeflatioNER)’s second premise and (NER): the claim that, as Bar-On and Simmons couch it, “there is no substantive property of truth shared by all and only [those] things we (properly) call true” (Bar-On and Simmons 2007, p. 83). Nothing in the line of argument given above suggests that, for the concept of truth to be able to do its explanatory work in the explanation of (T-), truth has to be a substantial property in the sense hinted at (and rejected) by Bar-On and Simmons. The only metaphysical assumption—the only one concerning the question of what truth is, the nature of truth—that is at work in the preceding line of argument is that truth is a property. This assumption is consistent with what most contemporary deflationists (apart from some prosententialists) have to say, by way of metaphysics, about the nature of truth. What my argument casts doubt on is not the deflationist’s metaphysics of truth but her proposed account of the concept of truth. Accepting an indispensable explanatory role for that concept in the context of understanding and explaining (T-) is consistent with conceding everything the deflationist has to say about the dispensability of truth talk in stating or expressing the normative content of (T+) (for a similar point see Bar-On and Simmons 2007). In expressing the normative point of (T+) all reference to the truth or untruth of can be skipped. We can express that content by means of (T-). But in explaining why (T-) holds we cannot skip reference to the truth or untruth of . Contrary to what

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many deflationists suggest, a good case can be made for the claim that appeal to ’s capacity to (fail to) exemplify the property of truth is indispensable in explaining why the relevance of the existence or nonexistence of vicious dogs is relevant to the normative status of assertions of . To be sure, the explanation of (T-) in terms of (B) involves the concept of truth in a rather local explanatory project. But it is an explanatory project none the less. So if (B) is a counterexample to (NER), then many deflationists should reshuffle their commitments.

References Armour-Garb, B. 2012, “Challenges to Deflationary Theories of Truth”, Philosophy Compass 7 (4), 256 Armour-Garb, B. and J.A. Woodbridge 2010, “Why Deflationists Should Be Pretense Theorists (and Perhaps Already Are)”, in Wright, C.D. and N.J.L.L. Pedersen (eds.) 2010, 59 Bar-On, D. and K. Simmons 2007, “The Use of Force Against Deflationism: Assertion and Truth”, in Greimann D. and G. Siegwart (eds.) 2007, Truth and Speech Acts, London: Routledge, 61 Blackburn, S. 2013, “Deflationism, Pluralism, Expressivism, Pragmatism”, in Pedersen, N.J.L.L. and C.D. Wright (eds.) 2013, 263 Brandom, R.B. 2002, “Explanatory vs. Expressive Deflationism About Truth”, in Schantz, R. (ed.) 2002, 103 Damnjanovic, N. 2005, “Deflationism and the Success Argument”, The Philosophical Quarterly 55 (218), 53 —. 2010, “New Wave Deflationism”, in Wright, C.D. and N.J.L.L. Pedersen (eds.) 2010, 45 Dodd, J. 2013, “Deflationism Trumps Pluralism”, in Pedersen, N.J.L.L. and C.D. Wright (eds.) 2013, 298 Field, H. 2001, Truth and the Absence of Fact, Oxford: Clarendon Press —. 2008, Saving Truth from Paradox, Oxford: Oxford University Press Grover, D. 2002, “On Locating Our Interest in Truth”, in Schantz, R. (ed.) 2002, 120 Horsten, L. 2009, “Levity”, Mind 118 (471), 555 —. 2010, “On a Necessary Use of Truth in Epistemology”, in Czarnecki, T. et. al. (eds.), The Analytical Way, London: College Publications, 371 —. 2011, The Tarskian Turn. Deflationism and Axiomatic Truth, Cambridge, MA: MIT Press Horwich, P. 1998, Truth, 2nd edition, Oxford: Oxford University Press —. 2001, “A Defense of Minimalism”, Synthèse 126 (1/2), 149 —. 2002, “Norms of Truth and Meaning”, in Schantz, R. (ed.) 2002, 133

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—. 2010, “What Is Truth?”, in Horwich, P., Truth—Meaning—Reality, Oxford, New York: Oxford University Press, 1 Jenkins, C.S. 2008, “Romeo, René, and the Reasons Why: What Explanation Is”, Proceedings of the Aristotelian Society 108 (1), 61 Künne, W. 2003, Conceptions of Truth, Oxford: Oxford University Press Pedersen, N.J.L.L. and C.D. Wright (eds.) 2013, Truth and Pluralism: Current Debates, Oxford: Oxford University Press Schantz, R. (ed.) 2002, What is Truth?, Berlin, New York: de Gruyter Schnieder, B. 2006, “Truth-Making without Truth-Makers”, Synthèse 152, 21 —. 2011, “A Logic for ‘Because’”, The Review of Symbolic Logic 4 (3), 445 Soames, S. 1999, Understanding Truth, Oxford: Oxford University Press Williams, M. 1999, “Meaning and Deflationary Truth”, The Journal of Philosophy 99 (11), 545 —. 2002, “On Some Critics of Deflationism”, in Schantz, R. (ed.) 2002, 146 —. 2007, “Meaning, Truth and Normativity”, in Greimann, D. and G. Siegwart (eds.) 2007, 377 Wright, C.D. and N.J.L.L. Pedersen (eds.) 2010, New Waves in Truth, New York: Palgrave Macmillan

Notes 1

For helpful discussions of, and comments on, previous versions of this paper I am grateful to an audience at the SIFA congress “The Answers of Philosophy” (Alghero, September 2012), to the participants of the Cogito workshop “Gaps, Gluts, and Truth” at the University of Padova (May 2013) and, in particular, to Stefano Caputo. 2 “” is short for “the proposition that some dogs are vicious” (see Horwich 1998, p. 10). Since I will make heavy use of the proposition that some dogs are vicious (even though it will nowhere be asserted in what follows), I will most of the times abbreviate further and write “” as short for “the proposition that some dogs are vicious”. 3 I take explanatory “because”-statements to have the form “EXPLANANDUM because EXPLANANS”, where the two words in small capitals are placeholders for declarative sentences. For some differences between genuinely explanatory and other “because”-statements see Schnieder (2011, p. 447). 4 To my knowledge, the last point—which might reasonably be taken as signalling a problem in deflationary accounts of the role and function of propositionally revealing truth talk—has so far not been addressed in the debate on deflationism. Note also that if the extra content of revealing truth ascriptions may sometimes be part of the relevant content, then an analogous point should be expected to hold for

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unrevealing contexts of truth talk as well. This may spell trouble for the “exhaustively” in (b). For what follows, however, these observations can be set aside. 5 See, for instance, Field (2001, p. 153): “[T]here is nothing in deflationism that prevents the use of ‘true’ in explanations as long as its only role there is as a device for generalisation”. See also Williams (2002, 2007). 6 Arguments along these lines can, for instance, be found in Horwich (2001, 2002), Dodd (2013), Williams (2002) and Blackburn (2013). 7 Here is a representative statement by Horwich (2001, p. 160): “Clearly our commitments to norms like this one [to norms like (T-), B.R.] have nothing to do with the concept of truth; for that concept is completely absent from their articulation.” 8 A variant of the same move, along disquotationalist lines, can be found in Field (2001, pp. 120-121). 9 A no-explanatory-role claim along the lines of (NERcausal) is made in Field (2001, p. 29, p. 152, p. 173).

CHAPTER THREE FAULTLESS DISAGREEMENT AND THE EQUAL VALIDITY PARADOX ANNALISA COLIVA AND SEBASTIANO MORUZZI

1. Relativism and the Equal Validity Paradox When it comes to disputes of inclination–i.e. disputes about what is tasty, beautiful and morally right–we seem to witness the existence of cases where contrary opinions are in good standing and subjects aren’t at fault–this phenomenon is known in the philosophical literature as “faultless disagreement” (see Kölbel 2003). A well-known problem, perhaps the main difficulty, for any account of the concept of faultless disagreement is that a genuine dispute seems incompatible with the idea of equal validity1–namely with the idea that the opinions involved in the dispute are equally valid. Suppose one maintains P and the other not-P. The Law of Non-Contradiction2 (henceforth “LNC”) tells us that they can’t both be right (or wrong). Hence any account that is meant to take seriously the idea of faultless disagreement cannot be coherently formulated. Let 's call this problem the Equal Validity Paradox. To get clear about the structure of the paradox, let's illustrate it in details by means of the following utterances: Mary: “Ginger is tasty”. Jane: “Ginger is not tasty”.

Call these the yes-no utterances, and let's conceive an utterance as an ordered couple where s is the uttered sentence and c the context of use of the sentence relevant for the utterance. The yes-no-utterances are the class of those utterances expressing disputes with the appearance of faultless disagreement. Intuitions related to the subjective character of

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taste discourse–i.e. to the fact that the yes-no utterances give rise to the appearance of faultless disagreement–suggest that these utterances must be equally valid. Suppose their equal validity entails that their semantic status is on a par. The hypothesis that they are both false doesn’t seem relevant here: the notion of equal validity seems to exclude the falsity of these utterances since any faultless assertion requires the truth of the asserted content. Hence if their semantic status is on a par and if they aren’t both false, it seems to follow that they are both true. Suppose thus that they are both true. Given the schema connecting utterance truth and propositional truth–if an utterance says that p then is true only if p is true in the circumstances of evaluation of the context of utterance c –, and given what the yes-no utterances say, the proposition that ginger is tasty and the proposition that ginger is not tasty are both true in the relevant circumstances of evaluation. Now given that these contents are taken to represent those aspects of reality against which the proposition is evaluated and given that both contexts seem to involve the same representationally relevant aspects–i.e. the same possible world–we can infer that the two utterances are related to the same circumstances of evaluation. Hence the proposition that ginger is tasty and the proposition that ginger is not tasty are both true in the same circumstances. However, by LNC these two latter propositions cannot both be true in the same world, hence, by reductio, we can conclude that, despite the equal validity intuition, the yes-no utterances cannot both be equally valid. Conclusion: the equal validity intuition cannot find a coherent formulation when it comes to the yes-no-utterances involved in a dispute of inclination.3 4 Before presenting the argument in a more detailed form, let's make some notational stipulations. The expressions “” and “” denote two utterances constituting an arbitrary instance of yes-no utterances. We employ two notions of truth: absolute utterance truth and relative propositional truth. To express utterance truth we use the predicate “x is true” that applies to utterances; this predicate is meant to express the property of an utterance of being true simpliciter. To express relative propositional truth, we use the relational expression “x is true relative to y”; this relational expression expresses the truth of propositions relative to the circumstances of evaluation. Sometimes we use the expression “circ()” to identify the circumstances relevant for this notion of relative propositional truth. Very simply, the expression “circ()” denotes the circumstances of evaluation relevant for the truth of the utterance . Hence “p is true relative to ” means that the proposition p is true relative to the circumstances of evaluation identified by the utterance . Given these stipulations, here is the formal

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structure of the Equal Validity Paradox:5 (1) and are equally valid (Ass. - Equal Validity). 2 (2) If and are equally valid, then and have the same semantic status (Ass.–Semantic Equal Validity). 3 (3) If and have the same semantic status, then they are both true (Ass–Truthfulness). 1,2,3 (4) and are both true (Modus ponens:6 1,2,3). 5 (5a) If expresses a proposition–p–then ( is true only if p is true in circ() ) (Ass–Utterance-Propositional Truth Schema). 5 (5b) If expresses a proposition–q–then ( is true only if q is true in circ() ) (Ass–Utterance-Propositional Truth Schema). 6 (6a) expresses a proposition–p (Ass- Propositionality). 6 (6b) expresses a proposition–q (Ass- Propositionality). 5,6 (7a) is true only if p is true in circ() (Modus ponens: 5a, 6a). 5,6 (7b) is true only if q is true in circ() (Modus ponens: 5b, 6b). 1,2,3,5,6 (8a) p is true in circ() (Modus ponens: 4, 7a). 1,2,3,5,6 (8b) q is true in circ() (Modus ponens: 4, 7b). 9 (9) The circumstances of -circ()- are the circumstances of –circ(). (Ass–Sameness of circumstances). 10 (10) The proposition that q is the proposition that not-p (Ass Contradiction). 1,2,3,5,6,9,10 (11) not-p is true in circ() (Substitutions of Identicals: 8b,9,10). 1,2,3,5,6,9,10 (12) p is true in circ() and not-p is true in circ (Introduction of conjunction: 8a, 11). 13 (13) Necessarily, for any circumstance C and proposition P, P and not-P are not both true in C (Ass - Law of Non-Contradiction). 1,2,3,5,6,9,10,13 (14) Contradiction (12,13).

The Equal Validity Paradox is thus generated by the following theses: (I) the yes-no utterances are equally valid (Equal Validity); (II) if the yes-no utterances are equally valid, then they have the same semantic status (Semantic Equal Validity); (III) if the yes-no utterances have the same semantic status, then they are both true (Truthfulness); (IV) the alethic profile of the truth property for the yes-no-utterances and for the propositions expressed is governed by a scheme connecting utterance truth to propositional truth (Utterance-Propositional Truth Scheme); (V) the yes-no utterances express propositions (Propositionality); (VI) the yes-no utterances are related to the same circumstances of evaluation (Sameness of Circumstances);

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(VII) the propositions expressed by the yes-no-utterances are contradictory (Contradiction); (VIII) these propositions cannot both be true in the same circumstances (LNC).

Before moving to the possible solutions of the paradox, let us clarify the exact nature of the paradox and its relationship with the phenomenon of faultless disagreement. First, notice that the paradox kicks start with an arbitrary instance of yes-no-utterances. Remember that we have defined these utterances as those utterances that give rise to the phenomenon of faultless disagreement. So the first feature of the paradox is that it is structurally intertwined with the phenomenon of faultless disagreement. Second, notice that the eight principles that we have listed in relation to the paradox have different natures. A first set of principles is specifically related to faultless disagreement and equal validity: Equal Validity, Truthfulness, Propositionality, Sameness of Circumstances and Contradiction belong to this first set. Equal Validity amounts to the claim that the class of utterances that give rise to the appearance faultless disagreement–i.e yesno-utterances–and that are equally valid is not empty. Semantic Equal Validity articulates the notion of equal validity for these utterances imposing a semantic reading. Truthfulness claims that if these utterances have the same semantic status they are true. Propositionality claims that these utterances are part of an area of discourse that expresses truthconditional contents–i.e. propositions. Sameness of Circumstances claims that whenever two utterances give rise to the appearance of faultless disagreement they share the same circumstances of evaluation. Contradiction claims that utterances that give rise to the appearance of faultless disagreement express contradictory propositions. The remaining two principles are totally general and not specifically related to the yes-noutterances. Utterances-Propositional Truth Schema articulates a general principle holding between utterance truth and propositional truth, and LNC expresses a property of propositions . To sum up, we can say that a the Equal Validity Paradox is an argument that generates a contradiction by assuming that the set of utterances that give rise to the appearance of faultless disagreement can be non-empty (principle I), and by constraining the properties of these utterances by means of five principles specifically related to faultless disagreement and equal validity (principles II, IV, V, VI, VII) together with two general principles about the truth properties of utterances and propositions (principles III and VIII).7

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2. Eight Ways Out of the Paradox Corresponding to the eight theses that give rise to it, there are eight strategies for blocking the Equal Validity Paradox. Each strategy provides the basis for building an account of faultless disagreement. The first strategy amounts to a revisionist approach to faultless disagreement. By denying Equal Validity, it follows that it is not the case the yes-no-utterances are equally valid. One way to implement this strategy is to hold a rampant form of realism: one utterance is true whereas the other is not true. There is thus a fact of the matter about who is right and no sense in which both disputants are entitled to hold on to their view. Realism, however, is not mandated by a revisionist approach to faultless disagreement, for in general any account that dispels the appearance of faultless disagreement by denying Equal Validity involves a revisionist approach to faultless disagreement. In fact, we will argue that all the remaining attempts to solve the Equal Validity Paradox fall back onto a revisionist approach to faultless disagreement–more on this later. The second strategy tries to solve the Equal Validity Paradox by denying Semantic Equal Validity. A natural way to implement this strategy is to take a realist approach to disputes of inclination by holding that there is a fact of the matter about who is right in these disputes; however, contrary to the revisionism of the first strategy, the realist approach is coupled with an account intended to vindicate the intuition of epistemic faultlessness. On this reading, the appearance of faultless disagreement, which is the source of the paradox, is explained with reference to the epistemic or rational standing of the thinkers and not to the semantic status of the judged propositions. The third strategy is based on the denial of Truthfulness: the yes-no utterances have the same semantic status, but neither is true. Judgements related to disputes of inclination express propositions whose semantic value takes a third value other than true and false. The fourth strategy obliterates the straightforward connection between utterance truth and propositional truth expressed by the UtterancePropositional Truth Scheme. A yes-no-utterance can be correct and can expresses a proposition, but it may be unsettled which proposition is expressed, and thus, a fortiori, it may be unsettled whether there is a proposition–expressed by the utterance–that is true at the circumstances of the context of utterance. The fifth strategy abandons the idea that yes-no-utterances express propositional contents. This strategy is typically developed by means of a non-cognitive approach to disputes of inclination such as expressivism. To

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exemplify, assertions about what is tasty do not express propositions and lack truth-conditions, moreover no attitude of belief is involved with respect to these utterances but rather a non-cognitive attitude such as, for example, gustatory appreciation. The sixth strategy is the route to a robust relativistic approach to faultless disagreement. If propositional truth is relativized to extra nonstandard parameters–e.g. standards of taste–, the same proposition can be true and false in the same world since it can be the case that two utterances in the same world identify different circumstances–e.g. different standards of taste. By denying Sameness of Circumstances two contradictory propositions can thus be both true since the circumstances of evaluation differ. The penultimate route, the seventh, exemplifies another popular strategy, usually called “indexical contextualism”. According to this strategy the yes-no-utterances do not express contradictory propositions since different contextual aspects related to the two utterances partially determine the contents which are semantically expressed. To exemplify: when Mary utters “Ginger is tasty” hidden indexical elements related to taste-aspects, e.g. the standard of taste salient for Mary, determine the expression of the proposition that ginger is tasty relative to Mary's standards. Since Jane's context of utterance selects a different standard, her utterance expresses the proposition that ginger is not tasty relative to Jane's standards, a proposition whose truth isn’t incompatible with the truth of the proposition expressed by Mary's utterance. The last route is to abandon the law of non-contradiction thus allowing the possibility that the yes-no-utterances express contradictory propositions that are nonetheless both true at the same circumstances. This route can be developed by means of the adoption of a paraconsistent account of subjective discourse that allows for the possibility of true contradictions in the actual world–i.e. dialetheism. The plan of the paper is the following. In the next sections we provide the bare-bones of each of these eight routes and we sketch the main challenges that each of these strategies face. Strategies (ii), (vi) and (vii) (realism, relativism and contextualism) are well-known options to block the Equal validity paradox, and their respective problems have been extensively explored in the relevant recent literature, we will thus be very brief in commenting on them and we will refer the reader to the relevant literature. The remaining strategies–i.e. (i), (iii), (iv), (v) and (viii)–have definitely received less attention (if any) and we will treat them more extensively since they open new interesting routes to block the Equal

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validity paradox. More specifically, strategy (i) (revisionism) allows to clarify an important, so far neglected, distinction between two basic opposite approaches to faultless disagreement: revisionary vs. descriptive approaches. Whereas a descriptive approach provides a “happy face solution”8 to the Equal Validity Paradox by identifying one (or more) faulty premise and by providing an explanation of its falsity and of its appearing otherwise, an “unhappy-face solution” to the Equal Validity Paradox reveals glitches of the concepts involved in the notion of faultless disagreement rather than mistakes in the articulation of this notion by means of some faulty premise. In fact, given that five9 of the principles involved in the Equal Validity Paradox articulate conceptual connections related to faultless disagreement, a potential reaction in relation to the paradox is to claim that the paradox shows that the concept of faultless disagreement is incoherent. Strategies (iii), (iv), (v) and (viii) require the abandonment of some orthodox principles about truth and meaning: strategy (iii) denies bivalence for utterance truth or for propositional truth by assigning a third semantic status to yes-no-utterances; strategy (iv) renounces a straightforward connection between utterance truth and propositional truth by invoking an indeterminacy in the semantic content expressed by the yes-no-utterances; strategy (v) denies that yes-no-utterances express a truth-conditional content that is the object of the attitude of belief; finally, (vi) calls for an exception to the law of non-contradiction. These latter strategies are less discussed in the recent literature on faultless disagreement, so we will provide a slightly more extensive treatment for them. Our main thesis is that each of strategies (ii)-(viii) faces the challenge of being, contrary to its official aim, a form of revisionary approach to faultless disagreement. None of the following considerations in framing the revisionist challenge for each of these strategies is of course meant to provide a knock-down argument against these proposals taken as descriptive accounts of faultless disagreement. Our aim is simply to show that there is a well-grounded suspect that these solutions to the Equal Validity Paradox cannot but end up being revisionary with respect to the phenomenon of faultless disagreement, despite their official pronouncements.

3. Revisionism Strategy (i)–i.e. the denial of the equal validity for the yes-no utterances–amounts to a revisionist approach to the problem of faultless disagreement. According to the revisionist approach disputes of inclination

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are motivated by a misguided view on the subject matter: when confronted with disputes of taste, for example, we wrongly think that both views are legitimate whereas the truth of the matter is that they are not. As a consequence, these disputes cannot be rationally sustained. At least one of the participants is wrong, even though we may not be able to tell which one is wrong. Hence, the only rationally responsible attitude would be to abstain from disputing. The Equal Validity Paradox would thus receive an unhappy-face solution, since the concept of faultless disagreement would turn out to be a defective and thus an empty concept. Of course not all empty concepts are defective. For example, it was believed that witches and ether existed, we no longer believe so now. It is plausible to claim that neither is a case of a defective concept, more simply we got the description of reality wrong. Now, according to the revisionist, the concept of faultless disagreement belongs to a different class of concepts: it is a concept that is empty in virtue of the very conceptual connections that (partly) constitute it. Given that these very same conceptual connections give rise to a contradiction–as exemplified by the Equal Validity Paradox–the concept is incoherent and thus defective and empty. Such a radical stance on the problem denies the phenomenon we wish to account for, i.e. the intuition that subjective discourse has its own specificity. Of course, such a radical departure from our practice needs an articulated motivation. Strategies (ii)-(viii) can then be seen as attempts to solve the paradox without renouncing the idea that the concept of faultless disagreement is coherent; for each of these latter strategies a descriptive account of faultless disagreement is possible–or so it seems. Before turning to the remaining solutions to the paradox it is important to stress the dialectical role of the Equal Validity Paradox with respect to the problem of faultless disagreement. Going revisionary means that the Equal Validity Paradox shows that our practice of judging and disputing about subjective domains is intrinsically misguided: no account can make sense of it as a rational practice. The paradox is thus taken as a symptom of the defectiveness of the practice because a defective concept informs it. In contrast, a descriptive account saves the phenomenon by rejecting some theoretical thesis–i.e. theses (ii)-(viii)–involved in the paradox without thereby renouncing the coherence of the practice. The challenge to any descriptive solution to the Equal Validity Paradox thus consists in the request to show that the theoretical revision invoked can indeed avoid a revisionary approach to the phenomenon. In other words, the challenge amounts to the request of showing that the rejection of any of theses (ii)-(viii) is consistent with the claim that yes-noutterances are genuine cases of faultless disagreement. As already

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mentioned, the revisionary approach can be moderate or radical. Moderate revisionism invokes a “weak unhappy-face” solution to the Equal Validity Paradox, the revision amounts to the claim that there cannot be a happyface solution, but that it is nonetheless possible to provide a suitable consistent revision of the concept of faultless disagreement. This revision can be seen, for example, as analogous to the explications invoked by Rudolf Carnap: explications are revisions of some ordinary concepts that maintain some core conceptual traits of the explicatum as being at the same time more precise and, crucially, coherent (Carnap 1956, pp. 7-8). Tarski's definition of truth can be seen an example of explication of the notion of truth capable of avoiding semantic paradoxes (Tarski 1944). Radical revisionism, in contrast, invokes a strong unhappy-face solution to the Equal Validity Paradox: like its moderate cousin it holds that there cannot be a happy-face solution, but, contrary to the optimism of moderate revisionism, it denies that a consistent revision of the notion of faultless disagreement is possible. In this paper we aim to show that the Equal Validity Paradox is a genuine aporia for any attempt–like (ii)-(viii)–to fulfil a descriptive project with respect to the phenomenon of faultless disagreement.10

4. Realism Whereas realism can be employed as a way to implement the revisionist strategy, if coupled with an epistemic reading of faultlessness the realist doctrine can be used as a descriptive account of faultless disagreement. Such a descriptive project rejects semantic equal validity by accounting for equal validity in epistemic terms without abandoning classical semantics (Schafer 2011). The difficulties for a realist treatment are well-known: the price to pay in order to preserve classical semantics is to make room for the existence of a fact of the matter on subjective questions–such as whether or not a food is tasty–when it is hard to imagine what else, if not our judgements, could determine what counts as the correct answer. More importantly: if truth is so remotely connected to our practices of judgements, it is difficult to consider these practices as rationally sustainable when it comes to the activity of disputing (Wright 2001). If the truth-value of taste propositions is determined by aspects of reality that are beyond our ken, then it is hard to see how our judgements about taste matters can be responsibly held. At best we could say that we judge blamelessly because we cannot aspire to judge from a better epistemic position. But a blameless judgement is different from a justified judgement–or so it seems. So the challenge for the realist is to explain how

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propositions belonging to subjective discourse get their truth-value determined and why our judgements about them seem rational and justified.

5. Analetheism Strategy (iii) takes a different route to the solution of the Equal Validity Paradox by denying the truthfulness of the yes-no utterances. So yes-noutterances are not true but have nonetheless the same value. As we saw at the outset, if both utterances are false, there seems to be no prospect of making sense of the idea of faultlessness in semantic terms. Hence this route is committed to the idea these utterances enjoy a third semantic value different (and incompatible) with the polar values of truth and falsity. The attribution of a third semantic value is meant to be the expression of the thought that subjective discourse is semantically indeterminate and that the appearance of faultlessness springs from this indeterminacy. One way to develop the idea is to claim that the meaning of expressions featuring in subjective areas of discourse, together with nonsemantic facts, underdetermines their extensions: the thoughts and practices of competent speakers and non-linguistic facts aren’t enough to determine whether “Ginger is tasty” is true or false. A natural way to model this idea is to uphold a non-classical semantics that evaluates yesno-utterances as neither true nor false. Following an entrenched terminology11 we can call this strategy: analetheism (Beall 2006). A major problem for the analetheist strategy is the potential tension between the thesis of semantic under-determination and the thesis of equal validity. In fact, if we stick to the orthodox view that we ought to assert and believe only what is true, yes-no utterances (and their respective beliefs) would count as incorrect. But then it becomes unclear in what sense, if any, both views on a dispute of inclination are equally valid if they are both incorrect–the threat of ending up being revisionary thus emerges also for option (iii).12

6. Semantic indeterminacy Strategy (iv) is difficult to assess since the link between utterance truth and propositional truth is hard to deny. Perhaps, one way to make sense of this strategy is to maintain that an utterance can semantically express more than one proposition leaving indeterminate which one it actually expresses.13 It can thus be the case that there are at least two propositions that could be expressed by the utterance such that one is false while the

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other is true. The idea behind this strategy is thus the following: when Mary utters “Ginger is tasty” there are several candidate propositions that might be expressed and Mary's utterance is true when at least one of these propositions is true–call this proposition P. Notice that this claim does not amount to saying that no proposition is expressed (hence (6a) and (6b) hold good), rather there is no fact of the matter as to which proposition is expressed. Assuming the existence of a proposition that is true among the candidate propositions and the thesis that an utterance is correct when at least one of these candidate propositions is true,14 it follows that while it is true that the utterance expresses a proposition, it is not true that it expresses a specific true proposition since there is no fact of the matter about which proposition the utterance expresses–hence the failure of (5a) and (5b). Notice the difference between the indeterminacy invoked by strategy (iii) and the one adopted by strategy (iv): whereas the former is an indeterminacy regarding the truth-value of the proposition expressed, the latter is an indeterminacy concerning which proposition is expressed. Borrowing a piece of terminology from Eklund (2010) we can call the former first-level indeterminacy and the latter second-level indeterminacy. While analetheists claim that meaning facts, together with non-linguistic facts, determine a third semantic value for the sentence (and hence for the proposition expressed) since there is a best way to assign a semantic value–i.e. neither truth nor false–to the sentence; the follower of strategy (iv) claims instead that meaning facts do not determinate which proposition is expressed since there is no best way of assigning a semantic value to the sentence, for different ways of assigning a truth-value are admissible. One basic worry concerning this sophisticated strategy is that it has difficulties in making sense of the faultlessness of the beliefs of two thinkers engaged in a dispute of inclination. For, what do they believe according to this proposal? If we stick to the standard view that belief is a relation between a thinker and a proposition, since there is no proposition determinately expressed there is also no determinate belief, but if the belief has no determinate content, how can it be faultless to have it? It seems that the appropriate stance to have in this case is to withhold belief since any attempt to believe would fail to put the thinker in relation to a proposition among the available candidates. Agnosticism seems thus the mandated attitude in disputes of inclination, and thus the proposal seems to fall back onto a form of revisionism.

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7. Non cognitivism Strategy (v) rejects the idea that yes-no utterances express genuine propositions. This option falls naturally under a non-cognitivist account of subjective discourse. Traditionally, non cognitivism used to involve the semantic thesis according to which the targeted discourse isn’t truth-apt and the psychological thesis that the attitudes expressed by utterances in this area of discourse do not express belief but some other kind of attitude. More recent non cognitivist theories have weakened the traditional thesis allowing for the truth-aptness of the targeted discourse by means of a deflationary reading of truth15, and by allowing that, though the primary role of utterances is not to express belief, they can express beliefs as a secondary function. Historically expressivism has been the preferred route for escaping the Equal validity paradox: given that no genuine proposition is involved, there is no problem in making sense of the truth-conditions of yes-no utterances. So, in principle, a non-cognitivist could block the Equal Validity Paradox by denying Propositionality, while maintaining, at the same time, all other principles including Truthfulness–provided a deflationist understanding of truth is upheld.16 Well known problems afflict the non-cognitivist route. First and foremost the so-called Frege-Geach problem, namely the challenge of accounting for the validity of deductive arguments without using a truth-conditional semantics.17 A related problem, more pressing for any non-cognitivist account of faultless disagreement, is the possibility to express different embeddings of the negation operator relative to an expression of attitude.18 Here is a way to present the problem.19 Suppose the expressivist semantic analysis of “Ginger is tasty” is explained by the fact that it expresses gustatory appreciation: Ginger is tasty iff APP(ginger),

where “APP” expresses that attitude of gustatory appreciation. How should the expressivist analyse “Not-ginger is tasty”? Following the same line of analysis, it should be analysed as the gustatory appreciation of any food other than ginger: Not-ginger is tasty iff APP(not-ginger).

Now consider the sentence “Ginger is not tasty”, which is relevant for the analysis of the yes-no-utterances. If we follow the non-cognitivist story, this sentence expresses an attitude different from belief, but which one?

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Ginger is not tasty iff ?(ginger).

It does not seem that any attitude of appreciation can fill the gap. Hence “Ginger is not tasty” does not seem to express a state of appreciation. Since “Ginger is tasty” is inconsistent with “Ginger is not tasty”, it follows that the non-cognitivist cannot account for this inconsistency by means of the same type of conative attitude. It could be replied that we can find a second conative attitude to account for the inconsistency, for example, the attitude of disgust. “Ginger is not tasty” is thus analysed as the expression of disgust towards ginger: Ginger is not tasty iff DISG(ginger),

where “DISG” expresses the attitude of gustatory disgust. The non cognitivist can thus redeem the inconsistency by means of the idea of the incompatibility between two attitudes. Two problems affect this solution, however. The first one is that it is debatable to account for a logical notion such as inconsistency by means of a psychological notion such as the impossibility of having simultaneously gustatory appreciation and gustatory disgust towards a certain food. The second problem is that this way of accounting for the inconsistency commits the non-cognitivist to the existence of an infinity of attitudes. This latter fact stems from two facts: i) the fact that for each subjective predicate she needs to postulate ad hoc attitudes just like she has done for “tasty”; ii) the negation problem can be generalized. To explain this latter point we can represent the non-cognitivist solution to the negation as the postulation of a new attitude that would allow the noncognitivist to express these three different acceptances: Mary accepts that ginger is tasty iff Mary accepts APP(ginger). Mary accepts that not-ginger is tasty iff Mary accepts APP(not-ginger). Mary accepts that ginger is not tasty iff Mary accepts DISG(ginger).

Now, what about conjunction? Mary accepts that ginger is tasty and Mary accept that rhubarb is tasty iff Mary accepts APP(ginger) and Mary accepts APP(rhubarb). Mary accepts that ginger is tasty and that rhubarb is tasty iff Mary accepts? Mary accepts that ginger and rhubarb are tasty iff Mary accepts APP(ginger and rhubarb).

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The non-cognitivist analysis lacks enough structure to mirror the differences in the syntactic structure of the three sentences. To fill in the gap it seems she needs to postulate a new attitude related to the conjunction, say “APP&”. The pattern generalizes with all logical connectives; and given that logical connectives combine each other to give rise to more logically complex sentences, the dimension of the set of the postulated new attitudes explodes.20 To sum up, the challenge for any non-cognitivist solution to the Equal Validity Paradox is to abandon Propositionality without losing a grip on the notion of inconsistency and thus on that of disagreement. Without any account of why yes-no-utterances are cases of disagreement, noncognitivists fail to offer a descriptive account of faultless disagreement, thus falling back onto revisionism.21

8. Relativism Of the remaining ways out of the paradox, two of them–rejection of theses (VI) or (VII)–are well-known in the literature and for each of these routes there are well-founded doubts that they can actually avoid a revisionary approach to faultless disagreement. Rejection of Sameness of Circumstaces (strategy (vi)) involves some non-standard extra parametrization of the relation of propositional truth. The basic idea is that the truth of propositions involved in disputes of inclination requires some extra parameters in addition to possible worlds. The circumstances of evaluations involved in taste discourse, for example, involve standards of taste so that the proposition that ginger is tasty can be true relatively to Mary's standards of taste and false relatively to Jane's standards.22 The basic problem for this position is to account for the appearance of disagreement since a difference in the circumstances of evaluation seems to involve a difference in the what the yes-no utterances are meant to be about. If, as it seems, two utterances concerning the very same proposition with respect to different possible worlds cannot constitute a case of disagreement, it seems to follow that, if yes-no-utterances concern different circumstances of evaluation for assessing the same proposition, they cannot involve a genuine form of disagreement.23 But if no genuine disagreement is in place, it seems that the only way to explain the dispute is that subjects fail to realize what the actual circumstances of evaluation relevant for their utterances are. Hence, they fail to know the actual truthconditions of what they say. Assuming a truth-conditional account for meaning, they ignore the meaning of their utterances. The relativist strategy is then committed to a thesis of semantic blindness with respect to

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yes-no-utterances. It follows that the relativist strategy cannot offer a descriptive account of faultless disagreement, and that it is committed to a revisionary approach to the problem.

9. Contextualism Indexical contextualists deny that the same proposition is involved in disputes of inclination and thus uphold strategy (vii). According to them, the semantic content of evaluative expressions such as “tasty” is sensitive to the context of use. When Mary utters “Ginger is tasty” she is actually expressing the proposition that ginger is tasty relative to her standards of taste, whereas Jane is saying that ginger is not tasty relative to her standards, hence they are expressing compatible contents just like when two different people utter “I am cold” and “I am not cold”. Whether or not the contextualist semantic story is credible, the most pressing problem for this strategy is the difficulty of making sense of the idea of disagreement when the yes-no utterances are taken to express compatible contents. But if no genuine disagreement is in place, it seems that the only story that can explain a dispute is that subjects fail to realize what the relevant propositions expressed by their utterances are. Hence they fail to know the truth-conditions of what they say and, assuming a truth-conditional account of meaning, they end up ignoring the meaning of their utterances. Hence contextualists are in the same ballpark as relativists in being committed to a thesis of semantic blindness with respect to the yes-noutterances. It follows that also contextualism cannot offer a descriptive account of faultless disagreement, and that it is committed to a revisionary approach to the problem.24

10. Dialetheism Lastly, strategy (viii) calls for an exception to the law of noncontradiction: yes-no-utterances express propositions that are both true and false, hence being also true, they are both correct. The main problem for this option is to offer an account of assertion that explains the appearance of disagreement for the yes-no-utterances. A well-known way of making sense of the abandonment of LNC is the adoption of a paraconsistent logic. Paraconsistent systems deny the socalled Explosion principle (aka “Scoto’s law” or “Ex falso quodlibet”) according to which from a contradiction any proposition follows, thus allowing one to devise ways to “contain” or “limit” the effects of the contradiction. Paraconsistent systems can be broadly divided into two

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families: weak and strong ones.25 Weak paraconsistent logics do not allow for true contradictions, they simply tell us how to behave when confronted with a theory that contains a contradiction in order not to discard it completely (e.g. subvaluationism). Strong paraconsistent logics allow for the truth of contradictions, hence they properly deny LNC (e.g. dialethesim). Given that any account of faultless disagreement must deal with the truth of actual or possible utterances of contradictory propositions, strong paraconsistency seems required. Let's call this proposal dialetheist strategy. Whereas the driving thought of the analetheist strategy (strategy (iii)) is that subjective discourse is a case where yes-no-utterances take a third semantic value distinct from the polar values of truth and falsity, the dialetheist proposes a dual approach opting for idea that subjective discourse is a case where yes-no-utterances take both polar semantic values: it is both true and false that ginger is tasty, hence LNC fails for the subjective domain. Dialetheism allows one to say that in a dispute where A asserts P and B denies P, A’s and B’s opposite judgments are actually contradictory, so no hidden or extra parameters are invoked to relativize yes-no utterance truth (as opposed to contextualism or truth-relativism); moreover it allows one to maintain that A’s and B’s judgments are equally valid (true contradictions). Since the dialetheist does not face the problem of making sense of the idea that a contradiction is involved in a dispute on faultless disagreement, she avoids the problems that contextualist proposals face, while distancing herself also from truth-relativism by succeeding in making sense of equal validity. After all, truth-relativists have no way to make sense of the idea that relative to the same parameter, to which truth is relativized, both opinions are equally valid. However, the dialetheist strategy faces a certain number of problems. In the following we will raise one we consider more pressing: the absence of any relativisation opens the way to a revisionary challenge to the proposal. Consider again our yes-utterances: Mary says “Ginger is tasty” and Jane says “Ginger is not tasty”. Let us apply dialetheism to them. Incompatibility between these utterances would fail: both parties should admit that also their opponent is right, not just from his own point of view, but tout court. Hence, they couldn’t preserve disagreement and should thus admit that both contradictory propositions are true. They should therefore cease to quarrel since they should both recognize that also the other party is right. The dialetheist strategy thus collapses into a form of revisionary account of faultless disagreement instead of being a form of descriptive account.

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This latter argument can be presented as an elaboration of a familiar problem for dialetheism (Parsons 1990; Batens 1990 and Priest 2006b p. 106): ARGUMENT 1 1) Suppose P is a dialetheia–i.e. P is both true and false (and A and B have evidence for this). 2) Suppose A asserts P. 3) B’s typical way of disputing with A is by asserting not-P. 4) Yet if we accept dialetheism, B’s assertion of not-P does not prevent her from also accepting P (if P is a dialetheia, P and its negation are both true). 5) Incompatibility between A and B’s judgments is lost. 6) The dispute between A and B on P is not rationally sustainable.

In a nutshell the problem stemming from the unrelativized nature of the dialetheia is that once a proposition P is a dialetheia the mutual correctness of the acceptance of the proposition and of its negation does not seem to leave room for any substantial disagreement over P: both accepting P and rejecting P (meant as accepting not-P) are correct. So opposite attitudes over P do not seem to motivate any rational dispute. A possible way out of this problem is to distinguish four possible notions (Priest 2006a, pp.96-99; 2006b, p. 103): Acceptance of P = mental state of believing P. Assertion of P = the speech act expressing the act of accepting P (stronger than agnosticism). Rejection of P = mental state of refusing to believe P. Denial of P = the speech act expressing the act of rejecting P.

Here is how Graham Priest explains these distinctions: Someone who rejects A cannot simultaneously accept it any more than a person can simultaneously catch a bus and miss it, or win a game of chess and lose it. If a person is asked whether or not A, he can of course say “Yes and no”. However this does not show that he both accepts and rejects A. It means that he accepts both A and its negation. Moreover a person can alternate between accepting and rejecting a claim. He can also be undecided as to which to do. But do both he cannot (Priest 1989, p. 618).

Hence acceptance and rejection are exclusive (but not exhaustive, agnosticism is always a third possible stance). Familiar examples in which these distinctions are applied are gappy (i.e. neither true nor false) sentences: given that a sentence can be untrue without being false, having ground for the denial of P and not-P does not ground, respectively,

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acceptance of not-P and of P. And, importantly, glutty (i.e. both true and false) sentences: given that a sentence can be false without being untrue, having ground for the acceptance of P and not-P does not ground, respectively, denial of not-P and of P. Priest’s point (Vs Frege and Geach) is thus that denying P does not always involve asserting not-P. Now let’s assume this notion of rejection, at least for the sake of the argument, though it is unclear what the information that it is conveyed by a subject’s denial is (Grim 2004; Berto 2008). A way to express incompatibility between attitudes can then be regained: A’s acceptance of P excludes the correctness of B's rejection of P. Given this distinction, we can state three different normative principles for rationality (Priest 2006b, p.110): (Accept-T) One ought rationally to accept P if there is good evidence for the truth of P. (Accept-F) One ought rationally to accept not-P if there is good evidence for the falsity of P. (Reject-T) One ought rationally to reject P if there is good evidence for the untruth of P.26

Given these principles, can the challenge of collapsing onto revisionary relativism be met? Unfortunately, it seems that the argument for the collapse of dialetheist strategy into revisionary strategy can still be presented as an extension of the familiar problem we have encountered before: ARGUMENT 2 1) Suppose P is a dialetheia–i.e. P is both true and false (and A and B have evidence for this). 2) Suppose A asserts P. 3) B’s way of disputing with A is by rejecting P. 4) B’s rejection of P prevents her from also accepting P (Incompatibility satisfied). 5) But since P is a dialetheia, B is wrong in rejecting P and she ought rationally not to reject it (ditto for A). 6)The dispute between A and B on P is neither faultless nor rationally sustainable.

