THE METAPHYSICS OF LOGIC
Featur Feat urin ingg four fourte teen en new new essa essays ys from from an inte intern rnat atio iona nall team 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 looking rst 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 nal section tackles speci c issues such as glutty glutty theorie theories, s, contradi contradicti ction, on, the metaph metaphysi ysical cal concept conception ion 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 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 el e l o pe p e r u s h is Hono Honora rary ry As Asso soci ciat atee with with the the Scho School ol of
Philosophy 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 Scienti c Realism (forthcoming).
THE METAPHYSICS OF LOGIC edited by
PENELOPE RUSH University of Tasmania
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 ages cm Includes bibliographical references and index. isbn 978 978-1-107-03964 -3 (Hardback) 1. Logic. 2. Metaph Metaphys ysics ics.. I. Rush, Penelo Penelope, pe, 1972– editor. bc50.m44 2014 160–dc23 2014 201402 0216 1604 04 isbn 978 978-1-107-03964 -3 Hardback
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With thanks to Graham Priest for unstinting encouragement, and to Annwen and Callum – never give up.
Contents
List of contributors
page ix
Introduction
1
Penelope Rush part i 1
the main p osi tions
11
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 par t i i 8
hi story an d author s
Logic and its objects: a medieval Aristotelian view Paul Thom
vii
145 147
viii 9
Contents
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 p art i i i 12
sp e cific i ss ue s
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
Contributors
j o d y a z z o u n i , Professor,
Department of Philosophy, Tufts University.
j c b e a l l , 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.
e r m a n n o b e n c i v e n g a , Professor
of Philosophy and the Humanities,
University of California, Irvine. Professor of Mathematics and Philosophy, Emeritus, and Patrick Suppes Professor of Humanities and Sciences, Emeritus, Stanford University.
s o l o mo n
f e f e rm a n ,
Associate Professor, Department of Philosophy, University of Notre Dame.
c u rt i s
f r an k s,
m i c ha e l h u g h es , Department
of Philosophy and UCONN Logic Group, University of Connecticut.
gyula klima,
Professor, Department of Philosophy, Fordham University,
New York. Associate Professor, Department of Philosophy, McMaster University.
s a n d ra l a p o in t e ,
p e n e l o p e m a d d y , Distinguished
Professor, Department of Logic and Philosophy of Science, University of California, Irvine.
g r a ha m p r i es t , Distinguished
Professor, Graduate Center, CUNY, and Boyce Gibson Professor Emeritus, University of Melbourne.
p e n e l o p e r u s h , Honorary
Associate, School of Philosophy, University
of Tasmania.
ix
x
List of contributors
s t e w ar t s h a p i ro ,
Professor, Department of Philosophy, The Ohio
State University. Professor Emeritus of Philosophy, the Hebrew University of Jerusalem.
mark steiner,
Finnish Academy Research Fellow, Department of Philosophy, History, Culture and Art Studies, University of Helsinki.
tuomas e. tahko,
Honorary Visiting Professor, Department of Philosophy, The University of Sydney.
paul thom,
Department of Philosophy and UCONN Logic Group, University of Connecticut.
r o s s v a n d e gr i f t ,
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 eld 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 specic 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 ‘
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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 denitively 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 rst 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.
Introduction
3
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 eld are admirably explored in both Stewart Shapiro s and Graham Priest s chapters, though from quite di erent 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 specic lens a o rded 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 eld. The rst 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 rst 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 bene ts 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 di erent mathematical structures “have di erent logics”, giving rise to logical pluralism – conceived of as “[the] view that di erent 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, identied 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 o e rs a thorough, precise, and immensely valuable analysis of the central concepts, and claries 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 denite 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 denite concepts and totalities ”, but may itself be justied on the basis of a semi-intuitionist logic “that is sensitive to distinctions that one might adopt between what is de nite and what is not”. Feferman shows how allowing that “di erent judgements may be made as to what are clear/denite concepts”, a o rds the conceptual structuralist a straightforward, sensible and clear understanding of the role and nature of logic. Penelope Maddy, in the fth chapter, o ers 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.
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 o ers 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 falsications’. In the nal two chapters of the rst part, Curtis Franks questions the assumption underpinning any metaphysics of logic at all: namely that there is “a logical subject matter una e cted 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 identies the key di erence (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 di erentiate 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
Penelope Rush
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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 denition (in Part II) to that which Franks presents on behalf of Socrates. Lapointe argues that, for Bolzano, there is more to a de nition than merely xing its extension, whereas Franks argues that Socrates was right to prioritise the xing of an extension rst 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 di erent 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 speci c authors of historical import for our particular theme. In the rst 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 e ect 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 rst part. The range of possibilities considered o er a fascinating and fruitful look into the historical precedents of the questions ’
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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 di erent categories. Thom’s chapter goes on to o er 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 di erence between them is not so much their ontologies as their di erent conceptions of concepts, grounding their di erent 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 di erent 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. O ering 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 di erent 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 denition of logical consequence and in so doing, identies some central considerations to help further our understanding of this topic. The nal part of the book deals with the speci c 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 o ers 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
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 di erent repercussions of di erent 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 di erent 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 ramications of the role we a ord our metaphysical commitments and the ramications 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 di erences. Tuomas Tahko nishes the book by examining a speci c realist metaphysical perspective and suggesting it as another approach we might take to understand logic, especially to interpret logical truth. His case study o ers an interpretation of paraconsistency which contrasts nicely with that o ered 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 o ers an interesting perspective on logical pluralism. From Tahko ’s metaphysical perspective, pluralism may be understood as about subsets of possible worlds representing genuine possible congurations 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.
chapter 1
Logical realism Penelope Rush
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The problem
Logic might chart the rules of the world itself; the rules of rational human thought; or both. The rst 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 physicalworldoranabstractworld)independentofourhumanthoughtprocesses. 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. 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: that the facts would have been as they 1
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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 signicant 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. 13
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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 di erence, 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 nd 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 justi catory 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 ’
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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 stu being posited as independently real is not su ciently like any stu that we can know, and if it were, it would not be the sort of thing intended by the mathematical realist in the rst 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 a rm 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 di cult 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 rst 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 o er 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 nd 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 o 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’, ‘scientic’ attitude, which he saw as pervasive all of philosophy (again, barring his own, e.g. Husserl 1964 : 18–19). By contrast, phenomenology o ers a picture of entangled cognition wherein independent reality is inextricable from cognition itself. This sort of picture takes the rst 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 de ned 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) o ers 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 briey 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, specically 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 e ectively defended.
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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 innite number-series presents an objectively xed totality of general objects . . . so the matter also stands with the ideal, pure-logical units, the concepts, propositions, and truths – in short, the signi cations 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’, ‘denite’, 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 denition 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, 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: 10
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 clarication 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 di erently depending on each term’s specic 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 de nition 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 denes consciousness, 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’, ‘scientic’ perspective, complete with its consequent dilemma. That is, Husserl’s point: 11
The problem of the possibility of cognition
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is the
traditional realist dilemma
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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 de nes (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: 12
Is . . . the object that we intend by means of a noema 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 negrained 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.
Over ow
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 reected 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 ‘over ow ’, 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 a a irs, 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 13 ‘seen’ 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, over ows 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 o . . . we acquire a de nite system of predicates either form or material , determined in the positive form or left “indeterminate” – and these predicates in their modi ed conceptual sense determine the “content ” of a core identity. (Husserl 1983: 364 , italics mine)
It is within this ‘core identity ’ we nd that which gives the noema its ‘pregnant sense of self evidence’; that which makes what is ‘given to cognition’ over ow 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
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Shim (2005) nicely characterizes hyletic data as the ‘sensual stu ’ of experience. He gives the following helpful example of the process of ‘precisication’ 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 ‘ lled 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 precisications of my memory until confronted by the corrective perception . . . there is no sense in talking about the veridical import in the precisi cations 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 ‘over ow ’ 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 di cult 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 nds a place in phenomenology. 5.
McDowell
It is useful to compare what has so far been drawn from Husserl to a specic 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 (specically 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 re ection 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 di erentiation 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 o ’ 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 re ectively 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 reection 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-de nes that independence, it is hard to see how the specically 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 (reective) experience simply by at (or by initial (re)design), then it becomes dicult to see how such an account can help us understand the problem that inspired it in the rst 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 o ers no essential insight into the dichotomy and the problem with which we began. 6 .
E e ctively defending R ~
The important word in the preceding paragraph is “somehow ”. Expanding on the ‘somehow ’, we can nd 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 o e red 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 o ering 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 reectively accessible factive reasons . . . entails that the facts that one can know by re ection 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 reective 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 o ers the broad reason that consciousness per se is not possible – if we try to imagine such a thing, we nd 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 di erence 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 conict (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 o e red to believe R, might be used as a sort of rst principle, or established by at; 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 o ered 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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Logical realism
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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 8x 8y((x y)). 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 e ective 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 corri- gible 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. ~
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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 denes 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 18 but in a particular way: ‘natural’ problem, one’s rst 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 clarication of the true transcendental problem, which in the initial obscure unsteadiness of its sense makes one so very prone . . . to shunt it o 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 di erentiating ‘being in the (real) world’ from ‘envattedness’. This di erentiation 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 deciencies of a philosophy that disallows any perspective other than its own. What we know from the phenomenological attitude might resist reduction to naturalist/scientic knowledge, but it nonetheless can o er an insight into the items with which the scienti c/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 satis ed” (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.
chapter 2
A defense of logical conventionalism Jody Azzouni
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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 recon gured within their nonclassical connes. 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 reect 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,” reects well or badly how a part of the world is, so too, the Principle of Bivalence is true or false because it reects 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
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. 32
A defense of logical conventionalism
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incompatible with those principles having representational content. 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 innite regress assails only one horn of this dilemma, the horn that presupposes that the innitely 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 rst 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 speci c conventional truths, used all along, t 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 di cult 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 rmly accepted, when he characterizes them as true by convention in such a sense.
These challenges aren’t specically directed against the conventionality of logic but against tacit “conventions” of any sort. One challenge is concerned with making sense of when specic 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 di ering 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 “rmly accepts them.” So Quine’s two objections come apart neatly. There is, rst, 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 ocial conventions. Even if this rst 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.
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See (Quine 1970b) for a reiteration of the rst challenge with respect to linguistic rules. See (Lewis 1969: 78) – but I draw this characterization from (Burge 1975: 32 –33).
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There is at least one alternative R to R such that the belief that the others conformed to R would give almost everyone a good and decisive practical or epistemic reason to conform to R likewise; such that there is a general preference for general conformity to R rather than slightly-less-than-general conformity to R ; and such that there is normally no way of conforming to R and R both. (6) (1)–(5) are matters of common knowledge . 0
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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 rst, 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 di erent . . . the copy would have di ered 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 6 re. A tribal population may simply fail to realize that banging rocks together will work instead. Their practice of rubbing sticks to start a re 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 re. 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 “
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Epstein (2006), for example, is worried. My thanks to him for conversations (and email exchanges) about this topic that have inuenced the rest of this section. I draw this example from (Epstein 2006: 4 ).
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ecacy 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 e cacious 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 inict 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 denitively 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 xation 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 di erent 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 “rm acceptance”) is also emerging due to intensive scienti c 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 specically, 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 prexed cognitive language-structure which is multiply triggered to a nal state by specic 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 specic utterances for that child’s 9
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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 rst 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, specically those of syntax. See, e.g., (Chomsky and Lasnik 1995).
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language-organ” to xate on a nal 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 nal-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 nished suggesting that the notion of tacit convention may (empirically) nd 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
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See (Chomsky 2003), specically page 313. See (Millikan 2003), specically 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 nd convincing Millikan’s arguments against the empirical possibility of a (virtually nal) 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 o cial conventionality by individuating the language practices nely 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 denitively 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 nally stabilized (as of the middle of the last century). The standard view is that an advantage of rstorder 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 rst-order context. The equivalence of the rule-governed model and the substitution model is established by the existence of equivalent characterizations of rst-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 rst-order classical logic isn’t reected 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 irrationality 17; I’m pointing out, rather, that as we become more sophisticated in our study of the neurophysiological
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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 arti cial 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-specic, 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 o cially 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 rst 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 o ered 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 rst 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 rst-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 rst-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 “nite,” 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 rst 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 rst 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 i 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. But these points are sucient 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 i 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
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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 speci c 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 24 “legislate” the truth of a statement like this. 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” inuenced by Wittgensteinian Tractarian views) who are driven by epistemic 22 23
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See (Azzouni 2008b), especially sections V and VI. Two issues drive my choice of the quali cation-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 nd 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 a ected 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 i ( 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). “
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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 . 26 See (Kleene 1971), for lots of explicitly indicated examples. “
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A defense of logical conventionalism 8.
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How Walker and Sider beg the question against logical conventionalism
Much of the argument I’ve o ered here has involved technical details that have been deliberately kept o -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 a ected by how individual logical idioms synergistically interact. But an important warning is in order. Discussion of these issues – specically, 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) I cannot legislate-true ‘It is raining ’ (ii) I cannot legislate-true ‘It is not raining ’ (iii) If I cannot legislate-true j, nor can I legislate-true ψ, then I cannot legislate-true the disjunction ┌ j or ψ ┐. (iii) 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 speci c 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 o er (both onstage and o ), 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 super cially 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
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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 quantication. 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.
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 di erent logics for di erent 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 : “
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ðGRSÞ Y is relative to X : 49
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In other words, in order to formulate a relativistic proposal, one rst species 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): “
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An argument is valid if and only if, in every case in which the premises are true, so is the conclusion. x
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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 : “
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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. “
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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 di erent theories/ structures have di erent 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 di erent 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 rami cations concerning semantics. Briey, pluralism about a given subject, such as truth, logic, ethics, or etiquette, is the view that di erent 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. Dene 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.
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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 e ect.
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consistency of the axioms of geometry, since the axioms are all true (presumably of space). Hilbert replied: 2
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 dened 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 satis ability 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 satisable, 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 briey present one such here, smooth in nitesimal analysis, a sub-theory of its richer cousin, Kock –Lawvere’s synthetic di erential geometry (see, for example, John Bell 1998). This is a fascinating theory of innitesimals, but very di erent from the standard Robinsonstyle non-standard analysis (which makes heavy use of classical logic). Smooth innitesimal analysis is also very di erent 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 eld, and consider the collection of “nilsquares”, numbers n such that n = 0. Of course, in both classical and intuitionistic analysis, it is easy to show that 0 is the only nilsquare: if n = 0, then n = 0. But not here. Among the new axioms to be added, the most interesting is the principle 2
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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-a neness, 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 innitesimal region that can have an orientation, but is too short to be bent. 3 It follows from the principle of micro-a neness that 0 is not the only nilsquare: :ð8αÞðα2 ¼ 0 !
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Otherwise, the value d would not be unique, for any function. Recall, however, that in any eld, 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
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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 (8 x )( x = 0_ x ¼ 6 0), and so smooth in nitesimal analysis is inconsistent with classical logic. Indeed, :(8 x )( x = 0_ x 6 ¼ 0) is a theorem of the theory (but, since the logic is intuitionist, it does not follow that (9 x ):( x = 0_ x 6 ¼ 0)). Smooth innitesimal analysis is an elegant theory of innitesimals, 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 innitesimal 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
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It follows from the principle of micro-a neness that every function is di erentiable everywhere on its domain, and that the derivative is itself di erentiable, 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 in nitesimal 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 innitesimal analysis is an interesting eld 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 in nitesimal 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 di erent 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 rami cations 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.
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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 gure 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: “
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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 awed 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 claried and deployed. There seems to be an important di erence – a di erence 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 di erent 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 gure out whether a given stretch of discourse is objective, on this or that axis, one must do some logical reasoning or gure 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
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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 xed . . . by assigning them conditions of potentially evidence-transcendent 5
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. 6 Actually, if the background logic is intuitionistic, there is a di erence between the absence of unknowable truths and the truth of the biconditional: P P may be known. That di erence 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 a airs 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 reect 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 , and let P be a statement that 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 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, 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 outelder catching a y 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 specied (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 ne, 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 specied, at least implicitly. So the central question is whether there can be unknowable truths concerning whether the argument (form) is valid in a given logic L? In e ect, 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 is an instance of &-elimination in which the premise and conclusion each have, say, 10 characters. Then is valid in, say, classical logic, but no one can know that, since no one can live long enough to check that is an instance of &-elimination. So, to give epistemic constraint a chance of being ful lled, we have to idealize on the knowers. One sort of idealization is familiar. We assume that our knowing subjects have unlimited (but still nite) 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 rst-order logic, or intuitionistic logic, or most of the relevant logics, and is an arbitrary argument form (with nitely many premises and conclusions), then a statement that 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 e ective 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 e ectively enumerable. Nor are things so clear for statements that a given argument 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 xed, 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. “
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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 misidentied 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 gure 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 di erences 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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di erent 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 qualications 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 di erences 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 be a given argument form, and consider two folks who disagree – or seem to disagree – whether is valid. One says it is and the other says it is not. Our question breaks into two, depending on whether we x the logic. For our rst type of case, let 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 x the logic and ponder disputes concerning that logic. We imagine two folks who disagree – or seem to disagree – whether is valid in L, where L is, say, a particular relevant logic. We start with the second sort of case, disagreements that concern a xed 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 a eld, 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 is fully formalized. One of our characters says that is valid
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in the given logic L and the other says that 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 dened in terms of a deductive system or that there is a completeness theorem for it. So L can be classical rstorder logic, intuitionistic logic, or one of the various relevant and paraconsistent logics that are given axiomatically. So our disputants di er on whether there is a deduction whose undischarged premises are among the premises of and whose last line is the conclusion of . So, up to Church s thesis, our disputants di e r over a Σ -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 9 -sentences (or, equivalently, Π -sentences). Now suppose that our xed 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 has no premises and that its conclusion is, in e ect, the continuum hypothesis (see Shapiro 1991: 105). So is valid if and only if the continuum hypothesis is true. So, in e ect, our disputants di er over the truth of the continuum hypothesis. Is that dispute cognitively blame worthy? Surely, that would take us too far a eld (but see Shapiro 2000, 2007, 2011, 2012), and we will leave this case with the reduction. Let us briey consider the analogues of our question concerning cognitive command with our other examples of folk-relative predicates. Suppose that two judges di er 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 xed 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 di er 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 di erently. 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 xed. To focus on a specic example, let be an instance of the law of excluded middle (with no premises). Let h be a classicist who says that is valid and let b be an intuitionist who insists that 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 ’
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di erent, 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 xed, 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 briey 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 di erent in di erent 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 di erent from the content expressed by the same sentence, uttered by my wife at the same time. Intuitively, the rst one says that I am hungry (then) and the second says that she is hungry (then). Clearly, these are di erent 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 di erent in the two contexts. In the rst, 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 di erence between non-indexical contextualism and assessment-sensitive relativism is made manifest by the phenomenon of retraction. That di erence does not matter here, and we can lump non-indexical contextualism and assessment-sensitive 9
Nearly all terms have di erent extensions in di erent 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 is valid and b says that is not valid. Recall that 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 rst about the content of the argument . 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 e ect 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 a ront to which the challenge gives rise is quickly allayed by a resolve to take no notice. The challenger must mean something di erent 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 di erent perspective, W. V. O. Quine ( 1986: 81) also holds that the various connectives change their content in the di erent 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 specic language. Presumably, the meaning of the logical terms di e rs in the di erent languages. So the Dummett–Quine–Carnap perspective has it that we have a kind of indexical contextualism here. The logical terms themselves have di erent 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 :C Φ is valid, while b holds that Φ_I :I Φ 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 suciently open-minded, h and b might agree that Φ_C :C Φ is valid and that Φ_I :I Φ 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 di erence with Dummett– Quine–Carnap well: If accepting di erent logics commits one to accepting di erent 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 : AR 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 AB
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 quantiers). 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 is valid and b says that is invalid. On the option considered now, championed by Beall and Restall, we have that h and b mean the same thing by . 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 valid x if and only if, in every case x 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 lled 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 , 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 and we have to assume that they assign the same content to the word “valid”. Given that 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 x 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 x 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.
chapter 4
Logic, mathematics, and conceptual structuralism Solomon Feferman
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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 speci c ones the integer and real number systems are taken for granted, and among general ones, notions of nite and innite 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 speci c kinds of structures: groups, rings, elds, 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 (PA ), 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 specically with the natural numbers, PA 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 quantication. The case of PA requires an aside; in that system the quanti ers are applied to both the rst-order and second-order variables. But we must be careful to distinguish the logic of quantication over the second-order variables as it is applied formally within PA 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 rst-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 rst-order variables has some specied interpretation. As an applied second-order formal system, PA may equally well be considered to be a two-sorted rst-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 2
2
2
2
2
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of the form 9 X 8 x [ x 2 X $ A( x ,. . .)] where A is an arbitrary formula of the language of PA 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 rst-order. Now, it is a remarkable fact that all the formal systems that have been set up to model mathematical practice are in e ect based on rst-order logic, more specically 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 rst-order logic by, for example, modal operators, the purpose of such has been philosophical. These operators are not used by mathematicians as basic or de ned 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, nitism, predicativism, naturalism, and structuralism. Roughly speaking, in all of these except for constructivism, nitism, and formalism, classical rst-order logic is either implicitly taken for granted or explicitly accepted. In constructivism (of the three exceptions) the logic is intuitionistic, i.e. it di ers 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 nitism, the logic is restricted to quanti er-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 scientic 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
‘
“
’
”
1
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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.
2.
3.
4 .
5.
6. 7.
8.
9.
2
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. 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. 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. Some signicant 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. Basic conceptions di er in their degree of clarity or deniteness. 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 denite conceptions. What is clear in a given conception is time dependent, both for the individual and historically. Pure (theoretical) mathematics is a body of thought developed systematically by successive renement and reective expansion of basic structural conceptions. 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. 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.
This section is largely taken from Feferman ( 2009), with a slight rewording of theses 5 and 10.
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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 conicting conclusions are eventually ltered 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 ( N þ, 1, þ Sc , <), where N is generated from the initial unit 1 by closure under the successor operation Sc , and m < n if m precedes n in the generation procedure. Certain facts about this structure (if one formulates them explicitly at all), are evident: that < is a total ordering of N þ for which 1 is the least element, and that m < n implies Sc (m) < Sc (n). Reecting on a given structure may lead us to elaborate it by adjoining further relations and operations and to expand basic principles accordingly. For example, in the case of N þ, thinking of concatenation of tallies immediately leads us to the operation of addition, m þ n, and that leads us to m n as m added to itself n times. The basic properties of the þ and operations such as commutativity, associativity, distributivity, and cancellation are initially recognized only implicitly. We may then go on to introduce more distinctively mathematical notions such as the relations of divisibility and congruence and the property of being a prime number. In this language, a wealth of interesting mathematical statements can already be formulated and investigated as to their truth or falsity, for example, that there are innitely many twin prime numbers, that there are no odd perfect numbers, Goldbach s conjecture, and so on. The conception of the structure ( N þ, 1, Sc , <, þ, ) is so intuitively clear that (again implicitly, at least) there is no question in the minds of mathematicians as to the denite meaning of such statements and the “
”
’
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assertion that they are true or false, independently of whether we can establish them in one way or the other. (For example, it is an open problem whether Goldbach’s conjecture is true.) In other words, realism in truth values is accepted for statements about this structure, and the application of classical logic in reasoning about such statements is automatically legitimized. Despite the “subjective” source of the positive integer structure in the collective human understanding, it lies in the domain of objective concepts and there is no reason to restrict oneself to intuitionistic logic on subjectivist grounds. Further re ection on the structure of positive integers with the aim to simplify calculations and algebraic operations and laws leads directly to its extension to the structure of natural numbers (N , 0 , Sc , <, þ, ), and then the usual structures for the integers Z and the rational numbers Q . The latter are relatively re ned conceptions, not basic ones, but we are no less clear in our dealings with them than for the basic conceptions of N þ. At a further stage of reection we may recognize the least number principle for the natural numbers, namely if P (n) is any well-dened property of members of N and there is some n such that P (n) holds then there is a least such n. More advanced reection leads to general principles of proof by induction and de nition by recursion on N . Furthermore, the general scheme of induction, P ð0Þ ^ 8n½P ðnÞ ! P ðSc ðnÞÞ ! 8nP ðnÞ
,
is taken to be open-ended in the sense that it is accepted for any de nite property P of natural numbers that one meets in the process of doing mathematics, no matter what the subject matter and what the notions used in the formulation of P . The question – What is a de nite property? – requires in each instance the mathematician’s judgment. For example, the property, “n is an odd perfect number, ” is denite, while “n is a feasibly computable number ” is not, nor is “n is the number of grains of sand in a heap.” Turning now to the continuum, in Feferman ( 2009) I isolated several conceptions of it ranging from the straight line in Euclidean geometry through the system of real numbers to the set of all subsets of the natural numbers. The reason that these are all commonly referred to as the continuum is that they have the same cardinal number; however, that ignores essential conceptual di erences. For our purposes here, it is su cient to concentrate on the last of these concepts. The general idea of set or collection of objects is of course ancient, but it only emerged as an object of mathematical study at the hands of Georg Cantor in the 1870s. Given
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the idea of an arbitrary set X of elements of any given set D , considered independently of how membership in X may be dened, we write S (D ) for the conception of the totality of all subsets X of D . Then the continuum in the set-theoretical sense is simply that of the set S (N ) of all subsets of N . This may be regarded as a two-sorted structure, (N , S (N ), 2), where 2 is the relation of membership of natural numbers to sets of natural numbers. Two principles are evident for this conception, using letters ‘ X ’, ‘Y ’ to range over S (N ) and ‘n’ to range over N . I. Extensionality 8 X 8 Y [8n(n 2 X $ n 2Y ) ! X = Y ] II. Comprehension For any denite property P (n) of members of N , 9 X 8n½n 2 X $ P ðnÞ:
What is problematic here for conceptual structuralism is the meaning of “all” in the description of S (N ) as comprising all subsets of N . According to the usual set-theoretical view, S (N ) is a de nite totality, so that quantication over it is well-determined and may be used to express denite properties P . But again that requires on the face of it a realist ontology and in that respect goes beyond conceptual structuralism. So if we do not subscribe to that, we may want to treat S (N ) as indenite in the sense that it is open-ended. Of course this is not to deny that we recognize many properties P as denite such as – to begin with – all those given by rstorder formulas in the language of the structure (N , 0, Sc , <, þ, ) (i.e. those that are ordinarily referred to as the arithmetical properties); thence any sets dened by such properties are recognized to belong to S (N ). Incidentally, even from this perspective one can establish categoricity of the Extensionality and Comprehension principles for the structure (N , S (N ), 2) relative to N in a straightforward way as follows. Suppose given another structure (N , S 0 (N ), 20 ), satisfying the principles I and II, using set variables ‘ X 0 ’ and ‘Y 0 ’ ranging over S 0 (N ). Given an X in S (N ), let P (n) be the denite property, n 2 X . Using Comprehension for the structure (N , S 0 (N ), 20 ), one obtains existence of an X 0 such that for all n in N , n 2 X i n 2 X 0 ; then X 0 is unique by Extensionality. This gives a 1–1 map of S (N ) into S 0 (N ) preserving N and the membership relation; it is seen to be an onto map by reversing the argument. This is to be compared with the standard set-theoretical view of categoricity results as exemplied, for example, in Shapiro ( 1997) and Isaacson ( 2011). According to that view, the subject matter of mathematics is structures, and the mère structures of mathematics such as the natural numbers, the continuum (in one of its various guises), and suitable initial segments of the cumulative hierarchy of
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sets are characterized by axioms in full second-order logic; that is, any two structures satisfying the same such axioms are isomorphic.3 On that account, the proofs of categoricity in one way or another then appeal prima facie to the presumed totality of arbitrary subsets of any given set.4 Even if the deniteness of S (N ) is open to question as above, we can certainly conceive of a world in which S (N ) is a denite totality and quantication over it is well-determined; in that ideal world, one may take for the property P in the above Comprehension Principle any formula of full second-order logic over the language of arithmetic. Then a number of theorems can be drawn as consequences in the corresponding system PA 2, including purely arithmetical theorems. Since the truth denition for arithmetic can be expressed within PA 2 and transnite induction can be proved in it for very large recursive well-orderings, PA 2 goes in strength far beyond PA even when that is enlarged by the successive adjunction of consistency statements transnitely iterated over such well-orderings. What condence are we to have in the resulting purely arithmetical theorems? There is hardly any reason to doubt the consistency of PA 2 itself, even though by Gödel s second incompleteness theorem, we cannot prove it by means that can be reduced to PA 2. Indeed, the ideal world picture of (N , S (N ), 2) that we have been countenancing would surely lead us to say more, since in it the natural numbers are taken in their standard conception. On this account, any arithmetical statement that we can prove in PA 2 ought simply to be accepted as true. But given that the assumption of S (N ) as a denite totality is a purely hypothetical and philosophically problematic one, the best we can rightly say is that in that picture , everything proved of the natural numbers is true. ’
3
Those who subscribe to this set-theoretical view of the categoricity results may di er on whether the existence of the structures in question follows from their uniqueness up to isomorphism. Shapiro (1997), for example, is careful to note repeatedly that it does not, while Isaacson (2011) apparently asserts that it does (cf., e.g., Isaacson 2011, p. 3). In any case, it is of course not a logical consequence. 4 In general, proofs of categoricity within formal systems of second-order logic can be analyzed to see just what parts of the usual impredicative comprehension axiom scheme are needed for them. In the case of the natural number structure, however, it may be shown that there is no essential dependence at all, in contrast to standard proofs. Namely, Simpson and Yokoyama (2012) demonstrate the categoricity of the natural numbers (as axiomatized with the induction axiom in second-order form) within the very weak subsystem WKL0 of PA 2 that is known to be conservative over PRA (Primitive Recursive Arithmetic). By comparison, it is sketched in Feferman ( 2013) how to establish categoricity of the natural numbers in its open-ended schematic formulation in a simpler way that is also conservative over PRA. For an informal discussion of the categoricity of initial segments of the cumulative hierarchy of sets in the spirit of open-ended axiom systems, see D. Martin ( 2001, sec. 3 ).
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Incidentally, all of this and more comes into question when we move one type level up to the structure (N , S (N ), S (S (N )), 2 , 2 ) in which Cantor s continuum hypothesis may be formulated. A more extensive discussion of the conception of that structure and the question of its deniteness in connection with the continuum problem is given in Feferman (2011). We shall also see below how taking N and S (N ) to be denite but S (S (N )) to be open-ended can be treated in suitable formal systems. 1
2
’
4.
Where and why classical rst-order logic?
Logic, as I a rmed at the outset, is supposed to provide us with all those forms of reasoning that lead invariably from truths to truths, i.e. it is given by an essential combination of inferential and semantical notions. But from the point of view of conceptual structuralism, the classical notion of truth in a structure need not be applicable unless we are dealing with a conception (such as that of the structure of natural numbers) for which the basic domains are denite totalities and the basic notions are de nite operations, predicates, and relations. It is clear that at least the classical rst-order predicate calculus should be admitted both on semantical and inferential grounds, since we have Gödel s completeness theorem to provide us with a complete inferential system. But why not more ? For example, model-theorists have introduced generalized quantiers such as the cardinality quantiers (Q x )P ( x ) expressing that there are at least κ individuals x satisfying the property P , where κ is any innite cardinal; one could certainly consider adjoining those to the rst-order formalism. A much more general class of quanti ers dened by set-theoretical means was introduced by Lindström ( 1966); each of those can be used to extend rst-order logic with a model-theoretic semantics for arbitrary rst-order domains. But for which such extensions do we have a completeness theorem like that of Gödel s for rst-order logic? It is well known that no such theorem is possible for the quanti er (Q x )P ( x ) which expresses that there are innitely many x such that P ( x ). For, using that quanti er and thence its dual ( there are just nitely many x such that P ( x ) ) we can characterize the structure of natural numbers up to isomorphism, so all the truths of that structure are valid sentences in the logic. But the set of such truths is not e ectively enumerable, indeed far from it, so it is not given by an e ectively specied formal system of reasoning. Surprisingly, Keisler (1970) obtained a completeness theorem for the quantier (Q x )P ( x ) when κ is any uncountable cardinal; as it happens, ’
κ
’
ω
“
κ
”
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that has the same set of valid formulas as for the case that κ is the rst uncountable cardinal. In view of the leap over the case κ = ω, one may suspect that the requirement that the set of valid formulas be given by some e ective set of axioms and rules of inference is not su cient to express completeness in the usual intended sense. We need to say something more about how such axioms and rules of inference ought speci cally to be complete for a given quantier. The key is given by Gentzen s (1935) system of natural deduction NK (or sequent calculus LK) where each connective and quantier in the classical rst-order predicate calculus is specied by Introduction and Elimination rules for that operation only. Moreover, for each pair of such rules, any two connectives or quanti ers satisfying them are equivalent, i.e. they implicitly determine the operator in question. So a strengthened condition on a proposed addition by a generalized quantier Q to our rst-order language is that it be given by axioms and rules of inference for which there is at most one operator satisfying them. That was the proposal of Zucker (1978) in which he gave a theorem to the e ect that any such quantier is denable in the rst-order predicate calculus. In particular, that would apply to the Lindström quantiers. However, there were some defects in Zucker s statement of his theorem and its proof; I have given a corrected version of both in Feferman (to appear). To summarize: we have fully satisfactory semantic and inferential criteria for a logic to deal with structures whose domains are rst-order and that are completely denite in the sense described above, and these limit us to the standard rst-order classical logic. Let us turn now to conceptions of structures with second-order or higher-order domains, such as (N , S (N ), 2, . . .) where the ellipsis indicates that this augments an arithmetical structure on N such as (N , 0, Sc , <, þ, ). Again, if S (N ) is considered as a de nite totality, the classical notion of truth is applicable and the semantics of second-order logic must be accepted. But as is well known there is no complete inferential system that accompanies that, since again the arithmetical structure is categorically axiomatized in this semantics and in consequence the set of its truths is not e ectively enumerable. In any case, as I have argued above, S (N ) ought not to be considered as a denite totality; to claim otherwise, is to accept the problematic realist ontology of set theory. As Quine famously put it, second-order logic is set theory in sheep s clothing. Boolos (1975, 1984 ) tried to get around this via a reduction of second-order logic to a nominalistic system of plural quanti cation. This was incisively critiqued by Resnik in his article Second-order logic still wild : Boolos is involved in a circle: he uses second-order quantication to explain English plural ’
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quantication and uses this, in turn, to explain second-order quanti cation” (Resnik 1988, p. 83). Though the Lindström quantiers are restricted to apply to rst-order structures and thus bind only individual variables they may well be de ned using higher-order notions in an essential way, in particular those needed for the cardinality quanti ers. Another example where the syntax is rstorder on the face of it but the semantics is decidedly second-order is IF (“Independence Friendly ”) logic, due to Hintikka ( 1996). This uses formulas in whose prenex form the existentially quanti ed individual variables are declared to depend on a subset of the universally quantied individual variables that precede it in the pre x list. Explanation of the semantics of this requires the use of quantied function variables; over any given rst-order structure (D , . . .) those variables are interpreted to range over functions of various arities with arguments and values in D . Indeed, Väänänen (2001, p. 519) has proved that the general question of validity of IF sentences is recursively isomorphic to that for validity in full secondorder logic. Thus, as with the Lindström quantiers, the formal syntax can be deceptive. See Feferman (2006) for an extended critique of IF logic. 5.
