Rush - The Metaphysics of Logic

THE METAPHYSICS OF LOGIC

Featuring fourteen new essays from an international team of renowned contributors, this volume explores the key issues, debates, and questions in the metaphysics of logic. The book is structured in three parts, looking first at the main positions in the nature of logic, such as realism, pluralism, relativism, objectivity, nihilism, conceptualism, and conventionalism, then focusing on historical topics such as the medieval Aristotelian view of logic, the problem of universals, and Bolzano’s logical realism. The final section tackles specific issues such as glutty theories, contradiction, the metaphysical conception of logical truth, and the possible revision of logic. The volume will provide readers with a rich and wide-ranging survey, a valuable digest of the many views in this area, and a long overdue investigation of logic’s relationship to us and the world. It will be of interest to a wide range of scholars and students of philosophy, logic, and mathematics. p e n e l o p e r u s h is Honorary Associate with the School of Philosophy and Online Lecturer for Student Learning at the University of Tasmania. She has published articles in journals including Logic and Logical Philosophy, Review of Symbolic Logic, South African Journal of Philosophy, Studia Philosophica Estonica, and Logique et Analyse. She is also the author of The Paradoxes of Mathematical, Logical, and Scientific Realism (forthcoming).

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THE METAPHYSICS OF LOGIC edi t ed by PENELOPE RUSH University of Tasmania

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University Printing House, Cambridge cb2 8bs, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107039643 © Cambridge University Press 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloging in Publication data The metaphysics of logic / edited by Penelope Rush, University of Tasmania. pages cm Includes bibliographical references and index. isbn 978-1-107-03964-3 (Hardback) 1. Logic. 2. Metaphysics. I. Rush, Penelope, 1972– editor. bc50.m44 2014 160–dc23 2014021604 isbn 978-1-107-03964-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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With thanks to Graham Priest for unstinting encouragement, and to Annwen and Callum – never give up.

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Contents

List of contributors

page ix

Introduction

1

Penelope Rush

part i

the main positions

11

1 Logical realism

13

Penelope Rush

2 A defense of logical conventionalism

32

Jody Azzouni

3 Pluralism, relativism, and objectivity

49

Stewart Shapiro

4 Logic, mathematics, and conceptual structuralism

72

Solomon Feferman

5 A Second Philosophy of logic

93

Penelope Maddy

6

Logical nihilism

109

Curtis Franks

7 Wittgenstein and the covert Platonism of mathematical logic

128

Mark Steiner

part ii

history and authors

8 Logic and its objects: a medieval Aristotelian view Paul Thom

vii

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145 147

Contents

viii

9 The problem of universals and the subject matter of logic

160

Gyula Klima

10 Logics and worlds

178

Ermanno Bencivenga

11

Bolzano’s logical realism

189

Sandra Lapointe

part iii

specific issues

12 Revising logic

209 211

Graham Priest

13 Glutty theories and the logic of antinomies

224

Jc Beall, Michael Hughes, and Ross Vandegrift

14 The metaphysical interpretation of logical truth

233

Tuomas E. Tahko

References Index

249 264

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Contributors

jody azzouni, Professor, Department of Philosophy, Tufts University. jc beall, Professor of Philosophy and Director of the UCONN Logic Group, University of Connecticut, and Professorial Fellow at the Northern Institute of Philosophy at the University of Aberdeen. ermanno bencivenga, Professor of Philosophy and the Humanities, University of California, Irvine. solomon feferman, Professor of Mathematics and Philosophy, Emeritus, and Patrick Suppes Professor of Humanities and Sciences, Emeritus, Stanford University. curtis franks, Associate Professor, Department of Philosophy, University of Notre Dame. michael hughes, Department of Philosophy and UCONN Logic Group, University of Connecticut. gyula klima, Professor, Department of Philosophy, Fordham University, New York. sandra lapointe, Associate Professor, Department of Philosophy, McMaster University. penelope maddy, Distinguished Professor, Department of Logic and Philosophy of Science, University of California, Irvine. graham priest, Distinguished Professor, Graduate Center, CUNY, and Boyce Gibson Professor Emeritus, University of Melbourne. penelope rush, Honorary Associate, School of Philosophy, University of Tasmania.

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List of contributors

stewart shapiro, Professor, Department of Philosophy, The Ohio State University. mark steiner, Professor Emeritus of Philosophy, the Hebrew University of Jerusalem. tuomas e. tahko, Finnish Academy Research Fellow, Department of Philosophy, History, Culture and Art Studies, University of Helsinki. paul thom, Honorary Visiting Professor, Department of Philosophy, The University of Sydney. ross vandegrift, Department of Philosophy and UCONN Logic Group, University of Connecticut.

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Introduction Penelope Rush

This book is a collection of new essays around the broad central theme of the nature of logic, or the question: ‘what is logic?’ It is a book about logic and philosophy equally. What makes it unusual as a book about logic is that its central focus is on metaphysical rather than epistemological or methodological concerns. By comparison, the question of the metaphysical status of mathematics and mathematical objects has a long history. The foci of discussions in the philosophy of mathematics vary greatly but one typical theme is that of situating the question in the context of wider metaphysical questions: comparing the metaphysics of mathematical reality with the metaphysics of physical reality, for example. This theme includes investigations into: on exactly which particulars the two compare; how (if ) they relate to one another; and whether and how we can know anything about either of them. Other typical discussions in the field focus on what mathematical formalisms mean; what they are about; where and why they apply; and whether or not there is an independent mathematical realm. A variety of possible positions regarding all of these sorts of questions (and many more) are available for consideration in the literature on the philosophy of mathematics, along with examinations of the specific problems and attractions of each possibility. But there is as yet little comparable literature on the metaphysics of logic. Thus the aim of this book is to address questions about the metaphysical status of logic and logical objects analogous to those that have been asked about the metaphysical status of mathematical objects (or reality). Logic, as a formal endeavour has recently extended far beyond Frege’s initial vision, describing an apparently ever more complex realm of interconnected formal structures. In this sense, it may seem that logic is becoming more and more like mathematics. On the other hand, there are (also apparently ever more) sophisticated logics 1

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describing empirical human structures: everything from natural language and reason, to knowledge and belief. That there are metaphysical problems (and what they might be) for the former structures analogous to those in the philosophy of mathematics is relatively easily grasped. But there are also a multitude of metaphysical questions we can ask regarding the status of logics of natural language and thought. And, at the intersection of these (where one and the same logical structure is apparently both formal and mathematical as well as applicable to natural language and human reason), the number and complexity of metaphysical problems expands far beyond the thus far relatively small set of issues already broached in the philosophy of logic. As just one example of the sorts of problems deserving a great deal more attention, consider the relationship between mathematics and logic. Questions we might ask here include: whether mathematics and logic describe the same or similar in-kind realities and relatedly, whether there is a line one can definitively draw between where mathematics stops and logic starts. Then we could also ask exactly what sort of relationship this is: is it one of application (of the latter to the former) or is it more complex than this? Another central problem for the metaphysics of logic is that of pinning down exactly what it is that logic is supposed to range over. Logic has been conceived of in a wide variety of ways: e.g. as an abstraction of natural language; as the laws of thought; and as normative for human reason. But, what is the ‘thought’ whose structure logic describes; how natural is the natural language from which logic is abstracted?; and to what extent does the formal system actually capture the way humans ought to reason? As touched on above, a key metaphysical issue is how to account for the apparent ‘double role’ – applying to both formal mathematical and natural reasoning structures – that (at least the main) formal logical systems play. This apparent duality lines up along the two central, indeed canonical applications of logic: to mathematics and to human reason, (and/or human thought, and/or human language). In many ways, the first application suggests that logic may be objective – or at least as objective as mathematics, in the sense that, as Stewart Shapiro puts it (in this volume) we might say something “is objective if it is part of the fabric of reality”. This in turn might suggest an apparent human-independence of logic. The second application, though, might suggest a certain subjectivity or intersubjectivity; and so in turn an apparent human-dependence of logic, insofar as a logic of reason may appear dependent on actual human thought or concepts in some essential way.

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Introduction

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Both the apparent objectivity and the apparent subjectivity of logic need to be accounted for, but there are numerous stances one might take within this dichotomy, including a conception of objectivity that is nonetheless human-dependent. In Chapter 4, Solomon Feferman reviews one such example in his non-realist philosophy of mathematics, wherein “the objects of mathematics exist only as mental conceptions [and] . . . the objectivity of mathematics lies in its stability and coherence under repeated communication”. Others of the various positions one might take up within this broad-brush conceptual field are admirably explored in both Stewart Shapiro’s and Graham Priest’s chapters, though from quite different stand points: Shapiro explores the nuances and possibilities in conceptions of objectivity, relativity, and pluralism for logic, whereas Priest looks at these issues through the specific lens afforded by the question whether or not logic can be revised. There are, then, a variety of possible metaphysical perspectives we can take on logic that, particularly now, deserve articulation and exploration. These include nominalism; naturalism; structuralism; conceptual structuralism; nihilism; realism; and anti-(or non-)realism, as well as positions attempting to steer a path between the latter two. The following essays cover all these positions and more, as defended by some of the foremost thinkers in the field. The first part of the book covers some of the main philosophical positions one might adopt when considering the metaphysical nature of logic. This section covers everything from an extreme realism wherein logic may be supposed to be completely independent of humanity, to various accounts and various degrees in which logic is supposed to be in some way human-dependent (e.g. conceptualism and conventionalism). In the first chapter I explore the feasibility of the notion that logic is about a structure or structures existing independently of humans and human activity. The (typically realist) notion of independence itself is scrutinised and the chapter gives some reasons to believe that there is nothing in principle standing in the way of attributing such independence to logic. So any benefits of such a realism are as much within the reach of the philosopher of logic as the philosopher of mathematics. In the second chapter, Jody Azzouni explores whether logic can be conceived of in accordance with nominalism: a philosophy which might be taken to represent the extreme opposite of realism. Azzouni argues the case for logical conventionalism, the view that logical truths are true by convention. For Azzouni, logic is a tool which we both impose by convention on our own reasoning practices, and occasionally also to evaluate

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them. But Azzouni shows that although there seems to be a close relationship between conventionality and subjectivity, logic’s being conventional does not rule out its also applying to the world. Stewart Shapiro, in the third chapter, argues the case for logical relativism or pluralism: the view that there is “nothing illegitimate” in structures invoking logics other than classical logic. Shapiro defends a particular sort of relativism whereby different mathematical structures “have different logics”, giving rise to logical pluralism – conceived of as “[the] view that different accounts of the subject are equally correct, or equally good, or equally legitimate, or perhaps even (equally) true”. Shapiro’s chapter looks in some depth at the relationship between mathematics and logic, identified above as a central problem for our theme. But in particular, it investigates the extent to which logic can be thought of as objective, given the foregoing philosophy. He offers a thorough, precise, and immensely valuable analysis of the central concepts, and clarifies exactly what is and is not at stake in this particular debate. In the fourth chapter, Solomon Feferman examines a variety of logical non-realism called conceptual structuralism. Feferman shares with Shapiro a focus on the relationship between mathematics and logic, extending the case for conceptual structuralism in the philosophy of mathematics to logic via a deliberation on the nature and role of logic in mathematics. He draws a careful picture of logic as an intermediary between philosophy and mathematics, and gives a compelling argument for the notion that logic, as (he argues) does mathematics, deals with truth in a given conception. According to Feferman’s account, truth in full is applicable only to definite conceptions. On this picture, when we speak of truth in a conception, that truth may be partial. Thus classical logic can be conceptualised as the “logic of definite concepts and totalities”, but may itself be justified on the basis of a semi-intuitionist logic “that is sensitive to distinctions that one might adopt between what is definite and what is not”. Feferman shows how allowing that “different judgements may be made as to what are clear/definite concepts”, affords the conceptual structuralist a straightforward, sensible and clear understanding of the role and nature of logic. Penelope Maddy, in the fifth chapter, offers a determinedly secondphilosophical account of the nature of logic, presenting another admirably clear and sensible account, focusing in this case on the question why logic is true and its inferences reliable. ‘Second Philosophy’ is a close cousin of naturalism as well as a form of logical realism and involves persistently bringing our philosophical theorising back down to earth.

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Introduction

5

In Maddy’s words: “The Second Philosopher’s ‘metaphysics naturalized’ simply pursues ordinary science”. Thus Maddy investigates the question from this ‘ordinary’ perspective, beginning with a consideration of rudimentary logic, and gradually building up (via idealisations) to classical logic. On this account, logic turns out to be true and reliable in our actual (ordinary, middle-sized) world partly because that actual world shares the formal structure of logic (or at least rudimentary logic). Maddy gives an extensive account of some of the ways we might come to know of this structure, presenting recent research in cognitive science that supports the notion that we are wired to detect just such a structure. She then offers the (tentative) conclusion that classical logic (as opposed to any nonclassical logic) is best suited to describe the physical world we live in, despite the fact that classical logic’s idealisations of rudimentary logic are best described as ‘useful falsifications’. In the final two chapters of the first part, Curtis Franks questions the assumption underpinning any metaphysics of logic at all: namely that there is “a logical subject matter unaffected by shifts in human interest and knowledge”; and Mark Steiner unpicks Wittgenstein’s idea that “The rules of logical inference are rules of the language game”. Steiner points out that for Wittgenstein “There is nothing akin to ‘intuition’, ‘Seeing’ and the like in following or producing a logical argument. Instead we [only] have regularities induced by linguistic training”. So, Steiner argues, supposing that logic is grounded by anything other than the regularities that ground rule following (say by some objective ‘fact’ according to which its rules are determined), is engaging in a kind of ‘covert Platonism’. Steiner identifies the key difference (for Wittgenstein) between mathematics and logic as the areas their respective rules govern: whereas both mathematical and logical rules govern linguistic practices, (only) mathematical rules also govern non-linguistic practices. Interestingly, while Steiner argues that the line between mathematics and logic is thus more substantial than many may think, Franks argues that the line between maths and logic is illusory, based on a need to differentiate the patterns of reasoning we have come to associate with logic from other patterns of reasoning, which itself is grounded on nothing more than a baseless psychological or metaphysical preconception. Franks argues that logicians deal not with truth but with the “relationships among phenomena and ideas” – and agrees with Steiner that looking for any further ‘ontological ground’ is misconceived (note, though, that Steiner himself does not commit himself to the views he attributes to

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Wittgenstein. Rather he gives what he takes to be the best arguments in Wittgenstein’s favour). As something of a side note, it is interesting to compare Sandra Lapointe’s discussion of Bolzano’s notion of definition (in Part II) to that which Franks presents on behalf of Socrates. Lapointe argues that, for Bolzano, there is more to a definition than merely fixing its extension, whereas Franks argues that Socrates was right to prioritise the fixing of an extension first before enquiring after the nature or essence of a thing. Steiner’s discussion of the Wittgensteinian distinction between explanation and description is also relevant here. This debate touches on another important subtheme running throughout the book: the nature and role of intentional and extensional motivations of logical systems; and the related tension (admirably illustrated by Franks’ discussion of the development of set theory) between appeals to form/formal considerations and appeals to our intuitions. Both Steiner’s Wittgenstein and Franks agree that the image of logic as a kind of ‘super-physics’ needs to be challenged, even eliminated; but each takes a different approach to just how this might be achieved, with Franks arguing for logical nihilism, and Steiner going to pains to show how, for Wittgenstein, the rules of logic ought to be conceived as akin to those of grammar and as nothing more than this. The next part of the book gives an historical overview of past investigations into the nature of logic as well as giving insights into specific authors of historical import for our particular theme. In the first chapter of this section Paul Thom discusses the thoughts of Aristotle and the tradition following him on logic. Thom focuses particularly on what sort of thing, metaphysically speaking, the objects of logic might be. He traces a gradual shift (in Kilwardby’s work) from a conception of logic as about only linguistic phenomena, through a conception wherein logic is also understood as also being about reason, to the inclusion of ‘the natures of things’ as a possible foundation of logic. Kilwardby considers a view whereby the principal objects of logic: ‘stateables’, are not some thing at all (at least not in themselves), insofar as they do not belong to any of Aristotle’s categories. Kilwardby opposes this view on the basis of a sophisticated and complex argument to the effect that there may be objects of logic that are human dependent but also external to ourselves, and can be considered both things of and things about nature itself. These insights are clearly relevant to the modern questions we ask about the metaphysics of logic and resonate strongly with the themes explored in the first part. The range of possibilities considered offer a fascinating and fruitful look into the historical precedents of the questions

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Introduction

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about logic still open today: e.g. Thom notes that for Aristotle, the types of things that can belong to the categories are ‘outside the mind or soul’, and so Kilwardby’s analysis clearly relates to our modern question as to the possible independence and objectivity of logic. The complexity of that question is brought to the fore in Kilwardby’s detailed consideration of the various ‘aspects’ under which stateables can be considered, and according to which they may be assigned to different categories. Thom’s chapter goes on to offer a framework for understanding later thinkers and traditions in logic, some of which (e.g. Bolzano in Lapointe’s chapter) are also discussed in this part. His concluding section ably demonstrates that understanding the history of our questions casts useful light on the modern debate. Gyula Klima also discusses strategies for dealing with the two way pull on logic – from its apparent abstraction from human reason and from its apparent groundedness in the physical world. Klima focuses on the scholastics, comparing the semantic strategies of realists and nominalists around Ockham’s time. One of these was to characterise logic as the study of ‘second intentions’ – concepts of concepts. Klima points out that when logic is conceived of in this way, the core-ontology of real mindindependent entities could in principle have been exactly the same for “realists” as for Ockhamist “nominalists”; therefore, what makes the difference between them is not so much their ontologies as their different conceptions of concepts, grounding their different semantics. Klima argues that extreme degrees of ontological and semantic diversity and uniformity mark out either end of a “range of possible positions concerning the relationship between semantics and metaphysics, [from] extreme realism to thoroughgoing nominalism” and points out how the conceptualisation of the sorts of things semantic values might be varies according to where a given position sits within this framework. His chapter illuminates the metaphysical requirements of different historical approaches to semantics and the way in which the various possible metaphysical commitments we make come about via competing intuitions regarding diversity: whether we locate diversity in the way things are or in the way we speak of or conceptualise them. In the next chapter, Ermanno Bencivenga picks up a thought Thom touches on in his closing paragraph – namely that our modern conception of logic appears to have lost touch with the relevant ways in which actual human reason can go wrong other than by not being valid. Offering a Kantian view, Bencivenga suggests we adjust our conception of logic to that of almost any structure we impose on language and experience, just so

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long as it is a holistic endeavour to uncover how our language acquires meaning. In this way almost all of philosophy is logic, but not all of what we commonly call logic makes the grade. For Bencivenga, logic should focus on meaning: on the way language constructs our world. From this perspective, the relationship of logic to reason is just one of many connections between the world we create and the internal structure of any given logic. For example, while appeals to reason may motivate logic’s claims, so too do appeals to ethos and pathos. Sandra Lapointe looks at the sorts of motivations and reasons we might have for adopting a realist philosophy of logic, pointing out that these reasons may not themselves be logical and developing a framework within which different instances of logical realism can be compared. Lapointe examines Bolzano’s philosophy in particular and shows how his realism may best be thought of as instrumental rather than inherent: adopted in order to make sense of certain aspects of logic rather than as a result of any deep metaphysical conviction. Lapointe’s chapter shows how Bolzano’s works cast light on a wide array of issues falling under our theme, from his evocative analogy between the truths of logic and the spaces of geometry to his critique of Aristotle’s criteria for validity. Lapointe’s discussion of the latter is worth drawing attention to as it deals with the topic mentioned earlier – of the tension between external and intensional; and formal and non-formal motivations for logical systems. Lapointe compares the results of Bolzano’s motivations with those of Aristotle for the definition of logical consequence and in so doing, identifies some central considerations to help further our understanding of this topic. The final part of the book deals with the specific issues of the possible revision of logic, the presence of contradiction, and the metaphysical conception of logical truth. Graham Priest’s chapter deals with the question of the revisability of logic and in so doing also offers a useful overview of much of what is discussed in earlier sections and indeed throughout this book. Priest outlines three senses of the term ‘logic’ and asks of each whether it can be revised, revised rationally, and (if so) how. In some ways, Priest’s paper dovetails with Shapiro’s discussion of the possible criteria used to judge the acceptability of a theory, and draws a conclusion similar to that of Shapiro’s ‘liberal Hilbertian’: i.e. “[that] There is no metaphysical, formal, or mathematical hoop that a proposed theory must jump through. There are only pragmatic criteria of interest and usefulness” – which, for Priest, are judged against the requirements of

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Introduction

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its application(s) and by “the standard criteria of rational theory choice”. And like Shapiro’s, Priest’s chapter is an immensely valuable overview of the key concepts informing any metaphysics of logic. In the next chapter, Jc Beall, Michael Hughes, and Ross Vandegrift look at different repercussions of different attitudes toward “glutty predicates” – predicates which “in virtue of their meaning or the properties they express . . . [are] both true and false”. Their chapter shows how our various theories and attitudes about such predicates may motivate different formal systems. The formal systems in question here are Priest’s well-known LP and the lesser-known LA advanced by Asenjo and Tamburino. The upshot of the discussion is that the latter will suit someone metaphysically “commited to all predicates being essentially classical or glutty” and the former someone for whom “all predicates [are] potentially classical or glutty”. Thus, Beall et al. draw out some interesting consequences of the relationships between our intuitions and theories regarding the metaphysical, the material, and the formal aspects of logic. They highlight both the potential ramifications of the role we afford our metaphysical commitments and the ramifications of the particular type of commitments they might be. So while Beall et al. look in particular at a variety of metaphysical theories about contradiction, and the impact of these on two formal systems, their discussion also gives some general pointers to the way in which our metaphysical beliefs impact on other central factors in logic: crucially including the creation of the formal systems themselves and the evaluation of their differences. Tuomas Tahko finishes the book by examining a specific realist metaphysical perspective and suggesting it as another approach we might take to understand logic, especially to interpret logical truth. His case study offers an interpretation of paraconsistency which contrasts nicely with that offered in the penultimate chapter. Tahko’s approach is to judge logical laws according to whether or not they count as genuine ways the actual world is or could be. From this perspective, he argues, exceptions to the law of non-contradiction now appear more as descriptions of features of our language than of reality. Thus he argues that the realist intuition grounding logic in how the world is (or could be) gives us good reason to preserve the LNC. Tahko’s metaphysical interpretation of logical truth also offers an interesting perspective on logical pluralism. From Tahko’s metaphysical perspective, pluralism may be understood as about subsets of possible worlds representing genuine possible configurations of the actual world. Tahko’s chapter is a meticulous investigation into the links, both

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already in place and that (from this perspective) ought to be, between an interesting set of metaphysical intuitions and those laws of logic we take to be true. In all, this book ranges over a vast terrain covering much of the ways in which our beliefs about the role and nature of logic and of the structures it describes both impact and depend on a wide array of metaphysical positions. The work touches on and freshly illuminates almost every corner of the modern debate about logic; from pluralism and paraconsistency to reason and realism.

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chapter 1

Logical realism Penelope Rush

1. The problem Logic might chart the rules of the world itself; the rules of rational human thought; or both. The first of these possible roles suggests strong similarities between logic and mathematics: in accordance with this possibility, both logic and mathematics might be understood as applicable to a world (either the physical world or an abstract world) independent of our human thought processes. Such a conception is often associated with mathematical and logical realism. This realist conception of logic raises many questions, among which I want to pinpoint only one: how logic can at once be independent of human cognition in the way that mathematics might be; and relevant to that cognition. The relevance of logic to cognition – or, at the very least, the human ability to think logically – seems indubitable. So any understanding of the metaphysical nature of logic will need also to allow for a clear relationship between logic and thought.1 The broad aim of this chapter is to show that we can take logical structures to be akin to independent, real, mathematical structures; and that doing so does not rule out their relevance and accessibility to human cognition, even to the possibility of cognition itself. Suppose that logical realism involves the belief that logical facts are independent of anything human:2 that the facts would have been as they 1

2

Two things: note I do not claim we can or ought to show that logic underpins, describes, or arises from cognition. In fact I think the relationship between thought and logic is almost exactly analogous to that between thought and mathematics (see Rush (2012)), and I disagree with the idea that there is any especially significant connection between logic and thought beyond this. Two: while this chapter deals with the notion of ‘independence’ per se, it investigates this from the perspective of applying that notion especially to logic. That is, my main aim here is to indicate one way in which the realist conception of an independent logical realm might be considered a viable philosophical position but one primary way I hope to do this is by showing how attributing independence to logic need be no more problematic than attributing independence to anything else (e.g. by arguing that the realist problem applies across any ‘type’ of reality which is supposed to be independent). See Lapointe’s characterisation as IND in this volume.