The conclusion of argument 2 is thus worse than the conclusion of argument 1: beside being unable to make sense of the rationality of the dispute, B's rejection of P is actually incorrect. Hence, faultlessness is lost. Therefore, a dispute on matters of inclination cannot be presented as involving rejection. However, the mere acceptance of P and of its negation

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does not justify any sense of genuine disagreement between subjects. In conclusion, arguments 1 and 2 set up a dilemma for the dialetheist strategy: either a dispute between A and B involves A's acceptance of a proposition and B's acceptance of its negation without rejecting the proposition accepted by A; or else, it involves A's acceptance of a proposition opposed to B's rejection of it. If the former, the incompatibility between the correctness of subjects' attitude is lost and hence, if the dispute is rational, ignorance of the fact that the proposition is a dialetheia must be then imputed to subjects (they mistakenly take acceptance of the negation of the proposition to be equivalent to the rejection of it27). If, on the other hand, the dispute between A and B involves A's acceptance and B's rejection of a proposition, B's rejection could be rational only insofar as she ignored that P is a dialetheia, for recognition that P is a dialetheia involves recognition that both P and not- P are true and hence that rejection of P and rejection of not-P are both incorrect.28 In both cases the idea that the dispute is rationally conducted and sustainable is at odds with the assumption that subjects have evidence that the relevant proposition is a dialetheia. So the dialetheist strategy is committed to a form of revisionism about faultless disagreement.

11. Conclusions The idea that disputes related to subjective domains, expressed by the yes-no-utterances, are genuine cases of faultless disagreement leads to the Equal Validity Paradox. We have identified eight ways out of this paradox. Whereas one strategy (strategy (i)) is explicitly a revisionary account of faultless disagreement–i.e. an account that denies that the notion of faultless disagreement can have a non-empty extension–, the remaining seven strategies aspire to solve the Equal Validity Paradox by providing a descriptive account of faultless disagreement–i.e. an account that does justice to the idea that some disputes are genuine cases of faultless disagreement. However, we have argued that this latter aspiration is in danger of being frustrated by the fact that each of these seven putative descriptive accounts of faultless disagreement, once pressed, seem to face revisionary challenges. If these revisionary challenges cannot ultimately be met, the appearance of faultless disagreement must be recognized to be a an illusion we must free ourselves from. The illusion has indeed proved to be so powerful as to motivate an array of descriptive accounts each of which has served as agit prop for the preferred flamboyant metaphysical and semantic idiosyncrasies. Perhaps more sophisticated developments of

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these putatively descriptive accounts may avoid the revisionary challenges proposed thus far29. Perhaps not. Further reflection and debate will show whether our allegations will prove to be right. For now we think we have formulated a clear and significant challenge for any descriptive approach to faultless disagreement.

References Batens, D. 1990, “Against Global Paraconsistency”, Studies in Soviet Thought 39 (3-4), 209 Beall, J.C. 2006, “Modelling the ‫ޖ‬Ordinary View'” in Greenough P. and M. Lynch (eds.), Truth and Relativism, Oxford: Oxford University Press, 61 Beall, J. and D. Ripley 2004, “Analetheism and dialetheism”, Analysis 64 (281), 30 Berto, F. 2007, How to Sell a Contradiction: The Logic and Metaphysics of Inconsistency, London: College Publications —. 2008, “Adynaton and Material Exclusion”, Australasian Journal of Philosophy 86 (2), 165 Boghossian, P. 2006, Fear of Knowledge: Against Relativism and Constructivism, Oxford: Oxford University Press Carnap, R. 1956 (1947), Meaning and Necessity, University of Chicago Press, 2nd ed. Coliva, A. and S. Moruzzi ms-1, “Lost in disagreement?”, manuscript Coliva, A. and S. Moruzzi ms-2, “Lost in contradiction?”, manuscript Dorr, C. 2002, “Noncognitivism and Wishful Thinking”, Nous 36 (1), 97 Dreier, J. 2009, “Relativism (and Expressivism) and the Problem of Disagreement”, Philosophical Perspectives 23 (1), 79 Eklund, M. 2010 “Vagueness and Second-Level Indeterminacy” in Dietz R. and S. Moruzzi (eds.) 2010, Cuts and Clouds, Oxford: Oxford University Press Gibbard, A. 2003, Thinking How to Live, Cambridge, MA: Harvard University Press Greenough, P.M. 2011, “Truthmaker Gaps and the No-No Paradox”, Philosophy and Phenomenological Research 82 (3), 547 Grim, P. 2004, “What is a Contradiction?” in Priest, G., J. C. Beall and B. Armour-Garb (eds.) 2004, The Law of Non-Contradiction: New Philosophical Essays, Oxford: Oxford University Press, 49 Huevenes, T.T. 2014, “Disagreement Without Error”, Erkenntnis 79 (1),143 Horgan, T. and M. Timmons 2006, “Cognitivist Expressivism” in Horgan

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T. and M. Timmons (eds.) 2006, Metaethics after Moore, Oxford: Oxford University Press, 255 Kölbel, M. 2003, “Faultless Disagreement”, Proceedings of the Aristotelian Society 104 (1), 53 —. 2004, “Indexical Relativism Versus Genuine Relativism”, International Journal of Philosophical Studies 12 (3), 297 López De Sa, D. 2008, “Presuppositions of Commonality” in GarcíaCarpintero M. and M. Kölbel (eds.) Relative Truth, Oxford: Oxford University Press, 297 MacFarlane, J. 2005, “Making Sense of Relative Truth”, Proceedings of the Aristotelian Society 105 (3), 321 —. 2007, “Relativism and Disagreement”, Philosophical Studies 132 (1), 17 Mares, E.D. 2004, “Semantic Dialetheism”, in Priest, G., J. C. Beall and B. Armour-Garb (eds.) The Law of Non-Contradiction, Oxford: Oxford University Press, 276 Marques, T. 2014, “Doxastic disagreement”, Erkenntnis 79 (1), 121 Parsons, T. 1990, “True Contradictions”, Canadian Journal of Philosophy 20 (3), 335 Pravato, G. MS, “Normative Language and Semantic Indeterminacy”, manuscript Priest, G. 1989, “Reductio ad Absurdum et Modus Tollendo Ponens”, in Priest, G., R. Routley and J. Norman (eds.) 1989, Paraconsistent Logic, Munich: Philosophia Verlag —. 2006a, In Contradiction, Oxford: Oxford University Press, 2nd ed. —. 2006b, Doubt Truth to Be a Liar, Oxford: Oxford University Press Schafer, K. 2011, “Faultless Disagreement and Aesthetic Realism”, Philosophy and Phenomenological Research 82 (2), 265 Schiffer, S. 2003, The Things We Mean, Oxford: Oxford University Press Schroeder, M. 2008, Being For. Evaluating the Semantic Program of Expressivism, Oxford: Oxford University Press Smith, N. 2008, Vagueness and Degrees of Truth, New York: Oxford University Press Sorensen, R. 2001, Vagueness and Contradiction, Oxford: Oxford University Press Sundell, T. 2011, “Disagreements About Taste”, Philosophical Studies 155 (2), 267 Unwin, N. 1999, “Quasi-Realism, Negation and the Frege–Geach Problem”, Philosophical Quarterly 49 (196), 337 —. 2001, “Norms and Negation: A Problem for Gibbard’s Logic”, Philosophical Quarterly 51 (202), 60

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Tarski, A. 1944, “The Semantic Conception of Truth”, Philosophy and Phenomenological Research 4 (3), 341 Wright, C. 2001, “On Being in a Quandary. Relativism Vagueness Logical Revisionism”, Mind 110 (437), 45

Notes 1

We borrow the term “equal validity” from Boghossian (2006). It is well known that there are several ways to formulate LNC, in fact several aspects have to be fixed in order to state this principle properly. First, the nature of the law must be determined: whether syntactic, pragmatic, semantic or ontological. Secondly, it must be clarified the type of contradiction: whether explicit or implicit. Thirdly, it must be decided what the objects of contradiction are: whether sentences (token, types), statements, claims, propositions, or state of affairs; finally it must be said which type of negation is involved: whether classical negation or non-classical. These choices give rise to a potentially large number of formulations of LNC, Patrick Grim (2004) has in fact calculated that there are at least 240 possible formulations of LNC! Given the orthodox assumption that propositions are the objects of our attitudes and since we are primarily interested in the attitudes of disputing subjects, we will use the following semantic formulation:

2

LNC (semantic reading) propositions P and not-P cannot be both true in the same circumstances where circumstances are intuitively taken to be those aspects of reality against which the proposition is evaluated (possible worlds are the standard case). 3 The Equal Validity Paradox is closely related to Crispin Wright's “simple deduction” (Wright 2001, p. 56). The main differences are: i) Wright's argument is formulated only by reference to propositions; ii) Wright uses the notion of “no cognitive shortcoming” to unpack the idea of faultlessness, whereas we unpack the notion by means of a semantic reading of equal validity. More generally, the Equal Valdity Paradox can be seen as a theoretical translation of the Simple Deduction where the relevant principles involved are made explicit and where the relation between utterances, contents and attitudes is problematised. 4 Interestingly, the Equal Validity Paradox shares some analogy with the No-No Paradox. The No-No Paradox was first formulated in Sorensen (2001, pp. 175180), we refer to the following formulation present in Greenough (2011). “Consider the following sentences: The neighbouring sentence is not true. The neighbouring sentence is not true. Call these the no-no sentences. Symmetry considerations dictate that the no-no sentences must both possess the same truth-value. Suppose they are both true. Given Tarski’s truth-schema—if a sentence S says that p then S is true iff p—and

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given what they say, they are both not true. Contradiction! Conclude: they are not both true. Suppose they are both false. Given Tarski’s falsity-schema—if a sentence S says that p then S is false iff not-p—and given what they say, they are both true, and so not false. Contradiction! Conclude: they are not both false. Thus, despite their symmetry, the no-no sentences must differ in truth-value. Such is the no-no paradox.” (Greenough 2011, p. 547) Though there are substantial dissimilarities between the two arguments, the symmetry considerations play a crucial role in both cases. In fact, Equal Validity and Semantic Equal Validity are versions of symmetry theses. 5 We employ natural deduction for representing the argument. In particular we use Lemmon-style notation whereby the first numerals make reference to the assumptions on which the proposition depends, whereas the numerals in between the parentheses number the lines of the argument. 6 Strictly speaking, modus ponens is here applied twice (to 1 and 2 and then to the conclusion of the latter application and 3). For brevity, we mention just the principle and we refer to the relevant assumptions involved in two applications. 7 Of course there are other principles involved in the argument (e.g. modus ponens), but we leave them aside because they are irrelevant for the point of the argument. 8 The term and the concept are borrowed from Schiffer (2003, pp. 196-98). 9 See infra end of previous section. 10 Whether these putative descriptive projects have to fall back onto radical rather than moderate forms of revisionism is a question that we leave open in this work. 11 The terminology goes back to Beall & Ripley (2004). 12 Beall (2006, p. 67) revises the norm for belief as: “one ought (rationally) to believe what is at least not false”, thus allowing propositions with truth-value gaps as objects of belief. That said, it remains to be explained why one ought to believe an untrue proposition whose third semantic value is interpreted as “at least not false”. If the untruth of this proposition is due to the absence of facts in the world, it seems that we cannot say that in believing we are representing something correctly. For to represent correctly amounts, intuitively, to believe a proposition that represents correctly an aspect of of the world- say a state of affairs. Now, if indeterminacy is due to the meaning it shouldn't be the case that the propositions has determinate semantic status by assigning a third semantic value. So the assignment of a third semantic status seems to imply that the indeterminacy springs from non-linguistic facts, i.e. from the represented state of affairs (see also infra §5 the distinction between first-level indeterminacy and second-level indeterminacy and Smith 2008). As a consequence, if the state of affairs does not determinately hold, the representor -i.e. the belief- cannot be determinately correct. But if the belief is not determinately correct , why ought one to believe it? 13 Pravato (MS) exemplifies a similar strategy for normative discourse. 14 For those familiar with subvaluationist semantics: the idea is to frame the correctness with a mechanism analogous to subvaluations when it comes to utterances. However, contrary to subvaluationism, no non-standard compositional semantics is invoked for sentences.

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The locus classicus is Blackburn’s (1984) quasi-realism. Recent literature on faultless disagreement (Dreier 2009; Huevenes 2014; Marques 2014) is sympathetic to an expressivist (hence non cognitivist) account. 17 The problem in a nutshell is that sentences that express moral judgement, when they are embedded in sentences that are semantically complex, figure in non assertoric positions (e.g. antecedent of a conditional). However, expressivist theories cannot easily account for the contribution of sentences to the semantics of these complex sentences since the semantic value of sentences has been identified with the attitude expressed by that sentence; but when a sentence occurs unasserted in, for example, an antecedent of a conditional it seems wrong to analyse its semantic contribution as the expression of an attitude. A further challenge for the cognitivist, related to the account of logical validity, is offered by Dorr (2002). 18 The problem is raised in Unwin (1999, 2001). 19 The original formulation is related to moral discourse. 20 See also Schroeder (2008). Some non-cognitivists (Gibbard 2003; Horgan and Timmons 2006) bite the bullet and accept this proliferation of attitudes. 21 Schroeder (2008) has developed a sophisticated expressivist proposal capable of accounting for the negation problem. However, he distances himself from it due to the heavily implausibly complicated commitments that the proposal requires in logic and semantics. 22 The basic form of relativism has been developed in two different strands: a moderate form invoking only this relativisation of propositional truth (known also as “non-indexical contextualism, see Kölbel 2004) and a more radical form invoking also the relativisation of utterance truth to contexts of assessments, contexts whereby the same utterance can be correctly evaluated in different ways (MacFarlane 2005, 2007). 23 This objection concerns the relation between the circumstances of evaluation involved by the yes-no-utterances and questions whether these utterances involve a genuine form of disagreement. As mentioned in the previous footnote, there are at least two varieties of relativism–non indexical contextualism and assessmentrelativism–that have different consequences with respect to the correctness conditions of the yes-no-utterances. However, this difference is not relevant for our objection, since both forms of relativism are committed to the view that yes-noutterances involve different circumstances of evaluation. The difference between non-indexical contextualism and assessment-relativism has consequences with respect to the problem of accounting for the idea of faultlessness, in fact truthrelativism does not seem to have a non-metalinguistic way to express the idea of faultlessness since from each perspective each subject is right is criticizing the opponent's view, so, there is no perspective in which both speakers, uttering two yes-no-utterances, are right. The only way to recover faultlessness is by means of a semantic ascent: each utterance is correct relative to the perspective of the speaker. It is fair to say that MacFarlane (2007) recognizes this fact and seems prone to abandon the project of accounting for faultlessness. 24 López De Sa (2008) and Sundell (2011) offer sophisticated contextualist accounts of faultless disagreement. We argue in Coliva & Moruzzi ms-1 that even 16

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these sophisticated accounts are doomed to be revisionary. 25 See Berto (2007) for an introduction. 26 The opposition between acceptance of P and not-P and rejection of P expresses the dialetheist thought that falsity is not opposite to truth but a subspecies of it. 27 When the discourse is consistent rejection of a proposition is in fact equivalent to acceptance of its negation. 28 What about A? Given that the proposition is a dialetheia, her acceptance would be correct. Would her disputing attitude be nonetheless rational? If A took herself to be in opposition to B because she believes the proposition to be true she could be rational only insofar she ignored that the proposition is a dialetheia. If, on the other hand, took herself to be in opposition to B because she has recognized B's rejection, then her disputing attitude could be rational even if she knew that the proposition is a dialetheia. In this latter case, however, the point of her disputing would be to oppose the improper attitude of B. The dispute would thus be not over a content but over an attitude on that content. Notice, however, that the original idea was that faultless disagreement is manifested in a dispute over the truth-value of a given propositional content. 29 In Coliva & Moruzzi (ms 1) we argue that this is not the case for sophisticated forms of contextualism and relativism and in Coliva & Moruzzi (ms 2) we argue that this not the case for the dialetheist strategy.

CHAPTER FOUR WEAK INDEXICAL RELATIVISM1 CARLO FILOTICO

1. Introduction Indexical relativism is a prominent proposal in the present debate about the nature of judgement. According to this view, sometimes labelled as “contextualism”, there are at least some areas of discourse—namely aesthetics, matters of taste, perhaps ethics and other fields—in which speakers’ assertions must not be read literally: they must be interpreted as implicitly referring to something that is hidden in the literal meaning of the asserted sentences; so, in order to understand properly such assertions and to assign to them a truth value, we need some knowledge about the context of utterance (Harman 1975; Dreier 1990; see also López de Sa 2008). More specifically, the indexical relativist proposal suggests that when e.g. a speaker utters: “Matisse is better than Picasso”, the asserted sentence must be read as a sort of abbreviation for such sentences as: “Matisse is better than Picasso according to my own aesthetic standards” or: “I prefer Matisse to Picasso”. This view qualifies itself as relativist—in a quite harmless sense—because it tries to meet the intuition that when people are talking within such areas of discourse as e.g. matters of taste, the truth value of their assertions depends on the speaker. Moreover, this view can be labelled as «indexical» because it reads speakers’ assertions as implicitly employing such expressions as «I», which point directly to the speaker and so change their referent across contexts. The result of the whole picture is the following: two different utterances of e.g. “Matisse is better than Picasso” can get different truth values (one can be true while the other can be false) because they can express two different contents, depending on the speaker at issue. Since the thesis that content depends on context (even for some expressions containing no indexical expressions) calls for a deeper analysis of the relations between semantics and pragmatics, the view I am discussing has been often called

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“contextualism” and has been assessed within a more general debate in the philosophy of language. I prefer to adopt the term “indexical relativism” because I will concentrate my focus on the debate about the notion of truth to be adopted within prima facie subjective discourse: a debate in which this view has played somehow the role of an intermediate position between relativism and objectivism. Many opponents of the view I sketched above criticise it on the grounds that it requires a certain amount of revisionism in the understanding of ordinary speech (Wright 2001, pp. 51-52; Kölbel 2002, ch. 3; 2003, § IV; 2004, §§ 2-3; 2008, §§ 1.3, 1.5; Lasersohn 2005, § 3; MacFarlane 2007, § 1): according to indexical relativism, utterances of such expressions as “Matisse is better than Picasso” do not say indeed what they seem to imply. Furthermore, a new wave of bold relativist thinkers suggests that the topic of disagreement must be taken more seriously than indexical relativists do: such bolder relativists suggest that utterances of e.g. “Matisse is better than Picasso” and “Matisse is not better than Picasso” must be read respectively as affirming and denying the same content, although this content can get different truth values, depending on the speaker, or perhaps on the assessor of the dispute (Kölbel 2003, pp. 71-72; 2004, pp. 305-306; Lasersohn 2005, p. 662; MacFarlane 2007, § 2).2 My own work is an attempt to formulate the basic insight behind indexical relativism in a modified, weakened version, in order to avoid some well-known objections. In Section 2 I will recall the present debate, with an overview of both indexical relativism and its standard objections; in Section 3 I will formulate my own view and try to show how it can avoid the standard objections; in Section 4 I formulate some straightforward objections to my own view and I try to reply to them.

2. A summary of the present debate 2.1. Standard formulations of indexical relativism The basic idea behind indexical relativism is that some knowledge about the context of utterance is required in order to understand some assertions that do not employ explicitly any indexical expressions (such as personal pronouns or possessive adjectives). The question as to which assertions deserve such a treatment is still open, but the indexical relativist picture has been usually proposed for an account of speech within prima facie subjective areas of discourse: matters of taste, aesthetics, moral values and perhaps some more complex issues, such as asking whether a

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given belief is justified, or whether a given event is probable. A sketchy characterisation of the view at issue can be found in the following quotation: All versions of indexical relativism will involve [...] a claim to the effect that sentences of a certain sort, which may not be overtly indexical, are nevertheless propositionally equivalent to a related indexical sentence. Thus we can imagine a whole range of other sentences that might be proposed to be propositionally equivalent to “Matisse is better than Picasso”: (IR1) I prefer Matisse to Picasso. (IR2) My standard of taste rates Matisse above Picasso. (IR3) The standard of taste of my group rates Matisse above Picasso. (IR4) The standard of taste of the experts recognized in my society rates Matisse above Picasso. Each of these proposals will give rise to a distinct form of indexical relativism [...] (Kölbel 2003, p. 63).

A first issue that must be clarified here concerns the nature of the relation between two sentences like “Matisse is better than Picasso” and “I prefer Matisse to Picasso” in the indexical relativist picture. The most straightforward move for the indexical relativist is apparently to claim that assertions of the two sentences above express the same proposition, as the notion of propositional equivalence employed in the quotation above suggests (see also Kölbel 2004, p. 301). A different move, that perhaps would give rise to a weaker theory, could be to require just some kind of pragmatic equivalence, i.e. that the two sentences above are assertible in the same contexts. Another move could be to propose a regimentation of English and read “Matisse is better than Picasso” as an incomplete expression that should be rephrased in order to be properly understood, e.g. by connecting to the sentence the clause “for me” (see e.g. MacFarlane 2005, § 1.1). However, all the alternative moves above aim to be universal in scope: whatever account among (IR1-IR4) we choose, indexical relativism amounts to a general method for an analysis of prima facie subjective assertions, with no room for any exceptions whatsoever. Then, if it should turn out that this theory has some difficulties in accounting for the data, we might raise the question whether the theory should somehow drop its ambition of universal scope (see Section 3.1 of this essay). A second issue to be clarified concerns the choice of an item among (IR1-IR4), to be taken as equivalent to “Matisse is better than Picasso”.3 As a starting point, there seems to be a basic difference between (IR1) and (IR2) on one side, (IR3) and (IR4) on the other side: on the one hand, if we

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choose the first two alternatives, then the norms that constrain speakers’ assertions seem to have a quite idle role, because we may assume that speakers have some sort of privileged access to knowledge about themselves; on the other hand, the latter two alternatives seem to have a deeper normative content, because then speakers’ assertions must obey to collective standards that can be in conflict with the speakers’ individual attitudes; but the normative role of collective standards is quite hard to be characterised, because (IR3) and (IR4) contain questionable assumptions about the existence and the uniqueness of the community to which a given speaker is linked. 4 Then the indexical relativist philosopher can either choose only one item among (IR1-IR4) to account for all uses, or just require that for each prima facie subjective assertion there is a suitable choice between (IR1-IR4) that can account for that assertion. In the former case, there seems to be a problem of adequacy for the theory, because evidence from linguistic behaviour offers hints in each of the four directions; so it seems to me that the latter, less ambitious move is to be preferred; however, even the less ambitious choice is questionable because of some well-known reasons that I will sketch in the next Section.

2.2. Standard objections to indexical relativism There is a family of related objections that point in the same direction: indexical relativism misidentifies the content of speakers’ assertions, because it forces us to overlook many evidences coming from linguistic behaviour and is incompatible with well-established intuitions about the nature of disputes. In this Section I will try to clarify some of these objections and I will raise some preliminary remarks. I will not try to reply directly to these objections because I cannot see how indexical relativism (as I characterised it in Section 2.1) can escape them; so, in order to get the pars construens of my work, we need to wait until Section 3. 1st Objection: a basic conflict with our intuitions. We are not allowed to treat e.g. all assertions of “Matisse is better than Picasso” as equivalent—in any sense—to assertions of any sentence among (IR1-IR4), just because there is no evidence that whoever utters the former sentence is somehow speaking about herself, about her standards of taste, about the standard of a given group or of any established experts. According to this objection, the burden of proof is on the indexical relativist’s side, because linguistic behaviour suggests prima facie only that e.g. in the example above a speaker is just speaking about Matisse. This revisionary attitude seems to be an undesirable feature of indexical relativism, even in the simplest cases.5

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2nd Objection: “the problem of lost disagreement” (MacFarlane 2007, § 1.2). Indexical relativism treats prima facie disagreements as cases where there is no disagreement at all. An early formulation of this point can be found in the following passage: This, very familiar, relativistic move is still supported in recent philosophy [...]. If it were right, there would be an analogy between disputes of inclination and the “dispute” between one who says “I am tired” and her companion who replies, “Well, I’m not” (when what is at issue is one more museum visit). There are materials here, perhaps, for a (further) disagreement but no disagreement has yet been expressed. But ordinary understanding already hears a disagreement between one who asserts that hurt-free infidelity is acceptable and one who asserts that it is not (Wright 2001, p. 51).

According to the remark in the quotation above, the indexical relativist account of “disputes of inclination” is not acceptable because speakers’ behaviour suggests that the persons involved in the dispute feel involved in a genuine disagreement, while the same does not happen in such pseudodisputes as those about two persons’ being tired. The indexical relativist account of disputes overlooks a huge amount of data from linguistic behaviour, all showing cases where people are not satisfied by a discussion ending e.g. with: —I like Matisse—I don’t. People usually go on to discuss in such disputes, because even though they already know that there is a conflict between two different standards of taste, they may still feel that the opponent should somehow adopt a different standard. If we accept indexical relativism, then it seems that in many areas of discourse speakers’ disposition to ask and give reasons becomes a quite inexplicable phenomenon.6 3rd Objection: a misunderstanding of negative short answers. Indexical relativism fails to identify the content of those assertions in which someone prima facie denies directly what someone else is saying. Let’s consider the example in which a speaker A says: “Licorice is tasty” and a speaker B replies: “No, you are wrong there” or: “What you said isn’t true” (Kölbel 2002, p. 39; Wright 2001, p. 51; Lasersohn 2005, p. 649; MacFarlane 2007, p. 18). According to indexical relativism, the content of A’s assertion is just the content of: “I find licorice tasty” and so B’s reply (which directly asserts the falsity of the content of A’s assertion) becomes the same as the content of: “No, you don’t find licorice tasty”. The indexical relativist reading seems to miss the point of B’s assertion; but this is a serious problem, because evidence from linguistic behaviour shows that there is plenty of people acting like B.

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4th Objection: no account of selective attitudes towards conjunctions of propositionally equivalent sentences. Indexical relativism cannot provide a satisfactory account of the following conversation (for a similar example, see Kölbel 2004, p. 304): A—Matisse is better than Picasso and I prefer Matisse to Picasso. B—I accept the second thing you said, but not the first one.7

The indexical relativist reading of the two assertions above is the following: AI—I prefer Matisse to Picasso and I prefer Matisse to Picasso. BI—You prefer Matisse to Picasso, but you don’t prefer Matisse to Picasso.

On the one hand, indexical relativism reads A’s assertion as pragmatically odd (because the information provided by the assertion is redundant) but semantically legitimate (because the conjunction of a sentence with its rephrasing leads to no inconsistency); on the other hand, according to indexical relativism B’s assertion is semantically inconsistent and so it cannot be accepted as a legitimate assertion. Then it seems that indexical relativism is so revisionary to count as a policy-making account of language use, according to which e.g. B is not allowed to assert what she asserts. This result is difficult to accept, if this view aims to an account of data from speakers’ behaviour and not to the construction of an idealised language for scientific use.8 I am not aiming here to a complete survey of the objections that indexical relativism has received in the literature, but I hope that at least the main points to be assessed have been adequately clarified. Then, instead of trying to reply directly to the objections above, I will propose in the next Section a slight modification of the standard indexical relativist framework; this move should suffice, I hope, to account for the data discussed within the objections above.

3. My own view 3.1. A straightforward weakening of indexical relativism Let’s start from the basic intuitions that motivate any forms of relativism about truth, in order to understand better the questions at issue. The source of the whole debate about relativism and its alternatives seems to be just the fact that the behaviour of speakers shows different attitudes

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towards different kinds of disputes: there are areas of discourse where rational arguments usually play a quite strong role, while in some other areas such role is much weaker. More precisely, there are areas of discourse where the speaker’s behaviour shows the expectation that, despite differences of opinion among the discussants, there are right answers to be found for the questions at issue; some other areas of discourse show often the opposite attitude in people’s behaviour: speakers prefer to drop the discussion and take differences of opinion as a cul de sac for any further argument. Moreover, the picture is somehow fuzzy, because e.g. the field of ethics seems to show a prevalence of an argumentative attitude in speakers’ behaviour, while discussions about matters of taste more often obey to the ancient saying: De gustibus non est disputandum. Indexical relativism tries to point to some data from linguistic behaviour, showing that in those areas of discourse where rational discussion is more difficult (if not pointless), speakers that are asked to clarify what they mean with their assertions sometimes provide clarifications that encourage an indexical reading of the assertions at issue. Let’s consider the following examples: Example 1. CARLO—How was the film yesterday? MARIA—Quite good! CARLO—So, do you think that I should watch it? MARIA—I don’t know… Indeed I just wanted to say that I liked it! Perhaps you wouldn’t like it as much as I did. Example 2. MARIA—Do you need some more sugar in your coffee? CARLO—No, it’s perfect as it is! MARIA (after tasting Carlo’s coffee)—Hey! It’s disgusting! CARLO—Sorry, I didn’t mean to tell you that it would be perfect for you too… I just wanted to say that it’s perfect for me. Example 3. CARLO—No more sugar thanks, my coffee is perfect as it is. MARIA—Well, since I know that you like your coffee bitter, I guess that it’s disgusting! CARLO—Perhaps it is disgusting for you, but not for me. MARIA—That’s what I meant.

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On the one hand, I do not mean that all discussions about matters of aesthetics, taste and similar topics go along these lines; on the other hand, it seems to me hard to deny that some of them do (Iacona 2008, § 1). So it seems that indexical relativism (as I sketched it in Section 2) is implausible because it forces all disputes e.g. about aesthetics to be accounted for as cases like examples 1-4; however, data from cases like examples 1-4 should be taken into account too. Standard indexical relativism aims to reduce the interpretation of assertions to a straightforward task, because it aims to provide a theoretical framework that tells us in advance which assertions deserve an indexical reading; 9 however, the objections in Section 2.2 seem to show that the indexical relativist framework has a lot of counterexamples. So, even accepting the desideratum that a theory be as strong as possible, we should start to accept that standard indexical relativism is just too strong; but then, instead of throwing out the baby together with the bath water, we should ask whether indexical relativism can survive in a weakened form. The best way to escape the problems above seems to me to reduce indexical relativism just to a (double) existential claim: there are some assertions that deserve an indexical reading (while some of them do not). For instance, in order to understand an utterance of “Matisse is better than Picasso”, the weakened theory that I am suggesting tells us that the option of reading the utterance as equivalent to “I prefer Matisse to Picasso” is allowed in principle; 10 but the weakened theory allows also a literal reading, according to which the utterance just says that Matisse is better than Picasso tout court. The task of understanding whether the assertion is to be read literally or indexically is left to the interpreter, who can accomplish her job by collecting further information from the context of utterance.

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3.2. How weak indexical relativism can avoid standard objections I will recall the objections in section 2.2, in order to understand whether they affect my own proposal. Reply to 1st Objection (a basic conflict with our intuitions). As we saw from examples 1-4 in the previous section, there is at least some evidence for the thesis that some assertions can be read indexically; evidence for denying that the indexical reading works in all cases (within a given area of discourse) is a problem for standard indexical relativism, but not for my own, weaker proposal. Reply to 2nd Objection (lost disagreement). This objection doesn’t affect my view, because I am not claiming that all prima facie disagreements in any area of discourse are to be accounted for as pseudo-disagreements. My proposal is just that a conversation between e.g. two persons A and B asserting respectively: “Matisse is better than Picasso” and “Matisse is not better than Picasso” can have indeed four different readings. The first one is the indexical reading: AI—I prefer Matisse to Picasso. BI—I don’t prefer Matisse to Picasso.

which is quite implausible in many contexts, but should not be ruled out in principle, as we saw from Example 3 in the previous Section. The second reading is the literal one: AL—Matisse is better than Picasso. BL—Matisse is not better than Picasso.

Of course, the literal reading above requires a semantics for such expressions as «is better than», which can be provided either in relativistic terms (by considering the extension of the relational predicate as depending on the relevant judge) or in objective terms (more or less like other relational predicates). Since it seems to me that the point of indexical relativism is just to avoid radical relativism, I prefer to rule out the former option; so I must accept that some disputes about e.g. what is better in the field of art must be accounted for by means of an objective notion of truth. I am not embracing the thesis that all uses of “is better than” within disputes about works of art express an objective property, but in order to make room for taking seriously what speakers say within some prima facie subjective disputes, I must concede that some aesthetic judgements may deserve a standard treatment in terms of truth, because they may rely on

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some objective features of e.g. Matisse and Picasso (see also Section 4 of this essay (reply to Objection C). The third reading and the fourth one leave room for the hypothesis that speakers A and B are indeed making their assertions in two different ways: then we might interpret A’s assertion literally and B’s assertion indexically, or vice versa (see e.g. Example 2 in the previous Section of this essay). A situation of this kind would constitute somehow a case of unsuccessful communication; but a philosophical theory should provide an account for it and explain why it is unsuccessful. If we ruled out the third and the fourth option from the theory, we would ignore the cases of unsuccessful communication as something that does not happen: but ordinary uses seem to show plenty of counterexamples for this theoretical move. A final remark to be added to the discussion above is that the analysis of the dispute between speakers A and B can be improved by means of the data coming from possible subsequent scenarios (Iacona 2008, p. 92), such as the following ones: Scenario 1. A—Indeed, I just wanted to say that I prefer Matisse to Picasso… I didn’t want to say anything about what other people should prefer. Scenario 2. A—No, you’re wrong! I will show to you that Matisse is better!

Now, evidence from Scenario 1 suggests that A’s original speech is to be read indexically, while evidence from Scenario 2 suggests a literal reading. The information from both scenarios comes after the time of A’s original utterance, but I cannot see why an interpretation of A’s previous utterance should not benefit from new information that becomes available in a subsequent situation. Of course, these remarks apply also to the case in which the interpreter of A’s utterance is just the speaker B involved in the dispute. In both scenarios B’s subsequent speech is somehow constrained by A’s clarifications, in order for B to be counted as a rational speaker; but this is no problem for my view, because these constraints for speakers’ rationality come from speech itself and not from the theory that is supposed to account for the data. Reply to 3rd Objection (a misunderstanding of negative short answers). I recall that the objection started from an example in which a speaker A says: “Licorice is tasty” and a speaker B replies: “What you said isn’t true”. The point of my own proposal is that A’s and B’s assertions can be read either literally or indexically. Then there are indeed four possible

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interpretations of the conversation above, but they are not equally plausible. The first one is to read the conversation literally, in this way: AL—Licorice is tasty. BL—No, licorice is not tasty.

This reading is coherent with the hypothesis that B has understood A’s assertion and that her contribution to the conversation is pragmatically adequate. Of course, this reading presupposes that such words as «tasty» get their interpretation within a suitable semantics—a task that is not trivial, as we saw in the discussion of the 2nd Objection. However, this reading is perfectly adequate to account for speakers’ intentions in many cases. A second option is to read the conversation just as standard indexical relativism does: AI—I find licorice tasty. BI—No, you don’t find licorice tasty.

According to this reading, A is just describing her own standard of taste and B correctly understands what A says, although B’s reply is quite puzzling, at least in most situations.11 Then, since we may assume that speaker B behaves rationally and tries to provide a useful contribution to the conversation, in most cases we would rule out this reading—which is legitimate in principle, according to weak indexical relativism—as less plausible than the first one. The third option and the fourth one try to make room for the hypothesis that B is misunderstanding the sense of A’s assertion, either by reading it indexically while it is to be read literally, like in the following reading: AL—Licorice is tasty. BI—No, you don’t find licorice tasty.

or vice versa (a situation happening more often): AI—I find licorice tasty. BL—No, licorice is not tasty.

I cannot see any reason why a philosophical account of assertions should not leave room for explaining a conversation through the hypothesis that the speakers are indeed misunderstanding each other. The

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question whether A and B are indeed understanding each other should be left to the interpreter, who can collect information from the context in order to make her hypotheses. Reply to 4th Objection (no account of selective attitudes towards conjunctions). In the example above, a speaker A utters: “Licorice is tasty and I find licorice tasty” and B replies: «I don’t agree with the first thing you said, although I accept the second one». Now, since we are discussing this case within my weakened form of indexical relativism, we are allowed to choose between an indexical reading and a literal one. On the one hand, the most straightforward strategy of interpretation for the case at issue seems to be the literal one, which reads the conversation above in the following way: AL—Licorice is tasty and I find licorice tasty. BL—Licorice is not tasty, but you find it tasty.

This reading is to be preferred, if we have no clear hints for abandoning it, because it is coherent with the assumptions that A is pragmatically competent (i.e. that her speech is not redundant) and that B is interpreting correctly A’s assertion. On the other hand, the theory allows in principle also a different reading, which is the following: AI—I find licorice tasty and I find licorice tasty. BI—You don’t find licorice tasty, but you find it tasty.

This reading will be ruled out by the interpreter in most situations, because it corresponds to the hypothesis that speakers A and B are acting quite irrationally: A’s utterance is semantically legitimate, but pragmatically redundant, while B’s assertion is semantically inconsistent and so is not assertible at all.12

4. New objections and new replies Objection A. Although evidence from linguistic behaviour suggests that there are speakers who utter e.g. «Matisse is better than Picasso» with the intention to mean: «I prefer Matisse to Picasso», this evidence by itself does not show that such speakers do mean what they want to mean. So the thesis that at least some assertions deserve an indexical reading has not yet been justified. Reply. Yes, but the theory-laden speech that we saw in examples 1-4 (Section 3.1) belongs to ordinary uses of language, so ruling it out as non-

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competent speech would be an undesirable result for any theory. A denial of my existential claim that some assertions deserve an indexical reading would amount to a revisionary account of examples 1-4, according to which Carlo and Maria are not allowed to speak as they do; all this would lead to a policy-making account of speech, that is an undesirable result for any theory, just as undesirable as the consequences of standard indexical relativism that we saw in Section 2.2. Objection B. Weak indexical relativism is too weak, because it doesn’t say anything about how to understand those cases where the indexical reading doesn’t fit. Reply. Nothing prevents us to conjoin weak indexical relativism with other theories on the philosophical market. Of course, also these theories should be restricted to a limited range of case, in order to let weak indexical relativism play some role (for a more general defence of a mixed approach towards prima facie subjective discourse, see Filotico 2010, pp. 181-182). Objection C. Since weak indexical relativism cannot account for all prima facie subjective disputes, it requires at least a certain amount of objectivism with respect to such disputes: a thesis which, in turn, raises many well-known objections (for a survey, see Kölbel 2002, ch. 2). Reply. This is a bullet that I prefer to bite, somehow. Indeed, whenever contextual evidence from a dispute suggests that speakers’ assertions are to be read literally, we are left with the difficult task of taking the dispute quite seriously. This is a situation that obtains quite often within some domains, like those of ethics or aesthetics, quite seldom in such discussions as to which is the best meal or the best colour. I believe that the effort that people pay in taking seriously some disputes—even when they have no idea about how to find a satisfying answer—is a quite reliable indicator that a satisfying answer is to be searched for; so it seems to me that the concept of objective truth is still a useful resource for understanding such disputes. However, by conjoining this basic objectivist attitude together with weak indexical relativism, we could avoid the most undesirable consequences of radical forms of objectivism. Objection D. Weak indexical relativism has no philosophical scope, because it drops the ambition to provide a self-contained theoretical framework for an account of disputes within prima facie subjective areas: too much work is left to the interpreter. Reply. I cannot see why the philosophy of language should do more than providing the general theoretical resources for the assignment of a content and a truth value to each assertion. Weak indexical relativism expresses an option in the philosophical debate: namely, the option that

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indexical readings of prima facie subjective assertions are allowed in principle; then, asking the philosopher to tell us exactly when indexical readings are to be adopted seems to me asking too much of her. The fact that we have to look to the context of utterance in order to choose the best interpretation (the literal or the indexical one) is not, in my view, a defect of the theory: it is just the result of a sharp separation between the task of the philosophy of language, which is to provide the methods for interpretation, and the task of interpretation itself, which is to employ such methods, together with a careful look to contextual information, in order to individuate the content of assertions. Looking carefully to contexts is, in my view, a quite concrete task: it means to collect information even from non-verbal speech, to look at previous and subsequent talks in order to understand as best as possible what speakers say. So, if philosophers take care of interpretation, they must help interpreters to do their work as best as they can; but if the indexical relativist philosopher just declares by fiat that e.g. assertions within aesthetics are indexical and those within physics are not, then she doesn’t seem to help interpreters to get adequate results.

5. Concluding remarks I believe that the objections I collected in Section 2.2 provide sufficient evidence from linguistic behaviour to say that indexical relativism, as I characterised it in Section 2.1, is not adequate and should be rejected. Nonetheless, in Section 3 I have tried to recall some evidence to the effect that the intuitions underlying indexical relativism are not completely wrong. Then I formulated a quite weak theory that captures at least some of these intuitions. This theory is specifically useful only for the account of some data, while some other data are to be accounted differently; so the theory I defended here is to be conjoined with other theoretical resources for a general account of disputes in prima facie subjective areas. The discussion within Section 4 suggests also that my own proposal calls for a deeper discussion about the scope and limits of philosophical work within the field of enquiries about language. Although I couldn’t go much deeper into the latter topic, I hope that my essay can also contribute to stimulate further discussion about this broader issue.13

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References Dreier, J. 1990, “Internalism and Speaker Relativism”, Ethics 101 (1), 6 Filotico, C. 2010, “How Can We Formulate Relativism?”, in Forme e formalizzazioni. Atti del XVI Congresso della Società di Filosofia del Linguaggio, edited by G.P. Storari and E. Gola, 171, Cagliari: CUEC García-Carpintero, M. and M. Kölbel (eds.) 2008, Relative Truth, Oxford: Oxford University Press Harman, G. 1975, “Moral Relativism Defended”, The Philosophical Review 84 (1), 3 Iacona, A. 2008, “Faultless or Disagreement”, in García-Carpintero M. and M. Kölbel 2008, 287 Kölbel, M. 2002, Truth Without Objectivity, London—New York: Routledge —. 2003, “Faultless Disagreement”, Proceedings of the Aristotelian Society 104 (1), 53 —. 2004, “Indexical Relativism Versus Genuine Relativism”, International Journal of Philosophical Studies 12 (3), 297 —. 2008, “Introduction: Motivations for Relativism”, in García-Carpintero M. and M. Kölbel 2008, 1 Lasersohn, P. 2005, “Context Dependence, Disagreement and Predicates of Personal Taste”, Linguistics and Philosophy 28 (6), 643 López de Sa, D. 2008, “Presuppositions of Commonality: An Indexical Relativist Account of Disagreement”, in García-Carpintero M. and M. Kölbel 2008, 297 MacFarlane, J. 2005, “Making Sense of Relative Truth”, Proceedings of the Aristotelian Society 105 (1), 321 —. 2007, “Relativism and Disagreement”, Philosophical Studies 132 (1), 17 Marconi, D. 2007, Per la verità. Relativismo e filosofia, Torino: Einaudi Wright, C. 2001, “On Being in a Quandary. Relativism Vagueness Logical Revisionism”, Mind 110 (437), 45

Notes 1

This work is a development of a sketchy remark I already formulated in Filotico (2010, § I.1.3). There are some similarities between my project and the view advocated by Iacona (2008), but I hope that my own work can contribute to formulate sharply and defend through independent reasons those points that can be found also in Iacona’s work.