Where and why intuitionistic rst-order logic?
Now let us turn to the question which logic is appropriate to structural conceptions that are taken to lack some aspect of de niteness. O hand, one might expect the answer in that case to be intuitionistic logic, but the matter is more delicate. The problem is that there is not one clear-cut semantics for it; among others that have been considered, one has the so-called BHK interpretation, Kripke semantics, topological semantics, sheaf models, etc., etc. Of these, the rst is the most principled one with respect to the basic ideas of constructivity; it is that that leads one directly to intuitionistic logic but it does not determine it via a precise completeness result. By contrast, as we shall see, not only does Kripke semantics take care of the latter but it relates more closely to the question of dealing with conceptions of structures involving possibly indenite notions and domains. For the details concerning both of these I refer to Troelstra and van Dalen (1988), a comprehensive exposition of constructivism in mathematics that includes treatments of the great variety of semantics and proof theory that have been developed for intuitionistic systems. The BHK (Brouwer –Heyting –Kolmogoro ) constructive explanation of the connectives and quanti ers is described in Troelstra and van Dalen (1988, p. 9). It uses the informal notions of construction and constructive
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proof; for each form of compound statement C necessary and sucient conditions are provided on what it is for a construction to be a proof of C , in terms of proofs of its immediate sub-statements. Namely, a proof of A ^ B is a proof of A and a proof of B ; a proof of A _ B is a proof of A or a proof of B ; a proof of A ! B is a construction that transforms any proof of A into a proof of B ; and a proof of : A is a construction that transforms any proof of A into a proof of a contradiction ⊥, i.e. is a proof of A ! ⊥. In the case of the quanti ers, where the variables range over a given domain D , a proof of ( 8 x ) A( x ) is a construction that transforms any d in D into a proof of A(d ); nally, a proof of ( 9 x ) A( x ) is given by a d in D and a proof of A(d ). (D must be a constructively meaningful domain, so that it makes sense to exhibit each individual element of D and for constructions to be applicable to elements of D .) A statement A of the rst-order predicate calculus is constructively valid according to the BHK interpretation if there is a proof of A, independently of the interpretation of the domain D and the interpretation of the predicate symbols of A in D . The axioms of intuitionistic logic in any of its usual formulations are readily recognized to be constructively valid and the rules of inference preserve constructive validity. But since there are no precise notions of proof and construction at work here, we cannot state a completeness result for the BHK interpretation. Instead, the literature uses “ weak counterexamples” to show why it is plausible on that account that a given classically valid form of statement is not constructively valid. Thus, for example, to show that A _ : A is not constructively valid as a general principle one argues that otherwise one would have a general method for obtaining for any given statement A, either a proof of A or a proof that turns any hypothetical proof of A into a contradiction. But if we had such a universal method, we could apply it to any particular statement A that has not yet been settled, such as the twin prime conjecture, to determine its truth or falsity. Similarly, the method of weak counterexamples is used informally to argue against the constructive validity of many other such schemes, for example :: A ! A, though the converse is recognized to be valid.5 Let us turn now to Kripke semantics for the language of rst-order predicate logic (Troelstra and van Dalen 1988, Ch. 2.5–2.6). A Kripke model is a quadruple (K , , D , v ) , where (i) (K , ) is a non-empty 5
Various methods of realizability, initially introduced by Kleene in 1945, can be used to give precise independence results for such schemes, but are still not complete for intuitionistic logic. Cf. Troelstra and van Dalen (1988, Ch. 4 .4 ).
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partially ordered set, (ii) D is a function that assigns to each k in K a nonempty set D (k ) such that if k k 0 then D (k ) D (k 0 ), and (iii) v is a function into f0, 1g at each k in K , each n-ary relation symbol R in the language and n-ary sequence of elements of D (k ), such that if k k 0 and d ,. . .,d 2 D (k ) and v (k , R (d ,. . .,d )) = 1 then v (k 0 , R (d ,. . .,d )) = 1 . One motivating idea for this is that the elements of K represent stages of knowledge, and that k k 0 holds if everything known in stage k is known in stage k 0 . Also, v (k , R (d ,. . .,d )) = 1 means that R (d ,. . .,d ) has been recognized to be true at stage k ; once recognized, it stays true. The domain D (k ) is the part of a potential domain that has been surveyed by stage k ; the domains may increase inde nitely as k increases or may well bifurcate in a branching investigation so that one cannot speak of a nal domain in that case. The valuation function v is extended to a function v (k , A(d ,. . .,d )) into f0, 1g for each formula A( x ,. . ., x ) with n free variables and assignment (d ,. . .,d ) to its variables in D (k ); this is done in such a way that if k k 0 and d ,. . .,d 2 D (k ) and v (k , A(d ,. . .,d )) = 1 then v (k 0 , A(d ,. . .,d )) = 1. The clauses for conjunction, disjunction, and existential quanti cation are just like those for ordinary satisfaction at k in D (k ). The other clauses are (ignoring parameters): v (k , A ! B ) = 1 i for all k 0 k , v (k 0 , A) = 1 implies v (k 0 , B ) = 1; v (k , ⊥) = 0; and v (k , 8 x A( x )) = 1 i for all k 0 k and d in D (k ), v (k 0 , A(d )) = 1. As above, we identify : A with A ! ⊥; thus v (k , : A) = 1 i for all k 0 k , v (k 0 , A) = 0. We say that k forces A if v (k , A) = 1; i.e. A is recognized to be true at stage k no matter what may turn out to be known at later stages. A formula A( x ,. . ., x ) is said to be valid in a model (K , , D , v ) if for every k in K and assignment (d ,. . .,d ) to its free variables in D (k ), v (k , A(d ,. . .,d )) = 1. Then the completeness theorem for this semantics is that a formula A is valid in all Kripke models i it is provable in the rst-order intuitionistic predicate calculus. We shall see in the next section how Kripke models can be generalized to take into account di erences as to de niteness of basic relations and domains. Satisfying as this completeness theorem may be, there remains the question whether one might not add connectives or quanti ers to those of intuitionistic logic while retaining some form of its semantics. Though intuitionistic logic is part of classical logic, the semantical and inferential criterion above for classical logic doesn t apply because of the di erences in the semantical notions. But just as for the classical case, on the inferential side each of the connectives and quanti ers of the intuitionistic rst-order predicate calculus is uniquely identied via Introduction and Elimination rules in Gentzen s natural deduction system NJ. Even more, Gentzen rst 1
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formulated the idea that the meaning of each of the above operations is given by its characteristic inferences. Actually, Gentzen claimed more: he wrote that “the [Introduction rules] represent, as it were, the ‘denitions’ of the symbols concerned ” (Gentzen 1969, p. 80). Prawitz supported this by means of his Inversion Principle (Prawitz 1965, p. 33 ): namely, it follows from the normalization theorem for NJ that each Elimination rule for a given operation can be recovered from the appropriate one of its Introduction rules when that is the last step in a normal derivation. Without subscribing at all to this proposed reduction of semantics to inferential roles, we may ask whether any further operators may be added via suitable Introduction rules. The answer to that in the negative was provided by the work of Zucker and Tragesser ( 1978) in terms of the adequacy of what they call inferential logic , i.e. of the logic of operators that can simply be marked out by Introduction rules. As they show, every such operator is de ned in terms of the connectives and quanti ers of the intuitionistic rst-order predicate calculus. To be more precise, this is shown for Introduction rules in the usual sense in the case of possible propositional operators, while in the general case of possible operators on propositions and predicates – now in accord with the BHK interpretation – “proof ” parameters and constructions on them are incorporated in the Introduction rules, but those are eventually suppressed.6 6 .
Semi-intuitionism: the logic of partially open-ended structures
An immediate generalization of Kripke structures is to allow many-sorted domains, possibly in nite in number. Let I be a collection of sorts. Then the denition of Kripke structure is modi ed to have each of K , , and D indexed by I , and the valuation function modi ed to accord with the di erent sorts. Thus we deal with n-tuples k = (k 1,. . .,k ) where k is of specied sort i ; the relation then holds between such n-tuples if it holds term-wise. Of course the basic predicates come with speci ed arities to show what sorts of objects they relate, and the variables in the rst-order language over these predicates are always of a specied sort. Then the denition of the valuation function on arbitrary formulas for a manysorted structure (K , I , , D , v ) proceeds in the same way as above. Now an n
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n-ary relation R may be considered to be de nite if v (k, R (d ,. . .,d n)) = v (k 0 , 0) R (d ,. . .,d n)) whenever k k 0 . A domain D i is de nite if D i(k ) = D i(k for all k and k 0 in K i, otherwise inde nite or open-ended . While the formulas valid in the structure obey intuitionistic logic in general, one may apply classical logic systematically to formulas involving denite relations as long as the quantied variables involved range only over denite domains. This is illustrated by reasoning about the ordinary two-sorted structure (N , S (N ), 2, . . .) where (N , . . .) is conceived of as de nite with denite relations, while S (N ) is conceived of as open-ended. To treat this as a two-sorted Kripke structure, take I = f0, 1g where N is of sort 0 and S (N ) is of sort 1. We may as well take K to consist of a single element, while K could be indexed by all collections k of subsets of N , ordered by inclusion. Now the membership relation is denite because sets are taken to be de nite objects, i.e. if X is in both the collections k and k 0 then n 2 X holds in the same way whether evaluated in k or in k 0 . So classical logic applies to all formulas A that contain no bound set variables, though they may contain free set variables, i.e. A is what is usually called a predicative formula . But when dealing with formulas in general, only intuitionistic logic is justi ed on this picture. This leads us to the consideration of semi- intuitionistic (or semi-constructive ) theories in general, i.e. theories in which the basic underlying logic is intuitionistic, but classical logic is taken to apply to a class of formulas distinguished by containing denite predicates and quantied variables ranging over denite domains. A number of such theories have been treated in the paper Feferman (2010), corresponding to di erent structural notions in which certain domains are taken to be de nite and others inde nite. They fall into three basic groups: (i) predicative theories, (ii) theories of countable (tree) ordinals, and (iii) theories of sets. The general pattern is that in each case one has a semi-intuitionistic version of a corresponding classical system, and they are shown to be proof-theoretically equivalent and to coincide on the classical part. Moreover, the same holds when the semi-intuitionistic system is augmented by various principles such as the Axiom of Choice (AC) that would make the corresponding classical system much stronger. It is not possible here to explain the results in adequate detail, so only some of the ideas behind the formulations of the systems involved are sketched. The reader who prefers to avoid even the technicalities that remain can easily skim (or even skip) the rest of this section. 1
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Semi-intuitionistic predicative theories
Here the language of arithmetic is extended by variables for function(al)s in all nite types; following Gödel (1958, 1972) in his so-called Dialectica interpretation, we also add primitive recursive functionals in all nite types. In many-sorted intuitionistic logic, the system obtained is denoted HA . In the process of obtaining reduction to a quanti er-free system, Gödel showed that this system is of the same strength as Peano Arithmetic, PA; in fact the same holds for HA þ AC. Now the latter is turned into a semi-intuitionistic system by adding the Law of Excluded Middle for all arithmetical formulas. For the proof-theoretical work on that, it proves to be more convenient to add the least-number operator and an axiom () that says that when the operator is applied to a function f : N N for which there exists an n with f (n) = 0, it yields the least such n. Under this axiom, all arithmetical formulas become equivalent to quantier-free (QF) formulas, for which the LEM then holds. Thus one is led to consider HA þ AC þ (), which turns out to be proof-theoretically equivalent to PA þ QF-AC þ (), and both are equivalent to ramied analysis through all ordinals less than Cantor s ordinal ε . If one adds the Bar Rule for arithmetical orderings in both the semi-intuitionistic and the classical systems, we obtain systems of proof-theoretical strength full predicative analysis, i.e. ramied analysis up to the least impredicative ordinal Γ . (The Bar Rule on an ordering allows us to infer trans nite induction w.r.t. arbitrary formulas from well-foundedness of the ordering.) On the other hand, if in the basic system we restrict the primitive recursive functionals to those with values in N and restrict induction to QF formulas, we obtain a semi-intuitionistic system Res-HA þ AC þ () that turns out to be of exactly PA in strength. ω
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Semi-intuitionistic theories of countable tree ordinals
By countable tree ordinals one means the members of the open-ended collection O of countably branching well-founded trees. Add a sort for the members of O to the preceding systems; extend the higher type variables accordingly; add the operator of supremum that joins a sequence of trees f : N O into a single tree sup( f ) in O; add the inverse operator that takes each sup( f ) in O and n in N and produces f (n); and, nally, add operators for transnite recursion on O. The resulting system is denoted SOO in intuitionistic logic and CO O in classical logic; then SO O þ () is a semi-intuitionistic system intermediate between these two. The main !
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result in this case is that the following are of the same proof-theoretical strength: SOO þ AC þ (), COO þ QF-AC þ (), and ID , the theory of arbitrary arithmetical inductive de nitions. It is known that the latter has the same proof-theoretical strength in intuitionistic logic as in classical logic. 1
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Semi-intuitionistic theories of sets
We turn nally to the picture of the cumulative hierarchy structure, the standard classical view of which leads us to the system ZFC, i.e. ZF þ AC. However, if we identify denite totalities with sets then by Russell’s paradox, the “universe V of all sets” must be considered to be an openended indenite totality if we are to avoid contradiction. But in the Separation Axiom scheme for ZF, 8a 9b 8 x [ x 2 b $ x 2 a ^ A( x )], one allows the formula A to contain bound variables that range without restriction over V , and hence in general do not represent de nite properties; the same criticism applies to the formulas A( x , y ) in the Replacement Axiom scheme. By a formula is meant one in which all quanti ed variables are restricted, i.e. take the form 8 y ( y 2 x ! . . .) or 9 y ( y 2 x ^ . . .), written respectively ( 8 y 2 x )(. . .) and ( 9 y 2 x )(. . .). The system KP of Kripke–Platek set theory in classical logic has, like ZF, the axioms of extensionality, ordered pair, union, innity, and the scheme of trans nite induction on the membership relation. In place of the Separation Axiom scheme it takes -Separation, i.e. the Separation Axiom scheme restricted to formulas. And in place of the Replacement Axiom scheme, it takes what is called -Collection, i.e. the scheme that for each formula A, (8 x 2 a )9 yA( x , y ) ! 9b (8 x 2 a )(9 y 2 b ) A( x , y ). This implies the Replacement Axiom scheme for formulas. It is known that the system KP is of the same strength as ID . The system IKP is taken to be the same as KP but restricted to intuitionistic logic. It turns out that we can strengthen it considerably by adding a bounded form ACS of the Axiom of Choice, namely ( 8 x 2 a )9 yA ( x , y ) ! 9 f [Fun( f ) ^ (8 x 2 a ) A( x , f ( x ))], where Fun( f ) expresses that the set f is a function in the set-theoretical sense, and where now A is an arbitrary formula of the language of set theory. Under the assumption AC S we can infer Collection for arbitrary formulas and hence Replacement for arbitrary formulas. Finally, since sets are considered to be de nite totalities, we obtain a semi-intuitionistic system from IKP by adjoining the law of excluded middle for formulas. The main result of Feferman ( 2010) is that the semi-intuitionistic system IKP þ ACS þ -LEM is of the same 0
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proof-theoretical strength as KP and hence of ID 1 in its classical and intuitionistic forms. Moreover, if we add the Power Set Axiom (Pow) we obtain a system that is of strength between that of KP þ Pow and that of KP þ Pow þ (V = L).7,8 It is natural in the context of semi-intuitionistic theories T to say that a sentence A in the language of T is de nite (relative to T) if T proves LEM for A, i.e. A _ : A. A question in set theory that has caused considerable discussion in recent years is whether Cantor ’s continuum hypothesis CH is a de nite mathematical problem. One formulation of it is that every subset of S (N ) is either countable or in 1-1 correspondence with S (N ). Of course, that is denite in the theory IKP þ Pow þ 0-LEM, because quantication over subsets of S (N ) is bounded once we have existence of S (S (N )) [i.e., S (S (ω))] by the Power Set Axiom. That suggests – as I did in Feferman (2011) – considering the weaker system T = IKP þ Pow(N ) þ ACS þ 0-LEM, where Pow(N ) simply asserts the existence of S (N ) as a set. I conjectured there that CH is not de nite relative to that system.9 Of course, that would not show that CH is not a de nite mathematical problem, but it might be considered as an interesting bit of evidence in support of that. 7.
Conceptual structuralism and mathematical practice
One criterion for a philosophy of mathematics that is often heard is that it should accord with mathematical practice. It’s very hard to know just what that means since there are so many dimensions along which practice can be viewed. One particular interpretation of the criterion is that philosophers have no business telling mathematicians what does or doesn ’t exist. Famously, David Lewis wrote: I’m moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors, now that philosophy has discovered that there are no classes? (Lewis 1991, p. 59)10 7
There is a considerable literature on semi-intuitionistic theories of sets including the power set axiom going back to the early 1970s. See Feferman (2010, sec. 7.2) for references to the relevant work of Poszgay, Tharp, Friedman, and Wolf. 8 Mathias (2001) proved that KP þ Pow þ (V = L) proves the consistency of KP þ Pow, so the usual argument for the relative consistency of (V = L) doesn ’t work. 9 Michael Rathjen (2014 ) has recently veried this conjecture. 10 Curiously, this quote is from Lewis’ book, Parts of Classes , which o ers a revisionary theory of classes that di ers from the usual mathematical conception of such.
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But this is a caricature of what philosophy is after; philosophers take for granted that mathematicians have settled problematic individual questions of existence like zero, negative numbers, imaginary numbers, in nitesimals, points at in nity, probability of subsets of [ 0, 1], etc., etc., using purely mathematical criteria in the course of the development of their subject. The existence of some of these has been established by reduction to objects whose existence is unquestioned, some by quali ed acceptance, and some not at all. But what the philosopher is concerned with is, rather, to explain in what metaphysical sense, if any, mathematical objects exist, in a way that cannot even be discussed within ordinary mathematical parlance. Lewis could equally well have laughed at the idea that some general principles accepted in the mathematical mainstream such as the Law of Excluded Middle or the Axiom of Choice would be dismissed as false (or unjustied) for philosophical reasons. But again, the use of truth in ordinary mathematical parlance is deationary and the reasons for accepting such and such principles as true has either been made without question or for mathematical reasons in the course of the development of the subject. The philosopher, by contrast, is concerned to explain in what sense the notion of truth is applicable to mathematical statements, in a way that cannot be considered in ordinary mathematical parlance. Whether the mathematician should pay attention to either of these aims of the philosopher is another matter. Conceptual structuralism addresses the question of existence and truth in mathematics in a way that accords with both the historical development of the subject and each individual s intellectual development. It crucially identies mathematical concepts as being embedded in a social matrix that has given rise, among other things, to social institutions and games; like them, mathematics allows substantial intersubjective agreement, and like them, its concepts are understood without assuming rei cation. What makes mathematics unique compared to institutions and games is its endless fecundity and remarkable elaboration of some basic numerical and geometrical structural conceptions. To begin with, mathematical objects exist only as conceived to be elements of such basic structures. The direct apprehension of these leads one to speak of truth in a structure in a way that may be accepted uncritically when the structure is such as the integers but may be put into question when the conception of the structure is less denite as in the case of the geometrical plane or the continuum, and ’
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should be put into question when it comes to the universe of sets. One criticism of conceptual structuralism that has been made is that it ’s not clear/denite what mathematical concepts are clear/denite, and making that a feature of the philosophy brings essentially subjective elements into play.12 Actually, conceptual structuralism by itself, as presented in the theses 1–10, takes no specic position in that respect and recognizes that di erent judgments (such as mine) may be made. Once such are considered, however, logic has much to tell us in its role as an intermediary between philosophy and mathematics. As shown in the preceding section, one can obtain denitive results about formal models of di erent standpoints as to what is denite and what is not. Moreover, the results can be summarized as telling us that to a signi cant extent, the unlimited (de facto) application of classical logic in mainstream mathematics – i.e., the logic of denite concepts and totalities – may be justied on the basis of a more rened mixed logic that is sensitive to distinctions that one might adopt between what is denite and what is not. 13 In other words, once more they show that, at least to that extent, you can have your cake and eat it too. There are other dimensions of mathematical practice that reward metamathematical study motivated by the philosophy of conceptual structuralism. One, in particular, that I have emphasized over the years is the open-ended nature of certain principles such as that of induction for the integers and comprehension for sets. This accords with the fact that in the development of mathematics what concepts are recognized to be denite evolve with time. Thus one cannot x in advance all applications of these open-ended schematic principles by restriction to those instances denable in one or another formal language, as is currently done in the study of formal systems. This leads instead to the consideration of logical models of practice from a novel point of view that yet is susceptible to metamathematical study. One such is via the notion of the unfolding of open-ended schematic axiom systems , that is used to tell us everything that ought to be accepted if one has accepted given notions and principles. Thus far, denitive results about the unfolding notion have been obtained by Feferman and Strahm ( 2000, 2010) for schematic systems of non- nitist and nitist arithmetic, resp., and by Buchholtz ( 2013) for arithmetical 12
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In particular, this criticism has been voiced by Peter Koellner in his comments on Feferman ( 2011); cf. http://logic.harvard.edu/EFI_Feferman_comments.pdf. These kinds of logical results can also be used to throw substantive light on philosophical discussions as to the problem of quantication over everything (or over all ordinals, or all sets) such as are found in Rayo and Uzquiano (2006).
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inductive denitions. As initiated in Feferman ( 1996), I am optimistic that it can be used to elaborate Gödel s program for new axioms in set theory and in particular to draw a sharper line between which such axioms ought to be accepted on intrinsic grounds and those to be argued for on extrinsic grounds. ’
AC KNOWLE DG EME NTS
I would like to thank Gianluigi Bellin, Dag nn Føllesdal, Peter Koellner, Grigori Mints, Penelope Rush, Stewart Shapiro, and Johan van Benthem for their helpful comments on a draft of this essay.
chapter 5
A Second Philosophy of logic Penelope Maddy
What’s hidden in my hand is either an ordinary dime or a foreign coin of a type I’ve never seen. (I drew it blindfolded from a bin lled with just these two types of objects.) It ’s not a dime. (I can tell by the feel of it.) Then, obviously, it must be a foreign coin! But what makes this so? It’s common to take this query as standing in for more general questions about logic – what makes logical inference reliable? what is the ground of logical truth? – and common, also, to regard these questions as properly philosophical, to be answered by appeal to distinctively philosophical theories of abstracta, possible worlds, concepts, meanings, and the like. What I ’d like to do here is step back from this hard-won wisdom and try to address the simple question afresh, without presumptions about what constitutes ‘logic’ or even ‘philosophy ’. The thought is to treat inquiries about reliability of the coin inference and others like it as perfectly ordinary questions, in search of perfectly ordinary answers, and to see where this innocent approach may lead. To clarify what I have in mind here, let me introduce an unassuming inquirer called the Second Philosopher, interested in all aspects of the world and our place in it. She begins her investigations with everyday perceptions, gradually develops more sophisticated approaches to observation and experimentation that expand her understanding and sometimes serve to correct her initial beliefs; eventually she begins to form and test hypotheses, and to engage in mature theory-formation and con rmation; along the way, she nds the need for, and pursues, rst arithmetic and geometry, then analysis and even pure mathematics; and in all this, she often pauses to reect on the methods she’s using, to assess their 1
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The Second Philosopher is introduced in (Maddy 2007), and her views on logic detailed in Part III of that book. The discussion here reworks and condenses the presentation there (see also (Maddy to appear)). For more on the Second Philosopher ’s approach to mathematics, see (Maddy 2011). 93
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e ectiveness and improve them as she goes. When I propose to treat the question of the reliability of the coin inference as an ordinary question, I have in mind to examine it from the Second Philosopher ’s point of view. She holds no prior convictions about the nature of the question; she sees it simply as another of her straightforward questions about the world and her investigations of it. The rst thing she’s likely to notice is that neither the reliability of the coin inference nor the truth of the corresponding if –then statement3 depends on any details of the physical composition of the item in her hand or the particular properties that characterize dimes as opposed to other coins. She quickly discerns that what ’s relevant is entirely independent of all but the most general structural features of the situation: an object with one or the other of two properties that lacks one must have the other. In her characteristic way, she goes on to systematize this observation – for any object a and any properties, P and Q , if Qa -or-Pa and not-Qa , then Pa – and from there to develop a broader theory of forms that yield such highly general forms of truth and reliable inference. In this way, she ’s led to consider any situation that consists of objects that enjoy or fail to enjoy various properties, that stand and don’t stand in various relations; she explores conjunctions and disjunctions of these, and their failures as well; she appreciates that one situation involving these objects and their interrelations can depend on another; and eventually, following Frege, she happens on the notion that a property or relation can hold for at least one object, or even universally – suppose she dubs this sort of thing a 4 ‘formal structure’. Given her understanding of the real-world situations she’s out to describe in these very general, formal terms, she sees no reason to suppose that every object has precise boundaries – is this particular loose hair part of the cat or not? – or that every property (or relation) must determinately hold or fail to hold of each object (or objects) – is this growing tadpole now a frog or not? She appreciates that borderline cases are common and fully determinate properties (or relations) rare. Thinking along these lines, she ’s led to something like a Kleene or Lukakasiewicz three-valued system: for a given object (or objects), a property (or relation) might hold, fail, or be indeterminate; not-(. . .) obtains if (. . .) fails and is otherwise indeterminate; (. . .)-and-(__) obtains if both ( . . .) and (__) obtain, fails if one of them fails, and is otherwise indeterminate; and so on through the obvious 3 4
I won’t distinguish between these, except in the vicinity of footnote 5. In (Maddy 2007) and (Maddy to appear), this is called KF-structure, named for Kant and Frege.
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clauses for (. . .)-or-(__), (there is an x , . . .x . . .) and (for all x, . . .x . . .). A formal structure of this sort validates many of the familiar inference patterns – for example, the introduction and elimination rules for ‘not’, ‘and’, ‘or ’, ‘for all’, and ‘there exists’; the DeMorgan equivalences; and the distributive laws – but the gaps produce failures of the laws of excluded middle and non-contradiction (if p is indeterminate, so are p -or-not- p and not-( p-and-not- p).5 The subtleties of the Second Philosopher ’s dependency relation undercut many of the familiar equivalences: not-(the rose is red)-or-2 þ 2 = 4 , but 2 þ 2 doesn’t equal 4 because the rose is red. Fortunately, modus ponens survives: when both ( q depends on p) and p obtain, q can’t fail or be indeterminate. Suppose the Second Philosopher now codies these features of her formal structures into a collection of inference patterns; coining a new term, she calls this ‘rudimentary logic’ (though without any preconceptions about the term ‘logic’). She takes herself to have shown that this rudimentary logic is satised in any situation with formal structure. This is a considerable advance, but it remains abstract: what ’s been shown is that rudimentary logic is reliable, assuming the presence of formal structure. Common sense clearly suggests that our actual world does contain objects with properties, standing in relations, with dependencies, but the Second Philosopher has learned from experience that common sense is fallible and she routinely subjects its deliverances to careful scrutiny. What she nds in this case is, for example, that the region of space occupied by what we take to be an ordinary physical object like the coin does di er markedly from its surroundings: it contains a more dense and tightly organized collection of molecules; the atoms in those molecules are of di erent elements; the contents of that collection are bound together by various forces that tend to keep it moving as a group; other forces make the region relatively impenetrable; and so on. Similarly, she con rms that objects have properties, stand in relations, and that situations involving them exhibit dependencies. Now it must be admitted that there are those who would disagree, who would question the existence of ordinary objects, beginning with Eddington and his famous two tables: One of them is familiar to me from my earliest years. . . . It has extention; it is comparatively permanent; it is coloured; above all it is substantial . By 5
Here, briey, the distinction between logical truths and valid inferences matters, because the gaps undermine all of the former. Inferences often survive because gaps are ruled out when the premises are taken to obtain.
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So far, the Second Philosopher need have no quarrel; Eddington can be understood as putting poetically what she would put more prosaically: science has taught us some surprising things about the table, its properties and behaviors. But this isn’t what Eddington believes: Modern physics has by delicate test and remorseless logic assured me that my second scientic table is the only one which is really there. (Eddington 1928: xii)
The Second Philosopher naturally wonders why this should be so, why the so-called ‘scientic table’ isn’t just a more accurate and complete description of the ordinary table.6 In fact, it turns out that ‘substance’ in Eddington’s description of table No. 1 is a loaded term: It [is] the intrinsic nature of substance to occupy space to the exclusion of other substance. (Eddington 1928: xii) There is a vast di erence between my scientic table with its substance (if any) thinly scattered in specks in a region mostly empty and the table of everyday conception which we regard as the type of solid reality . . . It makes all the di erence in the world whether the paper before me is poised as it were on a swarm of ies . . . or whether it is supported because there is substance below it. (Eddington 1928: xi–xii)
Here Eddington appears to think that being composed of something like continuous matter is essential to table No. 1, that one couldn’t come to 6
Some writers reject the ordinary table on the grounds that its boundaries would be inexact. As we ’ve seen, the Second Philosopher is happy to accept this sort of ‘ worldly vagueness’.
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realize that its supporting the paper or resisting my elbow arise very di erently than I might have at rst imagined – that one couldn’t come to realize this, that is, without also coming to realize that there is no such thing as table No. 1. But why should this be so? Why should our initial conceptualization be binding in this way? For that matter, is it even clear that our initial conceptualization includes any account at all of how and why the table supports paper or resists elbows? The Second Philosopher sees no reason to retract her belief in ordinary macro-objects. 7 So let’s grant the Second Philosopher her claim that formal structure as she understands it does turn up in our actual world. This means not only that rudimentary logic applies in such cases, but that it does so regardless of the physical details of the objects ’ composition, the precise nature of the properties and relations, any particular facts of spatiotemporal location, and so on. This observation might serve as the rst step on a path toward the familiar idea, noted earlier, that questions like these are peculiarly philosophical: the thought would be that if the correctness of rudimentary logic doesn ’t depend on any of the physical details of the situation, if it holds for any objects, any properties and relations, etc., then it must be quite di e rent in character from our ordinary information about the world; indeed, if none of the physical details matter, if these truths hold no matter what the particular contingencies happen to be, then perhaps they ’re true necessarily, in any possible world at all – and if that’s right, then nothing particular to our ordinary, contingent world can be what’s making them true. By a series of steps like these, one might make one ’s way to the idea that logical truths re ect the facts, not about our world, but about a platonic world of propositions, or a crystalline structure that our world enjoys necessarily, or an abstract realm of meanings or concepts, or some such distinctively philosophical subject matter. Many such 7
Eddington’s two tables may call to mind Sellars ’ challenge to reconcile ‘ the scientic image’ with ‘ the manifest image’. In fact, the manifest image includes much more than Eddington ’s table No. 1 – ‘it is the framework in terms of which, to use an existentialist turn of phrase, man rst encountered himself ’ (Sellars 1962: 6) – but Sellars does come close to our concerns when he denies that ‘ manifest objects are identical with systems of imperceptible particles’ (Sellars 1962: 26 ). He illustrates with the case of the pink ice cube: ‘the manifest ice cube presents itself to us as something which is pink through and through, as a pink continuum, all the regions of which, however small, are pink ’ (Sellars 1962: 26 ), and of course the scientic ice cube isn’t at all like this. Here Sellars seems to think, with Eddington, that science isn’t in a position to tell us surprising things about what it is for the ice cube to be (look) pink; he seems to agree with Eddington that some apparent features of the manifest ice cube can’t be sacriced without losing the manifest ice cube itself. Indeed the essential features they cling to are similar: a kind of substantial continuity or homogeneity. The Second Philosopher remains unmoved.