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are regardless of whether or not humans comprehended them, or even had existed at all. A sturdy sort of objectivity seems guaranteed by this stance. Janet Folina captures this neatly: [If logical facts exist independently of the knowers of logic], there is a clear difference, or gap, between what the facts are and what we take them to be. (Folina 1994: 204)3

This sturdy objectivity is just one reason we might find logical realism appealing.4 There is, though, a well-known objection to the idea that we can coherently posit the independence of facts (including logical facts) from their human knowers (and human knowledge). Wilfrid Sellars formulated a version of this objection in 1956. Sellars argued that in order to preserve both the idea that there is something independent of ourselves and epistemological processes, and the idea that we can access this something (e.g. know truths about it), we seem to have either to undermine the independent status of that thing (by attributing to it apparently human-dependent features) or to render utterly mysterious the way in which any knowledge-conferring relationship might arise from that access. Sellars’ idea is that we cannot suppose that we encounter reality as it is independently of us, unless we suppose something like a moment of unmediated access. But, there can be no relevant relationship between independent reality and us (e.g. we can make no justificatory or foundational use of such a moment) unless that unmediated encounter can be taken up within our own knowledge. The obvious move is simply to say that this initial encounter is available to knowledge. But this move undermines itself by casting what was independent as part of what is known: i.e. it attributes an already in-principle knowability to a supposed fully independent reality (for more on Sellars’ argument, see Fumerton (2010), and Sellars (1962)). The broadly applicable Sellarisan objection bears comparison to Benacerraf ’s (1973) objection to mathematical realism, which extends, at least to a degree, to logical realism.5 Benacerraf argued that even our best theory of 3

4

5

Folina was talking about mathematical realism, but the sort of logical realism I want to examine here is directly analogous to mathematical realism in this respect. Lapointe (this volume) explores a variety of reasons that may play a role in holding some version of logical realism, so I won’t go into these in depth here. For more on the possible entities a logical realist might posit (e.g. meanings/propositions), see Lapointe (this volume). Regardless of which entities are selected and where these are situated on the

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knowledge could not account for knowledge of mathematical reality just so long as that reality was conceived of in the usual mathematical realist way: as abstract, acausal, and atemporal. Part of the problem, as Benacerraf saw it, was that the stuff being posited as independently real is not sufficiently like any stuff that we can know, and if it were, it would not be the sort of thing intended by the mathematical realist in the first place. Sellars’ objection can be understood as a generalisation of Benacerraf ’s: common to both is the idea that the fully independent reality posited by the realist is not the type of thing we can know, or if it is, then it is not the type of thing the realist says it is. Thus, even were the mathematical or logical realist to adjust his conception of mathematical or logical reality by ruling out one or all of its abstractness, atemporality, or acausality, the problem induced by its complete independence of humans and human consciousness would remain. Recall, the realist idea of independence I am interested in here is one which posits an in-principle or always possible separation between what independently is and what we as humans grasp. The basic idea is that were there no humans to experience or be conscious of it, logic would still be as it is. So it seems that being the type of thing which is experienced or known can be no part of what it (essentially) is.6 The problem can be expressed this way: how can independent reality be part of human consciousness and experience if our human consciousness and experience of it can be no part of independent reality? A putative solution, then, might show how independent reality could play a role in human consciousness, but such a solution would need also to affirm the necessary condition that being the object of our consciousness is no (essential) part of independent reality itself. This notion of independence, then, is not only the most problematic feature of any logical realism, it may be outright contradictory: A realist . . . is basically someone who claims to think that which is where there is no thought. . . . he speaks of thinking a world in itself and independent of thought. But in saying this, does he not precisely speak of a world to which thought is given, and thus of a world dependent on our relation-to-the-world? (Meillassoux 2011: 1)

6

abstract–physical scale of possible entities, just so long as the realist also posits IND (Lapointe, this volume), they’ll encounter some version of Benacerraf ’s or Sellars’ problem. For more on the nuances of ‘independence’ available to the realist, see Jenkins (2005) – I take essential independence to follow from modal independence, and I take modal independence as characteristic of the sort of realism I want to explore.

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Husserl characterised the realist problem of independence (which he also called ‘transcendence’) in various ways, one of which is as follows: [the problem is] how cognition can reach that which is transcendent . . . [i.e.] the correlation between cognition as mental process, its referent and what objectively is . . . [is] the source of the deepest and most difficult problems. Taken collectively, they are the problem of the possibility of cognition. (Husserl 1964: 10–15)

Each of the above characterisations of the realist’s situation turns on the central theme of how we can sensibly (and relevantly) conceptualise the role that a reality independent of human consciousness could play in the realm of that consciousness. Husserl’s characterisation of the problem already gives a clue as to his overall approach: rather than view the problem as bridging a gap of the sort Folina describes, Husserl suggests we view it as “the possibility of cognition”.

2. The potential of phenomenology I hope to show how Husserl’s approach potentially enables us to take independent reality in both of the ways sitting either side of the gap: i.e. both as what is and as what is not the end point of a reasonable epistemology. That is, I hope to use his approach to see how we might accommodate the idea that what is cognised, and what must (on a realist account) remain irreducibly external (or, in principle, separable) to what is cognised – can be one and the same thing, or (perhaps) more accurately, a dual thing.7 At first glance, this might seem simply to concede the contradiction Meillassoux graphically outlines. I want to take a second glance – illustrating how such a concession need be neither simple nor impotent but rather offer a way to conceptualise the elements underpinning the realist notion of independent reality and so begin, if not to resolve, then to make some sense of its intractability. That is, there are ways in which the Husserlian perspective can motivate us to find reasons and avenues by which we might begin to accommodate the independent reality the realist posits, even as potentially contradictory – rather than to take its inherent instability as reason enough to brush it off as impossible and therefore irrelevant. These ways all intersect at the possibility opened in the phenomenological 7

As will become clear, I have a very particular notion of duality in mind here – i.e. a (contradictory) duality of object: ‘one that is also two’ – rather than a duality of an object’s role, or aspect, or components, etc: ‘one that has two aspects/dimensions/components, etc’.

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perspective (admittedly most probably neither envisaged nor anywhere claimed by Husserl himself ); namely that the realist predicament is itself an essential ingredient for the possibility of cognition.8 All of the above ways of rendering the realist problem of independence (barring his own) – i.e. as an intractable and apparently unbridgeable dichotomy between reality and our knowledge of it – Husserl characterised as a product of the ‘natural’, ‘scientific’ attitude, which he saw as pervasive all of philosophy (again, barring his own, e.g. Husserl 1964: 18–19). By contrast, phenomenology offers a picture of entangled cognition wherein independent reality is inextricable from cognition itself. This sort of picture takes the first step toward accommodating both sides of the divide insofar as it introduces the idea that our internal perspective itself irreducibly incorporates the possibility, even the necessity, of there being something outside that perspective. To be clear, I reiterate that this is my own interpretation of Husserl and my own exploration of the possibilities his work suggests to me. I do not attribute these possibilities to Husserl. As I understand him, for Husserl, experience is always experience of – and so cannot begin to be defined without allowing (at least) a place or a role for something external toward which it is directed at the outset. For me, the promising bit is this: that this something is both somehow outside or external to (‘constituting’) experience and within it (‘being constituted’) at the same time. It is by examining and enlarging on this promising bit that I hope to explore one way in which phenomenology (potentially) offers a role for the realist predicament itself as the (contradictory) structure of our relationship to independent reality. I hope to sketch how accepting the predicament in this way might enable us to make sense of reality, cognition, and experience within a realist framework – to see the realist’s ‘predicament’ as a complex and interesting structure that these elements share, as opposed to an impossible riddle or a problem in need of a solution. In what follows, I’ll briefly unpack just a couple of aspects of Husserl’s account in order to show how we might use them to begin to open and explore this possibility, specifically regarding the idea of a realistically imagined independent logical structure.9 8

9

Caveat: I’d like to argue that the predicament can play this role just so far as the basic idea of an independent reality existing at all can. It is the latter that I see the framework in Husserl’s ideas as able to directly establish. Or, again, to illustrate how conceiving logic as an independent objective structure akin to mathematics need not be considered an especially problematic instance of the general idea of independent reality itself, once that idea is effectively defended.

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3. Key aspects of phenomenology 3.1 The Platonic nature of logic Husserl had a very broad concept of logic that embraces our usual modern idea of logic as well as something he called ‘pure logic’, which we can loosely characterise as something like ‘the fundamental forms of experience’. For Husserl, logic as formal systems (and so too ‘modern logic’; incorporating classical, modal, and all the usual non-classical structures), is to be accounted for in much the same way as is mathematics: by its relationship to these fundamental forms. This relationship is roughly that which holds between practice and theory – pure logic is the purely theoretical structure (or, perhaps, structures – I don’t think it matters much here) that accounts for logic as practised. For Husserl, the fundamental forms of pure logic are an in-eliminable part of experience: i.e. ‘experience’ encompasses direct apprehension of these inferential relationships. The apprehended structures are abstract and platonic; discovered, rather than constructed. Theory, empirical observation, and experience are in this sense fallible: they may or may not ‘get it right’ and reveal the actual independent structure of logic. In Husserl’s words: As numbers . . . do not arise and pass away with acts of counting, and as, therefore, the infinite number-series presents an objectively fixed totality of general objects . . . so the matter also stands with the ideal, pure-logical units, the concepts, propositions, and truths – in short, the significations dealt with in logic . . . form an ideally closed totality of general objects to which being thought and expressed is accidental. (1981: 149)

Thus both logic and mathematics, for Husserl, have a ‘pure’, ‘abstract’, ‘theoretical’, ‘definite’, and ‘axiomatic’ foundation. Further, Husserl believed that: one cannot describe the given phenomena like the natural number series or the species of the tone series if one regards them as objectivities in any other words than with which Plato described his ideas: as eternal, self-identical, untemporal, unspatial, unchanging, immutable. (Hartimo 2010a: 115–118, italics mine)

So, according to the prevailing view, both logic and mathematics as they are characterised by Husserl, should encounter the realist problem of independence – neither are the sort of thing we can simply take as part of human cognition; i.e. not without also accommodating

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the idea that what cognition accesses is in principle no part of what either mathematics or logic independently is.

3.2

Inextricability

As touched on above, one of Husserl’s most suggestive and promising ideas is that consciousness is not separable from consciousness of an object – intentionality is built into the structure of consciousness and experience itself. The leading idea is consciousness as consciousness of: the very definition of experience and consciousness as involving already what it is directed toward, or what it is conscious of. Of course, this idea is also what a great deal of the controversy in Husserlian scholarship centres on. One reason for the controversy, I think, is the ambiguity in the prima facie simple idea of an object (or realm, or reality) as an object of anything (including, for example, consciousness, intention, act, or perception). Even on the most subjectivist reading, the notion is ambiguous between the idea of objects in experience, and as experienceable. This ambiguity interplays in obvious ways with the tension underpinning the realist’s problem: that between the object as given to an epistemological human-dependent process, and the object as independent. In turn (as we’ve seen) this ambiguity itself centres on a distinction between ‘internal’ (what we take the facts to be), and ‘external’ (what the facts are). I suggest that the urge to disambiguate Husserl on this point should be resisted,10 since to disambiguate here would be to miss a large part of the potential of phenomenology. Indeed, Husserl himself seems at times to deliberately preserve ambiguity here (though whether he meant to or not is tangential to the point). For example: First fundamental statements: the cogito as consciousness of something . . . each object meant indicates presumptively its system. The essential relatedness of the ego to a manifold of meant objects thus designates an essential structure of its entire and possible intentionality. (Husserl 1981: 79–80) On the one hand it has to do with cognitions as appearances, presentations, acts of consciousness in which this or that object is presented, is an act of consciousness, passively or actively. On the other hand, the phenomenology of cognition has to do with these objects as presenting themselves in this manner. (Husserl 1964: 10–12) 10

Thanks to Curtis Franks for help with the expression of this point.

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In the above quotes, both the ‘presenting objects’ and ‘the manner in which they present’ give cognition its essential structure. It seems that Husserl resists resolving the ambiguity in these phrases one way or the other. Husserl’s “phenomenology of cognition” is accomplished through a prior conceptual step called the ‘phenomenological reduction’. This ‘reduction’ is related to Descartes’ method of doubt (e.g. in Husserl 1964: 23. A useful elaboration can be found in Teiszen 2010: 80). Teiszen argues that for Husserl the crucial thing about the phenomenological reduction was what remains even after we attempt, in Cartesian fashion, to doubt everything. Teiszen makes the point that if we take a (certain, phenomenologically mediated) transcendental perspective, we can uncover in what remains (after Cartesian doubt) a lot more than an ‘I’ who is thinking. In particular, we can uncover direct apprehension of “the ideal objects of logic and mathematics” (Teiszen 2010: 9) whose pure forms extend far further than what Descartes ended up allowing as directly knowable, and further than the knowable allowed for in Kant’s philosophy. Just as there is with what to make of the ‘consciousness as consciousness of’ idea, so too there is much controversy surrounding exactly what the phenomenological reduction is and involves. To say that there is disagreement here among Husserl scholars is something of an understatement. Indeed: “there seem[s] to be as many phenomenologies as phenomenologists” (Hintikka 2010: 91). But the clarification of exactly what Husserl may have meant is not relevant to my purpose here, which is to see if there are ideas we can draw from Husserl that might help a realist philosopher of logic. I pause to note, though, that Teiszen’s interpretation of the reduction as a “‘suspension’ or ‘bracketing’ of the (natural) world and everything in it” (Teiszen 2010: 9) is standard; and the ‘ideal objects’ recovered in Teiszen’s consequent ‘transcendental idealism’ (including their ‘constituted mindindependence’) are also standard for an established tradition of Husserl scholarship (adhered to by Føllesdal, among others). But these ‘ideal’ objects are very far from the realist mind-independent realm that I want to imagine has a place here (to hammer this point home, see Teiszen 2010: 18). Again, it is the (possibly resolute) ambiguity in Husserl’s account that allows for my alternate reading of phenomenology. Another case in point: “the description on essential lines of the nature of consciousness . . . leads us back to the corresponding description

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of the object consciously known” (Husserl 1983: 359). The phrase: “the object consciously known” is ambiguous. It can be read differently depending on each term’s specific interpretation and on which terms are emphasized: e.g. the ‘consciously known’ can be read as ‘the object as we know it’ (i.e. a strictly constituted – internal – object); or as ‘the object that is known’. It is the latter interpretation that opens the possibility of an ‘external element’ in the basic ingredients of the nature of consciousness. To reiterate: the interesting thing about Husserl for my purpose is that in his ideas we can discern a (at least potential) role for an independent objective other, while nonetheless focusing on experience and consciousness: my thought is that if we can argue that intending reality as it appears (i.e. in the case of the realist conception of logic: as objective and independent) is itself constitutive of cognition and even of the possibility of cognition itself; then we can see a way in which objective independent reality is (complete with its attendant predicament) already there, structuring the essential nature of consciousness and experience. For me, the phenomenological reduction, or ‘ruling out’ of all that can be doubted, and the subsequent re-discovery of the world (ultimately) demonstrates an important way that reality, in all of the ways it seems to us to be (including being independent of us), in fact cannot be ruled out. Thus, we can see in the basic elements of the phenomenological analysis how objective, independent reality enters the picture as objective, and independent – not only as an object of consciousness, but as constituting consciousness itself. This is the case even if (or, as Husserl would have it, especially if ) we try to focus only on ‘pure experience’ or ‘pure consciousness’. I’ll mention a couple of other perspectives that gesture in a similar direction to my own before moving on. From Levinas we get: the fact that the in itself of the object can be represented and, in knowledge, seized, that is, in the end become subjective, would strictly speaking be problematic . . . This problem is resolved before hand with the idea of the intentionality of consciousness, since the presence of the subject to transcendent things is the very definition of consciousness. (1998: 114, italics mine) [and] the world is not only constituted but also constituting. The subject is no longer pure subject; the object no longer pure object. The phenomenon is at once revealed and what reveals, being and access to being. (1998: 118)

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Once we get our heads around the idea that the presence of the subject to transcendent things defines consciousness,11 it is not a huge leap to see how this initial subjective/transcendent relationship (even if it’s just one of mutual ‘presence’) can incorporate the entire problematic outlined above: i.e. that the Sellars–Meillassoux contradiction is ‘built in’ just so far as it describes that relationship. Recall that Husserl equates that problematic with the problem of the possibility of cognition (p. 16 above): it should now be apparent how his equation can be understood as a means by which to understand (rather than resolve or dissolve) the ‘natural’, ‘scientific’ perspective, complete with its consequent dilemma. That is, Husserl’s point: ‘The problem of the possibility of cognition is the traditional realist dilemma’

need not be interpreted thus: ‘the problem of the possibility of cognition supplants the traditional dilemma’. Rather, it may be interpreted thus: ‘the traditional dilemma defines (in some way or other) the problem of the possibility of cognition’. Hintikka is another who seems to suggest that the contradictory relationship between the subject and external reality is a part of Husserl’s (along with Aristotle’s) philosophy. He asks: Is . . . the object that we intend by means of a noema12 out there in the real “objective” world? Or must we . . . say that the object “inexists” in the act?

He then points out: Aristotle [and Husserl] would not have entertained such questions. For him [/them] in thinking (intending?) X, the form of X is fully actualised both in the external object and in the soul. If we express ourselves in the phenomenological jargon, this shows the sense in which the (formal) object of an act exists both in the reality and in the act. (2010: 96)

My own point is that this characterisation of the relationship (one I agree Husserl himself advocates) does not automatically eliminate or supplant the traditional, ‘natural’ characterisation of the relationship, and so nor does it eliminate the problem as it arises for that ‘natural’ characterisation. I suggest that the phenomenological perspective is best understood as a re-conceptualisation of the same relationship that is characterised and 11

12

Note that this need not go the other way: we can retain the phenomenological insight without the inverse claim that the object itself depends on, or even is (either necessarily or always) present to, consciousness. Husserl’s name for something akin to Fregean ‘sense’, but also apparently akin to (though more finegrained than) Fregean ‘reference’ (for some interesting details on these subtleties, see Haddock 2010).

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problematised in the natural attitude; and so as capable of engaging directly with its key concepts (rather than as wholly re-interpreting, removing, or supplanting those concepts).

4. Overflow I want now to discuss the idea of the “pregnant concept of evidence” (Husserl 1964: 46). Husserl says: If we say: this phenomenon of judgement underlies this or that phenomenon of imagination. This perceptual phenomenon contains this or that aspect, colour, content, etc., and even if, just for the sake of argument, we make these assertions in the most exact conformity with the givenness of the cogitation, then the logical forms which we employ, and which are reflected in the linguistic expressions themselves, already go beyond the mere cogitations. A “something more” is involved which does not at all consist of a mere agglomeration of new cogitationes. (1964: 40–1)

Elsewhere, he notes: The epistemological pregnant sense of self-evidence . . . gives to an intention, e.g., the intention of judgement, the absolute fullness of content, the fullness of the object itself. The object is not merely meant, but in the strictest sense given. (Husserl 1970: 765)

The point I want to draw attention to is that Husserl takes both logical and physical/perceptual ‘objects’ as the sort of thing that in one sense or another ‘overflow’, or ‘go beyond’ what is given to cogitation. The word ‘object’ must . . . be taken in a very broad sense. It denotes not only physical things, but also, as we have seen, animals, and likewise persons, events, actions, processes and changes, and sides, aspects and appearances of such entities. There are also abstract objects . . . (Føllesdal, in Føllesdal and Bell 1994: 135)

Bearing in mind that in the phenomenological reduction, access to abstract logical forms is not treated in any especially problematic way, all of what is given to experience can be explained in much the same fashion: “sensuous intuition means givenness of simple objects. Categorical intuition . . . means givenness of categorical formations, such as states of affairs, logical connectives, and essences” (Hartimo 2010b: 117). The structure underpinning logic – the form and structure of experience – is constituted and ‘given’ in experience. It is ‘seen’13 analogously to the way physical objects are seen by perception. 13

Or rather, ‘intuited’, where ‘intuition’ is used in the sense of “immediate or non-discursive knowledge” (Hintikka 2010: 94).

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So, the object of genuine perception and, by the extension I want to make here, genuine categorical intuition, overflows what is given to the act of perception or comprehension itself. For this reason it is capable of being veridical, and is opposed to Hyletic data, which is not.14 This is because genuine perception and intuition involve noema that are both conceptual and objectual.15 It is because each noema is objectual that our conceptual grasp can never fully contain the whole noema: i.e. that this grasp is always ‘pregnant’. Note that Husserl does not commit to there being two noemata for each act of perception or comprehension, but neither does he commit to the idea that the conceptual and the objectual are simply two aspects of the one noema.16 Rather, his claims regarding objectual (or, to anticipate what’s to come: ‘non-conceptual’) phenomena and conceptual phenomena are in tension with one another. In every noema, Husserl says: A fully dependable object is marked off . . . we acquire a definite system of predicates either form or material, determined in the positive form or left “indeterminate” – and these predicates in their modified conceptual sense determine the “content” of a core identity. (Husserl 1983: 364, italics mine)

It is within this ‘core identity’ we find that which gives the noema its ‘pregnant sense of self evidence’; that which makes what is ‘given to cognition’ overflow cognition and any (e.g. formal) ‘agglomeration of new cognitiones’. Other terms Husserl uses for this ‘core identity’ include: “the object”; “the objective unity”; “the self-same”; “the determinable 14

15

16

Shim (2005) nicely characterizes hyletic data as the ‘sensual stuff ’ of experience. He gives the following helpful example of the process of ‘precisification’ to contrast memory or fantasy with genuine perception: “In remembering the house I used to live in, I can precisify an image of a red house in my head. The shape, the color and other physical details of that house must be ‘filled in’ by hyletic data. Now let’s say I used to live in a blue house and not a red house. There is, however, no veridical import to the precisifications of my memory until confronted by the corrective perception . . . there is no sense in talking about the veridical import in the precisifications of [the memory or] fantasy” (pp. 219–220). In the latter cases, we may mistake merely hyletic data for nonconceptual (or objectual) phenomena (p. 220). An analogous situation might be said (by a logical realist) to occur for logical intuition when we encounter counter examples or engage directly with the meaning of logical operators – in these situations we can see a genuine role for veridical input capable of correcting or ‘precisifying’ our intuition. On the other hand, perhaps analogously to what occurs in a fantasy or hallucination, we may mistake the mere manipulation of symbols for genuine (veridical) comprehension. Shim gives a sophisticated argument for the idea that what provides perceptual noemata with ‘overflow’ is that they have both conceptual and non-conceptual content. My idea is similar, but, as will be elaborated shortly, the duality I want to consider should not be rendered as (noncontradictory) aspects of one and the same object, but rather as a contradictory object; whereas I think that Shim means the duality he proposes to be interpreted in the former sense. Thanks to Graham Priest for pressing this point.

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subject of its possible predicates”; “the pure X in abstraction from all predicates”; “the determinable which lies concealed in every nucleus and is consciously grasped as self-identical”; “the object pole of intention”; and, best of all: “that which the predicates are inconceivable without and yet distinguishable from”. This is conceptually located in a similar variety of ways, including as: “set alongside [the noema]”; “not separable from it”; “belonging to it”; “disconnected from it”; and “detached but not separable [from it]” (all quotations, 1983: 365–367). I simply note here that some of these characterisations are contradictory. What I hope to indicate, in what follows, is that this is as it should be. To review and sum up: The main points I get from Husserl are these: that independent abstract ‘reality’ is no more difficult to accommodate than is independent physical reality; that conceptualising logical structures as similar to platonic mathematical structures does not preclude conceptualising either as immediately apprehendable objects of cognition; and thus that the idea of independent reality as (genuinely, problematically) independent finds a place in phenomenology.