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2 The stress on the role of the assessor is a specific feature of MacFarlane’s theory, according to which the relativity of truth to the speaker would not amount to a genuine form of relativism (MacFarlane 2005, pp. 325-328). 3 Of course, (IR1-IR4) do not exhaust the alternatives that can be made in the indexical relativist framework; however, it seems clear to me that an improvement of my survey here could only strengthen the reasons in favour of my remark at the end of the present paragraph. 4 Some hints about how to read and assess the existential assumptions above can be found in Kölbel (2004, p. 302). The assumption of uniqueness of the community to which each individual belongs is perhaps more questionable: even setting aside the fact that a given individual belongs to many unrelated communities that obey to conflicting standards, we must accept that communities are often articulated in their subsets, so it is not clear whether the standards of a given community are to be taken strictly or broadly (for some insightful remarks about these problems, see Marconi 2007, p. 131). 5 Indeed, even opponents of indexical relativism recognise that this objection is somehow question-begging, because it is based just on the assumption that the standard reading of assertions is correct—which is exactly the main point that indexical relativism denies (see Kölbel 2002, pp. 38; 2004, p. 303). Then we might put things in this way: in order to adopt the revisionary approach, we need further reasons making the theory so appealing to justify that we drop our primary intuitions. 6 This point is even clearer when we look to such more delicate issues like those in the field of ethics. Indeed, the indexical relativist picture seems to be more adequate to some areas of discourse than in other ones. For a deeper discussion about this point, see Filotico (2010, § I.1.3). 7 The situation depicted in the objection is not so odd as it might appear at first sight, because it can occur also in situations where just one single person is involved, through different stages of judgement during time: see Kölbel (2003, p. 64). 8 We could compare this situation with what happens in the debate about semantic paradoxes: those theories that rule out e.g. the sentence “This proposition is false” as ill-formed can be accepted perhaps as accounts of the way an idealised language should work; but nowadays most philosophers consider these theories as defective accounts of the way natural languages do work. 9 Moreover, since standard indexical relativism is a universal claim telling that all assertions within the areas of discourse of a given set S are to be read indexically, it cannot avoid the task of telling us exactly which areas of discourse belong to S and which ones do not. The latter task is not so necessary for less ambitious theories, namely the proposal I will sketch in § 3.1. 10 This claim goes along the lines of (IR1), but it wouldn’t be difficult to conjoin the claim above with parallel existential claims along the lines of (IR2), (IR3) and (IR4).

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Perhaps we might find evidences for such a reading in a case where e.g. B says: «No, what you said wasn’t true: it just seems to you that licorice is tasty for you, just because the stick you ate was covered with honey». 12 Again, we must leave room also for the hypothesis that B is misunderstanding the point of A’s assertion by reading it indexically while it is to be read literally, or vice versa. The analysis of the data (even from subsequent speech) will help the interpreter to understand which situation is exactly at issue. My own theory can help the interpreter to reject some hypotheses as implausible, but it does not rule out any reading in principle. 13 I must thank Andrea Bianchi for his helpful remarks on a previous draft of this paper, Chaz Pugliese for his stylistic revision of the text.



CHAPTER FIVE RADICAL RELATIVISM, RETRACTION AND “BEING AT FAULT” FILIPPO FERRARI AND DAN ZEMAN

1. Introduction Radical relativism was born with a promise: to account for certain phenomena that opposite views are unable to explain. One example is the phenomenon of “faultless disagreement”, according to which two people, while disagreeing, are not at fault in any substantive way. The phenomena of retraction and assessments of truth in cases of eavesdropping are others. All these phenomena have been claimed to pose serious problems for rival views and be best accounted for within a radical relativistic framework. While “faultless disagreement” and the notion of disagreement in general has benefited from extensive discussion in current debates over semantic content, retraction has not been in the spotlight that much. In particular, very few things have been said about what retraction exactly amounts to and how to conceive of its normative profile. This will be the focus of our paper. We will begin by giving an intuitive characterization of retraction by means of some examples (section 2). After presenting the basics of the radical relativist view in section 3, we move to investigating retraction, offering what we take to be some key elements for a substantial analysis of the phenomenon (section 4). Such analysis, we claim, has the virtue of making clear what the normative peculiarity of the notion of retraction isȄnamely, its retroactive efficacy. In section 5, we inquire into the sense of “fault” in which retractors are said to deem their former selves as not being at fault when making the retracted assertion (MacFarlane 2014). In this connection, we highlight an asymmetry between retractions involving predicates of personal taste and moral terms (section 6). After noting that the epistemic notion of “fault” used by MacFarlane’s cannot explain the asymmetry, in the following section (7) we offer our own explanation, by



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appealing to a less discussed dimension of assertion evaluation which we call “circumstance-accuracy”. In section 8 we provide support for such an explanation by taking a cue from the legal domain and show how an important distinction found there can be applied to the case of retracting assertions as well. We flag some issues that our paper opens up for further research in section 9.

2. Introducing retraction: some examples To a first approximation, retraction is that speech act that an agent performs when she claims “I take that back”–where “that” is taken to refer to a previously unretracted assertoric speech act made by the agent whose content is currently deemed as false. The most effective way to introduce retraction is by looking at some examples of ordinary conversational situations that are taken to illustrate the phenomenon. The examples involve conversational situations about different subject matters, such as epistemic modals, knowledge attributions, judgments of taste and moral judgments: (1) Epistemic Modals Sally: Joe might be in Boston. George: No, he can’t be in Boston. I just saw him an hour ago in Berkeley. Sally: Okay, then, scratch that. I was wrong (MacFarlane 2011, p. 148). (2) Knowledge Attributions Judge: Did you know on December 10 that your car was in your driveway? Sam: Yes, your honor. I knew this. Judge: Were you in a position to rule out the possibility that your car had been stolen? Sam: No, I wasn’t. Judge: So you didn’t know that your car was in the driveway, did you? Sam: No, I suppose I didn’t, your honor (MacFarlane 2005b, p. 213). (3) Judgments of Taste Angelina, in her childhood: Spinach is tasty. Tom, in present times: You said that spinach is tasty. What about that? Angelina, in present times: Spinach is not tasty. I was wrong. (4) Moral Judgments1 Albert in his childhood: Torturing mice for fun is not wrong.



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3. Radical relativism “Relativism” is a label that covers a variety of different views. Our interest is primarily on a specific kind of relativism–namely, alethic relativism. The general idea behind this kind of relativism is that the truth of statements within certain areas of discourse (e.g. moral, aesthetic, epistemic) is relative. In recent philosophy of language two main versions of alethic relativism have crystallized: a moderate version, championed by Max Kölbel, and a more radical one recently defended by John MacFarlane. The moderate version is construed as a slight departure from the framework pioneered by Kaplan (1989), which already contains relativization of truth to contexts of utterance and circumstances of evaluation–the latter thought of as comprising parameters for possible worlds and times. In the same way in which Kaplan introduced the time parameter in the circumstances of evaluation (although not necessarily for the same reasons), relativists of the moderate stripe urge us to introduce more “unorthodox” parameters such as various kinds of standards–e.g., standards of taste, moral standards, aesthetic standards, epistemic ones– each pertaining to a specific area of discourse.2 The second, more radical version of relativism holds that, besides relativization to contexts of utterances and circumstances of evaluation, truth needs to be relativized to a third factor, namely what MacFarlane (2003) has dubbed “contexts of assessment”. The proposal, in a nutshell, is that truth-value is a function of parameters fixed by the context of assessment. A context of assessment is, roughly, any context in which the content of a given sentence is evaluated for its truth or falsity. Like in moderate relativism, the contribution of a certain parameter to determining the truth-value of an utterance of a sentence containing the relevant expression comes via the circumstance of evaluation with respect to which such an utterance is evaluated.3 However, unlike moderate relativism, in the radical version the value of the relevant standard is not established (or, to use MacFarlane’s technical term, initialized) by the context of utterance, but by the context of assessment. The main contrast between the two relativisms is thus the following. In the moderate version an utterance has a once-and-for-all settled truth-value even though there might be variation in truth-value among utterances expressing the same content in different contexts. In the radical version this is not so: the truth-



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value of an utterance depends crucially on the values of the relevant parameter that are established by the context of assessment – the context of utterance is not enough to determine that value. Thus, not only is there a variation in truth-value of different utterances that express the same content in different contexts; there is also a variation in the truth-value of the same utterance from one context of assessment to another. It is in this sense that MacFarlane’s relativism is “more radical” than the Kölbelian version. As we have mentioned, retraction has been taken by radical relativists to favor their view over competitor views. Here we will illustrate how retraction has been thought to raise a problem for the moderate version of relativism. There are two points to be made in this respect. Despite their different definitions of utterance-truth, moderate relativism and radical relativism yield similar predictions when it comes to the assertion norms prescribed by each view: given that in performing an assertion the context of utterance and that of assessment coincide, no normative difference will result.4 But the difference comes out when we consider retraction: here the two views make different predictions with different normative import. The difference can be seen in their respective norms of retraction: (MRRN) An agent in context c2 is required to retract an (unretracted) assertion of p made at c1 if p is not true as used at c1 and assessed from c1 (RRRN) An agent in context c2 is required to retract an (unretracted) assertion of p made at c1 if p is not true as used at c1 and assessed from c2,

where c1 and c2 are the context of utterance and the context of assessment, respectively. We can easily see that moderate relativism is ill equipped to explain the peculiar normative profile of retraction. By (MRRN), one should retract at a context of assessment only if a previous assertion is not true as used and assessed at the context of utterance. In other words, such norm predicts that one ought to retract assertions that were already incorrect because in violation of the assertoric norm. In a slogan: according to (MRRN) you ought to retract what you should not have asserted in the first place. Moreover, abiding by (MRRN) will not force retractors to retract a previous assertion if its content is true relative to the circumstances in play at the context of utterance, even if it is false relative to the circumstances in play at the current context of assessment. The normative profile characteristic to retraction remains unexplained. Thus,



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for instance, according to (MRRN) Albert in present times would be required to retract his assertion made in his childhood that torturing mice for fun is not wrong just in case the content expressed by that assertion was false according to Albert’s moral standard when he was a child. But assuming that Albert’s moral standard in his childhood deemed torturing mice as morally permissible, (MRRN) predicts that Albert is currently under no obligation to retract his previous assertion, regardless of whether he now believes that it is immoral to torturing mice for fun. On the other hand, the radical version of relativism is well equipped to handle the normative profile of retraction. By (RRRN), one should retract a previously unretracted assertion at a context of assessment if the content it expresses is not true as used at the context of utterance and assessed at that context of assessment. This gives retraction an interesting normative role to play. Despite the fact that my previous asserting that p was correct, because in compliance with the assertion rule, I ought to retract that assertion if p is false as evaluated from my current context of assessment. According to MacFarlane this provides a strong reason, albeit not conclusive, to prefer radical relativism over its moderate rival.

4. Retraction: towards an analysis As we have noted above, the ability to account for retraction plays a dialectically crucial role for establishing the superiority of radical relativism over its rivals. However, despite its importance in the current debate, a thorough analysis of retraction is still lacking. Our aim in this section is to take a few steps towards such an analysis. A first thing to note is that the examples given in section 2 involve two subjects that dialectically interact in conformity to the following pattern: there is a challenge issued by one subject to an assertion performed in some past context by the other subject, who replies by accepting the challenge and by retracting the assertion. Insofar as retraction is a speech act performed by an agent with some conversational aim, it presupposes an audience with some common conversational background. However, the aspect of dialectical confrontation between two subjects, and the presence of an actual dispute among them, is not an essential feature of the phenomenon. It is rather a heuristic device that makes the confrontational aspect more explicit. The examples could have been formulated in such a way as to involve just one agent and a non-interactive audience in a context in which both the retracted assertion and the motivation the agent has for retracting are publicly available.5



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The conversational situations presented above have been taken by radical relativists to instantiate the very same phenomenon: what Sally, Sam, Angelina and Albert are all doing is to take back their previous assertions on the basis of the fact that their respective standards are different from the ones had while making them. And the way they do this is by explicitly admitting that there is something wrong with their previous assertion, a sense of wrongness that is characteristically associated with the act of retracting and which will be clarified in due time. The intuition is that what Sally, Sam, Angelina and Albert are doing in reply to the challenge posed by their respective interlocutors is the most natural and sensible thing to do, given the specific conversational circumstances they find themselves in. We will not take issue with whether this intuition is in fact as widespread as radical relativists claim.6 We just assume that the phenomenon is real. Our aim is instead that of providing some elements for its analysis.

4.1. Retraction and disagreement It is not unusual to hear philosophers in informal conversation about retraction claiming that such a phenomenon simply amounts to a disagreement with one’s former self. However, it’s not clear what this means exactly. A further step towards an analysis of retraction is to clarify what the relation between retraction and disagreement is. To this end, two preliminary remarks about disagreement are in order. The first is that, as many philosophers have recently observed, disagreement seems to consist in a variety of, probably heterogeneous, phenomena rather than neatly falling under a unique type.7 However, although the project of clarifying the relation between retraction and the various kinds of disagreement discussed in the literature is certainly worth pursuing, we will restrict our attention to only one variety of disagreement: doxastic disagreement. Roughly, doxastic disagreement involves two subjects, A and B, that either have the same doxastic attitude (say, belief) towards two logically incompatible propositions, or they have two incompatible doxastic attitudes (say, belief and disbelief) towards the same proposition.8 The second remark concerns an ambiguity in the term “disagreement” brought to the fore by Cappelen and Hawthorne’s (2009) distinction between disagreement as state and disagreement as activity. Thus, the word “disagreement” might refer to the act of disagreeing which is part of an actual dispute over a certain subject matter; alternatively, it might refer to the state of being in disagreement.



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Disagreement as a state does not require the presence of an actual dispute. It concerns the relation between the doxastic states of two or more subjects, independently of their actually having, or having had in the past, a dispute about the subject matter of the disagreement. If Thales believed that water is an element and Cavendish believed that water is not an element, then they state-disagree even though one was living in Ancient Greece, between the VI and the V century BC, and the other was living in England during the 18th Century. Disagreement in the act sense requires the existence of a dispute. Following Egan (2010), we might characterize a dispute in the following way: two parties to a conversation assertively utter two (at least prima facie) incompatible judgments which they take to be in conflict. Then they engage in a process of argumentation and negotiation with the aim of reaching a common opinion, which both parties are prepared to sincerely assert and to accept, while rejecting its negation. Now, it is important to note that the kind of incompatibility in question might be only prima facie, and thus that the associated sense of conflict (however we want to exactly characterize it) involved in the dispute might be entirely apparent. For instance, it might be caused by some features of the conversational situation they are in, or by the fact that they are using some of the relevant words in their statements in a different way. Thus we might distinguish between merely verbal dispute, where the conflict is only apparent, and genuine disputes, where the conflict is real–i.e. grounded in disagreement in state. Two subjects are disagreeing in the act sense just in case they are engaging in a genuine dispute. What is then the relation between retraction and disagreement? Roughly put, whenever a subject S is under a deontic requirement to retract an assertoric speech act with content p, it is required that S’s current doxastic state is in disagreement in the state sense concerning p with her previous doxastic state at the time in which the assertoric speech act was performed. However, and quite intuitively, there is no sense in which retraction involves disagreement in the act sense.

4.2. MacFarlane on retraction With these clarifications at hand we can now turn to the task of providing a minimal analysis of retraction. We will begin our analysis from some elements provided by MacFarlane’s (rather sketchy) characterization of retraction. He writes: By “retraction”, I mean the speech act one performs in saying “I take that back” or “I retract that.” The target of a retraction is another speech act,



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which may be an assertion, a question, a command, an offer, or a speech act of another kind. The effect of retracting a speech act is to “undo” the normative changes effected by the original speech act (MacFarlane 2014, p. 108).

In this paper we are primarily interested in the phenomenon of retraction as targeting assertoric speech acts, and we will thus leave the discussion of cases of retractions of questions, commands or offers for another occasion. Concerning the effect of retracting assertoric speech acts, MacFarlane writes: [I]n retracting an assertion, one disavows the assertoric commitment undertaken in the original assertion. This means, among other things, that one is no longer obliged to respond to challenges to the assertion (since one has already conceded, in effect), and that others are no longer entitled to rely on one’s authority for the accuracy of this assertion (MacFarlane 2014, p. 108).

Some observations could be made starting from these quotes. First, retracting is not mere refraining to re-assert. We can in fact distinguish between no longer being willing to assert that p and retracting an earlier assertion that p. I might in fact refrain to (re)assert a proposition in cases in which, for instance, I move from a conversational situation in which it is perfectly appropriate to assert that p, to a situation in which, for purely pragmatic reasons an assertion of p would be deemed as totally inappropriate. Thus, despite the fact that in the new context I still believe the proposition to be true and justified, I refrain to assert it because aware of the fact that some pragmatic factors pertaining to the new conversational situation would make an assertion of p inappropriate. However, in refraining to assert that p, I am not thereby retracting my previous assertoric speech act as made in a context where it was appropriate to assert that p. On the contrary, it is totally legitimate for me to stand by that assertion, as made in such context, and to keep all the normative commitments associated with it. As a consequence, a challenge issued in the new context of an assertion of p, if I were to perform it in there, wouldn’t necessarily pose a challenge to my previous assertion as made in a less demanding context, unless such challenge suggests that p is false. Second, retracting is not just re-assessing the content of an assertion as a result of a change of mind. It requires more than that. It requires an additional speech act in which we claim that we take that assertion back. In fact, it is only by performing the speech act of retraction that we



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disavow the normative commitments undertaken with our previous assertion. After all, we might simply ignore our obligation to retract especially if we think that nothing hinges on that. In addition, other important features of retraction could be discerned. For one thing, retraction always involves two contexts, or conversational situations: one in which the original assertoric speech act is made and another in which the retraction is performed. The specific assertoric act targeted by an act of retraction is performed at an earlier time. Thus, retraction is diachronic and retrospective. Moreover, retraction is generally autocentric.9 An act of retraction performed by S characteristically targets another speech act performed by S herself at an earlier time.10 Another important feature of retraction is that it needs to be public. In order for an act of retraction to be effective in disavowing the normative commitment undertaken with the previous assertion, the act needs to be made public; it requires an audience that formally acknowledges your intention to take distance from all the normative commitments you undertook with your previous assertoric speech act.

4.3. The retroactivity of retraction As MacFarlane says, in retracting an assertion, one disavows the assertoric commitment undertaken in the original assertion. This is taken to be the most significant trademark of the normative profile of retraction. But how should we understand this sense of disavowing the assertoric commitments? Asserting a proposition, qua public speech act addressed to a specific audience, is, among other things, making oneself responsible for the truth of that proposition (Peirce 1934). This sense of making oneself responsible is associated with a network of commitments that in making an assertion a subject undertakes. Without going into details here, such a network is constituted by: (i) a commitment to vindicating the assertion (by providing grounds for its truth, or perhaps by deferring to someone else who can) when it is appropriately challenged; (ii) a commitment to be held responsible if someone else acts or reasons on the basis of the assertion when it proves to be false; (iii) a commitment to promote the assertion over alternative (and incompatible) judgments, when appropriate; (iv) a commitment to step back from an assertion or endorsement of the assertion whenever the evidence available does not support it to a sufficient degree.11 It is, of course, possible that not all these commitments are always associated with



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any particular assertoric act. In fact, which commitments are associated with a particular act might depend on specific features of the situation of assertion as well as on features of the content of the assertion. One way we think the sense of disavowal involved in any genuine act of retraction could be clarified is by appealing to a concept employed in jurisprudence–i.e. the concept of retroactivity. The claim is that a fruitful way to understand the normative effect that an act of retraction possesses is to say that retracting has some kind of retroactive efficacy. Roughly, the idea of retroactivity is the following: a law (or an act) has retroactive efficacy if it alters the legal (or normative) status of acts that were performed before the new law (or the act of retraction) came into existence. Thus, in these general terms, an act of retraction of a previous assertoric act has retroactive efficacy with respect to that act insofar as it alters the normative status of it. As we have said, the effect of retracting a previously unretracted assertion is that of undoing the normative changes effected by the original assertoric speech act. This means, among other things, that the subject is no longer required to respond to any challenge to the assertion which has been retracted and also that she is no longer responsible for the consequences of acting on the basis of that assertion. To briefly illustrate this point: if you decide to go to the pub to meet Smith on the basis of my previous assertion that Smith is at the pub after I publicly retracted that assertion, and you do not find Smith, I am not to be deemed responsible for your being disappointed. In fact, my aim in retracting is exactly that of stepping back from such network of commitments. And this is exactly the sense in which retraction is retroactive. We will come back to this feature of retraction in section 8.

5. Retraction and “fault” In section 4.2 we have listed several characteristics of retraction that could be used in forging a more complete notion and have underlined what we take to be its most important feature: its retroactive character. In this section we explore further its normative character. How should we think of the normative character of retraction? A good place to start is to inquire into the sense in which MacFarlane uses the notion of previous assertors being or not being at fault. Although subjects retract previous assertions whose content is deemed false, MacFarlane argues, this “is not tantamount to conceding that one was at fault in making [them]” and “[r]etracting is not admitting fault” (MacFarlane 2014, p. 110). What notion of “fault” is at stake here and what exactly does being at fault amount to? In this



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section we will focus on precisely this question. A first step towards addressing this question is to have a closer look at the dialogues with which we exemplified retraction in section 2. The phrase that expresses the retractor’s attitude towards her previous assertion is “I was wrong”. Let’s try to see in what sense MacFarlane understands speakers to use this phrase in common speech. As a general rule, MacFarlane claims, [I]t is important (…) to distinguish retracting an assertion from claiming that one ought not to have made it in the first place. To say that one was wrong in claiming that p is not to say that one was wrong to claim that p. Sometimes it is right to make a claim that turns out to have been wrong (false) (MacFarlane 2011, p. 148).

And, commenting on the example involving epistemic modals given above, he says: If you find it implausible that Sally would say “I was wrong” in the dialogue above, make sure you’re not interpreting her as saying “I was wrong to say that.” Of course she wasn’t wrong to say what she did. But what she said was wrong, and that is what she is acknowledging (MacFarlane 2011, p. 148).

In these quotes MacFarlane distinguishes between two dimensions along which an assertion could be evaluated. One dimension concerns “what is said”, understood here as the truth-evaluable content of the assertion. Assertors are thus at fault along this dimension when what they say is false as assessed from the current context. Another dimension concerns the saying–the assertoric speech act–itself. Assertors are at fault along this dimension when there is something wrong with the assertoric act itself, regardless of whether the content is true. According to MacFarlane’s indications about how to interpret the locution “I was wrong”, in the example considered above, Sally judges her previous self as being at fault with respect to “what is said”, but not with respect to the saying itself. Distinguishing these two dimensions of being-at-fault certainly helps in clarifying the issue. However, a new question arises regarding the sense in which retractors are said not to be at fault with respect to their assertoric speech acts themselves. What exactly is then the sense of fault applied to assertoric speech acts that MacFarlane is operating with? The following passage may give us some additional hint: 



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Suppose one’s evidence all strongly suggests that Uncle Jack is coming to lunch, and on the strength of that evidence you assert that Uncle Jack is coming. A bit later, Aunt Sally calls to say that Uncle Jack has broken his leg. This makes it quite unlikely that he is coming, so you retract your assertion. Nonetheless, you were perfectly reasonable in making it, and cannot be criticized for having done so. Retracting it is not admitting fault (MacFarlane 2014, p. 110).

 Here the sense in which retractors are said to not be at fault is distinctively epistemic, having to do with one’s available evidence, asserting on the basis of which makes one reasonable (or rational) even if what is said turns out to be false. The epistemic sense of “fault” that MacFarlane is employing here is not what we want to take issue with in the reminder of the paper. We think it is indeed highly plausible that retractors might not be at fault in this sense.12 However, we wonder whether there are other dimensions of assertion evaluation along which retractors could be said to be (or not) at fault–fault, or lack thereof, that cannot be traced down to the epistemic notion used by MacFarlane. In particular, we think that the epistemic notion of fault is not enough to explain a certain asymmetry that has been found between retractions involving predicates of personal taste and retractions involving moral terms.

6. An unexplained asymmetry The asymmetry alluded to above between retractions involving predicates of personal taste and moral terms consists in a difference in the attitude retractors have towards their former selves in the two cases. Although this issue has not been paid much attention to,13 we find that pretty often an attribution of falsity to a certain moral claim contains a (mostly implicit) criticism of the very moral standard from which the claim is issued. Thus, in retracting a moral claim on the basis of its falsity (relative to the moral standard of the context of assessment) the retractor is implicitly criticizing the moral standard previously held in the context in which the assertion was made. On the other hand, we find that this doesn’t happen that often in the taste case. In retracting a taste claim on the basis of its falsity (relative to the taste standard of the context of assessment) the retractor need not implicitly commit to a criticism of the gustatory standard previously held in the context in which the retracted assertion took place.14 To make the difference in retraction between the moral and the taste case more vivid, we will try to pump intuitions by appealing to dialogues



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that sound natural to us. To start with, consider the following retraction involving a predicate of personal taste–a slight modification of the original dialogue presented in section 2: TASTE Angelina, in her childhood: Spinach is tasty. Tom, in present times: You said that spinach is tasty. What about that? Angelina, in present times: Spinach is not tasty. I was wrong. However, there’s nothing bad with liking spinach.

In this dialogue, Angelina uses the locution “I was wrong” to signal that she retracts her previous assertion. But, as the continuation makes clear, she refuses to cast fault on her former self for having the standard of taste she had when the assertion was made, despite the fact that she now holds a different one, which mandates her retraction in the first place. Now, contrast TASTE with the retraction below, involving a moral term–again, a slight modification of the initial dialogue presented in section 2: MORALITY Albert in his childhood: Torturing mice for fun is not wrong. Lucy, in present times: You thought that torturing mice for fun was fine. Albert, in present times: I was mistaken. Torturing mice for fun is wrong. No one should be that cruel to animals.

As Angelina in TASTE, Albert uses the locution “I was mistaken” to signal that he retracts her previous assertion. But, in contrast to TASTE, the continuation makes clear that Albert is disposed to cast fault on his former self for having the moral standard he had when the assertion was made. In fact, in many cases we would expect Albert to feel ashamed for having held such a judgment. How is this asymmetry to be explained? Before proceeding to offer our explanation, let us note that the epistemic dimension of assertionevaluation MacFarlane proposes is not able to explain the asymmetry. According to the epistemic notion of fault, neither Angelina in TASTE nor Albert in MORALITY should be judged as being at fault, since (we can stipulate) nothing went wrong in either case from an epistemic point of view. But if the dialogues presented above track a real phenomenon, the fault that Albert bestows upon his former self and the fault that Angelina refuses to bestow upon her former self cannot be explained by mere appeal to an epistemic notion of fault.15



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7. Circumstance-accuracy What notion would provide an explanation of the asymmetry? The fact that the epistemic notion of “fault” fails to offer an explanation shows that we need to make room for a different dimension of assertion evaluation that allows for a difference in the attitude retractors have towards their previous selves. As it is perhaps obvious in the examples above, such a dimension is one that has to do with the inter-contextual assessment of the values of the parameters in play at the context in which the retracted assertion has been made. Although retraction need not be accompanied by a great deal of reflection, it is nevertheless the case that many retractors will reflect on the circumstances in which the retracted assertion has been made. Such a reflective retractor, as we will call her, will access the circumstances in which the assertion was made in evaluating a previous assertion and thus will attempt to retrieve the specific value of the relevant parameter and assess it. Such an assessment of the specific value of the relevant parameter will be connected with the evaluation of both the assertion itself and its content. We call this dimension of assertionevaluation “circumstance-accuracy”.16 Now, the retractors’ assessment of the specific value of the relevant parameter in play at the circumstances in which the retracted assertion has been made is usually accompanied by a judgment of their previous selves as being at fault or not for endorsing such a value. In other words, in retractions done by reflective retractors (such as Angelina and Albert in TASTE and MORALITY above), besides evaluating the content of the assertion as false, a new dimension of evaluation opens up for retractors, one that might involve substantial criticism of the retractor’s previous self in virtue of the fact that the standard held when the assertion was made is judged to be “the wrong one to have”. More needs to be said about how exactly to understand the notion of “the wrong standard to have”, and thus to make the notion of circumstance-accuracy more precise, but here we rest content with giving this intuitive characterization. What is sure, and what constitutes the explanatory advantage of such a notion, is that it involves an attribution of a different kind of fault to the retractor’s previous self (clearly different from MacFarlane’s epistemic understanding of “fault”). We take this notion of fault to be characteristic of certain acts of retraction made by reflective retractors, and thus part and parcel of the phenomenon of retraction itself. To see in more detail how appealing to circumstance-accuracy helps account for the difference in the attitude the two retractors have towards their former selves, let us go back to the retractions in TASTE and



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MORALITY above. As we have seen, in TASTE Angelina uses the locution “I was wrong” to signal that she retracts her previous assertion. Being what we called above a “reflective retractor”, she accesses the circumstances in which the retracted assertion was made and attempts to retrieve the specific value of the relevant parameter and evaluate it – in this case, her previous standard of taste, according to which spinach falls in the set of tasty things. But, as the continuation makes clear, she refuses to cast fault on her former self for having that standard of taste, despite the fact that she now holds a different one, which mandates her retraction in the first place. There need not be any relation of superiority between that standard and her current one (and this is a case in which we take there to be none). Thus, there is no basis for her to evaluate her former self having that standard as being at fault. This is the sense of “fault” in which Angelina can be said not to be at fault when retracting a previous assertion. Whatever the other dimensions along which she might deem her former self as being at fault, she is not at fault when it comes to circumstance accuracy. Now, we have seen that in Angelina’s case the standard she held when the retracted assertion was made was found not to be “the wrong one to have”. But, as we have seen with MORALITY, this does not happen in all cases. Like Angelina in TASTE, in MORALITY Albert uses the locution “I was mistaken” to signal that he retracts his previous assertion. Being a reflective retractor, he accesses the circumstances in which the retracted assertion was made and attempts to retrieve the specific value of the relevant parameter and evaluate it–in this case, his previous moral standard according to which torturing mice for fun falls in the set of acts that are morally permitted. But, in contrast to TASTE, the continuation makes it clear that Albert is disposed to cast fault on his former self for having had that moral standard. In this case there is a relation of superiority between that standard and his current one, so Albert has a basis to evaluate his former self as being at fault–the standard he had is “the wrong one to have”. Not only does Albert find his former self at fault along other dimensions (truth, accuracy, etc.), but he also finds himself at fault when it comes to circumstance-accuracy. Thus, in short, appealing to circumstance-accuracy makes room for a different dimension of assertion evaluation, one that is different from the epistemic dimension used by MacFarlane and one within which the asymmetry between the moral and the taste case can be explained. To further support the thought that circumstance-accuracy is an important notion to have in the philosopher’s explanatory toolkit, in the next section we turn to a feature of retraction that we have highlighted in section 4:



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namely, its retroactive character. Taking a cue from the legal domain, we show how an important distinction found there can be applied to retractions as well. Not only we find this distinction illuminating for the study of the particular features pertaining to retractions in different domains, but we think the distinction buttresses the idea that circumstanceaccuracy is a stable feature of retractions in general.

8. Retraction and the law: strong and weak retroactivity In his paper “Retroactive Law”, Stephen Munzer (1977) analyses the concept of retroactivity in jurisprudence and draws a distinction between weak and strong retroactivity of a law. Under the weak interpretation, a retroactive law changes the normative status of a previous act, but does so only on a forward-looking basis. This means that the retroactive efficacy of the new law changes the legal (normative) status of an act performed before the enforcement of the law only from the time in which the law is officially enforced. To illustrate the idea of weak retroactivity, consider the following scenario: Case 1: S performs an act A at t1 that is lawful according to the law in force at t1. At t2 (t2> t1) a new law is promulgated the enforcement of which has retroactive efficacy which makes A unlawful. The act performed by S at t1 is from t2 onward unlawful, but S is not indictable for the consequences of that act prior to t2; only for those posterior to t2.

The new law has retroactive efficacy because it changes the legal status of an act that was executed before the time in which the new law was enforced. But it is weakly retroactive because its normative efficacy only concerns the consequences of that action posterior to the enforcement of the law. On the other hand, under the strong interpretation of retroactivity, a retroactive law changes the legal (normative) status of a previous act both on a backward-looking basis as well as on a forward-looking basis. This means that under the strong interpretation, once the retroactive law is enforced, all acts performed before the new law that are targeted by that law change their legal (normative) status not only from the time in which the new law is enforced but also with respect to the lapse of time inbetween the targeted act and the enforcement of the law. Consider the following example:



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Chapter Five Case 2: S performs an act A at t1 that is lawful according to the law in force at t1. At t2 a new law is promulgated with retroactive efficacy which makes A unlawful. The act performed by S at t1 changes its status and becomes unlawful not only from t2 onward but also with respect to the lapse of time between t1 and t2. Thus, S is indictable for the consequences of A prior to t2 as well as for those posterior to t2.17

The law has retroactive efficacy because it changes the legal status of an act performed before its enforcement. But it is strongly retroactive because the change is done both on a backward-looking basis as well as on a forward-looking basis. This means that its normative efficacy concerns the consequences of that action that are both posterior and anterior to the enforcement of the law. Let us consider a fictional example that might help grasping the difference between the two readings of the retroactivity of a law. Suppose that a pharmaceutical company in January 2013 advertises a new diabetes drug (call it “drug-X”) making unsupported safety claims over its product. Although the company was aware of the fact that such safety claims were quite unsupported, insofar as the regulations concerning the safety of pharmaceutical products at that time were rather loose, the commercialization of that product was deemed as legal according to the law in force in January 2013. The advertising campaign was extremely successful and many people affected by diabetes took drug-X. However drug-X had serious side-effects, causing long-lived health problems to many patients, who had to spend quite a lot of money on medical treatments to counter its side-effects. In January 2014, a new law is promulgated the enforcement of which has retroactive efficacy which makes the commercialization of drug-X unlawful. Drug-X is immediately withdrawn from the market. In addition the new law forces the company to pay for all the medical treatments that are demonstrably associated with drug-X. According to the weak interpretation of retroactivity, the company has to pay only for those treatments that patients who took drug-X still require after the new law was promulgated. On the other hand, under the strong reading of retroactivity, the company is forced to pay for both the treatments that are still needed by patients after the promulgation of the law, and all the treatments associated with drug-X that patients had to do in the lapse of time between the commercialization of the product and the enforcement of the new law.18 We think that the distinction between weak and strong retroactivity can be extended to the case of retraction. Thus, there is a kind of retraction with weak retroactive normative efficacy by means of which the agent intends to disavow the assertoric commitments associated with the



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targeted assertoric act only on a forward-looking basis. And there is a kind of retraction with strong retroactive normative efficacy by means of which an agent intends to disavow the assertoric commitments associated with the targeted assertoric act both on a forward-looking and a backwardlooking basis. The idea is, then, that this way of understanding retroactivity and the difference in normative reach that a retroactive law can have under the weak and strong interpretation might provide a good model not only for understanding the kind of normative efficacy that an act of retraction might have with respect to the kind of commitment the subject has undertaken with her original assertoric act, but it might also help us explaining the asymmetry between the two cases illustrated in the previous section. If we are right in claiming that, contrary to the moral case where an act of retraction is implicitly critical of the very moral standpoint from which the retracted assertion was made, retraction in the taste case need not have this feature of criticism of one’s previous sensibility, then it seems that the strong and weak interpretation of retroactivity might turn out to be particularly useful in modeling and understanding this asymmetry concerning the normative reach of an act of retraction in the two domains. Thus, the strong interpretation of retroactivity seems to be appropriate with respect to the moral case where in retracting a moral assertion we not only give up the normative commitments associated to the previous assertoric act from the time of the retraction onward, but also, in implicitly criticizing the standard held by the subject in the context where the assertion was made as the wrong one to have, we want to distance ourselves from all the normative consequences associated with the assertion even in the lapse of time between the asserting and the retracting. The thought is that that assertion was never in good shape because issued by a wrong standard. In this respect, a substantive attribution of fault seems to be involved in any act of retraction targeting a moral assertion. We blame ourselves for having asserted a proposition that is false because it is grounded in the wrong moral standard. And, in this respect, we want to distance ourselves from all the assertoric commitments associated with that assertion and withdraw our responsibility from all the consequences that follow from such commitments (both post and ante our retracting). In this respect the strong interpretation of retroactivity applies in the moral case. What we aim at doing is to distance ourselves from the normative commitments associated with the targeted assertion not only on a forwardlooking basis but also on a backward-looking basis. We want to say that making such an assertion was a bad thing to do – and not just because



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false from the current perspective, but because issued from a moral standard that we evaluate as the wrong one to have. The contrast with the taste case is sharp. Here the weak interpretation of retroactivity seems the most appropriate since no implicit criticism to the previous standard of taste is associated with an act of retraction. In retracting a previously unretracted assertoric speech act we want to distance ourselves from the assertoric commitments on a forward-looking basis, and this is because we evaluate the content of such assertion as false from our current perspective. But because we do not take our retraction to be implicitly critical of our previous standard of taste, there is no reason for us to distance ourselves from the assertoric commitment also on a backward-looking basis. After all we are not licensed to conclude that that assertion was the incorrect one to make. In fact, we should consider that assertion as perfectly permissible in the past since grounded on a perfectly legitimate taste sensibility. In this sense, retractors can deem their former selves as not being at fault in a stronger sense than the epistemic one preferred by MacFarlane.

9. Summary, conclusions and further issues Radical relativists have claimed that accounting for retraction is an advantage their view has over rival positions. Despite the dialectical importance of the phenomenon, not much attention has been paid to it in the literature. Our aim in this paper has been to fill this gap by offering some key elements that would form the basis of a more thorough analysis. We have also inquired into the sense of “fault” in which retractors can be said not to be at fault in retracting. In this connection, we have pointed towards an asymmetry between retractions involving predicates of personal taste and moral terms and have attempted to provide an explanation of this asymmetry. Our explanation has led us to a less explored dimension of assertion evaluation–circumstance accuracy– appeal to which provides an explanation of the aforementioned asymmetry. In the last part we sought to give support to the thesis that circumstance accuracy is the right notion to appeal to by borrowing a distinction found in jurisprudence between weak and strong retroactivity and by applying it to the case of retraction. There are some issues that our paper both contributes to and opens up for further research. Despite a common core of features that makes them what they are (which we attempted to unveil in section 4), retractions come in many forms that might vary with, among other factors, the type of discourse they appear in. We have seen this in the case of retractions in the



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moral and taste domains, but we have only scratched the surface here. A more thorough examination of the features of retraction pertaining to those, and other, domains19 well exceeds the scope of this paper. Nevertheless, we take our highlighting, discussion and explanation of the asymmetry between retractions in the moral and taste domains to contribute to this project – alongside with aiming to provide the basis for a more complete characterization of retraction in general. Other issues that might be fruitfully inquired into are connected with the notion of circumstance accuracy. Not only a deeper philosophical understanding of this notion is desirable, but, we think, it is also important to clarify the relation between this dimension of assertion evaluation and other such dimensions–for example, truth, accuracy, or the epistemic one used by MacFarlane. It would be interesting to see how this relation pans out in the case of each of the views that purport to account for retraction, and to compare the results. In particular, since we are dealing with radical relativism here, it would be interesting to see how circumstance accuracy relates with the (relativized) notions of truth and accuracy such a view employs. At a first glance, circumstance-accuracy seems to lack the exclusively assessor-oriented character those other notions have. Whether or not this is a problem for radical relativism is a question that we leave for another occasion. We hope that our characterization of the phenomenon of retraction and the explanation of the asymmetry between retractions in the moral and taste case will provide a fertile ground for further discussion.