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options spring up in the wake of this line of thought, but ordinary facts, ordinary information about our ordinary world has been left behind, and ordinary inquiry along with it – we’ve entered the realm of philosophy proper. But suppose our Second Philosopher doesn’t set foot on this path. Suppose she simply notices that nothing about the chemical makeup of the coin is relevant, that nothing about where the coin is located is relevant, – that only the formal structure matters to the reliability of the rudimentary logic she’s isolated. From here she simply continues her inquiries, turning to other pursuits in geology, astronomy, linguistics, and so on. At some point in all this, she encounters cathode rays and black body radiation, begins to theorize about discrete packets of energy, uses the quantum hypothesis to explain the photo-electric e ect, and eventually goes on to the full development of quantum mechanics. And now she’s in for some surprises: the objects of the micro-world seem to move from one place to another without following continuous trajectories; a situation with two similar particles A and B apparently isn’t di erent from a situation with A and B switched; an object has some position and some momentum, but it can ’t have a particular position and a particular momentum at the same time; there are dependencies between situations that violate all ordinary thinking about dependencies. 8 Do the ‘objects’, ‘properties’, ‘relations’, and ‘dependencies’ of the quantum-mechanical micro-world enjoy the formal structure that underlies rudimentary logic? The Second Philosopher might well wonder, and sure enough, her doubts are soon realized. In a case analogous to, but simpler than position and momentum, she nds an electron a with vertical spin up or vertical spin down, and horizontal spin right or horizontal spin left – (Ua or Da ) and (Ra or La ) – but for which the four obvious conjunctions – (Ua and Ra ) or (Ua and La ) or (Da and Ra ) or (Da and La ) – all fail. This distributive law of rudimentary logic doesn’t obtain! We’re now forced to recognize that those very general features the Second Philosopher isolated in her formal structures actually have some bite. Though it wasn’t made explicit, an object in a formal structure was assumed to be an individual, fundamentally distinct from all others; having a property – like location, for example – was assumed to involve having a particular (though perhaps imprecise) property – a particular location, not just some location or other. These features were so obvious as to go unremarked until the anomalies of quantum mechanics came along to 8
For more on these quantum anomalies, with references, see (Maddy 2007, §III.4 ).
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demonstrate so vividly that they can in fact fail. 9 Those of us who ventured down that path the Second Philosopher didn’t take were tempted to think that her formal structure is to be found in every possible world, but it turns out it isn’t present even in every quarter of our own contingent world! 10 Rudimentary logic isn’t necessary after all; its correctness is contingent on the very general, but still not universal, features isolated in the Second Philosopher ’s formal structure. We’ve focused so far on the metaphysics – what makes these inferences reliable, these truths true? – but there’s also the epistemology – how do we come to know these things? If we followed the philosopher ’s path and succeeded in dismissing the vicissitudes of contingent world as irrelevant well before the subsequent shocks dealt the Second Philosopher by quantum mechanics, then we might continue our reasoning along these lines: if logic is necessary, true in all possible worlds, if the details of our contingent world are beside the point, then how could coming to know its truths require us to attend to our experience of this world? 11 Again a range of options ourish here, from straightforward theories of a priori knowledge 9
In yet another twist in the tradition of Eddington and Sellars, Ladyman and Ross ( 2007) begin from this observation – that the micro-world doesn’t seem to consist of individual objects – then go on to classify the ordinary table, along with the botanist ’s giant redwoods and the physical chemist’s molecules, as human constructs imposed for ‘epistemological book-keeping ’ (p. 240) on an entirely objectless world. I suspect that this disagreement with the Second Philosopher traces at least in part to di ering pictures of how ‘naturalistic’ metaphysics is to be done. The Second Philosopher ’s ‘metaphysics naturalized’ simply pursues ordinary science and ends up agreeing with the folk, the botanist and the chemist that there are tables, trees and atoms, that trees are roughly constituted by biological items like cells, cells by chemical items like molecules, molecules by atoms, and so on. She doesn’t yet know, and may never know, how to extend this program into the objectless micro world, but she has good reason to continue trying, and even if she fails, she doesn’t see that this alone should undermine our belief in the objects of our ordinary world. In contrast, the ‘ naturalized metaphysics’ of Ladyman and Ross is the work of ‘naturalistic philosophical under-labourers’ (p. 242), designed to show ‘how two or more specic scientic hypotheses, at least one of which is drawn from fundamental physics, jointly explain more than the sum of what is explained by the two hypotheses taken separately ’ (Ladyman and Ross 2007: 37) – and it’s this project that delivers the surprising result that ordinary objects are constructed by us. From their perspective, the Second Philosopher ‘metaphysics naturalized’ is just more science: the botanist and the physical chemist make no contribution to ontology; metaphysics only begins when their hypotheses are uni ed with fundamental physics. From the Second Philosopher ’s perspective, there’s no reason to suppose that ordinary objects are human projections or to insist that assessments of what there is must involve unication with fundamental physics. Indeed, from her perspective, given our current state of understanding (see below), quantum mechanics is perhaps the last place we should look for ontological guidance! 10 This incidentally removes another sort of skeptical challenge to the Second Philosopher ’s belief in ordinary macro-objects, namely, the charge that an inquiry starting with objects with properties, etc., will inevitably uncover objects with properties. 11 An inference from necessary to a priori is less automatic in our post-Kripkean age, when many philosophers recognize a posteriori necessities, but logical truth seems a poor candidate for this sort of thing. In any case, what I ’m tracing here are tempting paths, not conclusive arguments.
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to complex accounts of how logic serves to constitute inquiry and thus can’t itself be conrmed. But let’s return to the Second Philosopher ’s more naïve inquiries, still well clear of the philosopher ’s path, and ask how she answers the simple question: how do we come to know that rudimentary logical inference is reliable? In general, the Second Philosopher ’s epistemological investigations take the form of asking how human beings – as described in biology, physiology, psychology, linguistics, and so on – come to have reliable beliefs about the world – as described in physics, chemistry, botany, astronomy, and so on.12 Work in psychology, cognitive science, and the like is primary here, but the Second Philosopher ’s focus is somewhat broader; not only does she study how people come to form beliefs about the world, she also takes it upon herself to match these beliefs up with what her other inquiries have told her about how the world actually is, and to assess which types of belief-forming processes, in which circumstances, are reliable. Though her epistemology is naturalized – that is, it takes place roughly within science – it ’s also normative. In the case of rudimentary logic, the Second Philosopher ’s focus is on formal structure: her other studies of the world have revealed the existence of many objects, with properties, standing in relations, with dependencies, and she now asks how we come to be aware of these worldly features. Here she recapitulates the work of an impressive research community in contemporary cognitive science. 13 The modern study of our perception of individual objects reaches back at least to the 1930s, when Piaget used experiments based on manual search behavior 14 to argue that a child reaches the adult conception of a permanent, external object by a series of stages ending at about age 2. Conicting but inconclusive indications from visual tracking suggested that even younger children might have the object concept, but it wasn’t until the 1980s that a new experimental paradigm emerged for testing this possibility: habituation and preferential looking. In such an experiment, the infants are shown the same event over and over until they lose interest, as indicated by their decreased looking time (habituation); they ’re then shown one or another of two test displays, one that makes sense on the adult understanding of an object, the other 12
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This is reminiscent of many of Quine ’s descriptions of his ‘epistemology naturalized’, but Quine also tends to fall back on more traditional philosophical formulations, asking how we manage to infer our theory of the external world from sensory data (see Maddy 2007, §I.6; for more). I can only give the smallest sampling of this work here. For more, with references, see (Maddy 2007, §III.5). E.g., does the child lift a cloth to nd a desirable object she’s seen hidden there?
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inconsistent with the adult understanding; if an infant is thinking like the adult, the inconsistent display should draw a longer gaze (preferential looking). So, for example, suppose a metal screen is attached to a long hinge that extends from left to right on a stage; the screen can lie at toward the viewer on the stage surface, and it can pivot through 180 arc to lie at away from the viewer. The infant is habituated to seeing the screen move through this range of motion. Then the screen is positioned toward the infant, a box is placed behind it, and the screen is rotated backwards. The consistent display shows the screen stopping when it comes to rest on the now-hidden box; the inconsistent display shows it moving as before and coming to rest on the stage surface away from the viewer. If the infant thinks the box continues to exist even when it s hidden by the screen, and that the space it occupies can t be penetrated by the screen, then the inconsistent display should draw the longest gaze. (Notice that the inconsistent display is exactly the one the infant has been habituated to, so its very inconsistency would be su ciently novel to overcome the habituation.) In this early use of the new paradigm, this is exactly what was observed in infants around ve months of age. Obviously this is only the beginning of the story. For example, does the infant understand the box as an individual object, as a unit, or just as an obstacle to the screen? Experiments of similar design soon indicated that infants as young as four months perceive a unit when presented with a bounded and connected batch of stu that moves together. Now imagine a display with two panels separated by a small space. An object appears from stage left, travels behind one screen, after which an object emerges from behind the second screen, and vanishes stage right. One group of four-month-olds is habituated to seeing an object appear in the gap between the screens, as if it moved continuously throughout; another group is habituated to seeing an object disappear behind the rst screen and an object emerge from behind the second screen without anything appearing in the gap. The test displays are then without panels, showing either one or two objects. The result was that the infants habituated with the apparently continuous motion looked longer at the two-object test display than the infants habituated with the scene where no object was seen in the gap. It seems an object is regarded as an individual if its motion is continuous. Of course there s much more to this work than can be summarized here, but the current leading hypothesis is that these very young infants conceptualize individual units in these terms: they don t think that such a unit ˚
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can be in two places at once or that separate units can occupy the same space, and they expect them to travel in continuous trajectories. In the words of Elizabeth Spelke, a pioneer in this eld, the infant’s objects are ‘complete, connected, solid bodies that persist over occlusion and maintain their identity through time ’ (Spelke 2000: 1233): Putting together the ndings from studies of perception of object boundaries and studies of perception of object identity, young infants appear to organize visual arrays into bodies that move cohesively (preserving their internal connectedness and their external boundaries), that move together with other objects if and only if the objects come into contact, and that move on paths that are connected over space and time. Cohesion, contact, and continuity are highly reliable properties of inanimate, material objects: objects are more likely to move on paths that are connected than they are to move at constant speeds, for example; and they are more likely to maintain their connectedness over motion than they are to maintain a rigid shape. Infants’ perception appears to accord with the most reliable constraints on objects. (Spelke et al. 1995: 319–320)
Partly because so much of this research depends on experiments conducted with habituation/preferential-looking and closely related designs, partly for other reasons,15 these conclusions can’t be taken as irrevocably established, but then the fallibility of ongoing science is an occupational hazard for the Second Philosopher. Let’s take the early emergence of a modest human ability to detect (some of ) the world’s individual objects as a tentative datum. As for properties and relations, the infants’ sensitivity to these plays a role in the habituation/preferential-looking studies mentioned earlier: habituating to green objects then preferentially looking at red ones must involve noticing those colors, likewise the spatial relations of objects and the screens. What’s surprising is that object properties aren ’t initially used to individuate them. Ten-month-old infants watched as a toy duck emerged from the left side of a single screen, followed by a ball emerging from the right side of the screen; one of the two test displays then showed the duck and the ball, the other just the duck – and no signicant di erence between their reactions was found! The same experiment run 15
E.g., Hateld (2003) argues that the ndings of Spelke and her collaborators only establish that young infants perceive ‘bounded trackable volumes’ not ‘individual material objects’. Of course, Spelke (e.g., in Spelke et al. (1995), cited by Hateld) does allow that the infant’s object concept continues to develop in early childhood, so there is room here for clarication of levels or degrees of object perception. A question like this would prompt the Second Philosopher to get down to sorting things out, but I’m not so idealized an inquirer and leave these further investigations to others.
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on slightly older one-year-olds delivered what an adult would have expected: the test display with the duck alone drew greater attention. On reection these results aren’t so bewildering. While infants begin with simple but highly reliable spatiotemporal constraints on object identity (as Spelke notes), property distinctions require more judicious application: a red ball can turn blue and still be the same object; a human can change clothes and still be the same person. Some experience and learning must be needed for the child to realize that ducks don ’t generally turn into balls, and considerably more to reach the full adult concept: We are inclined to judge that a car persists when its transmission is replaced, but would be less inclined to judge that a dog persists if its central nervous system were replaced. . . . Because we know that dogs but not cars have behavioral and mental capacities supported by certain internal structures, we consider certain transformations of dogs to be more radical than other, super cially similar transformations of cars. (Spelke et al. 1995: 302–303)
With this in mind, it ’s less surprising that the beginnings of the child ’s identication of objects by their properties comes a couple of months later than their identication by the more straightforward spatiotemporal means, and perhaps even that this new development apparently coincides with the acquisition of their rst words – property nouns like ‘ball’ and ‘duck ’! So as not to belabor this fascinating developmental work, let me just note that similar studies have shown that young infants detect conjunctions and disjunctions of object properties, the failure of properties or relations, simple billiard-ball style causal dependencies, and so on. It’s also notable that many of these abilities found in young infants are also present, for example, in primates and birds. This suggests an evolutionary origin, and clearly the advantages conferred by the ability to track objects spatiotemporally, to perceive their properties and relations, to notice dependencies, would have been as useful on the savanna as they are in modern life. All this leaves the Second Philosopher with two well-supported hypotheses: the ability to detect (at least some of ) the formal structure present in the world comes to humans at a very early age, perhaps largely due to our evolutionary inheritance; whether by genetic endowment, normal maturation, or early experience, the primitive cognitive mechanisms underlying this ability are as they are primarily because humans (and their ancestors) interact almost exclusively with aspects of the world that display this formal structure. From here it ’s a short step to the suggestion that the presence of these primitive cognitive mechanisms, all tuned to formal
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structure, is what makes the simpler inferences of rudimentary logic strike us as so obvious. Assuming that the Second Philosopher has this right – that the formal structure is often present, that we are con gured to detect it, and that this accounts for our rudimentary logical beliefs – then a suciently externalist epistemologist might count this as a case of a priori knowledge. An epistemologist of more internalist leanings might hold that the sort of a posteriori inquiry undertaken here would be required to support actual knowledge of rudimentary logic. The Second Philosopher isn’t condent that this disagreement has a determinate solution, isn ’t condent that the debate is backed by anything more substantial than the various handy uses of ‘know ’, so she’s content to o er a fuller version of the story sketched here, and to leave the decision about ‘knowledge’ to others. Notice, incidentally, that if this is right, if the Second Philosopher ’s formal structure is so deeply involved in our most fundamental cognitive mechanisms, this explains why it ’s so dicult for us to come up with a viable interpretation of quantum mechanics, where formal structure goes awry. But this observation raises another question: if formal structure and hence rudimentary logic are missing in the micro-world, and if these are so fundamental to our thought and reasoning, how do we manage to carry out our study of quantum mechanics? Some suggest that we should adopt a special logic for quantum mechanics, 16 but the question posed here is how we manage to do quantum mechanics now, apparently using our ordinary logic. I think the answer is fairly simple: what we actually have in quantum mechanics isn’t a theory of particles with properties, in relations, with dependencies, but a mathematical model, an abstract Hilbert space with state vectors.17 This bit of mathematics displays all the necessary formal structure – it consists of objects with properties, in relations, with (logical) dependencies – so our familiar logic is entirely reliable there.18 The deep problem for the interpretation of quantum 16 17
18
See the discussion of deviant logics below. As noted above (footnote 9), Ladyman and Ross (2007) argue on grounds similar to the Second Philosopher ’s that the micro-world doesn’t consist of objects. Given her account of how our cognition and our logic work, the Second Philosopher would predict that these authors should encounter some diculty when it comes to describing the subject matter of quantum mechanics, and in fact, what they say on that score is consistent with the line taken here: ‘it is possible that dividing a domain up into objects is the only way we can think about it ’ (Ladyman and Ross ( 2007: 155); ‘ we can only represent [the non-objectual structures of the micro-world] in terms of mathematical relationships’ (Ladyman and Ross 2007: 299). In fact, the mathematical world is in some ways more amenable to our logical ways (see footnote 19 below).
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mechanics is to explain how and why the mathematical model works so well, to gure out what worldly features it’s tracking, but in the mathematics itself, our natural ways of thinking and reasoning are on impeccable footing. Now for all its advertised virtues – reliability in a wide variety of worldly settings, harmony with our most fundamental cognitive mechanisms – rudimentary logic is in fact a rather unwieldy instrument in actual use. We’ve seen, for example, that the presence of indeterminacies eliminates the law of excluded middle, the principle of non-contradiction, and indeed all logical truths. An inference rule as central as reductio ad absurdum can be seen to fail: that ( q- and-not-q ) follows from p only tells us that p is either false or indeterminate. And the substantive requirements on dependency relations undercut most of our usual manipulations with the conditional. Though he’s speaking of a full Kleene system, with a truthfunctional conditional, I think Feferman’s assessment applies to rudimentary logic as well: ‘nothing like sustained ordinary reasoning can be carried on’ (Feferman 1984 : 95). Under the circumstances, a stronger, more exible logic is obviously to be desired. The Second Philosopher has seen this sort of thing many times: she has a theoretical description of a given range of situations, but that description is awkward or unworkable in various ways. To take one example, she can give a complete molecular description of water owing in a pipe, but alas all practical calculation is impossible. In hope of making progress, she introduces a deliberate falsication – treating the water as a continuous substance – that allows her to use the stronger and more exible mathematics of continuum mechanics. She has reason to think this might work, because there should be a size-scale with volumes large enough to include enough molecules to have relatively stable temperature, energy, density, etc., but not so large as to include wide local variations in properties like these. This line of thought suggests that her deliberate falsication might be both powerful enough to deliver concrete solutions and benign enough to do so without introducing distortions that would undercut its e ectiveness for real engineering decisions. She tests it out, and happily it does work! This is what we call an ‘idealization’, indeed a successful idealization for many purposes. (It would obviously be unacceptably distorting if we were interested in explaining the water ’s behavior under electrolysis.) In similar ways, we ignore friction when its e ects are small enough to be swamped by the phenomenon we ’re out to describe; we treat slightly irregular objects as perfectly geometrical when this does no harm; and so on.
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With the technique of idealization in mind, the Second Philosopher looks for ways to simplify and streamline her theoretical account of formal structure, that is, her rudimentary logic, in ways that make it more exible, more workable, and to do so without seriously undermining its reliability. To this end, she makes two key idealizations, introduces two falsifying assumptions – that there is no indeterminacy, that any particular combination of objects and properties or relations either holds or fails; and that dependencies behave as material conditionals 19 – and at a stroke, she transforms her crude rudimentary theory into our modern classical logic. There can be no doubt that full classical logic is an extraordinarily sophisticated and powerful instrument; the only open question is whether or not the required idealizations are benign. And as in the other examples, this judgment can be expected to vary from case to case. This is where some of the so-called ‘deviant logics’ come in. Proponents of one or another of the various logics of vagueness, for example, may insist that indeterminacy is a real phenomenon, 20 may condemn ‘the lamentable tendency . . . to pretend that language is precise ’ (J. A. Burgess 1990: 434 ). On the rst point, the Second Philosopher agrees – indeterminacy is real – but she views the classical logician ’s pretending otherwise as no di erent in principle than the engineer ’s pretending that water is a continuous uid; what determines the acceptability of either pretense isn’t the obvious fact that it is a pretense, but whether or not it is bene cial and benign in the situation at hand. Most logics of vagueness begin from a picture not unlike the Second Philosopher ’s, in which, for example a property can hold of an object, fail to hold, or be indeterminate for that object; there ’s also the problem of higher-order indeterminacy, that is, of borderline cases between holding and being indeterminate, between being indeterminate and failing. So far, I think it ’s fair to say that there is no smooth and perspicuous logic of vagueness, no such logic that escapes Feferman’s critique. It is, of course, true that classical logic can lead us astray in contexts with indeterminacy – this is the point of the sorites paradox – but at least for now the Second Philosopher ’s advice is simply to apply 19
20
Though these idealizations involve falsication in her description of the physical world, they are satised in the world of classical mathematics: excluded middle holds and the dependencies are logical. For more on the ontology of mathematics, see (Maddy 2011). There is serious disagreement between various writers over the source of the indeterminacy: is it purely linguistic or does the world itself include borderline cases and fuzzy objects? Here the Second Philosopher sides with the latter, but this shouldn’t a e ct the brief discussion here, despite the formulation in the quotation in the next clause above.
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classical logic with care,21 as one should any idealization, rather than switch to a less viable logic.22 Advocates of various conditional logics protest the Second Philosopher ’s other bold idealization: replacement of real dependencies with the simple material conditional. There are many proposals for a more substantial conditional, far too many to consider here (even if my slender expertise allowed it), but perhaps the conditional of relevance logic can be used as one representative example. The motivation here speaks directly to the falsi cation in question: the antecedent of a conditional should be ‘relevant’ to the consequent.23 To return to our earlier example, the redness of this rose isn ’t relevant to the fact that 2 þ 2 = 4 , despite the truth of the corresponding material conditional (if the rose is red, then 2 þ 2 = 4 ). Of course, as before, the Second Philosopher fully appreciates that the material conditional is a falsication, that the rose inference is an anomaly, but the pertinent questions are whether or not the falsi cation is benecial and benign, and whether or not the relevance logician has something better to o er. Again I think that for now, we do best to employ our classical logic with care. So we see that some deviant logics depart from the Second Philosopher ’s classical logic by rejecting her idealizations,24 and that our assessment then depends on the extent to which the falsications introduced are benecial and benign, and on the systematic merits of the proposed alternative. But not all deviant logics t this prole; some concern not just the idealizations of classical logic, but the fundamentals of rudimentary logic itself. Examples include intuitionistic logic – which rejects double negation elimination – quantum 21
Sorensen (2012) credits this approach to H. G. Wells: ‘ Every species is vague, every term goes cloudy at its edges, and so in my way of thinking, relentless logic is only another name for a stupidity – for a sort of intellectual pigheadedness. If you push a philosophical or metaphysical enquiry through a series of valid syllogisms – never committing any generally recognized fallacy – you nevertheless leave behind you at each step a certain rubbing and marginal loss of objective truth and you get deections that are dicult to trace, at each phase in the process’ (Wells 1908: 11 ). 22 Williamson ( 1994 ) also advocates retaining classical logic, but his reason is quite di erent: because there is no real vagueness, because apparent borderline cases really just illustrate our ignorance of where the true borderline lies. This strikes many, including me, as obviously false. 23 Relevance logicians are particularly unhappy with what they call ‘ explosion’, the classical oddity that anything follows from a contradiction. For related reasons, full relevance logic rejects even some rudimentary logical inferences not involving the conditional, like disjunctive syllogism, but I leave this aside here. (For a bit more, see Maddy 2007: 292 , footnote 24 .) 24 Some other deviant logics respond to idealizations of language rather than the worldly features of rudimentary logic: e.g., free logicians counsel us to reject the falsifying assumption that all naming expressions refer. Here, too, our assessment depends on the e ectiveness of the idealization and the viability of the alternative. In practical terms, leaving aside the various technical studies in the theory of free logics, I’m not sure using a free logic is readily distinguishable from being careful about the use of existential quantier introduction in the context of classical logic. In any case, our concern here is with worldly idealizations, not linguistic ones.
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logic – which rejects the distributive laws – and dialetheism – which holds that there are true contradictions. Given the connection of rudimentary logic with the Second Philosopher ’s formal structure, the challenge for each of these is to understand what the world is like without this formal structure, what the world is like that this alternative would be its logic.25 Of the three, intuitionistic logic comes equipped with the most developed metaphysical picture, but it’s suited to describing the world of constructive mathematics, not the physical world.26 Quantum logic at rst set out to characterize the non-formal-structure of the micro-world, but in practice it has not succeeded in doing so;27 the problem of interpreting quantum mechanics remains open. And dialetheism faces perhaps the highest odds: as far as I know, its defenders have focused for the most part on the narrower goal of locating a compelling example of a true contradiction in the world, perhaps so far without conspicuous success.28 The Second Philosopher tentatively concludes that rudimentary logic currently has no viable rivals as the logic of the world, and that classical logic likewise stands above its rivals as an appropriate idealization of rudimentary logic for everyday use. In sum, then, the Second Philosopher ’s answer, an ordinary answer to the question of why that coin must be foreign, is that the coin and its properties display formal structure and the inference in question is reliable in all such situations. This answer doesn ’t deliver on the usual philosophical expectations: the reliability of the inference is contingent, our knowledge of it is only minimally a priori at best. The account itself results from plain empirical inquiry, which may lead some to insist that it isn ’t philosophy at all. Perhaps not. Then again, if the original question – why is this inference reliable? – counts as philosophical – and it’s not clear how else to classify it – then the answer, too, would seem to have some claim to that honoric. But the Second Philosopher doesn ’t care much about labels. After all, even ‘Second Philosophy ’ and ‘Second Philosopher ’ aren’t her terms but mine, used to describe her and her behavior. In any case, philosophy or not, I hope the Second Philosopher ’s investigations do tell us something about the nature of that inference about the coin. 29 25
26 27 29
Our interest here is in the logic of the world, not the logic that best models something else, as, e.g., paraconsistent logic (a variety of relevance logic) might serve to model belief systems (see Maddy 2007: 293 –296). See the discussion of Creator Worlds in (Maddy 2007: 231 –233, 296) and (Maddy to appear, §II). 28 See (Maddy 2007: 276 –279, 296). See (Maddy 2007: 296 –297). My thanks to Patricia Marino for helpful comments on an earlier draft and to Penelope Rush for editorial improvements.
chapter 6
Logical nihilism Curtis Franks
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1
Introduction
The idea that there may be more than one correct logic has recently attracted considerable interest. This cannot be explained by the mere fact that several distinct logical systems have their scienti c uses, for no one denies that the logic of classical mathematics di ers from the logics of rational decision, of resource conscious database theory, and of e ective problem solving. Those known as logical monists maintain that the panoply of logical systems applicable in their various domains says nothing against their basic tenet that a single relation of logical consequence is either violated by or manifest in each such system. Logical pluralists do not counter this by pointing again at the numerous logical systems, for they agree that for all their interest many of these indeed fail to trace any relation of logical consequence. They claim, instead, that no one logical consequence relation is privileged over all others, that several such relations abound. Interesting as this debate may be, I intend to draw into question the point on which monists and pluralists appear to agree and on which their entire discussion pivots: the idea that one thing a logical investigation might do is adhere to a relation of consequence that is out there in the world, legislating norms of rational inference, or persisting some other wise independently of our logical investigations themselves. My opinion is that xing our sights on such a relation saddles logic with a burden that it cannot comfortably bear, and that logic, in the vigor and profundity that it displays nowadays, does and ought to command our interest precisely because of its disregard for norms of correctness. I shall not argue for the thesis that there are no correct logics. Although I do nd attempts from our history to paint a convincing picture of a relation of logical consequence that attains among propositions (or sentences, or whatever) dubious, I should not know how to cast general doubt “
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on the very idea of such a relation. By “drawing this point into question ” I mean only to invite re ection about what work the notion of a correct logic is supposed to be doing, why the debate about the number of logical consequence relations is supposed to matter to a logician, and whether the actual details of logic as it has developed might be di cult to appreciate if our attention is overburdened by questions about the correctness of logical principles. Rather than issue any argumentative blows, I propose merely to lead the reader around a bit until his or her taste for a correct logic sours. .
2
The law of excluded middle
Surely the most notorious bone of contention in the discussion of logical correctness is the law of excluded middle, f _ :f. Is this law logically valid, so that we know that its instances, “Shakespeare either wrote all those plays or he didn’t,” “Either the continuum hypothesis is true or it is not true,” “He’s either bald or he isn ’t,” etc., each are true in advance of any further information about the world? One hardly needs to mention that hundreds of spirited disavowals and defenses of lem have been issued in the last century. Many of these have even been authored by expert logicians. But let us turn our backs to these ideological matters and consider briey some of what we have learned about lem quite independently of any question about its correctness. The simplest setting for this is the propositional calculus. The intuitionistic propositional calculus (IPC) di ers from the classical propositional calculus (CPC) precisely in its rejection of lem. For a concrete and standard formalization of IPC one may take a typical Hilbert-style axiomatization of CPC and erase the single axiom for double negation elimination, ‘ ::f ⊃f, leaving the rules of inference as before. In fact, one of the rst things observed about lem is its equivalence with dne , which is most easily seen in the setting of natural deduction. Standard natural deduction presentations of CPC have a rule allowing one to infer any formula f from the single premise ::f. It is easy to derive in such a system the formula f _ :f. It is similarly easy to show that if we modify 1
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I emphasize that this really is a matter of perspicuity. One should not think that the phenomena described below are artifacts of peculiar features of propositional logic. They are nearly all consequences of decisions about lem that are invariant across a wide spectrum of logics. Consider: the structural subsumption of lem applies also to the predicate calculus; the admissible propositional rules of Heyting Arithmetic (and of Heyting Arithmetic with Markov ’s principle) are exactly those of IPC (Visser 1999); the disjunction property holds for Heyting Arithmetic (Kleene 1945) and for intuitionistic Zermelo–Fraenkel set theory (Myhill 1973).
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this calculus by disallowing dne but now introduce a new rule that allows one to write down any instance of lem in any context, f will be derivable in all contexts in which ::f is derivable. In (1934 –35), Gentzen observed to his own surprise that the sequent calculus presentation of CPC admits an even more elegant modi cation into a presentation of IPC than the one just described. One simply disallows multiple-clause succedents and leaves the calculus otherwise unchanged. Thus lem and with it the entire distinction between intuitionistic and classical logic is subsumed into the background structure of the logical calculus. All the inference rules governing the logical particles (^, _, ⊃, :) and all the explicit rules of structural reasoning (identity , cut , weakening , exchange , and contraction) are invariant under this transformation. Thus it appears that a duly chosen logical calculus allows a precise analysis of what had been thought of as a radical disagreement about the nature of logic. When classicists and intuitionists are seen to admit precisely the same inference rules, their disagreement appears in some ways quite minor, if more global than rst suspected. Exactly how minor, on closer inspection, is the di erence between these super cially similar calculi CPC and IPC? Not very. Even before Gentzen’s profound analysis, Gödel (1932) observed that IPC satis ed the “disjunction property ”: formulas such as f _ ψ are provable only if either f or ψ is as well. At rst sight this might appear to be no more than a restatement of the intuitionist ’s rejection of lem. After all, that rejection was motivated by the idea that instances of lem are un-warranted when neither of their disjuncts can be independently established. But, one might think, if any disjunction is warranted in the absence of independent verication of one of its disjuncts, those like f _ :f are, so rejecting lem should lead to something like the disjunction property. This reasoning strikes me as worthy of further elaboration and attention, but it should be unconvincing as it stands. For one thing, the formal rejection of lem only bars one from helping oneself to its instances whenever one wishes. The gap between this modest restriction and the inability ever to infer any disjunction of the form f _ :f at all, unless from a record of that inference one could e ectively construct a proof either of f or of :f, is a broad one. More, there are in nitely many formulas f unprovable in 2
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Gentzen described the fact that lem prescribes uses of logical particles other than those given by their introduction and elimination rules as “ troublesome.” The way that in the sequent calculus the logical rules are quarantined from the distinction between classical and intuitionistic logic he called “seemingly magical.” He wrote, “I myself was completely surprised by this property . . . when rst formulating that calculus” (Gentzen 1938: 259).