5. McDowell It is useful to compare what has so far been drawn from Husserl to a specific interpretation of McDowell. Neta and Pritchard in their (2007) article make a point that helps situate Husserl’s programme: they argue that one way to understand attempts (specifically McDowell’s, but their ideas extend to Husserl’s) to reach beyond our ‘inner’ world to an external realm is precisely by close examination of the assumptions we bring to the Cartesian evil genius thought experiment. The argument they present demonstrates links between a particular (perhaps ‘natural’) way of conceiving the distinction between ‘inner’ and ‘outer’, and the commonly held assumption that: (R): The only facts that S can know by reflection alone are facts that would also obtain in S’s recently envatted duplicate. (p. 383)

Neta and Pritchard argue that McDowell rejects R on the basis that there is something about our actual, embodied experience of the world that cannot be replicated by stimulus, no matter how sophisticated, experienced by a brain in a vat (compare this with Husserl’s differentiation between genuine ‘pregnant’ perception and hyletic/sensuous data). The clue as to how McDowell rejects (R) and to uncovering the similarities between his and

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Husserl’s approaches is in the concept ‘experience of the world’. For McDowell, experience of (the world) is experience as (humans in the world). The idea is that if indeed that is what we are talking about, then when we talk of ‘experience in the world’, we cannot, as it were, ‘slice off ’ the part that is us experiencing from the part that is being experienced. Neta and Pritchard outline McDowell’s position as follows: McDowell (1998a) allows . . . that one’s empirical reason for believing a certain external world proposition, p, might be that one sees that p is the case. Seeing that is factive, however, in that seeing that p entails p. However, McDowell also holds that such factive reasons can be nevertheless reflectively accessible to the agent – indeed, he demands . . . that they be accessible for they must be able to serve as the agent’s reasons. (p. 384, italics original)

Thus, for McDowell, ‘it is true that p’; or ‘it being so that p’, are internal to the knower’s ‘space of reasons’. But her ‘satisfactory standing’ in the space of reasons in which p is so, involves ‘seeing that p’, which entails p itself. McDowell’s ‘factive reasons’ are subtle things with clear similarities to Hintikka’s characterisation of the Aristotelian/Husserlian ‘object of an act’: they are knowable by reflection alone, but also entail objective ‘external’ states. I remember my then seven-year-old son once saying ‘I think the trees have faces’, and thinking that this is a nice way of explaining some of the ideas in McDowell’s Mind and World (1994), which I take as an attempt to argue that what is external and objectively so is nonetheless also accessible – available to us as conceptual content. But I think that the McDowellian/Husserlian sort of manoeuvre can only work if ‘what is experienced’ genuinely is the realist’s independent reality (at least as much as it is accessible content). To the extent that any account re-casts or re-defines that independence, it is hard to see how the specifically realist problem (which both McDowell and Husserl identify in the ‘natural attitude’) is the problem their accounts actually address. Put another way, if an account implicates the external in our human (reflective) experience simply by fiat (or by initial (re)design), then it becomes difficult to see how such an account can help us understand the problem that inspired it in the first place: i.e. the problem of the realist’s conception of independence as independence from human experience. McDowell’s and Husserl’s solution are of a kind, both answer the sceptic along the following general lines: you can’t take away reference to external reality (as in the sceptical scenario) just because what we experience has external reality somehow written into it. But if a position’s ‘inwritten’

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externality collapses into (even an interesting) aspect of what remains, strictly, internal, then that position offers no essential insight into the dichotomy and the problem with which we began.

6. Effectively defending ~R The important word in the preceding paragraph is “somehow”. Expanding on the ‘somehow’, we can find a sense in which neither McDowell nor Husserl escapes or resolves the traditional, ‘natural’ dilemma. Or rather, to the extent that they can be said to, their solutions do not address this original dilemma. Conversely, I want to suggest it is just to the extent that they don’t escape the dilemma that they may (via expansion on the ‘somehow’) be taken as having offered a sort of solution wherein what was unintelligible from the traditional/natural perspective, is made at least a little intelligible. That is, their sort of insight might be taken as offering a perspective from which the contradiction inherent in speaking of a reality independent of humans altogether need not automatically undermine the possibility of a relationship between the two. To see this, we need to start by outlining the ways in which both positions “clearly [challenge] the traditional epistemological picture that has (R) at its core”. Neta and Pritchard outline McDowell’s challenge to R this way: McDowell’s acceptance of reflectively accessible factive reasons . . . entails that the facts that one can know by reflection are not restricted to the “inner” in this way, and can instead, as it were, reach right out to the external world, to the “outer”. One has reflective access to facts that would not obtain of one’s recently envatted duplicate, on McDowell’s picture. If this is correct, it suggests that the popular epistemological distinction between “inner” and “outer” which derives from (R) should be rejected, or at least our understanding of it should be radically revised. (p. 386)

Not believing R is tantamount to taking a more sophisticated or more complex view of the original Cartesian experiment. To accept ~R, we need reasons to suppose that the thought experiment of ‘doubting everything’ is not simply or not only constructible along lines drawn from our ‘natural’ understanding of the ‘outer/inner’ distinction. Husserl offers the broad reason that consciousness per se is not possible – if we try to imagine such a thing, we find a sense in which independent reality got there before us: consciousness itself incorporates ‘potentialities’ that, in turn, cannot be reduced to wholly ‘subjective’ or ‘internal’ phenomena.

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Neta and Pritchard argue that the temptation to interpret McDowell either wholly internally or wholly externally rests on believing R. Believing R then, is very like holding fast to the possibility that in principle what is given to cognition and what cognition intends, can always be untangled. For Husserl, only a radically impoverished view of envattedness can deliver the sceptical conclusion: a closer, careful look at cognition “in general, apart from any existential assumptions either of the empirical ego or of a real world” (Husserl 1981: 60) returns the world in all of its “modes of givenness” (Husserl 1981: 59), as constituted and constituting that cognition. So I think it is reasonable to take Husserl similarly to McDowell on the question of envattedness: i.e. to take Husserl as committed to there being a difference between envatted and non-envatted states. But I want to take issue with Neta and Pritchard’s claim that: “Once (R) is rejected . . . these two aspects [internal and external] of the view are no longer in conflict” (Neta and Pritchard 2007: 38b). And, for the same reasons, I take issue with similar claims Husserl makes regarding phenomenology e.g.: “In . . . phenomenology . . . the old traditional ambiguous antitheses of the philosophical standpoint are resolved” (Husserl 1981: 34). A genuine resolution of the ‘traditional antithesis’ could come about only via an explicit defence of ~R in the original (‘traditional’) terms in which R itself was conceptualised. In short, a ‘resolution’ of the problem generated by the original dichotomy must directly address that dichotomy as a genuine dichotomy. There are various ways ~R and an alternative conceptualisation of the internal/external dichotomy might be defended, but only some of these ways can be said to address and so potentially resolve, the original realist dilemma. For example, ~R itself, or a set of key reasons offered to believe ~R, might be used as a sort of first principle, or established by fiat; then again, an approach might give a bunch of positive reasons or arguments for ~R (independent of the original reasons for R) in order to convince us that ~R (along with any attendant, independent positive reasons offered for ~R) ought to replace or provide an alternative perspective to the ‘traditional’ perspective. But neither of these cases can be said to resolve the original problem. They might be said to replace that problem, perhaps; or to render it irrelevant in the face of a potentially more compelling scenario, but not to resolve that problem. Any potential resolution would need to directly challenge the original ‘traditional antithesis’ itself, which cannot be done except by explicitly engaging with that antithesis on its own terms (for a more detailed defence of these ideas, see Rush 2005). That is, an explicit argument against

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R (accommodating the terms and spirit in which it was intended) has to defend one of the following claims: an internal phenomenon is also notinternal (i.e. that a phenomenon able to act as internal in the R thought experiment is also one able to act as not-internal in the thought experiment); an external phenomenon is also not-external (taking ‘internal’ phenomena as ‘not-external’); or, for each case, there is no straightforward either/or dichotomy (i.e. it is not the case that such phenomena are ‘either external or not-external’, or ‘internal or not-internal’). That is, in R, the concepts ‘external’ and ‘internal’ are explicitly (intended as) subject to both the law of excluded middle (LEM): ~A v A; and the law of non-contradiction (LNC): ~(~A & A).17 So accounts that rest on or incorporate ~R in some way must also directly challenge the applicability of these classical laws to the internal/ external dichotomy. One such challenge might argue that the point of ~R is that it gives us reason to doubt that the LEM should hold here. The relationship between the phenomenological and ‘natural’ perspectives might then be seen as analogous to the relationship between the intuitionist rendering of the continuum as viscous and the classical rendering of the continuum as discrete. From the intuitionist’s perspective, the continuum has characteristics it does not have from the classical perspective. To see the former, we need to allow the LEM to fail, in particular, for 8x8y((xy)). In much the same way, we could argue that to see the more complex characteristics of our human experience in the world, we need to allow the LEM to fail for 8x(Ix v ~Ix) (where I is ‘internal’) and/or for 8x (Ex v ~Ex) (where E is ‘external’). (For more on the intuitionists’ continuum, see Posy 2005, especially pp. 345–348.) Note that this means that the most effective defence of ~R challenges the universal applicability of the laws of (classical) logic. So knowledge of (external, independent) logical truths is guaranteed only by an explicit, rather drastic instance of the corrigibility of that knowledge. Thus, the knowledge of logic that survives the phenomenological reduction is corrigible knowledge – but this is perhaps what we should expect, given the independence of logical truth: its fundamental role in cognition does not and cannot guarantee the infallibility of our own intuition.

17

That is, I think arguments for the claim that Husserl’s and McDowell’s accounts do not ‘hypostatise’ ultimately fail (for examples of such arguments, see Hartimo 2010b, and Putnam 2003, particularly p. 178). Or, to the extent that they succeed, the accounts themselves are rendered largely irrelevant to the philosophical problem I am addressing here.

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Another such challenge could argue that in the case of independent, external and dependent, internal phenomena, we have an explicit exception to the LNC: there are occasions where each type of phenomenon both is and is not that type (for more on this idea see Rush 2005 and Priest 2009). Either way, these challenges undermine the notion that the LEM and/or the LNC apply to internal and external phenomena. My own opinion is that it makes more sense, for an account wishing to engage with the philosophical problem, to mount the latter challenge – i.e. to argue that the LNC does not apply here, (given that it could be argued that LEM defines the terms of the original thought experiment, R) – but the main point is that only an explicit argument against (or recognising an implicit rejection of ) either or both of these classical rules can make such accounts as Husserl’s and McDowell’s relevant to the original ‘natural’ problem. And I do think that Husserl was interested in addressing the original ‘natural’ problem,18 but in a particular way: one’s first awakening to the relatedness of the world to consciousness [i.e. the philosophical problem] gives no understanding of how the varied life of consciousness, . . . manages in its immanence that something which manifests itself can present itself as something existing in itself, and not only as something meant but as something authenticated in concordant experience. (1981: 28) [and] We will begin with a clarification of the true transcendental problem, which in the initial obscure unsteadiness of its sense makes one so very prone . . . to shunt it off to a side track. (1981: 27)

In Husserl’s account then, there is a duality (akin to McDowell’s) ‘within’ the constituted object itself, insofar as it is also ‘given’ as independent. That this duality is a genuine counterexample either to the 18

Shim, Teiszen, and others see the duality (which Shim renders as conceptual/non-conceptual) as residing strictly in the phenomenological attitude, and so Shim (2005) argues that the phenomenological ‘solution’ cannot neatly slot into a ‘natural’ answer to scepticism. But I think phenomenology is relevant to the natural answer to scepticism exactly insofar as it provides this explicit way of differentiating ‘being in the (real) world’ from ‘envattedness’. This differentiation disrupts a neat holistic story, and so its lesson, carried through to science and the natural attitude, is perhaps not a ‘categorical mistake’ (Shim 2005: 225), but an alert as to the deficiencies of a philosophy that disallows any perspective other than its own. What we know from the phenomenological attitude might resist reduction to naturalist/scientific knowledge, but it nonetheless can offer an insight into the items with which the scientific/philosophical attitude is concerned: e.g. reality, experience, and knowledge. It is exactly what makes the phenomenological perspective “both tempt and frustrate . . . the very philosophical desire it should have satisfied” (Shim 2005: 225), that can make it relevant to that ‘desire’, and can potentially stop a too quick, neat, sealed holist answer from gaining complete purchase.

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LEM or LNC (or both) is something Husserl seems at times to appreciate – recall the contradictions in his various accounts of the ‘location of pure x’ listed earlier. And it is just where it seems able to incorporate the rejection of the LNC for internal and external reality that phenomenology holds the most promise. On the other hand the preservation of the LNC in this case calls for resolution one way or the other and so renders an account open to being interpreted as wholly internal or wholly external, which I contend, would drastically impoverish it as an account of human experience. As it stands though, its own internal inconsistencies bear witness to the richness of the very idea of phenomenology: of the inescapable, paradoxical, yet entirely natural thought that our human experience is irreducibly constituted by the notion (itself inherently either incomplete or inconsistent) that we might know reality and logic as it independently is.19 19

Thanks to Graham Priest, Curtis Franks, Tuomas Tahko, Sandra Lapointe, and Jody Azzouni for helpful feedback on earlier drafts.

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chapter 2

A defense of logical conventionalism Jody Azzouni

1. Introduction Our logical practices, it seems, already exhibit “truth by convention.” A visible part of contemporary research in logic is the exploration of nonclassical logical systems. Such systems have stipulated mathematical properties, and many are studied deeply enough to see how mathematics – analysis in particular – and even (some) empirical science, is reconfigured within their nonclassical confines.1 What also contributes to the appearance of truth by convention with respect to logic is that it seems possible – although unlikely – that at some time in the future our current logic of choice will be replaced by one of these alternatives. If this happens, why shouldn’t the result be the dethroning of one set of logical conventions for another? One set of logical principles, it seems, is currently conventionally true; another set could be adopted later. Quine, nevertheless, is widely regarded as having refuted the possibility of logic being true by convention. Some see this refutation as the basis for his later widely publicized views about the empirical nature of logic. Logical principles being empirical, in turn, invites a further claim that logical principles are empirically true (or false) because they reflect well (or badly) aspects of the metaphysical structure of the world. Just as the truth or falsity of the ordinary empirical statement “There is a table in Miner Hall 221B at Tufts University on July 3, 2012,” reflects well or badly how a part of the world is, so too, the Principle of Bivalence is true or false because it reflects correctly (or badly) the world’s structure. I’ll describe this additional metaphysical claim – one that I’m not attributing to Quine (by the way) – as taking logical principles to have representational content. Most philosophers think logical principles being conventional is 1

The families of intuitionistic and paraconsistent logics are the most extensively studied in this respect. There is a massive literature in both these specialities.

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incompatible with those principles having representational content.2 I undermine the supposed opposition of these doctrines in what follows. That still leaves open the question whether logical principles do have representational content; but I also undermine this suggestion. That may seem a lot to do in under eight thousand words. Luckily for me (and for you too), most of the important work is already done, and I can cite it rather than have to build my entire case from scratch.

2. Quine’s dilemma It’s really really sad that almost no one notices that Quine’s refutation of the conventionality of logic is a dilemma. The famous Lewis Carroll infinite regress assails only one horn of this dilemma, the horn that presupposes that the infinitely many needed conventions are all explicit. Quine (1936b: 105) writes, indicating the other horn: It may still be held that the conventions [of logic] are observed from the start, and that logic and mathematics thereby become conventional. It may be held that we can adopt conventions through behavior, without first announcing them in words; and that we can return and formulate our conventions verbally afterwards, if we choose, when a full language is at our disposal. It may be held that the verbal formulation of conventions is no more a prerequisite of the adoption of conventions than the writing of a grammar is a prerequisite of speech; that explicit exposition of conventions is merely one of many important uses of a completed language. So conceived, the conventions no longer involve us in vicious regress. Inference from general conventions is no longer demanded initially, but remains to the subsequent sophisticated stage where we frame general statements of the conventions and show how various specific conventional truths, used all along, fit into the general conventions as thus formulated.

Quine agrees that this seems to describe our actual practices with many conventions, but he complains that (Quine 1936b: 105–106): it is not clear wherein an adoption of the conventions, antecedently to their formulation, consists; such behavior is difficult to distinguish from that in which conventions are disregarded . . . In dropping the attributes of deliberateness and explicitness from the notion of linguistic conventions we risk depriving the latter of any explanatory force and reducing it to an idle label. 2

Ted Sider, a contemporary proponent of the claim that logical idioms have representational content, represents the positions as opposed in just this way; he (Sider 2011: 97) diagnoses “the doctrine of logical conventionalism” as supporting the view that logical expressions “do not describe features of the world, but rather are mere conventional devices.”

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Jody Azzouni We may wonder what one adds to the bare statement that the truths of logic and mathematics are a priori, or to the still barer behavoristic statement that they are firmly accepted, when he characterizes them as true by convention in such a sense.

These challenges aren’t specifically directed against the conventionality of logic but against tacit “conventions” of any sort. One challenge is concerned with making sense of when specific behaviors are in accord with the proposed tacit conventions and when they’re not. One problem, that is, is this: if the conventions are explicit, we know what the conventions are – because they’ve been stated explicitly – and the behavior can be directly measured against them to determine deviations. But tacit conventions must be inferred from that very behavior, so the challenge goes, and therefore a lot of unprincipled play becomes possible because various conventions may be posited, these conventions differing in how far the practitioners’s behavior is taken to deviate from them. A second issue Quine raises is with the label “convention”; he wants to know what’s distinctive about tacit conventions that makes them stand apart from the simple “behavioristic” attribution that the population “firmly accepts them.” So Quine’s two objections come apart neatly. There is, first, a challenge to the idea that a set of rules can be attributed to a population in the absence of explicit indications like a set of official conventions. Even if this first challenge can be circumvented, the second worry is why the set of rules so attributed to a population should be called “conventions.”3 If we concede the requirement of explicitness to Quine, we’re forced to something like the Lewis account of convention:4 A regularity R, in action or in action and belief, is a convention in a population P if and only if, within P, the following six conditions hold: (1) Almost everyone conforms to R. (2) Almost everyone believes that the others conform to R. (3) This belief that the others conform to R gives almost everyone a good and decisive reason to conform to R himself. (4) There is a general preference for general conformity to R rather than slightly-less-than-general conformity – in particular, rather than conformity by all but anyone.

3 4

See (Quine 1970b) for a reiteration of the first challenge with respect to linguistic rules. See (Lewis 1969: 78) – but I draw this characterization from (Burge 1975: 32–33).

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(5) There is at least one alternative R 0 to R such that the belief that the others conformed to R 0 would give almost everyone a good and decisive practical or epistemic reason to conform to R 0 likewise; such that there is a general preference for general conformity to R 0 rather than slightly-less-than-general conformity to R 0 ; and such that there is normally no way of conforming to R and R 0 both. (6) (1)–(5) are matters of common knowledge. There are many problems with this approach – indeed, it’s no exaggeration to describe condition (6) as yielding the result that there are almost no conventions in any human population anywhere. But can Quine’s challenges be met? Are tacit conventions cogent?

3. Tacit conventions: Burge and Millikan. Suboptimality Since Quine’s challenges are directed towards tacit conventions of any sort, let’s look at what appears to be a less-complicated case: purportedly tacit linguistic conventions. Linguistic conventionality seem less complicated than logical conventionality if only because the intuitions that seem to accompany logical principles (ones about necessity, ones about aprioricity) aren’t present in the linguisitic case. As Burge (1975: 32) writes, “Language, we all agree, is conventional. By this we mean partly that some linguistic practices are arbitrary: except for historical accident, they could have been otherwise to roughly the same purpose.” He adds, “which linguistic and other social practices are arbitrary in this sense is a matter of dispute.” I’ll shortly show that this matters to the empirical question whether language is conventional (and in what ways) – the thing Burge tells us we all agree about. But first, notice something important that Burge is explicit about (although he doesn’t dwell on it): there are psychological mechanisms that enable these regularities. Burge (1975: 35) writes, “the stability of conventions is safeguarded not only by enlightened self-interest, but by inertia, superstition, and ignorance.” He makes this point rapidly, and in passing, because he’s instead intent on undercutting the explicitness assumption for conventions: “Insofar as these latter play a role, they prevent the arbitrariness of conventional practices from being represented in the beliefs and preferences of the participants.” Let’s focus on the important word “inertia.” This is an allusion to an – ultimately neurophysiological – mechanism of imitation. The point is made quite explicit some years later by Millikan when she characterizes

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“natural conventionality” in terms of patterns that are “reproduced.” Crucial to the idea (Millikan 1998: 2) is that “these [conventional] patterns proliferate . . . due partly to weight of precedent, rather than due, for example, to their intrinsically superior capacity to perform certain functions.” That is (Millikan 1998: 3), “had the model(s) been different . . . the copy would have differed accordingly.” Some may be worried about this characterization of conventional patterns.5 As I understand the characterization, for it to work we need to sharply distinguish between the patterns being conventional because they are proliferating partly due to the weight of precedent, and the patterns instead only being thought to be conventional because they’re thought to involve arbitrariness in our choice of a course of action. On the one hand, we can simply be wrong – thinking that arbitrariness is involved when it isn’t. On the other hand, there can be “arbitrariness” without our realizing it: there are other model-options we don’t know about, which, were they in place, would have been imitated instead. Consider the venerable practice of rubbing two sticks together to start a fire.6 A tribal population may simply fail to realize that banging rocks together will work instead. Their practice of rubbing sticks to start a fire is conventional despite their failing to realize this. Imagine, however, that they live where there are no such rocks, and where, presumably, there are available no other ways to start a fire. Then the practice isn’t conventional. Suppose (after many moons) the tribe migrates to an area where suitable rocks are located. Because of a change of location, a practice that wasn’t conventional has become conventional. (More generally, technological development can induce conventionality because it creates practical alternatives that weren’t there before.) There is a lot of work to do here (much of it empirical) detailing more fully the notion of “genuine practical alternatives” – what sort of background factors should be seen as relevant and which not – but the need for hard empirical work isn’t problematical for this characterization of tacit convention. Another worry. Many people believe (and some believe correctly) that some of their practices P are optimal. They engage (imitate) those practices (so they believe) precisely because they think these are optimal practices and not because of the weight of precedent. Conventional or not conventional? Well, beliefs about optimality aren’t relevant; only the 5

6

Epstein (2006), for example, is worried. My thanks to him for conversations (and email exchanges) about this topic that have influenced the rest of this section. I draw this example from (Epstein 2006: 4).

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efficacy of P ’s optimality to the spread of the practice through the population is relevant. Suppose alternative suboptimal practices would not have spread through the population, if instead they were the models, precisely because their suboptimality would have extinguished the practices (or the population engaging in them). Then P isn’t conventional. Otherwise it is. Superior optimality, of course, can be why a practice triumphs over alternatives. It’s an empirical question in what ways the optimality of a practice relates to its popularity, but I’m betting that superior optimality rarely counts for why a practice P spreads through a population.7 If a practice has enough optimality over other options to make its superior optimality efficacious in its spread, then it isn’t conventional. On the other hand, some superior optimality clearly isn’t enough to erase conventionality. Therefore: How much superior optimality is required to erase conventionality is an empirical question, turning in part on how much damage a suboptimal practice will inflict on its population, how fast this will happen, how fast this will be noticed, and so on. These empirical complications, although of interest, don’t make the notion of tacit convention problematical. One point in the previous paragraph must be stressed further because I seem to be definitively breaking with earlier philosophers on conventionality on just this point. This is that roughly equivalent optimality is invariably built into the characterization of conventionality: the alternative practices that render a practice conventional are ones that are reasonably equivalent in their optimality – this is built into Lewis’ approach by condition (3), that others conforming to such alternatives would give people “good and decisive” reasons for engaging in them as well – this is false if the alternative practices are suboptimal enough. It seems built into Millikan’s approach – at least when conventional patterns serve functions – because alternatives should serve functions “about as well” (Millikan 2005: 56). Unfortunately, as Keynes is rumored to have pointed out in a related context, in the long run we’re all dead. Anthropology reveals that seriously suboptimal practices are quite stable in human populations (and, to be 7

Is it conventional that we cook some of our food and don’t eat everything raw? I think it is. Is the alternative suboptimal? There is controversy about this, but I think it is: I think this is why the alternative eventually died out among our progenitors (after thousands of years, that is). On the other hand, some of the reasons for why the alternative died out (the greater likelihood of food poisoning, the inadvertent thriving of parasites in one’s meal, etc.) have been – presumably – eliminated by technical developments in food processing. So the practice of eating all food raw needn’t be as suboptimal as it once was.

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honest, a cold hard look at our own practices reveals exactly the same thing). Evolution takes a really long view of things – even the extinction of a population because it engages in a suboptimal practice may occur so slowly that the conventional fixation of that practice can occur for many generations, at least.8 How suboptimal a practice can be (in relation to alternatives) is completely empirical, of course, and turns very much on the details of the practices involved (and the background context they occur in); but optimality comparisons should play only a moderate role in an evaluation of what alternative candidates there are to a practice, and therefore in an evaluation of whether that practice is conventional and in what ways. (This will matter to the eventual discussion of the conventionality of logic: that alternative logics are suboptimal in various ways won’t bar them from playing a role in making conventional the logic we’ve adopted.) One last additional point about conventionality that I’ve just touched on in the last sentence. This is that it isn’t – so much – entire practices that are conventional, but aspects of them that are. “Minor” variations in a practice are always possible, minor variations that we don’t normally treat as rendering the practice conventional because we don’t normally treat those variations as rendering the practice a different one. There are many variations in how sticks can be rubbed together, for example. How we describe a practice or label it (how we individuate it) will invite our recognition of these variations as inducing conventionality or not. It’s conventional to rub two sticks together in such and such a way, but not conventional (say) to rub two sticks together instead of doing something else that doesn’t involve sticks at all (in a context, say, where there are no rocks). How we individuate “practices” correspondingly infects how and in what ways we recognize a practice to be conventional; but this is hardly an issue restricted to the notion of tacit convention, or a reason to think the notion has problems.