References Brandom, R. 1983, “Asserting”, Noûs 17 (4), 637 Brogaard, B. 2010, “Moral Contextualism and Moral Relativism”, Philosophical Quarterly 58 (232), 385 Cappelen, H. and J. Hawthorne 2009, Relativism and Monadic Truth, Oxford: Oxford University Press Egan, A. 2010, “Disputing about Taste” in Warfield, T. and R. Feldman (Eds.) Disagreement, Oxford: Oxford University Press Ferrari, F. 2014, Disagreement and the Normativity of Truth beneath Cognitive Command, PhD Thesis: University of Aberdeen Ferrari, F. ms, “Disagreement about Taste and Alethic Suberogation”, unpublished manuscript von Fintel, K. and A. Gillies 2008, “CIA Leaks”, The Philosophical Review 117, 77



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von Fintel, K. and A. Gillies 2011, “‘Might’ Made Right” in Egan, A. and B. Weatherson (Eds.), Epistemic Modality, New York: Oxford University Press, 108 Kaplan, D. 1989, “Demonstratives”, in Almog, J., J. Perry and H. Wettstein (Eds.), Themes from Kaplan, Oxford: Oxford University Press, 481 Kompa, N. 2005, “The Semantics of Knowledge Attributions”, Acta Analytica 20 (1), 16 Kölbel, M. 2004, “Indexical Relativism vs. Genuine Relativism”, International Journal of Philosophical Studies 12 (3), 297 Krabbe, E. 2001, “The Problem of Retraction in Critical Discussion”, Synthese 127 (1-2), 141 Lasersohn, P. 2005, “Context Dependence, Disagreement, and Predicates of Personal Taste”, Linguistics and Philosophy 28 (6), 643 MacFarlane, J. 2003, “Future Contingents and Relative Truth”, The Philosophical Quarterly 53 (212), 321 —. 2005a, “Making Sense of Relative Truth”, Proceedings of the Aristotelian Society 105 (3), 321 —. 2005b, “The Assessment Sensitivity of Knowledge Attributions”, Oxford Studies in Epistemology 1, 197 —. 2011, “Epistemic Modals Are Assessment-Sensitive”, in Egan, A. and B. Weatherson (Eds.), Epistemic Modality, New York: Oxford University Press, 144 —. 2014, Assessment-Sensitivity: Relative Truth and Its Applications, Oxford: Oxford University Press Munzer, S. 1977, “Retroactive Law”, The Journal of Legal Studies 6, 373 Peirce, C.S. 1934, Belief and Judgment, Cambridge (MA): Harvard University Press Recanati, F. 2007, Perspectival Thought: A Plea for Moderate Relativism. Oxford: Oxford University Press Sundell, T. 2011, “Disagreements about Taste”, Philosophical Studies 155, 267 Wright, C. 2007, “New Age Relativism and Epistemic Possibility: The Question of Evidence”, Philosophical Issues 17 (1), 262 —. 2012, “Replies–Part III: Truth, Objectivity, Realism and Relativism”, in Coliva, A. (ed.) Mind, Meaning, and Knowledge: Themes from the Philosophy of Crispin Wright, Oxford: Oxford University Press, 418



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Notes 

1

MacFarlane doesn’t consider moral judgments to be apt for relative truth. But we see no obstacle in extending the framework to cover also the moral domain. After all, relativism is a very well represented meta-ethical view, and much discussion about relativism in the last century or so comes exactly from that neighborhood. 2 Moderate versions of relativism have been proposed, among others, by Kölbel (2004) for evaluative predicates in general, by Kompa (2005) for “know”, by Recanati (2007) for meteorological verbs and by Brogaard (2008) for moral terms. 3 We are using here the term “utterance” in a loose and semi-technical way. For the sake of precision we should use the term “occurrence” instead and thus be sensitive to Kaplan’s distinction between utterances and occurrences of sentences (see Kaplan 1989, pp. 522-523). For the purpose of this brief exposition of Kölbel’s and MacFarlane’s proposals, however, we ignore this subtle distinction. 4 As Wright (2012, p. 441) points out, this can be contested on the basis that the two views make slightly different predictions about the possibility of expressing the faultlessness of the disagreement from within a committed perspective. 5 Can there be retraction at the level of thought? Our worry with such a phenomenon is that, lacking the public aspect of a paradigmatic act of retraction, it might turn out to be a different phenomenon than that investigated in this paper. We leave the question whether there are any genuine cases of “mental retraction” aside in this paper. 6 Retraction data haven’t been unanimously accepted in the literature. In the case of epistemic modals, for example, von Fintel and Gillies (2011) question the solidity of the retraction data by providing examples in which the speaker stands by her previous claim instead of retracting it. See, however, MacFarlane (2014, chapter 10) for a response to von Fintel and Gillies’ examples. For worries about retraction data in general and its philosophical significance, see Wright (2007). 7 Witness, for example, MacFarlane’s (2014) pluralist take on the issue. 8 The kind of incompatibility at issue here is what MacFarlane calls “noncotenability”: “I disagree with someone’s attitude if I could not coherently adopt that same attitude (an attitude with the same content and force) without changing my mind – that is, without dropping some of my current attitude.” (MacFarlane 2014, p. 121) 9 For the distinction between autocentric and exocentric uses see Lasersohn (2005). 10 This is not always the case though: take the case of a spokesperson that is in charge to make some claim on my behalf. He asserts that p and in so doing he undertakes some normative commitment on my behalf. I might legitimately retract that very speech act even in cases in which I’ve never asserted that p. But such cases are rather exceptional. 11 This account of assertoric commitment is broadly in line with that advocated in earlier work by MacFarlane (see, for example, MacFarlane 2005a), which is, in turn, inspired by Brandom’s (1983) account. 12 Note, though, that appeal to relative truth is not needed in order to explain this



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 sense of faultlessness. Any theory of truth that has the resources for distinguishing between truth and (non-idealized) justification would be in a position to account for it. We don’t insist on this point since nothing important for our project hinges on it. 13 For a detailed discussion of the various differences in terms of normativity between moral judgments, comedic judgments and judgments of taste see Ferrari (2014, chapter 3). 14 See Ferrari (ms) for a non-relativistic account of truth in the taste domain that is compatible with this non-critical attitude of retraction it the taste domain and, more generally, it is able to account for the main normative differences between disagreements as they occur in the moral and in the taste domain. 15 Nor can it be explained by appeal to the notion of accuracy that MacFarlane uses (for the radical relativist definition of accuracy, see MacFarlane 2014, p. 127). Since accuracy is very closely related to truth, a retractor will not only deem the content of an assertion false, but also inaccurate (relative to context of assessment occupied by the retractor). So, both Angelina and Albert’s retracted assertions will be deemed as inaccurate by the radical relativist and thus they will both be judged as being at fault. Accuracy is thus of no use in explaining the asymmetry between the moral and taste case. 16 We think that our notion of circumstance-accuracy bears some resemblance to a notion that is implicit in Sundell’s (2011) discussion of various types of disagreement–in particular, what he calls “context-disagreement”. However, we are not exploring here disagreement along the assertion dimension we dub “circumstance-accuracy”, nor are we claiming that ordinary disagreement involving a sentence and its negation should be interpreted as disagreement about the context the interlocutors are in. 17 As Munzer himself notes, the strong interpretation does not suppose that a retroactive law obliterates the past. Such a law does not mean that the preenactment status never existed–nor does it make the consequences of the applications of the previous law unlawful during the period in which it was in force. A retroactive statute making legal an act that was once criminal would rarely entitle a person to recover damages for false imprisonment. 18 It is important to not be distracted by the following disanalogy: in the law cases briefly sketched here the retroactivity of the law has the effect of attributing legal responsibility to agents for acts performed before the enforcement of the law; whereas an act of retraction has typically the effect of disavowing the normative commitments of a previous assertoric act. But the presence of this disanalogy is a mere artifact of the way in which we introduced and discussed the notion of retroactivity in the legal case. Although concrete examples of retroactive laws that have the legal effect of disavowing an agent from the legal responsibility of an act performed before the enforcement of the law are harder to find, they are certainly possible. 19 See Krabbe (2001) for a previous discussion of the different forms retraction takes across discourses.



CHAPTER SIX CONTRADICTIONS, DISAGREEMENT AND NORMATIVE ERROR SAMUELE IAQUINTO

1. Introduction The aim of this paper is to discuss some counterexamples to the following principle: (P) Necessarily, for every proposition p, for every cognitive agent S and for every cognitive agent S*, if S believes that p and S* believes that ¬p, then either S makes a normative error or S* makes a normative error.

Here, with “normative error” I refer to the violation of doxastic norms such as: (DN) S should believe that p only if it is not false that p.

Obviously, if we accept the principle of bivalence, then this norm and a norm such as: (DN*) S should believe that p only if it is true that p,

are the same. However, as I will show in the following section, in analysing principles such as (P) some philosophers might reject the principle of bivalence. Consequently, they might deny that the norms (DN) and (DN*) are the same. In those cases, holding (DN) could rather involve holding a norm such as: (DN**) If it is false that p, S should not believe that p.

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Note that (DN**) allows S to believe an indefinite proposition without making a normative error; it requires only that S avoids false beliefs. If we assume the identity between S and S*, then (P) regulates what I'm going to call psychological contradiction; conversely, if we assume the non-identity between S and S*, then the principle regulates cases of disagreement. Following Grim (2004), at least three kinds of contradiction can be distinguished: syntactic, semantic and ontological. Grim quotes some definitions of syntactic contradiction. For example, Kalish, Montague and Mar (1980) state that “a contradiction consists of a pair of sentences, one of which is the negation of the other” (Kalish, Montague and Mar 1980, p. 18). And Forbes (1994) states that “two formulae are explicitly contradictory if and only if one is of the form q and the other of the form ~q, that is, if one is the negation of the other” (Forbes 1994, p. 102, italics in the original). In sum, we can say that a contradiction is syntactic if and only if it can be formalized in the following way (see Haack 1978, p. 244): a. D ‫¬ ޔ‬D

Note that this formulation does not directly involve any truth-predicate. However, someone could offer a definition of contradiction by directly appealing to the notions of truth and falsity. Grim quotes, among others, two definitions. The first one is proposed by Prior (1967): Contradictory negation, or contradiction, is the relation between statements that are exact opposites, in the sense that they can be neither true together nor false together (Prior 1967, p. 458).

The other one is proposed by Wolfram (1989): Two statements are inconsistent with each other if they cannot both be true, and more specifically if the truth of one would entail the falsity (nontruth) of the other (Wolfram 1989, p. 163).

In light of these definitions, we can call semantic contradiction every formula such as (where “T” is a truth-predicate, while “D” is a metavariable): b. T(D) ‫ ޔ‬T(¬D).

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Note that hereinafter I assume that b expresses a genuine semantic contradiction only under the hypothesis that the principle of bivalence holds. b says that at the same time the proposition D is true and false. The third kind of contradiction is called by Grim ontological because it explicitly involves certain ontological notions, namely, objects and properties. An example of ontological definition is due to Routley and Routley (1985): “A contradictory situation is one where both B and ~B (it is not the case that B) hold for some B” (Routley and Routley 1985, p. 204). We can formalize it in the following way (see Berto 2006, p. 23): c. 䳭x䳭P (P(x) ‫¬ ޔ‬P(x)).

We have a contradiction because the formula describes a state of affairs in which the object x both has the property P and lacks the property P. Given these definitions of contradiction, I distinguish among three kinds of disagreement: syntactic disagreement, semantic disagreement and ontological disagreement. Generally speaking, I say that we have a case of syntactic disagreement when, given two cognitive agents, from the conjunction of their beliefs we obtain a formula such as a. We have a case of semantic disagreement when from the conjunction of their beliefs we obtain a formula such as b. Finally, we have a case of ontological disagreement when the conjunction of their beliefs entails a formula such as c. In this paper I will propose a comparison between a three-valued approach and a relativist approach in offering counterexamples to (P). In section 2 I will present some counterexamples by adopting the threevalued approach. I will show that such an approach requires the support of controversial metaphysical views. Furthermore, it allows us to analyse only cases of syntactic disagreement. In sections 3, 4 and 5 I will examine the version of semantic relativism discussed by Cappelen and Hawthorne (2009), showing that adopting it is preferable since, in offering counterexamples to (P), it does not commit us to hold the controversial views that I will present in section 2. Furthermore, it allows us to propose genuine counterexamples not only in cases of syntactic disagreement, but also in cases of semantic and ontological disagreement.

2. Syntactic contradictions and syntactic disagreement In this section I will analyse the approach by which we could obtain genuine counterexamples to (P) by adopting some form of three-valued logic.

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Someone might try to propose a counterexample with the following argument. Argument 1. Consider the two sentences: 1. Smith is bald 2. ¬(Smith is bald).

Of course, the conjunction of 1 and 2 is a syntactic contradiction. Let us suppose that Smith is a borderline case of baldness. In this scenario, following a truth-value gap theory (see Soames 1999), S could decide to adopt the three-valued logic proposed by Kleene (1952) (see also Tye 1994, pp. 194-202). Kleene states that, given a conjunction D if at least one conjunct of Dis indefinite, then D is indefinite as well. Obviously, adopting this kind of logic requires us to reject the principle of bivalence. So, S does not consider (DN) and (DN*) to be the same norm. Now, suppose that S decides to adopt the norm (DN**). If S evaluates both the proposition expressed by 1 and the proposition expressed by 2 as indefinite, then S treats also the proposition expressed by the conjunction of 1 and 2 as indefinite. According to (DN**), if S believes the proposition expressed by this conjunction, she does not make a normative error, contrary to the principle (P). Similar considerations can be easily applied to cases of syntactic disagreement. Let us consider another example. Argument 2. S has to evaluate the propositions expressed by: 3. There will be a sea-battle tomorrow 4. ¬(there will be a sea-battle tomorrow).

3 and 4 express future contingent propositions. Suppose that S believes that we cannot determine the truth value of future contingent propositions: she thinks that they are indefinite. To express this idea, she decides to adopt àukasiewicz (1920)'s three-valued logic. Given a proposition D, if Dis indefinite, then ¬D is indefinite. And the conjunction of two indefinite propositions is indefinite as well. Again, suppose that S is willing to adopt (DN**). Contrary to (P), she does not make a normative error by believing the proposition expressed by the conjunction of 3 and 4. Concerning this kind of arguments against (P), I think that it is not clear at all why satisfying (DN**) should be sufficient to preserve S from an error. More precisely, it is not clear unless we adopt a specific metaphysical standpoint.

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Consider the argument 1. Those who want to employ it have to explain the reasons why we should reject the principle of bivalence in borderline cases. In other words, they have to hold that there is no fact of the matter about whether or not Smith is bald. As I will show in section 5, semantic relativism allows us to offer counterexamples to (P) without adopting this controversial view. Furthermore, given that in the argument 1 we reject the principle of bivalence, we could propose only counterexamples to cases of syntactic contradiction and syntactic disagreement. Someone might object that the rejection of the principle of bivalence does not prevent us from offering counterexamples to cases of semantic contradiction and semantic disagreement by proposing the following argument. Given the disquotational principle “T(D) if and only if D”, from “D ‫¬ ޔ‬D” we can infer “T(D) ‫ ޔ‬T(¬D)”. We could hold that if D is indefinite, then also T(D) is indefinite. Therefore, “T(D) ‫ ޔ‬T(¬D)” is indefinite as well. Now, according to (DN**), a given cognitive agent could believe the proposition expressed by “T(D) ‫ ޔ‬T(¬D)” without making a normative error (similar considerations can be applied to cases of disagreement). And “T(D) ‫ ޔ‬T(¬D)” is exactly what you call semantic contradiction (see the formula b). My answer is that in this argument “T(D) ‫ ޔ‬T(¬D)” does not stand at all for what I call semantic contradiction. When I introduced the formula b, I assumed that it expresses a genuine semantic contradiction only under the hypothesis that the principle of bivalence holds. What about argument 2? In proposing the putative counterexample to (P), it is necessary to specify why the future contingents are supposed to be indefinite. Indeed, if we take the future contingents to be epistemically indefinite, the argument does not work. More precisely, someone might hold that, even though we cannot know how to determine the truth-value of a future contingent, either 3 expresses a true proposition or 4 does. To defend the argument 2 we should hold that today it is not ontologically determined whether or not there will be a sea-battle tomorrow. Again, I think that the relativist approach is preferable. As I will show in the next three sections, it allows us to offer counterexamples to (P) without buying the controversial view just discussed. Furthermore, it offers counterexamples involving not only syntactic disagreement, but also semantic and ontological disagreement.

3. Semantic contradictions and semantic disagreement Let us consider the case of disagreement (S  S*) in which S believes the proposition expressed by the sentence:

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while S* believes the proposition expressed by the sentence: 6. ¬(Martha is justified in believing that p).

Now we adopt the principle of bivalence. Then we have to reformulate (DN) in the following terms: (DN*) S should believe that p only if it is true that p.

Let us describe the role played by the epistemic standards in the semantic evaluation of 5 and 6 by adopting a form of semantic relativism. More precisely, hereinafter I present and discuss the form of semantic relativism described by Cappelen and Hawthorne (2009). According to this account, propositions do not instantiate the monadic property of truth simpliciter, but rather a property of relative truth. Given an index (i) = {world, time, location, epistemic parameter, aesthetic parameters, …}, where (i) is an n-tuple of parameters that are relevant to the assertability of a certain propositional content p, p is true (or false) at that n-tuple. Consider the sentence 5: its propositional content is not true or false simpliciter, rather it is true at various n-tuples {world, time, epistemic parameter} and false at others. The propositional content is supposed to be non-specific with respect to the parameter index (i)1. The parameters are provided by the circumstance of evaluation. Sentence 5 can express the same content across a wide range of contexts of utterance. In other words, the propositional content is supposed to be context-insensitive. In trying to generate counterexamples to (P), why should we adopt such a framework? Because it would allow us to describe cases of faultless semantic disagreement. Some philosophers have recently raised objections against the idea that relativism allows us to properly describe genuine cases of faultless disagreement (see in particular Rosenkranz’s (2008) objections against Kölbel (2004) and Lasersohn (2005). See also Moruzzi’s (2008) objections to MacFarlane’s (2007) account of disagreement). Even though I think that these objections deserve attention, I will not examine them here. Indeed, my goal is simply to compare the three-valued approach and the relativist approach under the hypotheses that there are in fact such cases and that relativism is able to properly describe them. In order to represent such cases, we have to introduce into our language a disquotational operator “it is true that”, by adopting the following principle (see Cappelen and Hawthorne 2009, p. 13):

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(T) It is true that p is true at an n-tuple if and only if the content p is true at that n-tuple.

It allows us to characterize the monadic truth predicate with the proper tools of semantic relativism. Indeed, given the principle (T), the sentence: 7. It is true that p if and only if p

expresses a proposition that is true at all n-tuples (cf. Cappelen and Hawthorne 2009, p. 13). And we can demonstrate that 7 entails: 8. It is true that p if and only if it is not true that not-p.2

Given that the propositional content is non-specific with respect to the index (i), we can properly describe cases of semantic disagreement. Consider, for example, the sentence: 9. The vegetarian soup is delicious.

According to semantic relativism, 9 expresses a semantic content which is not true or false sic et simpliciter, but true at various n-tuples {world, time, standard of taste} and false at others. Given the principle (T), the proposition expressed by: 10. It is true that the vegetarian soup is delicious if and only if the vegetarian soup is delicious,

is true at all n-tuples. But 10 entails: 11. It is true that the vegetarian soup is delicious if and only if it is not true that ¬(the vegetarian soup is delicious).

Given 11, it is easy to see that if a given cognitive agent S believes the proposition expressed by 9 and another one, S*, believes the proposition expressed by its negation, then one of them believes a true proposition, while the other one believes a false proposition. In describing such a case we do not reject the principle of bivalence. This is the reason why semantic relativism allows us to describe it in terms of semantic disagreement. The assumption that believing a false proposition is a normative error (see (DN*)) does not take into account the role played by the index (i) in

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the semantic evaluation of sentences like 5 or 6. Consequently, we have to adopt the new doxastic norm3: (RDN) S should believe that p in a context c only if it is true that p relative to the index (i) that is operative for S in c.

According to (RDN), in case of disagreement it is not necessary that at least one cognitive agent, S or S*, makes a normative error. Reconsider the sentences 5 and 6. S can evaluate the proposition expressed by 5 as true relative to an index (i) = {world, time, location, epistemic parameter x}, where “epistemic parameter x” stands for a flexible standard to attribute justification, while S* can evaluate the proposition expressed by 6 as true relative to an index (i*) = {world, time, location, epistemic parameter y}, where “epistemic parameter y” stands for a rigid standard to attribute justification. Consider another example. Take the sentence: 12. The Rokeby Venus is beautiful.

Its semantic content is supposed to be true at various n-tuples {world, time, aesthetic parameter} and false at others. Given the principle 8, if S believes the proposition expressed by 12 and S* believes the proposition expressed by its negation, then one of them believes a true proposition, while the other one believes a false proposition. But, again, according to (RDN) it is not necessary that at least one cognitive agent, S or S*, makes a normative error. In case of disagreement, S can evaluate the proposition expressed by 12 as true relative to a certain aesthetic standard, while S* can evaluate the proposition expressed by the negation of 12 as true relative to another one. So far I have considered the use of semantic relativism in analysing cases of disagreement. But what about cases of psychological contradiction (cases in which S and S* are taken to be the same agent)? Semantic relativism does not seem to be useful. It would allow us to offer a genuine counterexample to (P) only if we demonstrated that S evaluates two contradictory propositions relative to two different parameters at the same time. But the very possibility of such a case is highly debatable. In fact, it seems to be implausible from a cognitive point of view.

4. Ontological disagreement The relativist approach can be used also when the conjunction of two beliefs entails an ontological contradiction. Let us formalize 5 as:

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5*. J(m, p)

and 6 as: 6*. ¬J(m, p).

We know that it is possible to demonstrate that the conjunction of 5* and 6* is a semantic contradiction by using the disquotational operator introduced with (T). Since ex falso quodlibet, in second-order logic it is possible to deduce from this conjunction the ontological contradiction c. But, considering the semantic evaluations relative to the indexes (i) and (i*), believing the proposition formalized as 5* or the proposition formalized as 6* is not sufficient to violate (RDN). In other words, even though we can deduce an ontological contradiction from the conjunction of the beliefs entertained by S and S*, this is not evidence that at least one of them makes a normative error.

5. Conclusions: making the comparison In trying to find out genuine counterexamples to (P), I have analysed two different approaches. The first one is based on three-valued logic. In my opinion, neither argument 1, which employs the three-valued logic proposed by Kleene (1952), nor argument 2, which is based on the threevalued logic due to àukasiewicz (1920), seems to offer a convenient solution (when compared with the relativist solution). According to argument 1, given that Smith is a borderline case of a bald person, under the hypothesis that a cognitive agent S is willing to adopt both Kleene's logic and a doxastic norm such as (DN**), S could believe the proposition expressed by the conjunction of 1 and 2 without making a normative error. However, in order to show that adopting (DN**) is appropriate to treat borderline cases, we would hold a controversial metaphysical view, that is, the idea that there is no fact of the matter about whether or not Smith is bald. Furthermore, argument 1 allows us to analyse only syntactic contradictions and syntactic disagreement. By employing argument 2, we face an analogous problem. We can state that, since 3 and 4 express future contingent propositions, the conjunction of 3 and 4 is indefinite and that whoever adopts (DN**) could avoid the normative error in believing it. But, also in this case, whether or not S makes an error seems to depend on a controversial metaphysical assumption, which is required in order to exclude that S evaluates a false

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conjunction as indefinite. In addition, by using this argument we are able to consider only syntactic contradictions and syntactic disagreement. In sections 3 and 4 I have presented the relativist account discussed by Cappelen and Hawthorne (2009). According to it, propositions instantiate a property of relative truth. Given an n-tuple of parameters that are relevant to the assertability of a certain propositional content p, p is true at that n-tuple. This propositional content is supposed to be non-specific with respect to the n-tuple of parameters. By introducing into our language a disquotational operator “it is true that”, we can represent genuine cases of semantic and ontological disagreement (even though –for the reason presented above– semantic relativism does not allow us to offer proper counterexamples in cases of psychological contradiction). And by employing the doxastic norm (RDN) we can show that, contrary to (P), it is not necessary that, if S believes the proposition expressed by the sentence 5 and S* believes the proposition expressed by 6, then at least one of them makes a normative error. My idea is that, in offering counterexamples to (P), from a metaphysical point of view the relativistic framework is more convenient than the threevalued approach discussed above. Of course, generally speaking semantic relativism is not completely independent of some form of metaphysical background. In proposing what she calls “factual relativism”, for example, Einheuser (2008) correctly holds that her account is somehow based on metaphysical assumptions about facts concerning taste. A deep analysis of factual relativism –she underlines– involves two different kinds of question: On the one hand there is the metaphysical question of the nature of the facts in the area of interest, in our case the facts concerning taste. These are the things in the world that the truth of our statements is sensitive to. Second there is the semantic question of how the truth conditions of statements about the area of interest are to be given in terms of the facts of that area. Facts and meanings are theoretical concepts introduced to systematically account for our linguistic behavior and cognitive interaction with the world (Einheuser 2008, p. 189, italics in the original).

In presenting her proposal, she argues that in evaluating the proposition expressed by a sentence such as 9 we have to take into account the role played by what she calls subjective general taste facts. And an adequate development of this view –she continues– commits us to say something about the “nature” of these facts. In Einheuser's words: Subjective general taste facts are determined by a combination of two factors: The world itself, independent of any particular tasters, determines the physiochemical features of items on the bases of which tasters classify

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them as tasting good or bad. It also determines the physiochemical features of individual tasters which are responsible for their classifying certain items as tasting good or bad. The world thus uniquely determines the particular taste facts yet not the general ones. Call the contribution made by the world a substratum. The second contributing factor is a collection of physiological and psychological features of particular tasters. I will call the contribution of particular tasters a (taste-)perspective and say that a perspective induces taste facts over a substratum (Einheuser 2008, p. 190, italics in the original).

It is hard to imagine a well-developed relativistic account which is not somehow committed to buying certain metaphysical assumptions. However, my point is that the relativist approach –as I have just shown in sections 3 and 4– allows us to find counterexamples to (P) without buying the controversial views that there is no fact of the matter about whether or not a borderline case of a bald person is actually bald or that today it is not ontologically determined whether or not there will be a sea-battle tomorrow. Furthermore, given that the relativist approach adopts the principle of bivalence, contrary to the three-valued approach it allows us to offer genuine counterexamples to all the kinds of disagreement considered above: syntactic, semantic and ontological.4

References Berto, F. 2006, Teorie dell'Assurdo. I Rivali del Principio di Noncontraddizione, Roma: Carocci Cappelen, H. and J. Hawthorne 2009, Relativism and Monadic Truth, Oxford: Oxford University Press Einheuser, I. 2008, “Three Forms of Truth Relativism”, in GarcíaCarpintero, M. and M. Kölbel 2008, 187 Forbes, G. 1994, Modern Logic, Oxford: Oxford University Press García-Carpintero, M. and M. Kölbel (eds.) 2008, Relative Truth, Oxford: Oxford University Press Grim, P. 2004, “What Is a Contradiction?”, in Priest, G., J.C. Beall and B. Armour-Garb (eds.), The Law of Non-contradiction. New Philosophical Essays, Oxford: Clarendon Press, 49 Haack, S. 1978, Philosophy of Logics, Cambridge: Cambridge University Press Kalish, D., R. Montague and G. Mar 1980, Logic: Techniques of Formal Reasoning, New York: Harcourt Brace Jovanovich Kleene, S.C. 1952, Introduction to Metamathematics, Amsterdam: NorthHolland

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Kölbel, M. 2004, “Faultless Disagreement”, Proceedings of the Aristotelian Society 104 (1), 53 Lasersohn, P. 2005, “Context Dependence, Disagreement, and Predicates of Personal Taste”, Linguistics and Philosophy 28 (6), 643 àukasiewicz, J. 1920, “O Logice TrójwartoĞciowej”, Ruch Filozoficzny 5, 170-171, trans. “On Three-Valued Logic”, in Borkowski, L. (ed.) 1970, Selected Works by Jan àukasiewicz, Amsterdam: North–Holland, 87 MacFarlane, J. 2007, “Relativism and Disagreement”, Philosophical Studies 132 (1), 17 Moruzzi, S. 2008, “Assertion, Belief and Disagreement: A Problem for Truth-Relativism”, in García-Carpintero, M. and M. Kölbel 2008, 207 Prior, A.N. 1967, “Negation”, in Edwards, P. (ed.), 1967, The Encyclopedia of Philosophy, New York: Macmillan & Free Press, 458 Rosenkranz, S. 2008, “Frege, Relativism and Faultless Disagreement”, in García-Carpintero, M. and M. Kölbel 2008, 225 Routley, R. and V. Routley 1985, “Negation and Contradiction”, Revista Colombiana de Matemáticas 19, 201 Soames, S. 1999, Understanding Truth, Oxford: Oxford University Press Tye, M. 1994, “Sorites Paradoxes and the Semantics of Vagueness”, in Tomberlin, J. (ed.), Philosophical Perspectives 8: Logic and Language, Atascadero: Ridgeview, 189 Wolfram, S. 1989, Philosophical logic, London and New York: Routledge

Notes 1

Here, I do not consider some complications about sentences involving paradigm indexicals like “I”, “now”, or “here”. 2 Proof (“T” stands for a monadic truth-predicate, while “D” stands for a metavariable): 1. T(D) 䳓D [Premise] 2. D 䳓¬¬D [Premise] 3. ¬D 䳓¬T(D) [from 1, by contraposition] 4. ¬¬D ֞ ¬T(¬D) [from 3, by substitution of D for ¬D] 5. T(D) ֞ ¬T(¬D) [from 1, 2 and 4, by transitivity of the biconditional]. 3 Cappelen and Hawthorne (2009, p. 13) discuss a slightly different norm of assertion. 4 My thanks to the audience at The Answers of Philosophy. SIFA Twentieth Anniversary Conference (Alghero, September 13th-15th, 2012). My thanks as well to Marcello Frixione, Carlo Penco and Massimiliano Vignolo for the helpful discussions about the ideas presented here. Suggestions given by Stefano Caputo have been a great help in writing this paper.

CHAPTER SEVEN DO WE HAVE A DETERMINATE CONCEPT OF TRUTH? FREDRIK STJERNBERG

1. Introduction Truth can appear to be simple. If a sentence is true, what it says is the case. But right after this, we get into difficulties. There is a bewildering variety of conceptions of truth. Not only is there disagreement about what explains truth (correspondence or coherence, for instance), or if there even is anything that explains what truth is; there is also disagreement about which kinds of sentences are candidates for being true. We find disagreements about the truth-aptness of moral statements, mathematics, conditionals, statements about the colours of objects, and so on. We can also find disagreements about how the paradoxes of truth, notably the Liar, are best handled–which refinements or revisions in the notion of truth are needed to accommodate difficult cases like the Liar? Different solutions to that paradox have been proposed, and attitudes to suggested solutions tend to be polarized; disagreements are for instance to be found in reactions to Tarski’s theory of truth. It is perhaps naïve to think that all the above disagreements boil down to one single disagreement, but they do reflect widely differing conceptions of what truth is. The general question of interest for me is to what extent we can be said to have a determinate conception of truth, to what extent our conception of truth manages to pin down the nature of truth, to ensure that there is one thing we are talking about when we talk about truth. This question is too big to be handled in a single paper. Issues concerning determinacy can arise in different ways. Waismann argued that virtually all concepts had an open texture (Waismann 1945). This means that there will always potentially be cases where our definitions of a concept fail to settle whether something falls under the concept. Waismann held that this open texture held for most but perhaps

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not all empirical concepts, and he thought that for instance mathematical concepts were not afflicted by open texture. Waismann’s own examples were of ordinary middle-sized objects, such as chairs or cats. Open texture is when our basic definitions of a given object prove powerless to settle unexpected or more complicated cases. A further kind of indeterminacy is perhaps the best known, namely Quinean indeterminacy of translation (Quine 1960, ch. 2). This kind of indeterminacy rests on the claim that meaning cannot be fixed by speakers’ dispositions to linguistic behaviour, though linguistic behaviour is the only plausible candidate for fixing meaning. The dispositions cannot provide enough detail for determining meaning. These kinds of indeterminacy are potentially relevant for the general question about the determinacy of the concept of truth, but I will concentrate on examining one particular aspect of the question. I will be looking at a particular version of alethic pluralism, and defending it from an objection, the so-called Instability Challenge, which often is thought to be decisive.1 This will shed some light on my bigger issue, an issue best left for some other occasion. I will argue that the Instability Challenge doesn’t refute alethic pluralism, but assumes some version of alethic monism from the outset. So there is a defence of alethic pluralism here. But it is limited, since the view advocated here is in the long run not that different from other views of truth, thought to have been in conflict with alethic pluralism. The pluralist has some resources for a defence, though the defence comes at a cost. Central for the defence is the question about the determinacy of our notion of truth, and the remaining position may turn out to be hard to distinguish from a kind of deflationism about truth, a position usually held to be a kind of monism about truth. The defence of alethic pluralism is limited in another respect as well. There are several other arguments against alethic pluralism, and my paper is silent on those.2

2. Alethic pluralism According to alethic pluralism, there are many ways for sentences to be true; the nature of truth is not exhausted by saying that truth is correspondence with facts, or what your preferred theory of truth may be. It is uncontroversial that different kinds of sentences have different kinds of content. Alethic pluralism goes beyond this uncontroversial starting point, and locates a further difference in the way in which a sentence can be true. Other theories of truth are all monist, holding that the nature of truth is exhausted by a given characterization of truth. This means that for

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the monist, if correspondence theories of truth are correct, all sentences will ultimately have their truth explained by correspondence. The alethic pluralist instead thinks that truth need not be exhaustively characterized by for instance correspondence. Correspondence may be a good characterization of truth when we are trying to determine what it is that accounts for the truth of empirical statements, but it may be less helpful when we are judging statements about morals, fictional characters, abstract objects, or various kinds of suggested conceptual truths. Michael Lynch has stressed this rationale, calling it a “scope problem”: Assume that we present a substantial, robust, characterization of what the property of being true consists in. Then it appears that we always can find some proposition K that lacks this property but is intuitively true (Lynch 2009, p. 4; pp. 34-36). We might formulate alethic pluralism as the idea that there is more than one way in which something can be true, or that there is no universal truth predicate. Instead, what we have is just a bunch of local truth predicates, perhaps good only for limited areas of discourse. Then it will be possible to hold that both statements about ordinary things, as well as statements about mathematics or morals, can be true, though they are not true in the same sense. The statements about ordinary things are true because there is a truthmaker for them, or because there is a fact that corresponds with the statement, whereas the statements about morals may be true on account of something else, perhaps coherence. We need no longer look for Meinongian entities to make statements about fictional characters true; perhaps their truth is settled by coherence as well.3 There appears to be a grave problem with alethic pluralism, however, and many writers think that there is an argument which in effect refutes alethic pluralism. Following Pedersen (2010), we can call this argument the Instability Challenge.4

3. The Instability Challenge Assume, with the pluralist that there are many ways for statements to be true, so we should stop talking about truth tout court. Instead, what we have is a choice between different notions of truth, which we can call truth1, truth2, and so on. Against this, the monist about truth holds that it will in fact always be possible to define a universal truth predicate, TU, which will apply if and only if a sentence is true in (at least) one of the senses the pluralist wants to accept: (Universal)

TU ˩G(T1 ଻GT2 ଻G... ଻GTn ).

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And now we have, contra the pluralist, actually managed to define a universal truth predicate. Then this universal truth predicate provides us with the univocal sense of truth. So pluralism cannot work as intended, and alethic monism is vindicated.5 This availability of a universal truth predicate is often taken as a quite strong argument against pluralism. But the pluralist shouldn’t give in too easily. The instability challenge will only work if we accept that the pluralist’s list of acceptable ways in which something can be true is complete, that is, if the possibly long disjunction T1 ଻GT2 ଻G... ଻GTn is held to comprise all the different senses of truth that the pluralist accepted. But it is not clear that the pluralist should accept this. It can be argued that the notion of truth, no matter how we end up explicating it, will be more openended than this objection assumes. The first point is general. If the list of possible truth-predicates Tn is thought of as full and complete, then there may admittedly be a possible way to define a universal sense of truth, TU, in the way suggested. Whether the pluralist can accommodate this new universal sense of truth is a question which has been discussed.6 The pluralist can for instance hold that this disjunctive sense of truth doesn’t really qualify as a bona fide sense of universal truth—at least not as long as the disjuncts are not held to have much in common, and the pluralist will be denying that there would be any one underlying sense that connects these disjuncts. So even if we could construct a universal truth predicate, the pluralist will think of it more as a kind of technical trick, not showing anything of interest about what truth is. The new universal truth predicate TU is just such a piece of machinery. The introduction of this new predicate doesn’t correspond to anything of underlying interest, reflecting a common nature in the concept being defined. If we were to set out to explain this new universal predicate, the order of explanation would go from the disjuncts to the universal predicate, not the other way round—from the universal predicate to an explanation of what the various disjuncts meant. It would be like introducing a universal bank predicate, explained by saying that it is a bank-in-the-monetary-institution-sense or a bank-in-the-sloping-land-nearwater-sense. Nothing of interest results from the use of this new universal predicate. There is also another resource the pluralist should make use of. The pluralist need not agree that the suggested universal truth predicate U exhausts the possible senses of truth. The pluralist can argue that no list of disjunctive truth predicates that the universalist presents–T1, T2, …, Tn– exhausts the potential truth predicates. The notion of truth can always outrun any finite list we present, and without such a finite list, the

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universalist will be unable to introduce the universal truth predicate TU by equivalence with the disjunction of the various truth predicates. As a technical issue, this is perfectly clear–if there remains some Tn+1 outside the suggested definition of the universal truth definition, then the proposed definition fails. But we can leave this observation aside. Does the pluralist have any motivation for this response, and what does the response amount to? If the pluralist wants to say that there will be no exhaustive list of predicates making up the universal truth predicate, why should we say that these new predicates could lay claim to being precisely truth predicates? First, part of the pluralist’s motivation for thinking that the list of truth predicates is inexhaustible is a diagonalizing argument. It will always be possible to diagonalize out of the existing finite list of truth predicates, making the notion of truth indefinitely extensible. Consider first a theory of truth with only one sense of truth, for instance correspondence, and call this sense T1. Then we can have a sentence (1) (1) is not T1.

Is (1) true or false? This is not quite the old Liar paradox, since there is a way out of the problems with (1) that is not available in the Liar case. The simplest way out is to say that (1) is true, but in some sense of “true” different from that given by T1. Nothing paradoxical will result now. W e can think of this new sense of truth as T2. This process can of course be repeated, as many times as we care: for every finite collection of truth predicates we will always be in a position to construct a Liar-like sentence, stating that that very sentence is not true in any of the senses listed, prompting an introduction of a new truth predicate. What we have here is an indefinite extensibility of new meanings of truth.

4. Indefinite extensibility The diagonalizing argument indicated that the concept of truth is indefinitely extensible. A concept that is indefinitely extensible can be extended, if we can take a totality of objects which have the property, and produce something that also has the property, but is not to be found in the earlier totality. The first characterization of indefinite extensibility can be found in Russell (1906), but a fuller and more precise characterization is given by Dummett:

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Chapter Seven An indefinitely extensible statement is one for which, together with some determinate range or ranges of objects falling under it, we are given an intuitive principle whereby, if we have a sufficiently definite grasp of any one such range of objects, we can form, in terms of it, a conception of a more inclusive such range. … By the nature of the case, we can form no clear conception of the extension of an indefinitely extensible concept; any attempt to do so is liable to lead us into contradiction (Dummett 1993, p. 454).

This characterization is not a full definition, since the notions of determinateness and definiteness are used in the passage above, but the gist should be clear (compare also Dummett 1993, p. 441). The pluralist’s conception of truth is also indefinitely extensible in this way. If we try to come up with a collection of all the senses in which something can be true, we can immediately form a new sense of truth, which can be used to handle the Instability Challenge. Indefinite extensibility applies to many different concepts, such as truth, ordinal number, and knowledge. Another case can be seen if we compare with Berry’s Paradox. Consider the natural numbers. I can refer to many natural numbers, but there will be some I have never referred to. There will be one number which is the smallest number that is bigger than the biggest number I have referred to. Call this number Charlie. But then I have referred to Charlie, so Charlie will not be the smallest number that is bigger than the biggest number I have referred to. So what I mean by “referring to a number” will have to be fixed. Perhaps we want to distinguish between ways in which I refer to a number, so that there are direct ways, as when using numerals, and indirect ways, as when I have used the italicized phrase above. Then we could say that there is a number which is bigger than any number I have referred to directly, and call this number David. So far no paradox ensues. First paradox solved. But then I can create a new paradox: Let Evan be the smallest number that is larger than any number I have referred to, directly or indirectly. And now I have referred to Evan, and we have a revenge paradox. We will always be able to say that there is some number I haven’t referred to in any way of the ways we have listed, and thus create a reference to an apparently paradoxical number. But this is not a strange fact about the natural numbers: it is a fact about how “refer to” can shift its meaning if we continue to reapply the term. The concept of referring to is indefinitely extensible as well. Many terms of interest in philosophical logic are unstable, subject to paradoxes, paradoxes which can show a tendency to return, zombielike, in revenge form. We will not be able to foresee those problems in advance,

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and hence are forced to tinker with our main concepts as best we can. The indefinite extensibility of truth will mean that we can always come up with new difficult cases for our suggested understanding of what truth is. Such difficulties were not understood before we encountered them, and the best way out is determined when we come across the difficulty at hand. The concept of truth is made up, as we go along, and is not determinately contained in the understanding of truth with which we started (compare Williamson 1998, p. 18).

5. But is it truth? The freedom to introduce new truth predicates wards off the Instability Challenge, as long as the introduction of these new predicates really is the introduction of truth predicates, and not just some technical trick. This takes us to the second point. The pluralist can argue that the new truth predicates are precisely truth predicates, because they serve the same kind of general purpose that the old truth predicates do, and this sharing of a general purpose is all we need, and all we get, in order to say something about truth. They therefore have enough in common with the things we used to think of as truth predicates for us to be justified in calling them truth predicates. We can think of the notion of truth as if not defined, then at least delineated, by a series of things we tend to feel are more or less platitudinous about truth. Among such platitudes we can find things like the transparency claim, that the proposition that p is true if and only if p; that truth is bivalent; or that contradictions cannot be true. There can be good reasons to reject the platitudes on occasion, but this will require sustained argument. Claims about total transparency will get us into difficulties with Liar sentences, and some have seen fit to question bivalence or the law of non-contradiction.7 There is no strong a priori reason to think that the platitudes we connect with the notion of truth will settle one and only one concept of truth. There is also no strong reason to simply take for granted that an assortment of platitudes will be consistent. There may well turn out to be tensions between platitudes, some platitudes can turn out to be false or misleading, and some should perhaps be revised. But the platitudes we associate with the notion of truth indicate that it is precisely truth we are talking about, and that we are not replacing a good candidate for being a truth predicate with something else, such as perhaps the concept of being something everyone agrees on. The new addition to the earlier set of senses of truth, which we find forced upon us by the diagonalizing

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argument, is precisely an addition to the senses of truth, because it has enough of the original platitudes in common with the senses we started with. Saying that something should have enough of the original platitudes in common with the senses we started with is a loose measuring stick. But at times, a loose measuring stick is the best we can hope for. It is not clear what we should say about cases where a proposed contender will diverge from some of our platitudes about truth. Different ways of diverging from the platitudes can give rise to different concepts of truth. This is the basic motivation behind the alethic pluralist’s position–no matter how we try to pinpoint the nature of truth, there will always be a possibility that several different concepts will fulfil the proposed criteria equally well, or that different criteria will fulfil the criteria in different ways. There will therefore be no guarantee that there is a unitary notion of truth lying behind, or captured by, our different suggestions about how truth is to be understood. The platitudes work, because they are useful starts in attempts to get a fuller understanding of the notion of truth. Are they the only way to get such an understanding? Two other contenders are on the one hand some sort of formalized, axiomatic, treatment and on the other hand our intuitions about the concept of truth. In general, there are two main approaches to constructing a formalized theory of truth. Halbach (2011) dubs them “definitional” and “axiomatic” (pp. 3ff). The definitional tries to say explicitly what truth is, in other terms, or provide an explanation of truth, be it in terms of correspondence or something of the kind. The axiomatic approach takes truth as an undefined primitive term, and constructs axioms for how this term works, thereby showing how the paradoxes can have consequences for our understanding of truth or perhaps avoiding the paradoxes. An axiomatization would perhaps be nice as an alternative, but the axiomatization won’t take us all the way to a fully determinate concept of truth–there are several competing attempts to provide an axiomatization of the concept of truth (see Halbach 2011). There are big differences between these different axiomatizations. Disagreements between such axiomatizations are hard to settle. As long as the axiomatizations play nicely with the platitudes, there is little ground for saying that precisely one of them captures the notion of truth. There are for instance differences between typed and type-free axiomatizations of truth, and our assessment of such alternative views will in part depend on what the broader logical ramifications of the two approaches are. The alternatives lead to differences in logics (see Halbach 2011, ch. 10), and no brief appeal to axioms will settle anything. The axiomatic theories will have to be judged by the state of the entire system

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being constructed. This will leave space for a question about to what extent the proposed axiomatic systems single out one determinate concept of truth. So if axiomatizations can only give us a partial determinacy, would a straightforward appeal to intuitions fare better? By “intuitions” I here mean some supposedly infallible a priori source of insight about the nature of a concept.8 The very notion of intuition is highly problematic: there is no universally agreed sense of the term, not even among philosophers who are in favour of appeals to intuition. It is often used just as a way of hedging. Saying that something seems intuitively true will then not mean much more than that it strikes someone as prima facie true.9 Such uses will not have a status that is very different from appeals to platitudes as starting points for understanding true; the main difference is that appeals to intuitions here are used as a kind of assessment about individual cases, rather than used as starting points for further study of the concept of truth. Even if the concept of intuition is contested, it is clear that several philosophers use it in a sense that goes beyond the low-key hedging sense. It is not very clear how appeals to intuition are supposed to work in this case, however. Our intuitions about what truth is will be insufficient. They are insufficient in two ways. First, there is nothing to ensure that people will have the same intuitions, and no clear way to settle disagreements between persons, second, there is little reason to think that our intuitions would serve to cover all kinds of cases. Even if I had strong intuitions about what truth is, these intuitions will not by themselves settle issues about what kinds of statement are truth-apt, for instance whether statements about the colour of objects are truth-evaluable. If two persons disagree about their intuitions on this matter, it is hard to see what would serve to settle their disagreement. One example of a disagreement of the first kind concerning truth can be found in the differences between Field (2008) and Beall (2009) about how we should best revise our intuitions about truth in view of the paradoxes. Field and Beall both develop treatments of truth that are intended to handle the paradoxes while staying as close as possible to a more or less pre-theoretic notion of truth; they want to make as small changes as possible to what we tend to think about truth, while disarming the threat of the paradoxes. They are in agreement that doing this will have some consequences for our notion of truth, but they disagree about what changes should be made. They also disagree about which measures would lead to the smallest changes to our notion of truth.