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IPC, such that IPCþf is consistent but insucient to prove lem. Examples like : p _ :: p may well fuel suspicions that even in the shadow of global distrust of lem, some disjunctions f _ ψ are more plausible than any instance of lem even in the absence of resources su cient to derive f or ψ . Thus the disjunction property is a non-trivial consequence of the invalidity of lem. In fact, this situation exempli es a recurring phenomenon in logic, wherein from the assumption of a special case of some general hypothesis, that hypothesis follows in its full generality. This often happens even when, as in the present case, the general phenomenon does not appear, even in hindsight, to be a logical consequence of its instance in any absolute sense. The disjunction property has further consequences of its own, however, to which we can pro tably turn. In the approach to semantics known as inferentialism, the meaning of a logical particle should be identied with the conditions under which one is justied in reasoning one s way to a statement governed by that particle. From this point of view, which is given expression already in some of Gentzen s remarks, and owing to the separation in sequent calculus of lem from the logical rules, the meanings of the familiar logical particles might be said not to di er in intuitionistic and classical logic. However, the disjunction property gives rise to an alternative interpretation of the logical particles of IPC in which each theorem refers back to the notion of provability in IPC itself. For if f _ ψ is provable only if one of f and ψ is as well, then a candidate and interesting reading of the sentence ‘ IPC f _ ψ is Either ‘ IPC f or ‘ IPC ψ . Expanding on this idea, one might suggest that the provability of a conjunction means that each of its conjunctions is provable, that the provability of a conditional, f ⊃ ψ , means that given a proof of f one can construct a proof of ψ , and that the provability of :f means that a contradiction can be proved in the event that a proof of f is produced. As we shall see shortly, this so far informal interpretation of intuitionistic logic is riddled with ambiguities. All the same, some reection should bring home the idea that some disambiguation of this reading is a possible way to understand the theorems of IPC. When one compares the situation with CPC, where conditional truth comes so cheap and the disjunction property fails badly, one can only conclude that the di erence between these calculi is in some sense great after all, greater even than the debate over the validity of lem alone rst suggests. When one then recalls the earlier observation that these calculi can be presented so that their formal di e rences are slight and their rules “
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identical, the observation that radical di erences in meaning result from so subtle a change in form is striking. If we are to take seriously the idea that theorems of IPC refer back to IPC provability, then some care must be taken in making this interpretation precise. For if the provability of a conditional, f ⊃ ψ , “means” that ‘ IPC ψ in the event that ‘ IPC f , then one should expect ‘ IPC f ⊃ ψ in every situation in which the set of theorems of IPC is closed under the rule “from f, infer ψ .” However, the disjunction property implies that these expectations will not be met. To see this, consider the Kreisel–Putnam rule, “From :f ⊃ (ψ _ χ ), infer (:f ⊃ ψ ) _ (:f ⊃ χ ).” The only derivations of :f ⊃ (ψ _ χ ) in natural deduction are proofs whose last inferences are instances of ⊃-elim (modus ponens), ^-elim, dne , or ⊃-intro . It is easy to see that a proof ending in ⊃-elim or ^-elim cannot be the only way to prove this formula, that in fact any such proof can be normalized into a proof of the same formula whose last inference is an instance of one of the other two rules. If, further, we consider the prospects of this formula being a theorem of IPC, then dne is no longer a rule, and we may conclude that any proof necessarily contains a subproof of ψ _ χ from the assumption :f (to allow for ⊃-intro ). What might this subproof look like? Once more, dne is not an option, so again by insisting that the proof is normalized (so that it doesn’t end needlessly and awkwardly with ⊃-elim or ^-elim) we ensure that its last step is an instance of _-elim. But this means that an initial segment of this subproof is a derivation in IPC either of ψ or of χ from the assumption :f (it is here that the disjunction property rears its head), and in each case it is clear how to build a proof of ( :f ⊃ ψ ) _ (:f ⊃ χ ). Putting this all together, we see that whenever :f ⊃ (ψ _ χ ) is a theorem of IPC, so too is ( :f ⊃ ψ ) _ ( :f ⊃ χ ). When a logical system’s theorems are closed under a rule of inference, we say that the rule is “admissible” for that logic. The above argument established that the Kreisel–Putnam rule is admissible for IPC. One might expect that the rule is also “derivable,” that ‘IPC (:f ⊃ (ψ _ χ )) ⊃ ((:f ⊃ ψ ) _ (:f ⊃ χ ) ). However, it is not (Harrop 1960). This situation is dis-analogous to that of classical logic, where all admissible rules are derivable so that the distinction between admissibility and derivability vanishes. In the parlance, we say that CPC is “structurally complete” but that IPC is not. In fact much of the logical complexity of IPC can be understood as a residue of its structural incompleteness. For the space of intuitionistically valid formulas is far more easily navigated than the space of its admissible
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rules: although it is decidable whether or not a rule is admissible in IPC, there is no nite basis of rules that generates them all (Rybakov 1997). What ought one make of the structural incompleteness of IPC? One thing that can denitely be said is that reading the expression f ⊃ ψ as “There is a procedure for transforming a proof of f into a proof of ψ ” is problematic for both the classicist and the intuitionist, but for di erent reasons. This reading is wrong for the classicist, because the idea of procedurality simply does not enter into the conditions of classical validity. By contrast, procedures of proof transformation are central for the intuitionist. However, we now know that there are procedures for transforming a IPC-proof of f into a IPC-proof of ψ in cases where f ⊃ ψ is not a theorem of IPC. So at best one could say that IPC is incomplete with respect to this semantics, and more plausibly one should say that this reading of ‘IPC f ⊃ ψ is erroneous. Thus we see a sense in which the phenomenon of structural completeness is related to a sort of semantic completeness: a structurally incomplete logic will be incomplete with respect to the most naive procedural reading of its connectives. It also happens that structural completeness bears a precise relation to the phenomenon of Post-completeness, the situation in which any addition made to the set of theorems of some logic will trivialize the logic by making all formulas in its signature provable. To state this relationship, we refer to a notion of “saturation.” For a logical calculus L whose formulas form the set S , let Sb ( X ) be the set of substitution instances of formulas in X S and let CnL( X ) be the set of formulas f such that X ‘L f. L is saturated if for every X S CnL( X ) = CnL(Sb ( X )) for every X S . By a (1973) theorem of Tokarz, a Post-complete calculus is structurally complete if, and only if, it is saturated. For these and perhaps other reasons many authors have felt that the presence of non-derivable, admissible rules is a de ciency of systems like IPC. The very term “structural incompleteness” suggests that something is missing from IPC because correct inferences about provability in this logic are not represented as theorems in IPC. Rybakov ( 1997), for example, suggests that “there is a sense in which a derivation inside a [structurally incomplete] logical system corresponds to conscious reasoning [and] a derivation using [its] admissible rules corresponds to subconscious reasoning.” He faults such systems for having rules that are “valid in reality ” yet “invalid from the viewpoint of the deductive system itself ” (10–11). Structurally complete systems, by contrast, are “self contained” in the sense that they have the “very desirable property ” of being conscious of all the rules that are reliable tools for discovering their own theorems ( 476).
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It seems to me that this attitude derives from wanting to preserve the naive procedural understanding of the logical connectives. The situation ought rather, I counter, lead one to appreciate the subtlety of procedurality exhibited in intuitionism. For the logical lesson to be learned is that in the absence of lem the context of inference takes on a new role. Thus f ⊃ ψ means that given any background of assumptions from which f is provable, a proof of f can be transformed into a proof of ψ under those same assumptions, and this understanding does not reduce, as it does with logics insensitive to context, to the idea that any proof of f can be transformed into a proof of ψ . This irreducibility strikes me as a very desirable property for many purposes. I should like to know more about the conditions that lead to it. From this point of view, it is natural to ask whether there are logics that, unlike classical logic, admit a constructive interpretation but, like classical logic, are not sensitive in this way to context. Perhaps the constructive nature of IPC derives from its context-sensitivity. Surprisingly, Jankov s logic, IPCþ:f _ : :f, appears to undermine any hope of establishing a connection between these phenomena. Consider the Medvedev lattice of degrees of solvability. The setting is Baire space ω (the set of functions from ω to ω) and the problem of producing an element of a given subset of this space. By convention, such subsets are called mass problems, and their elements are called solutions. One says that one mass problem reduces to another if there is an e ective procedure for transforming solutions of the second into solutions of the rst. If one de nes the lattice of degrees of reducibility of mass problems, it happens that under a very natural valuation, the set of identities of corresponds to the set of theorems of Jankov s logic, so that the theory of mass problems provides a constructive interpretation of this logic.3 However, Jankov s logic is structurally complete (Prucnal 1976). Thus one sees that the so-called weak law of excluded middle preserves the context insensitivity of CPC despite, standing in the place of full lem, allowing for a procedural semantics. In my graduate student years, several of my friends and I were thinking about bounded arithmetic because of its connections with complexity theory and because the special diculty of representing within these theories their own consistency statements shed much light on the ne details of arithmetization. We had a running gag, which is that formal theories like PA are awfully weak, because with them one can t draw very many distinctions. The implicit punchline, of course, is that the bulk of “
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the distinctions one can draw in theories of bounded arithmetic are among statements that are in fact equivalent. Lacking the resources to spot these equivalences is no strength! Something perfectly analogous happens in the case of substructural logics. There are theorems of CPC that are unprovable in IPC, but not vice versa, so the latter logic is strictly weaker. Moreover, CPC proves all sorts of implications and equivalences that IPC misses. But if we stop believing for a moment, as the discipline of logic demands we do, and ask about the ne structure of inter-dependencies among the formulas of propositional logic, IPC delivers vastly more information. Consider just propositional functions of a single variable p. In CPC there are exactly four equivalence classes of such formulas: those inter-derivable with p, : p, p _ : p, and p ^ : p. In IPC the equivalence classes of these same formulas exhibit a complicated pattern of implications, forming the innite Rieger – Nishimura lattice. One thought one may have is that IPC should be considered an expansion of CPC: every classical tautology can be discovered with IPC via the negative translation of Gödel ( 1933) and Gentzen, so with IPC one gets all the classical tautologies and a whole lot more. (Gödel at times suggested something like this attitude.) But, of course, neither the negative translation nor the very idea of a classical tautology arises within intuitionistic logic. The thought that I encourage instead is this: The logician is loath to choose between classical and intuitionistic logic because the phenomena of greatest interest are the relationships between these logical systems. Who would have guessed that the rejection of a single logical principle would generate so much complexity – an r.e. set of admissible rules with no nite basis, an innite lattice of inter-derivability classes? The intuitionist and the classicist have very ne systems. Perhaps with them one gains some purchase on the norms of right reasoning or the modal structure of reality. The logician claims no such insight but observes that one can hold xed the rules of the logical particles and, by merely tweaking the calculus between single conclusion and multi conclusion, watch structural completeness come in and out of view. The same switch, he or she knows, dresses the logical connectives up in a constructive, context-sensitive interpretation in one position and divests them of this interpretation in the other. These connections between sequent calculus, constructive proof transformation, structural completeness, and lem are xtures from our logical knowledge store, but they cannot seriously be thought of as a network of consequences in some allegedly correct logic.
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Logic imposed and logic discovered
If I have conveyed my attitude successfully, then I will have inspired the following objection: You speak about an unwillingness to embrace any one or select few logical systems because of an interest in understanding all such systems and how their various properties relate to one another. But by making logical systems into objects of investigation, you inhabit an ambient space in which you conduct this investigation. It is legitimate to ask which logic is appropriate in this space. What is your metalogic ? “
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I shall explain that this objection rests on various preconceptions that I do not share. I hope the explanation functions to aid the reader in seeing these as misconceptions. If it does, then logical nihilism will be understood. If we agree that as logicians we are interested, not in factual truth, but in the relationships among phenomena and ideas, then the point of view we must hasten to adopt should be the one that assists us in detecting and understanding these relationships. Which relationships? Presumably, it has been suggested, those that accurately pick out grounds and consequences, those that answer the question What rests on what? But why stop here? What sort of purpose is served by simultaneously disregarding factual states of a a irs and pledging allegiance to factual relations of ground and consequence? I have an intuitive sense of what a fact is; I have no such sense of ontological grounding. Nor have I seen any reason to expect that the study of logic can foster such a sense in me. More appropriate seems to be a disregard for privileged relationships similar to our disregard for truth. Suppose that we are interested in detecting and understanding whatever relationships we can nd. Then we might wish not to be wedded to any point of view. We might, instead, try on a few hats until some interesting patterns appear where before there seemed to have been only disorder. We might nd that one hat helps time and again, but we will be well-advised not to forget that we are wearing it. For if we never take it o , then we risk forever overlooking logical relationships of considerable interest. Worse, we risk coming to think of the relationships we can detect as in the world, preconditions of thought, or some such thing. Allow me to illustrate this point with an example. At least since Aristotle it was expected that great complexity could be uncovered by a proper analysis of the quantiers that occur in natural language. Notably, quantiers allow us to reason in succinct strokes about in nite collections. About “
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a century ago, David Hilbert had the idea of analyzing quantiers with consistency proofs, and he devised an intricate calculus with “transnite axioms” which allows proof gures involving quantiers to be transformed (in principle) into gures without them. Hilbert’s idea was that reasoning about this transformation (which by its nature requires being attentive to constructibility) would expose the quanti ers as innocuous parts of our mathematical language and also make perspicuous their complexity.4 Hilbert conducted this investigation in the shadow of a great ideological quarrel about the validity of various logical and mathematical principles. In one quarter were dour skeptics who distrusted not only lem but other forms of innitary reasoning. Chief among them was Kronecker, against whom Hilbert (1922: 199–201) railed because he “despised . . . everything that did not seem to him to be an integer. ” Less famously, but perhaps more importantly, he faulted Kronecker also because “it was far from his practice to think further ” about what he did accept, “about the integer itself.” In a similar vein, Hilbert observed that Poincaré “ was from the start convinced of the impossibility of a proof of the axioms of arithmetic ” because of his belief that mathematical “induction is a property of the mind.” Thus Hilbert viewed these gures as short-sighted, not only in their rejection of mathematical techniques that he wished to defend, but also in their belief that the manifest validity of a principle precludes any hope of our analyzing it. Both attitudes, he cautioned, “block the path to analysis.”5 After only a decade of partial successes, it was discovered that the sort of consistency proofs Hilbert envisioned are not available. Speci cally, Gödel (1931) demonstrated that no proof of the consistency of a reasonably strong and consistent mathematical system could be carried out within that same system. Typically it is the recursion needed to verify that the proof transformation algorithm halts that cannot be so represented. This situation raises the question whether proving with such principles that a system is consistent is not obscurum per obscurius . In (1936) Gentzen made these circumstances much more precise by providing a perspicuous proof of the consistency of PA. Gentzen’s proof is carried out in the relatively weak theory PRA together with the relatively strong principle of trans nite induction up to the ordinal
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For details, see section 2 of (Franks 2014 ). For more on Hilbert’s view, see Professor Shapiro’s contribution to this volume.
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Together with Gödel s result, this proof demonstrates that transnite induction through 0 is unprovable in PA. All of this is well known. Most people familiar with the history of logic are aware that Gentzen proved also that transnite induction to any ordinal beneath 0, any ... n ordinal ω1 for n N, is in fact provable in PA. But Gentzen pressed even further. One can consider fragments of PA de ned by restricting the induction scheme to formulas with a maximum quantier complexity (call these the theory s class of inductive formulas). Gentzen showed in ( 1943) that the height of the least ordinal su cient for a proof of the consistency of such a fragment corresponds with the quanti er complexity of that theory s class of inductive formulas, and that trans nite induction to any smaller ordinal is provable in the fragment. So the number of quanti ers over which mathematical induction is permitted equals the number of exponentials needed to express the ordinal that measures the theory s consistency strength. One quanti er equals one exponential. Thus the theory of constructive proof transformations has turned up a precise mathematical analysis of the complexity of natural language quanti ers, a remarkable realization of Hilbert s original ambition. Why has so little attention been given to this result? The discussion of Gentzen s work has been dominated by debate about whether or not the proof of PA s consistency can really shore up our condence in this theory. To anyone who has witnessed a talk about ordinal analysis sabotaged by this debate, the scene will be familiar: someone reminds us that the proof uses a principle that extends the resources of PA. Someone else defends the principle despite this fact and points out that in every other way the proof is extremely elementary compared to the full strength of PA. Because the two theories PA and PRA þti 6 are in this basic way incomparable, the jury is out as to the gains made by reducing the consistency of one to that of the other. From the point of view of logic, however, this is all a distraction from what Gentzen actually achieved: he showed that the question of the consistency even of elementary theories can be formulated as a precise problem, and he showed that the solution to this problem requires new perspectives and techniques and carries with it unexpected insights about logical complexity. If philosophers did not harbor skepticism about PA, then they would likely not be interested in Gentzen s result ’
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one way or the other. Their disinterest in the analogous results about fragments of PA is just evidence that they harbor no skepticism about these theories. We recognize Gentzen s analysis of rst-order quantiers as one of the deepest results in the history of logic as soon as, and no sooner than, we stop believing. I now wish to respond more directly to the objection that opened this section. Of course it is true that in the study of logical systems one must engage in reasoning of some sort or another. This reasoning can possibly be described by one or a few select logical systems. But why should anyone assume that this amount of reasoning is anything more than ways of thinking that have become habitual for us because of their proven utility? Further, why should anyone assume that there is any commonality among the principles of inference we deploy at this level over and above the fact that we do so deploy them? To expand on the rst of these points, it may be helpful to draw an analogy between rudimentary logic and set theory. Often it is thought that decisions about which principles should govern the mathematical theory of sets should be made by appealing to our intuitions about the set concept and even about the cumulative hierarchy of sets. Doubtless such appeals have gured centrally in the development of set theory. But the history of the subject suggests that a complete inversion of this dynamic has also been at play. Kanamori (2012: 1) explains: ’
[L]ike other elds of mathematics, [set theory s] vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results . . . from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. ’
Whereas one can today nd endless phenomenological and metaphysical justications of, for example, the replacement axiom, Kanamori contends that set theory in fact evolved primarily by absorption of successful techniques, like transnite induction, devised to answer mathematical questions. It was von Neumann s formal incorporation of this method into set theory, as necessitated by his proofs, that brought in Replacement (33). In similar fashion, the power set existence assumption, which originally had many detractors, was not nally embraced in the wake of any argument or philosophical insight. It merely happened that iterated cardinal exponentiation gured prominently in Kurepa s proofs in innitary combinatorics, so that shedding deeper concerns the power set operation became further domesticated (46). The upshot? Set theory is “
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a particular case of a eld of mathematics in which seminal proofs and pivotal problems actually shaped the basic concepts and forged axiomatizations, these transmuting the very notion of set ( 1). The point is not that we have no reliable intuitions about the set concept, nor even that they should play no role in the development of formal set theory. The point is that those intuitions have evolved partly in response to our need to make sense of routinely counter-intuitive scienti c discoveries. If our intuitions have been at least partially, perhaps largely, shaped by developments in logic, then the fact that we appeal to them on occasion in order to re ne our denitions and techniques or in order to choose new axioms seems to lose its signi cance. Rather than develop a theory of sets that unpacks the fundamental truths we intuit, we have developed an intuition about sets that makes sense of mathematically interesting relations we have discovered. I need not argue that rudimentary logic has taken shape in a similar fashion rst because Professor Maddy has been persuasive on this same point in her contribution to this volume and second because it does not really matter for my purposes whether, in the end, this is true. I am content simply to reveal the picture that holds us captive when we begin to think about logic. For the objection we solicited was that if our modern science of logic thrives on an unprejudiced consideration of the full gamut of properties exhibited by logical systems and the relations among them, so that issues of correctness do not arise, then it has only smuggled those issues in through the back door, in the metalogic that makes the science possible. But it is a preconception that science is made possible by ahistorical norms of right reasoning. Once one considers the possibility that logic may be studied with patterns of thought adapted to what we learn along the way, it becomes hard to understand what special status the rudimentary principles we nd ourselves reexively appealing to are supposed to have. This brings us to the second of the points above, the idea that the principles that have found their way into our basic toolkit must presumably have some features in common that led them there. Even if these principles have been adopted over the course of time, the idea goes, there must be some reason for their being adopted instead of other principles. Perhaps this reason can be repackaged as an explanation of their being the true logical principles. In response to this suggestion, I wish only to expose the presupposition driving it. Whoever said that there must be some property of logical validity that some principles of inference enjoy and others do not? If we ”
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knew in advance of all evidence that some such property attains, then it might be reasonable to look for it in whatever classes of inference rules we happen to nd collected together. But we have no such foreknowledge and, in fact, the evidence suggests that the arrangement of our toolkit is a highly contingent matter. Are we not better o shedding this vestigial belief that among all the intricate and interesting consequence relations out there, some have a special normative status? Can we not get by with the understanding that principles of inference with a rather wide range of applicability di er from those suited only to speci c inference tasks only in having a wider range of applicability? Had this been the understanding that our culture inherited, would anything we have learned from studying logic lead us to question it? At the end of Plato s Phaedrus , Socrates explains that prior to investigating the essence of a thing, it is important to devise an extensionally adequate denition of that thing so that we will be in agreement about what we are investigating. This attitude seems right to me, and it seems to me that the familiar debates about, for example, where logic leaves o and mathematics begins violate this principle unabashedly. Suppose it were clear to everyone that some but not all patterns of reasoning are inescapable and furthermore that it were easy to tell which these are. Then we would have good reason to label this logic in distinction to patterns of reasoning that we all recognize as the province of some special science or particular application. It would be reasonable to wonder what accounts for the privileged role that these principles play in our lives. As things stand, however, many of us seem instead to assume that there simply must be patterns of reasoning that di er in kind from others. Typically, our minds are already made up about the psychological or metaphysical circumstances that underwrite this di erence. This is what Wittgenstein stressed with his observation that the crystalline purity of logic was not the result of an investigation, that instead it was a requirement (Wittgenstein 1953: §107). Driven by this assumption, we thrash about looking for some extensional denition that we can hang our ready-made distinction on. These denitions are simply unconvincing on their own. They can satisfy only people who cannot tolerate the thought that there is no line to be drawn. When Gentzen began his study of logic, he parted ways with his predecessors7 by not rst dening logical validity and then seeking out logical principles that accord with that denition. He simply observed that ’
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some patterns of reasoning abound. Consider that to de ne “predicate logic” he said only that it “comprises the types of inference that are continually used in all parts of mathematics” (1934 –35: 68). This homely denition set Gentzen on a task of empirically tallying the techniques used in mathematical proofs, ignoring those that are unique to geometry, arithmetic, and other specic branches of mathematics.8 Of course such a survey is by no means guaranteed to be exhaustively executable, and it was Gentzen’s good fortune that his subject matter happens to exhibit few instances. One should not, however, write o his success as purely a matter of luck. Gentzen devised an ingenious argument to the e ect that his tally was in fact exhaustive. This involved constructing an innovative type of logical calculus that is at once formal and patterned on the informal reasoning recorded in mathematical proofs. This scheme enabled him to do more than construct a serial tally of proof techniques, because the inference types identiable with it are extremely few and systematically arranged so that one can be sure that none have been overlooked. After all, if there are any inference types that went unnoticed, then for that very reason they fail to meet the criterion of ubiquity in mathematical practice that Gentzen imposed. This empirical “completeness proof ” bears little semblance to familiar conceptions of logical completeness and is interesting for this reason. I mention it now only to draw attention to the fact that while Gentzen ’s denition of predicate logic does pick out a well-de ned body of inferences,9 he did not concoct the de nition in the service of a preconceived notion of logical validity. He did not, for example, rst stipulate a semantic notion of logical consequence based on his own intuitions and then ask whether his calculi adequately capture this notion. Gentzen simply proposed that the intuitions guiding mathematicians in their research would be worth isolating and studying, and he therefore modeled a logical calculus on the inferences mathematicians actually make.10 Of course mathematicians also deploy proof techniques that are less universal, and the only observable di erence between these and the ones that meet Gentzen’s criterion of ubiquity is their relative infrequency. Mathematicians do not report a feeling that arithmetical reasoning is less 8 9
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valid or valid in some other way than general reasoning, and even if they did, we should be inclined to ignore these reports if they did not re ect in mathematical practice. For these reasons Gentzen could never bring himself to describe the distinction between inference rules that appear in the predicate calculus and those that belong speci cally to arithmetic as a distinction between the logical and the non-logical. He only thought that he had designated a logical system, one that by design encodes some of the inferences he was bound to make when reasoning about it but whose logical interest derives solely from what that reasoning brings to light. Contrast this with one of the more famous attempts to demarcate the logical: Quine’s defense of rst-order quantication theory. Second-order quantication, branching quantiers, higher set theory, and such can each be dissociated from logic for failing to have sound and complete proof systems, for violating the compactness and basic cardinality theorems, and other niceties. There is even a ( 1969) theorem, due to Lindström, to the e ect that any logic stronger than rst-order quantication theory will fail to exhibit either compactness or the downward Löwenheim –Skolem theorem. Second-order logic, Quine concluded, is just mathematics “in sheep’s clothing ” because by using second-order quantiers one is already committed to non-trivial cardinality claims (Quine 1986: 66). How true will these remarks ring to someone who doesn ’t know in advance that they are expected to distinguish logical and mathematical reasoning? Quine’s consolation is telling: “ We can still condone the more extravagant reaches of set theory, ” he writes, “as a study merely of logical relations among hypotheses” (Quine 1991: 243). I should have thought that this accolade, especially in light of the intricate sorts of logical relations that set-theoretical principles bear to one another and that set theory bears to other systems of hypotheses, would be used rather to enshrine a discipline squarely within the province of logic. For if we never suspected that among the plenitude of logical relations are a privileged few that capture the true inter-dependencies of propositions, what else would we mean by “logic” than just the sort of study Quine described? As to the properties that characterize rst-order quantication theory, it should now go without saying that from our perspective Lindström’s theorem, far from declaring certain formal investigations extra-logical, exempli es logic. So too do results of Henkin ( 1949) and others to the e ect that second-order quantication theory and rst-order axiomatic set theory each are complete with respect to validity over non-standard models. For a nal example, I can think of none better than the recent result of Fan Yang that Väänänen ’s system of dependence logic (with
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branching quantiers), extended with intuitionistic implication (the context sensitive and constructive operator that results from the denial of lem), is equivalent to full second-order quanti cation theory (Yang 2013). Wittgenstein (§108) expected his readers to recoil from the suggestion that we shed our preconceived ideas by “turning our whole examination around.” Rather than impose our intuitions about logic on our investigations by asking which principles are truly logical, let us rst ask if a close look at the various inference principles we are familiar with suggests that some stand apart from others. If some do, then let us determine what it is that sets them apart. But the question he puts in our mouths – “But in that case doesn’t logic altogether disappear?” – suggests that we know deep down that our empirical investigation is bound to come up empty. Various criteria will allow us to demarcate di erent systems of inference rules to study, but when none of these indicate more than a formal or happenstance association we will nd ourselves hard pressed to explain why any one of them demarcates “the logical.” I am more optimistic than Wittgenstein. The conclusion that I expect my reader to draw from the absence of any clear demarcation of the logical is not that there is no such thing as logic. Let us agree instead that no one part of what logicians study, contingent and evolving as this subject matter is, should be idolized at the expense of everything else. Logic outstrips our preconceptions both in its range and in its depth. 4.
Conclusion
Traditional debates about the scope and nature of logic do not do justice to the details of its maturation. In asking whether certain inferential practices are properly logical or more aptly viewed as part of the special sciences, for example, we ignore how modern logic has been shaped by developments in extra-logical culture. Similarly, questions about whether logic principally traces the structure of discursive thought or the structure of an impersonal world presuppose a logical subject matter una e cted by shifts in human interest and knowledge. I mean, by saying this, not just to suggest that the principles of rudimentary logic are contingent, not di erent in kind from principles that we use only some of the time or very rarely and only for speci c tasks. I do urge this attitude. But the caution against mistaking our default, multi-purpose habits of reasoning for something monumental is only preparation for a second, more valuable reaction. One should warm up to the trend of identifying logic with the specialized scienti c study of the
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relationships among various systems and their properties. This is, after all, how logicians use the word. Our preference to ignore questions about a logic’s correctness stems not only from an interest in exploring the properties of possible logical systems in full generality but also from an appreciation, fostered by the study of logic, that no one such system can have all the properties that might be useful and interesting. In closing, let me re-emphasize that the idea of a true logic, one that traces the actual inter-dependencies among propositions, is unscathed by all I have said. Part of the di culty in questioning that idea is that it is a moving target: argue against it, you feel it again in that very argument; close the door, it will try the window. But this very circumstance only underlines the fact that the idea is a presupposition, nothing that emerges from any discovery made in the study of logic. For the same reason that we can marshal no evidence against it, we see that if we can manage to forget it our future discoveries will not reveal to us that we have erred. This realization, coupled with the observation that a xation on the true logical relationships “out there” hinders the advancement of logic, certainly recommends nihilism on practical grounds. The question that remains is whether we are capable of sustaining a point of view with no direct argumentative support. The proper antidote to our reexive tendencies will surely extend an analysis of modern logic and include a rehearsal of the subject ’s history. I cannot o er that here. I can only mention that logic as a discipline has evolved often in deance of preconceived notions of what the true logical relations are. Logic has been repeatedly reconceived, not as a fallout from our better acquaintance with its allegedly eternal nature, but in response to the changing social space in which we reason. There is reason neither to expect nor to hope that logic will not be continually reconceived. Such reconceptions have been and likely will again be fundamental, so that what makes the moniker “logic” apt across these diverse conceptions is not an invariable essence. In these pages I have indicated instead logic ’s modern contours, highlighting the fact that the deepest observations logic has to o er come with no ties to preconceptions about its essence. The richness of logic comes into view only when we stop looking for such an essence and focus instead on the accumulation of applications and conceptual changes that have 11
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made current logical investigations possible. The study of logic might be the best practical antidote to the view of it that we have inherited. In his Logic of 1780, Condillac wrote: “People would like to have had philosophers presiding over the formation of languages, believing that languages would have been better made. It would, then, have required other philosophers than the ones we know ” (237–8). Our interest in a better made language is an interest in a language that traces a pre-existing logical structure. Like Condillac, Wittgenstein warned that presupposing such a structure fosters dismissive attitudes about the languages we have: “ When we believe that we must nd that order, must nd the ideal, in our actual language, we become dissatised with what are ordinarily called ‘propositions,’ ‘ words,’ ‘signs’” (§105). When we stop believing for a moment, as the discipline of logic demands we do, the structures we nd immanent in our several, actual languages command our interest more than anything we could have devised in the service of our ideal.
chapter 7
Wittgenstein and the covert Platonism of mathematical logic Mark Steiner
(I use the following abbreviations: PI = Philosophical investigations (Wittgenstein 2009), RFM = Remarks on the foundations of mathematics (Wittgenstein 1978), LFM = Wittgenstein s lectures on the foundations of mathematics, Cambridge, 1939 (Wittgenstein 1976), PG = Philosophical grammar (Wittgenstein and Rhees 1974).) By the end of the 1930s, Wittgenstein s thought on mathematics had undergone a major, if often undetected, change. 1 The change had to do with the relationship between arithmetic, including elementary number theory and geometry,2 and empirical regularities, including behavioral regularities that are induced by training. During the rst part of the decade, Wittgenstein continued to regard mathematical theorems as akin to grammatical rules. As such, there was no need to seek a general theory of mathematical applicability, as Frege did.3 The applications, he repeated, take care of themselves. (E.g. PG, III, 15: 308.) After all, grammatical rules have no applications outside grammar itself, being norms, not descriptions of nonlinguistic objects or processes. This does not imply, to be sure, that the environment in which language operates has no e ect on which rules we use in language to describe the environment. In a 1939 lecture at Cambridge (LFM XX: 194) Wittgenstein remarked there is, in all the languages we know, a word for all but not for all but one . ’
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See (Steiner 2009). In this chapter, for the most part, I will not address the complicated question of to what extent Wittgenstein s ideas were intended to describe advanced mathematics, and to what extent he actually succeeded in describing advanced mathematics. Hence, we will focus upon arithmetic and elementary number theory, and Euclidean geometry. 3 As Dummett points out (Dummett 1991b), for arithmetic, Frege s theory of application involved (a) rendering all arithmetic statements in second-order logic, universally quantied, where the predicate letters range over concepts. Then to apply an arithmetic proposition, all one needs to do is to perform universal instantiation, replacing each predicate variable with a constant predicate that expresses a particular concept. To apply arithmetic to empirical situations, for example, all we need to do is instantiate empirical predicates for the universally quanti ed second-order variables. Note that mathematical applicability is in this account the same thing as logical applicability. 2
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He certainly meant to say that this is true because in our world it is most convenient to have a universal quantier, not because logic is itself empirical. I believe that this part of what he says there, though not all of it, can be attributed to him well before 1939.4 Even in 1939, Wittgenstein told the class: To say “ A reality corresponds to ‘2 þ 2 = 4 ’” is like saying “ A reality corresponds to ‘two’.” It is like saying a reality corresponds to a rule, which would come to saying: “It is a useful rule, most useful – we couldn’t do without it for a thousand reasons, not just one .”5 (LFM: 249)
Such a view of mathematics places it on a par with the rules of logic. Both are grammatical rules, the di erence being which vocabulary the rules govern. This is not to say, with Frege and Russell, that mathematics is logic. The rules of logic are used to prove mathematical theorems, to be sure, but this does not make mathematics into logic: logic is used in every discourse. During the period 1936–7, Wittgenstein began to study in earnest the concept of rule-following which was to loom so large in his Philosophical investigations . The connection between rules and regularities (Regelmäßigkeit ) becomes manifest to those who study his notebooks. Rules are norms which evaluate what happens or what is done by people; regularities are what happen most of the time, or what people do most of the time – when they are trained the same way. Rules label the deviations from these regularities “mistakes,” “abnormalities,” “perturbations”6 (in 4
Wittgenstein goes on to say: This is enormously important: this is the sort of fact which characterizes our logic. “ All but one” seems to us a complex idea – “all”, that’s a simple idea. But we can imagine a tribe where “all but one” is the primitive idea. And this sort of thing would entirely change their outlook on logic.