4. Empirical evidence for tacit conventions More than a serviceable notion of tacit convention is needed to respond to Quine. Recall his worry about evidence, that “in dropping the attributes of 8

A nice example, probably, is the arrangement of the lettered keys on computer keyboards. No doubt the contemporary distribution of letters is suboptimal compared to alternatives; it’s clearly an inertial result of the earlier arrangement of the keys on typewriters – which was probably also suboptimal in its time and relative to its context at that time. I’m not suggesting, of course, that keyboard conventions are contributing to a future extinction event – although I have no doubt that a number of conventions that we currently use are doing precisely that.

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deliberateness and explicitness from the notion of linguistic convention we risk depriving the latter of any explanatory force and reducing it to an idle label.” As it turns out, and this is an empirical discovery, for conventional patterns to even be possible for a human population requires neurophysiological capacities and tendencies in those humans. These are currently being intensively studied, and preliminary results reveal how human children have a capacity to imitate that’s largely not shared with other animals.9 The recently discovered mirror-neuron system is crucial to this capacity (but is hardly the whole story). My point in alluding to this empirical literature is to indicate how a systematic response to Quine’s challenges has emerged: Not only is a decent characterization of tacit conventionality – as noted above – now in place, but an explanation of the capacity for imitation that underwrites tacit conventions in this sense (and one that goes far beyond sheer behavioral facts about “firm acceptance”) is also emerging due to intensive scientific study.10 Of course, Millikan (and Burge) seem to largely assume that language is “conventional” in the appropriately tacit sense. But this (on their own views) should be an entirely empirical question – patently so now that the neurophysiological mechanisms of imitation are being discovered. It’s an empirical question, for one thing, whether these mechanisms (mirror neurons, etc.) are involved in language acquisition – more specifically, it’s an empirical question how they’re involved in language acquisition. Imagine (instead) that something like Chomsky’s principles and parameters model is at work in language acquisition.11 Then the picture is this: the child starts language-acquisition with a massive prefixed cognitive language-structure which is multiply triggered to a final state by specific things the child hears. Imagine (what’s surely false, but will make the principle of the point clear) that there are (say) only three thousand and seventeen human languages that are possible, so that the child has only to hear a relatively small number of specific utterances for that child’s 9

10

11

See the introduction to (Hurley and Chater 2005a&b) for an overview of work as of that date. See the various articles in the volumes for details. The first sentence of the introduction (Hurley and Chater 2005a: 1) begins, dramatically enough, with this sentence: “Imitation is often thought of as a low-level, cognitively undemanding, even childish form of behavior, but recent work across a variety of sciences argues that imitation is a rare ability that is fundamentally linked to characteristically human forms of intelligence, in particular to language, culture, and the ability to understand other minds.” It’s important to stress how recent these discoveries are – only within the last couple of decades. One almost shocking development is that the study of these mechanisms is successfully taking place at the neurophysiological level, and not at some more idealized (abstract) level – as is the case with most language studies to date, specifically those of syntax. See, e.g., (Chomsky and Lasnik 1995).

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“language-organ” to fixate on a final language. A mechanism like this, even if it helps itself to the neurophysiological imitation mechanisms to enable the child to imitate the initial triggers, may leave very little of actual language as conventional simply because the child’s final-state competence would leave practically nothing for the child to subsequently learn.12 To respond to Quine, notice, what’s needed are both subpersonal mechanisms that allow alternative imitations (on the part of a population) as well as feasible alternative practices made available by the contextual background a population of humans is in. Without appropriate subpersonal imitation mechanisms (as opposed to say, subpersonal mechanisms of the parameter/principles type), the apparent alternatives don’t render the current practice conventional – because members of the population are actually incapable of imitating those alternatives. But if the feasible alternative practices are absent from the contextual background then the practice is rendered nonconventional because of this alone.

5. Three theories of logical capacity I’ve just finished suggesting that the notion of tacit convention may (empirically) find almost no foothold in language, despite the appearance of massive contingency, because the mechanisms of imitation – crucial to tacit convention – may play only a minimal role in language acquisition.13 This is an empirical question, unresolved at the moment. But what about logic? Despite the subject matter of logic (in some sense) being so ancient, the actual principles of logic don’t become explicit until the very end of the nineteenth century. I now attempt to show that – possibly unlike 12

13

See (Chomsky 2003), specifically page 313. See (Millikan 2003), specifically pages 37–38. This empirical question is the nub of their disagreement, as Millikan realizes (Millikan 2003: 37): “If [the child’s language faculty] reaches a steady state, that will be only if it runs out of local conventions to learn.” I don’t find convincing Millikan’s arguments against the empirical possibility of a (virtually final) steady-state for the language faculty: They seem to turn only on the sheer impression that there’s always more language conventions for adults to acquire. But given that the empirical question is about what actual subpersonal mechanisms are involved in language acquisition and also in the use of the language by adults who have acquired a language, it’s hard to see why sheer impressions of conventionality deserve any weight at all. One can always introduce the appearance of massive official conventionality by individuating the language practices finely enough – e.g., minor sound-variations in the statistical norms of utterances determining the individuation of utterance practices (recall the last paragraph of Section 3); but I’m assuming this trivial vindication of the “conventionality” of language isn’t what either Burge or Millikan have in mind when they presume it as evident that natural language is full of conventionality.

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the case of language, and rather surprisingly – tacit convention has a genuine place in the characterization of logic. There are at least three (at times competing) historical characterizations of logic. The absence, until relatively recently, of explicit logical principles enables the insight that these models of logic are, strictly speaking, general theories of the basis of human logical capacities, and not a priori characterizations of what logic must be. The earliest model, arguably, is the substitution one. Syllogistic reasoning especially, but also contemporary reconstruals of logic in terms of schemata, invites the thought that logical principles require an antecedent segregation of logical idioms. Logical truths are then characterized as all the sentences generated by the systematic substitution of nonlogical vocabulary for nonlogical vocabulary within what can be characterized as a recursive set of logical schemata or argument forms. Such a characterization also allows the view that logical principles can be recognized by their general applicability to any subject area: logical principles are “formal,” as it’s sometimes put, or “topic neutral.”14 A second model is the content-containment one. Here a notion of “content” is hypothesized, and the central notion of logic – consequence (or implication) – is characterized in terms of content-containment: the content of an implication Im is contained in the content of the statements Im is an implication of. An intensional version of this model is clearly at work in Kant’s notion of analytic truth, and in notions of a number of earlier thinkers as well. An extremely popular contemporary version of the content-containment approach externalizes the notion of content of a statement – taking it to be the possible situations, models, or worlds in which a statement is true. A deductive (intensional) construal of “content” understands the content of a statement to be all its deductive consequences. Yet a third model emerged only in the middle of the last century: what I’ll call the rule-governed model of logical inference. This is that logical deduction is to be characterized in terms of a set of rules according to which logical proofs must be constructed. Part of the reason this model emerged so late for logic is that it required the extension of mathematical axiomatic methods to logic, something achieved definitively only by Frege.15 14 15

See (Sher 2001) for discussion and for citations of earlier proponents of this approach to logic. Although the axiomatic model anciently arose via Euclidean geometry, it’s striking that it wasn’t generally recognized – when Euclidean geometry was translated entirely to a language-based format – how gappy those rules were. An early view was that a nonethymematic mathematical proof was one without “missing steps” or gaps. But this view, based as it was on a picture of a

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A requirement, it might have been thought, is that any model of logic must be adequate to mathematical proof. For mathematical proof – right from its beginning – exhibited puzzling epistemic properties: We seemed to know that the conclusion of a mathematical proof had to be true if the premises were: this was one important ground of the impression of the “necessity” of mathematical results. This phenomenon seemed to demand a logical construal, at least in terms of one of the underlying models of logic I’ve just given: content-preservation. Substitution criteria seem irrelevant to mathematical proof, and so did explicit rules, since the practice of mathematical proof – apart from isolated occurrences until the twentieth century – occurred largely in the absence of explicit rules but instead in terms of the perceived semantic connections between specialized (explicitly designed) mathematical concepts.16

6. A case for the conventionality of logic Let’s grant the suggestion that what logic is has finally stabilized (as of the middle of the last century). The standard view is that an advantage of firstorder logic over alternative logics is that all three models of logic can be applied to it – and arguably, all three models converge as equivalent in the first-order context. The equivalence of the rule-governed model and the substitution model is established by the existence of equivalent characterizations of first-order logical truths in terms either of sentence-axioms or in terms of axiom-schemata. The equivalence of these characterizations in turn with the content-containment model is enabled by Gödel’s completeness theorem, subject to the model-theoretic characterization of the content-containment model via models (in a background set theory). This sophisticated theoretical package of first-order classical logic isn’t reflected in the psychological capacities of the humans who adhere (collectively) to this model of reasoning. In saying this, I’m not alluding to the rich and developing literature on human irrationality17; I’m pointing out, rather, that as we become more sophisticated in our study of the neurophysiological

16

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conceptual relationship between the steps in a mathematical proof, remained purely metaphorical (or, at best, promissory) until the notion of algorithm in the context of artificial languages emerged at the hands of Turing, Church, and others in the twentieth century. See (Azzouni 2005: 18–19). It should be noted that this dramatic aspect of informal-rigorous mathematical proof is still with us despite the presence of formal systems that are apparently fully adequate to contemporary mathematics. That is, informal-rigorous mathematical proof continues to operate largely by conceptual implication – supplemented, of course, with substantial computational bits. Nicely popularized by one of the major researchers in the area: See (Kahneman 2011).

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basis for our capacities for mathematics, and for reasoning generally, there is no echo in our neuropsychological capacities to reason, and to prove, of the semantic/syntactic apparatus the contemporary view of logic (and even its competitors) provides.18 That apparatus is an all-purpose topic-neutral piece of algorithmic machinery; how we actually reason, by contrast, involves quite topic-specific, narrowly applied, highly componentalized, mental tools. This means that the role of formal logic can only be a normative one; it has emerged as a reasoning tool that we officially impose on our ordinary reasoning practices and that we (at times) can use to evaluate that reasoning.19 The foregoing, if right, makes the conventionality of logic quite plausible even if it’s an optimal logic, compared to competitors.20 The foregoing, if right, also makes plausible the emergence of classical logic as explicitly conventional in the twentieth century; and it makes plausible its role as tacitly conventional (at least in mathematical reasoning) for earlier centuries – before sets of rules for logic became explicit. I turn now to discussing some of the reasons philosophers have for denying logic such a conventional status. The first kind of objection I’ll consider turns on how the notion of truth is used in the characterization of validity; next I’ll evaluate certain arguments that have been offered for why logical principles have a (metaphysical) representational role.

7. Criticisms of the truth-preservation characterization of logic We philosophers are all pretty familiar with the apparent truism – the apparent explanatory cliché, the apparently essential characterization – of 18

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See (Carey 2009), especially chapter 4 – also see (Dehaene 1997) – for good introductions to this remarkable and important empirical literature. I’ve argued that this role of formal logic has emerged in the course of the twentieth century; it first occurred in mathematics but has spread throughout the sciences in large part because of the mathematization of those sciences. See (Azzouni 2013), chapter 9, as well as (Azzouni 2005) and (Azzouni 2008a) for discussion. I should stress that there are several psychological and historical contingencies that seem involved in why the tacit employment of logical consequence in mathematical practice turned out to be in the neighborhood of a first-order and classical one: one of those, I suggest (Azzouni 2005), is the psychological impression (on a case-by-case basis) that the introduction and elimination rules for the logical idioms (“and,” “or,” “not,” etc.) are contentpreserving, an impression that isn’t sustained for even quite short inference patterns, such as modus ponens or syllogism. Some philosophers argue that classical first-order logic isn’t optimal because of its representational drawbacks: proponents of higher-order logics (e.g., Shapiro), on the one hand, think that it can’t represent mathematical concepts such as “finite,” proponents of one or another paraconsistent approach (e.g., Priest) think it can’t represent certain global concepts, e.g., “all sentences.” Although I’ve weighed in on these debates, they don’t matter for the issue of whether logic is conventional precisely because it’s been established in Section 3 of this chapter that suboptimality in relation to competitors doesn’t bar a practice from nevertheless being an alternative candidate.

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deductive reasoning preserving truth: If the premises are true then the conclusion is true (must be true) as well. Many philosophers have taken truth-preservation to be a characterization of classical logical principles. If the notion of truth, in turn, is a correspondence notion, then it would seem to follow that classical logic is semantically rooted in metaphysics, in what’s true about the world. And, it might be thought that what follows from this is that logic cannot be conventional. This argument-strategy fails for a large number of reasons; for current purposes, I’ll focus on only three of its failures. The first is that a characterization of deduction as “truthpreserving” fails to single out any particular set of logical rules – it fails to even require that a set of logical principles be consistent! The second is that, in any case, even if a characterization of logic in terms of truthpreservation singled out only classical logic (and not its alternatives), that wouldn’t rule out the conventionality of classical logic: suboptimality of alternatives is no bar to their rendering a practice conventional. The last reason is that truth, in any case, is too frail an idiom to root logic semantically in the world. This is because it functions perfectly adequately in discourses that bear no relationship to what exists. The first claim is easy to prove. Relevant is that the truth idiom is governed by Tarski biconditionals: given a sentence S and a name of that sentence “S,” “S ” is true iff S. Also relevant is that this condition can’t be supplemented by adding conditions to either wing of the Tarski biconditionals that aren’t equivalent to the wings themselves.21 But these points are sufficient to make the truth idiom logically promiscuous: it’s compatible with any logical principles whatsoever. Let R be any set of logical principles. And supplement R with the following inference schema T: S ‘ T“S ”, and T“S ” ‘ S. If the original set of rules is syntactically consistent (as, e.g., Prior’s “tonk” isn’t), then so is the supplemented version. That R is “truthpreserving” follows trivially, regardless of whether R is consistent or not: If U ‘ V according to R, then, using T, we can show: U ‘ V iff T“U ” ‘ T“V ” holds in [R, T]. Notice that a characterization of a choice of logic being “legislatedtrue” is licensed by the foregoing: Start by choosing one’s logic, and then supplement that choice with the T-schema. The resulting logic has been “legislated-true.” It might be thought that more substantial uses of the truth idiom, in semantics and in model theory, can’t be executed in the context of a nonclassical logic. But this isn’t true either. In particular,

21

See (Azzouni 2010), 4.8 and 4.9.

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a model theory – characterized metalogically using intuitionistic connectives in the metalanguage – is homophonic to classical model theory.22 The second point has already been established: Imagine (contrary to what has just been shown) that a population adopts a suboptimal set of logical principles – ones strictly weaker than ours. (Intuitionist principles, for example.) Then one possible result would be a failure to know all sorts of things, both empirically and in pure mathematics, that we proponents of classical logic know. Let’s say that this is suboptimal;23 but this is hardly fatal. And so the conventionality of logic isn’t threatened by the presumed suboptimality of other candidates. Lastly, a number of philosophers have thought that the Tarski biconditionals all by themselves characterize “truth” as a correspondence notion. There are many reasons to think they are wrong about this. Among them is the fact that if a consistent practice of using nonreferring terms, such as “Hercules” or “Mickey Mouse” is established, such a practice remains consistent if it’s augmented with the T schema. Regardless of whether the truth idiom functions as a correspondence notion for certain discourses, it won’t function that way in this discourse. That shows that talk of truth has to be supplemented somehow to give it metaphysical traction. All by itself, it doesn’t do that job. The point generalizes, of course. In trying to determine whether logic is conventional, some philosophers focus on specific statements like “Either it is raining or it is not raining,” and worry about whether this statement is about the world or not; more dramatically, some philosophers worry about whether the supposed conventionality of logic yields the result that we “legislate” the truth of a statement like this.24 But this misses the point. The claim that logic “is conventional” is orthogonal to the question of whether “logical truths” have content (worldly or otherwise), or (equivalently?) whether they are or aren’t “about the world.” No doubt some philosophers have thought these claims linked – especially philosophers (like the paradigmatic “positivists” influenced by Wittgensteinian Tractarian views) who are driven by epistemic 22 23

24

See (Azzouni 2008b), especially sections V and VI. Two issues drive my choice of the qualification-phrase: “let’s say.” First, mathematical possibilities are richer in the intuitionist context than they are in the classical context – that could easily count against the supposed suboptimality of intuitionistic mathematics. Second, there are a lot of results that show that the apparent restrictions of intuitionist mathematics – and constructivist mathematics, more generally – in applied mathematics can be circumvented. See (Sider 2011: 203–204).

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motives to deprive logical principles of “content.” The issue, to repeat, isn’t whether particular logical truths are or aren’t about the world, but instead whether our current set of logical principles lives in a space of viable candidate alternatives. In addition, the claim that logical principles (or truth) are about the world isn’t to be established by ruling out such worldly content on the part of statements like “Either it is raining or it is not raining,” but by ruling in such worldly content on the part of statements like “Either unicorns have one horn or unicorns don’t have one horn.” I’d like to close out this section with a couple of remarks about the curious project of trying to find individual representational contents for logical idioms, such as disjunction, conjunction, and so on.25 One extremely natural way to try going about this is to give such notions content on an individual basis via introduction and elimination rules. We then understand the content of “and” (“&”) to be characterized by the rules, for all sentences U and V: U & V ‘ U, U & V ‘ V, U, V ‘ U & V, and so on (familiarily) for the other idioms. An evident danger with this approach is that the holistic nature of “logical content” emerges clearly when it’s recognized, for example, that intuitionistic logic can be characterized by exactly the same introduction and elimination rules, with the one exception of negation. That logical truths not involving negation are nevertheless affected is an easy theorem.26 We can instead attempt to capture the individualized contents of the connectives “semantically,” via truth tables for example. The problem here is that truth tables are simply descriptions of truth conditions in neatly tabular form: e.g., A or B is true iff (A is true and B is not true) or (A is false and B is true) or (A is true and B is true). As noted earlier in this section, such an approach simply amounts to a characterization of logical principles (in a metalanguage) using those very same logical principles plus the T-schema. The holism problem therefore is still with us. The appearance that we are semantically characterizing logical idioms on an individual basis, that is, is still the same illusion that we experience when we approach the project directly by attempting to characterize the content of logical idioms individually, using natural deduction principles (for example). 25

26

Although the discussion is murky (or perhaps just metaphorical), this seems to be part of the project undertaken by (Sider 2011), when he speaks of “joint-carving logical notions,” e.g., on page 97. See (Kleene 1971), for lots of explicitly indicated examples.

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8. How Walker and Sider beg the question against logical conventionalism Much of the argument I’ve offered here has involved technical details that have been deliberately kept off-stage. That was a necessity because technical details in a paper restricted to eight thousand words must largely be kept in the background for purely spatial reasons: to describe these technical details in even terse self-contained detail would expand the paper greatly – e.g., details about the role of the truth idiom in metalanguages when characterizing a set of logical principles, or details about how the consequence relation is holistically affected by how individual logical idioms synergistically interact. But an important warning is in order. Discussion of these issues – specifically, the issue of the conventionality of logic – often takes place at an informal level that masks the fact that relevant technical points are being overlooked. I’ll close with an illustration. Sider (2011: 104) argues against the idea that logical principles can be legislated-true, that in particular, the statement “Either it is raining or it is not raining,” can be legislated-true. Here is the argument: (i) (ii) (iii) (iii)

I cannot legislate-true ‘It is raining’ I cannot legislate-true ‘It is not raining’ If I cannot legislate-true j, nor can I legislate-true ψ, then I cannot legislate-true the disjunction ┌ j or ψ ┐. is obviously the key premise. Sider writes (2011: 104),

In defense of iii): a disjunction states simply that one or the other of its disjuncts holds; to legislate-true a disjunction one would need to legislatetrue one of its disjuncts. . . . It is open, of course, for the defender of truth by convention to supply a notion of legislating-true on which the argument’s premises are false. The challenge, though, is that the premises seem correct given an intuitive understanding of “legislate-true.” One of the oldest (but still quite popular) ways of begging the question against proponents of alternative logics (as well as a popular way of begging the question against logical conventionalism) is to implicitly adopt a lofty metalanguage stance, and then use the very words that are under contention against the opponent. That doing this is so “intuitive” evidently contributes to the continued popularity of the fallacy. Some readers may be tempted to deny that this is a fallacy. They may want to speak as Walker (1999: 20) does: Anyone who refuses to rely on modus ponens, or on the law of noncontradiction, cannot be argued with. If they insist on their refusal there

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Jody Azzouni is therefore nothing to be done about it, but for the same reason there is no need to take them seriously.27

But this argument is just awful. Even if an opponent refuses to rely on modus ponens as a law of logic, this doesn’t mean that opponent won’t be able engage in a debate using specific inferences that fall under classical modus ponens. This is because all it means to deny modus ponens as a logical principle is to claim that it has exceptions. That can nevertheless leave enormous common ground for debate – that is, for arguments that both debaters take to be sound.28 Even if the reader who has gotten this far in the paper isn’t (or isn’t fully) convinced by the details of the intricate philosophical argument on offer (both onstage and off ), I can at least hope the following take-away message is convincing: This is that the issue of whether or not logic is “conventional” is a subtle and intricate (and interesting) philosophical question that can’t be successfully adjudicated by merely superficially rehearsing Quine’s old arguments against “truth by convention,” and supplementing that rehearsal with a semantic argument for the representational content of logic that blatantly presupposes the very logical idioms under dispute. Also pertinent (or so I would have thought) is a discussion of the philosophical literature on tacit conventionality that has emerged subsequent to Quine, including the relevant empirical results. I also think (and have tried to illustrate) that needed as well is a moderately deep discussion of whether and in what ways the attribution of “representational” contents to logical idioms does or doesn’t contradict the supposed conventionality of logic. 27

28

See (Lewis 2005), where a similar refusal on similar grounds to debate the law of non-contradiction is expressed; see (van Inwagen 1981) for the same maneuver directed towards substitutional quantification. As I said: it’s a popular maneuver with many illustrious practitioners. Metalogical debates, in particular, are ones where proponents can easily debate one another on common ground, as many clearly do in the philosophical literature. See (Azzouni and Armour-Garb 2005) for details.

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chapter 3

Pluralism, relativism, and objectivity Stewart Shapiro

I have been arguing of late for a kind of relativism or pluralism concerning logic (e.g., Shapiro 2014). The main thesis is that there are different logics for different mathematical structures or, to put it otherwise, there is nothing illegitimate about structures that invoke non-classical logics, and are rendered inconsistent if excluded middle is imposed. The purpose of this chapter is to explore the consequences of this view concerning a core metaphysical issue concerning logic, the extent to which logic is objective. In the philosophical literature, terms like “relativism” and “pluralism” are used in a variety of ways, and at least some of the discussion and debate on the issues appears to be bogged down because the participants do not use the terms the same way. One group of philosophers uses the word “relativism” for what another group calls “contextualism”. So, in order to avoid getting lost in cross-purposes, we need a brief preliminary concerning terminology. The central sense of “relativism” about a given subject matter Φ is given by what Crispin Wright (2008) calls folk-relativism. The slogan is: “There is no such thing as simply being Φ”. If Φ is relative, in this sense, then in order to get a truth-value for a statement in the form “a is Φ”, one must implicitly or explicitly indicate something else. A major discovery of the early twentieth century is that simultaneity and length are relative, in this sense. To get a truth-value for “a is simultaneous with b”, one needs to indicate a frame of reference. Arguably, so-called predicates of personal taste, such as “tasty” and “fun” are also folk-relative, at least in some uses. To get a truth-value for “p is tasty”, one must indicate a judge, a taster, a standard, or something like that. This folk notion of “relativism” seems to be the one treated in Chris Swoyer’s 2003 article in the Stanford Internet Encyclopedia of Philosophy. Swoyer suggests that discussions of relativism, and relativistic proposals, focus on instances of a “general relativistic schema”: ðGRSÞ Y is relative to X : 49

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In other words, in order to formulate a relativistic proposal, one first specifies what one is talking about, the “dependent variable” Y, and then what that is alleged to be relative to, the “independent variable” X. So, according to special relativity, the dependent variable is for simultaneity and other temporal or geometric notions like “occurs before”, and phrases like “has the same length as”. The independent variable is for a reference frame. For predicates of personal taste, the independent variable is for a given taste notion and the dependent variable is for a judge or a standard (depending on the details of the proposal). The main thesis of Beall and Restall (2006) is an instance of folkrelativism concerning logical validity. They begin with what they call the “Generalised Tarski Thesis” (p. 29): An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion.