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Field retains as much as possible of the naïve truth theory, in particular transparent truth, but revises the allegiance to the law of excluded middle, and revises the background logic. Beall thinks that we instead should retain other aspects of a naïve notion of truth, and that his suggested amendment of the naïve notion is that which leads to the smallest changes, even if Beall embraces a version of dialetheism. Since there is no straightforward way to measure the size of a proposed change in a notion of truth, we cannot expect that this issue would be simple to settle; there would presumably be room for disagreement here. Beall says as much when discussing Field: “Indeed, … I do not know of any terribly strong arguments against Field’s approach. My main reason for preferring my own account comes down, I’m afraid, to a fairly fuzzy sense that Field’s approach not only misses an essential feature of negation, but might also be more complicated than we need” (Beall 2009, p. viii). Field’s attitude towards this issue is also summed up quite early in his book. Our views on truth and related concepts lead to paradox. Hence, they are inconsistent, and something must be changed. There does not seem to be any clear sense in saying that certain amendments or revisions of our starting notions of the relevant concepts are impossible to change, since they would amount to changing the meaning of the concept. This is simply not the most fruitful way to handle the problems, since they rest on a problematic idea about “meaning constitutive” or “analytic” principles. “Perhaps there are subtle techniques of linguistic analysis that would enable us to discover that certain of these principles involved in the paradoxes aren’t really constitutive of the ‘meanings’ that English speakers attach to their words in the way that the rest of the principles are, but I’m skeptical” (Field 2008, p. 17). But if there is no “subtle technique of linguistic analysis” that can settle the issue, there will be no quick way to judge the amount of revision imposed by a proposed solution. We cannot handle the paradoxes without making changes somewhere. The Beall-Field discussion seems to have arrived at a curious kind of impasse. Both are presenting theories of truth intended to amend the naïve truth theories as little as possible, hence to stay as close as possible to our starting notion of truth. They are both doing this as a suggested way to handle the paradoxes of truth. Yet they arrive at different results, and there seems to be very little to help us to choose between their views. They seem to agree about this, as well. Beall is explicit, while Field, who doesn’t discuss Beall’s views explicitly, at least makes room for an acknowledgement like this when he provides a general defence of his position. None of them will appeal to a silver bullet, something that is

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uncontrovertibly meaning-determining for truth. In the absence of a silver bullet of that kind, differences would have to be settled in some other way. The second sort of disagreement was about how to handle disagreements over in what areas it is in order to talk about sentences being true. One such example concerns colour talk. Boghossian and Velleman (2008) think that it is false that ripe tomatoes are red. Not because they are some other colour, but because objects really aren’t coloured at all. They think that “all statements attributing colours to external objects are false” (Boghossian and Velleman 2008, p. 311). They readily acknowledge that their view, an error view of colour talk, strikes most people as false, or even nonsensical, and they address two worries. First, if all colour talk is false, how can we still distinguish between “good” and “bad” uses of colour predicates? Even if it is false, on their theory, that ripe tomatoes are red, saying this is much better than saying that they are blue. What kind of difference, between true statements and merely “good” statements, is this? Second, how can an error theorist account for the seeming indispensability of colour talk? They argue that there can be a viable difference between “good” and “bad” colour statements, since statements about what colour something appears to have under normal conditions will give better guidance to appearances in other, non-standard, conditions. All objects are dark or black in the dark, so we cannot draw conclusions about their colour in daylight conditions from their apparent colours in the dark, but reasoning from daylight appearances to nighttime appearances is fine. So some colour statements are better, because they are more informative (while still not being true). They defend the usefulness of colour talk by an analogy with talk about the sun’s movements. We say that the sun rises, but this is strictly speaking false. The sun’s movements are merely apparent. It is still in good order to say that the sun rises, because it would cause greater disturbances in our web of everyday beliefs if we started denying that the sun rises. Denying that the sun rises, and instead insisting that the horizons move, is misleading, and serves no communicative purpose. “Only an undue fascination with the truth could lead someone to reform ordinary discourse about the sun” (Boghossian and Velleman 2008, p. 312). This kind of reasoning will only provide us with a very patchy defence of the usefulness of some kinds of statements. We can distinguish between good and bad colour talk, but this will not carry over to distinguishing good and bad talk in other problematic areas. Each area will have to be judged on its own merits.

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As a general methodological point, we should be very reluctant to posit massive production of falsehoods on the part of normally competent speakers. Boghossian and Velleman are too heavily invested in a view that sees the role of talk about truth as exclusively concerned with a particular form of fact-stating. The alethic pluralist will instead view talk about truth as a kind of open texture concept, where there will be several different ways for something to be true. This will mean that there is no principled reason to deny that competent colour talk can at times be a good candidate for producing truths. Talk about intuitions guiding us in saying that certain kinds of statements are not even candidates for being true will not help us much here. If there can be no principled settlement of disagreements about intuitions, what consequences should we draw? It seems that this situation reflects an underlying indeterminacy in our understanding of the concept of truth. We are therefore back with the starting platitudes. If neither axiomatization nor intuitions will give us everything we want to ensure determinacy of the concept of truth, we have to rest content with working from a suitable assortment of platitudes, and scale back our desire for a fully determinate concept of truth. The platitudes we put forward concerning the role of the notion of truth will not be sufficient to determine what kind of more detailed notion of truth that is meant, but they will indicate that some kind of more substantial backing is needed for the sentence to be true. Next section considers a few existing attempts to combine a platitude-based approach to the concept of truth with a certain amount of pluralism.

6. Other views There have been a few earlier attempts to develop a view of truth that combines general platitudes about truth with more substantial ideas about what it is that makes for the truth of a given statement in some area. Wright (1992) combines a thin, minimalist, view of truth with a more robust, substantial view. The minimalist sense is given by the platitudes concerning truth, the substantivalist senses are given by another set of claims about truth. The former are precisely platitudes, perhaps not very exciting, but expected to be accepted by everyone who claims to be talking about truth and not some other concept. The latter are substantial, nonobvious and not platitudes. They will hold, if at all, for limited areas of discourse. These two approaches to a notion of truth are then combined, so that the platitudinous notion of truth holds together the various substantial notions.

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Lynch (2009) first considers seeing truth as a kind of second-order property. It is a second-order property, because it is a common property to all the things that are suggested to make a sentence true, even if these different things by themselves don’t have that much in common. On this view, there is a single property of truth, but it is a second-order property: the property of having some property satisfying the various platitudes about truth. Lynch finds that this view of truth as a second-order property fails, since this second-order property will not satisfy the platitudes we started with (Lynch 2009, pp. 64ff). According to Lynch, if truth as a second-order property is not enough, we should take an extra step, and view truth as a functional property: truth is one property, which is multiply realizable. The role of truth is given by the platitudes. This is intended to be a parallel to functionalism in the theory of mind, or at least to the variant called analytic functionalism (Levin 2013, sec. 3.3). For the analytical functionalist, the property of being in pain is a second-order property which is realized by different first-order properties in different organisms. It is pain if the second-order property has the right kind of relations to other states of the organism. An organism tries to avoid pain, finds pain disagreeable, and so on. The various platitudes connected with pain serve to settle the identity of the kind of mental state, distinguishing it from anger, pleasure, and so on. Lynch is not very clear on what happens if a suggested candidate for being a second-order property satisfies some, but not all the suggested platitudes. This possibility should leave some room for various secondorder concepts of truth here, introducing yet another indeterminacy. Such possibilities seem to arise when we try to handle the paradoxes. Sher (2013) combines a correspondence view of truth with a kind of deflationism in a different way. In her view, which she calls “pluralism within the bounds of correspondence” (Sher 2013, Abstract and sec. II), the original notion of truth as correspondence is liberalized in a way that makes it possible to talk about truth in mathematics as explained by correspondence, without falling prey to the problems in understanding what mathematical objects are, or what correspondence to mathematical entities amounts to. The correspondences are created by an “intricate” mind, capable of developing new concepts for new areas of discourse (sec. V). We should not expect that there is one single route from one thought aiming at truth to reality; instead, such routes are multifaceted, constantly evolving, and can very well be different for different areas. The grasp we have of truth is something like a notion of truth in a more technical sense, not unlike that presented in Richard (2008, pp. 104ff). A “notion” is that which is common to different uses of a given term, and

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which makes it reasonable to say that we are using the term with the same meaning on different occasions. This will leave room for some indeterminacy. The main difference, compared with these views, is that the present view leaves more indeterminacy in the notion of truth. Truth, on this account, cannot be exhaustively characterized, and this will have consequences for certain kinds of disagreement. Questions about contested areas of discourse, such as colour, normative statements, statements about fictions, and others, will be left undecided, given the view of truth in these different areas. Whether it will be useful to talk of truth in such areas depends mainly on two things. First, what kind of development of the notion of truth we come up with, second, how thought in and about this area develops. Talk about truth can be useful for certain areas, and there is no delimiting in advance which areas are out of bounds.

7. Conclusion The Instability Challenge against alethic pluralism does have some force, but not the force the objectors think. It doesn’t show that alethic pluralism is wrong; it only serves to highlight that “true” will not have a fixed sense or extension. This is perhaps bad news, but not specifically for the alethic pluralist. It is instead, something that ties in well with general features of our command of concepts. There is normally not as much determinacy as we would like to think in our command of concepts. Finding some indeterminacy in our talk about truth is therefore nothing new. And this is where the “limited” part of the defence kicks in. It is not entirely clear to what extent this indefinite extensibility of the truth predicate is that much different from a universal truth predicate, which can encompass all kinds of truth predicates that we can come up with. One of the centrally important platitudes about truth is the transparency property, which is also at the heart of the deflationist’s view of truth. The present proposal still differs from ordinary deflationism in holding that the deflationist view of truth doesn’t have to be the final word on why a given sentence is true–truth of a sentence can be backed up by some substantial property of that particular sentence. From the outset, several possible varieties of indeterminacy of possible relevance for our understanding of the notion of truth were sketched. Not all were relevant. We have seen two kinds of indeterminacies connected with the use of a truth predicate. The first was that it could be argued that there is an indefinite extensibility to the notion of truth: no attempt to

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provide a finite list of truth predicates will by itself suffice to determine a universal truth predicate. In this way, the alethic pluralist can resist the Instability Challenge. The second type of indeterminacy was that none of our usual ways of delineating the sense of truth will in itself guarantee that we have managed to get hold of a single, determinate, concept. This is a more general observation, which goes back to Waismann’s views on the open texture of concepts. This is something that we have managed to live with in other areas, and we should accept that this holds for the notion of truth as well. Our notion of truth is not fully determinate. It is not sufficiently determinate to settle all the kinds of differences concerning the notion of truth that were our starting point, but this is in itself not a great problem. Our understanding of the notion of truth can come under pressure in difficult cases, as when we consider the paradoxes or disagree about which kinds of statement that are truth-apt. But this leaves the notion of truth untouched in many cases–there is enough determinacy for practical purposes. Trouble comes when we ask for, or expect, more determinacy.

References Beall, Jc. 2009, Spandrels of Truth, Oxford: Oxford University Press Boghossian, P. 2008, Content and Justification, Oxford: Oxford University Press Boghossian, P. and D. Velleman 2008, “Colour as a Secondary Quality”, in Boghossian, P. 2008, 293 (originally appeared in Mind 98, 1989, 81) Cappelen, H. 2012, Philosophy Without Intuitions, Oxford: Oxford University Press Dummett, M. 1993, The Seas of Language, Oxford: Oxford University Press Field, H. 2008, Saving Truth from Paradox, Oxford: Oxford University Press Halbach, V. 2011, Axiomatic Theories of Truth, Cambridge: Cambridge University Press Levin, J. 2013, “Functionalism”, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/fall2013/entries/functionalism Lynch, M. 2004, “Truth and Multiple Realizability”, Australasian Journal of Philosophy 82, 384 —. 2009, Truth as One and Many, Oxford: Oxford University Press —. (ed.) 2001, The Nature of Truth: Classical and Contemporary Perspectives, Cambridge, Mass.: MIT Press

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Pedersen, N. 2010, “Stabilizing Alethic Pluralism”, The Philosophical Quarterly 60 (238), 92 Pedersen, N. and C.D. Wright 2013, “Pluralist Theories of Truth”, The Stanford Encyclopedia of Philosophy (Spring 2013 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/spr2013/entries/truth-pluralist/ Pedersen, N. and C.D. Wright (eds.) 2013, Truth & Pluralism: Current Debates, Oxford: Oxford University Press Priest, G. 2006, In Contradiction. A Study of the Transconsistent, Oxford: Oxford University Press, 2d ed. (first ed. 1987) Pust, J. 2012, “Intuition”, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/archives/win2012/entries/intuition/ Quine, W.V.O. 1960, Word and Object, Cambridge, Mass.: MIT Press Richard, M. 2008, When Truth Gives Out, Oxford, Oxford University Press Russell, B. 1906, “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society 4, 29 Sher, G. 1999, “On the Possibility of a Substantive Theory of Truth”, Synthese 117 (1), 133 —. 2013, “Forms of Correspondence. The Intricate Route from Thought to Reality”, in Pedersen N. and C. Wright (eds.) 2013, 157 Tappolet, C. 1997, “Mixed Inferences: a Problem for Pluralism about Truth Predicates”, Analysis 57 (3), 209 —. 2000, “Truth Pluralism and Many-Valued Logics: A Reply to Beall”, Philosophical Quarterly 50 (200), 382 Waismann, F. 1945, “Verifiability”, Proceedings of the Aristotelian Society, Suppl. Vol. 19, 119 Williamson, T. 1998, “Indefinite Extensibility”, Grazer Philosophische Studien 55, 1 Wright, C. 1992, Truth and Objectivity, Cambridge, Mass.: Harvard University Press —. 2001, “Minimalism, Deflationism, Pragmatism, Pluralism”, in Lynch 2001, 751

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Notes 1 Wright (1992) gives an influential presentation of the case for alethic pluralism. See Pedersen (2010), Tappolet (1997, 2000) concerning the Instability Challenge. 2 The problem of mixed inferences is one such issue. See Pedersen & Wright (2013, sec. 4.4). 3 A good overview of alethic pluralism is to be found in Pedersen & Wright (2013). 4 Pedersen (2010), Tappolet (1997, 2000). Versions of this argument can be found in for instance Sher (1999) and Lynch (2004). 5 See Pedersen (2010) for a recent discussion of this problem, suggesting a way for the pluralist to meet the objection. 6 See Pedersen & Wright (2013). 7 As in Priest (2006) concerning the law of non-contradiction. 8 See Cappelen (2012, secs. 3-5), and Pust (2012) for some different philosophical uses of “intuition”. 9 Philosophers’ uses of this kind can be found in Cappelen (2012, sec. 4.2).

CHAPTER EIGHT ALETHIC PLURALISM AND LOGICAL PARADOXES MICHELE LUBRANO

1. A problem for pluralism A recent contribution by Stewart Shapiro (Shapiro 2011) has driven the attention of some supporter of alethic pluralism to a problem that their theory of truth faces when it comes to logical paradoxes. As is widely known, alethic pluralism is a kind of theory of truth which maintains that there isn’t a unique way for a proposition to be true. Some propositions are true because they somehow correspond to a fact (typically propositions about physical entities and their qualities); other are true in virtue of their property of being warranted at some stage of inquiry and remaining so at every successive stage; yet other are true because of their coherence with a system of beliefs. 1 We can distinguish between two different kinds of alethic pluralism: strong alethic pluralism and weak alethic pluralism. The former claims that it doesn’t exist a unique property that all and only true propositions possess, while the latter claims that there exist a (minimal or functional) property of being true which supervenes on the possession of a property that realizes truth. Both versions maintain that a necessary condition that a property should respect, in order to confer (or realize) truth, is the fulfilment of the well-known T-schema: T-SCHEMA (PROPOSITIONAL): The proposition that p is true if and only if p.

In this respect it’s remarkable that Michael P. Lynch, in his Truth as One and Many, maintains that every theory of truth should explain the TSCHEMA, or, otherwise, it should explain it away. Either all the instances of the T-SCHEMA are consequences of the axioms of our theory of truth

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or this schema must be explicitly refuted by our theory. One of the virtues which Lynch ascribes to his approach is the fact that this schema is entailed by his very definition of the functional role of truth. Nevertheless the T-SCHEMA is the source of some known difficulties. For the sake of expositions, we are going to consider the sentential version of it (which is, moreover, the original one; see Tarski 1933). So consider this: T-SCHEMA (SENTENTIAL): ଢpଣ is true if and only if p.

To run up against a paradox it suffices to consider that, if we have a method which allows us the assignment of a unique name to each sentence, then it’s easy to construct a sentence which says of itself to be false. Suppose that O is such a sentence and its name is ଢOଣ; thus we have: O l ൓T(ଢOଣ).

But the T-SCHEMA allows us to replace O with T(ଢOଣ). Therefore we obtain: T(ଢOଣ) l ൓T(ଢOଣ).

It would be quite unsound to think that this kind of paradox is irrelevant for pluralist because they often set up propositions (and not sentences) as truth bearers since, presumably, there exists a proposition which says of itself to be false. The derivability of such a paradox has led many philosophers to develop various kind of solutions. For instance Tarski (1933) claimed that we ought to distinguish between objectlanguage and metalanguage: the former doesn’t contain any semantic predicate that can be applied to its own sentences, the latter instead contains semantic predicates applicable to the sentences of the objectlanguage. Not even the metalanguage has the resources necessary to express semantic claims about its own sentences. The property of containing its own semantic predicates was called, by Tarski (1944), “semantic closure” and no language should have it, on pain of giving rise to paradoxes. Shapiro (2011) claims that alethic pluralism, at least as it’s presented by its main supporters, seems to be a theory affected by this very semantic closure; in his opinion, Lynch could be accused of claiming that each domain of discourse includes its own truth predicate. This

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inclusion leads necessarily to contradiction.2 If we want to make a set of sentences paradox-free, we ought to impose it some restrictions. It seems that no chunk of propositions can have the resources to characterize the truth-realizer for those very propositions, provided that those resources are to have the requisite instances of T-SCHEMA as consequences, as Lynch seems to require. [...] what makes for truth in any given chunk of discourse is not a matter of that chunk of discourse to decide (Shapiro 2011, p. 43).

Cotnoir (2013) remarks that is quite surprising that none of the supporters of alethic pluralism has previously paid attention to this issue. Following Shapiro’s remarks, we could say that this negligence might be explained by the fact that many pluralists are looking for the best metaphysical theory of truth; a metaphysician of truth usually deals with truth as a property owned by some particular entities (propositions, beliefs and whatever is apt to play the role of truth-bearer) and not with truth as a (potentially dangerous) predicate of language. Therefore, as Shapiro observes, problems related with consistency are frequently left aside. 3 Nevertheless, given that truth paradoxes are a problem for every theory of truth and therefore alethic pluralism is under their threat too, even a pluralist metaphysician of truth should try to find a suitable way to save his theory from them.

2. Cotnoir’s solution Since now the only pluralist who has tried to reply to Shapiro’s charges is Aaron Cotnoir. In a recent essay (Cotnoir 2013), he claimed that at least strong alethic pluralism has internal resources enough to dispose of paradoxes. To understand Cotnoir’s solution it is important to bear in mind that what distinguishes strong alethic pluralism from other forms of pluralism is the endorsement of the following thesis: NO UNIQUE TRUTH: No truth predicate can apply to every domain of discourse.

Cotnoir’s solution has the virtue of being specifically pluralist; indeed it employs conceptual resources which are typical of alethic pluralism. Thus, his solution could hardly be used within different theoretical frameworks. Another remarkable merit is that it avoids any reference to

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non-classical logics (while other strategies require it). Let’s start our analysis from this key passage: The derivation [of a paradox] depends crucially on the assumption that the O1-liar is actually in domain1. But the pluralist, of course, is free to reject that O1 is in domain1. If O1 is not a sentence of domain1, then we do not have to commit to T1(ଢȜ1ଣ) l Ȝ1. Thus we do not arrive to the paradoxical consequence that ٟ™T1(ଢȜ1ଣ) l T1(ଢȜ1ଣ). Here is another way of stating the point. As strong pluralists we are free to claim that Ȝ1 is not true1. Of course, Ȝ1 actually says of itself that it is not true1. And so intuitively, it ought to be true! But if Ȝ1 is actually in domain2, it may very well be true2. We can endorse T2(ଢȜ1ଣ) without paradox. Of course, we will be able to define a new liar, Ȝ2, by diagonalization using T2. Ȝ2: ൓T2(ଢȜ2ଣ). But notice that Ȝ2 is not the same sentence as Ȝ1. Indeed, the two use different truth predicates. Here again, the pluralist is free to reject that Ȝ2 is in domain2, but rather in, say, domain3. The process can continue and the result is that the pluralist can consistently endorse (TS) [the T-SCHEMA] for Ti over domaini for every natural number i (Cotnoir 2013, pp. 341-342).

The passage is a bit tricky but the main idea seems to be sufficiently clear.4 Consider the sentence O1, saying of itself of being false. We can represent the situation, as usual, by writing O1: ™T1(ଢO1ଣ). Now, Cotnoir claims that nothing prevents a pluralist from considering Ȝ1 as included in a domain different from domain1, say domain2. So, from Ȝ1 (and from the T-SCHEMA) we cannot deduce T1(ଢO1ଣ), because the truth predicate that holds in domain2 is T2 and not T1. So the truth of O1 ought to be expressed by T2(ଢO1ଣ). A paradox could arise only if we obtain T1(ଢO1ଣ), but, since we can deduce T2(ଢO1ଣ) but not T1(ଢO1ଣ), the considered portion of language (domain1 plus domain2) is paradox free. Moreover, this move of “shifting the domain” which O belongs to can be iterated over every domain of

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discourse, making the entire language paradox-free. It seems to me that here there’s a problem with Cotnoir’s solution: he cannot deny that both the two predicates T1 and T2 are allowed to have ଢO1ଣ as argument; indeed the content of O1 can be expressed by ™T1(ଢO1ଣ). This is not compatible with the previous claim according to which Ȝ1 is not included in domain1. Cotnoir’s solution can work only if O1 is not included in domain1, but this is not the case, because of the very content of O1. Such a content entails that T1 can be applied to a sentence like O1 and, consequently that O1 is included in domain1. For this reason I think that Cotnoir’s solution should be abandoned or, at least, modified.

3. A different approach Nevertheless the idea of giving a solution to logical paradoxes which takes into account the peculiarities of pluralist theories of truth is interesting and it deserves to be further explored. This is exactly what I’m going to do. My proposal is essentially Tarskian in spirit: it’s an attempt to make the Tarskian machinery exploitable for supporters of pluralist theories of truth. I share with Cotnoir the idea that strong alethic pluralism seems to be better equipped to dispose of Liar-like paradoxes. Thus I assume NO UNIQUE TRUTH plus a “Tarskian” idea: every domain has a truth predicate which applies to its sentences, but this predicate is not included in that very domain. For example, in domain1 you could never find a sentence saying that a certain sentence belonging to domain1 is true (or false). Our theoretical proposal requires the addition, to NO UNIQUE TRUTH, of these two principles: COMPLETENESS: For each domain of discourse there exists a truth predicate which applies to all the sentences belonging to it. NO SELF-REFERENCE: No domain of discourse includes a truth predicate which applies to its own sentences.

My aim is to propose, in a semi-formal way, an algebra capable of representing adequately these ideas and, through it, to deduce some important (and apparently puzzling) consequences for strong alethic pluralism. We define D as the set of all domains of discourse and ī as the set of all truth predicates. To each member of ī is associated an index, consisting in an ordered couple of sets. Such an ordered couple is composed by two sets which are both subsets of D; the former is the set of the domains

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which include the truth predicate corresponding to the index, the latter is the set of domains to which the corresponding truth predicate applies. For convenience’s sake we will call dwelling of a truth predicate the former; the latter will be called scope of a truth predicate. In general: if the truth predicate Tk correspond to the index , this means that the domains i1, i2, ..., in contain Tk (they constitute its dwelling) and that Tk is the truth predicate of all and only the sentences belonging to j1, j2, ..., jn (which constitute its scope). If we define / as the set of all indexes, we can say that there is a function ij: īo/ which associates an index to each truth predicate. It’s not necessary to require the function to be injective or surjective. There is no evident reason to think that two different predicates cannot correspond to the same index; two (or more) different truth predicates can belong to the same domains i1, i2, ..., in and apply to the same domains j1, j2, ..., jn and nevertheless denote different properties. 5 Moreover there is no reason to think that each index is associated to, at least, one truth predicate. This notation allows us to represent a wide spectrum of situations. For example we can represent: 1) a situation in which a certain domain contains more than one truth predicate (it suffices that the symbol that stands for that domain occurs in more than one dwelling); 2) a situation in which more than one truth predicate apply to the same domain (it suffices that the symbol that stands for that domain occurs in more than one scope), and so on. Our notation allows us to reformulate the previous three principles in a more straightforward way: NO UNIQUE TRUTH: There is no truth predicate such that its scope is identical to D. COMPLETENESS: For each i belonging to D, there is a truth predicate T such that i is a member of its scope. NO SELF-REFERENCE: There is no truth predicate T such that the intersection of its own dwelling with its own scope is not empty.

These three conditions characterize what we will call Tarskian Strong Pluralism. These conditions are not enough to assure the consistency of a theory of truth, because, as is well known, self-reference is not the only mechanism originating paradoxes. We can obtain a paradox even via a crossed-reference mechanism. Suppose we have two domains, i and j such that i contains a truth predicate which applies to the sentences of j (its index is something like ) and j contains a truth predicate which applies to the sentences belonging to i (its index is something like

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). Then suppose that, within i, there is a sentence p saying that the sentence q, belonging to j, is true and that the sentence q says that p is false. The result is that both p and q ought to be true and false. If we want to prevent a theory of truth from allowing situations like this, we need to endorse a further principle: NO SIMMETRY: Every couple of truth predicates, Tk and Tl, is such that if the intersection of the scope of Tk with the dwelling of Tl is not empty then the scope of Tl and the dwelling of Tk are disjointed.

The adoption of this principle should allow us to avoid the paradoxes whose existence is due to crossed-reference mechanism; if a domain can contain sentences expressing the truth or the falsity of sentences of another domain, then this latter domain cannot contain sentences expressing the truth or the falsity of sentences of the former domain. Nevertheless, even if we endorse such a principle, there is still room for paradoxes to arise. Consider the following situation: we have three truth predicates Tk, Tl, Tm, whose indexes are respectively , , . Moreover they respect NO SIMMETRY. Nothing rules out the possibility of having these three sentences: ¾ a sentence p, belonging to the domain a, saying that q is true. ¾ a sentence q, belonging to the domain b, saying that r is true. ¾ a sentence r, belonging to the domain c, saying that p is false. In that case we would find us in a situation in which p, q, and r ought to be true and false. Therefore we need a further principle stopping this kind of cyclic-referential mechanisms. In order to do this, we ought to define the notion of priority. We say that a truth predicate Ti is strictly prior to truth predicate Tj (in symbols Ti!Tj) if and only if the intersection of the scope of Ti with the dwelling of Tj is not empty. If Ti is strictly prior to Tj they constitute a segment of chain. There is no limit to the length of segments of chains: if Ti>Tj and Tj>Tk then Ti, Tj and Tk constitute a segment of chain (and so on). If Ti and Tk are such that there is a segment of chain which connects them and which starts from Ti, we can say that Ti is prior to Tk (in symbols Ti•Tk). The principle that should prevent the previously mentioned contradiction from arising is: NO LOOPS: If a truth predicate Ti is prior to Tj, then the scope of Tj and the dwelling of Ti are disjointed.

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As it’s easy to see, this principle makes NO SIMMETRY redundant. NO LOOPS states that the relation of “being prior to” is not symmetric. Moreover it’s easy to acknowledge that it’s transitive and not reflexive; so (ī, •) is a partial order. We cannot say that it’s a total order because, given two truth predicates Ti and Tj, it’s not true that in every case either Ti•Tj or Tj•Ti. Does NO LOOPS allow us to avoid all kind of truth paradoxes? Maybe not. There is a kind of paradoxes, discovered by Yablo (1993), which makes no appeal to cyclic-referential mechanisms. 6 Consider an infinite sequence of sentences p0, p1, p2, p3, ... each saying that all the sentences below are false: p0: for every n bigger than 0, pn is false. p1: for every n bigger than 1, pn is false. p2: for every n bigger than 2, pn is false. p3: for every n bigger than 3, pn is false. p4: ... ... Suppose that p0 is true; then p1 is false. But p1 says that, for every n bigger than 1, pn is false and this is true in virtue of p0 being true. Then p1 is true (contradiction). Now suppose p0 is false; then at least one of the ensuing sentences is true. Suppose pk is a true sentence belonging to the series; then Pk+1 is false. But Pk+1 says that, for every n such that n is bigger than k+1, pn is false and this is true in virtue of pk being true. Then pk+1 is true (contradiction). Does Tarskian Strong Pluralism allow the formulation of Yablo Paradox? The answer is not straightforward, so let’s analyse the situation in detail. Suppose that p0 belongs to the domain i0 and that, within i0, there is a truth predicate T0 such that, for every n bigger than 0, in belongs to the scope of T0. Then we iterate the same move at every successive stage: p1 belongs to the domain i1 and, within i1, there is a truth predicate T1 such that, for every n bigger than 1, in belongs to the scope of T1, and so on. It’s easy to see that there is no violation of NO LOOPS in doing so; the series of truth predicates T0, T1, T2, ..., whose indexes are respectively , , , ... , is a perfectly acceptable segment of chain. Let’s proceed another step: suppose p0, which says “for every n bigger than 0, ¬T0(pn)”, is true; hence ¬T0(p1), ¬T0(p2), ¬T0(p3) and so on. Now, the paradox arises only if we maintain that ¬T0(p2), ¬T0(p3), ¬T0(p4), ... entails T0(p1). But p1 is true if and only if ¬T1(p2), ¬T1(p3), ¬T1(p4), ..., since p1 says “for every n bigger than 1, ¬T1(pn)”. Therefore the paradox arises only if we assume a principle like this: for every p and every Ti and Tj, if p belongs to a domain which is in the scope of both truth predicates,

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Ti(p) l Tj(p). Such a principle would allow us to deduce, from ¬T0(p2), ¬T0(p3), ¬T0(p4), ... the consequence ¬T1(p2), ¬T1(p3), ¬T1(p4), etc. I prefer to avoid any claim about the intrinsic plausibility of it. It’s enough to know that we can dispose of Yablo’s paradox by explicitly rejecting it. We can do this via the following principle: RELATIVITY: given any couple of truth predicates Ti and Tj and whatever sentence p belonging to a domain which is in the scope of both truth predicates, Ti(p) doesn’t entail Tj(p) and vice versa.

This claim can be easily accepted by pluralists, because there are some convincing examples which seem to support it. For example, consider a sentence that says something on the actual number of stars in the universe. Its truth, in the sense of its being correspondent to a fact, doesn’t entail its truth in the sense of being superwarranted. At this stage we can take stock: if we adopt Tarskian Strong Pluralism, we need to accept NO LOOPS and RELATIVITY. As we are going to see now, the endorsement of the first of these two principles involves a significant consequence. Suppose we have a segment of chain of truth predicates starting from Tn, a truth predicate whose index is . Our theory of truth, and COMPLETENESS in particular, requires that for each i, belonging to D, there is a truth predicate T such that i is a member of the scope of T. Now, NO LOOPS requires that none of the domains contained in the dwelling of Tn is contained in the scope of one of the truth predicates that follow Tn in the segment of chain (since Tn is prior to any truth predicate that follows it in the segment of chain). Therefore we are compelled to claim the existence of a further truth predicate whose scope includes the dwelling of Tn, or, alternatively, to claim the existence of further various truth predicates such that the union of their scopes includes the dwelling of Tn. But, obviously, these predicates require the existence of other predicates whose scopes contain their dwellings, and so on. Our theory entails that a segment of chain cannot be finite since COMPLETENESS must be respected; if a segment of chain is such that every domain which feature in the dwelling of a predicate features in the scope of another predicate we can say that it is complete. A complete segment of chain is a chain.7 The above discussion demonstrates that every chain is infinite. An immediate consequence is: ī-INFINITY: ī has infinite members (namely, there exist infinite truth predicates).

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Another important consequence is the infinity of domains. Indeed, any predicate Tn which is prior to Tn+k contains in its dwelling domains which doesn’t occur in the scope of any of the following members of the chain. Given the infinity of any chain we are compelled to claim: D-INFINITY: D has infinite members (namely, there exist infinite domains of discourse).

Thus, the endorsement of Tarskian Strong Pluralism commits us to the claim of the infinity of truth predicates and of the infinity of domains of discourse. This could seem an heavy commitment for a pluralist. Indeed usually we don’t think there are infinite different properties on which truth supervenes; philosophers has identified only a small number of these: correspondence, coherence, assertability under ideal epistemic conditions, superassertability and few others. Each of these property amounts to the property of being true in the corresponding domain of discourse: the domain of objective discourses for the property of being correspondent to a fact, the domain of epistemically constrained discourses for the property of being superassertable, and so on. Neither truth-conferring properties, nor domains of discourse seem to be infinite. Nevertheless it should be noticed that it’s not mandatory to consider as legitimate truth conferring properties only those which are independent of each other. Nothing prevents us from considering, for example, “being correspondent to a fact” and “being true in virtue of a certain sentence being correspondent to a fact” as two different (although not independent) truth-conferring properties. Consider a domain of discourse m including all and only the sentences of chemistry. Suppose that the truth predicate which holds for the sentences of m has the index . It’s not necessary to claim that the domain of discourse l ought to differ from m in virtue of its sentences being about a completely different and independent topic (e.g. ethics); the domain l could be constituted entirely of sentences expressing semantic properties of the sentences of m (“s1 is true”, “s2 is false”, etc., where s1, s2, ..., sn are sentences belonging to m). If we accept this, there’s no reason to be concerned about the infinity of truth-conferring properties and of domains of discourse, because such infinity doesn’t require, for example, the existence of infinite different and independent topics a discourse can be about. In the end, we can maintain that even a pluralist theorist of truth should make the effort of finding a solution to Liar-like paradoxes. Such a solution should take into account the peculiarities of pluralist theories of truth. Cotnoir’s theory is an interesting attempt, but there are compelling reasons to refuse it. Nevertheless an alternative is available; such an

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alternative is Tarskian in spirit and suitable for alethic pluralism at the same time.

References Beall, Jc. 2001, “Is Yablo’s Paradox Non-Circular?”, Analysis 61 (3), 176 Cotnoir, A. 2013, “Pluralism and Paradox”, in Pedersen, N.J.L.L. and C.D. Wright (eds.), Truth and Pluralism: Current Debates, New York: Oxford University Press, 339 Lynch, M.P. 2009, Truth as One and Many, Oxford: Oxford University Press Priest, G. 1997, “Yablo’s Paradox”, Analysis 57 (4), 236 Shapiro, S. 2011, “Truth, Function and Paradox”, Analysis 71 (1), 38 Tarski, A. 1933, “The Concept of Truth in the Languages of the Deductive Sciences” (Polish), Prace Towarzystwa Naukowego Warszawskiego, Wydzial III Nauk Matematyczno-Fizycznych 34, Warsaw; expanded English translation in Tarski 1983, 152 —. 1944, “The Semantic Conception of Truth”, Philosophy and Phenomenological Research 4 (3), 341 —. 1983, Logic, Semantics, Metamathematics: Papers from 1923 to 1938, Corcoran J. (ed.), Indianapolis: Hackett Publishing Company Yablo, S. 1993, “Paradox without Self-Reference”, Analysis 53 (4), 251

Notes 1

Here we are only talking of atomic propositions; complex and quantified propositions raise further problems which won’t be at issue here. 2 For simplicity’s sake, here we ignore the fact that, for a given portion of language to talk of itself, it’s necessary, not only to have a truth predicate, but also to have a name for each sentence. No pluralist have, since now, expressed herself about the possibility, for a domain of discourse, to include names of sentences. Nevertheless, given the paucity of expressive means which we need, to allow a language to do that, it’s reasonable to suppose that alethic pluralists wouldn’t deprive any portion of language of such means. 3 In this regard see what M. P. Lynch says in Lynch (2009, p. 6). 4 For Cotnoir’s solution to work is also necessary to assume that infinite disjunctions are impossible (see Cotnoir 2013, § 3), but here we are not going to deal with this aspect. 5 I think there is nothing in alethic pluralism which prevents us from thinking so. If someone finds this point controversial he is free to consider ij as injective. Relatively to our discourse it makes no difference.

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6 The fact that Yablo paradox is self-reference-free is still controversial; see Priest (1997) and Beall (2001). 7 Notice that, here, my use of the term “chain” is very different from the usual settheoretic use.

CHAPTER NINE DEFLATION AND REFLECTION: ON TENNANT’S CRITICISM OF THE CONSERVATIVENESS ARGUMENT CIRO DE FLORIO1

1. The Conservativeness Argument In the debate between the deflationists and the advocates of a substantial conception of truth, Stewart Shapiro (1998, 2002) and Jeffrey Ketland (1999) independently developed a somewhat relevant argument: the conservativeness argument. Since this paper is in the context of formal theories of truth, it is necessary to offer some further introductory words.2 In the axiomatic theories of truth one usually assumes that the theory of truth is constituted by a list of axioms which determine the meaning of the predicate “is true”. A straightforward example of a truth-axiom is as follows: (1) For every proposition p and for every proposition q, it is true (p and q) if and only if it is true p and it is true q.

Axiomatic theories of truth attribute truth to certain objects or “truthbearers”; these are parts of a suitable base theory. A “base theory” is any axiomatic theory describing a certain object domain. From the axioms of geometry, for instance, we obtain theorems describing the properties of geometrical entities. The theory of truth for geometry must state the conditions according to which a sentence belonging to (the language of) geometry is true. In the previous example the truth-bearers are propositions; however, in the widely discussed axiomatic theories of truth, there are other candidates willing to be truth-bearers, such as sentences for instance, which can be intended as sentence-types or sentence-tokens.3 In this paper we will follow the literature and use an arithmetic theory as a

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base theory.4 Therefore the following is a theorem of our base theory (henceforth B): (2) 2 + 2 = 4.

In addition the following is a theorem of a hypothetical theory of truth T for the base theory B: (3) “2 + 2 = 4” is true.

By adding the axioms of the theory of truth T to the base theory B we get what is commonly referred to as an extension of B. Specifically, the extension is considered conservative if every proposition belonging to the language of the base theory which is derivable from the base theory, together with the theory of truth that extends the base theory, is already derivable from the base theory alone. In other words if a theory is a conservative extension of a base theory it does not increase our preexisting knowledge of the base theory domain. Going back to our example, if the theory of truth T is a conservative extension of the arithmetic base theory B, we do not obtain any new knowledge of arithmetic facts from B+T as compared to what we have already achieved from B alone. Conservativeness seems to be a good formal translation of the deflationary requirement, which states that truth must be insubstantial. Therefore, for Ketland and Shapiro, the deflationists should be committed to a theory of truth being conservative on the base theory. Ketland calls this requirement conservation constraint. A deflationary conception of truth should be committed to some sort of conservation constraint for its favoured truth axioms (Ketland 2005, p. 77).

However this is not the only requirement that the deflationists must satisfy; according to Ketland a theory of truth must also satisfy the reflective adequacy constraint. Namely, a theory of truth—beyond its conservativeness—must be reflectively adequate. But what does this mean? Ketland recalls Tarski’s intuition. In establishing the semantic conception of truth, Tarski states that a good definition of truth should necessarily be materially adequate.5 Generally, according to Tarski, a definition is materially adequate if it is able to prove, given a sentence ࢥ, that ࢥ is true if and only if ࢥ; “analogously, reflective adequacy is the requirement on T that from a theory B we can prove ‘all theorems of B are true’” (Ketland 2010, p. 424). Therefore a reflectively adequate theory of truth for B has to prove that:

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In summary, according to Ketland and Shapiro, a deflationary theory of truth must fulfil the two requirements above: it must be both conservative and reflectively adequate. As the argument continues, though, these two requirements are not, in principle, able to be satisfied together. The ensuing reason depends on Gödel’s theorems which prove that if a formal theory able to express arithmetic elementary facts is consistent there exist sentences of that theory that are neither provable nor refutable.6 The structure of Gödel’s undecidable sentence is that of a sentence that states its own underivability. Furthermore, it is possible to show that this sentence is equivalent to one stating the consistency of the theory. It means that no consistent mathematical theory is able to prove that it is consistent. However, if a theory T can prove a generalisation as the one previously considered, i.e., that all theorems of the theory B are true, the theory T proves the consistency of B. Indeed if a theory is inconsistent it proves a contradiction; yet a contradiction cannot be true. Therefore if T proves that all theorems of B are true, it also proves that B does not contain contradictions, namely, that it is consistent. Thus if a theory T is reflectively adequate, it proves the consistency of the base theory B. In literature this generalisation (4) is usually called the reflection principle for B.7 Now, based on the conservation constraint, the theory of truth T must be a conservative extension of B; it cannot prove anything expressible in the language of B that is already provable in B. However, according to what was previously stated, T is able to prove that B is consistent; therefore B should prove to be consistent in itself as well. However, because of Gödel’s theorem, this is not possible. Moreover, thanks to the arithmetization procedure (invented by Gödel), it is possible to “code” the syntax of the theory B through natural numbers; but the sentence stating the consistency of the theory B belongs to the language of B. It follows that not every reflectively adequate theory of truth can be conservative. Otherwise, if the theory of truth is conservative, it cannot prove the reflection principle and it is consequently not reflectively adequate. Shapiro and Tennant’s conclusion is the following: deflationism is threatened as it cannot satisfy the two requirements that seem to be essential for the deflationary conception. Beginning with a reflection on the Gödel Phenomena, a non-deflationary conception of truth seems plausible.