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This further idea, expressed, as I say, in 1939, I would not want to attribute to Wittgenstein in the early 1930s (which is what I am discussing here) – and I will discuss it below, in the context of (what I will call) his revolution of 1937. It is reminiscent of Nelson Goodman ’s relativism concerning natural kinds (“grue”). The idea that the relationship between the empirical world and mathematical propositions is that the former makes the latter useful is not replaced in 1939, but augmented by a much deeper connection between mathematical propositions and empirical reality, which we will discuss later. I don’t mean to say that the mathematical technique of perturbation theory is “normative.” I bring the subject of perturbations in because Wittgenstein himself does: Suppose we observed that all stars move in circles. Then “ All stars move in circles” is an experiential proposition, a proposition of physics – Suppose we later nd out they are not quite circles. We might say then, “ All stars move in circles with deviations ” or “ All stars move in circles with small deviations.” (LFM IV: 43 ) If it is a calculation we adopt it as a calculation – that is, we make a rule out of it. We make the description of it the description of a norm – we say, “This is what we are going to compare things with.” It gives us a method of describing experiments, by saying that they deviate from this by so much. (LFM X: 99)
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physics) and the like. There are a number of possible explanations of the utility of stigmatizing deviations in this way, at least in the areas of language, logic, and mathematics.7 Society has an interest in rendering certain practices as uniform as possible, and adding negative and positive incentives may do the trick.8 This account is plausible in the areas of language and mathematics, which is our topic here. The so-called “rule-following paradox,” as Wittgenstein himself labels it in PI, is (and is intended to be) a paradox only for academic philosophers. I9 use this term to refer to those who take the goal of philosophy to explain10 human practices like rule-following – i.e., almost all philosophers besides Wittgenstein. The explanations emerge from diverse philosophies, from mentalism to physicalism, but all agree that there must be some fact about a person, beyond the regularities of his behavior, in virtue of which we can say he is following a specic rule at a specic time and not another. The explanation that follows here follows that of Saul Kripke, though, as will become clear, it di ers from his in some crucial details.11 Kripke attributes a “skeptical argument” concerning rule-following to Wittgenstein, and has drawn much criticism on this account. I agree with Kripke that Wittgenstein did construct a skeptical argument, but I hold that the argument is supposed to be valid only for academic philosophy, as distinct from Wittgenstein ’s own philosophy. 7
Wittgenstein never attempted to found an account of ethics on this basis – there are regularities in the way people treat one another, and moral norms arise from stigmatizing deviational behavior – and it is an interesting question why. 8 “ When someone whom I am afraid of orders me to continue the series, I act quickly, with perfect certainty, and the lack of reasons does not trouble me.” (PI: 212 ) I believe that this passage reects actual occurrences in Wittgenstein’s life. After the publication of the Tractatus, Wittgenstein left academics and went into school teaching. Ray Monk (Monk 1991, pages 195 –196, 232 –233) reports that Wittgenstein used to inict corporal punishment on his pupils if he thought they were not applying themselves to the arithmetic lessons he was giving them. Not enough has been said about the connection between the rule-following arguments in Philosophical investigations and Wittgenstein’s short-lived experience as a schoolteacher, which came to an end, when one of his pupils lost consciousness as a result of being struck by Wittgenstein. 9 Wittgenstein himself referred to “academic philosophy ” in a letter, but not in his published or (so far as I know) unpublished works. Felix Mühlhölzer draws my attention to the following passage from Zettel , 299 : We say: “If you really follow the rule in multiplying, it MUST come out the same.” Now, when this is merely the slightly hysterical style of university talk, we have no need to be particularly interested. . . 10
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Since it turns out that there is no such “fact” which can serve as an explanatory criterion for rule-following, academic philosophers are faced with a paradox since it now follows that there is no such thing as rulefollowing at all.12 This could be considered a skeptical argument, though an ad hominem one.13 Let us now examine the arguments Kripke brings to support the contention (on behalf of Wittgenstein) that there is no fact in virtue of which somebody is following a rule. Kripke adduces two arguments for this, of which only one is actually in Wittgenstein. This is the “normative” argument: to ascribe rule-following to someone is to assert that someone is acting according to a norm, i.e. following the rule “correctly ” – as we say. The gap between “is” and “ought” then implies that no fact or state of the person at time t could be identied with following the rule at t . The situation is di e rent in other cases of explanation and reduction in science. 14 Disposition terms (e.g. “solubility ”) can in principle be reduced to “state descriptions” of a substance, which actually replace the disposition term. This is the so-called “place holder ” theory of dispositional terms. Given the normative nature of rule-following, i.e. its social nature, it applies to what interests society: behavior. Thus it cannot be reduced to, or identi ed with, but only correlated with, an underlying state description either of the mind or of the brain; that is Wittgenstein ’s argument. Kripke has another argument, the so-called “innity ” argument, according to which any state of the brain, 15 for example, is necessarily nite; while rule-following commits the trainee to innitely many 12
The case is formally similar to Frege’s foundation of arithmetic upon “logic.” When Russell’s paradox showed that Frege’s “logic” is inconsistent, Frege overreacted by saying “arithmetic totters.” 13 Kripke compares Wittgenstein’s “skeptical argument” to that of David Hume, and the comparison is just, but not in the way that Kripke imagines: both Wittgenstein and Hume use skeptical arguments to dispose of various kinds of academic philosophy, without themselves being skeptics. As I have argued in the text above, Hume’s skeptical argument disposes of necessary connections between events, which are used in rationalist explanations of causal reasoning. It is a skeptical argument only for them, because they hold that, without the necessary connections there is no causal reasoning at all. See here (Steiner 2009: 26 ). 14 I am here o ering my own opinions, not those of Wittgenstein. In fact, I am not at all sure that Wittgenstein distinguished clearly between dispositions in science and abilities in humans, since in Philosophical investigations , 193–194 , he claims that academic philosophers make the same kind of mistakes in discussing the abilities of humans with dispositions of machines. See also Philosophical investigations , 182, where he compares “to t” (said of bodies in holes) and “to be able,” “to understand,” said of humans. 15 As my colleague Oron Shagrir has cogently argued, Kripke seems to be thinking of a brain state as the physical realization of a nite digital computer.
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applications of the rule – as in the rule “add two.” Not only is this argument not in Wittgenstein, it couldn ’t be, as I will argue below.16 The “rule-following paradox ” is, then, a paradox only for academic philosophy. For Wittgenstein himself there is no paradox to begin with. For paradox to loom, our ordinary discourse about some topic must be seen to lead to catastrophe. For example, Zeno ’s paradoxes began with ordinary conceptions of motion and showed that they lead to inconsistency, or to the conclusion that no motion is possible. Wittgenstein ’s account of rule-following involves the claim that all rules are supervenient upon regularities. In the case of the rule “add 2” which surfaces in Philosophical investigations – how do we know that our trainee is following this rule if he manages to go 2, 4 , 6, 8, . . ., 1000 – or another rule which says that after 1000 one starts “adding 4 ”? We don’t know , but our experience, both as students AND as teachers, is that almost all who produce this series go on the same 17 way: to 1,002 and not to 1,004 . (The claim is not that we re ect upon these regularities – that would be a misinterpretation – but that the regularities make our practice in this regard possible and coherent.) This regularity allows us to attribute the rule “add two” (the rule which is “hardened” from just this regularity) with great condence to our trainee, and to call his response “erroneous” or perhaps “provocative” if he says next 1,004 . (We may lter out frivolous responses by warning the trainee that he will be severely sanctioned if he doesn’t give the right answer.) Wittgenstein expressed this idea quite clearly in 1939: Because in innumerable cases it is enough to give a picture or a section of the use, we are justied in using this as a criterion of understanding, not making further tests, etc. (LFM I: 21)
The position I am attributing to Wittgenstein is not that of Kripke. Kripke seems to regard as a criterion that our trainee is following the rule þ2, if he 16
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In fairness, however, I must add that my own presentation of Wittgenstein ’s “normative” argument that there is no “fact” in virtue of which somebody is following a rule, also “ improves” the argument somewhat. The distinction I draw between disposition terms – which are in principle reducible to state descriptions – and rule-following ascriptions – which are not – is not in Wittgenstein. On the contrary, Wittgenstein tends to see rule-following as precisely a disposition, but denies that disposition terms are reducible to underlying state descriptions. Since I think Wittgenstein is mistaken here, not only about the issue itself, but also about how to make his own argument (a malady which many philosophers are prone to), I have made the necessary adjustments. Kripke ’s “innity ” argument, on the other hand, is one which in fact contradicts basic Wittgensteinian insights and has no place even in an “improved” Wittgensteinian corpus. I will deal below, page 133 , with the objection that there is no objective meaning to “the same” and hence that the claim is circular.
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goes on to give the same responses that “ we” would give, i.e. agreement with society. On the contrary, society is itself predicated on empirical regularities of its members. The criterion, then, is simply that our trainee has successfully followed the rule þ2 up to now. It is true that without behavioral regularity upon training, this criterion would have no point. But to apply the everyday criterion, one does not have to re ect on this regularity, or even know about it. Wittgenstein argues in Philosophical investigations that the notion of a rule and that of regularity are picked up simultaneously as a result of our common training, so that there is no circularity in saying that a rule is founded on regularity even though detecting a regularity requires an ability of ours to follow rules. The same training teaches the concept of “the same.” For this reason our previous statement that “People trained the same act the same” is not susceptible to skeptical doubt. 18 In the passages of Philosophical investigations we are discussing, though not necessarily in RFM, Wittgenstein is employing a very simple concept of “applying ” a rule. We may understand Wittgenstein as saying that applying a rule is simply following it (correctly). Since rules are grounded in regularities, it is the ability to continue the series by doing what almost everybody does when placed in the same situation, which grounds the ability to apply the rule. In principle, there is no di erence between rules of logic and grammatical rules. The “hardness of the logical must” is a kind of projective superstition, much as the superstition that Hume thought he had exposed in the idea there is “necessary connection” between causes and their e ects. It is similar to the superstition of thinking that when one is reading he is having a characteristic experience of being “inuenced” or being “guided” by the text. (PI, §§ 170 .) Since rules are norms, there is no equivalence between saying that somebody is following a rule and saying that his behavior falls under the underlying regularity. Saying that somebody is following a rule is simply evaluating his behavior, not describing it – even though the evaluation results from observing his previous behavior and responding to it in light of our own training in following rules, and the regularities that are instilled by that training. In other words, we can describe the criteria that a teacher is using to evaluate the student ’s behavior as successfully following a rule, even though the teacher may not be aware of these criteria. In fact one of the purposes of Wittgenstein’s analysis in
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Philosophical investigations is to unearth these criteria in support of his view of philosophy as being purely descriptive. To the question “How does the rule follower himself know that he is following a rule” Wittgenstein answers that the exclamation “Now I can go on” is often not a description at all, certainly not of his own mental state,19 though Wittgenstein does not deny that there are such states. It is certainly no form of self-knowledge, and given what Wittgenstein says in On certainty (1974 ) , the conviction on the part of the rule follower who exclaims “Now I can go on” actually rules out knowledge.20 In summation, logic consists of rules governing the use of logical expressions like “and,” “or,” “if . . .then,” “everything,” etc. As Wittgenstein himself put it, even in the 1940s, “The rules of logical inference are rules of the language-game.” (RFM, VII: 401) There is nothing akin to “intuition,” “seeing,” and the like in following or producing a logical argument. Instead we have regularities induced by linguistic training, which in hindsight are interpreted, or misinterpreted, by us as some kind of determination. Deviation from this regularity is labeled by society as “incorrect” reasoning. Wittgenstein ’s aim is to demystify logic and logical necessity, just as Hume’s aim was to demystify causation by eliminating the alleged “necessary connection” between events. The image of logic as a kind of “super-physics” is what needs to be debunked. Philosophical investi- gations contains a number of references to this mysti cation of logic: 89. With these considerations we nd ourselves facing the problem: In what
way is logic something sublime? For logic seemed to have a peculiar depth, a universal signi cance. Logic lay, it seemed, at the foundation of all the sciences. 97. Thinking is surrounded by a nimbus.
Its essence, logic, presents an order: namely, the a priori order of the world; that is, the order of possibil- ities , which the world and thinking must have in common. –
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. . .The preconception of crystalline purity [in logic] can only be removed by turning our whole inquiry around. (One might say: the inquiry must be turned around, but on the pivot of our real need.)
Wittgenstein devoted a great amount of thought to this topic in LFM. Concerning the “law of contradiction” (actually the law of “noncontradiction”) he stated: 19
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Let us go back to the law of contradiction. We saw last time that there is a great temptation to regard the truth of the law of contradiction as something which follows from the meaning of negation and of logical product and so on. Here the same point arises again. (LFM: 211).
The expression “follows from” is circular here, Wittgenstein is pointing out, since logic itself is the criterion of “ what follows.” The term “follows from the meaning,” is incoherent, since meaning is tied to use, and it does not make sense to speak of “following from” use. The “temptation” of which Wittgenstein speaks here, is the attraction of the academic philosopher (and the early Wittgenstein, of course) to a covert Platonism. Wittgenstein ’s own “demystied” view is that logical laws are a special case of rules that are based on regularities of speakers of language – i.e. rules of “grammar.” As in the general case of rule following, in which the rules are grounded in regularities, and nothing more, so are logical laws the “application” of training in rules to new cases. In another lecture, Wittgenstein said: 21
If we give a word one particular partial use, then we are inclined to go on using it in one particular way and not in another. “Not” could be explained by saying such things as “There’s not a penny here” or saying to a child “Must not have sugar ” (preventing him). We haven ’t said everything but we have laid down part of the practice. Once this is done, we are inclined when we go on to adopt one step and not another – for example, double negation being equivalent to a rmation. (LFM: 242–243)
We can say, then, that logical laws arise in a two step process. First, the child is trained in the use of words like “not.” The training induces a regularity in this use, a regularity which society reinforces as “correct” usage. Within this regularity, however, there arises a subregularity, when the rules for using “not” are to be applied to special cases like double negation. Most trainees nd themselves using double negation as they would a rmation. This regularity is then “put in the archives” as a law of logic. Something similar happens in arithmetic, according to Wittgenstein. In applying the rules for division to 1/7, most procient students nd themselves repeating the sequence 0.142857142857. . . In fact, most pro cient students in dividing m by n always get a nite decimal or a repeating decimal. This subregularity is then converted into a rule in itself, a law or 22
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This is a point that Quine also made during the very same period. See (Quine 1936a), reprinted in (Quine 1976). See LFM, p. 123 .
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proposition of arithmetic, as a result of a proof. This view is in accordance with Wittgenstein’s arguments in PI (186 . ) that rule-following is grounded, and grounded only, in the actual behavioral regularities of individuals and group. The covert Platonist wishes to say more: that, for example, there is an objective fact of the matter by which the theorem about repeating decimals is “determined,” in some further – perhaps metaphysical – sense by the rules for division, once they are accepted. The search for a fact like that, however, Wittgenstein has argued, collapses into paradox. It would be an error, however, to conclude that the only di erence between arithmetic and logic is that they control di erent vocabularies: that arithmetic controls numerical terms and logic controls sentential connectives and quantiers. The year 1937 saw a revolution in Wittgenstein’s view of arithmetic, and mathematics in general: arithmetic propositions remained rules as always – it was the nature of the rules that changed. Mathematical rules were to govern nonlinguistic practices as well as linguistic ones. Arithmetic propositions, in Wittgenstein ’s post-1937 thought, are rules that govern our practice of counting. Geometrical propositions are rules that govern our practice of measuring. 23 Not only does the application no longer “take care of itself,” it is the very heart of the mathematical proposition. The canonical application is now precisely the regularity of counting or measuring which is “hardened” into a rule. The applications of arithmetic and geometry are outside mathematics; they are empirical applications. The applications of logic remain, as before, within logic: an application of modus ponens, for example, is simply an inference of the form “If A, then B; A; therefore B. ” To see the di erence between these two kinds of applications, consider an example Wittgenstein loves to use: the game of chess. When we apply the rules of chess, we are only playing chess. The rules can apply to in nitely many “chess sets,” which are unlimited in their physical composition and also shapes. However, although chess is essentially a war game between two “kingdoms,” there are no applications of chess outside the game itself, even to war itself. The more abstract theory of games, which is real mathematics, does have such “external” applications. Wittgenstein even ventured the idea in 1939 that set theory is not mathematics at all, because it has only imaginary applications. In the
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This is not Wittgenstein’s only interpretation of geometry: see (Mühlhölzer 2001) for another one.
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1940s he stated that the meaning of a mathematical proposition, as well as
of a mathematical concept, is determined by the application. 24 In this scheme, proofs bring the mathematician to convert regularities to rules. Wittgenstein came up with the idea that they do so by being schematic pictures of these nonmathematical regularities, something like owcharts in computer science or schematic diagrams in electronics: “
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What is interesting is that some of these new rules function as rules that determine whether previous rules have been followed. (Since, by Wittgenstein s rule-following considerations, there is no fact by which the previous rule has been followed, the idea as such does not harbor any contradiction. For Wittgenstein, the rule-following paradox is not only not a paradox, but it bolsters his account of mathematics.) Consider again the theorem that: ’
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One might think that the in nite expansion of 1/7 is determined25 from the beginning by the rules for division that are learned in school (or were once learned in school). But the rules cannot outstrip the regularities that are their basis, and the regularities, being regularities of human beings cannot go on forever, and in fact, at some nite point, the regularities will peter out: the deviation will increase to the extent that no rule could be founded on human practice. Mathematics to the rescue of mathematics: the theorem gives a sche- matic picture of doing the division. Using a pigeonhole principle it is clear that the algorithm will run out of remainders, and thus that the “
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remainder, 3, will recur, and thus that the whole cycle will start again. This induces the mathematician and the rest of us to label as “ wrong ” any calculation which does not lead to a repeating decimal; it overrules the naïve use of the school rules. This kind of proof is characteristic of mathematics: It is just the same with 1:7 = 0.142857142. . . You say, “This must give so-and-so.” Suppose it doesn’t. Suppose what doesn’t? Here I am adopting a new criterion for seeing whether I divide this properly – and that is what is marked by the word “must”. But it is a criterion which I need not have adopted. For just as bricks measured with all exactness might give a curve ( ‘space is curved’), so 1 : 7 = 0. . . . looked through with all exactness might give something else. But it hardly ever does [my italics – i.e., we have noticed an empirical regularity]. And now I ’ve made up a new criterion for the correctness of the division. And I have made it up because it has always worked. If di erent people got di erent things, I’d have adopted something di erent. (LFM XIII: 129)
In fact, Wittgenstein remarked in RFM that one should not regard calculations with very large numbers as simple applications of the rules for the operations which we learned on small numbers: We extend our ideas from calculations with small numbers to ones with large numbers in the same kind of way as we imagine that, if the distance from here to the sun could be measured with a footrule, then we should get the very result that, as it is, we get in a quite di erent way. That is to say, we are inclined to take the measurement of length with a footrule as a model even for the measurement of the distance between two stars. (RFM, Part III: 147)
The reader should not be surprised to nd Wittgenstein in a somewhat ambivalent attitude towards “nitism”:26 If one were to justify a nitist position in mathematics, one should say just that in mathematics “innite” does not mean anything huge. To say “There’s nothing in nite” is in a sense nonsensical and ridiculous. But it does make sense to say we are not talking of anything huge here. (LFM: 255 ) 26
By “nitism” Wittgenstein always means what is now called “strict nitism,” according to which it is incorrect or false to assert “There are innitely many natural numbers.”
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We now can see where Kripke went wrong in attributing the innity argument to Wittgenstein. The argument was supposed to defeat the idea that following a rule is identiable with some (perhaps dispositional) state of the brain. When we say that somebody is following the rule þ2 or even plus, we are saying that he is committed to in nitely many (correct) responses to the question, What is . . . þ2 ? But the brain, being nite, cannot produce in nitely many answers to questions of this kind. Kripke discusses a number of possible responses to this argument and nds fault with them all. He does not realize, however, that the major premise of his argument is in direct con ict with a basic feature of Wittgenstein s account of arithmetic: the idea that adopting an algorithm like plus determines in some physical, mental, or metaphysical way one s response to innitely many exercises is nothing but covert Platonism, in many ways worse than the Platonism of objects. These reections reect on the application of logic to arithmetic. By the application of logic to arithmetic I mean simply the substitution of arithmetic propositions in the variables (or schematic letters, if you prefer) of logical rules or truths. Consider the law of the excluded middle, a law of the Propositional Calculus, p_ p. An application of this would be: Either the Goldbach conjecture is true or its negation is true. The Goldbach conjecture states that every even number greater than 2 is the sum of two primes (e.g. 8 = 5 þ 3). The conjecture has been shown to hold for very large numbers, and there are corollaries of the conjecture which have been proved. But no proof of the full conjecture has been given, though most mathematicians are persuaded that it is true. (There are pseudo-probabilistic arguments for this, based on the fact that as the numbers get larger, the probability that a given number can be partitioned into two primes rises monotonically, since the number of the partitions themselves rises.) The intuitionists hold that it is a form of metaphysics to assert the law of excluded middle for such a case. To assert it here is to presuppose that the natural numbers form a closed totality, or what Aristotle called an actual innite, so that we can say that either there is, or is not, a counterexample to the Goldbach conjecture in this closed totality. If we think of the natural numbers through the metaphor of becoming, rather than being, then the present absence of a proof or of a refutation of the Goldbach conjecture means only that the truth of the conjecture is not determined, and the law of the excluded middle cannot be asserted. As an invalid rule of inference, it is thus banished from classical mathematics. “
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Let us now apply Wittgenstein’s ideas to the Intuitionist program. Wittgenstein agrees entirely with the Intuitionist critique of the law of excluded middle. For the Goldbach conjecture to be “true” in the sense of classical mathematics, we have to say that the operations of arithmetic determine in advance that every even number, no matter how large, can be partitioned into two primes. Wittgenstein agrees that this is not mathematics, but metaphysics: a statement like this cannot be grounded on the behavioral regularities inculcated in grade school. A statement true of all the natural numbers can be based only upon a theorem – which lays down a new norm (on the basis of a proof ) which labels any deviation from the Goldbach conjecture a “mistake.” Hence, Wittgenstein agrees with the Intuitionists that one cannot regard the law of excluded middle for the Goldbach conjecture as a theorem of mathematics. It cannot be regarded as the “hardening ” of a regularity. How, then, are we to square this with Wittgenstein ’s explicit disavowal of Intuitionism (“Intuitionism is all bosh,” he said, “entirely ” (LFM XXIV: 237))? There are two explanations available. The rst has to do with the connection of Intuitionism with . . . intuition. Brouwer writes as if the numbers themselves are mental constructions, and the law of excluded middle does not apply to mental constructions, which can never produce a closed totality. Wittgenstein is an implacable opponent of the concept of mathematical intuition – he believes, among other things, that it has no explanatory value, and hence its only rationale fails. From this point of view, Intuitionism is a form of mentalism, the other side of the coin from Platonism. Both are unacceptable foundations of mathematics. It should be noted, however, that Michael Dummett (Dummett 1975) championed a non-metaphysical version of Intuitionism, one which has little or nothing to do with mathematical intuition. According to this point of view, which is presumably heavily inuenced by Wittgenstein’s thought, truth in general is associated with “assertibility.” And since mathematical propositions are asssertible only when provable, Dummett thinks,27 one cannot assert an instance of the law of excluded middle at time t unless we can show at t that one of the two alternatives can be proved. Thus a proof of the following form is invalid at t , despite the acceptance of it by almost all mathematicians: 27
I actually deny this, and have given examples of mathematical propositions that were assertible even when there was no proof of them in (Steiner 1975). But I will take for granted that Wittgenstein agrees with Dummett on this point, an agreement that has a solid basis in the corpus.
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If the Goldbach conjecture is true, then T If the Goldbach conjecture is false, then T Therefore, T.
For Wittgenstein, to blacklist mathematical theorems, on the basis of the Intuitionist attack on the law of excluded middle means: to revise mathematics on the basis of a philosophical argument. Wittgenstein is quite explicit on this point in one of the most famous passages of Philosophical investigations : no mathematical discovery is relevant to philosophy, and no philosophical argument can revise accepted mathematical practice. Philosophy describes practice; and the only reason we need philosophy is that we have a strong tendency to misdescribe it (i.e., practice). We now seem to have reached an impasse: Wittgenstein upholds the behavioral/empirical basis of the mathematical propositions, or rules. At the same time he refuses to revise mathematical practice on the basis of Dummett’s arguments, themselves based on Wittgensteinian ideas! The resolution of this “paradox ” is based on another Wittgensteinian idea: that mathematics is a “motley ”28 of proofs. The idea that mathematical theorems are “hardenings” of regularities was never meant to be a characterization of the “essence” of mathematics. The philosopher who emphasized so strongly the idea that the referents of certain terms (and really all terms) are related only by a “family resemblance” did not become an “essentialist” suddenly when he studied mathematics. And in fact, Wittgenstein told the students in his 1939 Lectures at Cambridge that the law of excluded middle in the in nite case (i.e. either all natural numbers have property P or not all natural numbers have property P) should be regarded as a “postulate” and was used as such in mathematics. Presumably the postulate should be judged by its usefulness in mathematics, though Wittgenstein, ironically, rejected the most celebrated attempt (Hilbert 1983) to “ justify ” the law of excluded middle – namely, by showing – without using the law of excluded middle – that the law of excluded middle does not lead to contradiction, when applied to “innitary ” statements: “Either all numbers have property P, or there is a number that does not have property P. ” A consistency proof can be compared to theorems to the e ect that, in chess, a forced checkmate is not possible from a certain position. And the attempt to nd one is associated with David Hilbert ’s programmatic “On 28
Mühlhölzer protests this translation, which has become entrenched in the philosophical Wittgenstein discourse, and insists that the right phrase is “multi-colored.”
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the Innite” – the program which is almost universally thought to have been refuted by Gödel’s second theorem – which states that arithmetic cannot prove its own consistency even if the law of excluded middle is used, to say nothing of the kind of combinatorial, “metamathematical,” proof Hilbert had in mind. At the risk of digressing, I would now like to discuss in a little more detail why a consistency proof for Wittgenstein is not what we are seeking in showing the usefulness of the law of excluded middle. Wittgenstein’s rejection of Hilbert’s program had nothing to do with Gödel’s theorem, which he in any case regarded with suspicion. On the contrary, he regarded Gödel’s theorems as part and parcel of what was wrong with the program to begin with – the concept of “metamathematics.” Nor did he regard the search for consistency proofs for mathematics as having anything to do with showing the usefulness of the “postulate” of the law of excluded middle as he saw it. Wittgenstein’s discussion of contradictions and consistency is of a piece with his theory of rule-following in general. How do we get convinced of the law of contradiction? – In this way: We learn a certain practice, a technique of language; and then we are all inclined to do away with this form – on which we do not naturally act in any way, unless this particular form is explained afresh to us. (LFM: 206)
He later explained this point this way: This simply means that given a certain training, if I give you a contradiction (which I need not notice myself ) you don’t know what to do. This means that if I give you orders I must do my best to avoid contradictions; though it may be that what I wanted was to puzzle you or to make you lose time or something of that sort.
The Law of Contradiction is thus nothing but the hardening of a linguistic practice into a rule. From this it follows that the concept of a “hidden contradiction” does not have a clear meaning. There is always time to deal with a contradiction when we get to it. When we get to it, shouldn ’t we simply say, “This is no use – and we won’t draw any conclusions from it”? (LFM: 209)
At this point, Alan Turing (who attended Wittgenstein’s lectures at Cambridge in 1939) remarked that the problem could arise in the applica- tions of logic and mathematics.
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The real harm will not come in unless there is an application, in which case a bridge may fall down or something of that sort. (LFM: 211)
Wittgenstein responded to Turing ’s attack in the following way: The question is: Why should they be afraid of contradictions inside mathematics? Turing says, “Because something may go wrong with the application.” But nothing need go wrong. And if something does go wrong – if the bridge breaks down – then your mistake was of the kind of using a wrong natural law. (LFM: 217)
For as long as an actual inconsistency does not turn up, Wittgenstein held, we need not worry that the “bridges will fall down.” Like any other mathematical proposition, inconsistency is either a rule, or nothing. As long as it is not a rule, i.e. a proven theorem, physical applications go on as before. But let’s look at this a little closer. Wittgenstein discusses whether a bridge could fall down because somebody divided by zero. This is certainly possible; consider the equation x 2 = x . 93 percent of precalculus students at City College of the City University of New York, in a recent test, divided by x and got the (only) answer x = 1.29 Not knowing about the solution x = 0 could, in some scenarios, indeed cause a bridge to fall down. Much more sophisticated cases could be constructed in which somebody does not know he is dividing by zero. Is this a case, however, of an inconsistency of a formal system, or is it just a simple mistake in informal mathematics? One could imagine a case of teaching students an axiomatic number theory in which cancellation of zero is possible, in other words an inconsistent system. The students might not even notice that ac = bc a = b yields 1 = 2 if we allow c to be zero, because they have little cause to divide by zero. But it is hard to think of an actual case in which a hidden contradiction in a formal axiomatic system caused “bridges to fall down.” A good example of this quandary is the theory of quantum electrodynamics (QED), pioneered by, among others, Schwinger and Feynman. 30 The calculations a o rded by this theory are remarkably accurate, but nobody knows how to base the calculations in a consistent axiomatic mathematical system. In fact, there are mathematical physicists who think it cannot be done. One reason is as follows. In calculating the probability of events in !
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quantum mechanics, according to the usual rules, we come upon in nite integrals. To renormalize these integrals, which are intended to be (Feynman 1985) functions of the basic constants of physics, we change the rules: instead of theoretical values like e , the charge of the electron, we substitute the observed value of the electronic charge, a value derived from experiment, not theory. But now we have a new problem. While the new rules work very well for observed values at the low scales of energy with which we are familiar, as the energies get higher the observed electronic charge becomes larger and larger, so that even the new rules are not valid. In fact, mathematical physicists think that this charge may become in nite at some nite high energy, so that QED is not de ned at all as a universal theory of light and matter. 31 For Wittgenstein, this would just show what he was claiming all the time: that the ideal of a formal system does not t the reality of mathematical physics.32 This would be a perfect example of The disastrous invasion of mathematics by logic. (RFM, V: 281) It is also plausible that the inconsistency which appears in the in nite integrals has its source in the physics, not the mathematics, exactly as Wittgenstein says. But, further, the physicists managed to eliminate the troublesome integrals, albeit by tweaking the rules for calculations in QED by using, as we said, not the naked magnitudes that appear in the Hamiltonian of the system, such as e , but the dressed magnitudes as measured in the laboratory. This artice works, and no physicist worries that a possible inconsistency (which is suspected though not proved) could somehow spoil the calculations we make at familiar energies. Coming back to the law of excluded middle, we see that the problem is not that it has no formalist justi cation in terms of a combinatorial consistency proof. It is rather that the law of excluded middle cannot be regarded as a hardened regularity in cases in which we are applying it to a putative innite totality. But precisely because of this, there is no direct comparison possible between empirical observations and mathematical theorems in this type of proof. That is what Wittgenstein means by a postulate. The justication of such a postulate would be, in Quine s pithy words, where rational, pragmatic. 33 It would seem that Wittgenstein, in accommodating classical mathematics and rejecting the intuitionist revisionism, ends up where Quine began: in holism. 34 “
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part ii
History and Authors
chapter 8
Logic and its objects: a medieval Aristotelian view Paul Thom
What is logic about? According to one familiar account logic tells us when arguments are valid; logic is thus about arguments. On another account logic tells us which propositions are (unconditionally) necessary; logic is thus about propositions (Smith 2012). Less familiar than either of these accounts is the Aristotelian tradition of thinking about logic. Aristotelians have standardly thought of logic as being about terms, as well as propositions and arguments. Let us call propositions and arguments, and whatever else logic has been supposed to be about, the objects of logic. The general question that interests me is: What are the metaphysical types to which the objects of logic belong? More specically, I will look at the way this question has been addressed in the Aristotelian tradition. I will not be dealing with answers to our question proposed by Platonists or with the Stoic concept of lekta . I use the expression the Aristotelian tradition to cover the writings of Aristotle himself as well as those over time who have broadly sympathised with his views. The latter include the ancient Greek commentators, a multitude of medieval logicians writing in Arabic or Latin, and a smaller number of later thinkers (notably Bernard Bolzano). But my main focus will be on just one of these, the thirteenth-century philosophical logician Robert Kilwardby (d. 1279). Kilwardby dealt with our question at some length, and his discussion is also useful in that it considers several views other than his own. Let us begin with Aristotle s own ideas on our question. ‘
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First, the Categories makes a remark about statements: Statements and beliefs, on the other hand, themselves remain completely unchangeable in every way; it is because the actual thing changes that the contrary comes to belong to them. For the statement that somebody is sitting remains the same; it is because of a change in the actual thing that it comes to be true at one time and false at another. Similarly with beliefs. Hence at least the way in which it is able to receive contraries – through a change in itself – would be distinctive of substance, even if we were to grant that beliefs and statements are able to receive contraries. However, this is not true. For it is not because they themselves receive anything that statements and beliefs are said to be able to receive contraries, but because of what has happened to something else. For it is because the actual thing exists or does not exist that the statement is said to be true or false, not because it is able itself to receive contraries. No statement, in fact, or belief is changed at all by anything. So, since nothing happens in them, they are not able to receive contraries. (Aristotle 1963: 4 a 5)
Here Aristotle leaves two positions open: either statements do not change truth-value at all, or else any change in their truth-value is due to a change in something external to them, namely the things which the statements are about. Second, in the De Interpretatione we nd Aristotle apparently proposing a general semantic theory according to which the meaning of spoken and written utterances is to be found in the existence of mental items that somehow correspond to them: Now spoken sounds are symbols of a ections in the soul, and written marks symbols of spoken sounds. . . . Just as some thoughts in the soul are neither true nor false while some are necessarily one or the other, so also with spoken sounds. For falsity and truth have to do with combination and separation. (Aristotle 1963: 16a 2)
Here, the meaning of spoken and written language is derived from ‘a ections in the soul ’, and truth and falsity are seen as residing primarily in the combination or separation of mental items. Third, there is a remark in the Posterior Analytics which, again, seems to point to the soul as the locus of truth and demonstration. By contrast, it is always possible to nd fault with ‘external’ arguments (i.e. spoken or written ones): For demonstration is not addressed to external argument – but to argument in the soul – since deduction is not either. For one can always object to external argument, but not always to internal argument. (Aristotle 1994 : 76b23)
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Finally, there is a remark in Metaphysics Book 6 which again locates truth and falsity in the soul rather than in external reality: But since that which is in the sense of being true, or is not in the sense of being false, depends on combination and separation, and truth and falsehood together are concerned with the apportionment of a contradiction (for truth has the a rmation in the case of what is compounded and the negation in the case of what is divided, while falsity has the contradictory of this apportionment – it is another question, how it happens that we think things together or apart; by ‘together ’ and ‘apart’ I mean thinking them so that there is no succession in the thoughts but they become a unity – ; for falsity and truth are not in things – it is not as if the good were true, and the bad were in itself false – but in thought; while with regard to simple things and essences falsity and truth do not exist even in thought): – we must consider later what has to be discussed with regard to that which is or is not in this sense; but since the combination and the separation are in thought and not in the things. (Aristotle 1993: 1027b30)
In sum, Aristotle thinks that 1.
statements, as the bearers of truth and falsity, are in the soul and are either unchanging or any change in them is due to a change in something else; 2. the meaning of written and spoken language is to be explained by reference to what goes on in our minds; 3. truth and falsity belong in the rst instance to combinations and separations that occur in our minds. These are scattered remarks. Aristotle doesn ’t show how they could be combined in a coherent theory of terms, propositions and arguments. We do not nd such a theory in Aristotle; we nd only some materials that seem to have the potential for theoretical development. An interpreter of Aristotle, faced with this situation, might try to develop a theory in one of two ways. One option would be to enlist elements drawn from Aristotle ’s metaphysics or his account of scienti c knowledge. Another would be to import non-Aristotelian ideas. We will see that both approaches were used by later Aristotelians in their e orts to esh out Aristotle ’s sketchy remarks. One obvious place to look for theoretical help in this enterprise is the Philosopher ’s division of all beings into the ten categories (substances, quantities, relatives, qualities etc). From the standpoint of the theory of the categories, our question becomes: Do the objects of logic belong to any of the Aristotelian categories, and if they do, to which category or categories
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do they belong? This question was explicitly posed by a number of thinkers in the twelfth and thirteenth centuries.