For Beall and Restall, the variable x ranges over types of “cases”. Classical logic results from the Generalized Tarski Thesis if “cases” are Tarskian models; intuitionistic logic results if “cases” are constructions, or stages in constructions (i.e., nodes in Kripke structures); and various relevant and paraconsistent logics result if “cases” are situations. So Beall and Restall take logical consequence to be relative to a kind of case, and the General Relativistic Schema is apt. For them, the law of excluded middle is valid relative to Tarskian models, invalid relative to construction stages (Kripke models). Beall and Restall call their view “pluralism”, eschewing the term “relativism”: we are not relativists about logical consequence, or about logic as such. We do not take logical consequence to be relative to languages, communities of inquiry, contexts, or anything else. (p. 88, emphasis in original)

It seems that Beall and Restall take “relativism” about a given subject matter to be a restriction of what we here call “folk relativism” to those cases in which the “independent variable” ranges over languages, communities of inquiry, or contexts (or something like one of those). Of course, those are the sorts of things that debates concerning, say, morality, knowledge, and modality typically turn on. Here, we do not put any restrictions on the sort of variable that the “independent variable” can range over. However, there is no need to dispute terminology. To keep things as clear as possible, I will usually refer to “folk-relativism” in the present, quasi-technical sense.1 1

John A. Burgess (2010) also attributes a kind of (folk) relativism to Beall and Restall: “For pluralism to be true, one logic must be determinately preferable to another for one clear purpose while determinately inferior to it for another. If so, why then isn’t the notion of consequence simply

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I propose below, and elsewhere, a particular kind of folk-relativism for logic. The dependent variable Y is for validity or logical consequence, and the independent variable X ranges over mathematical theories or, equivalently, structures or types of structures. The claim is that different theories/ structures have different logics. Once it is agreed that a given word or phrase is relative, in the foregoing, folk sense, then one might want a detailed semantic account that explains this. Are we going to be contextualists, saying that the content of the term shifts in different contexts? Or some sort of full-blown assessment-sensitive relativist (aka MacFarlane (2005), (2009), (2014))? Questions of meaning, our present focus, thus come to the fore, and will be broached below. But, as construed here, folk-relativism, by itself, has no ramifications concerning semantics. Briefly, pluralism about a given subject, such as truth, logic, ethics, or etiquette, is the view that different accounts of the subject are equally correct, or equally good, or equally legitimate, or perhaps even (equally) true (if that makes sense). Arguably, folk-relativism, as the term is used here, usually gives rise to a variety of pluralism, as that term is used here. All we need is that some instances of the “independent variable” in the (GRS) correspond to correct, or good, versions of the dependent variable. Define monism or logical monism to be the opposite of logical relativism/pluralism. The monist holds that there is such a thing as simply being valid – full stop. The slogan of the monist is that there is One True Logic.

1. Relativity to structure Since the end of the nineteenth century, there has been a trend in mathematics that any consistent axiomatization characterizes a structure, one at least potentially worthy of mathematical study. A key element in the development of that trend was the publication of David Hilbert’s Grundlagen der Geometrie (1899). In that book, Hilbert provided (relative) consistency proofs for his axiomatization, as well as a number of independence proofs, showing that various combinations of axioms are consistent. In a brief, but much-studied correspondence, Gottlob Frege claimed that there is no need to worry about the purpose relative” (p. 521). Burgess adds, “[p]erhaps pluralism is relativism but relativism of such a harmless kind that to use that word to promote it would dramatise the position too much.” The present label “folk-relativism” is similarly meant to cut down on dramatic effect.

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consistency of the axioms of geometry, since the axioms are all true (presumably of space).2 Hilbert replied: As long as I have been thinking, writing and lecturing on these things, I have been saying the exact reverse: if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by them exist. This is for me the criterion of truth and existence.

The slogan, then, is that consistency implies existence. It seems clear, at least by now, that this Hilbertian approach applies, at least approximately, to much of mathematics, if not all of it. Consistency, or some mathematical explication thereof, like satisfiability in set theory, is the only formal criterion for legitimacy – for existence if you will. Of course, one can legitimately dismiss a proposed area of mathematical study as uninteresting, or unfruitful, or inelegant, but if it is consistent, or satisfiable, then there is no further metaphysical, formal, or mathematical hoop the proposed theory must jump through before being legitimate mathematics. But what of consistency? The crucial observation is that consistency is a matter of logic. In a sense, consistency is (folk) relative to logic: a given theory may be consistent with respect to one logic, and inconsistent with respect to another. There are a number of interesting and, I think, fruitful theories that invoke intuitionistic logic, and are rendered inconsistent if excluded middle is added. I’ll briefly present one such here, smooth infinitesimal analysis, a sub-theory of its richer cousin, Kock–Lawvere’s synthetic differential geometry (see, for example, John Bell 1998). This is a fascinating theory of infinitesimals, but very different from the standard Robinsonstyle non-standard analysis (which makes heavy use of classical logic). Smooth infinitesimal analysis is also very different from intuitionistic analysis, both in the mathematics and in the philosophical underpinnings. In the spirit of the Hilbertian perspective, Bell presents the theory axiomatically, albeit informally. Begin with the axioms for a field, and consider the collection of “nilsquares”, numbers n such that n2 = 0. Of course, in both classical and intuitionistic analysis, it is easy to show that 0 is the only nilsquare: if n2 = 0, then n = 0. But not here. Among the new axioms to be added, the most interesting is the principle 2

The correspondence is published in Frege (1976) and translated in Frege (1980). The passage here is in a letter from Hilbert to Frege, dated December 29, 1899.

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of micro-affineness, that every function is linear on the nilsquares. Its interesting consequence is this: Let f be a function and x a number. Then there is a unique number d such that for any nilsquare α, f (x þ α) = f x þ d α.

This number d is the derivative of f at x. As Bell (1998) puts it, the nilsquares constitute an infinitesimal region that can have an orientation, but is too short to be bent.3 It follows from the principle of micro-affineness that 0 is not the only nilsquare: :ð8αÞðα2 ¼ 0 ! α ¼ 0Þ:

Otherwise, the value d would not be unique, for any function. Recall, however, that in any field, every element distinct from zero has a multiplicative inverse. It is easy to see that a nilsquare cannot have a multiplicative inverse, and so no nilsquare is distinct from zero. In other words, there are no nilsquares other than 0: ! " ð8αÞ α2 ¼ 0 ! ::ðα ¼ 0Þ , which is just " ! ð8αÞ α2 ¼ 0 ! :ðα 6¼ 0Þ :

So, to repeat, zero is not the only nilsquare and no nilsquare is distinct from zero. Of course, all of this would lead to a contradiction if we also had (8x)(x = 0_x 6¼ 0), and so smooth infinitesimal analysis is inconsistent with classical logic. Indeed, :(8x)(x = 0_x 6¼ 0) is a theorem of the theory (but, since the logic is intuitionist, it does not follow that (9x):(x = 0_x 6¼ 0)). Smooth infinitesimal analysis is an elegant theory of infinitesimals, showing that at least some of the prejudice against them can be traced to the use of classical logic – Robinson’s non-standard analysis notwithstanding. Bell shows how smooth infinitesimal analysis captures a number of intuitions about continuity, many of which are violated in the classical theory of the reals (and also in non-standard analysis). Some of these intuitions have been articulated, and maintained throughout the history of philosophy and science, but have been dropped in the main contemporary account of continuity, due to Cantor and Dedekind. To take one 3

It follows from the principle of micro-affineness that every function is differentiable everywhere on its domain, and that the derivative is itself differentiable, etc. The slogan is that all functions are smooth. It is perhaps misleading to call the nilsquares a region or an interval, as they have no length.

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simple example, a number of historical mathematicians and philosophers followed Aristotle in holding that a continuous substance, such as a line segment, cannot be divided cleanly into two parts, with nothing created or left over. Continua have a sort of unity, or stickiness, or viscosity. This intuition is maintained in smooth infinitesimal analysis (and also in intuitionistic analysis), but not, of course, in classical analysis, which views a continuous substance as a set of points, which can be divided, cleanly, anywhere. Smooth infinitesimal analysis is an interesting field with the look and feel of mathematics. It has attracted the attention of mainstream mathematicians, people whose credentials cannot be questioned. One would think that those folks would recognize their subject when they see it. The theory also seems to be useful in articulating and developing at least some conceptions of the continuum. So one would think smooth infinitesimal analysis should count as mathematics, despite its reliance on intuitionistic logic (see also Hellman 2006). One reaction to this is to maintain monism, but to insist that intuitionistic logic, or something even weaker, is the One True Logic. Classical theories can be accommodated by adding excluded middle as a (non-logical axiom) when it is needed or wanted. The viability of this would depend on there being no theories that invoke a logic different from those two. Admittedly, I know of no examples that are as compelling (at least to me) as the ones that invoke intuitionistic logic. For example, I do not know of any interesting mathematical theories that are consistent with a quantum logic, but become inconsistent if the distributive principle is added. Nevertheless, it does not seem wise to legislate for future generations, telling them what logic they must use, at least not without a compelling argument that only such and such a logic gives rise to legitimate structures. One hard lesson we have learned from history is that it is dangerous to try to provide a priori, armchair arguments concerning what the future of science and mathematics must be. If a set Γ of sentences entails a contradiction in classical, or intuitionistic, logic, then for every sentence Ψ, Γ entails Ψ. In other words, in classical and intuitionistic logic, any inconsistent theory is trivial. A logic is called paraconsistent if it does not sanction the ill-named inference of ex falso quodlibet. Typical relevance logics are paraconsistent, but there are paraconsistent logics that fail the strictures of relevance. The main observation here is that with paraconsistent logics, there are inconsistent, but nontrivial theories.

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If we are to countenance paraconsistent logics, then perhaps we should change the Hilbertian slogan from “consistency implies existence” to something like “non-triviality implies existence”. To transpose the themes, on this view, non-triviality is the only formal criterion for mathematical legitimacy. One might dismiss a proposed area of mathematical study as uninteresting, or unfruitful, or inelegant, but if it is non-trivial, then there is no further metaphysical, formal, or mathematical hoop the proposed theory must jump through. To carry this a small step further, a trivial theory can be dismissed on the pragmatic ground that it is uninteresting and unfruitful (and, indeed, trivial). So the liberal Hilbertian, who countenances paraconsistent logics, might hold that there are no criteria for mathematical legitimacy. There is no metaphysical, formal, or mathematical hoop that a proposed theory must jump through. There are only pragmatic criteria of interest and usefulness. So are there any interesting and/or fruitful inconsistent mathematical theories, invoking paraconsistent logics of course? There is indeed an industry of developing and studying such theories.4 It is claimed that such theories may even have applications, perhaps in computer science and psychology. I will not comment here on the viability of this project, nor on how interesting and fruitful the systems may be, nor on their supposed applications. I do wonder, however, what sort of argument one might give to dismiss them out of hand, in advance of seeing what sort of fruit they may bear. The issues are complex (see Shapiro 2014). For the purposes of this chapter, I propose to simply adopt a Hilbertian perspective – either the original version where consistency is the only formal, mathematical requirement on legitimate theories, or the liberal orientation where there are no formal requirements on legitimacy at all. And let us assume that at least some non-classical theories are legitimate, without specifying which ones those are. I propose to explore the ramifications for what I take to be a longstanding intuition that logic is objective. One would think logic has to be objective, if anything is, since just about any attempt to get at the world, as it is, will depend on, and invoke, logic.

2. What is objectivity? Intuitively, a stretch of discourse is objective if the propositions (or sentences) in it are true or false independent of human judgment, 4

See, for example, da Costa (1974), Mortensen (1995), (2010), Priest (2006), Brady (2006), Berto (2007), and the papers in Batens et al. (2000). Weber (2009) is an overview of the enterprise.

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preferences, and the like. Many of the folk-relative predicates are characteristic of paradigm cases of non-objective discourses. Whether something is tasty, it seems, depends on the judge or standard in play at the time. So taste is not objective (or so it seems). Whether something is rude depends on the ambient location, culture, or the like. So etiquette is folk-relative and, it seems, not objective. Etiquette may not be subjective, in the sense that it is not a matter of what an individual thinks, feels, or judges, but, presumably, it is not objective either. It is not independent of human judgment, preferences, and the like. One would be inclined to think that simultaneity and length are objective, even though both are folk-relative, given relativity. As is the case with much in philosophy (and everywhere else), it depends on what one means by “objective”. We are told that whether two events are simultaneous, and whether two rods are of the same length, depends on the perspective of the observer. Does that undermine at least some of the objectivity? But, vagueness and such aside, time and length do not seem to depend on anyone’s judgment or feelings, or preferences. A given observer can be wrong about whether events are simultaneous, even for events relative to her own reference frame. One might say that a folk-relative predicate P is objective if, for each value n of the independent variable, the predicate P-relative-to-n does not depend on anyone’s judgment or feelings. For example, if a given subject can be wrong about P-relative-to-n, then the relevant predicate is objective. However, even an established member of a given community can be wrong about what is rude in that community. But one would not think that etiquette is objective, even when restricted to a given community. Clearly, to get any further on our issue, we do have to better articulate what objectivity is, at least for present purposes. Again, objectivity is tied to independence from human judgment, preferences, and the like. There is a trend to think of objectivity in straightforward metaphysical terms. It must be admitted that this has something going for it. The idea is that something, say a concept, is objective if it is part of the fabric of reality. The metaphor is that the concept cuts nature at its joints, it is fundamental. Theodore Sider (2011) provides a detailed articulation of a view like this, but the details do not matter much here. Presumably, taste and etiquette are not fundamental; tastiness and rudeness do not cut nature at its joints (whatever that means). Does logic, or, in particular, logical validity cut nature at its joints? It is hard to say, without getting beyond the metaphor. What are the “joints” of reality? Does it have such “joints”? How does logic track them?

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One might argue that there can be at most one logic that is objective, in this metaphysical sense. Sider does argue that at least parts of logic are fundamental. As it happens, the logic he discusses is classical, but, so far as I know, there is no argument supporting that choice of logic. It might be compatible with his overall program that, say, parts of intuitionistic logic or a relevant logic are fundamental instead. But perhaps two distinct logics cannot both be fundamental. Contraposing, if the present folk-relativism about logic is correct, then logic is not objective, in the foregoing metaphysical sense. For what it is worth, I would not like to tie objectivity to such deep metaphysical matters as Sider-style fundamentality. First, things that are not so fundamental can still be objective. Intuitively, the fact that the Miami Heat won the NBA title in 2012 is objective (like it or not), but (I presume) it is hardly fundamental. One can call a proposition objective if it somehow supervenes on fundamental matters, but that requires one to accept a contentious metaphysical framework, and to articulate the relevant notion of supervenience. More important, perhaps, several competing philosophical traditions have it that there simply is no way to sharply separate the “human” and the “world” contributions to our theorizing. Protagoras supposedly said that man is the measure of all things. On some versions of idealism, not to mention some postmodern views, the world itself has a human character. The world itself is shaped by our judgments, observations, etc. Perhaps such views are too extreme to take seriously. A less extreme position is Kant’s doctrine that the ding an sich is inaccessible to human inquiry. We approach the world through our own categories, concepts, and intuitions. We cannot get beyond those, to the world as it is, independently of said categories, concepts, and intuitions. On the contemporary scene, a widely held view, championed by W. V. O. Quine, Hilary Putnam, Donald Davidson, and John Burgess, has it that, to use a crude phrase, there simply is no God’s eye view to be had, no perspective from which we can compare our theories of the world to the world itself, to figure out which are the “human” parts of our successful theories and which are the “world” parts (see, for example, Burgess and Rosen 1997). On such views, the world, of course, is not of our making, but any way we have of describing the world is in human terms. As Friedrich Waismann once put it: What rebels in us . . . is the feeling that the fact is there objectively no matter in which way we render it. I perceive something that exists and put

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Stewart Shapiro it into words. From this, it seems to follow that something exists independent of, and prior to language; language merely serves the end of communication. What we are liable to overlook here is the way we see a fact – i.e., what we emphasize and what we disregard – is our work. (Waismann 1945: 146)

This Kant–Quine orientation may suggest that there simply is no objectivity to be had, or at least no objectivity that we can detect. Perhaps objectivity is a flawed property, going the way of phlogiston and caloric, or witchcraft. If this is right, then there simply is no answering the question of this paper – folk-relativism or no folk-relativism. Logic is not objective, since nothing is. Despite having sympathy with the Kant – Quine orientation, I would resist this rather pessimistic conclusion. There may not be such a thing as complete objectivity – whatever that would be – but it still seems that there is an interesting and important notion of objectivity to be clarified and deployed. There seems to be an important difference – a difference in kind – between statements like “the atmosphere contains nitrogen” and statements like “the Yankees are disgusting”. The distinction may be vague and even context dependent, but it is still a distinction, and, I think, an important one. Our question concerns whether the present folk-relative logic falls on one side or the other of this divide (or perhaps on or near its borderline). Crispin Wright’s Truth and objectivity (1992) contains an account of objectivity that is more comprehensive than any other that I know of, providing a wealth of detailed insight into the underlying concepts. Wright does not approach the matter through metaphysical inquiry into the fabric of reality, wondering whether the world contains things like moral properties, funniness, or numbers. He focuses instead on the nature of various discourses, and the role that these play in our overall intellectual and social lives. As Wright sees things, objectivity is not a univocal notion. There are different notions or axes of objectivity, and a given chunk of discourse can exhibit some of these and not others. The axes are labeled “epistemic constraint”, “cognitive command”, “the Euthyphro contrast”, and “the width of cosmological role”. In a previous paper, (Shapiro 2000), I argue that logic easily passes all of the tests. The conclusion is (or was) that, on each of the axes, either logic is objective (if anything is) or matters of logic, such as validity and consistency, lie outside the very framework of objectivity and non-objectivity, since most of the tests presuppose logic. That is, to figure out whether a given stretch of discourse is objective, on this or that axis, one must do some logical reasoning or figure out what is

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consistent, or the like. So it is hard to even apply the framework to matters of logic. The main target of Shapiro (2000) was Michael Resnik’s (1996), (1997) non-cognitivist account of logical consequence, a sort of Blackburn (1984)-style projectivism, which would make logic non-objective at least on the intuitive conception of objectivity. According to Resnik, to call an argument valid, or to call a theory consistent, is to manifest a certain attitude toward the theory.5 The present relativism/pluralism was not on the agenda then. The plan here is to return to the matter of objectivity with the present folkrelativism concerning logic in focus. Sometimes we will concentrate on general logical matters, such as validity and consistency, as such, and sometimes we will deal with particular instances of the folk-relativism, such as classical validity, intuitionistic consistency, and the like. We will limit the discussion to Wright’s axes of epistemic constraint and cognitive command.

3. Epistemic constraint Epistemic constraint is an articulation of Michael Dummett’s (1991a) notion of anti-realism. According to one of Wright’s formulations, a discourse is epistemically constrained if, for each sentence P in the discourse, P $ P may be known: ðp: 75Þ

In other words, a discourse exhibits epistemic constraint if it contains no unknowable truths.6 It seems to follow from the very meaning of the word “objective” that if epistemic constraint fails for a given area of discourse – if there are propositions in that area whose truth cannot become known – then that discourse can only have a realist, objective interpretation. As Wright puts it: To conceive that our understanding of statements in a certain discourse is fixed . . . by assigning them conditions of potentially evidence-transcendent 5

6

It is perhaps ironic (or at least interesting) that Resnik argues against pluralism and relativism about logic. He claims that there “ought to be” but one logic; the logic he favors is classical. Actually, if the background logic is intuitionistic, there is a difference between the absence of unknowable truths and the truth of the biconditional: P $ P may be known. That difference does seem to bear on Wright’s argument that if epistemic constraint fails – in the sense that there are, or could be, unknowable propositions in that area – then the discourse is objective, but we will not pursue this further here.

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Stewart Shapiro truth is to grant that, if the world co-operates, the truth or falsity of any such statement may be settled beyond our ken. So . . . we are forced to recognise a distinction between the kind of state of affairs which makes such a statement acceptable, in light of whatever standards inform our practice of the discourse to which it belongs, and what makes it actually true. The truth of such a statement is bestowed on it independently of any standard we do or can apply . . . Realism in Dummett’s sense is thus one way of laying the essential groundwork for the idea that our thought aspires to reflect a reality whose character is entirely independent of us and our cognitive operations. (p. 4)

In other words, if epistemic constraint fails for a given discourse, then it is objective, and that is the end of the story. The other axes of objectivity – cognitive command, cosmological role, and the Euthyphro contrast – are then irrelevant; they do not track a sense of objectivity (if the axis can be applied at all). Or so Wright argues. So what of logic? Are there, or could there be, unknowable truths concerning logical consequence, consistency, and the like? The present folk-relativism concerning logic pushes that question in a certain direction. Consider a given argument, or argument form Δ, and let P be a statement that Δ is valid. Could something like P be an unknowable truth? Not as it stands, but that is because, absent context, P is not a truth (or a falsehood) at all. According to the present folk-relativism, in order to get a truth-value for P, one must specify something else, such as a particular mathematical theory, a structure, or perhaps just a logic. We have to ask separately whether Δ is valid in classical logic, in intuitionistic logic, in various relevant logics, etc. So it seems to me that in order to ask whether logic is epistemically constrained, we have to consider statements of validity and the like with the logic made explicit. We must consider statements in the form, Δ is valid in logic L, where L is one of the logics that can go in for the dependent variable in the general relativistic scheme. To push the analogy, consider, again, relativity. Let p and q be two events. Say that p is a runner in baseball leaving third base, and q is an outfielder catching a fly ball. Consider the statement S that p occurred before q (which an umpire sometimes has to adjudicate). According to relativity, we cannot get a truth-value for S without specifying a frame of reference. So, a fortiori, we cannot even ask if there is an unknowable truth for a statement about what happened before what without indicating a reference frame. If a reference frame is specified (implicitly or explicitly), then, it seems, there can be unknowable truths in this area. For example, it

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may be unknowable whether the runner left base before the ball was caught, from the perspective of the home plate umpire. For example, it may be too dark or no human can see distinctions that fine, or whatever. So matters of temporal order, from a given reference frame, are not epistemically constrained. And, intuitively, matters of temporal order are objective, vagueness aside. The point here is that with folk-relative discourses, we can only ask about epistemic constraint for statements that have the relevant parameters fully specified, at least implicitly. So the central question is whether there can be unknowable truths concerning whether the argument (form) Δ is valid in a given logic L? In effect, this matter was dealt with in my earlier paper (Shapiro 2000), and also in Shapiro (2007), which concerns mathematics. Classical logic was in focus then, but to some extent, the argument generalizes. Whether there are unknowable truths in this area depends on what one means by “unknowable”. If we do not idealize on the knowers, then of course there can be unknowable truths. Suppose that our argument Δ is an instance of &-elimination in which the premise and conclusion each have, say, 10100 characters. Then Δ is valid in, say, classical logic, but no one can know that, since no one can live long enough to check that Δ is an instance of &-elimination. So, to give epistemic constraint a chance of being fulfilled, we have to idealize on the knowers. One sort of idealization is familiar. We assume that our knowing subjects have unlimited (but still finite) time, attention span, and materials at their disposal, and that they do not make any simple computation errors. These idealizations are common throughout mathematics, and we take them to be conceptually unproblematic (and thus we set aside issues concerning rule-following, as in, say Kripke (1982)). Then, if L is classical first-order logic, or intuitionistic logic, or most of the relevant logics, and Δ is an arbitrary argument form (with finitely many premises and conclusions), then a statement that Δ is valid in logic L is true if and only if that fact is knowable (by one of our ideal agents). That is because each of those logics has an effective and complete deductive system. Things are not so clear if the logic in question is classical second-order logic, since its consequence relation is not effectively enumerable. Nor are things so clear for statements that a given argument Δ is not valid in one of the aforementioned logics. Invalidity is not recursively enumerable, and so checking invalidity is not a matter of running an algorithm. So if we are to insist that all matters of logic are epistemically constrained, once the logic

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is fixed, we have to attribute to our knowers the abilities to decide membership in non-recursive sets. Things get vexed here. It is not at all clear what the relevant modality is for the key phrase “knowable”. Moreover, as noted, the issues are essentially the same as with monism concerning logic. So I propose to just take it as given, for the sake of argument, that the relevant discourse is epistemically constrained, in at least some relevant sense, so that we can move on to another of Wright’s axes of objectivity.

4. Cognitive command Assume that a given area of discourse serves to describe mind-independent features of a mind-independent world. In other words, assume that the discourse in question is objective, in an intuitive, or pre-theoretic sense. Suppose now that two people disagree about something in that area. It follows that at least one of them has misrepresented reality, and so something went wrong in his or her appraisal of the matter. Suppose, for example, that two people are arguing whether there are seven, as opposed to eight, spruce trees in a given yard. Assuming that there is no vagueness concerning what counts as a spruce tree and no indeterminacy concerning the boundaries of the yard, or whether each tree is in the yard or not, it follows that at least one of the disputants has made a mistake: either she did not look carefully enough, her eyesight was faulty, she did not know what a spruce tree is, she misidentified a tree, she counted wrong, or something else along those lines. The very fact that there is a disagreement suggests that one of the disputants has what may be called a cognitive shortcoming (even if it is not always easy to figure out which one of them it is that has the cognitive shortcoming). In contrast, two people can disagree over the cuteness of a given baby or the humor in a given story without either of them having a cognitive shortcoming. One of them may have a warped or otherwise faulty sense of taste or humor, or perhaps no sense of taste or humor, but there need be nothing wrong with his cognitive faculties. He can perceive, reason, and count as well as anybody. The present axis of objectivity turns on this distinction, on whether there can be blameless disagreement: A discourse exhibits Cognitive Command if and only if it is a priori that differences of opinion arising within it can be satisfactorily explained only in terms of “divergent input”, that is, the disputants working on the basis of

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different information (and hence guilty of ignorance or error . . .), or “unsuitable conditions” (resulting in inattention or distraction and so in inferential error, or oversight of data, and so on), or “malfunction” (for example, prejudicial assessment of data . . . or dogma, or failings in other categories . . . (Wright 1992: 92)

Intuitively, cognitive command holds for discourse about spruce trees (vagueness and indeterminacy aside) and it fails for discourse about the cuteness of babies and the humor of stories. Later in the book, Wright (1992: 144) adds some qualifications to the formulation of cognitive command, meant to deal with matters like vagueness. A discourse exerts cognitive command if and only if It is a priori that differences of opinion formulated within the discourse, unless excusable as a result of vagueness in a disputed statement, or in the standards of acceptability, or variation in personal evidence thresholds, so to speak, will involve something which may properly be described as a cognitive shortcoming.