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2. Tennant’s Objection Tennant is not a deflationist, and he explicitly admits it: “I am not a deflationist. I believe that truth and falsity are substantial” (Tennant 2005, p. 89). However he argues against Ketland and Shapiro’s argument and favours a deflationary conception of truth. Tennant’s criticism is extremely interesting from a philosophical point of view, even if it is not so easy to understand. Briefly, Tennant shows how to provide a particular extension of the base theory B—let us call it B*—which is reflectively adequate and deflationistically acceptable, even if not conservative on B. Tennant’s specific contribution to the debate on the conservativeness argument lies in the thesis according to which the extension, albeit nonconservative, is nevertheless unproblematic for a deflationist. However he does not directly formulate his criticism of Ketland and Shapiro’s thesis; rather the charge against the conservativeness argument is a consequence of Tennant’s (2002) wider discourse where he maintains that it is possible to account for the truth of Gödel’s sentence without adopting a substantive conception of truth. Let us briefly consider Tennant’s general argument; then we will analyse the aspects that best fit our objective, that is, his criticism of the conservativeness argument. As stated before Gödel’s theorems prove the existence of formally undecidable sentences within any non-trivial and consistent mathematical theory. Even more interestingly, these sentences, although unprovable, are intuitively true.8 It is possible to expound an argument that exploits the very structure of the undecidable sentence to show that it is true. Since the notion of truth in the following argument is highly relevant, we shall refer to it as the semantical argument. Semantical Argument G is a universally quantified sentence (as it happens, that is a universal quantification of a primitive-recursive predicate). Every numerical instance of that predicate is provable in the system S. (This claim requires a sub argument exploiting Gödel numbering and the representability in S of recursive properties.) Proof in S guarantees truth. Hence every numerical instance of G is true. So, since G is simply the universal quantification over those numerical instances, it too must be true (Tennant 2002, 556).

This argument is particularly interesting for us as it applies the notion of truth in the standard model of natural numbers. It is not by chance that Tennant calls this argument “semantical”; now, from Tennant’s point of view, the problem is the following: is it necessary to resort to a substantive conception of truth to establish the truth of Gödel’s sentence? Those who

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answer affirmatively adhere, in Tennant’s words, to the substantialist dogma: The way in which the semantical argument establishes the truth of the Gödel sentence requires that the notion of truth be substantial (Tennant 2002, p. 557).

Tennant’s long paper aims to show that it is possible to reject the substantialist dogma; it is possible, that is, to provide an argument that allows the establishment of the truth of Gödel’s sentence in a deflationistically acceptable way. Here we pay no attention to the technicalities of Tennant’s strategy; the crucial point is that if we give up the substantial notion of truth, we must assume a principle whereby we can prove that G is “true”. The method for doing so is easy and it depends on Gödel’s theorem: if it were possible to show that G is true through an argument that can be formulated within the theory B we would infringe the theorem itself according to which G is neither provable nor refutable within B. Indeed any argument proving the truth of Gödel’s sentence for a theory B is not representable within the theory B; alternatively it must go beyond the formal resources of B. We will return to this point. Now, Tennant formulates his argument in the theory that B*, according to what we have maintained before, is not a conservative extension of B. Then, how is it possible to extend B so that this move will be plain for a deflationist? Clearly it is not possible to extend B by adding the axioms of a substantive theory of truth9 as this would entail ipso facto the acceptance of the substantialist dogma. After evaluating several alternatives, Tennant returns to constructing B* by means of a particular reflection principle called RFN(B). As we have previously said, the general form of the reflection principle for the theory B is the following: (Ref(B)) All theorems of the theory B are true.

Now it is possible to reformulate it without using any truth predicates: (Ref’(B)) For any sentence p, if p is a theorem of B, then p.

Actually Tennant is more precise and refers to numerical instances of arithmetic formulas. Let ࢥ(x) be an open formula belonging to the language of B; since this paper is in the context of arithmetic theories, x varies on numbers. Therefore: (Ref’’(B)) For every x, if it is provable in B that ࢥ(x), then ࢥ(x).

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As we can see, the left-hand side of the reflection principle is related to the concept of provability, while the right-hand side states, in a certain way, the truth—or a surrogate of it—of what has been proven. In other words the reflection principles for B, whatever form they may take, are statements of the soundness of the theory B, since they guarantee the soundness of the proof procedures in B. Moreover, as we have seen in Ref’’(B), we refer to formula ࢥ belonging to the language of the theory B; the strength of the reflection principle directly depends on how wide the considered range of formulas is: the more complex they are, the more informative the principle. The most informative is the reflection principle whereby for every formulas ࢥ of the language of the theory B and for every x, if it is provable in B that ࢥ(x), then ࢥ(x). However, for Tennant’s purposes such a strong reflection principle is not required; a rather notable restriction is enough, more specifically, a restriction of primitive recursive formulas, which are formulas that have minimal logical complexity and express elementary relations among the objects of B.10 Therefore, the final form of Tennant’s reflection principle is: (RFN(B))  PrB(Χࢥ(‫)ܪ‬Ψ) ĺxࢥ(x).11

If for every x, it is provable in B that ࢥ(x), then for every x, ࢥ(x); where ࢥ is primitive recursive. In fact the reason that restriction is not difficult to understand is that the arithmetic predicate “x is the code of a proof in B of the sentence Į” is representable by a primitive recursive formula. Tennant’s formulation of the argument can be summarised as follows: Let us assume that m is the code of a proof of G in B. B is able to represent the primitive recursive relations and, so, in B it is provable that m is the code of a proof of G in B. However, since G is itself unprovable, it follows that m cannot be the code of proof of G in B. Therefore, in B, we can show that given a number m, this is not the code of proof of G in B. Hence, since m is arbitrary, this result can be extended to any n. Therefore, for every n, it may be proven in B that n is not the code of proof of G in B. Now let us extend B to B* by assuming the reflection principle RFN(B). For every x, if ࢥ(x) is provable in B, then ࢥ(x). Let us substitute ࢥ(x) with the formula that says that it may be proven in B that n is not the code of proof of G in B. Thus, by applying RFN(B) in B*, we derive that there is no proof of G in B.12

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Compared to the previously mentioned semantical argument, this reformulation makes no use of the notion of truth and, a fortiori, of the substantive conception of truth. As we have just seen this version of the argument is expressible within an extension of the base theory B, B*, based on the uniform reflection principle restricted to primitive recursive formulas. Finally, as the most important point for our purposes, Tennant charges the conservativeness argument: there is no need for a substantive theory of truth to prove that all theorems of the theory B are true (here, “true” must be construed in an acceptable, deflationary sense). It is enough to use the logical device of the reflection principles—specifically of the principle RFN(B)—that turn out to be clearly acceptable from a deflationary point of view, as they do not employ the notion of truth.

3. Ketland’s Reply Ketland (2005) replies to Tennant’s charge. The crucial point of Ketland’s argument is the following: On Tennant’s proposal, instead of proving the reflection principles in the manner proposed by Feferman, Shapiro and myself, the deflationist may simply assume them. As far as I can see, in the absence of the sort of truththeoretic justification given by Feferman, Shapiro and myself, Tennant’s proposal is that the deflationist may assume these principles without argument. No reason, argument or explanation for adopting the reflection principles is given by Tennant (Ketland 2005, p. 85).

In other words, according to Ketland, Tennant does not provide any justification for the assumption of RFN(B). Let us compare the two strategies: (Ketland) Base theory B + theory of truth for B. The resulting theory (let us call it T) allows for proving the reflection principle for B; i.e., it is able to prove that every theorem of B is true. Of course the theory is not conservative as further proof that truth is a substantial notion. (Tennant) Base theory B + extension of B based on RFN(B). The resulting theory (let us call it B*) allows for proving the fact that every theorem of B is true, and from that follows the consistence of B. Even in this case, B* is not a conservative extension of B.

Tennant’s reply is rather sharp: No further justification is needed for the new commitment made by

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expressing one’s earlier commitments. As soon as one appreciates the process of reflection, and how its outcome is expressed by the reflection principle, one already has an explanation of why someone who accepts [B] should also accept all instances of the reflection principle (Tennant 2005, p. 92).

It seems that Tennant is saying two different things: first, the justification of the reflection principle does not have to be external with respect to the intuitions underlying the formation of the base theory B; second, accepting the reflection principle is inevitable once some minimal epistemic conditions are satisfied. Let us consider Tennant’s conditions more in detail.

4. Reflection on Procedures Tennant claims that once the following conditions are satisfied, accepting a theory B entails, ipso facto, accepting the principle RFN for the theory B: (i) We reflect on the procedures of proof; (ii) The outcome of this process is expressed by the reflection principle.

Tennant’s position has an epistemic flavour: we are within the theory B and from B we can draw certain inferences, i.e., proofs from the axioms of B. This activity of proving can then become the subject of a reflection process the outcome of which is expressed by the reflection principle. Specifically, the acceptance of a given theory B is characterised by the fact that we “trust” the logical machinery of B. That is, if in B we prove a certain sentence ࢥ, ࢥ has to be true. Therefore, the reflection principle is the—rigorous—translation of the epistemic confidence attitude in the logical resources of B. From this perspective Tennant’s reply also becomes meaningful: it is obvious that we do not have to require further justification for RFN(B). Accepting the reflection principle for B is immediate once one accepts the theory B; as such, the following interpretation of Tennant’s thought seems plausible: to accept a theory means to accept its logical machinery, too. This is expressed by the reflection principle for B. Tennant’s and Ketland’s positions seem to be at a stalemate. By Gödel’s theorem the fact that all theorems of a given theory B are true is not, in turn, a theorem of B, even though it may be expressed in the language of B. In other words, the soundness of a theory—which indicates its epistemic reliability—is connected to the

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acceptance of the very theory; it would be completely irrational to accept the theory B and believe, at the same time, that some consequences of the axioms of B are not true. Nevertheless, the sentence expressing the soundness—the reflection principle—is not provable from the theory axiom. This entails what Ketland calls conditional epistemic obligation. Conditional epistemic obligation If one accepts a mathematical base theory [B], then one is committed to accepting a number of further statements in the language of the base theory (and one of these is the Gödel sentence G) (Ketland 2005, p. 79).

According to Ketland the theory of truth explains conditional epistemic obligation (CEO) through the notion of truth (see pp. 79-80). Tennant, on the other hand, thinks it is possible to explain CEO through the reflection principle, that is, the formal expression of epistemic confidence in the base theory B. Therefore, although these two extensions are not logically symmetrical—the theory of truth is a stronger assumption, both logically and conceptually, than Tennant’s extension—the state is symmetrical from the point of view of their justification. Ultimately, we have, on the one hand, truth-theoretic principles grasped by the theory of truth, and on the other, the reflection on the proving procedures grasped by the reflection principle. Both extensions are not conservative and both are able to account for CEO. If this is correct then a stalemate situation may occur; the feasibility of Ketland’s position is directly connected to intuitions concerning a classic, realistic concept of truth, which we cannot analyse further here. The opposing stance, advocated by Tennant, is connected to the possibility of assuming the principle of reflection and its justification. In the next paragraph we shall offer a deeper insight into this question.

5. On Tennant’s Principle of Reflection As we have described previously Tennant’s proposal hinges on the adoption of a particular reflection principle for the base theory B. Tennant, then, claims that further external justifications for this principle are not necessary as it is the straightforward consequence of the acceptance of the proving procedure drawn from the theory B. In other words, the principle RFN simply states that proof in B is, in a sense, sound and therefore reliable. Tennant’s thesis holds or dissolves depending on the possibility of arguing this conception of soundness. Below we intend to illustrate three points: first, the conception of soundness at stake cannot be the classical one; second, we will argue that Tennant’s idea of proof relies on a rather strong form of evidence that may not be completely formalised within

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theory B and third, we will ask whether this epistemic assumption is really compatible with a general deflationary stance. As far as the first point is concerned it is clear that Tennant cannot consider a classic conception of soundness, since it would entail, to some extent, a conception of truth and, in this way, Tennant would justify the principle of reflection through the notion of truth. Instead Tennant must consider the soundness of the deductive procedures in a non-classical— that is, constructivist—sense.13 However, some possibilities unfold here. Some scholars—faithful to the original Hilbert program—require that proof, in order to be reliable from an epistemic point of view, must be connected to finitist evidence.14 Now, one could surmise that Tennant’s optimum is to adopt a completely justified theory B*, namely a reliable theory. This, though, is an unachievable ideal because of Gödel’s very theorem; whatever extension is adopted to prove that base theory theorems are true must be deductively stronger, that is, it must be a non-conservative extension. From the epistemic point of view this means that if the reliability of a procedure of proof is standard the theory through which the soundness of the procedure is made explicit must go beyond the formal resources of the base theory. In other words, this theory is less reliable than the base theory. Tennant is aware of that: One accepts with equanimity [the process of reflection] by formulating that commitment (to one’s earlier methods) as a new principle, one has thereby taken on essentially new commitments. For that is the lesson of the Gödel phenomena. The more we try to express our confidence in our licit methods of proof, the more we extend them (Tennant 2005, p. 92).

Therefore, we can aim to interpret Tennant’s strategy as follows. If for every natural number n, it is evidently provable in B the sentence ࢥ(n), then the universal closure of that sentence is evident, that is, xࢥ(x). This would guarantee that the reliability for any single concrete case is transmitted to all infinite cases. That is, it is quite compatible with the reflections made before and, mainly, the notion of truth is nowhere in sight. Instead we have a particular form of evidence and it becomes relevant here to recall Tennant’s restriction to his reflection principle. RFN(B) is restricted to primitive recursive formulas, i.e., formulas endowed with the smallest logical complexity. This is a meaningful point since the arithmetic predicate “n is a proof of ࢥ” is primitive recursive; hence, the procedure of proof in B is reliable. In summary, we can say that:

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However, there is one last point worthy of mention. Arithmetic theories are usually interpreted in the universe of natural numbers. This structure is called the standard model. The standard model is constituted by the sequence 0, 1, 2, 3 and so on, to infinity. Nevertheless, regarding the model theory, it is a known fact there are infinite non-standard models that make the axioms of the arithmetic theories true but that are not structurally identical to the standard model. Very roughly, these models contain alien elements, that is, infinite numbers that are not reachable through the operation of adding 1 to 0.15 Now, as we have seen before, the reflection principle is epistemically justified as long as it encodes in some way the acceptance of our procedure of proof within the theory B. That is, the evidence at stake should permit the move from a potentially infinite series of cases to a universal generalisation. The consequence of the reflection principle is a universal quantification. It follows that it is important to know which range is of the universal quantifier. Intuitively we would answer: if ࢥ may be proven for each numerical instance, then ࢥ holds for every number. That is, the principle RFN(B) is justified based on the reflection of deductive procedures if an inference with the following structure holds: (ȍ) ࢥ(0), ࢥ(1), ࢥ(2), … Ÿxࢥ(x).

The inference (ȍ) seems to be intuitive; nevertheless (ȍ) is valid only if it is interpreted on a domain that contains standard natural numbers, exclusively. The reason is quite clear. Let us assume the domain to include a non-standard element c, which is not reachable through the application of the successor function to 0. Let us assume that ¬ࢥ(c). Clearly, the infinite premises of (ȍ) will be satisfied, as opposed to the conclusion— since x¬ࢥ(x). Therefore (ȍ) is valid only if the referring domain does not contain alien elements. However this means that the inference is valid only in the standard model of natural numbers. Tennant’s justification argument—

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according to which an external justification is not required for the adoption of the reflection principle—implicitly refers to an inference that is valid only in the standard model of natural numbers. Thus, we can add one last condition, which is: (v) The process of reflection on the procedures of proof must be restricted to the standard model of natural numbers.

6. Conclusion Let us summarise our process. Tennant’s strategy is justified if, in turn, the assumption of that particular kind of reflection principle for the base theory B is justified. Now, we have shown that the justification of RFN(B) does not have to be external to B, meaning that RFN(B) expresses our epistemic confidence in the logical machinery of B. Nevertheless this is not sufficient to establish the feasibility of Tennant’s strategy. In our opinion it is important to deepen the very meaning of the reflection principle and the role it plays in the debate between the advocates of a substantive conception of truth (Ketland and Shapiro) and those who argue for a deflationary conception of truth (Tennant).16 The point clarified in the last paragraph is that RFN(B) is justifiable if we have a strong form of evidence at our disposal, one able to guarantee the soundness of the deductive procedures restricted to the standard model of natural numbers. However, here we have a last question about Tennant’s proposal. Broadly speaking we can say that deflationism aims for the smallest commitment towards the concept of truth. In Shapiro’s inspired words, for the deflationists, truth is “metaphysically thin”. Now, it is clear that this sobriety regarding the concept of truth could not be reached through another notion that, in turn, is not deflationary. An example can be made in order to shed some light on the matter. Shapiro (1998) discusses a possible answer to the conservativeness argument, which hinges on higher-order logic.17 By adopting a second-order logic system the formal results are extremely interesting. For instance, it is possible to prove that Peano’s arithmetic is categorical, i.e., it only admits models that are structurally identical to the standard model. Second-order arithmetic is short of the wealth of non-standard models since we have the resources to describe the standard model of natural numbers univocally. It is difficult to treat the second-order logical consequence. One can prove that second-order predicate calculus is not semantically complete. This means there are logical consequences that are not provable; equivalently, there are

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consistent sets of second-order formulas that do not have models. All this information is relevant to the discussion on deflationism; one can show that the theory of truth for B, expressed in a second-order language, is semantically conservative on B. Roughly, it may be proven that every model that proves true the axioms of B can be extended to a model that proves true the axioms of T. This means that for every sentence ࢥ belonging to the language of B, if ࢥ is true in all models of T, it is true in all models of B. This seems to be particularly close to the deflationist desiderata: no new semantic consequence of the theory T is not already a consequence of the base theory B. Now is this a viable strategy for the deflationists? There seems to be a certain conceptual tension: on the one hand it is possible to avoid a substantive notion of truth; on the other, however, one assumes an equally “robust” notion, such as the secondorder logical consequence. As Shapiro puts it: The point here is that no matter what one’s views are on the status of second-order logic, a critic of deflationism might respond to the secondorder manoeuvre by arguing that the deflationist is hiding the robustness of truth in the second-order consequence relation. If the consequence relation is itself robust, then the contemplated manoeuvre fails to show that truth is thin and has no nature (Shapiro 1998, p. 510).

In other words, the passage to a stronger logic to advocate deflationism is, at least in Shapiro’s opinion, a deflationary suspicious passage. Is Tennant’s case analogous? First, we have to stress that in this debate the dichotomy of “robust” (or substantive) as opposed to “thin” (or deflationary) is not characterised in a precise way, and it is very much left to one’s intuition. In the discussion on the conservativeness argument this dichotomy has been characterised through the logical property of conservativeness; however, in that case, the conservativeness is purely syntactical, while it is possible to refer to other kinds of conservativeness, such as semantic. Indeed Tennant must resort to some form of evidence in order to justify the reflection principle. If our exegesis is correct this evidence must guarantee the epistemic reliability of the procedures of proof in B. However, what features is this evidence supposed to have? As we have seen before the evidence leading to the proving procedures hinges on a series of concrete instances—the numerical instances—to generalise all cases. For example, one has evidence that ࢥ(0), then ࢥ(1), then ࢥ(2) and so on until we can conclude that ࢥ holds for all numbers. The reliability of this procedure concerns standard numbers, that is, the “inhabitants” of the

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standard model. It is impossible to reach non-standard numbers through this procedure. By applying the successor function to a standard number, we always get a standard number. So, let us summarize our reflections on Tennant’s general strategy: -

-

The justification of Tennant’s reflection principle requires that, at least from the epistemic point of view, a form of omega-rule holds; that is, if it is evident that a certain property holds for 0, 1, 2, and so on, then it is evident that this property holds for every natural number. That justification requires the exclusion of the non-standard models of arithmetic; in other terms, we have to confine ourselves to the only standard model of natural numbers.

Now, is the notion of standard model troublesome from a deflationary point of view? This is a crucial question. As is well known to catch the standard model of natural numbers we have to pass to a second order characterization; but in this case, we are in Shapiro’s case: -

The second-order logical consequence is problematic from a deflationary point of view Then, the justification of Tennant’s reflection principle is troublesome too.

Therefore, we conclude that Tennant’s case is analogous to Shapiro’s: the move is suspicious from a deflationary point of view, since we earn the non-substantial truth at the price of introducing another substantial notion. Obviously Tennant could reply and argue that, as a deflationist, he is committed to the thesis whereby the truth is metaphysically thin, though not every notion has to be the same. In this particular case we are not committed to “metaphysically cumbersome” second order logical consequence, but only to the evidence of the omega-rule. But, the deflationist should exhibit independent reasons to differentiate the substantive notion (which he used) from the notion of truth. Tennant’s case is, indeed, peculiar. By arguing in favour of deflationism he expounds his argument without making use of the concept of truth. A fortiori—he concludes—a deflationist may adopt his strategy. Let us postulate a deflationist to assume Tennant’s view and add a disquotational theory of truth18 (let us call it DT). From a deflationary point of view this would normally be considered unproblematic. The justification for extending the base theory B* is then grounded in the reflection on the procedures of proof drawn within B, whilst the notion of truth introduced by DT is simply a linguistic device. The resulting theory B*+DT can prove that all

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theorems of B are true and, therefore, it is not conservative on B. What is required, now, is an independent reason to distinguish in B+DT* the truththeoretic deflationist aspect—connected to the list of Tarskian biconditionals—from the proof-theoretic substantial aspect—connected to the reflection principle. The justification, in other words, cannot simply lie in the claim that truth is not substantial; as in this case the manoeuvre does not seem entirely acceptable for a deflationist.

References Field, H. 1999, “Deflating the Conservativeness Argument”, The Journal of Philosophy 96 (10), 533 Halbach, V. 2001, “How Innocent is Deflationism?”, Synthese 126 (1), 167 —. 2011, Axiomatic theories of truth, Cambridge: Cambridge University Press Kaye, R. 1991, Models of Peano Arithmetic, Oxford: Clarendon Press Ketland, J. 1999, “Deflationism and Tarski’s Paradise”, Mind 108 (429), 69 —. 2005, “Deflationism and the Gödel Phenomena: Reply to Tennant”, Mind 114 (453), 75 —. 2010, “Truth, Conservativeness, and Provability: Reply to CieĞliĔski”, Mind 119 (474), 423 Serény, G. 2011, “How Do We Know that the Gödel Sentence of a Consistent Theory Is True?”, Philosophia Mathematica 19 (1), 47 Shapiro, S. 1998, “Proof and Truth: Through Thick and Thin”, The Journal of Philosophy 95 (10), 493 —. 2002, “Deflation and Conservation” in Halbach, V. and L. Horsten (eds.), Principles of Truth, Frankfurt: Ontos Verlag Smith, P. 2013, An Introduction to Gödel’s Theorems, Cambridge: Cambridge University Press Smorynski, C. 1977, “The Incompleteness Theorems” in Barwise, J. (ed.), Handbook of Mathematical Logic, Amsterdam: Elsevier Tait, W. 1981, “Finitism”, The Journal of Philosophy 78(9), 524 Tarski, A. 1983, Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Corcoran, J. (ed.)., Indianapolis: Hackett Publishing Tennant, N. 2002, “Deflationism and the Gödel Phenomena”, Mind 111(443), 551 —. 2005, “Deflationism and the Gödel Phenomena: Reply to Ketland”, Mind 114 (453), 89

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Notes 1

I would like to thank Stefano Caputo, Sergio Galvan and an anonymous referee for their comments. 2 In the following, we use logical symbols as little as possible, as well as crossreferencing to more technically detailed works. We hope that the general philosophical meaning of the discussion is not affected. An excellent work on the formal theories of truth is Halbach (2011). 3 See Halbach (2011, p. 8). 4 The choice of the base theory could be relevant depending on the purposes; in our case, we will see that the conservativeness argument can be formulated independently of the base theory chosen—at least if we limit ourselves to firstorder theories. Things are slightly different in the case of pure logical calculus. It can be shown, indeed, that by extending first-order logic with the list of Tarskian bi-conditionals (namely, the list constituted by all instances of the schema T(ࢥ) ļ ࢥ), it is possible to prove that there exist two objects, xy(x  y). It is a known fact that the existence of two objects cannot be proven using pure logic, so the extension is not conservative over the logic. See Halbach (2001). 5 See Tarski (1983, pp. 187-188). 6 See, for instance, Smith (2013). 7 See Smorynski (1977, pp. 821-865). As we will see, an essential part of Tennant’s proposal depends on the adoption of a particular reflection principle. 8 A brief remark is in order: in logic, one never speaks of truth in an absolute sense. Normally, a sentence (such as, for example, Gödel’s sentence) is true in a model, that is, in a particular interpretation of the theory’s signs in a determinate structure. 9 An axiomatic theory of truth normally regarded as substantive is constituted by Tarskian axioms. Field (1999) does not think so. Generally, if the theory of truth is Tarski’s classical theory of truth, then it is possible to prove the consistency of the base theory B. 10 Technically, the primitive recursive formulas—or PR formulas—are formulas in which quantifiers do not occur. 11 The expression “PrB(Χࢥ(‫)ܪ‬Ψ)” means that the open formula ࢥ(x) is provable in B. For details, see Smorynski (1977). 12 See Tennant (2002, p. 577). 13 According to this view, proof is correct because it is epistemically reliable; this happens because proof is an epistemic construction, which preserves the evidence from premises to conclusion. 14 Although it is an open question, it is held that finitist evidence is connected to theoretical objects that do not refer to infinite domains. According to this approach, the reliable procedures of proof do not entail form of reasoning to infinity. See, for instance, Tait (1981) for a defence of the finitist stance. 15 On the logical properties of the models of arithmetic, see classical Kaye (1991). 16 As previously stated, Tennant is not a deflationist, but in Tennant (2002) he “was playing devil’s advocate. The devil […] was the deflationist” (Tennant 2005, 89). 17 Shapiro (1998, 508-510).

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By disquotational theory of truth, we mean the theory of truth composed by all (and only) instances of Tarski’s bi-conditionals. If B is our base theory, DT is made up of all instances of T(ࢥ) ļ ࢥ, where ࢥ belongs to the language of B.

CHAPTER TEN HOW SIMPLE IS THE SIMPLICITY OF TRUTH? RECONCILING THE MATHEMATICS AND THE METAPHYSICS OF TRUTH ANDREA STROLLO

Metaphysica sunt, non leguntur! Mathematica sunt, non leguntur! (G.Frege)

1. Introduction The notion of truth is a central subject both in Philosophy and Mathematical Logic. The logical approach on the one side and the philosophical one on the other, however, mostly deal with problems which, apparently, require different tools to be tackled. In this paper I argue that such a separation can and should be overcome, and, in order to build a bridge, I focus on the philosophical issue of the insubstantiality of truth, which is a crucial topic to distinguish inflationist from deflationist proposals. Elaborating on the interpretation of insubstantiality in terms of the sparse/abundant classification of properties, I put forward a refined version in which certain flaws afflicting other formulations are solved. Then, I show how, using this improved variant, the philosophical notion of abundance can be fruitfully related to the formal notion of expandability of models, if a logical framework is adopted. Among other virtues, the obtained link can shed new light on the debate on deflationism and conservativity.1

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2. Insubstantiality as the philosophical mark of deflationism When deflationism is at stake, it is worth reminding that there is not such a thing as The deflationary conception of truth; instead, we have several variegate positions that may also be rather different. Nevertheless, we are often allowed to speak of deflationism as a general ideology because each member of the family holds a bunch of fundamental theses and is inspired by the same metaphysical hostility. It is now customary to use “deflationism” to refer to a typical version holding three tenets: 1. Tr[A] and A, (where “Tr” is a truth predicate, “A” a sentence and “[A]” a name for “A”) are intersubstitutable in all (non-opaque) contexts,2 and this is enough to explain everything about truth; 2. the truth predicate only serves a logical role giving us a disquotational device apt to express, through finite generalizations, infinite conjunctions and disjunctions; 3. truth is an insubstantial property. Two remarks should be added. First of all, point 2. is normally reconstructed as a consequence of point 1.; we are entitled to use the logical predicate to perform very useful logical roles, forming new sorts of expressions, because of intersubstitutability. Secondly, while such two claims concern the truth predicate, point 3. regards the property of truth. These are, though, two sides of the same coin. On the one hand, the property should be regarded as insubstantial because the corresponding predicate is only used to serve logical functions and not descriptive purposes; on the other hand, one can argue that it is exactly because truth has an insubstantial nature that we can apply the predicate to achieve purely logical goals, without committing ourselves to some deep picture of reality. The logical nature of the truth predicate, thus, could perhaps be clarified not by reflecting on its syntactical behaviour-as already discussed and criticized by Edwards (2013a)-but on semantical features (as Tarski's invariance under permutation, for example).3 We would have a logical predicate in the language, thanks to the fact that the property in the world is a logical property.4 It is important to stress that insubstantiality is crucial to characterize deflationism in a satisfactory way. Although intersubstitutability and the logical function have been the privileged way to argue in favour of the anti-metaphysical stance, they are neither the core of deflationism nor the unique reasons supporting it. In spite of that, intersubstitutability and the T-schema are frequently thought to be enough to capture a deflationary position, and the importance of insubstantiality is often neglected. If the logical function is clearly not necessary, since it could be dismissed-as the

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redundancy theory5 of truth shows-, not even intersubstitutability is essential.6 Despite being often viewed as the core feature of deflationism, intersubstitutability is not a sufficient condition, because insubstantiality is always needed. In fact, a proposal embracing both intersubstitutability (and possibly the thesis of the logical function), but rejecting insubstantiality at the same time, would lose the deflationary gist. The reason is that the Tschema and intersubstitutability can equally be combined with other inflationist conceptions as well.7 This frequently neglected point undermines the chance of sorting deflationism out simply by means of T-sentences or their cousins. Indeed, we can include the T-schema in a correspondence theory, for example, or we could simply accept intersubstitutability, while, at the same time, holding that the truth property is a robust substantial property, in the face of deflationism. In other words, we could turn deflationism into a refined version of (metaphysical) primitivism defending the idea the truth is a substantial and robust, yet simple and un-analysable, property. Thus, we would have a detailed account of how the truth predicate works, but a traditional metaphysics attached to it. Simplicity of truth, however, is not insubstantiality, and deflationism must be separated from a primitivist metaphysics. One might object that if intersubstitutability and insubstantiality are strictly connected, we are not free of espousing whatever metaphysics we want, and, perhaps, once intersubstitutability is bought, insubstantiality automatically follows. Even in this case, however, insubstantiality would still be the real reason to contrast primitivism. A different question that can be wondered is whether deflationism could not be distinguished from primitivism by the claim that the Tschema exhausts the theory of truth. A primitivist, in fact, might claim that truth is an ineffable notion, so that, besides being indefinable, it could not be axiomatized or adequately characterized, let alone exhausted, by any kind of principle. If so, the T-schema could be enough to identify deflationism, and insubstantiality would have no role to play. However, even if such a version of ineffable primitivism were legitimate, a deflationary theory should keep its distance from the primitivist idea according to which the truth property is a fundamental constituent of reality.8 Insubstantiality is necessary to account for this metaphysical difference. This does not mean that deflationism and primitivism do not share any important idea. They do, actually. Both approaches hold, i.e., that the notion of truth is un-analysable, and, accordingly, possibly an axiomatic treatment preferable; nevertheless, their metaphysical disagreement is harsh. In other words, what is at stake in this debate is the metaphysical pedigree of a philosophical ideology; it is the particular defended

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metaphysics what primarily marks a position as deflationary.9 If we embrace an inflationist position toward the metaphysical nature of the property of truth we stop being deflationists and it is no use invoking the T-schema or similar principles.10 Insubstantiality may be insufficient, but it is certainly necessary to identify deflationism.

3. How to understand insubstantiality Given such a pivotal role of the issue, what “insubstantiality” means is an urgent preliminary point. The problem is hard and deflationists have been shrewd to loiter and avoid the due explanation. In recent times, however, the topic received attention also in the debate concerning the alleged plurality of truth properties. In such a context significant efforts can be found. In particular, in a recent article, Edwards (2013a) discusses the principal options. The opacity conception, according to which insubstantiality could be understood in terms of our capacity of grasping the nature of a property simply by grasping its concept, is the first to be put aside. The second excluded optionņthe logical property conceptionņtries to identify insubstantiality with the logicality of the property of truth. The constitution conception, holding that substantiality could be explained by a definition specifying the constituents of truth, is also dismissed. Finally, the sceptic temptation, which considers all such efforts doomed, is rejected. Edwards offers a systematic discussion of each position, eventually arriving at a very promising proposalņindependently devised also by Asay (2014)ņwhich I am going to follow. Edwards and Asay broaden the discussion to the general question of the nature of properties applying a distinction originally proposed by David Lewis (1983). After all, although our interest is in truth, what we are looking for is not restricted to it. Notoriously, the biggest division between properties-theorists is the grand fight between nominalists and realists. According to a prominent version defended by the former, properties should be identified with classes. In this sense, being a table is just being a member of the class of tables. Since there is not any severe restriction upon the admissibility of classes, we have a great abundance of class-properties; there is a class-property for any meaningful predicate of the language, irrespectively of its cumbersome form and apparent outlandish nature. For example, we certainly have the property of being triangular or Jack the ripper or not the second chapter of this book. It is simply the class of things which are triangular or identical to Jack the ripper or not a chapter of this book; a rich class indeed. It is customary to call such kind of properties “abundant”. Note that, crucially, according to

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this approach, things are members of a class not because they exhibit a certain property; it is the other way around: things have properties because they belong to a certain class.11 On the other side of the battlefield, we have realists about properties. According to an influential version (going from Plato to Armstrong), properties are not mere classes of particulars, they are universals particulars instantiate or not. If properties are so understood, justice can be done to objective resemblance of objects, because now it is in virtue of exemplifying a certain universal that an individual falls under a determinate class. Universals enter individuals to constitute them; they are found in reality, and not just a collection stipulated by gathering entities already there. Thus, realists go in the opposite direction of classnominalists. For similar considerations, universalsņcontra properties as classesņare also well suited to play causal-explanatory roles. Given such premises, it is clear that we might hardly be willing to associate a universal to every meaningful predicate. It would be puzzling, for example, to admit that being triangular or Jack the ripper or not the second chapter of this book could ground objective resemblance or be required in causal explanations. We must have fewer universals than meaningful predicates. Opposite to the abundance of class-properties, thus, universals are sparse. Sparse properties carve reality at joints while abundant properties carve reality at joints-and everywhere as well.12 It is important to stress that austere realism and class nominalism are certainly not the only options to make sense of properties, and the abundant/sparse distinction is not forcefully a nominalist/realist opposition, though this is a natural option. I only presented it this way for the sake of simplicity and make the distinction sharp. Indeed, the distinction between sparse and abundant properties is legitimate and useful, regardless of how one understands the nature of properties.13 The distinction can be very useful both for a nominalist and a realist, as Lewis (1983) showed. Edwards and Asay noticed that the sparseness/abundance distinction is arguably what we have been looking for in the case of the (in)substantiality of truth. Sparseness, in fact, superbly explains what being metaphysically substantial should be according to inflationary intentions; while abundance can be identified with insubstantiality. If truth is sparse, it is a genuine ingredient of reality in a precise sense: it is a property needed to explain objective resemblance among certain things (the truths) and possibly grounding causal-explanatory roles. Deflationists are forced to deny the sparseness of truth, because their aim is exactly that of rejecting the idea that the truths exhibit some significant similarity, or

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that truth grounds any causal-explanatory power. However, they still want to claim that truth is a property of some sort to make sense of the truth predicate. Thus, they had better insist that sparse properties do not exhaust the viable options. What they need is an account of properties according to which the truths can be gathered together to form a suitable extension for the truth predicate, without being committed to any sort of robust grounding. Clearly, this is just what abundant properties are intended for. For opposite reasons, abundance is inopportune to the inflationist.

4. A problem Lewis' framework is clearly well suited for our goal but, nevertheless, it has its problems. The weak point is that, according to Lewis (1983) and Armstrong (1978), sparse properties are just those properties that are required by fundamental Physics. It is Physics what teaches us which universals are to admit in the ultimate ontological inventory of reality. If our conception implies such a super-sparse ideology, however, we cannot identify the substantiality of truth with sparseness, since there is little hope that truth will be classified as a basic property in fundamental particlePhysics. Thus, we are just forced to deny that truth is a fundamental sparse property, and we are back with our problem of insubstantiality. Edwards, who is aware of the difficulty, proposes a way out. In Lewis' account, in fact, sparse and abundant properties are not two rigidly separated groups; instead, properties should be arranged along a scale from sparser to more abundant ones. The degree of sparseness, then, could be determined by the length of the definition of a certain property from the fundamentally sparse ones. With this in mind, Edwards proposes a suitable reformulation: deflationary truth is insubstantial in the sense that it is in a fairly higher level of the scale than inflationists think it to be. This idea of a hierarchy of sparseness/abundance degrees is a simple option but it is still quite objectionable. For example, in order to posit a certain property on the scale, we need a precise reduction of this property in terms of the fundamental ones, but, even putting the huge problems of reduction aside, there is no big hope that we can find how to define the property of being amusing and not an economic crisis if between parentheses, a rather abundant property, in terms of Physics. More worryingly, we could imagine a world (possibly our world), call it “Onion”, with infinite complexity, so that every property would have infinitely many sparser predecessors.14 In this case the length of the definitions would always be infinite.15 Perhaps such problems might be solved with adequate discussions, but

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I believe that, with respect to the insubstantiality of truth, the approach is impracticable. The obstacle lies in the idea that sparse/abundance properties come along a unique continuous scale. The option is effective to trace some distinction between inflationists and deflationists in principle, but it is actually a bad option if we try to work with it concretely. If we take the scale seriously, even putting Onion aside, we should think that at the most basic level there are purely physical properties (like, to say, being a boson), at a higher level we find imperfectly natural properties (like being metallic), and as we go up we find things like being a table, being a grammatical English sentence, being triangular or Jack the ripper or not the second chapter of this book and so forth. Now, apart from the obstacle of giving the appropriate definition in each case, how could we decide the degree of not trivially classifiable properties? Is being a grammatical English sentence sparser than being the masterpiece of Rembrandt? Is being amusing and not an economic crisis if between parentheses sparser than being deflationary true? I do not see any straightforward or easy way to settle these cases. It seems that in each particular comparison we should enter some despairing debate where the crucial role is played by our intuitions and our metaphysical opinions. If so, what might we ever answer to the question: in which precise sense is inflationary truth sparser than deflationary truth? Certainly it is not enough to propose objective resemblance or something like that, because this is exactly the point. Even if we insisted that they have different degrees of sparseness/abundance, while being unable to determine them, the sceptic might insist: “how much significantly discrepant are their degrees to make such a classification relevant?”. The problem, in fact, is that in both cases we are exceptionally far from the fundamental properties of micro-Physics. Whatever their different degrees might be, both inflationary and deflationary truth would have more similarities than differences with respect, to say, being metallic or being triangular or Jack the ripper or not the second chapter of this book. We could simply doubt that between the two conceptions there is enough difference to make a difference. In both cases we have a rather abundant property of truth.16 Perhaps, we could speculate on what makes one slightly more abundant than the other, but to do so, since there is no precise procedure available, we should again evaluate intuitions and our underlying metaphysics. If this is the result, we are dragged back to our starting problem: what does (in)substantiality of truth consist in? If there is a significant difference here, the proposed sparse/abundance scale is not very helping or decisive. There is another solution, however. In fact, the fundamental view, according to which the only sparse properties are those recognised by

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micro-Physics, is not the unique option. Another view, the scientific view, which was defended for different reasons by Schaffer (2004), holds that we should regard as sparse exactly those properties admitted by some specific science, not only by fundamental Physics.17 Schaffer gives a number of different arguments supporting the scientific view as opposed to the fundamental view,18 but what we care about is just that, if we embrace his position, we still have a scale of sparseness, though not along a unique continuous line.19 When we claim that truth is (in)substantial, we should mean that it is a sparse property with respect, not to particle-Physics, but to the science relevant for the level of truth bearers, something like, to say, Linguistics or Syntax. In these terms we can keep both the valuable sparseness/abundance distinction and, at the same time, avoid the difficulties above. Thus, speaking of sparser or more abundant properties makes more intuitive sense, since the distance from the root is shorter and even reasonably recognizable. Of course, one still has the problem of making such a hierarchy precise (see below), but the issue is now handleable and the framework will not force possible differences into irrelevance.

5. A neglected option Good as the improved sparse/abundant proposal may be, the entire manoeuvre is deficient in one respect. The discussion above, in fact, ignores formal approaches to truth and is a pure, yet efficient, case of philosophical investigation that keeps separating the two fields. This is already unpleasant, but the worst aspect is that, doing so, it completely forgets a very prominent notion that has been discussed, in a formal framework, to make sense of the insubstantiality of truth: conservativity.20 Did we fall victim of ignoratio elenchi wasting all our subsequent efforts, or can this formal option be somehow reconciled with our philosophical account? I shall argue for the latter. In particular, while refining the conservativity setting, I will build some useful bridge between metaphysical and formal treatments of truth. Suppose that B is a formal (first order) theory in the language LB and a new theory T in LT is added, yielding B U T. Despite the new vocabulary, in some cases the theory T might be a quite innocent addition to B. For instance, it might be the case that B U T does not prove more sentences in the original language LB than those already provable by B alone. In other words, viewed from the perspective of B, the addition of T is (in this sense) useless: concerning the entities B describes, T gives no new contribution. When this is the case we say that T is conservative over B.

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Since a conservative theory of truth does not deliver new theorems apart from those concerning semantical matters (namely sentences involving the truth predicate), the notion seems a good way to specify why truth should be considered an insubstantial property.21 To apply the concept of conservativity, however, a formal framework is needed and a mathematical counterpart of deflationism required. Such an approach can be rephrased in more general terms. The basic reason to do so is that, given the motivations of deflationists, a stronger concept can be proved to be more appropriate. Furthermore, once such a reformulation is available, it can be reconciled with our Lewisian analysis. The basic idea is that of replacing the notion of conservativity with that of expandability of models.22 This is a slightly technical concept, but it is a very basic tool in model theory and I shall introduce it avoiding technicalities.