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Robert Kilwardby
The thirteenth-century English philosopher and churchman Robert Kilwardby commented extensively on Aristotle s logic, as well as composing a treatise On the origin of the sciences and a set of questions on the four books of Peter Lombard s Sentences . Over the course of his career he showed a continuing interest in the nature of the objects of logic, and indeed the nature of logic itself. In his early question-commentary on the Prior Analytics Kilwardby takes the view that logic is one of the language-related sciences along with grammar and rhetoric. In the work s rst sentence he adopts Boethius s characterisation of logic as an art of discoursing (Kilwardby 1516: 2 ra).1 He goes on to consider the meaning of the words proposition [ propositio , Aristotle s protasis ] and syllogism [syllogismus ] as they occur in Boethius s translation of Aristotle s text, distinguishing propositions from statements [enuntiationes ]. A statement is put forward on its own account, a proposition on account of the conclusion it is intended to support. A statement expresses what is in the speaker s soul, and accordingly is dened as that which is either true or false since truth and falsity reside in the soul (Kilwardby 1516: 4 rb).2 In his other writings Kilwardby will generally preserve this distinction, reserving the term proposition for the premise of an argument. He asks whether a syllogism should be de ned as a kind of process, rather than a kind of discourse (following Aristotle s denition). He agrees that there is a sense in which a syllogism is a mental process, but says that this is a metaphorical sense (Kilwardby 1516: 4 vb).3 And it must indeed be regarded as a transferred usage for someone whose starting-points are Aristotle s usage of syllogism to mean a kind of discourse and Boethius s characterisation of logic as a science of language. In his later work On the rise of the sciences logic is no longer characterised purely as a linguistic science, and the syllogism is no longer a purely linguistic phenomenon. Logic is there presented under two guises. It is a science of reason as well as being a language-related science: ’
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It is called a science of reason not because it considers things belonging to reason as they occur in reason alone, since in that case it would not properly be called a science of discourse, but because it teaches the method of reasoning that applies not only within the mind but also in discourse, and because it considers the things belonging to reason as the reasons why things set forth in discourse can be reasoned about by the mind. . . .. It is, therefore, a ratiocinative science, or science of reason, because it teaches one how to use the process of reasoning systematically, and a science of discourse because it teaches one how to put it into discourse systematically. (Kilwardby 1988: 265)
On this view, which places mental reasoning on a par with reasoning in words, it cannot be right to dismiss as merely metaphorical a conception of the syllogism as a mental process. He raises the issue of the basis or foundation of logic, declaring that there are three di e rent kinds of basis on which a body of scienti c knowledge can be founded. The science might be based on things that actually exist. Or it might be based on potentialities, even when they are unactualised. Finally, a science might be based not on potentialities but on aptitudes of things. These are incomplete potentialities, such as the aptitude for sight which exists even in a blind eye. Now, even though speech passes away as soon as it is uttered, something remains, namely certain natural principles wherein potentialities or aptitudes reside. Because of these, speech contains enough on which to found a science, even when no-one is speaking (Kilwardby 1976: 429). Later in On the rise of the sciences he adds that an art of reasoning has a sucient foundation in the natures of things through which they are susceptible to a rational account. Among these natures he mentions antecedents, consequents, incompatibles, universality, particularity, middles, extremes, gure and mood (Kilwardby 1976: 463). This interest in the foundations of the art of logic is even more evident in a late theological work, Kilwardby s questions on Peter Lombard s four books of the Sentences . Question 90 on Book One of the Sentences contains a detailed exposition of the metaphysical status of the objects of logic. It seems that Kilwardby himself attached some importance to this exposition, for in the alphabetic index which he compiled of the matters covered in his questions, he refers on four occasions to question 90 on Book One. 4 It is therefore worth examining his exposition in detail. ’
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Here is his question: The next question is about divine knowledge in respect of things of reason, which namely are in the human reason and are brought about by reason – things such as propositions, syllogisms and the like, and all manner of complex and incomplex things insofar as they concern reason. And the rst question about these is whether they are something, in such a way that they are things in one or more of the categories. (Kilwardby 1986: 1, q.90: 1)
Although he refers here to ‘propositions’, he proceeds to discuss instead what he calls ‘stateables’ (enuntiabilia). This is no doubt partly because of the distinction he had made earlier between propositions and statements; but this doesn’t explain why he talks about stateables rather than statements. Christopher Martin takes the expression enuntiabile in earlier authors to refer to a statement ’s content rather than to the statement itself (C. Martin 2001: 79). But I will argue later that there is reason for doubting that this is Kilwardby ’s meaning. Concerning the nature of the objects of logic, Kilwardby mentions a view according to which stateables cannot be assigned to any of the Aristotelian categories. Among the arguments he mentions in favour of this view, two rest on Aristotelian texts. First, there is the chapter of the Categories where the ten categories are presented as a classi cation of ‘things said without any complexity ’; stateables on the other hand, if they are things at all, are things possessing complexity. The second Aristotelian text is the one we noted above from Metaphysics book 6. Here, says Kilwardby, the ten categories are presented as being truly outside the mind or soul, whereas composition and division are said to belong to cognition, not to external things (Kilwardby 1986: 1, q.90: 57). Views denying categorial status to stateables or similar quasi-entities were not uncommon in the twelfth and early thirteenth centuries. Peter Abelard, for one, in using the word dictum to refer to a that-clause, or an accusative and innite construction in Latin, thought that the question of what sort of things these dicta are simply does not arise: they are not things at all (King 2010).5 5
King 2010: ‘ Abelard describes this as signifying what the sentence says, calling what is said by the sentence its dictum (plural dicta). To the modern philosophical ear, Abelard ’s dicta might sound like propositions, abstract entities that are the timeless bearers of truth and falsity. But Abelard will have nothing to do with any such entities. He declares repeatedly and emphatically that despite being more than and di erent from the sentences that express them, dicta have no ontological standing whatsoever. In the short space of a single paragraph he says that they are “no real things at all ” and twice calls them “absolutely nothing.” They underwrite sentences, but they aren’t real things. For although a sentence says something, there is not some thing that it says. The semantic job of sentences is to say something, which is not to be confused with naming or denoting some thing. It is instead a matter of proposing how things are, provided this is not given a realist reading. ’
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Again, the twelfth-century Ars Burana denies that enuntiabilia belong to any of the Aristotelian categories.They exist, but belong to a category of their own ( Ars Burana, 208).6 But Kilwardby doesn’t have these versions in mind when he refers to the view that the stateables are not to be found in any Aristotelian category. Rather, he is thinking of the version of the view advanced by the English theologian Alexander of Hales (Hales 1951–1957: 1 d.39 n.1). Alexander held that the ontological type to which a statement belongs depends on whether the statement expresses an essential or an accidental predication. In the former case the statement is nothing other than its subject, and thus belongs to the same Aristotelian category as its subject. Thus the statement ‘Fido is a dog ’ is a substance, and is the very same substance as Fido. In the case of accidental predications, the statement can be reduced to the Aristotelian categories in one of two ways: either it reduces to the category in which its accidental predicate is located, or partly to that category and partly to the category of the subject. Thus ‘Fido is white’ turns out either to be a quality (and then it is the quality of whiteness) or partly a quality and partly a substance (and then it is partly Fido and partly whiteness) (Kilwardby 1986: 1 q.90: 70). In opposition to this view, Kilwardby holds that compositions, stateables and the other objects of logic can be assigned to the Aristotelian categories in their own right without having to be reduced to the categories to which their subjects and predicates belong. His view involves a complex reduction to the Aristotelian categories. Every thing, he declares, is either divine or human. The products of nature he includes among the divine, along with things that issue from God by himself. Human things, in his parlance, do not include what issues from humans solely in virtue of their existence as natural beings, but only what comes about through human activity in the form of industry or skill. He classes the objects of logic, not among divine things, but among human things in this narrow sense (Kilwardby 1986: 1 q. 90: 102). Among such things he distinguishes those that are internal to a human and those that are external. The former include actions of combining, dividing or reasoning, as well as the corresponding acts which he calls 6
Anon 1967: ‘If you ask what kind of thing it is, whether it is a substance or an accident, it must be said that the sayable [enuntiabile ], like the predicable, is neither substance nor accident nor any kind of other category. For it has its own mode of existence [suum enim habet modum per se existendi ]. And it is said to be extracategorial, not, of course, in that it is not of any category, but in that it is not of any of the ten categories identied by Aristotle. Such is the case with this category, which can be called the category of the sayable [ praedicamentum enuntiabile ].’
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combinations, divisions, reasonings etc. The human things that are external include utterances, the making of works and the works made (e.g. the making of a house, and the house that is made). This distinction between what is internal to the human and what is external appears to rest on a distinction between doing and making. While making can be considered as a kind of doing, it can also be distinguished from other kinds of doing insofar as it involves the production of something, or at least a process aimed at the production of something. Thus when we mentally combine or separate concepts, or when we reason in our heads , we do not thereby produce anything external to ourselves: we have done something but we haven t made anything. But when we utter something, or build a house, we do produce something external, we make something. If this is what Kilwardby means, then the acts which he distinguishes from actions, and which he also considers to be internal, cannot be products of those actions. Being purely internal, they have no product. It is clear that the relation between acts and actions should be similar to the relation between works and the making of works. But works stand to the making of works in more than one relation. The relation of product to process is one such relation, but it is of no use to us here because doings which are not makings have no products. There is, however, another relation connecting works to their making: the relation of completion. All actions, in principle, have completions; and it is these completions, I believe, that Kilwardby refers to as acts or things-done. Thus the human things that are the objects of logic include completed acts of stating and reasoning, as well as the actions that have those acts as their completions. According to Kilwardby, all of these are things of reason. They are secondarily in a category, because they are founded on things of nature in one of two ways. In the case of makings and actions of reason, they are founded on things of nature in the sense that the latter constitute their subject matter. In the case of things-done or made by reason and art, they are founded on things of nature in the sense that they are certain relations or accidental conditions of things of nature. Kilwardby takes both of these senses to indicate that the things of reason and art have things of nature as their subjects; and he means here the metaphysical subject that underlies these things of reason and of art. Thus, while it is the things of nature that are primarily and of themselves in the categories, the things of reason and art can be assigned to the categories in a secondary sense, via the things of nature that are their metaphysical subjects (Kilwardby 1986: 1 q.90: 111). Stateables and arguments, whether completed or incomplete, may exist in writing, in speech or merely in thought; and Kilwardby applies the ‘
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above analysis to all three types of case. A written statement or argument, he says, is a string of characters whose order is in accordance with the rules of some art (he has in mind the arts of grammar and logic), and which has the purpose of communicating knowledge of something through visual perception. The characters are, let us say, written in ink; and their metaphysical subject is then the primary substances which these blobs of ink constitute. The written argument or the stateable is not these blobs of ink; it is constituted by certain relations and accidental features of the ink. Entities satisfying this complex description can be considered under more than one aspect; and accordingly they will belong to di erent categories, depending on the aspect under which they are considered. Considered as signs they belong to the category of relatives. Considered as an ordering of characters they could be assigned to the category of location or the category ‘ Where’. Considered as exhibiting a certain syntactic form they can be assigned to the fourth species of quality. In all these ways the relations or properties which constitute the rational entity in question are accidental features of the underlying subject: it is not essential to the ink that it be a sign, nor is it essential to the characters that they be so ordered as to make propositions. Similar treatments can be given of spoken and mental statements and arguments. Whether spoken or merely thought, these are signs and thus belong to the category of relatives. As spoken they are qualities. As thought they are dispositions of the mind – either states or passions and qualities. Equally, the basic mental components which are combined or separated – Aristotle’s ‘passions of the soul’, and Kilwardby ’s intentiones or concepts – can be considered either as qualities residing in the soul, or as relatives insofar as they are signs of external things (Kilwardby 1986: 1 q.90: 163). In sum, Kilwardby holds that stateables and the other objects of logic have the following features: 1. 2. 3. 4 . 5.
They are human things. Some of them are spoken, some written, some mental. They are things of reason. They are grounded in things of nature. Considered as signs, they t into the category of relatives.
Let us return to the meaning of ‘stateable’ in the light of this overall picture. Whatever Kilwardby means by this word, it is evident that stateables must satisfy the above ve conditions. They must also satisfy the terms in which question 90 was framed: they have to be ‘in the human reason and are brought about by reason’. Given these things, it is
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impossible to suppose that Kilwardby meant stateables to be propositional contents, in the modern sense of eternal abstract objects. If he meant stateables to be the contents of statements, he would have to have meant it in a sense that complies with the above constraints. Now, it might be proposed that a suitable notion of content can be devised, according to which contents exist only when the things of which they are the contents also exist. Such a notion, it might be argued, complies with the above constraints. Alternatively, using the Aristotelian notion of potentiality, we could say that a stateable is just a potential statement. The second of these approaches, unlike the rst, allows for the possibility that some stateables are not (yet) actually stated. How well does Kilwardby ’s account of logic and its objects t the sketch given by Aristotle? Aristotle envisaged two possible answers to the question whether statements are immutable. His rst suggestion (that they are entirely immutable) does not gure in Kilwardby ’s account. The objects of logic, on his account, are human things and thus subject to change. And if stateables are potential statements then they change when their potentiality is actualised. However, Aristotle’s alternative suggestion, that statements might be such that any change in them is really a change in other things, is consistent with Kilwardby ’s account. Mental compositions, considered as signs, are relative to that of which they are signs. Moreover, theirs is a special kind of relativity – a kind that gives rise to Cambridge change. I can change from being on your left to being on your right simply because you walk around to my other side while I remain stationary; and similarly the stateable that Socrates is sitting can change from being false to being true simply because Socrates sits down. The mentalistic semantics sketched in the De interpretatione is also consistent with Kilwardby ’s account of the objects of logic, as is his account of composition and separation as located in the mind. But only the second of the ve points listed above is found explicitly in Aristotle. Kilwardby ’s specic conception of logic as an art – an art that deals with human things which are grounded in things of nature – is not to be found in Aristotle. It is Kilwardby ’s way of turning Aristotle’s sketchy account into a theory of the objects of logic. Notwithstanding its departures from Aristotle ’s own remarks on the nature of the objects of logic, Kilwardby ’s account is wholly Aristotelian in its motivation. But the Aristotelian ideas on which he draws do not belong in logic itself; they belong in natural philosophy and metaphysics. His account is thus, to use the terminology of Sandra Lapointe (this volume), an external one.
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Later thinkers
Jakob Schmutz argues that scholastic ideas were transmitted to the early modern period along two paths. The rst of these paths, which he calls the idealistic main road , took the subject-matter of logic to be the activity of the mind. The second path, the realistic by-pass , took logic to deal with independent objects and structures (Schmutz 2012: 249). We have seen a version of the rst path in the writings of Kilwardby. Kilwardby was a moderate realist. But other versions of this path can be found in nominalists like William Ockham, for whom the objects of logic are individual written, spoken, or mental tokens. 7 Walter Burley, who opposed Ockham s views in most matters, appears to be working within the second path: for him, propositions are either complexes depending on mental acts of composition and separation, or intentional complexes existing in the mind, or complexes existing outside the mind, which are signi ed by those mental complexes . These extra-mental propositions [ propositiones in re ] are the causes of truth of mental propositions (Cesalli 2007: 234 ). The second path is taken up in the nineteenth century by Bernard Bolzano then by Frege. Bolzano believed in propositions in themselves (Sätze an sich ), and held that it is the job of logicians to describe these entities and their properties (Lapointe, this volume). He outlines his notion of a proposition as follows: ‘
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One will gather what I mean by proposition as soon as I remark that I do not call a proposition in itself or an objective proposition that which the grammarians call a proposition, namely, the linguistic expression, but rather simply the meaning of this expression, which must be exactly one of the two, true or false; and that accordingly I attribute existence to the grasping of a proposition, to thought propositions as well as to the judgments made in the mind of a thinking being (existence, namely, in the mind of the one who thinks this proposition and who makes the judgment); but the mere proposition in itself (or the objective proposition) I count among the kinds of things that do not have any existence whatsoever, and never can attain existence. (Bolzano 2004 : 40)
The objects of logic, on Bolzano s view, are not human things and are not grounded in the things of nature. As Rusnock and George say, It should be possible, [Bolzano] thought, to characterize propositions, ideas, inferences, and the axiomatic organization of sciences without reference to a thinking subject (Rusnock and George 2004 : 177). ’
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For Bolzano, propositions are not human things, they do not exist in the mind or in language or in any way at all, and they are objective not relative. His view is designed to pare down our conception of the objects of logic to a bare minimum so that propositions are understood simply as that which is true or false, and arguments are understood as congurations of propositions.
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Concluding remarks
In his essay in the present volume Graham Priest asks whether logic can be revised, whether this can be done rationally, and if so how. And he distinguishes logic as something that is taught, logic as something that is used, and logic as the correct norms of reasoning (Priest, this volume). I would like to add a few comments on Priest ’s questions. The history of logic contains plenty of examples of logicians proposing to revise what hitherto had been accepted as the correct norms of reasoning. Some of the great logicians – Abelard and Ockham along with the well-known greats of the nineteenth century – saw themselves as not just revising but reforming logic. Sometimes these reforms are motivated by a sense that accepted logics are erroneous or in other ways inadequate to accepted ideals of what logic should be. And sometimes what motivates a reforming logician is a new vision of what logic should be. I think that the major reformers of the nineteenth century had this sort of motivation. Looking at the ‘traditional’ logic of their day, which was a watered-down version of medieval logic, usually along the lines of Schmutz’s ‘idealist road’, they worked with a vision of logic as an objective science. We bene t today from the fruits of that vision. But it can be salutary occasionally at least to look back to the di erent aims of the ‘idealist’ logicians of the high Middle Ages. The reason why Kilwardby and other ‘idealist’ medieval logicians conceived of the objects of logic as human things is to be found in the aims which they thought logic should have. In treating logic as an art, they were committed to thinking that it should teach us how to construct good denitions, divisions and arguments. So the objects of logic had to include human activities of dening, dividing and arguing. Everyone agrees that an argument is faulty if it allows the conclusion to be false while the premises are true; and accordingly any good logical theory has to include among its norms that one should not argue from truths to a falsehood. Faults and norms go together.
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But in the opinion of most medieval logicians arguments were subject to a variety of faults besides invalidity. Kilwardby is expressing a commonly held view when he says that Aristotle identies two of these faults in his denition of the syllogism: rst when he species that the conclusion conclusion must be other than the premises, second when he requires that the conclusion follow from the premises that are explicitly stated. According to Kilwardby, the rst of these specications rules out begging the question, and the second excludes the fallacy of stating as a reason what is not a reason [ non 1516: 4 vb). causa ut causa ] (Kilwardby 1516 vb).8 In order to see why begging the question, and failing to state explicitly what premises on which the conclusion relies, are faults in reasoning, one has to look at what the point point of reasoning is. Many of the medievals believed that it is the function [opus [ opus ] of the activity of reasoning to make something known by proving it: the function of the syllogism is to prove and make known. (Kilwardby 9 1516, 12rb)
If a particular argument is not suitable for making its conclusion known by proving it, then it is faulty in the way that a functional object is faulty when it is incapable of performing its function. fu nction. And if a form of reasoning is not suitable for making anything known, then it is faulty. Kilwardby s idea idea here here is that that form formss of reas reason onin ingg which which are intr intrin insi sica cally lly quest questio ionnbegging, or which include redundant material, cannot perform the function that belongs to reasoning: ’
And it is to be said that there isn t always a demonstration when the conclusion follows of necessity, but there has to be proof of the conclusion and it has to be made known, and further it is required that the premises are apt to prove the conclusion conclusion and to make it known. But this is lacking when 10 the question is begged. (Kilwardby 1516 1516: 72vb) ’
In turning an art of human reasoning into an objective science, modern logic has made enormous gains in comprehension and rigour. But it has lost its connection with a conception of reasoning as an activity whose point in human a a irs makes it subject to other faults than invalidity. airs 8 9 10
Robert Kilwardby, Notule Kilwardby, Notule libri Priorum Lectio 4 dubium 1 . Kilwardby, Notule Kilwardby, Notule libri Priorum Lectio Priorum Lectio 11 dubium 3 . Kilwardby, Notile Kilwardby, Notile libri Priorum Lectio 67 dubium 3 .
chapter 9
The problem of universals and the subject matter of logic Gyula Klima
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Introd Introduct uction ion:: the the subje subject ct matt matter er of of logic logic and the problem of universals
It might seem that the problem of universals should have little to do with the issue of the subject matter of logic. After all, in (formal) logic we deal with the deductive validity of arguments based on their formal structure, whereas the problem of universals, at least in one of its possible formulations, is the question of what corresponds to the universal terms of our language, which constitute precisely the material part of arguments, the part we disregard or abstract from in formal logic. However, upon a closer look, there is a certain certain connection. connection. On the semantic conception conception of validity validity (which is also the intuitive motivation for syntactic rules of inference in deductive systems), a formally valid argument has to be truth-preserving, i.e., the truth of the premises has to guarantee the truth of the conclusion. In a formal semantic system, this notion of truth-preservation is spelled out in terms of the idea of compositionality, namely, the idea that the semantic values of complex expressions are a function of the semantic values of their components. Given this idea of compositionality and the range of all possible evaluations of the components of the propositions constituting an argument, the semantic notion of validity can be spelled out by saying that an argument is valid just in case there is no possible evaluation of the primitive components of its propositions that would, based on the composition of these components, render the premises true and the conclusion false. Obviously, this notion of validity presupposes that we have a pretty clear idea of what the range of all possible semantic values of the primitive components in question are and how those determine the truth and falsity of propositions based on their compositional structure. But then, when we deal with predicate logic, some of those possible semantic values are precisely the correlates of our universal terms, the bone of contention in the problem of universals. “
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So, in the end, the semantic notions of truth and logical validity in predicate logic, being dependent on what the correlates of our universal terms are, demand at least a certain semantic clari cation of the issue of universals. Contemporary conventional wisdom that we can glean from ordinary logic textbooks would tell us that those correlates are sets, the “extensions” or “denotations” of common terms. (See, e.g., Hurley 2008: 82–84 ) And if we press the issue of what sets are, then we are told that they are possibly completely arbitrary collections of just any sorts of things, yet somehow they are “abstract entities”. Clearly, ordinary logic text books can just stop there. After all, a ll, they th ey are not supposed to go into the metaphysical problems of “abstract entities”: qua logic logic texts, they are just supposed to provide some validity-checking machinery, and need not worry about the possible ontological qualms of metaphysicians these machineries involve, just like elementary math texts, as such, need not worry about the ontological status of “mathematical entities” when they concern themselves only with providing reliable methods of calculation or construction. This sort of attitude of the logician toward the metaphysical issues raised by his subject is almost as old as the subject itself, as is testi ed by Porphyry ’s famously raising the fundamental questions concerning universals just in order to set them aside as pertaining to “deeper enquiries”, but not to logic. (Spade 1994 : 1 ) And of course it is one of the famous ironies ironies of the history of ideas that it was precisely on account of these questions that medieval logicians got so much involved in these “deeper enquiries” that 2009: 111–116) had John of Salisbury in his Metalogicon (John of Salisbury 2009 to complain about how his contemporaries ’ endless debates over these issues confuse, rather than instruct, their students of introductory logic. But despite the pedagogical validity of John’s objection to this practice, one cannot really blame those logicians who get involved in these issues; after all, as we shall see, the answers to Porphyry Porphyry ’s questions determine to a large extent the construction of logical semantics in general, and thus the understandin understandingg of the relationship relationship between the subject matters matters of logic and metaphysics metaphysics in particular. particular.
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Realis Realism, m, nomina nominalis lism, m, concep conceptua tualis lism m
Apparently, the primary issue concerning universals is ontological: are there universal entities? After all, nobody in their right mind would doubt whether we have universal words, i.e., words that on account of their meaning apply to a multitude, indeed, to a potential in nity of entities. However, the question then is: how come we can have such universal
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terms at all? Plato s realist answer, namely, that the di erence erence between universal and singular terms hinges on the ontological di erence erence between the kinds of entities these terms primarily name, rests on a relatively simplistic understanding of the semantic relations of these terms: i.e. the notion that their meaning consists in naming these di erent erent kinds of entities in the same way. In fact, generalizing on this idea we might say that on a realist conception semantic di erences erences are accounted for in terms of the ontological di erences erences of the semantic values of syntactical items of di erent erent categories, and not in terms of the di erences erences in the semantic functions of these items themselves: on this approach, in realism we can have semantic uniformity at the expense of ontological diversity . By contrast, those medieval thinkers who were convinced by Aristotle s and Boethius s arguments against platonic universals (by John of Salis 2013a: n. 27)) would account for bury s time practically everybody (Klima 2013 the semantic diversity of singular and common terms not on the basis of the ontological di erences erences of the kinds of entities these terms denote, but how they denote the same same kind of entities, namely, rather in terms of how individuals, the only kind of real entities there are. Thus, on this understanding of the Aristotelian view, we view, we can have ontological uniformity on the basis of semantic diversity . As we shall see, the two formulae just italicized can be regarded as the two extremes of a whole range of possible positions concerning the relationship between semantics and metaphysics, ranging from extreme realism to thoroughgoing nominalism. Indeed, let me call the theoretical extreme of extreme extreme realism the realism the position that holds that all semantic di erences erences are ontological di erences: erences: di erent erent items in semantically di erent erent syntactical categories di er er in what kinds of entities their semantic values are and not in what kinds of semantic functions relate them them to thei theirr sema semant ntic ic value values. s. By contr contras ast, t, on the the other other theor theoret etic ical al extreme we have the position of extreme extreme nominalism, nominalism, which would hold that all di erent erent items in semantically semantically di erent erent syntactical categories di er er only in the kinds of semantic functions that relate them to their semantic values, but all those semantic values are ontologically ontologically of the same kind, the same, single kind of entities (or just the one single entity) there is. But in orde orderr to see see how how actu actual al hist histor oric ical al posi positi tion onss can can be arra arrang nged ed on this this theoretical scale, we should get into some further details concerning each extreme. On the platonic view, as we could see, the semantic relation semantic relation between common and singular terms and their semantic values would be of the same same kind kind:: name namely ly,, deno denoti ting ng a sing single le enti entity ty.. What What wo woul uld d make make the the di eren e rence ce wo woul uld d be just just the the furt furthe her r ontological ontological re relat latio ion n of the the enti entity ty ’
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denoted by the common term, the universal, to its singulars as their exemplar. It is only on account of this ontological relation that we can use these terms to denote secondarily the singulars imitating or participating in their exemplar, but what the terms truly and primarily denote is the exemplar itself. So, on this platonic understanding, the semantic function function of universal terms would be the same as that of singular terms, namely, denoting a single entity, just like the representative function of a portrait is to represent a single individual. However, However, just as the portrait of a monarch monarch can stand for a whole nation and thus can identify someone as a member of that nation (say, in a passport), so the name of the universal can stand for a whole kind and thus identify any individual participating in it as a member of that kind. On the the Ar Aris isto tote teli lian an view view,, on the the othe otherr hand hand,, univ univer ersa sall term termss are are universal precisely because they apply to a multitude of singular entities, the same singular entities we can denote by their proper names, but di erently , namely, in a universal fashion, in fashion, in abstraction from their from their individual di erences. erences. So, on this conception, what accounts for universality is abstraction, abstraction, a mental activity, the activity of the Aristotelian agent intellect (nous poietikos, intellectus agens ), ), which by this activity produces the rst universal representations, the so-called intelligible so-called intelligible species out out of the singular representations of sensible singulars stored in sensory memory, the socalled phantasms . The inte intell llig igib ible le speci species, es, howev however, er, altho althoug ugh h they they are are universally representing mental acts, generally were not regarded as the universals Porphyry meant to consider in his work. An intelligible species on this conception is rather an acquired disposition enabling the receptive intellect (nous (nous pathetikos, intellectus possibilis ) to form a universal concept in actual use. For example, once I acquire the intelligible species of circles, that enables me to form actual thoughts about circles in general, but that does does not not mean ean that that I am think hinkin ingg of circ circle less all all the time. ime. Thus Thus,, in possession of the intelligible species my mind still needs to form time and again another mental act, the so-called formal concept , to form an actual thought, thought, as when I actually think that all circles touch a straight straight line in one point. However, this mental act is still not the universal. It is a universally representing singular act of a singular human mind; so, my universal concept of circles is not the same item as your universal concept of circles, even if those concepts are exactly alike in their representational content, just like my dance moves I perform with my body are not the same items you perform with yours, even if we are making exactly the same kinds of moves, say, in a chorus line. What is What is the the universal in the intended sense is the common representational content of both your concept and
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mine, on account of which we can be said to have the same concept, despite the individual di erences of the mental acts whereby we have it, just like we can be said to make the same dance moves, despite the individual di erences of our bodies whereby we make them. Therefore, this commonly intended object, the universal representational content of both of our individual mental acts, was rightly called by later scholastic thinkers the objective concept or intention, both because it is the universal representation of the ultimately intended objects, namely, all singulars of the same kind from some of which the intelligible species giving rise to this concept was abstracted in the rst place, and because it is the common objective content of the formal concepts of all those individual human minds that are capable of thinking this objective concept at all. Now, even if this notion of a universal (as the objective representational content of individual mental acts representing a natural kind of singulars in an abstract fashion) may seem to be rather contrived from a contemporary perspective, it should be clear that the conception that treats universals as objective concepts, the universality of which is the result of the intellectual activity of abstraction, does not allow in its “core ontology ” the sort of “abstract objects” Plato entertained. On this view, the intellect can form universal objects of thought, but those objects of thought are not objects or things absolutely speaking. Since they are the results of a mental activity, they are ontologically posterior to that activity. (Although Scotus and his followers would insist that among individuals of a certain kind there is a certain less-than-numerical unity that is ontologically prior even to this activity, and even Aquinas would admit a certain formal unity among individuals of the same kind prior to any activity of the intellect (Klima 2013a: n. 39)). As Averroes was often quoted by medieval authors: intellec- tus facit universalitatem in rebus – it is the understanding that generates universality among things. 3.