So what of logic, again assuming the correctness of the foregoing folkrelativism? Let Δ be a given argument form, and consider two folks who disagree – or seem to disagree – whether Δ is valid. One says it is and the other says it is not. Our question breaks into two, depending on whether we fix the logic. For our first type of case, let Δ be an instance of excluded middle or double-negation elimination, and consider the “dispute” between advocates of classical logic and advocates of intuitionistic logic. The inference is valid in classical logic, invalid in intuitionistic logic. For the other sort of case, we fix the logic and ponder disputes concerning that logic. We imagine two folks who disagree – or seem to disagree – whether Δ is valid in L, where L is, say, a particular relevant logic. We start with the second sort of case, disagreements that concern a fixed logic. I would think that there is room for blameless disagreement concerning how a given argument, formulated in natural language, should be rendered in a formal language. However, such issues would take us too far afield, broaching matters of the determinacy of meaning, the slippage between logical terms and their natural language counterparts, and the intentions of the arguer. It is not so clear whether a disagreement in how to render a natural language argument is “excusable as a result of vagueness . . . or in the standards of acceptability, or variation in personal evidence thresholds” or the like. So let us set such matters aside, and just assume that our target argument Δ is fully formalized. One of our characters says that Δ is valid

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in the given logic L and the other says that Δ is invalid in that logic L. Do we know (a priori) that at least one of them has a cognitive shortcoming? Suppose that the logic L is either defined in terms of a deductive system or that there is a completeness theorem for it. So L can be classical firstorder logic, intuitionistic logic, or one of the various relevant and paraconsistent logics that are given axiomatically. So our disputants differ on whether there is a deduction whose undischarged premises are among the premises of Δ and whose last line is the conclusion of Δ. So, up to Church’s thesis, our disputants differ over a Σ1-sentence in arithmetic, one in the form (9n)Φ, where Φ is a recursive predicate. So our question concerning cognitive command for this logic L reduces to whether cognitive command holds for these simple arithmetic sentences. I would think that cognitive command does hold here. One of our disputants has made (what amounts to) a simple arithmetic error, and that surely counts as a cognitive shortcoming. But I will rest content with the reduction. Cognitive command holds in this case if and only if it holds for 91-sentences (or, equivalently, Π1-sentences). Now suppose that our fixed logic L is not complete. Say it is secondorder logic, with standard, model-theoretic semantics. In that case, the question at hand reduces to set theory. Suppose, for example, that our target argument Δ has no premises and that its conclusion is, in effect, the continuum hypothesis (see Shapiro 1991: 105). So Δ is valid if and only if the continuum hypothesis is true. So, in effect, our disputants differ over the truth of the continuum hypothesis. Is that dispute cognitively blameworthy? Surely, that would take us too far afield (but see Shapiro 2000, 2007, 2011, 2012), and we will leave this case with the reduction. Let us briefly consider the analogues of our question concerning cognitive command with our other examples of folk-relative predicates. Suppose that two judges differ on whether a certain event a occurred before another event b from the same frame of reference (putting aside the fact that this discourse is not epistemically constrained). Assume, for example, that the two judges are in the same reference frame. Then, unless the disagreement is “excusable as a result of vagueness . . . or in the standards of acceptability, or variation in personal evidence thresholds”, at least one of them exhibits a cognitive shortcoming. She did not look carefully enough, or did not time the events properly, or forgot something. So cognitive command holds, and, of course, matters of temporal order from a fixed frame of reference are intuitively objective. The same goes for matters of length.

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Now consider two folks who disagree over whether a given food is tasty for one and the same subject. Suppose, for example, that Tom and Dick differ over whether licorice is tasty to Harry. To keep things simple, assume that neither Tom nor Dick is Harry. Tom’s and Dick’s judgments would presumably be based on what Harry has told them and their observations of his reactions when eating licorice. We should assume that Tom and Dick have exactly the same body of such evidence (since otherwise one of them has the cognitive shortcoming of lacking relevant evidence). And we should set aside matters of “vagueness . . . standards of acceptability, [and] variation in personal evidence thresholds”. Tom and Dick may have come to opposite conclusions because they weighed certain pronouncements or reactions differently. In this case, perhaps, neither of them has a cognitive shortcoming – each is cognitively blameless. If so, cognitive command fails.7 I take it that talk about taste in general – concerning what is tasty (simpliciter) – is a paradigm of a non-objective discourse, but I am not sure whether discourse about Harry’s taste is objective, intuitively speaking. Maybe we have a borderline case. Returning to matters logical, I’ve saved the hardest sort of situation for last. That is on prima facie disagreements when the logic is not held fixed. To focus on a specific example, let Δ be an instance of the law of excluded middle (with no premises). Let h be a classicist who says that Δ is valid and let b be an intuitionist who insists that Δ is not valid. Is this a disagreement that is (cognitively) blameless? If so, then cognitive command fails here, and this aspect of logic falls on the non-objective side of this particular axis (assuming that cognitive command tracks a sort of objectivity). According to the foregoing folk-relativism, h and b are both right. Each has spoken a truth, and so presumably there is nothing to fault either of them. So each is (cognitively) blameless, at least concerning this particular matter. The only question remaining is whether they disagree. Here we encounter a matter that is treated extensively in the philosophical literature, and I must report that the issues are particularly vexed. There does not seem to be much in the way of consensus as to what makes for a disagreement. John MacFarlane (2014), for example, articulates several 7

A referee for Shapiro (2012) suggested that the failure of cognitive command does not distinguish cases which are not objective from those in which evidence is scant. The situation sketched above, with Tom and Dick, is not that different (in the relevant respect) from cases in science where available evidence must be evaluated holistically – say in cosmology. Two scientists might both be in reflective equilibrium, having assigned slightly different weights to various pieces of evidence. Cognitive command might fail there, too, despite science being a paradigm case of objectivity.

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different, and competing senses of “disagreement”. We will keep things at a more intuitive level, as far as possible. One thesis, perhaps, is that a necessary condition on disagreement is that the parties in question cannot both be correct. If so, and continuing to assume our folk-relativism concerning logic, we have that h and b do not disagree. A fortiori, we do not have a case of blameless disagreement. We can still maintain that cognitive command holds when the logic is held fixed, as above, and so logic passes this test for objectivity. The thesis that in a disagreement both parties cannot be correct is controversial. It is sometimes taken as a criterion of being non-objective that parties can disagree and both be correct (see, for example, Barker 2013). Suppose that Harry announces that licorice is tasty, and Jill responds, “no it is not; licorice is disgusting”. That looks like a disagreement; Jill uses the language of disagreement, apparently denying Harry’s assertion. And yet, one might say, both are correct. Or at least one might say that both are correct. To make any progress here, we have to get beyond the loose characterization of folk-relativism and address matters of semantics. I’ll briefly sketch the framework proposed by John MacFarlane (2005), (2009), (2014) for interpreting expressions in a folk-relative discourse. The terms used by other philosophers and linguists can usually be translated into this framework, though sometimes with a bit of loss. Indexical contextualism about a given term is the view that the content expressed by the term is different in different contexts of use. The clearest instances are the so-called “pure indexicals”, words like “I” and “now”. The content expressed by the sentence “I am hungry”, when uttered by me on a given day, is different from the content expressed by the same sentence, uttered by my wife at the same time. Intuitively, the first one says that I am hungry (then) and the second says that she is hungry (then). Clearly, these are different propositions; they don’t say the same thing about the world – not to mention that one might be true and the other false. Although very little is without controversy in this branch of philosophy of language, words like “enemy”, “left”, “right”, “ready”, and “local” seem apt for indexical contextualist treatments.8 Suppose, for example, that Jill, sitting at a table says that the salt is on the left while, at the same time, Jack, who is sitting opposite her, says that the salt is not on the left (since it is on the right). Intuitively, Jack and Jill do not disagree with each other, and the propositions they express are not contradictories. The reason is 8

Of course, this is not to say that these terms are like the standard indexicals in every manner.

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that the content of the word “left” is different in the two contexts. In the first, it means something like “to the left from Jill’s perspective” and in the second it means “to the left from Jack’s perspective”. And they can both be correct – intuitively one of them is correct just in case the other is. Non-indexical contextualism, about a given term, is the view that its content does not vary from one context of use to another, but the extension can so vary according to a parameter determined by the context of utterance.9 Suppose, for example, that a graduate student sincerely says that a local roller coaster is fun, and her Professor replies “No, that roller coaster is not fun, it is lame”. According to a non-indexical contextualism about “fun” (and “lame”), each of them utters a proposition that is the contradictory of that uttered by the other – so they genuinely disagree. Yet, assuming both are accurately reporting their own tastes, each has uttered a truth, in his or her own context. For the graduate student, at the time, the roller coaster is fun, since it is fun-for-the-graduate-student. For the professor, the roller coaster is not fun, since it is not fun-for-the-professor. Indeed, it is lame-for-the-professor. Finally, assessment-sensitive relativism, sometimes called “relativism proper”, about a term agrees with the non-indexical contextualist that the content of the term does not vary from one context of use to another, and so, in the above scenario, the relativist holds that the graduate student and the professor each express a proposition contradictory to one expressed by the other. However, for the assessment-sensitive relativist, the term gets its extension from a context of assessment. Suppose, for example, that a third person, a Dean, overhears the exchange between the graduate student and professor and, assume that the roller coaster is not fun-for-the-Dean. Then, from the context of the Dean’s assessment, the student uttered a false proposition and the professor uttered a true one. And, from the graduate student’s context of assessment, the Professor uttered a false proposition, and from the Professor’s context of assessment, the student uttered a false proposition. According to MacFarlane, the difference between non-indexical contextualism and assessment-sensitive relativism is made manifest by the phenomenon of retraction. That difference does not matter here, and we can lump non-indexical contextualism and assessment-sensitive 9

Nearly all terms have different extensions in different possible worlds. That is not the sort of contextual variation envisioned here. For terms subject to non-indexical contextualism, the relevant contextual parameter is for a judge, a time, a place, etc.

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relativism together. If we go with a contextualist treatment of the dispute between our logicians h and b, then they do not disagree, and so cognitive command is saved. If we opt for a non-indexical contextualist or an assessment-sensitive interpretation, we do have a disagreement – in the sense that each of them accepts a content that is the contradictory of that accepted by the other. As above, the disagreement is blameless (since both are correct), and so cognitive command fails. Recall that h says that our (fully formalized) argument Δ is valid and b says that Δ is not valid. Recall that Δ is an instance of excluded middle Φ_:Φ, with no premises. There are two places to look here, but both deliver the same range of verdicts. We can ask first about the content of the argument Δ. Do h and b mean the same thing by the disjunction “_” and by negation “:”? We thus broach the longstanding question of whether the classicist and the intuitionist (or, indeed, advocates of any rival logics) are talking past each other. Michael Dummett (1991a: 17) argues that the “disagreement” is merely verbal: The intuitionists held, and continue to hold, that certain methods of reasoning actually employed by classical mathematicians in proving theorems are invalid: the premisses do not justify the conclusion. The immediate effect of a challenge to fundamental accustomed modes of reasoning is perplexity: on what basis can we argue the matter, if we are not in agreement about what constitutes a valid argument? In any case how can such a basic principle of rational thought be rationally put in doubt? The affront to which the challenge gives rise is quickly allayed by a resolve to take no notice. The challenger must mean something different by the logical constants; so he is not really challenging the laws that we have always accepted and may therefore continue to accept.

Dummett goes on to argue that the classicist has no coherent meaning he can assign to the connectives, but we can set that aside here (as inconsistent with the foregoing folk-relativism). From a very different perspective, W. V. O. Quine (1986: 81) also holds that the various connectives change their content in the different logical theories. Concerning the debate over paraconsistent logics, he wrote: My view of this dialogue is that neither party knows what he is talking about. They think they are talking about negation, “!”, “not”; but surely the notation ceased to be recognizable as negation when they took to regarding some conjunctions in the form “p.!p” as true, and stopped

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regarding such sentences as implying all others. Here, evidently, is the deviant logician’s predicament: when he tries to deny the doctrine he only changes the subject.

And Rudolf Carnap (1934: §17): In logic, there are no morals. Everyone is at liberty to build his own logic, i.e. his own form of language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments.

Again, the key idea is that each logic is tied to a specific language. Presumably, the meaning of the logical terms differs in the different languages. So the Dummett–Quine–Carnap perspective has it that we have a kind of indexical contextualism here. The logical terms themselves have different contents for our characters h and b. Using a subscript-C to indicate a classical connective and a subscript-I for the corresponding intuitionistic connective, we have that h holds that Φ_C:CΦ is valid, while b holds that Φ_I:IΦ is invalid. This is the same sort of situation as with Jack and Jill and the salt. There is no disagreement between h and b unless it be over whether the other has a coherent meaning at all. If they are sufficiently open-minded, h and b might agree that Φ_C:CΦ is valid and that Φ_I:IΦ is invalid. So we do not have a failure of cognitive command. The Dummett–Quine–Carnap perspective is not shared by all. Beall and Restall (2006), for example, insist that their “pluralism” concerns the notion of validity for a single language, with a single batch of logical terms. So there is not, for example, a separate “_C” and “_I”. There is just “_”. Restall (2002: 432) puts the difference with Dummett– Quine–Carnap well: If accepting different logics commits one to accepting different languages for those logics, then my pluralism is primarily one of languages (which come with their logics in tow) instead of logics. To put it graphically, as a pluralist, I wish to say that A, :A ‘C B, but A, :A⊬R B A and :A together, classically entail B, but A and :A together do not relevantly entail B. On the other hand, Carnap wishes to say that A, :C A ‘ B, but A, :R A⊬B A together with its classical negation entails B, but A together with its relevant negation need not entail B.

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So (Beall and) Restall reject an indexical contextualism concerning the connectives (and quantifiers). Either there is no folk-relativism at all for the connectives – each has a single, uniform content – or we have a nonindexical contextualism or an assessment-sensitive view. Recall that h says that Δ is valid and b says that Δ is invalid. On the option considered now, championed by Beall and Restall, we have that h and b mean the same thing by Δ. What about “valid”? Does that have the same content in the two pronouncements? Recall Beall and Restall’s (2006: 29) “Generalised Tarski Thesis”: An argument is validx if and only if, in every casex in which the premises are true, so is the conclusion.

I presume that Beall and Restall did not intend to make a claim about the semantics of an established term of philosophical English. However, the presence of the subscript x in the statement of the thesis might indicate that the word “valid” has a sort of elided constituent, a slot where a logic can be filled in. This suggests a sort of indexical contextualism about the word “valid”. The same idea is suggested by the use of subscripts in the above passage from Restall [2002], when he is using his own voice. He says that, for him: A, :A ‘ C B, but A, :A ⊬R B:

So the technical term “‘” seems to have an elided constituent, and that suggests a kind of contextualism. So, on the Beall and Restall view – as on the opposing Dummett– Quine–Carnap view – our logicians h and b do not have a genuine disagreement. They are in the analogous situation as Jack and Jill with the salt. Beall and Restall insist that h and b give the same content to the argument Δ, but not to “valid”. For h, it is “classically valid”, “‘C”, and for b it is “intuitionistically valid”, “‘I”. So, once again, we do not have a failure of cognitive command. To get cognitive command to fail, we have to assume that our logicians h and b assign the same content to the terms in the argument Δ and we have to assume that they assign the same content to the word “valid”. Given that Δ has the same content, “valid” must be folk-relative (since both h and b are correct). The options for that term are thus non-indexical contextualism and assessment-sensitive relativism. I do not know of anyone who explicitly defends that combination of views, and I won’t consider how plausible it is (but see Shapiro 2014). To summarize and conclude, Wright’s criterion of epistemic constraint concerns the possibility of unknowable truths. Given the present

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folk-relativism, statements of validity do not get truth-values unless one somehow indicates a particular logic. If a particular logic is so indicated, then it depends on how much idealization goes into the notion of “knowable”. If we fix a particular logic, then either cognitive command holds trivially, or, at worst, the question is reduced to one concerning mathematics which is, I would think, almost a paradigm case of objectivity. If we do not fix a particular logic, and consider statements of validity simpliciter, then the question of cognitive command depends on some delicate, and controversial semantic theses concerning both the logical terminology and the word “valid”. Prima facie, it might seem strange that matters of cognitive command, and indirectly, matters of objectivity, should turn on semantics. After all, we are concerned with validity and not with the meanings of words, like “or”, “not”, and, indeed “valid”. However, the notion of cognitive command depends on the notion of disagreement and, as we saw, that does turn on notions of meaning. Recall the Kant–Quine thesis articulated above, that there is no way to sharply separate the “human” and the “world” contributions to our theorizing (perhaps with some emphasis on “sharply”). So we might expect some tough, borderline cases of objectivity. Add to the mix some widely held, but controversial views that meaning is not always determinate, involving open-texture, and the like (e.g., Waismann 1945, Quine 1960, Wilson 2006). Then perhaps the connection between objectivity and semantics is not so surprising.

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chapter 4

Logic, mathematics, and conceptual structuralism Solomon Feferman

1. The nature and role of logic in mathematics: three perspectives Logic is integral to mathematics and, to the extent that that is the case, a philosophy of logic should be integral to a philosophy of mathematics. In this, as you shall see, I am guided throughout by the simple view that what logic is to provide is all those forms of reasoning that lead invariably from truths to truths. The problematic part of this is how we take the notion of truth to be given. My concerns here are almost entirely with the nature and role of logic in mathematics. In order to examine that we need to consider three perspectives: that of the working mathematician, that of the mathematical logician, and that of the philosopher of mathematics. The aim of the mathematician working in the mainstream is to establish truths about mathematical concepts by means of proofs as the principal instrument. We have to look to practice to see what is accepted as a mathematical concept and what is accepted as a proof; neither is determined formally. As to concepts, among specific ones the integer and real number systems are taken for granted, and among general ones, notions of finite and infinite sequence, set and function are ubiquitous; all else is successively explained in terms of basic ones such as these. As to proofs, even though current standards of rigor require closely reasoned arguments, most mathematicians make no explicit reference to the role of logic in them, and few of them have studied logic in any systematic way. When mathematicians consider axioms, instead it is for specific kinds of structures: groups, rings, fields, linear spaces, topological spaces, metric spaces, Hilbert spaces, categories, etc., etc. Principles of a foundational character are rarely mentioned, if at all, except on occasion for proof by contradiction and proof by induction. The least upper bound principle on bounded sequences or sets of real numbers is routinely applied without mention. Some notice is paid to applications of the Axiom of Choice. To a 72

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side of the mainstream are those mathematicians such as constructivists or semi-constructivists who reject one or another of commonly accepted principles, but even for them the developments are largely informal with little explicit attention to logic. And, except for some far outliers, what they do is still recognizable as mathematics to the mathematician in the mainstream. Turning now to the logicians’ perspective, one major aim is to model mathematical practice – ranging from the local to the global – in order to draw conclusions about its potentialities and limits. In this respect, then, mathematical logicians have their own practice; here I shall sketch it and only later take up the question how well it meets that aim. In brief: Concepts are tied down within formal languages and proofs within formal systems, while truth, be it for the mainstream or for the outliers, is explained in semantic terms. Some familiar formal systems for the mainstream are Peano Arithmetic (PA), Second-Order Arithmetic (PA2), and Zermelo–Fraenkel set theory (ZF); Heyting Arithmetic (HA) is an example of a formal system for the margin. In their intended or “standard” interpretations, PA and HA deal specifically with the natural numbers, PA2 deals with the natural numbers and arbitrary sets of natural numbers, while ZF deals with the sets in the cumulative hierarchy. Considering syntax only, in each case the well-formed formulas of each of these systems are generated from its atomic formulas (corresponding to the basic concepts involved) by closing under some or all of the “logical” operations of negation, conjunction, disjunction, implication, universal and existential quantification. The case of PA2 requires an aside; in that system the quantifiers are applied to both the first-order and second-order variables. But we must be careful to distinguish the logic of quantification over the second-order variables as it is applied formally within PA2 from its role in second-order logic under the so-called standard interpretation. In order to distinguish systematically between the two, I shall refer to the former as syntactic or formal second-order logic and the latter as semantic or interpreted second-order logic. In its pure form over any domain for the first-order variables, semantic second-order logic takes the domain of the second-order variables to be the supposed totality of arbitrary subsets of that domain; in its applied form, the domain of first-order variables has some specified interpretation. As an applied second-order formal system, PA2 may equally well be considered to be a two-sorted first-order theory; the only thing that acknowledges its intended second-order interpretation is the inclusion of the so-called Comprehension Axiom Scheme: that consists of all formulas

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of the form 9X8x[x 2 X $ A(x,. . .)] where A is an arbitrary formula of the language of PA2 in which ‘X ’ does not occur as a free variable. Construing things in that way, the formal logic of all of the above-mentioned systems may be taken to be first-order. Now, it is a remarkable fact that all the formal systems that have been set up to model mathematical practice are in effect based on first-order logic, more specifically its classical system for mainstream mathematics and its intuitionistic system for constructive mathematics. (While there are formal systems that have been proposed involving extensions of first-order logic by, for example, modal operators, the purpose of such has been philosophical. These operators are not used by mathematicians as basic or defined mathematical concepts or to reason about them.) One can say more about why this is so than that it happens to be so; that is addressed below. The third perspective to consider on the nature and role of logic in mathematics is that of the philosopher of mathematics. Here there are a multitude of positions to consider; the principal ones are logicism (and neo-logicism), “platonic” realism, constructivism, formalism, finitism, predicativism, naturalism, and structuralism.1 Roughly speaking, in all of these except for constructivism, finitism, and formalism, classical first-order logic is either implicitly taken for granted or explicitly accepted. In constructivism (of the three exceptions) the logic is intuitionistic, i.e. it differs from the classical one by the exclusion of the Law of Excluded Middle (LEM). According to formalism, any logic may be chosen for a formal system. In finitism, the logic is restricted to quantifier-free formulas for decidable predicates; hence it is a fragment of both classical and intuitionistic logic. At the other extreme, classical second-order logic is accepted in set-theoretic realism, and that underlies both scientific and mathematical naturalism; it is also embraced in in re structuralism. Modal structuralism, on the other hand, expands that via modal logic. The accord with mathematical practice is perhaps greatest with mathematical naturalism, which simply takes practice to be the given to which philosophical methodology must respond. But the structuralist philosophies take the most prominent conceptual feature of modern mathematics as their point of departure.

2. Conceptual structuralism This is an ontologically non-realist philosophy of mathematics that I have long advanced; my main concern here is to elaborate the nature and role of 1

Most of these are surveyed in the excellent collection Shapiro (2005).

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logic within it. I have summarized this philosophy in Feferman (2009) via the following ten theses.2 1. The basic objects of mathematical thought exist only as mental conceptions, though the source of these conceptions lies in everyday experience in manifold ways, in the processes of counting, ordering, matching, combining, separating, and locating in space and time. 2. Theoretical mathematics has its source in the recognition that these processes are independent of the materials or objects to which they are applied and that they are potentially endlessly repeatable. 3. The basic conceptions of mathematics are of certain kinds of relatively simple ideal-world pictures that are not of objects in isolation but of structures, i.e. coherently conceived groups of objects interconnected by a few simple relations and operations. They are communicated and understood prior to any axiomatics, indeed prior to any systematic logical development. 4. Some significant features of these structures are elicited directly from the world-pictures that describe them, while other features may be less certain. Mathematics needs little to get started and, once started, a little bit goes a long way. 5. Basic conceptions differ in their degree of clarity or definiteness. One may speak of what is true in a given conception, but that notion of truth may be partial. Truth in full is applicable only to completely definite conceptions. 6. What is clear in a given conception is time dependent, both for the individual and historically. 7. Pure (theoretical) mathematics is a body of thought developed systematically by successive refinement and reflective expansion of basic structural conceptions. 8. The general ideas of order, succession, collection, relation, rule, and operation are pre-mathematical; some implicit understanding of them is necessary to the understanding of mathematics. 9. The general idea of property is pre-logical; some implicit understanding of that and of the logical particles is also a prerequisite to the understanding of mathematics. The reasoning of mathematics is in principle logical, but in practice relies to a considerable extent on various forms of intuition in order to arrive at understanding and conviction. 2

This section is largely taken from Feferman (2009), with a slight rewording of theses 5 and 10.