6. Insubstantiality as expandability of models The strategy of the renewed approach is simple: we should just shift from focusing on the impact of a theory of truth over a (not semantical) base theory to the impact of a theory of truth over the models of such (not semantical) base theory. In the simplest case, a model M of a theory B in the language LB is just a pair where we have a domain D (a set of objects with a certain structure) and an interpretation I of the language LB (a function giving the semantics for the language of the theory) such that, if LB is interpreted according to I, then the theory B is true of D. D may be viewed as the set of entities we want to speak about. The function I, instead, gives the semantics of our language: it turns LB from a purely syntactical language into an interpreted one. Finally, we have B, which is a collection of correct assertions about what D is like (M is assumed to be a model of B). When we add a new theory T to B, getting B U T, the base model M can be adapted (when possible) in two ways: or by expanding the interpretation I (so that even LT will be interpreted in D) or, when this is not enough, by extending the universe D (so that new elements will be added to the domain or its structure modified according to what T requires). Clearly at least an expansion is mandatory insofar the new theory is formulated in a new language, because a connection between the new expressions and the domain is needed. On the contrary, an extension is not always necessary; sometimes we can leave the original domain unchanged and make it apt to make also the new theory T true, provided that we interpret the new language LB conveniently. When an extension of the domain is forced, though, we have to change the base universe D. These basic model theoretic concepts relate to conservativity in a direct

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way. In fact, we have that: if every model M of a theory B in LB can be expanded to a model M* of the extended theory B U T in LB U LT , then T is conservative over B (the other direction fails).23 Now, the point is that interpreting the insubstantiality of truth in terms of expandability is easier and more natural than interpreting it in terms of conservativity. Indeed, the latter may be seen as a mere consequence, or a particular case, of the former. If every model M of a (not semantical) base theory B can be expanded to a model of B plus a theory of truth T, then the addition of a truth predicate is such that it only affects the language. The relevant aspect is the semantical interpretation I and the strength of truth does not exceed that of being a linguistic device. The entities characterized by the (not semantical) theory B, which form the domain of M, are not touched by the introduction of a truth predicate and the corresponding property (they are only affected when also an extension is imposed). In particular, if any model of B can be expanded to a model of B U T, then the truth predicate can be given an extension in every domain of B, irrespective of its structure. In this sense, truth does not discriminate among all possible domains the base theory B describes. On the contrary, if a mere expansion is not enough and an extension of the model is required, then truth has an import beyond semantics and is far from being an evanescent presence. In this case truth does not supervene over the extra-semantical entities in the domain, but enters it forcing them to take a certain shape: if a truth predicate must be semantically interpreted, then also the domain is required be so and so. Truth, when an extension of the model is at stake, is not confined into a semantic region, it crosses the boundaries of language and enters the realm of extra-semantical things the language describes. In this precise sense, we can say that extension of models implies a substantial metaphysics of truth.24 Things can be put in more general terms. Usually, a first order theory B does not characterize one single model (up to isomorphism) so that in the standard case we have the theory B and the class of its models Mod(B).25 If every model M of B can be expanded to a model of B U T, we have that Mod(B) Ž Mod(B U T),26 but the more expandability fails, the more B U T is able to exclude models in Mod(B). This class dwindles according to the metaphysical richness of T. Since, on the market, there is a number of different axiomatizations of the truth predicate, with highly different strength, then the whole range of axiomatic theories can be arranged in a map,27 by classifying them from the thinnest (those which leave Mod(B) unchanged) to the thicker ones (those which drastically restrict Mod(B)).28 If expandability can be identified with insubstantiality, as I am proposing, then we have a straightforward metaphysical classification of formal

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theories of truth and an open door to their philosophical evaluation.29

7. Expandability and Abundance The philosophical considerations on the problem of the insubstantiality of truth, which led us to confront with a general account of the nature of properties, can be now reconciled with the formal reflections sketched above. An abundant property can give an extension to any possible predicate, so that it is natural to see it as a mere collection of things, such that their membership is not grounded in an exemplification of further determinate features. Instantiating an abundant property leaves a thing basically unaltered. Giving an extension is just a matter of gathering entities in some way, without forcing them to exhibit any new trait.30 Suppose that such things have been characterized by specifying which sparse properties they instantiate, thus the introduction of an abundant property over them is harmless, it does not improve them any further. Sparse properties, in fact, are assumed to be all is needed to ground objective resemblances and causal behaviours. Per se, sparse properties are supposed, by definition, to give a complete account of reality. On the contrary, abundant properties make no serious contribution to mark things. Exactly the same happens, in a formal counterpart, when expandability is at stake. We have a theory describing a certain model; if this model can be expanded to a new model of an enriched theory, then subsets of the old domain giving extensions for the new predicates can be found.31 If such sets are always available (in any model of the base theory), then giving a semantical interpretation for the new predicates is just a matter of putting things in a suitable class, without forcing them to instantiate any new particular way of being. The added theory T does not theorize new sparse properties with respect to those already described by the base theory. Therefore, abundance can be red in terms of expandability of models when a formal framework is available: expandability = abundance = insubstantiality. On the other hand, if we could give an extension to a predicate only forcing the entities in the domain to instantiate some further specific feature, such a predicate would not just call for a semantical interpretation, it would impose some new robust constraint onto the objects it applies to too. A sparse property would be added. Sparseness, in turn, can be identified with lack of expandability. Given the reformulation of the conservativity explanation in terms of expandability of models and the correlation of abundance with expandability, we can now put everything together. First of all, according to the scientific view, we argued that the properties that should be

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considered sparse are those required by the relevant science. In the case of a truth theory, the right level can be individuated in something like a theory of syntax. What should be checked, then, is whether a theory of truth is able to impose a new sparse property onto the domain of syntactical objects, or, on the contrary, its contribution is dispensable and under this respect the relevant science is enough. In our formal counterpart, then, we will start with a formal theory of syntax (like, to say, Peano Arithmetic or a Concatenation theory),32 and assess whether a certain axiomatic truth system admits or not, and in what measure, expandability of models.33

8. Conclusion Once we have identified insubstantiality with abundance and read abundance in terms of expandability of models, we have a powerful bridge to go back and forth from philosophical considerations to formal evaluations. The proposed path is hard and many philosophers will probably dislike it. However, we had better not confuse difficulty with irrelevance. If it is true that there is not a mathematical substitution for Philosophy, this is not a reason to reject the valuable services of Mathematics, when available. Unless we prefer avoid precision and answering hard questions.

References Asay, J. (2014), “Against Truth”, Erkenntnis 79 (1), 147 Armstrong, D. 1978, Universals and Scientific Realism, Cambridge: Cambridge University Press Damnjanovic, N. 2010, “New Wave Deflationism”, in Pedersen, N. and C.D. Wright (eds.) New Waves in Truth, New York: Palgrave Macmillan, 45 Edwards, D. 2013a, “Truth as a Substantive Property”, Australasian Journal of Philosophy 91 (2), 279 —. 2013b, “Naturalness, Representation, and the Metaphysics of Truth”, European Journal of Philosophy 21 (3), 384 Halbach, V. 2001, “How Innocent Is Deflationism?”, Synthese 126 (1-2), 167 —. 2011, Axiomatic Theories of Truth, Cambridge: Cambridge University Press Hodges, W. 1993, Model Theory, Cambridge: Cambridge University Press Horsten, L. 2009, “Levity”, Mind 118 (471), 555 Horwich, P. 1998, Truth, Oxford: Clarendon Press

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Ketland, J. 1999, “DeÀationism and Tarski’s Paradise”, Mind 108 (429), 69 Lewis, D. 1983, “New Work for a Theory of Universals”, Australasian Journal of Philosophy 61 (4), 343 —. 1999, Papers in Metaphysics and Epistemology, Cambridge: Cambridge University Press Schaffer, J. 2004, “Two Conceptions of Sparse Properties”, Pacific Philosophical Quarterly 85 (1), 92 Shapiro, S. 1998, “Proof and Truth: Through Thick and Thin”, Journal of Philosophy 95 (10), 493 Sider, T. 1995, “Sparseness, Immanence, and Naturalness”, Noûs 29 (3), 360 Strollo, A. 2010, Subtle Truths. A Formal Investigation into Deflationism and Conservativeness, Doctoral Dissertation, University of Torino —. (in preparation), “Deflationism and the Invisible Power of Truth”

Notes 1

The debate started with Shapiro (1998) and Ketland (1999). For a critical survey see Strollo (2010). 2 This is what, in recent discussions, tends to replace the more familiar appeal to Tarskian biconditionals, which can be seen as a particular case of intersubstitutability when the law of identity is available. If the underlining logic proves AĺA (for each sentence A), then, by intersubstitutability, we immediately gain AļTr[A]. 3 A permutation of a collection of objects is a one-one mapping from the collection onto itself. Logical constants have been thought to be indifferent to the particular nature of objects we are speaking about, so that it should make no difference if some are switched with others. Permutation invariance is intended to specify this idea. 4 I find this line of thought particularly appealing, but I will not proceed in this direction. Basically, because the problem of the nature of logical constants is an highly debated one and, if I should rely on such a dubious proposal, I could hardly say to have solved the riddle. Moreover, I think that another strategy can be squared with the proposal of Edwards (2013a) and Asay (2014), as, hopefully, the second half of the paper will show. 5 I am using here the term “theory” to refer to an informal proposal. I hope the reader is not confused by this use and shall be able to disambiguate the term when used, instead, to refer to a formal logical system. 6 According to a redundancy theory of truth, truth would ground intersubstitutability, the property of truth would be insubstantial and still we would not get any special logical tool. Indeed, a truth predicate would be absolutely redundant. I should remind that a redundancy theory holds that a truth predicate can be dropped from the language without expressive loss. Strictly speaking

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neither a pro-sentential theory nor Ramsey's conception (if carefully described) are redundancy theories. 7 Suppose we had a proposal defending the position according to which truth is an insubstantial property, but, for logical reasons, intersubstitutability fails (for example in the case of paradoxical sentences) and consequently no logical instrument is gained. Given the fact that it would reject the basic thesis of inflationism, it would be hard not to see this as a version of deflationism. Notably, Horwich's (1998) proposal can be viewed as a significant example of such a case. 8 For a critique of ineffability as a criterion, however, see Edwards (2013a, section 3). What I am here calling “ineffability” could be understood, in Edwards terms, as irrecoverably opacity. 9 This point is usually acknowledged by those working on deflationism (see Damnjanovic 2010, Edwards 2013a), so that mine is not a peregrine position, though often neglected. 10 Of course one might argue that the difference between a primitivist and a deflationist metaphysics is not such a great one, and possibly the difference is only apparent. Several theorists, (for example Halbach 2001 and Horsten 2009), have tacitly thought that they were free to pass from a deflationary position to a primitivist one without great celebrations. I believe this is wrong and underestimated. Along this way deflationism is simply pushed back into a traditional position. 11 On this view, abundant properties are highly undiscriminating: “any two things share infinitely many properties and fail to share infinitely many others” (Lewis 1999, p. 346). 12 See Sider (1995) for a discussion. 13 See Asay (2014). 14 See Sider (1995). 15 We could certainly try to embrace an infinitary language, but even in this case many definitions will have the same length. 16 The distinction could not be recovered by pointing to the possible reduction of inflationist truth to sparser properties. In this way, in fact, insubstantiality would be clarified not with the notion of sparseness, but embracing the constitution conception, an option Edwards already rejected. 17 Perhaps Edwards (2013b, section 3.1) is close to this when discussing representational theories of truth. In this case the relevant science should probably be Psychology or Neuroscience. 18 For instance the problem of Onion is easily solved. 19 Notice that this approach is neutral with respect to ontology, since there is still room to argue that the ontologically basic and perfectly natural properties are those admitted by the most basic science (if any) and the higher ones somehow related to the lower by reduction or supervenience. Sparse properties would be those we should reduce at each level. 20 I guess that the reason is purely contingent. Edwards develops his proposal focusing on pluralism; most philosophers in this area did not take part in the conservativity discussion and vice versa. Moreover, while the latter involves a

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rather rich application of logical tools, the former is a more traditional exercise of speculative philosophy. In any case, I believe this mutual negligence to be serious; the two debates should stop ignoring each other. 21 For an extensive discussion see Shapiro (1998), Ketland (1999), Strollo (2010). 22 I exploited this identification to enlighten different problems in Strollo (in preparation). 23 This is a basic fact in Model Theory, see Hodges (1993), chapter 2. The fact that conservativity and expandability are not equivalent concepts has not trivial philosophical consequences. See Strollo (in preparation). 24 For a more extended discussion see Strollo (in preparation). 25 Roughly speaking, two models are isomorphic if there is a one-to-one mapping which preserves the relations existing among the elements in each domain. In other words, two isomorphic models are indistinguishable by means of L. 26 This way of writing is incorrect since on the one side we have models of B and, on the other side, models for B U T, which are a different sort of structures (i.e. signatures and interpretations are different). This might be made more precise speaking, for instance, of the LB-reducts of the models in Mod(B U T), however this is a case where I prefer perspicuity to technicality. 27 The map does not need to be a strict hierarchy. 28 There are problems against a generalized application of expandability. First of all it is not clear when an axiomatic system could be considered mirroring a semantic construction (see Halbach 2011). Secondly, there are semantical constructions for which an axiomatic system can not be devised (Revision theory for example is not axiomatizable). In such cases, expandability is no use. However, we should not abandon the effort of joining philosophical and formal approaches together. To this aim we can directly focus on semantical constructions and correlate insubstantiality not with expandability but with set theoretic complexity. These two strategies are strictly tied, since set theoretic complexity is strongly correlated with proof strength of axiomatic systems. Such refined approach might be viewed as the most valuable one but, unfortunately, for problems of space, I can not adequately extend my treatment to it. 29 The difference with a mere application of conservativity is twofold. First of all, this model theoretic approach can be efficiently related to a metaphysical treatment (i.e. via expandability/abundance); secondly, it allows a finer grained classification of formal theories, at least in this context. 30 Apart from being member of the class itself, which, however, should be viewed as an extrinsic feature. 31 We could have many different expansions. 32 Notice that this approach fits surprisingly well with the standard practice followed in the conservativity debate. 33 This presentation gives at least a sufficient description of one direction: formal truth theories can be metaphysically evaluated. In order to mathematically test philosophical theories we need formal counterparts. This is an hard task, but there is no reason to think it impossible. In the case of deflationism this happened, and the exchange has been deeply fruitful.

CHAPTER ELEVEN THE GENERAL MISSING FROM THE HIERARCHY* ELIA ZARDINI

1. Semantic Paradox and Generality As shown by the semantic paradoxes, a language with enough expressive resources whose background logic is classical cannot contain a truth predicate validating every instance of the truth schema: (T) “P” is true iff P.1

For example, consider the Liar sentence Ȝ0 “Ȝ0 is not true”. The relevant instance of (T) is “‘Ȝ0 is not true’ is true iff Ȝ0 is not true”. Since “Ȝ0 is not true” is identical with Ȝ0, by indiscernibility of identicals the relevant instance of (T) entails “Ȝ0 is true iff Ȝ0 is not true”,2 a classical absurdity. Keeping henceforth fixed the expressive resources required to achieve the referential loops triggering the semantic paradoxes, these then present us with the choice between classical logic and unrestricted (T) (also known as naive truth),3 and solutions to the semantic paradoxes can correspondingly be classified according to whether they preserve classical logic and revise naive truth or, instead, preserve naive truth and revise classical logic. It is not an easy choice.4 Sometimes, solutions that revise classical logic are faced with the charge of “crippling” (sustained) ordinary and scientific reasoning with logical operations like negation:5 such solutions are supposed to make trouble for an utterly innocent logical principle like, to take as example a prominent kind of revision of classical logic, “There are infinitely many twin primes or it is not the case that there are infinitely many twin primes” (for that principle has the same form as other principles that such solutions reject). But, just so, solutions that revise naive truth can be faced with the charge of crippling ordinary and scientific reasoning with semantic properties like truth: such solutions can

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be supposed to make trouble for an utterly innocent semantic principle like, to take as example a prominent kind of revision of naive truth, “If there are infinitely many twin primes, then ‘There are infinitely many twin primes’ is true” (for that principle has the same form as other principles that such solutions reject). Often, it is conceded that the clash between classical logic and naive truth can be restricted not only to discourse about truth, but, more specifically, to discourse about truth of sentences that are, in some vague sense that would definitely need to be made more precise, involved in a referential loop triggering a semantic paradox (I’ll henceforth call such sentences “ungrounded”). The concession is crucial first in that it forces the area in which solutions that revise classical logic envisage an effective failure of classical logic to be no larger (and no smaller) than the area in which solutions that revise naive truth envisage an effective failure of naive truth—the initial advantage enjoyed by the latter solutions of only affecting discourse about truth rather than any kind of discourse involving logical operations evaporates. Moreover, the concession is crucial also in that it operates a restriction that is much less stringent than it may at first appear to be, since, given the variety of referential loops in discourse about truth, many statements about truth with enough generality will be apt to involve ungrounded sentences—within the resulting area, it is far from clear that the restrictions on ordinary and scientific reasoning imposed by solutions that revise naive truth are in any sense less crippling than the restrictions imposed by solutions that revise classical logic. Given such concession, then, solutions that revise classical logic are still faced with the charge of crippling ordinary and scientific reasoning about ungrounded truth: such solutions are still supposed to make trouble for a logical principle like, to stick to one style of example of the last paragraph, “Everything the Sultan says is true or it is not the case that everything the Sultan says is true” (for that principle has the same form as other principles that such solutions reject, and may well involve an ungrounded sentence). But, just so, solutions that revise naive truth can still be faced with the charge of crippling ordinary and scientific reasoning about ungrounded truth: such solutions can still be supposed to make trouble for a semantic principle like, to stick to the other style of example of the last paragraph, “If it is not the case that everything the Sultan says is true and ‘It is not the case that everything the Sultan says is true’ is everything the Grand Vizier says, then everything the Grand Vizier says is true” (for that principle has the same form as other principles that such solutions reject, and may well involve an ungrounded sentence).

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With these facts in view, it is thus at least far from clear that the restrictions on ordinary and scientific reasoning in effect imposed by solutions that revise naive truth are in any sense more natural or less hampering than the restrictions in effect imposed by solutions that revise classical logic. My two cents is that, in typical cases, it will actually be solutions that revise naive truth that impose less natural and more hampering restrictions on ordinary and scientific reasoning, for, in the area of ungrounded truth, while solutions that revise classical logic still preserve a wealth of principles for logical operations (e.g. the very useful principles of adjunction, modus ponens, De Morgan etc.), solutions that revise naive truth only preserve precious few principles for truth (e.g. the very useless principle that a conjunction is true only if both conjuncts are plus a couple of other principles depending on the details of the solution). To make this vivid, notice that, while, given only naive truth and a bunch of other plausible assumptions, it is a child’s play to establish the complex quantificational claim “If the Grand Vizier contradicts something the Sultan says and everything the Sultan says attributes truth to something the Mother Sultan says, then everything the Grand Vizier says is true only if something the Mother Sultan says is not true” (which, contrary to “Everything the Sultan says is true or it is not the case that everything the Sultan says is true”, remains naturally compelling in the presence of ungroundedness and is the typical kind of thing that ordinary and scientific reasoning about ungrounded truth delivers), it is arguably impossible to establish it in the presence of classical logic without falling into triviality (or without rejecting the principle that what is provable is true, which remains naturally compelling in the presence of ungroundedness and is a crucial tool of ordinary and scientific reasoning about ungrounded truth).6

2. Hierarchy Be that as it may with respect to the big debate between solutions to the semantic paradoxes that revise naive truth and solutions that revise classical logic, in this paper I’ll focus on whether and how a particular approach falling into the first camp can deal with some of the issues concerning certain uses of truth involving generality. The approach in question, going back to Tarski (1933), is the one consisting in replacing the property of truth with a hierarchy of properties that are nevertheless, in the sense that will emerge in the next sentence, “truth-like”. The rough idea (which, for our purposes, we needn’t make fully precise) is to introduce a series of ordinal-indexed truth-like predicates (and, consequently, of ranks of sentences), such that, for every finite ordinal n,7 the truth-like

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predicate “truen”, when replaced for “true”, makes correct the (T)-schema as restricted to sentences of rank d n.8 A bit less roughly, we start with a base set of sentences free of truth-like predicates (rank 0); we form rank 1 by adding “true0”, which, when replaced for “true”, is supposed to make correct the (T)-schema for sentences of rank 0 (let’s call the resulting schema “the (T0)-schema”); we form rank 2 by adding “true1”, which, when replaced for “true”, is supposed to make correct the (T)-schema for sentences of rank 0 or 1 (let’s call the resulting schema “the (T1)schema”); we form rank 3 by adding “true2”, which, when replaced for “true”, is supposed to make correct the (T)-schema for sentences of rank 0, 1 or 2 (let’s call the resulting schema “the (T2)-schema”) etc. (we can also add that no truth-like predicate applies correctly to anything that is not a sentence). Even given a complete evaluation for the sentences of rank 0, these stipulations about the hierarchical theory (henceforth, “the Hierarchy”) do not even suffice to fix the correct application of “true0” over sentences of rank 1 and higher (although they do suffice to fix its correct application over sentences of rank 0), and thus they do not even suffice to fix the correct application of “true1” over sentences of rank 1. To fill that gap, I’ll assume the version of the Hierarchy according to which “truen” does not apply correctly to any sentence of rank > n. This assumption has the consequence that the hierarchy is strictly decreasing in strength, so that a sentence is truen only if it is truem [m: m > n]9 whereas the converse needn’t hold (we’ll see two counterexamples in the next paragraph). To get a concrete sense of the workings of the resulting hierarchy, let’s go through a couple of examples. Since “‘Snow is white’ is true0” has rank 1, it is not true0. However, since “Snow is white” has rank 0, and since snow is white, by the (T0)-schema “Snow is white” is true0, and, since “‘Snow is white’ is true0” has rank 1, by the (T1)-schema “‘Snow is white’ is true0” is true1. Or, since the Liar sentence Ȝ1 “Ȝ1 is not true0” has rank 1, it is not true0, and so, by the (T1)-schema, it is true1. The second example is particularly telling, as it explicitly contradicts a widespread understanding of the Hierarchy according to which it involves some sort of weird syntactic or semantic prohibition to apply “truen” to any sentence of rank > n (see e.g. Sainsbury 1995, pp. 118–121). On the alternative understanding of the Hierarchy I’m recommending, it is not as though Ȝ1 is a syntactically or semantically bad thing to do; quite the contrary, Ȝ1 is perfectly well-formed and perfectly meaningful. Indeed, Ȝ1 is actually truen [n: n > 0], which, with a staggering abuse of the concepts available in the Hierarchy, one might take to imply that the Hierarchy thinks that Ȝ1 is “true” (after all, a valid law like “‘Snow is white’ is true0

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or it is not the case that ‘Snow is white’ is true0” is also not true0 but truen [n: n > 0], and one might take it that the Hierarchy thinks that such law is “true”). More generally, one might take it that the Hierarchy thinks that Liar sentences are “not true up to their rank” and “true at all higher ranks”, and hence that they are “true”, each of them in effect amounting to a particular a instance of the general correct point that every sentence of rank n+1 is not truen. One might thus take it that the Hierarchy thinks that Liar sentences offer “true” partial descriptions of an important structural feature of the Hierarchy, and so that Liars speak the “truth” about the Hierarchy. Rather than jumping to the subverting conclusion that the Hierarchy must then be a lie, I emphasise that the particular way I’ve put things constitutes a “staggering abuse of the concepts available in the Hierarchy” since the Hierarchy does not envisage any property of truth, only an infinite series of properties that behave like truth only with respect to a finite number of ranks and do not behave at all like truth on the remaining infinite number of ranks (nor does the Hierarchy envisage the quantification into subscript position implicit in the third last sentence). Obviously, this circumstance does give rise to a cluster of worries concerning the Hierarchy’s ability to account for generalising uses of truth, worries which will entertain us from section 3 onwards.10 A more immediate worry concerns however the Hierarchy’s ability to account even for non-generalising uses of truth. What we may call “specifying non-generalising uses”, as, for example, in “‘‘Snow is white’ is true’ is true”, are easy to account for: in the example, the sentence referred to must really be something like “‘Snow is white’ is truen”, and then one can choose a higher rank (say, n+1) guaranteed to do the job of producing a sentence that is correct iff “‘Snow is white’ is truen” is. What we may call “blind non-generalising uses”, as, for example, in “The first sentence uttered by a Pole in 1933 is true”, are more difficult to account for: in the example, at least without further information on which sentence is the first sentence uttered by a Pole in 1933 one can choose no rank guaranteed to do the job of producing a sentence that is correct iff the first sentence uttered by a Pole in 1933 is (whichever rank n one chooses, there is a risk that the first sentence uttered by a Pole in 1933 has rank, say, n+1).11 It is this problem with blind non-generalising uses of truth that has at least in part motivated contextualist versions of the Hierarchy (a paradigmatic reference is Burge 1979), which, among other things, typically postulate “true” and its likes in natural language to function in such a way as to be guaranteed to pick out in each context an appropriately high rank even if the speaker does not know exactly which rank that is.

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This postulation is however problematic in the light of what seem to be general features of the contextual interpretation of adjectives. For example, it does not seem that “tall” works in such a way as to be guaranteed to apply correctly to whichever person in a context the speaker tries to denote by uttering “The tall person next door is Polish” (assuming that there is exactly one person next door). But that would be the broad analogue for “tall” of the contextualist postulation for “true”, with “true” picking out a rank n high enough for the (Tn)-schema to apply to the first sentence uttered by a Pole in 1933 (and so, as per the intention of a typical utterance of “The first sentence uttered by a Pole in 1933 is true”, say something that is correct iff the first sentence uttered by a Pole in 1933 is correct) comparing to “tall” picking out a height threshold low enough for “the tall person next door” to denote the person next door (and so, as per the intention of a typical utterance of “The tall person next door is Polish”, say something that is correct iff the person next door is Polish).12 Having noted this, I’ll henceforth assume for the sake of argument that the kinds of pragmatic mechanisms postulated by the contextualist Hierarchy are acceptable. With this background, we can turn to a wellknown worry concerning the Hierarchy’s ability to account for generalising uses.

3. Truth Meets Politics An example due to Kripke (1975, pp. 696–697) is alleged to show that the Hierarchy cannot account for certain generalising uses of truth. We consider Dean and Nixon, each of whom wants to deny everything the other says,13 by uttering, respectively: (D) Everything Nixon says is not true; (N) Everything Dean says is not true.

Let ran(ij) be the rank of ij. Then, by the linear ordering of the ordinals, either ran((D)) < ran((N)) or ran((N)) < ran((D)) or ran((D)) = ran((N)). In the first case, Dean fails to deny at least one sentence said by Nixon, namely (N) itself: for (N) does not satisfy the (Tran((D))-1)-schema, and so “(N) is not trueran((D))-1” does not entail that it is not the case that everything Dean says is not trueran((N))-1 (in fact, for every n [n: n t ran((D))], it is truen purely in virtue of the structural features of the Hierarchy, and so truen independently of whether everything Dean says is not trueran((N))-1). In the second case, it is Nixon who fails to deny at least one sentence said by

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Dean, for a reason completely symmetric to the one given in the last case. In the third case, they both fail to do so for reasons similar to those given for the two last cases. Thus, no matter which ranks their truth-like predicates pick out, it seems that not both Dean and Nixon can succeed in denying everything the other says, and so it seems that the Hierarchy cannot account for this generalising use of truth.14,15

4. Runners-Up I’d like first to stave off some natural proposals for “solving” the DeanNixon controversy (that is, for so understanding Dean’s utterance of (D) and Nixon’s utterance of (N) so that Dean and Nixon succeed in denying everything the other says). The first proposal is to allow for unrestricted quantification into subscript position: (UQSP) By uttering (D), Dean asserts “For every n, everything Nixon says is not truen” (ditto for Nixon).16

Although it might achieve the desired expressive purpose, (UQSP) is off mark, as unrestricted quantification into subscript position is simply not well-behaved in the Hierarchy. For suppose on the contrary that it is, and consider then the sentence Ȝ2 “For every n, Ȝ2 is not truen”. Suppose that Ȝ2 is truem. Then, ran(Ȝ2) d m, and so, by the (Tm)-schema, for every n, Ȝ2 is not truen, and hence, by universal instantiation, Ȝ2 is not truem. By reductio, Ȝ2 is not truem, and so, by universal generalisation, for every n, Ȝ2 is not truen. However, for every l [l: l t ran(Ȝ2)], by the (Tl)-schema Ȝ2 is truel, and so, by universal instantiation and contraposition, it is not the case that, for every n, Ȝ2 is not truen. Contradiction.17 Notice that this does not show that no form of restricted quantification into subscript position is well-behaved in the Hierarchy (in fact, many such forms are straightforwardly definable in the Hierarchy). Unfortunately, (UQSP) requires unrestricted quantification into subscript position, for otherwise not both Dean and Nixon can succeed in denying everything the other says. When full-blooded quantification is unavailable, a usually good idea is to resort to schemata. The second proposal is in effect to allow for unconditional schematic assertion: (USA) By uttering (D), Dean asserts every instance of “Everything Nixon says is not truen” (ditto for Nixon).

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In spite of its pertinent label, (USA) does not achieve the desired expressive purpose. For Dean may well assert only sentences whose rank is at least 1 (the only thing we know is that Dean asserts every instance of “Everything Nixon says is not truen”, and the lowest rank such an instance can have is 1, considering the “true0”-instance), in which case some sentence asserted by Nixon (namely, “Everything Dean says is not true0”) is correct, and so Dean should not after all be willing to deny everything Nixon says. Notice that the problem would simply represent itself one rank higher up if Nixon only asserted every instance of “Everything Dean says is not truen” except for “Everything Dean says is not true0”: for, by the symmetry between Dean and Nixon, Dean should do the same, but then Dean may well assert only sentences whose rank is at least 2 (the only thing we now know is that Dean asserts every instance of “Everything Nixon says is not truen” except for “Everything Nixon says is not true0”, and the lowest rank such an instance can have is 2, considering the “true1”instance), in which case some sentence asserted by Nixon (namely, “Everything Dean says is not true1”) is correct, and so Dean should not after all be willing to deny everything Nixon says. Notice also that the problem cannot be obviated by having Dean additionally assert some trivially correct sentence of rank 0, for it would then be Nixon that should not after all be willing to deny everything Dean says.18 If an assertion is problematic because it pronounces on unintended cases, a natural move is to hedge it by conditionalisation. The third proposal is in effect to allow for conditional schematic assertion: (CSA) By uttering (D), Dean asserts every instance of “For everything Nixon says, if it has rank n, it is not truen” (ditto for Nixon).

(CSA) avoids the specific problem, affecting (USA), that Nixon may correctly pronounce all sentences asserted by Dean to be not true0 (as they may all have at least rank 1), for some sentences asserted by Dean do not have rank 0 and Nixon’s conditional schematic assertion does not get to pronounce those sentences to be not true0. However, in accordance with its anachronistic label, it is quite obvious that (CSA) manages to avoid that problem only by opening up another possibility (of a different kind) for some of the sentences asserted by Nixon to be correct. For, by the properties of material implication, “If it has rank 0, it is not true0” correctly applies to every sentence asserted by Dean if every such sentence has rank > 0 (which it may well have). The argument can then proceed exactly as in the case of (USA).19,20 If a general conditional assertion results in a bunch of annoyingly vacuous instances, these can be avoided by opting instead for carefully

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chosen particular unconditional assertions. The fourth proposal is in effect to allow for this, and has two main components. Firstly, the proposal allows for schematic restriction: the asserted instances of a schema may be restricted to those having a certain property. It is uncontroversial that we usually employ schematic restriction: for example, “If n exists, so does its successor” is usually understood as so restricted that the referent of the instance of “n” is a natural number. Secondly, the proposal allows for schematic dependence: the asserted instances of a schema containing more than one schematic symbol may be restricted to those where the instances of the schematic symbols stand in a certain relation. There is evidence that sometimes schematic restriction comes together with schematic dependence: for example, it is arguable that “If x is bald and y has less hair than x, then y is bald” is usually understood as so restricted that the referent of the instance of “y” has roughly the same hair properties as the referent of the instance of “x”. Anyhow, the device of schematic dependence is fully intelligible and may well be introduced in the language to achieve the desired expressive purposes. The fourth proposal, then, is in effect to allow for restricted schematic assertion with schematic dependence: (RSASD) By uttering (D), Dean asserts every instance of “‘P’ is not truen” so restricted that: (RSASD1) The instance of “P” is asserted by Nixon; (RSASD2) The referent of the instance of “n” is the rank of the instance of “P” (ditto for Nixon).

Dean can thus finally achieve the desired expressive purpose. By uttering (D) as so understood, for every sentence asserted by Nixon of rank n Dean will assert that it is not truen and will assert nothing more; as wished, he will thereby only assert sentences [that are correct] iff Nixon only asserts sentences that are incorrect.21,22 Unfortunately, though it might achieve the desired expressive purpose, (RSASD) is problematic because of its crucial reliance on (RSASD1). Firstly, and less importantly, one might object to such restrictions as imposed by (RSASD1) that, in the light of general features of reference, they do not seem possible if the speaker does not know exactly which instances she is restricting to (as Dean might be stipulated to be, since Dean might be stipulated not to know exactly which sentences Nixon

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asserts). For example, if the speaker does not know the height of the person next door, it would not seem possible for her to assert every instance of “The height of the person next door is exactly r” so restricted that the referent of the instance of “r” is exactly the height of the person next door.23 Secondly, and more importantly, one should object to such restrictions as imposed by (RSASD1) that they make the use of a truth or truth-like predicate completely superfluous in our context, since, for example, Dean may then simply assert every instance of “It is not the case that P” so restricted that the instance of “P” is asserted by Nixon and thereby already achieve the desired expressive purpose (ditto for Nixon).

5. (SCSA) ’n’ Politics My own proposal tries to improve on the deficiencies of the previous proposals. Instead of having context do the unlikely work of picking out exactly which sentences a speaker asserts, my proposal appeals to context only for doing the work that it is supposed to do anyways in the contextualist Hierarchy, namely that of picking out appropriately high ranks. More precisely, my proposal is in effect to allow for selected conditional schematic assertion: (SCSA) Dean asserts every instance of “For everything Nixon says, if it has rank n, it is not truen” such that the referent of the instance of “n” is the rank of some sentence asserted by Nixon (ditto for Nixon).

Dean can thus achieve the desired expressive purpose. By uttering (D) as so understood, for every maximal non-empty set of sentences asserted by Nixon of rank n Dean will assert something to the effect that every member of it is not truen and will assert nothing more; as wished, he will thereby only assert sentences [that are correct] iff Nixon only asserts sentences that are incorrect. I’d like to present and discuss an important objection against (SCSA) (as well as against all the other schematic proposals we’ve been considering). According to the objection, one significant expressive limit of schemata is that they can only be used to make assertions, but cannot be embedded under logical operations. This would be detrimental to the generality of (SCSA), since examples analogous to the Dean-Nixon controversy can be devised in which the schema would need to be embedded, only to give two staple examples, under negation or as antecedent of a conditional. It is important to see that the objection is solid, in the sense that, in our context, it cannot be addressed by suggesting that schemata can be embedded under sentence-yielding

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logical operations. For, no matter how appealing such approach may be in other contexts, under plausible assumptions in our context such embedding would suffice to simulate the effects of unrestricted quantification into subscript position that we’ve seen in section 4 to be fatal for (UQSP). For example, consider the sentence Ȝ3 “It is not the case that it is not the case that Ȝ3 is not truen” (where the embedded schema is “Ȝ3 is not truen”): plausibly, the schema-embedding sentence “It is not the case that Ȝ3 is not truen” is correct iff some instance of “Ȝ3 is not truen” is not, and so Ȝ3 itself is correct iff every instance of “Ȝ3 is not truen” is—that is, iff, for every n, Ȝ3 is not truen. The argument proceeds then similarly to the case of (UQSP). Suppose that Ȝ3 is truem. Then, ran(Ȝ3) d m, and so, by the (Tm)schema and the plausible interpretation of Ȝ3 just sketched, it in effect follows that Ȝ3 is not truem. By reductio, Ȝ3 is not truem. But, plausibly, if one should assert every instance of a schema, one should also assert the schema-embedding double negation of the schema, and so one should assert Ȝ3. Then, for every l [l: l t ran(Ȝ3)], by the (Tl)-schema Ȝ3 is truel. Contradiction with the above conclusion that, for every m, Ȝ3 is not truem. Rather than trying to make schemata more similar to sentences, we should understand logical operations as having a wider domain than the set of sentences, and in particular as extending to (schematic) speech acts. A rough sketch of a natural way of implementing this plan would go as follows. There are two fundamental types of speech acts: assertion and denial. The atomic speech act is atomic (possibly selected) assertion, which is assertion of every (possibly selected) instance of a schema (and which is sound iff every (possibly selected) instance is correct). (Assertion of a sentence can be seen as a limit case thereof.) The molecular speech acts are molecular assertion, which is assertion of a set of speech acts (and which is sound iff every speech act in the set is sound), and molecular denial, which is denial of a set of speech acts (and which is sound iff some speech act in the set is not sound). Utterances of schema-embedding expressions are then interpreted by assigning, to an embedded schema, the relevant atomic assertion; to ™H (if directly or indirectly embedding a schema), the molecular denial of the set of speech acts assigned to H; to H0 H1 (if directly or indirectly embedding a schema), the molecular assertion of the union of the sets of speech acts assigned to H0 and to H1 (notice that, while molecular assertion and denial are defined over arbitrary sets of speech acts, on the pragmatics just sketched the set of speech acts assigned to a schema-embedding expression is always a singleton).24 To take an example from a far more sophisticated political arena, an utterance of “If, [for everything Berlusconi says, if it has rank n, it is truen], then Dudù is our only hope” (where the embedded schema is

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“For everything Berlusconi says, if it has rank n, it is truen”) is treated—as usual—as tantamount to an utterance of “It is not the case that, [[for everything Berlusconi says, if it has rank n, it is truen] and it is not the case that Dudù is our only hope]”, and the pragmatics just sketched yields that such utterance is a molecular denial of the molecular assertion of {the atomic selected assertion of “For everything Berlusconi says, if it has rank n, it is truen”, the atomic assertion of “It is not the case that Dudù is our only hope”}. The compositional clauses for soundness of speech acts just sketched yield then that the original utterance is sound iff either some selected instance of “For everything Berlusconi says, if it has rank n, it is truen” is incorrect or “It is not the case that Dudù is our only hope” is incorrect, the intuitively required result.25 Formally, this approach may well boil down to a near notational variant of embedding schemata under sentence-yielding logical operations, but, philosophically, it is in a completely different ballpark: an utterance of “If, [for everything Berlusconi says, if it has rank n, it is truen], then Dudù is our only hope” is not a simple assertion—straightforwardly evaluable in the narrow terms of truth—involving a single sentence whose content features a concept in effect equivalent with the concept of truth (a concept which is unavailable in the Hierarchy), but a complex speech act—only evaluable in the broader terms represented by the soundness clauses given in the last paragraph—involving every selected instance of “For everything Berlusconi says, if it has rank n, it is truen” (plus “Dudù is our only hope”) whose contents only feature the concepts of truth0, of truth1, of truth2 etc. Because of this, on this approach, the attempt at considering the sentence Ȝ4 “It is not the case that it is not the case that Ȝ4 is not truen” fails. For, on this approach, that string of symbols is a schema-embedding expression (where the embedded schema is “Ȝ4 is not truen”) rather than a sentence, and hence cannot be named as such or be sensibly attributed truthn: by uttering such expression, one performs a complex speech act involving as only sentences the instances of “Ȝ4 is not truen”—there just is no appropriate sentence for “Ȝ4” to refer to. And, if one insists that “Ȝ4” is then to refer to the speech act itself, or to the schema-embedding expression itself, since these are not even truthn bearers every instance of “Ȝ4 is not truen” is correct, and so the molecular denial performed by uttering “It is not the case that it is not the case that Ȝ4 is not truen” is sound (even if of course not truen), and that’s the end of it.26

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6. From Politics to Semantics Having thus defended my proposal for solving the particular DeanNixon controversy, I wish to leave for another occasion a more rigorous and systematic development of the idea behind it that the Hierarchy can “scheme out” of many of its alleged expressive limits (for example, see note 25 for a direction for future research). I wish to close instead by developing what I think is a less tractable worry concerning the Hierarchy’s ability to account for generalising uses of truth. To locate this less tractable worry in our conceptual territory, I start with conjecturing that, not only does (SCSA) work for the Dean-Nixon controversy, but also variations thereof will indeed allow the Hierarchy to account for all those generalising uses— often, but not only, found in ordinary discourse—in which, roughly, one uses predications of “true” as mere means to arrive at certain other propositions typically about the non-semantic world (what we may call “non-semantic generalising uses”, a paradigmatic example of which is a typical assertion of “Every axiom of PA is true”, which means to say something about natural numbers rather than about a certain axiom system). But some generalising uses— often, but not only, found in theoretical discourse—do not fit into this pattern, as they are uses in which, roughly, one uses predications of “true” as ends in themselves that attribute a semantic (i.e. world-relating) property to certain sentences (what we may call “semantic generalising uses”, a paradigmatic example of which is a typical assertion of “Everything in today’s edition of La Gazzetta dello Sport is true”, which means to say something about the reliability of the newspaper rather than about Italian football). I’d like to take as running example of semantic generalising use of truth a typical assertion of: (ALEM) All instances of: (SLEM) P or it is not the case that P are true.