Scholastic conceptualisms “
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To see this issue in a little more detail, we should see exactly how the pieces of the theory presented so far t together in this tradition of medieval logic, which I like to call “via antiqua semantics”, in contrast to a radically di erent medieval logical tradition that emerged from the works of William Ockham, John Buridan, and their fellow nominalists, which I refer to as “via moderna logic” (Klima 2011a, 2013a). As we shall see, both of these approaches to logical semantics are basically variations on what may still be called conceptualism; however, they are based on radically di erent
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conceptions of what concepts are and how they are related to their objects, and accordingly give rise to very di erent constructions of logical semantics. The easiest way to make this contrast is through the analysis of an example. Take one of the staples of scholastic lore: Every man is an animal . This is an a rmative, universal categorical proposition (in the medieval sense of proposition , meaning sentence-token), both terms of which are common or universal terms, joined by a copula and determined by a universal sign of quantity (a universal quanti er, as we would say). On the common via antiqua analysis, the subject and predicate terms of this proposition, its categorematic terms, have their semantic property of signifying human and animal natures, respectively, on account of being subordinated to the respective concepts our minds abstracted from their individuating conditions in the humans and animals we have been exposed to. Thus, although whatever it is on account of which I am a man (i.e., a human being, regardless of gender) is a numerically distinct item from whatever it is on account of which you are a man, the concept we abstracted from humans we have been exposed to in forming our concept of man abstracts from any individual di erences ( individuating conditions ). This is precisely the reason why this concept will represent not only the humans we have been exposed to, but any past, present, future and merely possible humans, that is to say, whatever it is that did, does, will or can satisfy the condition of being human, whatever this condition is, and whatever means we have (or don t have) for verifying the satisfaction of this condition (which would be a question of epistemology and not of semantics). Accordingly, the corresponding term ( man in English or homo in Latin) can stand for any of these individuals in a proposition. Indeed, this is what it does in this proposition: it stands or (to use the Anglicized form of the scholastic technical term commonly used in the secondary literature) supposits for all human beings that presently exist. (For an overview of scholastic theories of properties of terms , including supposition, see Read 2011) The reason why this term supposits only for presently existing humans is the present tense of the copula, which restricts the supposition (reference) of the term to present individuals that actually satisfy the condition of its signi cation, namely, those individuals that actually have human nature signied in general by this term. By contrast, with di erent tenses or modalities, or when construed with verbs and their derivatives that signify acts of the cognitive soul (i.e., sensitive or intellective, as opposed to the purely vegetative, soul) that are capable of targeting objects beyond the presently existing ones (such as memory, imagination, “
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anticipation, abstract thought, etc.), the supposition of this term would be extended, or ampliated , to use the Anglicized form of the scholastic term, to past, future, or merely possible humans. (Klima 2001a, 2014 ) Since medieval philosophers did not equate ontological commitment with quantication à la Quine, they did not nd any special ontological di culty in talking about non-existent objects , that is, objects of our cognitive faculties beyond the objects directly perceived in our present environment. In fact, even the ontologically most squeamish nominalists would not hesitate to quantify over mere possibilia , simply because the exibility of their theory of quanti cation and reference, namely, the theory of suppos- ition coupled with the theory of ampliation, allowed them to contend that these mere objects of thought (and of other cognitive acts) are simply nothing, and so to inquire into their nature and ontology would be just a wild goose chase, amounting to nothing. (Cf. Klima 2014 , 2009: c. 10.) The nominalists, however, did have a bone (or two) to pick with via antiqua semanticists on other aspects of their theory. In the rst place, and perhaps most fundamentally, the medieval realists (practically anybody before Ockham), even if they did not buy into Plato s stratied ontology of universals vs. singulars, and had a much more sophisticated semantic theory than the uniform naming relation between di erent kinds of words and correspondingly di erent kinds of things, they did preserve some sort of semantic uniformity at the expense of some sort of ontological diversity. As we have seen, the signication of common terms, based on the idea of words being subordinated to concepts to inherit their natural semantic features, coupled with the Aristotelian theory of abstraction, led to a peculiar theory of predication within this framework, often referred to in the literature as the inherence theory of predication. The theory is simple enough: the predication x is F is true, just in case the F-ness of x actually exists, or equivalently, just in case F-ness, the form or property signi ed by the predicate F in the individual x actually inheres in x. The problems start when we consider all sorts of substitution instances of F. For then we start realizing that, apparently, by the lights of via antiqua semantics, as Ockham put it a column is to the right by to-the-rightness, God is creating by creation, is good by goodness, just by justice, mighty by might, an accident inheres by inherence, a subject is subjected by subjection, the apt is apt by aptitude, a chimera is nothing by nothingness, someone blind is blind by blindness, a body is mobile by mobility, and so on for other, innumerable cases (Ockham 1974 : I, 51). In short, to the nominalists, starting with Ockham, it appeared that their realist opponents (in the case “
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of Ockham, especially John Duns Scotus and Walter Burleigh) generate metaphysical problems where there shouldn’t be any, simply on account of a misconception of semantics, because their conception would “multiply beings according to the multiplicity of terms ” (Ockham 1974 : I, 51). To be sure, the “realists” did make a number of metaphysical distinctions between the types and modes of being depending on the substitution instances of F to avoid apparent metaphysical absurdities (such as a thing undergoing change without losing or acquiring a property, action at a distance, non-beings undergoing change, etc. – cf. Klima 1999), but for Ockham and his ilk, that is precisely the problem: to maintain a certain type of semantic uniformity, the realists introduce ontological diversity where there shouldn’t be any, since the di e rence is not in the things signied by our di erent terms, but in the di erent concepts signifying the same things in di erent ways (Klima 2011a). To illustrate the sort of semantic uniformity and the requisite ontological diversity in the via antiqua approach to semantics, let us brie y return to the via antiqua analysis of the meaning and conditions of truth of ‘Every man is an animal ’. The two categorematic terms both have the same type of signicative function, namely, signifying the individualized natures of individuals represented by their corresponding concepts in an abstract, universal fashion. The subject term, in turn, has the function of standing for those individuals that actually have this nature at the time connoted by the tense of the copula, whereas the predicate has the function of attributing the nature it signi es to the individuals thus picked out. And since the universal sign in front of the subject indicates that the truth of the entire proposition requires that all these individuals have the nature signied by the predicate in actuality at the time connoted by the copula, the propositional complex, variously called dictum, enuntiabile , or complexe signi cabile , resulting from the combination of subject and predicate by the copula as further determined by the universal sign, will be actual just in case all individuals supposited for by the subject actually have the nature signied by the predicate. As can be seen, the via antiqua analysis of this single proposition apparently requires an extremely complex, multilayered ontology; however, the payo in the end is the simple, uniform semantic criterion of truth originally proposed by Aristotle: a proposition is true just in case what it signi es exists. But it is instructive to take a closer look at the ontological status of the items required by this analysis. In the rst place, the analysis requires the existence of some ordinary primary substances, namely, humans. However, for something to count as
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a human, it has to have humanity. And since humans are essentially rational animals, their existence also requires the existence of their rationality and animality. Furthermore, on this conception, if something exists in actuality, its existence also has to be in actuality, and it is this actual existence that is supposed to be signi ed by the copula. But the copula also co-signies the union of what is signi ed by the predicate and by the subject, thereby indicating that the existence of the thing signi ed by the predicate is also the existence (whether substantial or accidental existence, but in the case of the proposition at hand, it is the substantial existence) of the thing supposited for by the subject, which is actual at the time connoted by the tense of the copula. Now, these are just the real, mind-independently existing items required for the truth of this proposition. However, as we could see, these items can be picked out by the relevant syntactical items from reality only on account of these syntactical items being subordinated to their respective concepts that renders them meaningful in the rst place. So, on this analysis, all propositions require a further ontological layer , as it were, the layer of concepts. But, as we could see, concepts come in two necessarily connected sorts, namely, formal and objective concepts. The formal concepts are real, inherent, individualized qualities of the individual minds that form them. The objective concepts, on the other hand, are the direct objects of these individual mental acts, some of which represent extra-mental individuals in a universal manner, but without representing the sorts of universal things imagined by Plato. Thus, these objective concepts form another ontological layer, the layer of beings of reason, which in the strict sense are mere objects of thought (the representational contents of formal concepts), but with a more or less remote foundation in reality (as opposed to mere gments). In the case of universal concepts, this foundation in reality consists in the individualized natures of the things from which these objective concepts derive in the process of abstraction and concept formation (through the generation of intelligible species). But the objective concepts do not occur to the mind in isolation. They enter into the composition of complex thoughts, which are formed by means of syncategorematic concepts, such as the copula, which, as we could see, besides its syncategorematic function of joining the concepts of subject and predicate also has the categorematic function of signifying the existence of what is signi ed by the predicate in the relevant supposita (referents) of the subject, the relevant supposita being determined by the syncategorematic concept of the sign of quantity, in the present case the “
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universal sign (quantier). The propositional complex thus formed is another being of reason with a foundation in reality. (Klima 2011b, 2012) Indeed, the obtaining of this complex is conditioned both on the side of the mind forming it and on the side of the things serving as its foundation. For the state of a a irs (dictum, enuntiabile , etc.) that every man is an animal actually obtains just in case there are humans and each of them actually has its animality. But then, providing the rules of composition for all types of thought, based on the syntactical structure of the proposition expressing it, one can provide the uniform Aristotelian criterion of truth for all types of propositions, and based on that, the uniform criterion for the formal validity of an inference or consequence. In fact, since in this framework truth is dened in terms of the content of propositions, a stronger entailment relation is also de nable, in terms of the more negrained notion of content-containment, as was proposed by some authors in this tradition (Martin 2010, 2012; Read 2010; for comparison, an interesting contemporary development of the idea of entailment based on content- or meaning-containment can be found, for instance, in Brady and Rush 2009). However, as we could see, this could be obtained only in terms of the multi-layered ontology that provoked Ockham s and his fellow-nominalists charges. Nevertheless, we should also emphasize that Ockham s and his followers charges were not entirely justied, and, accordingly, the ultimate di erence between late-medieval realists and nominalists did not lie simply in their di erent ontologies or simply in their di erent semantics that allowed them to handle their ontological problems in rather di erent ways, but rather in their di erent conceptions of concepts underlying even their semantic di erences. The Ockhamist charge of multiplying entities with the multiplicity of terms was unjustied for several reasons. In the rst place, even realists had at their disposal at least two di erent kinds of strategies to reduce the ontological commitment of their semantics: ( 1) the identication of the semantic values of terms belonging to di erent categories, and (2) attributing a reduced form of existence to some of the semantic values of some terms in some categories. The rst strategy could rely already on the authority of Aristotle, who in his Physics identied action and passion with the same motion, but several original considerations allowed scholastic thinkers to identify relations with their foundations (i.e., with entities in the absolute categories of substance, quantity and quality), and in general entities in the remaining six categories with those in the rst three. ’
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The second strategy, as we could already see, was based on the idea that the mind s di erent ways of conceiving of mind-independent entities of external reality produces certain mind-dependent, intentional objects, the objective concepts , the information contents of our mental acts, by means of which we variously conceive ultimately those mind-independent objects that satisfy the criteria of applicability set by these objective concepts, or intentions . This is precisely why in this tradition the subject matter of logic was generally characterized as the study of second intentions , that is, of concepts of concepts (such as the concepts of subject, predicate, proposition, negation, or the ultimately targeted notion of valid consequence). So, the core-ontology of real mind-independent entities could in principle have been exactly the same for these realists as for Ockhamist nominalists . In fact, both late-medieval realists and nominalists were conceptual- ists , but based on a rather di erent conception of concepts and their role in logic, semantics, and epistemology. In this connection, it is informative to compare Ockham s earlier, ctum-theory of universals with that of the via antiqua conception discussed so far. For the important di erence between the two is that even if Ockham s cta are ontologically on the same footing as the objective concepts of the realists (they are beings of reason), and they would be best characterized in the same way, namely, as the objective information content of individual mental acts, they do not have the same role in Ockham s theory. In fact, as prompted by the arguments of his confrere, Walter Chatton, Ockham came to realize that cta did not play any signi cant role in his logic at all, and so, grabbing his famous razor, he painlessly cut them out from his ontology. The reason why Ockham could do so is that for him the universality of universal representations (whether cta or universally representing mental acts) consists merely in their indi erent representation of a number of individuals (in the case of a natural kind, all past, present, future, and merely possible individuals of the same kind). However, this indi erent representation is due not to some abstracted condition of having a certain nature that individuals of a given kind satisfy, but, as a matter of brute fact, to the indi erence of the causal impact of one individual or another of the same natural kind on the human mind. Accordingly, for Ockham, there is no question whether there is a real distinction between the nature of an individual represented by a universal concept and the individual itself (as this emerged as a metaphysical question in the via antiqua ), because what these concepts indi erently represent are just the individuals themselves. Therefore, for him, the ’
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supposita of the terms subordinated to these concepts are not the individuals that actually have these natures relative to the time connoted by the copula (as was conceived in the via antiqua ), but simply the individuals represented by the concept that are actual at that time. As a result, terms in the predicate position do not signify inherent natures either, so Ockham and his followers endorse the identity-theory of predication, as opposed to the inherence-theory . According to the identity-theory, an a rmative predication is true, just in case the terms of the proposition supposit for the same thing or things. But this is obviously not a general “denition of truth”. In order to achieve a truth-de nition on this approach, one should provide similar “satisfaction clauses” for all logically di erent proposition-types, such as negatives, universals, particulars, not to mention the propositional complexes, such as conjunctions, disjunctions, etc. (Klima 2009: c. 10). As can be seen, on this nominalist approach, just as terms do not gure into the calculation of truth-values with their intensions, but their extensions, so too, the truth of propositions themselves is not determined in terms of their intension or signi cation, but solely by the extensions (sets of supposita ) of their categorematic terms. Accordingly, nominalist semantics as such has no use for enuntiabilia or complexe signi cabilia , as is brilliantly illustrated by the logic of John Buridan. On Buridan’s theory, propositional signi cation is simply the set of all signi cata (and connotata ) of a proposition ’s categorematic terms, which of course yields a very coarse-grained conception of propositional signi cation. In fact, on this conception, contradictory propositions must signify the same, although di erently, on account of the concept of negation included in the one, but not in the other of the contradictory pair of propositions (Klima 2009: c. 9). However, Buridan does not have to care much. On his account, truth is not a function of signi cation, so, two propositions of the same signication can have opposite truth-values. Thus, when he needs a more ne-grained semantics of propositional signication (as in intentional contexts) he can always refer to the diversity of the corresponding propositions on the mental level, where, of course, in line with his nominalist ontology, the mental propositions in question are just inherent qualities, individual acts of individual human minds, just as are the concepts entering into their semantic make-up (Klima 2009: c. 8). So, nominalist semantics can a o rd to be based on an entirely homogeneous, parsimonious ontology (containing only two or three distinct categories of entities, namely, substances, quantities – sometimes identi ed with substances or qualities, as by Ockham – and qualities). However, this
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parsimonious homogeneity is achieved at the “expense” of massive semantic diversity, assigning some of these entities various, distinctive semantic functions, especially on the mental level. However these semantic functions are always de ned in terms of the extensions of these mental items, the formal concepts inherent in individual human minds, to the exclusion of items in the “ontological limbo” of the objective concepts of the older model. Still, even the nominalist version of scholastic conceptualism could maintain that logic is the study of second intentions without lapsing into subjectivism, conventionalism, or skepticism, let alone psychologism – features that in a modern context are so often associated with conceptualism. Well, how come? Actually, answering this question will allow us to draw some general conclusions concerning both major versions of scholastic conceptualism sketched out here, and some general lessons we can learn from these scholastic theories concerning the subject matter of logic and metaphysics. 4.
Conclusion: the lessons
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In the rst place, it should be quite obvious that the objective concepts of the via antiqua conception are objective not only in the medieval sense, i.e., in the sense that they are the objects of individual mental acts inherent in individual human minds as their individualized forms (the formal concepts), but also in the modern sense of being intersubjectively accessible and the same for all. For an objective concept is the common, abstract information content of any formal concept that carries this information, and any formal concept that does not carry the same information is just not a formal concept of the same objective concept. Thus, in this frame work it simply cannot happen that you and I have di erent concepts of the same kind of entities as such, or the same concept of entities of di erent kinds. If I have the concept of H O and you have the concept of XYZ, then we are just not talking about the same thing, no matter that we use the same word in our miraculously matching English idioms of Putnam ’s Twin Earth scenario. (Putnam 2000: 422) Since I use the term ‘ water ’ as subordinated to my concept and you use it as subordinated to yours, we use our words equivocally, no matter how phenomenally similar the two kinds of things are, and how similarly we would describe their phenomenal properties. Therefore, if I say ‘This is water ’ pointing to a glass of H O and you say, pointing to the same, ‘No, that is not water ’, we actually do not contradict each other, although, of course, it can take a while until we 2
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gure out just why. But that is an issue of epistemology, not semantics. As
far as meaning is concerned, on this conception you can have the same concept as I do only if our individual mental acts latch onto the same kind of objects in the same way, carrying exactly the same information. To be sure, one of us may have a better understanding of the nature of the thing or things thus conceived, on account of being more aware of the relationships among this concept and others, picking out the same nature di erently, on account of other, more speci c or more generic information, as when one of us knows the genuine quidditative denition of the kind of thing in question. But regardless of whether either of us has this de nition in mind or knows what it would be, we can be said to have a concept of this kind of thing as such , only if we managed to form the objective concept of its essence, which must be the same for both of us, or we just do not have this concept at all (Cf. Aquinas 2000: Sententia Metaphysicae , lib. 9. l. 11. n. 13.). From this it should also be clear that these objective concepts are nonconventionally objective. For what determines the information content of our abstracted, simple concepts is what kinds of things they are abstracted from, that is to say, the nature or essence of those things themselves. To be sure, we can construct complex concepts out of these simple ones as we wish and agree to express them by words we wish (ad placitum, as the scholastics said), but whether the concept we both abstracted from samples of H O will apply to all and only samples of that kind of thing (even if we cannot infallibly identify all such samples in all possible scenarios) is clearly not a matter of our wishes. Finally, even if our psychological mechanisms require that when we form these simple concepts and their combinations our minds work with their own individual, subjective mental acts, their formal concepts; the logical relations among these mental acts are not a matter of the causal or other psychological relations among them, but a matter of the relations of their objective semantic contents, the relations among their objective concepts. So, no wonder scholastic thinkers working in this tradition would identify the subject matter of logic as those second intentions or objective concepts of our objective concepts that express precisely these objective conceptual relations. (Schmidt 1966; Natalis 2008) Therefore, it should also be clear that the laws of logic in this framework are supposed to be fundamentally di erent from the laws of psychology. For while the former are the laws of the logical relations among objective concepts, the latter are the laws of the causal relations among formal concepts. Thus, whereas logic can be normative, prescribing the laws of 2
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valid inference, cognitive psychology can only be descriptive, describing and perhaps explaining those psychological mechanisms that can, for instance, make us prone to certain types of logical errors. But similar observations can be made about the nominalist approach, although with some interesting, and from a modern perspective, especially instructive modications. As for the issue of objectivity, the nominalist authors, after Ockham had dropped his cta as being ontologically bothersome and theoretically unnecessary, still insisted on our simple mental concepts being externally determined and not just subjectively made up by us, grounding this belief in the generally reliable natural mechanisms of sense perception and universal concept formation (Panaccio 2004 ; Klima 2009: c. 4 ). So, for the nominalists there are no longer quasi-entities in mere objective being: our concepts are anchored in extramental reality through the objective (in the modern sense, meaning mind-independent) laws of natural causality. However, there is a slight, but very signi cant shift in the way nominalists conceived of this anchoring , as opposed to their realist counterparts. For realists, what did the anchoring was a certain formal unity , the sameness of the information content that was encoded in the mental acts carrying this information and that was realized in the very nature of the things these concepts represented to the subjects having them. This conception of formal unity provided a much stronger anchoring for the via antiqua conception than what is available in the nominalist via moderna . The via antiqua conception, as we could see, builds the identity of the nature of the represented objects into the identity-conditions of a concept itself, hence tying the identity of its objects by logical necessity to the identity of the concept in question. By contrast, the via moderna conception ties the identity of the concept by mere natural, causal necessity to the identity of its objects. In a medieval theological context, however, this di erence amounts to the di erence of what could and could not be done by divine absolute power. Therefore, it should come as no surprise that anticipations of Descartes famous Demon argument crop up precisely in this context, once the nominalist conception opened up at least the logical possibility of a cognitive subjects having exactly the same concepts planted in his mind by a deceptive God without the mediation of these concepts adequate objects; i.e., the subject having exactly the same phenomenal consciousness, regardless of whether any items of it are veridical, faithful representations of reality or not (Klima and Hall 2011; Karger forthcoming). “
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As for conventionalism, the nominalists usually laid even more emphasis on the conventionality of spoken and written languages than their realist counterparts; however, they equally strongly emphasized the nonconventional, natural character of the language of thought or mental language, based on the xed laws of nature. So, even for the nominalists, our simple concepts, anchored in natural kinds by causality, are not made up by us at will in the way the words we express them by are. Nevertheless, as we could see, the nominalist conception still allows for the possibility of “supernatural skepticism”, providing a whole range of di erent reactions to this possibility, but perhaps most typically o ering a “reliabilist” solution, dismissing the overblown certainty-criteria of the skeptic in terms of di erent sorts of reliability-criteria for our various cognitive powers and mechanisms utilized di erently in di erent cognitive scenarios (Aristotle providing again a good authority by his remark that one should not expect mathematical certainty in all elds of inquiry) (Klima 2009: c. 12). In any case, in deductive logic, nominalists still required the same, highest form of certainty as in mathematics. For despite their conception of concepts as being simply individualized mental acts tied to their objects by mere natural necessity, the nominalists did not take logic to collapse into psychology. Perhaps, the best illustration of this fact comes from Adam Wodeham ’s “thought experiment” concerning the presumed “perfect telepathy ” of human minds uncorrupted by original sin (of which now we have no actual example) and of good angels (the ones that did not fall with Lucifer). These minds, according to Wodeham, are perfectly capable of intuiting each other ’s thoughts, however, this would still not amount to communication, because they would not be able, simply on account of this intuition, to decode the contents of those thoughts (pretty much like brain scans can give us some information about some sort of brain activity, but not about what that activity is about). Accordingly, based on these observations, these minds could come up with a natural science describing the regularities of these mental activities, but that science would tell us nothing about the content, let alone the validity of the thought processes couched by these activities, which would be the concern of a di erent science, namely, logic. (Karger 2001: 295–6) So, what conclusions can we draw for ourselves from this however sketchy, general comparison of the two main scholastic approaches to the problem of universals and the subject matter of logic? The traditionally recognized alternative answers to the problem of universals come in many shades and colors. But especially in their
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sophisticated medieval versions, they are primarily di erentiated erentiated not by their di erent erent ontologies, but rather by their di erent erent concept conception ionss of concepts, concepts, determining determining di erent erent kinds of constructions of logical semantic theories. These di erent erent theories theories can be arranged arranged on a theoretical scale , ranging from extreme realism to extreme extreme nominalism, nominalism, meaning maximal semantic uniformity along with maximal ontological diversity on the realist end (every linguistic item has the same type of semantic function, say, naming some entity, while these items di er er semantically on the basis of what type of entity they name), and maximal ontological uniformity with maximal semantic semantic diversity on the nominalist nominalist end (having just one ontological ontological type of entities, while all semantic di erences erences consist in the di erent erent semantic functions of some of these entities of the same ontological type). Even if, perha perhaps ps,, no actual actual histo histori rica call theor theoryy can be placed placed on eith either er extrem extremee (although, if Parmenides had had one, it would have probably been close to the extreme nominalist end, whereas if Plato had had an articulated semantic theory, then it might have been close to the extreme realist end, as probably so would be Wittgenstein s caricature of Augustine s theory), the actual, well-articulated theories can better be understood as variously removed from either extreme on account of various elements of variety introduced either on the side of ontology, by multiplying the semantically relevant distinct categories of entities, or on the side of semantics, by multiplying the di erent erent types of semantic relations that map the syntactical categories of language onto the ontological categories distinguished by the theory. In the the scho schola last stic ic theo theori ries es disc discus usse sed d here here,, thes thesee di eren e rentt type typess of semantic relations were understood in terms of how our di erent erent kinds of concepts relate our words to things in our ontology. Here, in the via can be under underst stoo ood d rather rather loos loosely ely for any any antiqua framework things can obje object ct or quas quasii-ob obje ject ct of our our thou though ght, t, wher wherea eass in the the via moder moderna na fram framew ewor ork, k, they they wo would uld be restr restric icte ted d to really really exist existin ingg enti entiti ties es in the the category of substance and quality (or for Buridan and his followers also in the category of quantity). It is very telling, however, that the core ontology (i.e., the categories of real entities to the exclusion of beings of reason) of the via antiqua framework could be just as parsimonious as the nominalist core ontology was (as is illustrated, for instance, by the ontology of the late-scholastic Domingo Soto). Furthermore, even the via framework could have in principle reduced its ontology to one moderna framework homogeneous category, had it not been for certain theological worries “
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concerning the Eucharist, and metaphysical worries concerning atomism, widely taken to have been refuted by Aristotle. Finally, in view of the foregoing comparative analysis of how both medieval viae medieval viae would would avoid, in their own ways, contemporary worries about conceptualism in general, we can conclude that such comparisons can be especially useful for rening our understanding of the implications of the various theories that can be arranged on the theoretical scale set up in this chapter. Such renements in the end will allow us to overcome certain modern theoretical reexes (nominalism entails skepticism, conceptualism leads to psychologism, psychologism, etc.) by shining shining a new light on the historical historical origins of these reexes themselves. themselves.
chapter 10
Logics and worlds Ermanno Bencivenga
There are no specically logical objects. A logic is a theory of the logos , of meaningful discourse: a theory of how discourse acquires the meaning it does. Traditionally, logics have investigated the behavior of syncategorematic words like and, which contribute to the meanings of their contexts while having ha ving no meaning in isolation, and hence have h ave studied the contrast, say, between and as it occurs in sentences like “
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(1) They They are are marr marrie ied d and and have have a chil child d (2) They They got got marr marrie ied d and and had had a chi child. ld. But there is no reason why they should not also study the relation, say, between (1) and (3)
They They are are marr marrie ied d and and have have a pet pet;;
in particular, why they should not inquire into whether ( 1) and (3) are interchangeable. A logic that studied the contrast between ( 1) and (2) would be (among other things) a logic of and (a theory of the meaningful meaningful use of and ); a logic that studied the relation between ( 1) and (3) would be (among other things) a logic of child (and of pet ). Nothing other than greater generality, and attendant lesser detail, is gained by concentrating on logics of conjunctions, adverbs, and pronouns; and at no level of generality does it make any sense to grace a word (or maybe a diacritical sign, like a parenthesis) with the label logical constant. If you ever get sidetracked into a Quinean fatuous search for the nonexistent Eldorado of pure logic, I recommend a refreshing immersion into Buridan s subtle, perceptive study of the innite nuances of signi cation and supposition. supposition. But, if there is no speci c ontological realm for logic, there de nitely are ontological questions pertaining to it. Two questions, primarily: Do logical laws (the laws bringing out the meaningful behavior of various words) have “
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an objective status? And, if so, how do they acquire it? No answers to such questions can be attempted without a substantive view of what objectivity, and an ontology, are. Since this is not the place to defend my Kantian, transcendental-idealist position on the matter, I will simply state it before moving on. (Though I must note that, here, the matter dealt with is not innocent: an ontology is a logic of being, hence what ontological status a logic has is not independent of what logic it is.) A transcendental philosophy as described and practiced by Kant is itself a logic. It is not intended to decide such factual questions as whether there is a God or humans are free, but to address semantical issues like what the meaning of “God” or “freedom” is. The reason why the formidable epithet attach ched ed to it is prec precis isel elyy the the misu misund nder erst stan andi ding ng “transcendental” is atta I alluded to above: if you think that logic only deals with (some) conjunctions, adverbs, and pronouns, then you are forced to qualify this narrow concern as “general logic” and to conjure up some other name for the full line of business. Within the semantical space where the (transcendental) logical enterprise is located, one can take di erent erent words as primitives and establish a network of semantical relations and dependencies based on those primitives and involving involving other words, words, each time resulting resulting in (the beginning of ) a di eren erentt tran transc scen enden denta tall philos philosoph ophy/ y/log logic ic;; as more more such such stru struct ctur uree is exposed, exposed, the meanings meanings of the words involved will become correspondin correspondingly gly better established and clearer. If we want, we can even talk about “concepts”: clusters of largely interchangeable words resonating with a common theme, not necessarily spoken but suggestively intimated by the resonance. A transcende trans cendental ntal realism real ism (TR) is a transcenden transc endental tal philosophy philo sophy/logic /logic that takes a cluster of largely interchangeable words including “object,” “substance, ” “thing, ” and “existence ” as primitives, and then turns to the (hopeless) task of de ning words like “experience ” or “knowledge” on that basis. A transcendental idealism (TI) – my chosen course – is a tran transc scen ende dent ntal al phil philos osop ophy hy/l /log ogic ic that that take takess its its cue cue from from a di erent erent cluster including including “experience,” “representation, ” and “consciousness, ” and then moves to de ning “object” and “existence. ” Not surprisingly, a TI has a lot more to say about objectivity – what makes an object an object – than a TR: of primitives we will forever be dumb and, though occasionally that incapacity is depicted as mystical depth, the bottom line is that no interesting account of what primitives mean is forthcoming. In a TI, however, objectivity belongs to a derived cluster; hence its 2
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derivation provides for a rich and complex contribution to the logos – one we need to work out now. Representations (or experiences, or consciousness) are always of something: their so-called intentional objects, which despite their name are not objects yet, indeed never will be. And neither representations nor their intentional objects can be objective object ive in in isolation: they can only be objective to the extent that they are categorially connected – that they are mutu mutuall allyy cons consis iste tent nt;; that that ther theree are are relat relatio ions ns of mutu mutual al dete determ rmin inat atio ion n among them; that there is a de nite fact of the matter of how many of them there are, hence how they are identical with, or distinct from, one anot another her.. The grad gradual ualit ityy sign signale aled d by the the locut locutio ion n “to the extent that ” would only be redeemed at the limit: by a system of representations to which nothing further could be added and where each member were fully determined to be what it is by its relations with all others. Within that system (suddenly, as soon as completeness were reached), all representation tionss wo woul uld d be obje object ctiv ivee and and all all thei theirr inte intent ntio iona nall obje object ctss wo woul uld d be obje object cts, s, perio period: d: exis existe tent nt obje object cts. s. The The limi limitt canno cannott be exper experie ienc nced, ed, in the strongest sense of “cannot”: it would be contradictory (antinomical) to suppose otherwise, hence all intentional objects will forever stay that way – remain appearances . But this conclusion is only going to worry those who reduce a TI to a series of empirical claims about what takes place (or can take place) in a mind. As none of that is implied here, for what is in question is rather the semantics of “objectivity,” we have all we could ask for: a regulative idea that orients our everyday, always fallible vicissitudes, signaling the direction in which we are likely to nd more objectivity and the standards we must enforce to maximize it (coherence, agreement, agreement, inclusivenes inclusiveness, s, mathematical mathematical structure), structure), inevitably inevitably staying staying clear of such a complete realization of the idea as would make ( per ( per impossibile ) the whole project fall apart. What the system of representations envisioned at the limit would represent is (as one might expect) a system of objects to which nothing further could be added and where each member were fully determined to be what it is by its relations with all others. I call (and Kant calls) this system a world world . No one ever experiences a world, though most everyone (everyone but severely disturbed people) ordinarily presumes herself to be experiencing part of one, and sometimes goes through the catastrophe of seeing what she took to be part of a world explode into incoherence and disconnectedness. When such unfortunate events take place, we try hard to blame them on contingent occurrences (on misreadings of data) while keeping faith with the semantical laws that organize our logic. What I saw
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in the corner was not an elephant; it was an armchair; but the meanings of “elephant” and “armchair ” are not disrupted by this mishap. And yet, it is not always that easy; for, what about the semantical necessity, until circa 1905, that a wave is not a particle? And about the logical clash that ensued when we were forced to deny that necessity: a clash whose logical character would be missed by more parochial characterizations of logic? If, on the other hand, you want to insist on a parochial characterization and attribute that clash to the empirical realm, I urge you to consider what happened a few years earlier, when the very logic of sets blew up in people ’s face. As very unfortunate happenings of this kind can never nally be ruled out, logics align themselves with worlds, in the following way: A logic cannot be a theory of meaning less discourse (of alogos ). But any word we use can only be meaningful if our whole discourse is meaningful: if all words we use belong in an ideal complete dictionary that sets consistent, connected relations among them – once again, it is an all-ornothing a air. Only a logic associated with this kind of dictionary would be objective in the sense of possibly describing a world of objects (would be a real , not an apparent , logic), independently of the data that gave empirical content to its entries. As no such dictionary can ever be at hand, we are never in possession of a logic but only of something we presume to be a fragment of one, and which is always at risk of dissolving into the stu dreams are made of. A logic (like a world) is worse than a territory constantly under threat of being conquered by enemies: it is constantly under threat of vanishing into thin air. That being the case, a major consequence follows, of a sign opposite to the Quinean puritanism mentioned earlier. Just as, in the absence of a complete system of representations or a complete world, we are to maximize the consistency, connectedness, and inclusiveness of what systems of representations or of intentional objects we do have, in the absence of a logic we are to maximize our closeness to one, walking away from the depopulated citadels of the propositional and the predicate calculi toward a ner and ner appreciation of the logical distinctions between “crowd” and “mob,” or “magenta ” and “scarlet.” In a true Kantian vein, completeness will be not actual but set as a task, so logic will graduate from a tenseless doctrine into a concrete practice ready to uncover semantical treasures under any rock, and carry semantical threads around any corners. The routine of jotting down a few axioms, “formally interpreting ” them by translating them into the stock language of set theory, and “vindicating ” them by proving a completeness theorem will be shunned in favor of the completeness that really matters: the one that is never
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achieved but demands that we trace more and more connections, across a larger and larger eld. The two most obvious examples of this search for an objective logic, in our tradition, come from Aristotle and Hegel 3 – or, I should say, from Hegel and from Aristotle as interpreted by Hegel. For the o cial story about Aristotle is that logic for him is an organon, a neutral tool to be prexed to research proper, indeed to be done with exhaustively before embarking in any research. If that were true, Aristotle ’s would no more be a logic than quantication theory or S 4 are: it would be an abstract, uninformative, and ultimately irrelevant repertory of (logical) platitudes. But, fortunately, such is not the case (as Hegel points out): every segment of Aristotle’s philosophy (and science) deepens and widens his logical analysis4 – his logic as analysis, his analytic logic. Whether he is talking about the challenge sea-anemones bring to the logical distinction between animals and plants, or he is illuminating through a careful examination of courage or friendship the relation between focal and extended/analogical meanings, Aristotle is reshaping his dictionary (including what it is to be a dictionary) every step of the way. So this is Hegel’s Aristotle I am talking about. It is also Kant’s. Aristotle’s text could be used as a prime example of a commitment to a TR. But one can also see it as a major avenue that is open to us when we, within a TI, get to the point of spelling out categorial connectedness; more precisely, when we spell out the part that has to do with counting objects, hence identifying and distinguishing them. If we go with Aristotle there (with the Aristotle that maintains a not-entirely-comfortable presence inside Kant) then the issue is simple: as soon as we face a contradiction between two representations, or their intentional objects, a distinction must be made – there must be at least two things. The Aristotelian world is structured by contraries : by what cannot be true together and invokes a splitting. If waves can cause interference phenomena and particles cannot, then waves are not particles, and for a denition of light we have only two choices: we can either have particles or waves but not both , or give up on 3 4
For a systematic account of the contrast between Aristotelian and Hegelian logic, see my 2000. See the following passages from Hegel’s 1995: “in his metaphysics, physics, psychology, etc., Aristotle has not formed conclusions, but thought the concept in and for itself ” (p. 217; translation modied); “it must not be thought that it is in accordance with . . . syllogisms that Aristotle has thought. If Aristotle did so, he would not be the speculative philosopher that we have recognized him to be” (p. 223 ); “Like the whole of Aristotle’s philosophy, his logic really requires recasting, so that all his determinations should be brought into a necessary systematic whole” (p. 223). While he thus acknowledged the comprehensive character of Aristotle’s logic, however, Hegel did not see it as an alternative to his own, as I do, but rather as a step toward the latter.