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10. The objectivity of mathematics lies in its stability and coherence under repeated communication, critical scrutiny, and expansion by many individuals often working independently of each other. Incoherent concepts, or ones that fail to withstand critical examination or lead to conflicting conclusions are eventually filtered out from mathematics. The objectivity of mathematics is a special case of intersubjective objectivity that is ubiquitous in social reality.

3. Two basic structural conceptions These theses are illustrated in Feferman (2009) by the conception of the structure of the positive integers on the one hand and by several conceptions of the continuum on the other. Since our main purpose here is to elaborate the nature and role of logic in such structural conceptions, it is easiest to review here what I wrote there, except that I shall limit myself to the set-theoretical conception of the continuum in the latter case. The most primitive mathematical conception is that of the positive integer sequence as represented by the tallies: |, ||, |||, . . . From the structural point of view, our conception is that of a structure (Nþ, 1, Sc, < jφ ! ψjv ¼ 1 > > > :2 1

n1 o if jψjv ¼ 0 and jφjv 2 ,1 2 o n 1 1 if jψjv ¼ and jφjv 2 ,1 2 2 otherwise

The resulting logic, which we call LA!, enjoys a detachable conditional. In particular, defining ‘LA! as above (no interpretation designates the premise set without designating the conclusion), we have: φ, φ ! ψ ‘LA! ψ:

The trouble, however, comes from Curry’s paradox. Focusing on the settheoretic version (though the truth-theoretic version is the same), Meyer et al. (1979) showed that, assuming standard structural rules (which are in place in LP and LA! and many other logics under discussion), if a conditional detaches and also satisfies ‘absorption’ in the form φ ! ðφ ! ψÞ ‘ φ ! ψ

then the given conditional is not suitable for underwriting naïve foundational principles. In particular, in the set-theory case, consider the set c ¼ fx : x 2 x ! ⊥g

which is supposed to be allowed in the Asenjo and Tamburino (and virtually all other) paraconsistent set theories.9 By unrestricted comprehension (using the new conditional, which is brought in for just that job), where $ is defined from ! and ^ as per usual, we have c 2 c $ ðc 2 c ! ⊥Þ:

But, now, since the Asenjo–Tamburino arrow satisfies the given absorption rule, we quickly get c2c!⊥ 9

Throughout, ⊥ is ‘explosive’ (i.e., implies all sentences).

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which, by unrestricted comprehension, is sufficient for c’s being in c, and so c 2 c:

But the Asenjo–Tamburino arrow detaches: we get ⊥, utter absurdity. The upshot is that while LA may well be sufficient for standard firstorder connectives, the ‘remedy’ for non-detachment (viz., moving to LA!) is not viable: it leads to absurdity.10 Other LP-based theorists, notably Priest (1980) and subsequently Beall (2009), have responded to the nondetachability of LP by invoking ‘intensional’ or ‘worlds’ or otherwise ‘nonvalue-functional’ approaches to suitable (detachable) conditionals. We leave the fate of these approaches for future debate.11

6. Closing remarks Philosophy, over the last decade, has seen increasing interest in paraconsistent approaches to familiar paradox. One of the most popular approaches is also one of the best known: namely, the LP-based approach championed by Priest. Our aim in this chapter has been twofold: namely, to highlight an important predecessor of LP, namely, the LA-based approach championed first by Asenjo and Tamburino, and to highlight the salient differences in the logics. We’ve argued that the differences in logic reflect a difference in both background philosophy of logic and background metaphysics. LA is motivated by a material approach to logical consequence combined with a metaphysical position involving antinomic predicates, while LP is compatible with both a formal and material approach to consequence and can be combined with a large host of metaphysical commitments (including few such commitments at all).12 10

11

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We note that Asenjo himself noticed this, though he left the above details implicit. We have not belabored the details here, but it is important to have the problem explicitly sketched. We note, however, that Beall has recently rejected the program of finding detachable conditionals for LP, and instead defends the viability of a fully non-detachable approach (Beall (2013)), but we leave this for other discussion. We note that Priest’s ultimate rejection of LP in favor of his non-monotonic LPm (elsewhere called ‘MiLP’) reflects a move ‘back’ in the direction of the original Asenjo–Tamburino approach, where one has ‘restricted detachment’ and the like, though the latter logic (viz., LA) is monotonic. We leave further comparison for future debate. For some background discussion, see Priest (2006, Ch. 16) and Beall (2012) for discussion.

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chapter 14

The metaphysical interpretation of logical truth Tuomas E. Tahko

1. Two senses of logical truth The notion of logical truth has a wide variety of different uses, hence it is not surprising that it can be interpreted in different ways. In this chapter I will focus on one of them – what I call the metaphysical interpretation. A more precise formulation of this interpretation will be put forward in what follows, but I wish to say something about my motivation first. Part of my interest concerns the origin or ground of logic and logical truth, i.e., whether logic is grounded in how the world is or how we (or our minds) see the world.1 However, this is not my topic here. Rather, I will assume that logic is grounded in how the world is – a type of realism about logic – and examine the status of logical truth from the point of view of logical realism. The upshot is an interpretation of logical truth that is of special interest to metaphysicians.2 My starting point is the apparent difference between what we might call absolute truth and truth in a model, following Davidson (1973). The notion of absolute truth is familiar from Tarski’s T-schema: ‘Snow is white’ is true if and only if snow is white – in the world and absolutely. Instead of being a property of sentences as absolute truth appears to be, truth in a model, that is relative truth, is evaluated in terms of the relation between sentences and models.3 Davidson suggested that philosophy of language should be interested in absolute truth exactly because relative truth does not yield T-schemas, but I am not concerned with this proposal here.4 1

2

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For a recent discussion on this topic, see Sher (2011), who examines the idea that logic is grounded either in the mind or in the world, and defends that it is grounded in both – hence logic has a dual nature. See also the opening chapter of this volume. See Chateaubriand Filho (2001, 2005) for a version of the metaphysical interpretation of logical truth partly similar to mine. ‘Models’ are to be interpreted in a wide sense: they may for instance be interpretations, possible worlds, or valuations. We will return to this ambiguity concerning ‘model’ below. I should mention that I will omit discussion of Carnap and Quine on logical truth, as their debate is not directly relevant for my purposes. However, see Shapiro (2000) for an interesting discussion of Quine on logical truth.

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To clarify, relative truth is an understanding of logical truth in terms of truth in all models. One can be a realist or an anti-realist about the models, hence about logical truth. But there are choices to be made even if one is realist about the models, as the models can be understood interpretationally or representationally, along the lines suggested by John Etchemendy (1990). We will discuss the difference between these views in the next section, but ultimately none of these alternatives are expressive of the metaphysical interpretation of logical truth. Instead, we need a way to express absolute truth, which is not possible without spelling out the correspondence intuition, to be discussed in a moment. Given the topic of this chapter, one might expect that Michael Dummett’s view would be discussed, or at least used as a foil, but I prefer not to dwell on Dummett. The primary reason for this is that Dummett’s methodology is entirely opposite to the one that I use. Here is a summary of Dummett’s method: My contention is that all these metaphysical issues [questions about truth, time etc.] turn on questions about the correct meaning-theory for our language. We must not try to resolve the metaphysical questions first, and then construct a meaning-theory in the light of the answers. We should investigate how our language actually functions, and how we can construct a workable systematic description of how it functions; the answers to those questions will then determine the answers to the metaphysical ones. (Dummett 1991a: 338)

Since I am analyzing logical truth from a realist, metaphysical point of view, Dummett’s methodology is obviously not going to do the trick. In my view, there is a bona fide discipline of metaphysics and I am interested in finding a use for logical truth within that discipline. I doubt there is enough initial common ground to fruitfully engage with Dummett. Let me briefly return to Davidson and Tarski before proceeding. When considering the distinction between absolute and relative truth, an initial point of interest is absolute truth’s characterization by the T-schema. One question that emerges is the connection between the T-schema and metaphysics. A likely approach is to explicate this connection in terms of correspondence. However, at least according to one reading, Tarski (1944) considered truth understood as a semantic concept to be independent of any considerations regarding what sentences actually describe, that is, independent of issues concerning correspondence with the world. Indeed, the T-schema is now rarely considered to play a crucial role in correspondence theories of truth, despite the appearance of a correspondence relation

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between sentences and the world.5 Yet, Tarski’s (1944: 342–343) initial considerations on the meaning of the term ‘true’ explicitly take into account an ‘Aristotelian’ conception of truth, where correspondence with the world is central. Davidson (1973: 70) as well seems to have some sympathy for the idea that an absolute theory of truth is, in some sense, a ‘correspondence theory’ of truth, although he insists that the entities that would act as truthmakers here are ‘nothing like facts or states of affairs’, but sequences (which make true open sentences). I will not aim to settle the status of the correspondence theory here, but it will be necessary to discuss it in some more detail. I suggest adopting an understanding of the correspondence relation which is neutral in terms of our theory of truth. It is this type of weak correspondence intuition that I believe central to the metaphysical interpretation of logical truth. But it should be stressed that the correspondence intuition itself is not necessarily expressive of realism (Daly 2005: 96–97). For instance, Chris Daly’s suggested definition of the intuition is simply that a proposition is true if and only if things are as the proposition says they are. Daly explains the neutrality of (his version of ) the correspondence intuition as follows:6 Consider the coherence theorist. He may consistently say ‘If is true, it has a truthmaker. corresponds to a state of affairs, namely the state of affairs which consists of a relation of coherence holding between and the other members of a maximal set of propositions’. Consider the pragmatist. He may consistently say, ‘If is true, it has a truthmaker. corresponds to a state of affairs, namely the state of affairs of ’s having the property of being useful to believe’. It is controversial whether there exist states of affairs. Let that pass. My point here is that the coherence theory and the pragmatic theory are each compatible with the admission of states of affairs. Furthermore, each of these theories is compatible with the admission of states of affairs standing in a correspondence relation to truths. (Daly 2005: 97)

A neutral version of the correspondence intuition is desirable because I do not want to rule out the possibility of different approaches to truth, despite assuming realism in the present context. A central appeal of the correspondence intuition is, I suggest, its wide applicability. However, a slightly 5

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Furthermore, the idea that the T-schema or the correspondence theory are somehow expressive of realism has been forcefully disputed. See for instance Morris (2005) for a case against the connection between realism and correspondence; in fact Morris argues that correspondence theorists should be idealists. See also Gómez-Torrente (2009) for a discussion about Tarski’s ideas on logical consequence as well as on Etchemendy’s critique of Tarski’s model-theoretic account. The angled brackets describe a proposition, following Horwich (1998).

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better formulation than Daly’s can be found by following Paolo Crivelli (2004), who interprets Aristotle as an early proponent of the correspondence theory. Crivelli defines correspondence-as-isomorphism as follows: ‘a theory of truth is a correspondence theory of truth just in case it takes the truth of a belief, or assertion, to consist in its being isomorphic with reality’ (Crivelli 2004: 23).7 This type of view, which Crivelli ascribes to Aristotle, is expressive of the correspondence intuition, but avoids mention of propositions, or indeed states of affairs.8 Hence, we may define the correspondence intuition as follows: (CI) A belief, or an assertion, is true if and only if its content is isomorphic with reality.

This formulation preserves Daly’s idea. ‘Reality’ in CI may consist, say, of what it is useful to believe, as the pragmatist would have it, so neutrality is preserved. If we accept that CI is neutral in terms of different theories of truth, then we can characterize the issue at hand as follows. There is an apparent and important difference between truth understood along the lines of CI, and truth understood as a relation between sentences and models. I take this to be at the core of Davidson’s original puzzle concerning absolute and relative truth. We ought to inquire into these two senses of truth before we give a full account of logical truth. This is exactly what I propose to do, arguing that the metaphysical interpretation of logical truth must respect CI. Tarski and the model-theoretic approach may have made it possible to talk about logical truth in a manner seemingly independent of metaphysical considerations, but important questions about the metaphysical status of logical truth and the interpretation of models remain. One thing that makes this problem topical is the recent interest in logical pluralism, or pluralism about logical truth (e.g., Beall and Restall 2006). In the second section I will assess the metaphysical status of the notion of logical truth with regard to the two senses of truth familiar from Davidson. The third section takes up the issue of interpreting logical truth in terms of possible worlds and contains a case study of the 7

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Crivelli also defines a stricter sense of correspondence, which can be found in Aristotle. But sometimes Aristotle’s view on truth is also considered as a precursor to deflationism about truth, so we shouldn’t put too much weight on the historical case. For a more historically inclined discussion, see Paul Thom’s chapter in this volume. Admittedly, once we explicate isomorphism, reference to propositions, states of affairs or something of the sort could easily re-emerge. This shouldn’t worry us too much, because it is likely that we want a structured mapping from something to reality. The reason to opt for isomorphism here is merely to keep the door open for one’s preferred (structured) ontology.

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law of non-contradiction. A brief discussion of logical pluralism will take place in the fourth section, before the concluding remarks.

2. Reconciling the two senses of truth Can we reconcile the two senses of truth familiar from Davidson, the absolute and the relative? As Etchemendy (1990: 13) notes, the obvious way to attempt this would be in terms of generalization: if absolute truth is a monadic predicate of the form ‘x is true’, then it may be helpful to analyze it in terms of a relational predicate of the form ‘x is true in y’, for instance ‘x is a brother’ could be analyzed by first analyzing ‘x is a brother of y’, thus using the generalized concept of brotherhood. However, this does not apply to truth: ‘[C]learly the monadic concept of truth, the concept we ordinarily employ, is no generalization of any of the various relational concepts. A sentence can be true in some model, yet not be true; a sentence can be true, yet not be true in all models’ (1990: 14). Accordingly, generalization will not help in reconciling the two senses of truth. Another alternative that Etchemendy considers is to interpret absolute truth as a specification of truth in a model, namely, absolute truth could be considered equivalent to truth in the right model, the model that corresponds with the world. This maintains the correspondence intuition expressed by CI above, but note that ‘correspondence with the world’ already suggests a realist theory of truth, so the neutrality of the formulation is in question.9 However, there are good reasons to think that the notion of ‘model’ is not entirely appropriate when discussing absolute truth, as it is closely associated with relative truth. Hence, interesting as Etchemendy’s characterization may be, it is unlikely to result in a metaphysical account of logical truth. Still, Etchemendy’s account may help pinpoint the issue; consider the following passage: Once we have specified the class of models, our definition of truth in a model is guided by straightforward semantic intuitions, intuitions about the influence of the world on truth values of sentences in our language. Our criterion here is simple: a sentence is to be true in a model if and only if it would have been true had the model been accurate – that is, had the world actually been as depicted by that model. (Etchemendy 1990: 24) 9

Note that the question concerning which model is ‘right’ is not, strictly speaking, a question for the logician. For instance, as Burgess (1990: 82) notes, it is the metaphysician’s task to determine the correct modal logic, as this depends on our understanding of (metaphysical) modality. In contrast, the question about the ‘right’ sense of logical validity remains in the realm of logic.

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There is an important requirement in the passage above, namely, it must be the case that the model could have been true. How do we interpret the modality in effect here? If we understand it as saying that it must be the case that the world could have turned out to be like the model depicts, then this supports the case for a metaphysical interpretation of logical truth, for it introduces as a requirement for the notion of ‘model’ that it is a possible representation of the world. This representational approach, or ‘representational semantics’ can be contrasted with ‘interpretational semantics’, which Etchemendy discusses later on: [I]n an interpretational semantics, our class of models is determined by the chosen satisfaction domains; our definition of truth in a model is a simple variant of satisfaction. (Etchemendy 1990: 50)

Etchemendy claims that the Tarskian conception of model-theoretic semantics is of the ‘interpretational’ kind, although his interpretation of Tarski can certainly be questioned (e.g. Gómez-Torrente 1999). But I do not wish to enter the debate about Tarski or interpretational semantics. According to Etchemendy, in the representational approach models must represent ‘genuinely possible configurations’ of the world, and I am interested in the correct understanding of these possible configurations (cf. Etchemendy 1990: 60). However, instead of developing Etchemendy’s representational account, I will propose a pre-theoretic account of absolute truth, which aligns nicely with Etchemendy’s analysis. The biggest complication is the interpretation of the modal content in Etchemendy’s picture; we will need to return to this issue later (in the next section). What I propose to draw from Etchemendy is that once we have specified the class of ‘genuinely possible configurations’, we can define relative truth according to Etchemendy’s suggestion. In this regard, my analysis will not follow that of Etchemendy’s, as the case for absolute truth will come before Etchemendy’s account. Etchemendy’s representational approach notwithstanding, the notion of ‘model’ is not ideal for this task, as it is strongly reminiscent of relative truth.10 Instead of ‘models’, I propose to resort to talk of ‘possible worlds’. What I have in mind is interpreting possible worlds as metaphysical possibilities. 10

It has been suggested to me (by Penny Rush) that relative truth may be problematic because of its underlying metaphysical commitment to relativism, rather than not being up to the job of giving a metaphysical interpretation of logical truth at all. This may indeed be the case. I have attempted to preserve ontological neutrality while at the same time making it clear that I am presently only interested in putting forward a realist interpretation of logical truth. But I will set this issue aside for now, whether or not it is possible to combine relative truth and realism.

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This is, of course, somewhat controversial, but as we will see, there are reasons to think that only metaphysical modality is fitting for the task. In any case, more needs to be said about how the space of metaphysical possibilities is restricted. We will return to this in the next section. We are now in the position to define a provisional sense of logical truth which I propose to call metaphysical: (ML) A sentence is logically true if and only if it is true in every genuinely possible configuration of the world.

ML leaves open the criteria for a ‘genuinely possible configuration of the world’. But it does preserve CI and it provides us – via the possible worlds jargon – a ‘metaphysician friendly’ interface to the notion of logical truth. It is time to see if we can actually work with that interface.

3. Genuinely possible configurations and the case of the law of non-contradiction The puzzle can now be expressed in the following form: What sort of criteria can be established to evaluate whether a given possible world is a genuinely possible configuration of the world, i.e., could have turned out to correspond with the actual world? Let me approach the problem with a case study. Take, arguably, one of the most fundamental laws of logic, the law of non-contradiction (LNC). When I say that the law of noncontradiction is true in the ‘metaphysical sense’, I mean that LNC is true in the sense of absolute truth, i.e., it is a genuine constraint on the structure of reality. The metaphysical formulation of LNC takes a form familiar from Aristotle (Metaphysics 1005b19–20), although my proposed formulation is somewhat weaker, defined as follows: (LNC) The same attribute cannot at the same time belong and not belong to the same subject in the same respect and in the same domain.

The above formulation differs from Aristotle’s only with regard to the qualification regarding ‘the same domain’ – here the domain is the set of genuinely possible configurations of the world. How do we know whether LNC is true in this sense? I have previously argued (Tahko 2009) that we do have a good case for the truth of LNC in the metaphysical sense – the primary opponent here is Graham Priest (e.g., 2005, 2006b).11 I will not 11

See also Berto (2008) for an attempt to formulate a (metaphysical) version of LNC which even the dialetheist must accept. Berto’s idea, to which I am sympathetic, is that LNC may be understood as

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repeat my arguments here, but it may be noted that this is not strictly a question for logic. For instance, Priest’s most celebrated arguments in favor of true contradictions (in the metaphysical sense) concern the nature of change and specifically motion, the paradoxical nature of which is supposedly demonstrated by Zeno’s well-known paradoxes. Although these paradoxes can quite easily be tackled by mathematical means, the relevant question is whether change indeed is paraconsistent.12 The answer to this question requires both metaphysical and empirical inquiry. I will return to this point briefly below, but first I wish to say something about the methodology of logical-cum-metaphysical inquiry. In terms of ML, demonstrating the falsity of LNC would first require a genuinely possible configuration of the world where LNC fails. That is, it is not enough that we have a model where LNC is not true, such as paraconsistent logic, but we would also need to have some good reasons to think that the world could have been arranged in such a way that the implications of the metaphysical interpretation of LNC do not follow. This point deserves to be emphasized, for it would be much easier to show that a paraconsistent model can be useful in modelling certain phenomena, or interpreted in such a way that it is compatible with all the empirical data. But what is required here is that LNC, fully interpreted in the metaphysical sense, can be shown to fail. Note that we may also ask whether LNC is necessary, i.e., are there any possible worlds in which LNC does not hold – even if we did have a good case for its truth in the actual world? In fact, this is the question we should begin with, since if LNC is necessary, then it could not fail in the actual world either. However, it is not clear how we could settle this question conclusively, given that we are dealing with the metaphysical interpretation of LNC. Moreover, I do think that there could (in an epistemic sense) be possible worlds in which LNC fails, and hence I take the debate about LNC seriously. Yet, I am uncertain about whether such a paraconsistent possible world is in fact a genuinely possible configuration, as I will go on to explain.13 In any case, if a possible world in which LNC is not true

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a principle regarding structured exclusion relations (between properties, states of affairs, etc.), and the world is determinate insofar as it conforms to this principle. For discussion regarding Zeno’s paradoxes, see for instance Sainsbury (2009: Ch. 1). It is worth pointing out here that in my proposed construal, the distinction between absolute truth and truth in a model is not quite so striking for dialetheists. The idea, which I owe to Francesco Berto, is that the world cannot be a model, because it contains everything, and there’s no domain of everything, on pain of Cantor’s paradox. The result is that something can be a logical truth in the sense of being true in all models, without being true in the absolute sense, for the world is not a model. My proposed treatment of this issue proceeds by understanding absolute truth in terms of

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were genuinely possible, then LNC would obviously not be necessary. This should be relatively uncontroversial, but I should finally say something more about ‘genuine possibility’. As was mentioned in the previous section, there are reasons to understand genuine possibility in terms of metaphysical possibility, as only metaphysical modality could secure the correspondence between a possible world and the structure of reality – this is also what CI requires. The relevant modal space must consist of all possible configurations of the world and only them. Logical modality cannot do the job because it is not sufficiently restrictive. This can be demonstrated with any traditional example of a metaphysical, a posteriori necessity, such as gold being the element with atomic number 79. Assuming that it is indeed metaphysically necessary that gold is the element with atomic number 79, we must be able to accommodate the fact that gold failing to be the element with atomic number 79 is nevertheless logically possible. But since we are interested in genuinely possible configurations of the world, we ought to rule out metaphysically impossible worlds, such as the world in which gold fails to be the element with atomic number 79. The upshot is that if we accept the familiar story about metaphysical a posteriori necessities of this type, then there are necessary constraints for the structure of reality which logical necessity does not capture.14 The only other viable alternative in addition to metaphysical and logical modality is conceptual modality, i.e., necessity in virtue of the definitions of concepts. Nomological modality is already too restrictive, as we sometimes need to consider configurations of the world that are nomologically impossible but at least may be genuinely possible (e.g., superluminal travel). However, conceptual modality is too liberal, quite like logical modality, as it also accommodates configurations of the world which are not genuinely possible, such as violations of the familiar examples of metaphysical a posteriori necessities. If we accept these examples, then neither definitions of concepts nor laws of logic rule out things like gold failing to be the element with atomic number 79. Accordingly, if one accepts that there are metaphysical necessities that are not also conceptually and logically necessary – something that most metaphysicians would accept – the only available interpretation of genuine possibility is in terms of metaphysical possibility.

14

metaphysical modality, but the dialetheist could, in principle, endorse paraconsistent set theory and posit that absolute truth is just truth in the world-model – the model whose domain is the world. I should add that cashing out these constraints is, I think, a much more complicated affair than the traditional Kripke–Putnam approach to metaphysical a posteriori necessities suggests.