Notice that, as every other suitable sentence, (ALEM) too also admits of a non-semantic generalising use—a use whose point is to say that snow is white or it is not the case that snow is white, and grass is green or it is not the case that grass is green, and the sky is blue or it is not the case that the sky is blue…—and that that can easily be accounted by the Hierarchy, in this case with the selected conditional schematic assertion of “For every instance of (SLEM), if it has rank n, it is truen” (where the referent of the

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instance of “n” is the rank of some sentence that is an instance of (SLEM), of course a selection that is in this case vacuous). What now interests us is however a semantic generalising use of (ALEM)—a use whose point is to say that the law of excluded middle (LEM) has a certain nice property (I leave it intentionally open whether LEM itself is to be identified with (ALEM), (SLEM) or something else). There is no question that, for every instance L of (SLEM), according to the Hierarchy L has some “semantically nice” property. For “L is truen” [n: n t ran(L)] (and so “‘L is truen’ is truem” [m: m > n]) is something semantically nice that the Hierarchy can say about L (as per section 2 and the last paragraph). However, the battery of truth-like properties that the Hierarchy must envisage makes it also the case that “L is not truen” [n: n < ran(L)] (and so “‘L is not truen’ is truem” [m: m > n]) is on the contrary something semantically non-nice that the Hierarchy must say about L (as per section 2). So, what’s the overall status of L in the Hierarchy? A brute answer would be to say that, in the Hierarchy, the overall status of L is positive because the relevant semantically nice properties that the Hierarchy attributes to it are infinitely many while the relevant semantically non-nice properties that the Hierarchy attributes to it are finitely many. However, such a brutely quantitative observation is in itself totally unilluminating: here as elsewhere, size in itself doesn’t matter. A slightly more sophisticated answer would be to say that, in the Hierarchy, the overall status of L is positive because the truth-like predicates that allow one to attribute semantically non-nice properties to it are defective in that each of them, for some n, applies correctly only to sentences whose rank is at most n. However, this observation shoots itself in the foot, as, in the Hierarchy, absolutely every truth-like predicate—including those that allow one to attribute semantically nice properties to L—is defective in the sense just specified. A much more sophisticated answer would be to say that, in the Hierarchy, the overall status of L is positive because the truthlike predicates that allow one to attribute semantically non-nice properties to it are not those that are really sensitive to whether L is correct. However, the observation is smoke and mirrors. In the Hierarchy, “correct” can only be identified either with “true0” or with “true1” or with “true2” etc. (or with finite Boolean combinations thereof). Suppose now that L is “‘Snow is white’ is true0 or it is not the case that ‘Snow is white’ is true0”. Then, taking “correct” to be “true1”, it is indeed the case that “true0” is “not really sensitive to whether L is correct”. But that is so simply in the boring sense that L is true1 but is not true0. For all that has been said by this observation, one can equally legitimately take “correct” to be “true0”, and

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then it is “true1” that is “not really sensitive to whether L is correct” (in the same boring sense that L is true1 but is not true0).27 Therefore, while it is the case that, for every instance of (SLEM), according to the Hierarchy there is some truth-like predicate that correctly applies to it, it is extremely unclear how significant that is in the overall semantic picture offered by the Hierarchy (to go back to an issue emerged in section 2, it is not just that it is a staggering abuse of the concepts available in the Hierarchy to take it that the Hierarchy thinks that L is “true”, it is even extremely unclear whether there is any good excuse for committing the abuse in the first place).28,29 It is now time to observe that, in any event, that does not even imply the Barcanian consequence that, according to the Hierarchy, for every instance of (SLEM) there is some truth-like predicate that correctly applies to it. For, again, in the Hierarchy “correct” can only be identified either with “true0” or with “true1” or with “true2” etc. (or with finite Boolean combinations thereof). But it’s easy to see that no instance of “For every instance of (SLEM), there is some truthlike predicate that trulyn applies to it” is correct. The Hierarchy’s thoughts are so blind as to lead to such a stunning failure of the relevant instance of the Barcan formula, and so to a failure of the Hierarchy of accepting an apparent consequence of what it thinks. Although the issue is admittedly somewhat vague, I think that we can safely draw from these and similar supporting considerations also the more general conclusion that it is not even the case that, according to the Hierarchy, every instance of (SLEM) has a semantically nice property, and so, a fortiori, the conclusion that it is not the case that, according to the Hierarchy, there is a semantically nice property had by all instances of (SLEM) (where the particular quantifier is now swapped with the universal one).30 Therefore, there is no semantically nice property which, substituted for truth, the Hierarchy can deploy to vindicate a semantic generalising use of (ALEM). It is tempting to reply that, although the Hierarchy violates the letter of a semantic generalising use of (ALEM), its spirit is respected by the Hierarchy in virtue of the already observed fact that, for every instance L of (SLEM), according to the Hierarchy L has some semantically nice property. But, on reflection, once that fact has been stripped of its natural implications (as has been done in the last paragraph), it’s hard to see that what’s left has much to do with even the spirit of (ALEM). For what’s basically left is, when all is said and done, the bunch of sundry assertions of every instance of “‘P or it is not the case that P’ is truen” (where the referent of the instance of “n” is the rank of the instance of “P”), and what, in themselves, such assertions achieve is merely to attribute to different instances of (SLEM) different semantically nice properties, which (as has

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been noted in the last paragraph) in this case does not even imply an assertion to the effect that every instance of (SLEM) has some semantically nice property. However, whatever LEM exactly is, clearly to say something semantically nice about it, rather than (specifically or nonspecifically) about some proper subset of the instances of (SLEM), requires asserting that there is some semantically nice property had by all instances of (SLEM), or at least asserting that every instance of (SLEM) has some semantically nice property—it is clearly not enough, for every instance of (SLEM), to attribute a semantically nice property to it. Don’t say that, for example, attributing the property of being true0 to one instance of (SLEM) and attributing the property of being true1 to another instance of (SLEM) naturally implies attributing the property of either being true0 or being true1 to both. For, while that may be alright as far as it goes, its required generalisation covering all instances of (SLEM) would involve the property of either being true0 or being true1 or being true2…, and that’s in a different guise just the infamous property of, for some n, being truen which has been discussed in connection with (UQSP) in section 4. Therefore, in themselves, the bunch of sundry assertions of every instance of “‘P or it is not the case that P’ is truen” do not achieve to say anything semantically nice about LEM.31 If such instances are then the semantically nicest thing that the Hierarchy can offer about all the instances of (SLEM) (and they are), the Hierarchy certainly goes against not only the letter, but also the spirit of (ALEM), and, more generally, against the minimal requirement on a semantic theory to say something semantically nice about LEM. Thus, by being unable to account for the relevant semantic generalising use, the Hierarchy does not really say anything about LEM32—in a sense, as far as the Hierarchy is concerned it is as though as LEM did not exist.33 The point obviously generalises to many other general principles: by being unable to account for the relevant semantic generalising uses of truth, the Hierarchy does not really say anything about them—as far as the Hierarchy is concerned, it is as though as such principles did not exist. Many general principles can only be detected by hierarchy-transcending semantic generalising uses. And the point is even amplified in the case of general principles whose very formulation involves a semantic generalising use (which may not be the case for LEM). Consider, for example, the general principle about negation: (NEG) The negation of a sentence is true iff the sentence is not true.

Again, what’s basically left of (NEG) in the Hierarchy is, when all is said and done, the bunch of sundry assertions of every instance of “‘It is

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not the case that P’ is truen iff ‘P’ is not truen” (where the referent of the instance of “n” is the rank of the instance of “P”). However, (NEG) is a principle about negation, and clearly to say something semantically interesting about negation, rather than (specifically or non-specifically) about some proper subset of the instances of “It is not the case that P”, requires asserting that there is some semantically interesting property had by all instances of “It is not the case that P”, or at least asserting that every instance of “It is not the case that P” has some semantically interesting property—it is clearly not enough, for every instance of “It is not the case that P”, to attribute a semantically interesting property to it. Therefore, in themselves, the bunch of sundry assertions of every instance of “‘It is not the case that P’ is truen iff ‘P’ is not truen” do not achieve to say anything semantically interesting about negation. If such instances are then the semantically most interesting thing that the Hierarchy can offer about all the instances of “It is not the case that P” (and they are), the Hierarchy certainly goes against not only the letter, but also the spirit of (NEG),34 and, more generally, against the minimal requirement on a semantic theory to say something semantically interesting about negation. Thus, by being unable to account for the relevant semantic generalising use of truth, the Hierarchy does not really say anything about negation—in a sense, as far as the Hierarchy is concerned it is as though as negation did not exist.35 Many general principles and notions can only be detected by hierarchy-transcending semantic generalising uses of truth. By appealing to selected conditional schematic assertion and its scheming likes, the Hierarchy may be able to account for the kind of generality required by non-semantic generalising uses, and so able to account for political controversies; but the Hierarchy is in any case unable to account for the kind of generality required by semantic generalising uses, and so unable to account for LEM or negation. I like to think that, in a good sense, for a semantic theory to preserve classical logic involves, for example, saying something semantically nice about LEM. In that sense, the Hierarchy does not after all preserve classical logic.

References Burge, T. 1979, “Semantical Paradox”, The Journal of Philosophy 76 (4), 169 —. 1982, “The Liar Paradox: Tangles and Chains”, Philosophical Studies 41 (3), 353

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Goldstein, L. 1992, “‘This Statement Is Not True’ Is Not True”, Analysis 52 (1), 1 Kripke, S. 1975, “Outline of a Theory of Truth”, The Journal of Philosophy 72 (19), 690 Makinson, D. 1965, “The Paradox of the Preface”, Analysis 25 (6), 205 Restall, G. 2005, “Multiple Conclusions”, in Hájek, P., L. Valdés Villanueva, and D. Westerståhl (eds.), Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, London: College Publications, 189 Sainsbury, M. 1995, Paradoxes, 2nd edition, Cambridge: Cambridge University Press Skyrms, B. 1984, “Intensional Aspects of Semantic Self-Reference”, in Martin, R. (ed.), Recent Essays on Truth and the Liar Paradox, Oxford: Clarendon Press, 119 Sorensen, R. 2001, Vagueness and Contradiction, Oxford: Oxford University Press Strawson, P. 1952, Introduction to Logical Theory, London: Methuen Tarski, A. 1933, PojĊcie prawdy w jĊzykach nauk dedukcyjnych, Warsaw: Nakáadem Towarzystwa Naukowego Warszawskiego Zardini, E. 2008, “Truth and What Is Said”, Philosophical Perspectives 22 (1), 545 —. 2011, “Truth without Contra(di)ction”, The Review of Symbolic Logic 4 (4), 498 —. 2012, “Truth Preservation in Context and in Its Place”, in Caret, C. and O. Hjortland (eds.), Insolubles and Consequences, London: College Publications, 249 —. 2014a, “The Opacity of Truth”, Topoi, forthcoming —. 2014b, “The Role of Utterances in Bradwardine’s Theory of Truth”, Recherches de théologie et philosophie médiévales, forthcoming

Notes * I’ve drastically changed my mind a couple of times on the topics of this paper, and as a consequence I’ve been working on it for a long while (or is it vice versa?). Earlier versions of the material in the paper have been presented in 2006 at the 5th Conference of the Spanish Society of Logic, Methodology and Philosophy of Science (University of Granada); in 2007, at the Arché Reading Party in Raasay (University of St Andrews); in 2011, in a course on Formal Theories of Truth at UNAM; in 2012, in a lecture at the University of Barcelona; in 2013, in a course on Philosophy of Logic (co-taught with Sven Rosenkranz) at the University of Barcelona, in a lecture at the Institute of Catalan Studies and in a course on Semantic Paradoxes at the University of the Azores; in 2014, at the LOGOS

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Philosophy of Logic Seminar (University of Barcelona) and in a course on Paradoxes at the University of Maltepe. I’d like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Philipp Blum, Zekiye Kutlusoy, Josep Macià, Nasim Mahoozi, José Martínez, Sergi Oms, Carlos Romero Castillo, Sven Rosenkranz, Rui Sampaio da Silva, Chiara Tabet, Pilar Terrés, Kurt Wischin and Crispin Wright. I’m also grateful to the editors Fabio Bacchini, Stefano Caputo and Massimo Dell’Utri for inviting me to contribute to this volume and for their support and patience throughout the process, as well as to Silvia Negroni for copy-editing the paper. At different stages during the writing of the paper, I’ve benefitted from an AHRC Doctoral Research Fellowship, from a UNAM Postdoctoral Research Fellowship and from the FP7 Marie Curie IntraEuropean Research Fellowship 301493 on A Non-Contractive Theory of Naive Semantic Properties: Logical Developments and Metaphysical Foundations (NTNSP), as well as from partial funds from the project CONSOLIDERINGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP), from the FP7 Marie Curie Initial Training Network 238128 on Perspectival Thoughts and Facts (PETAF), from the project FFI2011-25626 of the Spanish Ministry of Science and Innovation on Reference, Self-Reference and Empirical Data and from the project FFI2012-35026 of the Spanish Ministry of Economy and Competition on The Makings of Truth: Nature, Extent, and Applications of Truthmaking. 1 Throughout, I use bold-faced symbols for schematic expressions. I’ll assume that a schema is uttered, but the sentences that are thereby said, asserted, denied etc. (but not themselves uttered) are the schema’s relevant instances (I’ll introduce new objects of assertion and denial in section 5). Even so, throughout I consider the collective sayings achieved by utterances of schemata as speech acts in their own right (and, for simplicity’s sake, I sometimes conflate these and similar illocutory acts with the locutory acts that utterances strictly speaking are). 2 In the following, analogous indiscernibility-of-identicals steps will be left implicit (see Skyrms 1984 for a problematisation thereof). 3 Some solutions to the semantic paradoxes include as crucial component a shift from sentences to utterances as the operative truth bearers (see e.g. Goldstein 1992), and they are sometimes presented as having the virtue of preserving both classical logic and naive truth (see e.g. Sorensen 2001, pp. 181–182). Let’s set aside the grave problem for such utterance-based solutions consisting in the fact that they are subject to revenge paradoxes of utterance truth (see Zardini 2008, pp. 561–566). And let’s also set aside the other grave problem for utterance-based solutions consisting in the fact that, in the case of the semantic paradoxes, we clearly have notions of truth (maybe not the fundamental or central ones) applying to coarse-grained entities like sentences and propositions, and that one can develop semantic paradoxes employing those notions which offer considerable resistance to the strategies pursued by utterance-based solutions (see Zardini 2014b). What should be stressed here instead is that, for some meaningful and noncontext-dependent sentence “P” (like e.g. “The Liar utterance is not true”), standard utterance-based solutions accept that P but also accept of a certain

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utterance of “P” that it is not true. Willy-nilly, that is a violation of the natural analogue of (T) for utterance truth, and so utterance-based solutions do not after all preserve naive (utterance) truth. (This having been noted, throughout I assume that the operative truth bearers are sentences.) 4 I myself have chosen to go the second way (see e.g. Zardini 2011), although I am no friend of extreme understandings of naive truth (see Zardini 2008, pp. 550–561; 2012; 2014a). 5 Stylistically, it may be interesting to observe that such charges typically come rife with grisly anatomical metaphors (for example, “mutilate” and its relatives are another favourite). I’m not sure it’s particularly illuminating to think of reasoning or logic as having, say, feet, and to think of revisions of classical logic as damaging or chopping off such feet, and I suspect that the abundance of rhetoric stands in for the scarcity of something else. 6 Reason for the impossibility: to establish the complex quantificational claim in the text, one will arguably need to appeal in effect to the general principle that, if M contradicts \ and \ attributes truth to F, M is true only if F is not. But, considering a sentence like Ȝ0 (whose contradictory is “Ȝ0 is true”), it’s easy to see that that principle together with the further general principle that what is provable is true are inconsistent in classical logic. It is interesting to observe that, while, under plausible assumptions, these principles are entailed by naive truth, even together they fall very much short of entailing anything like naive truth (see Zardini 2014a for further details and discussion of these issues). 7 The construction can be extended into the transfinite, but, for our purposes, that would only bring unnecessary complications (see also note 17). 8 I’ll often help myself to something like the concept of truth (typically expressing it with “correct” and its relatives to avoid confusion with truth-like predicates), although this is strictly speaking unavailable in the hierarchical theory. I do this for ease of expression; whether or not the same point can always be made using a construction available in the theory is however moot, and in fact I’ll argue in section 6 that sometimes this is not the case. 9 I’ll often help myself to unrestricted quantification into subscript position, although this is strictly speaking unavailable in the Hierarchy (see the discussion of (UQSP) in section 4). I do this for ease of expression; whether or not the same point can always be made using a construction available in the Hierarchy is however moot, and in fact I’ll argue in section 6 that sometimes this is not the case. 10 Thanks to Pilar Terrés for pushing me on the issues discussed in this paragraph. 11 The distinction between non-generalising and generalising uses of truth is really meant to be the distinction between, very roughly, uses only involving attributions of truth such that all the relevant sentences can be assumed to have, for some rank n, rank d n, and uses involving attributions of truth for which this is not the case. So understood, the non-generalising/generalising distinction is orthogonal to the referential/quantificational distinction (as examples of the not totally obvious combinations, the non-generalising “Every sentence uttered by John Sobieski on 12/09/1683 is true” is quantificational, and the generalising “These are true”— pointing at the infinite series of sentences “Snow is white”, “‘Snow is white’ is

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true0”, “‘Snow is white’ is true1”…—is referential). Thanks to Sergi Oms for urging this clarification. 12 Burge (1979, p. 193) tries to make the postulation of the contextualist Hierarchy plausible by pointing to the example of a road sign saying “(You) slow down”. Setting aside whether one is really doing a favour to the contextualist Hierarchy by resorting to such atypical and puzzling uses of language in explaining typical and non-puzzling blind non-generalising uses of truth, the example would seem anyways to be scarcely relevant, as it is an example in which, on Burge’s preferred (and controversial) interpretation, it is an element of the context (i.e. the audience) that supplies the materials for interpreting “you”. But, in the case of an utterance of “The first sentence uttered by a Pole in 1933 is true”, there is no element of the context that supplies the materials for interpreting “true”: for what supplies such materials is the first sentence uttered by a Pole in 1933, but not everything that is quantified over in a context is an element of that context (assuming standard understandings of definite descriptions and of contexts). More relevantly, there is at least one kind of case in which it would seem that “tall” is guaranteed to apply correctly to whichever person in a context the speaker tries to denote by uttering “The tall person next door is Polish”: namely, the case of contrastive use in which the speaker, knowing that there are exactly two people of substantially different heights next door, intends to denote the taller one. However, the postulation of the contextualist Hierarchy obviously needs to concern many cases that are not cases of contrastive use. Indeed, in the case of a contrastive use in which the speaker, knowing that there are exactly two correct sentences of substantially different ranks uttered next door, tries to denote the one of higher rank by uttering “the true sentence uttered next door”, she clearly fails to do so. Thanks to Carlos Romero Castillo for conversations on some of these issues. 13 Throughout, by “deny” and its relatives, I generally mean something along the lines of “rule out”, so that a denial of ij is something along the lines of ruling out ij. Fortunately, in a classical framework, that can be taken to be equivalent with asserting “It is not the case that ij” (the connection between denial and assertion of the negation becomes looser in many non-classical frameworks, see Restall 2005). 14 In fact, the same point could have been made using “true” rather than “not true” in the Dean-Nixon “controversy” (I’m not quite sure why Kripke chose to use “not true”, but I’ll follow him in this choice to show that it does not create additional problems). 15 Burge (1979, p. 194; 1982, pp. 360–361) does consider the example but oddly seems to rest content with showing how, given various ways of filling in the details of the Dean-Nixon controversy, his versions of the Hierarchy “deftly” provide allegedly “intuitively sound” evaluations in terms of their truth-like predicates. He thereby seems to overlook completely the crucial point of Kripke’s example that I’ve insisted on in the text (and that Kripke 1975, p. 696 himself takes pains to emphasise), which rather concerns the apparent fact that, by the Hierarchy’s lights, not both Dean and Nixon can succeed in denying everything the other says. Curiously, Burge (1979, pp. 191–192) does mention the possibility of interpreting

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certain utterances of certain sentences containing “true” as performing schematic speech acts, but does not apply this insight to the case at hand. 16 Throughout, I only consider proposals for solving the Dean-Nixon controversy that treat Dean and Nixon symmetrically. 17 The contradiction could be avoided by introducing new ranks beyond the finite ordinals, and postulating that ran(Ȝ2) is one such. However, to achieve the desired expressive purpose, (UQSP) should then understand the relevant quantification into subscript position to encompass such ranks, and an analogous argument would show that that quantification is simply not well-behaved in the Hierarchy. 18 Thanks to Philipp Blum for suggesting to consider this last move. 19 A variation on (CSA) would rephrase it in terms of a binary universal quantifier taking the pairs of the properties expressed by coordinated instances of “It has rank n” and “It is not truen” respectively (rather than, as in (CSA), a unary universal quantifier taking the properties expressed by instances of “If it has rank n, it is not truen”). The argument would then appeal to the usual feature of the binary universal quantifier of always yielding a correct sentence when taking as first property one that is not had by anything (if such feature is denied, it still is at least very unclear that Dean should be willing to assert every instance of the resulting schema—certainly, he should not be so willing if, as is sometimes suggested (see e.g. Strawson 1952, pp. 173–179), the binary universal quantifier always yields an incorrect sentence when taking as first property one that is not had by anything). 20 It is the possibility of entangling a blind non-generalising use of truth in, for example, a Dean-Nixon-like controversy that convinces me that such use cannot be understood in terms of an unrestricted or of a conditional schematic assertion doing away with the postulation of the contextualist Hierarchy discussed in section 2 (for such understanding would be affected by problems analogous to those affecting (USA) and (CSA)). Indeed, my own schematic proposal in section 5 for solving the Dean-Nixon controversy will crucially appeal to that postulation. Thanks to José Martínez for raising this issue. 21 Throughout, I use square brackets (among other things) to disambiguate constituent structure. 22 It’s easy to see that, contrary to the previous proposals, (RSASD) has the plausible consequence that neither Dean nor Nixon assert in fact anything if the only utterances made by them are those of (D) and (N) respectively. An analogous comment applies to (SCSA) in section 5. 23 One might grant the intuition that it is not possible to impose such restriction in the example just considered in the text, but still think that the example is relevantly different from the Dean-Nixon controversy, since the restriction that (RSASD1) imposes in that case does not refer to the semantic properties of the target instances (such as those referred to by “The referent of the instance of ‘r’ is exactly the height of the person next door”), but only to non-semantic ones (such as pragmatic ones as those referred to by “The instance of ‘P’ is asserted by Nixon”). However, the relevance of such semantic/non-semantic distinction would seem to be undermined by the very examples I’ve used to support the ideas of schematic restriction and schematic dependence. Those examples, as well as many other

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considerations, would rather seem to point to the relevance of some sort of particular/general distinction (with respect to which (RSASD1) would seem to fall on the wrong side). 24 A variation on this idea would also envisage as atomic speech act atomic (possibly selected) denial, which is denial of some (possibly selected) instance of a schema (and which is sound iff some (possibly selected) instance is incorrect). Utterances of schema-embedding expressions would then be interpreted by assigning, to an embedded schema or its negation, the relevant atomic assertion or atomic denial (it may then be natural to modify along similar lines also the rest of the pragmatics just sketched in the text). The differences, if any, between these two approaches do not matter much in our context (but see note 26). 25 This indicates how, on this approach, it is also possible to assert or deny “what is said” by utterances of schema-embedding expressions (scare quotes being used since such utterances are not simple assertions). For the compositional clauses for soundness of speech acts provide in effect a definition of what it is, for every such well-behaved expression H, to be an “H-good” set of sentences—that is, a set of sentences whose correctness makes an utterance of H sound. One can then assert “what is said” by an utterance of H by uttering “For some H-good set X, for every M in X, M is truen” (where the embedded schema is “M is truen”). (This requires a natural extension of the approach to cover quantification while consequently allowing for the selection of the instances of a schema to depend on the value of the variable bound by the schema-embedding quantifier; interestingly, such extension would also lead to a simplification of (SCSA).) And, given that such schema-embedding expression is fully further embeddable, one can also use it to deny, suppose, disjoin etc. “what is said” by an utterance of H. Indeed, although the details lie beyond the scope of this paper, I expect that, under plausible assumptions (including natural clauses dealing with referentially looping schemaembedding expressions), the approach can be so developed as to offer a systematic account of the kind of generalising use of truth focussed on in sections 3–5 in terms of the schema-embedding expression just introduced (an account that would basically reduce to (SCSA) in the case of the Dean-Nixon controversy). Thanks to José Martínez for pressing me on this. 26 It might be felt that my reply to the objection requires too sharp a distinction between truth-like properties of sentences (and of assertions thereof) and soundness of speech acts. Haven’t I myself just ended up giving—with my compositional clauses for soundness of speech acts—a perfectly natural explanation of truth of speech acts? I think that such explanation would actually be very far from natural. For, arguably, a molecular assertion of a set reduces to the conjunctive fact that one performs all the members of the set (with an atomic assertion of a schema reducing to the conjunctive fact that one asserts every instance of the schema), and a molecular denial of a set reduces to the conditional fact that, if one performs all the members of the set but one assertion (denial), one performs the opposite denial (assertion). (This requires the notion of atomic denial as per note 24, with an atomic denial of a schema reducing to the conditional fact

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that, if one asserts every instance of the schema but one instance, one denies that instance.) It is very far from natural to attribute truth or truth-like properties to such facts. 27 It might be protested that what is meant by saying that “true0” is “not really sensitive to whether L is correct” is simply [that [“Snow is white” is true0 or it is not the case that “Snow is white” is true0] but L is not true0]. But then, since it is completely trivial (at least from section 2) that [“Snow is white” is true0 or it is not the case that “Snow is white” is true0], and since anyways that in itself involves no mention of L (let alone of its correctness), what is meant is just, in effect, that L is not true0—hardly something that tells against the significance of the fact that L is not true0! 28 In this dialectic, I freely switch, according to what is expositorily more convenient, between talk of properties and their being had, on the one hand, and talk of predicates and their correctly applying, on the other hand. 29 This point affects also the ability of the Hierarchy to account for semantic (specifying or blind) non-generalising uses of truth (notice that, in my discussion of non-generalising uses in section 2, I was implicitly focussing on non-semantic ones). 30 For example, a similar supporting consideration is to the effect that, although, according to the Hierarchy, the semantic property of being, say, [true79 if of rank 79] is had by all instances of (SLEM), that is not a semantically nice property, as it is had also, for example, by the semantically totally ugly “‘Snow is white’ is true79 and it is not the case that ‘Snow is white’ is true79” (I insist that the latter is semantically totally ugly: the fact that is not of rank 79 does not make it semantically any better than “Snow is white and it is not the case that snow is white”). 31 This situation is usefully compared with the one envisaged by a common position on the preface paradox (see Makinson 1965 for the introduction of the paradox), according to which, roughly, although the author of a book believes of each statement in the book that it is correct, she does not believe that the book as a whole is correct. But, while that common position is attractive, the one taken by the Hierarchy on LEM is much less so, since LEM, contrary to typical empirical theories, can be assumed to be certain. Notice that the author might still believe that the book as a whole contains some, or even a lot of, correct statements. But the analogous beliefs in our context (for example, the belief that (SLEM) has some true79 instances) are semantically nice beliefs (non-specifically) about some proper subset of the instances of (SLEM) rather than about LEM (by the standards of our context, it is not enough for a principle to have a semantically nice property that some, or even a lot, of its instances have some such property, for in that sense affirming the consequent, or enumerative induction, would have just as semantically nice a property as LEM). 32 “Really” because, as per note 30, the Hierarchy does say, for example, that the semantic property of being, say, [true79 if of rank 79] is had by all instances of (SLEM); however, for the reason pointed out in that note, that is not something

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semantically nice (nor something semantically non-nice), and so not really something relevant to say. 33 One might picture the present dialectic thus. EZ: “Hierarchy, what do you think about LEM?”; Hierarchy (taking a big breath): “‘Snow is white or it is not the case that snow is white’ is true0; ‘‘Snow is white’ is true0 or it is not the case that ‘Snow is white’ is true0’ is true1; ‘‘Snow is white’ is true1 or it is not the case that ‘Snow is white’ is true1’ is true2—”; EZ: “Stop, stop—I meant about LEM itself”; Hierarchy (taking time): “Well, in that case, I would say that every instance of (SLEM) is, hum,”—silence. 34 Ironically, Tarski (1933, p. 197) famously took to task certain theories of truth for failing to prove a close kin of (NEG). Tu quoque Alfrede! 35 Points similar to those made in notes 30–33 for LEM apply for negation.

ABSTRACTS

Annalisa Coliva and Sebastiano Moruzzi, Faultless Disagreement and the Exual Validity Paradox The putative phenomenon of faultless disagreement gives rise to the Equal validity Paradox. The Equal Validity Paradox is an argument that generates a contradiction by assuming that the set of utterances that give rise to the appearance of faultless disagreement can be non-empty, and by constraining the properties of these utterances by means of five principles specifically related to faultless disagreement and equal validity together with two general principles about the truth properties of utterances and propositions. The paradox allows for eight solutions. Some of these solutions are well known (realism, contextualism and relativism), whereas others are less explored (if at all) in the literature on faultless disagreement (analetheism, dialetheism, semantic indeterminism and non-cognitivism). We argue that each of these solutions is, ultimately, a revisionist approach to faultless disagreement: instead of accounting for faultless disagreement, the paradox is a symptom of glitches in our conception of faultless disagreement.

Ciro De Florio, Deflation and Reflection. On Tennant’s Criticism of the Conservativeness Argument In the debate concerning deflationism the question originated from the so-called conservativeness argument (see Ketland 1999 and Shapiro 1998, 2002) is rather relevant. According to the conservativeness argument, the deflationists are forced to face the following dilemma: a deflationary theory of truth should fulfil two requirements: i) the conservation constraint and ii) the reflection constraint. But if it fulfils (i) it cannot meet (ii) as well, and, vice-versa, if it meets (ii) it cannot fulfil (i). In the following, we will examine the debate on the conservativeness argument between Ketland (1999) and Shapiro (1998, 2002), on one side, and Tennant (2002, 2005) on the other. We shall analyse its philosophical relevance for the general conception of truth. In order to do that, we will informally present the conservativeness argument (§1); then, we will

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consider Tennant’s objection (§2) and Ketland’s reply (§3) and delve into whether and to what extent is Tennant’s proposal justified (§§4-5). Finally, we will briefly highlight a shortcoming of Tennant’s strategy.

Pascal Engel, The Norm of Truth: a Dialogue Pilate's question might have been different from the famous “What is truth?”. Perhaps he was actually interested in asking Jesus what the value of truth was, and whether truth is the norm of belief. In a surprising anticipation of contemporary debates Pilate and his Epicurean interlocutor discuss these issues, some 50 years after J.C., and ask whether truth, justification or knowledge is the main norm of belief. They discuss the form that the norm has to take in order to guide our beliefs, and the nature of the normativity which is involved in our epistemic attitudes. They conclude, to their satisfaction, that the true norm of belief is knowledge.

Filippo Ferrari and Dan Zeman, Radical Relativism, Retraction and “Being at Fault” Radical relativists have claimed that accounting for retraction is an advantage their view has over rival positions. Despite the dialectical importance of the phenomenon, little attention has been paid to it in the literature. Our aim in this paper is to fill this gap by offering some key elements for an analysis of retraction and its normative profile. We also inquire into the sense of “fault” in which retractors can be said not to be at fault in retracting their previous assertions and, in this connection, we highlight an asymmetry between retractions in the moral and the taste domain. We then provide an explanation of this asymmetry by appealing to a less discussed dimension of assertion evaluation which we call “circumstance-accuracy”. We support the idea that such a dimension is the right explanation of the asymmetry by drawing upon a distinction found in jurisprudence between weak and strong retroactivity that also further contributes to the analysis of the normative profile of retraction.

Carlo Filotico, Weak Indexical Relativism According to indexical relativism, speakers’ assertions within some areas of discourse—namely aesthetics, matters of taste, perhaps ethics and other fields—implicitly refer to a feature of the speaker, so their content depends on the context of utterance. For instance, utterances of «Matisse is better than Picasso» must be read as assertions of «Matisse is better than

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Picasso according to my own aesthetic standards» or «I prefer Matisse to Picasso». According to this picture, two assertions of a sentence containing no indexical terms can get different truth values just because they can express different contents in different contexts. Opponents of indexical relativism point out that this view seems to misidentify the content of the assertions in many cases, namely those in which evidence suggests that speakers are genuinely disagreeing about the same content. My own work is an attempt to formulate and defend a weakened version of indexical relativism, allowing both the indexical and the literal reading of assertions such as «Matisse is better than Picasso». I try to show that my proposal can avoid the standard objections, then I formulate some straightforward objections to my own view and I try to reply to them.

Samuele Iaquinto, Contradictions, Disagreement and Normative Error My aim is to discuss some counterexamples to the following principle: (P) Necessarily, for every proposition p, for every cognitive agent S and for every cognitive agent S*, if S believes that p and S* believes that ¬p, then either S makes a normative error or S* makes a normative error.

In trying to offer counterexamples to (P), I will compare two different approaches: a three-valued approach and a relativist approach. I will argue that adopting the latter is preferable, since, contrary to the former, in offering counterexamples to (P) it does not commit us to hold controversial metaphysical views. Furthermore, it allows us to propose genuine counterexamples not only in cases of syntactic disagreement, but also in cases of semantic and ontological disagreement.

Michele Lubrano, Alethic Pluralism and Logical Paradoxes In this contribution I will examine Cotnoir’s (2013) solution to the problems that alethic pluralism faces when it comes to logical paradoxes. I will argue that his proposal fails to be a viable option and I will put forward an alternative approach, more Tarskian in spirit, but with the same “pluralist” trait of Cotnoir’s solution. Such an alternative approach is based on the idea that each truth predicate can be associated with an index that fully describes its relations with other truth predicates. I will show how such indexes must be constrained in order to avoid logical paradoxes.

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Abstracts

Boris Rähme, An Explanatory Role for the Concept of Truth Advocates of different deflationary accounts of truth agree that the concept of truth has no explanatory role to play in philosophy. They do not deny that truth talk is sometimes useful for the purposes of formulating and expressing explanations; but, they insist, such talk does not and cannot contribute any genuinely explanatory content to the explanations which we formulate with its help. The article offers and discusses a counterexample to this no-explanatory-role claim.

Fredrik Stjernberg, Do we Have a Determinate Concept of Truth? There are many disagreements about the nature of truth. It will be argued that the concept of truth is indeterminate, in at least two ways. First, the concept of truth is indefinitely extensible, second, none of our usual ways to understand the notion of truth will ensure determinacy of the concept of truth. The main ways to get an understanding of the concept of truth are via formal treatments or through some unspecified intuitions about the nature of truth. None of these alternatives will provide us with a determinate concept of truth. This remaining indeterminacy is normally not problematic. The upshot of this dual indeterminacy is a kind of alethic pluralism. The view will be briefly compared with other related views.

Andrea Strollo, How Simple is the Simplicity of Truth? Reconciling the Mathematics and the Metaphysics of Truth The notion of truth is a central subject both in Philosophy and Mathematical Logic. The logical approach on the one side and the philosophical one on the other, however, mostly deal with problems which, apparently, require different tools to be tackled. In this paper I argue that such a separation can and should be overcome, and, in order to build a bridge, I focus on the philosophical issue of the insubstantiality of truth, which is a crucial topic to distinguish inflationist from deflationist proposals. Elaborating on the interpretation of insubstantiality in terms of the sparse/abundant classification of properties, I put forward a refined version of it in which certain flaws afflicting other formulations are solved. Then, I show how, using this improved variant, the philosophical notion of

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abundance can lucratively be related to the formal notion of expandability of models, if a logical framework is adopted. Among other virtues, the obtained link can shed new light on the debate on deflationism and conservativity.

Elia Zardini, The General Missing from the Hierarchy After some preliminary remarks concerning the debate on whether, given the semantic paradoxes, to revise naive truth or classical logic, I zoom in on a particular theory that revises naive truth by replacing the property of truth with a hierarchy of truth-like properties, and focus on a cluster of worries concerning the theory’s ability to account for generalising uses of truth. I argue that, by making a judicious use of schematic assertions and of other resources that have been appealed to by its defenders, the hierarchical theory can account for one kind of generalising use of truth, which is exemplified in S. Kripke’s well-known Dean-Nixon controversy and which essentially consists in using truth as an expressive device. I also argue, however, that the hierarchical theory cannot account for another kind of generalising use of truth, which essentially consists in using truth to attribute a unifying feature to all instances of certain kinds of sentences, and, by doing so, to be able to speak for the first time about general principles (like the law of excluded middle) and notions (like negation)—as far as the hierarchical theory is concerned, it is as though such principles and notions did not exist.

CONTRIBUTORS

Annalisa Coliva is Associate Professor at the University of Modena and Reggio Emilia and Deputy Director of Cogito Research Centre in Philosophy. She has published widely in philosophy of mind, epistemology and history of analytic philosophy. Her most recent publications include I modi del relativismo (Laterza 2009), Scetticismo. Dubbio, paradosso e conoscenza (Laterza 2012), Moore and Wittgenstein. Scepticism, Certainty and Common Sense (Palgrave 2010). As editor, she has published, among several other works, The Self and Self-Knowledge (Oxford UP 2012) and Mind, Meaning and Knowledge. Themes from the Philosophy of Crispin Wright (Oxford UP 2012). Ciro De Florio is Assistant Professor at the Università Cattolica del Sacro Cuore, Milan. His areas of interest are logic, philosophy of logic and mathematics, theories of truth. He wrote Categoricità e Modelli intesi, devoted to the ontological relevance of second-order logic and La forma della verità, about the relations between the Tarskian conception of truth and deflationism. Pascal Engel is Directeur D’Études at EHESS, Paris, and Full Professor at the University of Geneva. His present work is mostly in epistemology, especially on the nature of belief and epistemic norms. His most recent book is Volontà, credenza e verità, Jouvence, Milano 2014. Filippo Ferrari is a Research Fellow at the Northern Institute of Philosophy, University of Aberdeen. His research focuses primarily on the normativity of truth in connection with the notion of disagreement. He is also interested in topics concerning alethic pluralism, deflationism and skepticism. Carlo Filotico completed a Ph.D. in Philosophy at the State University of Milan with a dissertation on the deflationary conception of truth. He has been appointed as Lecturer in Philosophy of Language at the University of Parma, Italy, and the State University of Milan. His areas of interest are the philosophy of language and the philosophy of logic; his present research focuses mainly on the theory of truth.

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Samuele Iaquinto is a Ph.D. student at the University of Milan (Department of Philosophy) and he was a visiting research student at the King's College London. His present research focuses on normative epistemology, formal epistemology and metaphilosophy. Michele Lubrano is currently a PhD student at University of Turin; his present research focuses on the issue of abstraction principles in the foundations of arithmetic. His main area of interests are philosophy of language, ontology, philosophy of mathematics, theories of truth. Sebastiano Moruzzi is Assistant Professor at the Department of Philosophy and Communication of the University of Bologna, he is also member of Cogito Research Centre in Philosophy. His areas of research include philosophy of language, philosophy of logic and epistemology. He is presently working on vagueness and relativism. He has published a book on vagueness (Vaghezza, Laterza 2012) and edited (together with Richard Dietz) a collection of essays on vagueness (Cuts and Clouds, Oxford: OUP 2012). Boris Rähme is Researcher at the Bruno Kessler Foundation (FBK), Trent. He was previously Researcher and Lecturer at the Department of Philosophy, Freie Universität, Berlin. His publications include a book on epistemic conceptions of truth (Wahrheit, Begründbarkeit und Fallibilität, Heusenstamm: Ontos 2010) and articles in epistemology, philosophy of language and theory of truth. Fredrik Stjernberg is Professor of philosophy at Linköping University, Sweden. He has worked in the philosophy of language, especially Fitch’s knowability paradox and related issues, philosophy of logic and the philosophy of mind. Andrea Strollo got a PhD in Philosophy of Language and Philosophy of Mind at the University of Torino, and a Post Doc at the University of Helsinki. He is a Research Fellow at the Scuola Normale Superiore, Pisa. His research focuses on topics lying at the intersection among logic, philosophy of language and metaphysics.

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Contributors

Elia Zardini did his undergraduate studies in philosophy, mathematics and history at the Ca’ Foscari University of Venice and at the Technical University of Berlin. He obtained his PhD in philosophy from the University of St Andrews in 2008. He has since held postdoctoral fellowships at the University of St Andrews, the University of Aberdeen and the National Autonomous University of Mexico. He is currently a Marie Curie Intra-European Fellow in the Department of Logic, History and Philosophy of Science at the University of Barcelona and an Associate Fellow at the University of Aberdeen. Dan Zeman is a Postdoctoral Fellow at the University of the Basque Country. He is interested in semantics, especially the issue of contextsensitivity (broadly construed). He wrote several articles on relativism about various expressions (knowledge attributions, predicates of personal taste etc.), temporalism and disagreement. .

INDEX OF NAMES Armstrong D., 165-166 Armour-Garb B., 15, 32 Asay J., xii, 164-165, 173-174 Bar-On D., 34 Batens D., 54 Beall J.C., 47, 60, 123-124, 143 Belnap N.D., vii Berker S., 13 Berto F., 55, 62, 105 Blackburn S., 37, 61 Boghossian P., 13, 59, 125-126 Brandom R.B., 15, 101 Brogaard B., 101 Burge T., 180, 196 Bykvist K., 13 Camp J.L., vii Cappelen H., 85, 105, 108-109, 112, 114, 131 Carnap R., 46 Chrisman M., 8 Cicero Marcus Tullius, 10 Coliva A., 62 Cotnoir A., xi, 134-136, 141-142 Damnjanovic N., 16, 18, 174 De Florio C., xi, xii Dodd J., 15, 37 Dorr C., 61 Dreier J., 61, 63 Dummett M., viii, 119-120 Edwards D., xii, 162-166, 172-174 Egan A. , 86 Einheuser I., 112-113 Eklund M., 48 Engel P., ix, 11, 13 Epicurus, 3, 5, 7 Ferrari F., x, 102 Field H., vii, 15, 32, 37, 123-124, 159 Filotico C., x, 75, 77-78 Forbes G., 104

France A., 1, 13 Frege G., 49, 55, 161 Geach P., 49, 55 Gettier E., 13 Gibbard A., 14, 61 Gibbons J., 14 Gigante M., 13 Gillies A., 101 Glüer K., 13 Gödel K., viii, 146-148, 151-153, 159 Goldstein L., 194 Greenough P.M., 59-60 Grim P., 55, 59, 104-105 Grover D., vii, 15 Haack S., 104 Halbach V., 122, 159, 174-175 Harman G., 63 Hattiangadi A., 13, 14 Hawthorne J., 85, 105, 108-109, 112, 114 Hilbert D., 153 Hodges W., 175 Horgan T., 61 Horsten L., 16, 174 Horwich P., vii, 15, 17-18, 25, 2930, 36-37, 174 Huevenes T.T., 61 Hume D., 13 Iacona A., 70, 72, 77 Iaquinto S., x Jenkins C.S., 19 Kalish D., 104 Kaplan D., 82, 101 Kaye R., 159 Ketland J., xi, 144-147, 150-152, 155, 173, 175 Kleene S.C., 106, 111 Kölbel M., vii, 38, 61, 64-65, 67-68, 75, 78, 82, 101, 108, 114 Kompa N. , 101

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Index of Names

Kornblith H., 8 Krabbe E., 102 Kripke S., xi-xii, 181, 196 Lasersohn P. , 64, 67, 101, 108 Levin J., 127 Lewis D., xii, 164-166, 174 López De Sa D., 61, 63 Lubrano M., xi àukasiewicz, 106, 111 Lynch M.P., 117, 127, 131-134, 142 MacFarlane J., vii, x, 61, 64-65, 67, 78, 80-84, 86-94, 98-99, 101102, 108 Makinson D., 199 Mar G., 104 Marconi D., 78 Marques T., 61 Millar A., 8 Montague R., 104 Moruzzi S., ix, 62, 108 Munzer S., 95, 102 Parsons T., 54 Peano G., 155, 172 Pedersen, 117, 131 Peirce C.S., 88 Philodemus, 1, 13 Plato, vii, 165 Pravato G., 60 Priest G., 54-55, 131, 143 Prior A.N., 104 Pust J., 131 Quine W.V.O., vii, 116 Rähme B., ix Ramsey F.P., vii, 174, Recanati F., 101 Restall G., 196 Richard M., 127 Ripley D., 60, Rosenkranz S., 108, 193 Routley R., 105 Routley V., 105 Russell B., 119 Sainsbury M., 179 Schafer K., 46 Schaffer J., 168 Schiffer S., 60,

Schnieder B., 26, 33, 36 Schroeder M., 49, 61 Shapiro S., viii, xi-xii, 132-134, 144-147, 150, 155-157, 159, 173, 175 Sher G., 127, 131 Sider T., 174 Simmons K., 34 Skyrms B., 194 Smith N., 60 Smith P., 159 Smithies D., 11 Smorynski C., 159 Soames S., 15, 17, 106 Sorensen R., 59, 194 Sosa E., 13 Spinoza B., 13 Steglich-Petersen A., 13 Stjernberg F., xi Strawson P.F., 197 Strollo A., xii, 173, 175 Sundell T., 61, 102 Tait W., 159 Tappolet C., 131 Tarski A., viii, 46, 59-60, 115, 133, 145, 159-160, 162, 178, 200 Tennant N., xi, xii, 146-157, 159 Teroni F., 13 Timmons M., 61 Tye M., 106 Unwin N., 61 Velleman D., 125-126 Vesperini P., 13 von Fintel K., 101 Waismann F., xi, 115-116, 129 Wikforss Å., 13 Williams M., 15-16, 34, 37 Williamson T., 121 Wolfram S., 104 Woodbridge J.A., 32 Wright C., vii, 46, 59, 64, 67, 101, 126, 131, 194 Wright C.D., 131 Yablo S., xi, 139-140, 142 Zardini E., xii, 194-195, Zeman D., x