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There is, however, a way to understand logical modality which may do a better job in capturing the relevant sense of logical truth. This type of understanding has been proposed by Scott Shalkowski, who suggests that ‘logical necessities might be explained as those propositions true in virtue of the natures of every situation or every object and property, thus preserving the idea that logic is the most general science’ (Shalkowski 2004: 79). On the face of it, this suggestion respects the criteria for genuine possibility. According to this approach, logical modality concerns the most general (metaphysical) truths, such as the law of noncontradiction when it is considered as a metaphysical principle (as in Tahko 2009). In this view, logical relations reflect the relations of individuals, properties, and states of affairs rather than mere logical concepts. Indeed, this understanding effectively equates metaphysical and logical modality. The idea is that the purpose of logic is to describe the structure of reality and so it is ‘the most general science’. As Shalkowski (2004: 81) notes, denying the truth of LNC would, in terms of this understanding, amount to a genuine metaphysical attitude instead of, say, the fairly trivial point that a model in which the law does not hold can be constructed. Do we have any means to settle the status of LNC in the suggested sense? A simple appeal to its universal applicability may not do the trick, but the burden of proof is arguably on those who would deny LNC. One might even attempt to distill a more general formula from this: logical principles – which are presumably reached by a priori means – are prima facie metaphysically necessary principles. They may be challenged and sometimes falsified even by empirical means, but merely the fact that we can formulate models in which they do not hold is not enough to challenge their truth; it will also have to be demonstrated that there are possible worlds which constitute genuinely possible configurations of the world. However, this approach seems biased towards historically prior logical principles, the ones that were formulated first. It is not implausible that the reason why they were formulated first is because they are indeed the best candidates for metaphysically necessary principles: for Aristotle, the law of non-contradiction is ‘the most certain of all principles’ (Metaphysics 1005b22). But this is admittedly quite speculative – we ought to be allowed to question even the ‘first’ principles. It would certainly be enough to challenge the metaphysical necessity of LNC, or other logical principles, if empirical evidence to the effect that the principle is not true of every situation or every object and property would

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be found.15 This is what Priest has attempted to show with the case of change and Zeno’s paradoxes, but I remain unconvinced. As I have argued (Tahko 2009), Priest’s examples can all be accounted for in terms of semantic rather than metaphysical dialetheism – a distinction developed by Edwin Mares (2004). The idea is that there may be indeterminacy in semantics, but this does not imply that there is indeterminacy in the world. Only the latter type of indeterminacy would corroborate the existence of a genuinely possible paraconsistent configuration of the world. Since I have not seen a convincing case to the effect that such a configuration is genuinely possible, I take it that LNC is a good candidate for a metaphysically necessary principle. If I am right, this means that a paraconsistent possible world could not have turned out to accurately represent the actual world. The fact that there are paraconsistent models has no direct bearing on this question. I do not claim to have settled the status of LNC once and for all, but I think that a strong empirical case for the truth of LNC can be made, on the basis of the necessary constraints for the forming of a stable macrophysical world, i.e., the emergence of stable macrophysical objects. I have developed the preceding line of thought before with regard to the Pauli Exclusion Principle (PEP) (Tahko 2012), and electric charge (Tahko 2009). For instance, as PEP states, it is impossible for two electrons (or other fermions) in a closed system to occupy the same quantum state at the same time. This is an important constraint, as it is responsible for keeping atoms from collapsing. It is sometimes said that PEP is responsible for the space-occupying behavior of matter – electrons must occupy successively higher orbitals to prevent a shared quantum state, hence not all electrons can collapse to the lowest orbital. Here we have a principle which captures a crucial constraint for any genuinely possible configuration of the world that contains macroscopic objects. Whether or not there are genuinely possible configurations that do not conform to PEP is an open question, but it seems unlikely that such a configuration could include stable macroscopic objects. Consider the form of PEP: it states that two objects of a certain kind cannot have the same property (quantum state) in the same respect (in a closed system) at the same time. Compare this with Aristotle’s formulation of LNC: ‘the same attribute cannot at the same time belong and not belong to the same subject in the same respect’ (Metaphysics 1005b19–20). LNC is of course a much more general criterion than PEP – it concerns 15

I have in mind concrete objects in the first place; see Estrada-González (2013) for a case to the effect that there are abstracta which violate LNC in this sense.

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one thing rather than things of a certain kind – but its underlying role is evident: if any fermion were able to both be and not be in a certain quantum state at the same time, then PEP would be violated and macroscopic objects would collapse. If LNC is needed to undergird PEP, then we have a strong case in favor of the metaphysical interpretation of LNC in worlds that contain macrophysical objects, given the necessity of PEP for the forming of macrophysical objects. This is of course not sufficient to establish the metaphysical necessity of either principle, but it is an interesting result in its own regard.

4. Pluralism about logical truth Now that we have a rough idea about the metaphysical interpretation of logical truth, we can consider the implications of this interpretation in a wider context. Here I would like to focus on the topic of logical pluralism, which has lately received an increasing amount of attention. Perhaps the most influential form of logical pluralism derives from pluralism about logical consequence, i.e., the view that there are models in which the logical consequence relation is different, and irreconcilably so. Beall and Restall have formulated and defended this type of pluralism: Given the logical consequence relation defined on the class of casesx, the logicalx truths are those that are true in all casesx. If you like, they are the sentences that are x-consequences of the empty set of premises. The logicalx truths are those whose truth is yielded by the class of casesx alone. Since we are pluralists about classes of cases, we are pluralists about logical truth. (Beall and Restall 2006: 100)

If this is indeed what pluralism about logical truth amounts to, then it appears that anyone who accepts multiple classes of cases is a pluralist about logical truth. But what does ‘being true in a case’ mean? On the face of it, one might think that it means exactly the same as ‘being true in a model’, that is, we are talking about a type of relative truth familiar from Davidson. This would imply that anyone who accepts multiple classes of models will also be a pluralist about logical truth. Pluralism about logical truth would then mean only that there are multiple models, and we can talk about logical truth separately in each one of these models. But this would be a rather uninteresting sense of logical pluralism, at least from the point of view of the metaphysical interpretation of logical truth. However, as Hartry Field has recently pointed out, this cannot be what Beall and

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Restall have in mind. Moreover, Field suggests two reasons why modeltheoretic accounts are irrelevant to logical pluralism: One of these reasons is that by varying the definition of ‘model’, this approach defines a large family of notions, ‘classically valid’, ‘intuitionistically valid’, and so on; one needn’t accept the logic to accept the notion of validity. A classical logician and an intuitionist can agree on the modeltheoretic definitions of classical validity and of intuitionist validity; what they disagree on is the question of which one coincides with genuine validity. For this question to be intelligible, they must have a handle on the idea of genuine validity independent of the model-theoretic definition. Of course, a pluralist will contest the idea of a single notion of genuine validity, and perhaps contend that the classical logician and the intuitionist shouldn’t be arguing. But logical pluralism is certainly not an entirely trivial thesis, whereas it would be trivial to point out that by varying the definition of model one can get classical validity, intuitionist validity, and a whole variety of other such notions. (Field 2009: 348)

And the second reason: [I]f we were to understand ‘cases’ as models, then there would be no case corresponding to the actual world. There is no obvious reason why a sentence couldn’t be true in all models and yet not true in the real world. This connects up with the previous point: the intuitionist regards instances of excluded middle as true in all classical models, while doubting that they are true in the real world. (Field 2009: 348; italics original)

Field goes on to suggest that Beall and Restall must have meant that there is an implicit requirement for interpreting ‘truth in a case’, namely, that truth in all cases implies truth. Field then argues that this will not produce an interesting sort of logical pluralism as the pluralist notion of logical consequence suggested by Beall and Restall does not capture the normal meaning of ‘logical consequence’. But it should be noted that Beall and Restall (2006: 36 ff.) do say something about the matter. Specifically, they suggest that on one reading of ‘case’ (the TM account), Tarskian models are to be understood as cases. Another reading (the NTP or necessary truthpreservation account) takes possible worlds to be cases. Beall and Restall (2006: 40) add that the existence of a possible world that invalidates an argument entails the existence of an actual (abstract) model that invalidates the argument. So, it is not clear that Field’s critique is accurate, as Beall and Restall do suggest that there is a case that corresponds with the actual world – on the TM account it is a Tarskian model and on the NTP account it is a possible world. The latter is of immediate interest to us, given that the metaphysical

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interpretation of logical truth also makes use of the possible worlds jargon. Yet, Beall and Restall do not provide an interpretation of possible worlds, so it is not quite clear what the connection, if any, between the NTP account and the metaphysical interpretation of logical truth is. Connecting all this with the analysis provided in the previous section, one might suggest that classes of cases are sets of metaphysically possible worlds, distinguished in terms of logical truths that are true in each set of possible worlds. Only one possible world is actual, but the logical truths that are true in the actual world will also be true in all worlds which are in the same set of possible worlds, i.e., these worlds may differ in other regards, but they are close to the actual world in the sense that all the logical truths are shared. Accordingly, pluralists about logical truth, in the metaphysical sense, hold that there are distinct sets of possible worlds in which different logical truths hold. The metaphysical interpretation of logical truth can accommodate this sense of logical pluralism, provided that possible worlds are interpreted appropriately – this also enables us to preserve CI.16 However, accommodating pluralism in the metaphysical interpretation of logical truth does require a revision in our original definition (ML), which defined a sentence as logically true if and only if it is true in every genuinely possible configuration of the world. Since in this view of logical pluralism there can be proper subsets of genuinely possible configurations with different laws of logic, we must revise ML as follows: (ML-P) A sentence is logically true if and only if it is true in every possible world of a given subset of possible worlds representing genuinely possible configurations of the world.

ML-P can of course also accommodate the situation where the laws of logic are the same across all subsets of genuinely possible configurations, i.e., logical monism – in that case the relevant subset of possible worlds would not be a proper subset of the genuinely possible configurations. An alternative formulation of ML-P is possible, dismissing subsets altogether. We could understand logical pluralism by giving different interpretations to ‘genuinely possible configurations’.17 This formulation 16

17

Why is interpreting logical truth on the basis of metaphysical possibility the only way to preserve CI? Because we’ve seen that only by restricting our attention to metaphysically possible worlds can we preserve a sense of correspondence between logical truth and genuinely possible configurations of the world. Only metaphysically possible worlds are sufficiently constrained to take into account all the governing principles such as metaphysical a posteriori necessities. Thanks to Jesse Mulder for suggesting this type of formulation.

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could be developed by adopting a line of thought from Gillian Russell (2008). Russell suggests that we can distill a sense of pluralism by understanding logical validity as the idea that in every possible situation in which all the premises are true, the conclusion is true (2008: 594), where possibility is ambiguous between logical, conceptual, nomological, metaphysical, or other senses of modality, hence producing a similar ambiguity concerning validity. A friend of the metaphysical interpretation of logical truth could accept this idea, but only provided that we prioritize the reading where possible situations reflect metaphysical possibility, as CI is preserved only in this reading. Nevertheless, there may still be room for a type of pluralism concerning metaphysical possibility and hence genuinely possible configurations. Unfortunately I have no space to develop this approach further. It may be noted that since I have been discussing logical pluralism only with regard to the law of non-contradiction, the resulting sense of pluralism is limited. Given that I consider there to be strong reasons to think that LNC holds in the actual world, we can define a set of possible worlds in which the law of non-contraction holds, call it WLNC. The assumption is that WLNC includes the actual world. But since I have made no mention of any other laws of logic that hold (in the metaphysical sense) in WLNC, the sense in which we can talk of a logic may be questioned. In other words, it may be wondered if the resulting sense of logical pluralism is able to support a rich enough set of logical laws to constitute a logic. However, I suspect that the case can be extended beyond LNC. That is, we can extend the metaphysical interpretation to other laws of logic as well in such a way that a subset of WLNC may be defined. This is not quite as straightforward in other cases though. Very briefly, consider modus ponens (A ^ (A ! B)) ! B. If thought of as a rule, it is not obvious that modus ponens can be applied to the world in the sense that I have suggested with regard to LNC. Yet, there are clear cases of physical phenomena that feature a modus ponens type structure. As a first pass, causation might be offered as a candidate of ‘real world modus ponens’, but there are obvious complications with this suggestion, as it depends on one’s theory of causation. However, there are better candidates. Take the simple case of an electron pair in a closed system, where two electrons occupy the same orbital. As we’ve already observed, two electrons in a closed system are governed by the Pauli Exclusion Principle. In particular, since the electrons cannot be in the same quantum state at the same time, we know that the only way for them to occupy the same orbital (i.e., having the same orbital quantum numbers) is for them

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to differ in spin (i.e., to have different spin quantum numbers). Accordingly, when we observe electron A having spin-up, we immediately know that any electron, B, in the same orbital as A must have spin-down. Moreover, there can be only two electrons in the same orbital and they must always have opposite spin. If cases such as the one for a ‘real world modus ponens’ can be found, then we may indeed have a rich enough set of logical laws to constitute a logic, enabling the suggested interpretation of logical pluralism. The resulting subset could be called WLNCþMP. This hardly exhausts the debate about logical pluralism, but it appears that there are ways, perhaps several ways, to accommodate pluralism about logical truth within the metaphysical interpretation.

5. Conclusion In conclusion, I have demonstrated that there is a coherent metaphysical interpretation of logical truth, and that this interpretation has some interesting uses, such as applications regarding logical pluralism. It has not been my aim to establish that this interpretation of logical truth is the correct one, but only that it is of special interest to metaphysicians. I have assumed rather than argued for a type of realism about logic for the purposes of this investigation, but I contend that for realists about logic, one interesting interpretation of logical truth is the one sketched here.18 18

Thanks to audiences at the University of Tampere Research Seminar and the First Helsinki-Tartu Workshop in Theoretical Philosophy, where earlier versions of the paper were presented. In particular, I’d like to thank Luis Estrada-González for extensive comments. In addition, I appreciate helpful comments from Franz Berto and Jesse Mulder. Thanks also to Penny Rush for editorial comments. The research for this chapter was made possible by a grant from the Academy of Finland.

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Index

admissible inference rule, 110, 113–114, 116 analyticity, 195–196, 200, 202 Aristotelian, 147, 204 categories, 149–150, 152–153 Aristotle, 6–8, 22, 54, 117, 139, 147–150, 205, 207, 212–213, 239, 242–243 Crivelli interpretation of, 236 notion of validity, 204, 206–207, See Bolzano Axiom of Choice, 72, 86, 88, 90 Beall, Jc, 229–230, 232 Beall and Restall, 245 Beall and Restall’s pluralism, 50, 69–70, 236, 244–246, See Restall, Greg Beall, Hughes and Vandegrift, 9 Priest and Beall, 232 bounded arithmetic, 116 Burgess, J.A., 50–51, 57, 237 Carnap, R, 69, 233 Dummett-Quine-Carnap, 69–70 classical validity, 59, 114, 245 cognitive command, 58–60, 62–63, 65–66, 68–71 completeness, 80–83, 114, 116, 123, 180–181 theorem, 42, 64, 80, 84, 181, See Gödel Condillac, 122, 127 conditional logic, 107 material, 106–107, 218 the, 105, 107, 231 consistency, 51–52, 55, 58–60, 79, 115, 118–119, 141–142, 144, 188 constructive, 82–83, 115–116, 119, 125 mathematics, 74, 108 semi-constructive, 86 constructivism, 74, 82 contextualism, 49, 66–67, 69–70, 257 continuum, 29, 54, 64, 76–80, 90, 97, 105 Continuum Hypothesis, 89, 110

contradiction, 8–9, 16, 27, 53–54, 72, 83, 107–108, 134–135, 137, 141–143, 149, 182–183, 213, 224, 228, 240 convention, 3, 33, 39, 115 Lewis account of, 34–35 tacit, 34, 36–38, 40–41 truth by, 32, 34, 47–48 conventionalism linguistic, 35 logical. See logical conventionalism conventions, 33–35, 45, 191 explicit, 33–35 logical, 32–33 optimality of, 36–38, 44–45 tacit, 34–35 correspondence, 44–45, 195, 234–237, 241, 246 1-1, 89 criteria for validity, 8 criterion for a philosophy of mathematics, 89 legitimacy, 52 mathematical legitimacy, 55 rule-following, 131 validity, 169 Darapti, 213 Davidson, D, 57, 233–234, 236, 244 dialetheism, 108 metaphysical, 243 semantic, 243 disjunction property, 110–113 Dummett, M, 59–60, 68–70, 128, 140–141, 234 Eddington, A.S., 95–97, 99 epistemic constraint, 58–61, 70 Etchemendy, J, 234–235, 237–238 Explosion, 107, 213–214, 228 Feferman, S, 3–4, 75–77, 79–82, 86, 88–89, 91–92, 105–106

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Index Field, H, 211, 244–245 first-order logic, 42–43, 61, 64, 74, 80, 82, 213, 226 Føllesdal, D, 20, 92 formalism, 74 Frege, G, 1, 41, 94, 128, 131, 157, 195, 214 and Russell, 129, 211 correspondence with Hilbert, 51–52 Fregean, 22 realism, 196 Gentzen, G, 111–112, 116, 122–124 meaning, 85 proof, 118–120 system of natural deduction, 81, 84 geometry, 8, 52, 93, 123, 128, 136, 208 application, 136, 215 axioms, 52 Euclidean, 41, 77, 128, 215 non-Euclidean, 186, 215 Gödel, K, 87, 92, 111, 116, 118–119, 142, 191 completeness theorem, 42, 80 incompleteness theorem, 79 Goldbach conjecture, 76, 139–141 Hatfield, G, 102 Hilbert Hilbertian, 8 Hilbert, D, 51–52, 110, 118–119, 211 Hilbertian, 52, 55 Hilbert’s program, 142 space, 72, 104 Husserl, E, 17, 25–28, 30 cognition, 22 concept of evidence, 23–24 conception of logic, 18–19 Husserlian, 26 logical realism, 190 phenomenological reduction, 19–21 transcendence, 17 idealization, 71, 106–107 classical logic, 106–108, See logic, classical of rudimentary logic, 5, See logic, rudimentary technique of, 105–106 throughout mathematics, 61 implication, 41, 73 intuitionistic, 125 incompleteness, 113–114 theorem, 79, See Gödel independence, 3, 13 conceptions of, 26 essential and modal, 15 human-, 2, 15 IF Independence Friendly, 82

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mind-, 20, 56 of facts, 14 of logic, 7 of logical truth, 29 proofs, 51 realist, 3, 15–18 results, 83 intuitionism, 115, 140 intuitionist validity, 245 intuitionistic, 69, 74, 112 analysis, 52, 54 consistency, 59 intuitionistic logic, 116 intuitionistically, 45, 70, 113 logic, 46, 50, 52, 54, 57, 60–61, 63–64, 74, 77, 82–83, 108, 111–112 predicate calculus, 84–86 propositional calculus, 110 semi-, 86, 89, See logic, intuitionist semi-intuitionism, 85 Jankov’s logic, 115 Kant, 20, 41, 57, 94, 180, 183, 187, 195, 203, 208, 213 Anti-, 198 ethics, 184–185 Kantian, 7, 179, 181, 183 Kant-Quine, 58, 71 KF-structure, 94 Ladyman, J, 99 Ladyman, J and Ross, D, 99, 104 language acquisition, 40, 103 law of excluded middle (LEM), 29, 50, 65, 74, 87–88, 90, 139–142, 144 weak, 115 law of non-contradiction (LNC), 9, 29, 48, 239, 242 Lewis Carroll regress, 33 logic applied, 215 canonical application, 2, 215–216, 220 classical, 4–5, 42–45, 50, 52–53, 77, 81, 84, 86–88, 107, 111–113, 115, 211, 214, 216, 218, 228 application to mathematics, 91 idealization, 106–108 rise of, 214 valid in, 60–61, 63 conditional, 107 content-containment model of, 42 deviant, 69, 104, 106–107 intuitionist, 215–216, 219 mathematical, 72–73, 212, 214, 217

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Index

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logic (cont.) medieval, 147, 158–159, 161, 164, 212–214 Megarian, 212 non-classical logic, 5, 49, 211, 218 paraconsistent, 50, 54–55, 64, 68, 108, 211, 215, 225, 240, See paraconsistency Port Royale, 213 pure, 18, 178, 215–216 relevance, 54, 107 rudimentary, 5, 95, 97–100, 104–108, 120–121, 125 rule-governed model of, 41–42 semi-intuitionist, 4 substitution model of, 42 traditional, 214, 217 logica docens, 212–216, 218, 220, 223 logica ens, 212, 216, 220, 223 logica utens, 212, 218–219, 223 logical connectives, 23, 115–116, 222 consequence, 8, 51, 59–60, 79, 109–110, 112, 123, 235 Beall and Restall’s, 50, 244–245 Bolzano, 203–204, 207 in mathematical practice, 43 material approach to, 232 Read’s defense of material, 228 traditional definition of, 192 conventionalism, 3, 33, 47, 190 inference, 5, 41, 93, 100, 134 pluralism, 4, 9, 217, 237, 244–248 realism, 4, 8, 13–15, 189–192, 195–197, 208, 233 schemata, 41 logical validity, 50, 56, 121–123, 161, 237, 247 logicism, 74 MacFarlane, J, 51, 65–67 Maddy, P, 5, 121 mathematical objects, 1, 90, 221 proof, 42, 120, 123 realism, 14 reality, 1, 15 McDowell, J, 25–30 meaning of a mathematical proposition, 137 of ‘all’, 78 of logical operations, 24, 85 of logical particles, 112 of logical predicates, 225, 228 of logical terms, 69, 135 of ‘proposition’, 150 of spoken and written utterances, 148–149 of ‘stateable’, 155 Medvedev lattice, 115 metalogical, 45

metalogical debates, 48 mirror neuron, 39 model theory, 44–45, 226–227 model-theoretic, 42, 64, 80, 204, 207, 235–236, 238, 245 account of validity, 221–223 modus ponens, 43, 48, 95, 113, 136, 189–190, 214, 228, 230, 247–248 monism, 51, 54, 62, 217, 246 naturalism, 3–4, 74, 189 necessity, 42, 159, 186, 204–207, 241, See possibility causal, 174 epistemic, 208 follows of, 205 logical, 35, 134, 174, 241 metaphysical, 242, 244 natural, 175 semantical, 181 non-realist, 3–4, 74 norms of reasoning, 158 objectivity, 3, 14, 56–60, 65–66, 71, 174, 179–180, 184, 186 axes of, 62 criteria of, 183 of logic, 7 of mathematics, 76 open-texture, 71 paraconsistency, 9–10, See logic, paraconsistent Peano Arithmetic, 73, 87 Piaget, 100 Plato, 18, 122, 162, 164, 166, 168, 176, 184 platonic, 18, 25, 74, 97, 162 platonism, 5, 135, 139–140 platonist, 136, 147, 221 pluralism, 9, 49, 51, 69, 189, 247, See logical: pluralism possibility, 70, 197, 228, 247, See necessity genuine, 241 logical, 174 metaphysical, 241, 247 of cognition, 13, 16–17, 21–22 of logic, 32 Priest, G, 3, 9, 158, 225–226, 229–230, 232 arguments against LNC, 240, 243 principle of bivalence, 32 principles and parameters model, 39 quantum mechanics, 98–99, 104, 108, 144 Quine, W.V.O. on second-order logic, 124 substitutional procedure, 195

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Index Quine, W.V.O., 32, 81, 100, 124, 135, 166 challenge, 32–35, 38–40, 48 Dummett-Quine-Carnap, 68–69 holism, 144 Kant-Quine, 58, 71 Putnam and Davidson, 57 Quinean, 178, 181, 216 rationality, 168, 184 relativism, 49, 51, 129, 190, 220, 238 folk-, 49–51, 57–60, 63, 65–66, 68, 70–71, 190 logical, 49, 51 proper, 67–68, 70 rule-following, 61, 129–132, 137, 142 second-order logic, 61, 64, 73, 79, 81, 128 classical, 74 full, 79, 82 Quine on, 81, 124 semantics, 81 Sellars, W, 22, 97, 99, 196 objection, 14–15 set theory, 6, 64, 81, 88, 92, 124, 136, 181, 231 background, 42 development of, 120–121 Kripke-Platek, 88 paraconsistent, 241 satisfiability in, 52 Zermelo-Fraenkel (ZF), 73 Shapiro, S, 2–4, 9, 43, 61, 78–79, 118, 189, 233 smooth infinitesimal analysis, 52–54 Spelke, E, 102–103 structuralism, 3, 74 conceptual, 3–4, 78, 80, 90–91 in-re, 74 modal, 74 Tarski, A, 195, 202, 234–236, 238 biconditionals, 44–45 definition of logical consequence, 203 Generalised Tarski Thesis, 50, 70 T-schema, 233

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-type, 204 Tarskian conception, 238 Tarskian model, 50, 207, 245 theory choice, 9, 216, 223 truth absolute, 233–234, 237–239, 241 by convention, 32, 47–48 in a model, 233 logical, 3, 9, 29, 41, 46, 95, 99, 233, 240, 242 all, 105 first-order, 42 ground of, 93 interpretation, 233–234, 248 metaphysical, 8–9, 233, 235–239, 244–247 pluralism about, 244–245 Quine on, 233 realist, 238 reflecting facts, 97 preservation, 44, 160, 222, 245 preserving, 44, 160 relative, 234, 236–238, 244 truth tables, 46 T-schema, 44, 46, 233–235 vagueness, 56, 61–65 logics of, 106 real, 107 worldly, 96 Waismann, F, 57, 71 Wason Card Test, 218–219 Wittgenstein, L, 5, 125, 144, 195, 219 and physics, 144 means by ‘postulate’, 144 on mathematics, 128–129 on rule-following, 129–132, 143 rejection of Hilbert’s program, 142 Steiner on, 6 Wittgensteinian, 45, 132, 141 Wright, C, 49, 58–60, 62–63, 70 Zermelo-Fraenkel set theory, 73

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