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both and think of something else entirely. That Kant s criteria of objective identity be spatiotemporal shows him committed to this Aristotelian route: one and the same thing cannot be at two di erent locations at the same time. But, when it comes to the semantics of regulative ideas, including the ones that determine the criteria of objectivity, his inclination seems to be proto-Hegelian, witness his derivation of positive from negative freedom, or of reciprocal action from simultaneity. 5 With Hegel, on the other hand, contradiction is not a threat: it is an opportunity. When the semantics of a word faces a bifurcation between contradictory options, its fate is to take both, and its job is to evolve in such a way that both options be present in a dialectical overcoming of their contrast. Light is both particles and waves: the two are complementary descriptions of one and the same complex reality, indeed belong to the very substance of that reality, which is nourished (adds to its concreteness, Hegel would say) by their antagonism. Therefore the world that no one will ever experience but of which everyone takes herself to be experiencing a portion is a monistic one: as not even contradictions can divide, no two things are radically divided; all divisions are but chapters of one story. And the very unfortunate events that might bring this logic to a crisis will not be the surfacing of contradictions, as is the case with its analytic counterpart. It will rather be the confronting of occurrences (the Holocaust, say) that simply cannot be integrated within one and the same comprehending, rationalizing, spiritual narrative. I said that these are the two most obvious examples of the search for a logic. There are countless, less obvious, others; except that they are not to be found where one would be most likely to look for them. As I pointed out already, individual calculi cannot be regarded as logics, unless they are part of an ambitious program that extends over a substantial area of experience, indeed potentially all of it. But, whereas most of what falls under the academic discipline of logic does not qualify as logic for me, a lot of traditional philosophy does. Transcendental philosophy is not a new way of doing philosophy initiated by Kant: it is a new way of looking at what philosophy has always done, without much awareness and hence with considerable self-deception. Of course pre-Kantian, and many postKantian, philosophers typically took themselves to be establishing factual claims like the existence of God or human freedom, but the way they did ’
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this was by launching apodeictic demonstrations, that is: by trying to prove that the existence of God or human freedom were more than facts – that they were necessities. Therefore, it was not facts about God or freedom they were directly addressing: it was the meanings of “God” and of “freedom,” and facts only insofar as they were inescapable consequences of what those meanings were. Virtually every one of them was doing, largely unbeknowst to himself, transcendental philosophy, which is to say: (transcendental) logic . The logic of the State and of justice, if they were Plato or Hobbes; the logic of art, if they were Plotinus or Schiller; the logic of economic exchanges, if they were Ricardo or Marx. Far from being just an organon of philosophy, logic constitutes the very body of it: all philosophical theses, arguments, and theories are but logical matters – pages of an ideal dictionary by which we try to make sense of experience. And, as I showed earlier with Aristotle, most of these theses, arguments, and theories, though often grown on TR soil, can be put to pro table use in developing a TI, specically in dening objectivity for various TI philosophies/logics. If each logic (within a TI framework) develops its own de nition of objectivity, one important feature of logic as ordinarily understood turns out to be mistaken – and one more way emerges in which traditional philosophers were deluded about their own work. The feature I refer to is that logic, one often believes, allows us to conclusively refute an opponent by proving him conceptually confused, or to conclusively establish our position by proving it sustained by necessary argument; which is just what philosophers traditionally took themselves to be doing when they o ered the apodeictic demonstrations I mentioned. Here too it might help to get to our point by a digression through Kant, this time through Kant ’s ethics. (In what follows, for the sake of simplicity, I will adopt an analytic, Aristotelian perspective, hence speak of inferences, that only have currency within an analytic framework. Hegelian, dialectical logic has other conversational and confrontational modes – which explains the repeated failure of attempts at coming up with a dialectical theory of inference. But mutatis mutandis what I say could be extended to the Hegelian camp.) According to Kant, ethics is rationality, 6 and never mind at the moment that rationality, like objectivity, cannot be de nitively established. Suppose we pronounce an ethical judgment that, on as solid grounds as we can manage, stigmatizes a certain behavior as immoral. Assuming the grounds to be as solid as they appear, this is a case of reason itself speaking, and one 6
For a detailed discussion of this thesis, see my 2007.
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would imagine that, when reason speaks, everyone will stop and listen. Not so: behavior occurs in empirical reality, Kant thinks, subject to empirical laws, so only empirical factors like temperament, education, or emotions can have motivating force there. Reason has none. One can only hope (if one takes reason ’s side) that those empirical factors will promote what reason would want to see done; that moral feeling, say, will ally itself with rational judgment and make the agent move the “right” way – in this case, take his distance from the stigmatized behavior. If the agent decides otherwise, there is nothing reason can do. It can call the agent irrational, even deny him the status of an agent; but the (non?)agent need not be impressed by any of this. In fact, he can appropriate words like “reason” and “rational” and provide them with his own semantics; and there will be no forcing him to recognize that as an error. Reason ( whatever that is) is playing its own game and, however consistent and connected the game might be, one can always, simply, opt out of it. Same thing here. Every transcendental philosophy/logic sets out its own game, to be played by its own set of rules. Now suppose that, by the rules current in a particular game, I prove that, than which nothing greater can be thought, necessarily to exist. If I am a believer, I rejoice in thus seeing my faith conrmed and I generously broadcast my proof to all others, so that they can see the light also. And I am puzzled when many of those others, instead of coming to a harmonious, reasonable agreement with me, use their disbelief as the premise of a modus tollens and start looking for what is wrong with my proof. Eventually they might focus on something I took to be included in the semantics of “greater ”: that existing, say, is greater than not existing. And they might deny it: adopt an alternative account of “greater.” What can I do then? Clearly, they are playing a di erent game; and, no matter how loud I protest, there is no convincing them that my game is the one they should be playing. This last judgment is internal to my game, and of course from that internal perspective it looks irrefutable. From the outside, it just looks like something else one could say. In and by itself (more about this quali cation shortly), logic has no persuasive force. Despite the metaphors of constraint that are invariably brought up in its wake, it can constrain no one. If anything, the practice of logic (as opposed to the often deceptive theory of it) has a liberating e ect. You felt constrained to making a certain inferential step (say, from something being necessary to its being necessarily so); but, when you bring logical acuity and attention to bear upon it, you realize that it was a matter of habit, that you can dislodge yourself from that straightjacket and make
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the step not a forced but an optional one – that you are free to go either way, you have a choice in the matter that you had missed at rst. Think of the story long told about Girolamo Saccheri: 7 of how he wanted to rm up, once and for all, the necessity of Euclid ’s fth postulate (to withdraw the option of having it, or not having it, as an independent assumption) and ended up unwittingly freeing thought from Euclidean fetters. What, then, is the use of a logic? How is the search for its objectivity ever going to pay o ? Its value judgments, I said, are internal to the game the logic is playing; it is only from within that game that certain principles appear secure, certain inferential steps apodeictical, certain objections untenable. So it is only internally that a logic, in and by itself, has a use. The development of my transcendental philosophy/logic will be like the development of an organism: a realization of its own potential and a functional interaction of all its components. Repeatedly, I will come upon theoretical options, and the game I am playing (I decided to play, I committed myself to playing) will sometimes determine my choice of one of them, in which case I will “naturally ” accept that choice, and sometimes not, in which case I will re ect on what else I want to add to the rules of the game in order to have it cover more ground, to make it more delicately responsive to the rugged terrain on which I must travel. And the places I get to by traveling on that terrain will retroact on my initial commitments: I will regard those commitments as conrmed to the extent that I approve of my destination; I will correct them to the extent that I nd it unwelcome. There will even be surprises along the way: locations I never thought I would reach but my rules irresistibly take me to, either to be more powerfully reassured that I am on the right track, or more anxiously aware that I must be doing (assuming) something wrong. A logic is a self-organizing structure, self-enclosed and self-referential, that provides the bare sca o lding of a world and, if given enough data, even a large part of its actual construction. (So, as anticipated earlier, a logic includes its own ontology.) Luigi Pirandello called it a “corrosive”, infernal “little machine”8 because from any imagined variation in the existing circumstances it could engineer, one step after the other, the most horrid outcomes; and he considered it something to be afraid of. For me, the fear at issue here is the one that always accompanies freedom. 7
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A Jesuit priest and professor of mathematics at Pavia, who published in 1733 Euclides ab omni naevo vindicatus , a presumed reductio proof of Euclid’s fth postulate from his other assumptions, long regarded as the rst (unintended) development of a non-Euclidean geometry. Pirandello 1990, pp. 1108–1109.
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Whatever the outcome, whether horrid or even benevolent, logic is revealing of our powers: of the creative process by which we shape our world and hence of the responsibility that follows from it. Decades of existentialist thought (starting with Kant!) have made it clear that none of that is taken lightly, or should be. In the Rhetoric , Aristotle discusses three means of persuasion: ethos (an appeal to the speaker ’s character, intended to suggest authority and cause respect), pathos (an appeal to the public’s emotions), and logos (an appeal to reasoning and argument). On the face of it, this taxonomy would seem to contradict my (and Kant’s) claim that reason has no motivational force, indeed it might look like the premise of an edifying call to exercising personal and emotional restraint and letting the austere business of logos take control of public exchanges and safely guide the community to perfectly reasonable outcomes. In light of what I said so far, this attitude would be delusional: if in fact a speaker were to convince an interlocutor to switch to her side by the use of logical argument, it would only be by cleverly hiding the optional character of her principles and the controversial nature of her inferential steps, and that itself would happen, most likely, because of the competence, hence the authority, the interlocutor attributes to her, hence ultimately because of her implicit use of ethos , her implicit appeal to her superior ability (and honesty) in dealing with these matters.9 Rhetoric a cionados are fond of making some such point, and of collapsing logos into a fraudulent mannerism, which will succeed (when it does) by couching in impressive, authoritarian pseudorational garb sub jective (and often repressive) opinions and policies. This extreme, unwarranted stance issues from a gut reaction to the equally extreme, and equally unwarranted, claim that logic, in and by itself, can persuade anyone; and we have now prepared the ground for a more plausible and balanced posture – and for nally explaining what I have meant by the qualication “in and by itself. ” When creationists say that evolution is only a theory, they are saying something clearly, even trivially, true; and their opponents ’ angry retorts that evolution is a fact are only signs of bad faith. But that evolution (and creation as well) be a theory is the beginning of a story, not the end of one, as some theories are better than others along a signi cant list of parameters: they are more detailed, more discriminating, more resourceful, more ingenious. And, when compared with creation, evolution is all that. One can imagine that, at the limit of becoming more and more detailed, 9
Also, crucially, because of her interlocutor ’s deference to her authority. See the following footnote.
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discriminating, etc., a theory could be judged to be the factual description of a world; as that is one more limit that cannot be reached, we will make do with what approximations we can reach, and develop theories that weave a ner and ner texture of a presumed reality. Then we will throw our theories onto the marketplace of ideas and defend them as best we can, hoping that others will buy them. Some theories (creationism, for example) will have a clear advantage in terms of pathos : they will line up the support of strong feelings – the fear of death, the already mentioned fear of freedom. Others will have more modest, though no less genuine emotions on their side: intellectual curiosity, the fascination of complex solutions, the aesthetic satisfaction of seeing things fall into place. And they will stir such emotions more, the more detailed, discriminating, resourceful, and ingenious they are, that is : the more reason structures them internally – the reason that is forever asking pointed questions and expecting relevant answers, the reason Socrates taught us how to use. A logic is a highly ambitious theory: one that attempts to construct a universal language. In and by itself , this theory will be found persuasive only by those who are already committed to the particular view it expresses and articulates. But, the more the view is articulated, the more material it includes and makes t in a well-organized, thoroughly sensible structure, the more it will look to others like the groundwork for a majestic cathedral, and the more they might nd it attractive. Despite the attraction, they might never leave the hovels they are used to, since those give them more comfort and reassurance; still, however slim a chance logos has of winning over fear, by eliciting the waner passions germane to itself, this is a chance, that in happy (safe, relaxed, sociable) circumstances might well come to fruition. Forcefully asserting our axioms and proudly marching to the tune of our proofs will never get us but a reputation for arrogance; patiently working out a thing of beauty and making it a paradigm (an example, that is) of internal richness and consistency might make a few others want to play with us. Not because they have to; but because that internal richness and consistency – the logos that internally paces it – might make them feel that it would be fun to do. And so they might, if perhaps only for a while, come to inhabit our world. 10
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Similar points could be made about ethos . The character and competence of a speaker, in and by themselves, will have no power to persuade an audience unless the latter feels respect for them. So, as appropriate to a discussion of persuasion – that is, of how an audience can be manipulated – , it is always pathos (the emotions the speaker is able to instigate in the audience) that works if anything does; and the real distinction is among the emotions that are in play. Both logos and ethos will only be successful if the speaker can raise the emotions akin to them, and if these emotions, under the circumstances, prevail over conicting ones.
chapter 11
Bolzano s logical realism ’
Sandra Lapointe
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1
Framework
The term ‘logical realism’, as it is commonly understood, picks out a family of views that are committed to at least two theses. The rst, let us call it ‘(LF)’, is that there are “logical facts”. Here (LF) is construed in the widest possible sense to include any theory that assumes that there is a fact of the matter when it comes to the truth-value of claims about logic . (LF) can thus be cashed out in more or less robust terms. Take for instance the putatively true claim that modus ponens is a valid principle of inference. The realist may be committed to there being “something ” – whatever this turns out to imply – that makes the claim that modus ponens is valid true. Or she may understand the idea that the validity of modus ponens “is a fact” to mean merely that the corresponding claim is true . Both interpretations of (LF), and every other one in between, raise a number of questions that go beyond the scope of the present chapter. (For instance: what is truth? What is it for a fact to “make true” a truth?) What is relevant here is the following: whatever she understands “logical facts” to be, what makes the adherent to (LF) a realist about logic is a further assumption (IND), that logical facts are independent of our cognitive and linguistic make-up and practices ; they are independent of our minds and languages. In this sense, for the logical realist the truth or falsity of logical claims is “objective”. History o ers a number of theoretical alternatives to logical realism. What’s common to nihilism, pragmatism and pluralism, for instance, is the fact that they deny (LF). By contrast, the proponents of naturalism (of which there are many variants, including logical psychologism) and 1
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My thanks to Matt Carlson, Nicholas F. Stang, Penny Rush, David Sanson, Ben Caplan and Peter Hanks, Julie Brumberg and Teresa Kouric for their input on previous versions of this chapter. This characterization of logical realism draws on Resnik ’s 2000: 181. I revert to the denition of “pluralism” given by Stewart Shapiro in the chapter included in this collection. 1
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conventionalism need not reject (LF). A psychologistic logician – think of John Stuart Mill – need not disagree with the idea that there is a fact of the matter as to whether or not ‘ Modus ponens is valid’ is true. Rather, he might be denying that the truth of this claim can be established independently of psychological knowledge and therefore independently of certain facts concerning our mind. Likewise, the conventionalist may assume that there are determinate facts concerning our (linguistic) practices that determine whether, for instance, the claim that modus ponens is valid ought to count as true or false. What logical psychologism and conventionalism share is the fact that they reject (IND). As I’ve characterized it so far, logical realism is compatible with certain kinds of relativism. In the chapter included in this volume, Shapiro describes the view he calls “logical folk-relativism”. While one who holds this view assumes (i) that the truth-value of ‘y is a consequence of x ’, for instance, varies from one logical framework to another; she also admits (ii) that there is a fact of the matter as to whether y is a consequence of x in a given framework (i.e., LF); and (iii) that this fact is objective (i.e., IND). For the purpose of this chapter, I will use ‘logical realism’ in a narrower sense that does not include relativism of this sort; the type of logical realism I will be discussing below is “monistic”. The ontological questions that underlie logical realism – e.g. what kinds of “facts”, if any, ground the truth or falsity of logical claims? – are to be strictly separated from the types of concerns that arise when explaining how we come to know the truth of a claim about logic. The distinction between questions about the epistemology of logic and questions about its metaphysics is important, among other reasons, for assessing the consistency of some theories. Take Edmund Husserl, for instance. At least in the rst edition of the Logical Investigations (1900–1901), he adopts a form of logical realism of the more robust kind. What makes claims about, say, validity, true according to the Logical Investigations are certain features of abstract entities that exist independently of us: “Bedeutungen”. Nonetheless, Husserl believed that the only way to know the truth-value of logical claims is to engage in certain (admittedly rather esoteric types of ) psychological analyses. Whatever its other merits, Husserl’s theory is not inconsistent. The ontological position according to which there are mind-independent “logical facts” need not be at odds with the epistemological position according to which we can only discover the truths of logic through an investigation of the mind. More generally, logical realists, while they hold that a claim about logic, if it is true, is true independently of what we believe or do, may also believe that the recognition of the
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’
truth-value of such claims require us to investigate the way our brain or mind works and/or reect upon our cognitive abilities, psychological dispositions, linguistic conventions or other uses and practices. The alternative would presumably be to assume that we come to recognize the truths of logic through some sort of immediate logical grasp . And while this cannot be excluded a priori, it is an assumption that might seem dubious to anyone who has ever taught introductory logic to college students.3 Logical realism raises a number of interesting metaontological questions. Consider two simple toy semantic theories. Let us assume that the content of both theories is dened as the set of true instances of: “
(1)
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s means p
where s is taken to stand for a sentence of natural language and p for the meaning of this sentence, say the proposition that p. To the extent that one holds that at least some instances of ( 1) are true, both theories commit one to there being sentences and, more controversially, to there being propositions. What makes the two theories di erent theories may be a variety of things: they may diverge on which instances of ( 1) are true, they may rest on di erent accounts of what a sentence is, or have di erent views on what propositions consist of (e.g. structured entities, sets of possible worlds). Or they may agree on all this and still not be identical. When it comes to comparing types of logical realism, di erences that reside in the metatheory, and in particular in the kinds of grounds that underlie commitment to the existence of proposition-like entities, can be especially enlightening. I want to take the notion of ground in a broad, intuitive sense: Agent A s belief that x is a ground for her belief that y if her holding x to be true has explanatory value when it comes to accounting for A s belief that y. The notion of explanation used here is to include the case in which y follows from x (in a sense of follows to be specied) as well as a range of other cases I will discuss below. What s peculiar about all these cases is the fact that the relation between y and x is to be construed in epistemic terms. As I use the terms ground and explain here, whether y objectively follows from x is not ultimately what matters when it comes ‘
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The theory of eidetic variation and Wesensschau Husserl eventually committed to is an instance of this kind of epistemology, and this explains in good part why, in many circles, his theories eventually fell into disrepute. Gödel adopted a similar view, and one directly inspired by Husserl. (Cf. Kennedy 2012.) The idea that the realist might be bound to adopt an epistemology seems to be a common objection to the doctrine as a whole. In her chapter in this volume Penny Rush argues for the potential of phenomenology as regards this problem. “
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to determining whether A ’s belief that y can be explained by A ’s belief that x. It is su cient in order for A ’s belief that x to have explanatory value (to be a ground) in the relevant sense when it comes to accounting for A ’s belief that y that A e ectively believes that y is a consequence of x. This qualication is important if we are to account for the fact that grounds that are unclear, implausible or otherwise mistaken nonetheless have explanatory value when it comes to understanding an agent ’s motivation for certain claims. If it makes sense to say that A holds the belief that y because A holds the belief that x then A ’s belief that x – and the corresponding claim – is a ground for A ’s belief that y. There are at least two kinds of ground to adhere to (LF) and (IND) and, accordingly, two main types of realism in logic. The proponent of logical realism may have “external” grounds to assume that there are putative logical facts, even if these grounds are implicit, unconvincing or otherwise awed. In the context of logical realism, what I mean by “external grounds” are grounds that arise out of a concern that is not itself for logic. While it might be di cult to dene precisely what counts as a logical concern, the idea that some concerns pertain to logic while others don ’t is uncontroversial enough. On the contemporary understanding the de nition of validity and logical consequence belongs to logic – construed widely enough to include semantics. The investigation of what is involved in perception and cognition, what moral principle(s) we should abide by and what there is in the world, by contrast, do not. The (more or less well dened) boundaries between the various philosophical subdisciplines are not hermetic and indeed are often such that the grounds we have to hold a belief in one, are e ectively driven by another. For instance, there’s nothing that forbids that a logical realist ’s grounds to commit to (LF) and (IND) be external to the extent that they are driven by metaphysical concerns i.e. concern for what there is in the world in addition to rocks and chairs (assuming that there are such things). But this sort of “metaphysical” realism in logic is uncommon – if it exists at all – and the kinds of grounds that underlie realist commitments, when they are external, are typically not metaphysical. The realist’s grounds for positing mind and language-independent logical facts, when they are external, are typically driven by other aspects of her philosophical theory altogether. One may be a logical realist on epistemological grounds, for instance. Take Leibniz’s arguments that truths exist eternally in the mind of God, 4 4
But it will be further asked what the ground is for this connection, since there is a reality in it which does not mislead. The reply is that it is grounded in the linking together of ideas. In response to this “
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and that God “displays” (some of ) these truths to us.5 The former commits Leibniz to truths that are independent of human minds (and language). And taken together, these two assumptions explain how human knowledge is possible on Leibniz ’s view. Leibniz’s primary concern in introducing propositio is not for what there is, but for how we acquire knowledge. Leibniz’s grounds to commit to the existence of proposition-like entities are thus (in part) that the supposition of such entities – and the further assumption that a benevolent God exists! – is required to provide a coherent theory of knowledge. Similarly, Popper ’s grounds for thinking that there is “objective knowledge” – in (Popper 1968) for instance – whatever their merit, is that this allegedly explains certain features of the sciences, such as the relatively autonomous character of scienti c theories and problems. Whether they are epistemological or otherwise, as long as the logical realist’s grounds for believing in the existence of logical facts are not themselves logical, I will call the kind of realism she adopts ‘external’ or ‘extra-semantic’. One’s grounds to subscribe to (LF) and (IND) and to the idea that there are proposition-like entities, in particular, need not be external. What often underlies one’s commitment to logical facts may correspond to (implicit) theoretical desiderata or aims. Desiderata and aims are types of grounds in the relevant sense: they have explanatory value when it comes to accounting for the ontological commitments that come with a logical theory. Let us call the kind of logical realism that would underlie such a theory ‘internal’. Historically, many instances of realism in logic have been internal. The exact nature of the grounds that underlie the internal realist ’s commitment to logical facts vary. It may be that the logician desires to see certain “intuitions” satised or certain epistemic “purposes” fullled by the logical theory. Why precisely these intuitions and purposes ought to be satised by the theory is bound to be a matter of contention, but there ’s a case to be made to the e ect that they pervade logic and its philosophy. 6 What I call “intuitions” here correspond to certain claims that seem more certain, more epistemically salient or otherwise accessible to the
it will be asked where these ideas would be if there were no mind, and what would then become of the real foundation of this certainty of eternal truths. This question brings us at last to the ultimate foundation of truth, namely to that Supreme and Universal Mind who cannot fail to exist and whose understanding is indeed the domain of eternal truths . . .That is where I nd the pattern for the ideas and truths which are engraved in our souls. ” IV.xi.447 (my emphasis added). 5 IV.v.397. I wish to thank Chloe Armstrong for an informative discussion concerning this point. 6 The exact nature of the distinction between intuitions and purposes would benet from a closer investigation, but this would go beyond the scope of the present chapter.
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agent (though there might not be a fact of the matter as to whether they really are). The logical realist may be convinced, for instance, that truth, whatever it is, is immutable , in the sense that it cannot be changed or destroyed and she won t regard the theory as adequate unless the immutability of truth is a consequence of it. Since she is also likely to hold the belief that individual sentence- and thought-tokens do not persist inde nitely (for they don t) and thus cannot be the fundamental bearers of truth (and falsity), she might deem it necessary to introduce ontologically robust abstract entities, precisely in order to satisfy this intuition. If that is the case, then A s (desire to satisfy this) intuition has explanatory value when it comes to accounting for her commitment to proposition-like entities: there is a denite sense in which A believes that there are propositions because she believes that truth is immutable. Quantifying over meanings may also serve certain more or less clandestine purposes within the theory. The logical realist may, for instance, be guided by the fact that systematically including instances of (1), above, in a semantic theory (surreptitiously) introduces a paraphrastic procedure that can be used to clarify natural language sentences or make them more exact .7 If one s motive, be it explicitly or not, in introducing the semantic operator means and in quantifying over propositions are the (stealthy) epistemic gains that come from translations of this type, one s grounds to commit to propositions are subservient to the semantic theory and the type of realism they embrace is internal. Admittedly, in certain cases, it could be unclear whether one s grounds are internal or external. Take the case in which A s belief that there are logical facts is the consequence of certain assumptions concerning the relation between language and the world. A may believe that there are objective logical facts because A believes (TM): “
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(TM) The truth of a claim implies its correspondence to something that makes it true (or the existence of a truth-making relation), whatever this turns out to be. “
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See (Lepore and Ludwig 2006). They write: The assignment of entities to expressions, which was to be the key to a theory of meaning, turns out to have been merely a way of matching object-language expressions with metalanguage expressions thought of as used (in referring to their own meaning), so that we are given an object-language expression and a matched metalanguage expression we understand, in a context which ensures that they are synonymous (Lepore and Ludwig 2006: 31). “
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precisely the type of “full-bodied” correspondence Wittgenstein’s Tractatus was meant to put into question. 8 But is (TM) an “external” ground to subscribe to logical facts? After all (TM) is a claim that belongs to the metaphysics of logic and one could argue that logic does include its own metaphysics. This raises a question – what is the scope of logic/semantics – which I am inclined to answer liberally but which I leave open for now. It is sucient for our purposes that this question has an answer in principle even if it is a di cult one.
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2
Bolzano s internal realism in logic ’
Bolzano’s Theory of Science ( 1837) presents the rst explicit and methodical espousal of internal logical realism. It also contains a formidable number of theoretical innovations. They include (i) the rst account of the distinction between “sense” (Sinn, Bedeutung ) and “reference” (or “objectuality ”: Gegenständlichkeit ) , (ii) de nitions of analyticity and consequence, i.e. “deducibility ” ( Ableitbarkeit ) based on a new substitutional procedure that anticipates Quine’s and Tarski’s, respectively, and (iii) an account of mathematical knowledge that excludes, contra Kant, recourse to extraconceptual inferential steps and that is rooted in one of the earliest systematic reections on the nature of deductive knowledge. (i) –(iii) all assume the existence of mind- and language-independent entities Bolzano calls “propositions and ideas in themselves” (Sätze an sich ). Take (i) for instance. Appeal to propositions in themselves in this context serves Bolzano ’s antipsychologism in logic: according to Bolzano, the sense (Sinn) of a sentence – the proposition it expresses – is to be distinguished from the mental act in which it is grasped. Just like what is the case in Frege, a sentence has the semantic properties it has (e.g. truth) on Bolzano’s account derivatively, by virtue of its relation to mind-independent entities: the primary bearers of semantic properties are the propositions that constitute their Sinne . Bolzano’s version of logical realism is among the more robust. It yields a unique form of “semantic descriptivism”: there are objective,9 immutable10 entities, “propositions” (for short), that bear certain properties and relations, which it is the task of logicians to describe . The rst two books of the Theory of Science (together they make up the Theory of Propositions and 8 10
9 On this, see (Mulligan, Simons and Smith 1984 : 289). Cf. (Bolzano 1837, §21: 84 ). Cf. (Bolzano 1837, § 125: 7 ). That truth is immutable – that ‘is true’ is not a relativized predicate – is thus an intuition Bolzano’s theory seeks to satisfy and one of the grounds that motivates his commitment to logical realism.
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Representations in Themselves ) are divided into chapters whose headers include reference to propositions’ and representations’ “general characteristics”, “properties”, “relations” (among themselves, to objects) and “internal constitution ”. Bolzano even devotes entire sections of the book to the analysis of claims in which such properties and relations are ascribed to representations and propositions. As Bolzano sees it, truths of logic – including denitions of such notions as meaning, analyticity and apriority – amount to descriptions in which (often multifaceted) properties or relations are ascribed to propositions in themselves, their parts or classes thereof. Bolzano’s descriptive approach to logic is both original and noteworthy. Nonetheless it comes with an explicit commitment to the existence of certain kinds of non-natural entities which, because it is explicit and indeed unequivocal, is perhaps somewhat perplexing: one could be left with the impression that Bolzano’s ontology of logic is a more direct target for standard naturalistic objections than some other varieties of logical realism. There are at least two grounds why this impression is misleading. First, to the extent that ontological commitments cannot be measured on a scale and that all logical realists subscribe to (LF) and (IND), all variants of realism are equally ontologically “candid” from a naturalistic standpoint. There is in principle nothing more ontologically damning about Bolzano ’s semantic descriptivism than about the kind of realism Frege will eventually put forward in (1918). Second, while ontological commitments do not come in degrees, metatheoretical considerations are not irrelevant and some kinds of “grounds” for positing non-natural entities may be more palatable to the naturalist than others. The naturalistic criticism of logical realism is typically motivated by a concern for metaphysical economy (Is it consistent to postulate entities that do not exist in the causal realm?) or by related epistemological reservations (What does it mean to “grasp” or cognize or be epistemically related to something that does not exist causally?). For this reason, one who has independent (external) metaphysical or epistemological grounds to subscribe to (LF) and (IND) is a more direct target for naturalistic criticism. But Bolzano’s commitment to the existence of non-natural entities, while it is uncompromising, is also clearly motivated by the kind of internal grounds that makes him least susceptible to the naturalistic concern. 11
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As Bolzano sees it, the main reason for positing propositions is their “usefulness” for certain theoretical purposes, in particular for the purpose of reaching satisfactory de nition of logical notions (understood broadly): The usefulness of the distinction [between propositions in themselves and thought propositions] manifests itself in tens of places and in the most surprising way in that it allows the author to determine objectively a number of concepts that had not been explained before or that were explained incorrectly. For instance, the concept of experience, a priori , possibility, necessity, contingency, probability, etc. (1839: 128)
Bolzano is clear that the positing of propositions should not be considered to have bearing outside of logic. As Bolzano sees it, logicians should be allowed to appeal to entities that may reveal themselves to be inconsistent with paradigmatic metaphysical and/or epistemological theories: Thus, to give another example, the logician must have the same right to speak of truths in themselves as the geometrician who speaks of spaces in themselves (i.e., of mere possibilities of certain locations) without thinking of them as lled with matter, although it is perhaps possible to give metaphysical reasons why there is no, and cannot be any, empty space. (1837, §25: 113–14 )
What’s perhaps most remarkable about Bolzano ’s internal logical realism is the fact that, while he argues that he needs to posit what he calls ‘propositions’ to arrive at satisfactory de nitions, and while he assumes that the bearers of the logical properties and relations he de nes in fact bear this name: Bolzanian denitions of logical notions are in principle compatible with a number of di erent ontologies. This idea – it amounts to claiming that logic is “topic neutral” – emerges from a series of remarks Bolzano makes in a text he published some years after the Theory of Science , the Wissenschaftslehre (Logik) und Religionswissenschaft in einer beurtheilenden Uebersicht ( 1841) whose (failed) purpose was to arouse the public ’s interest for Bolzano’s theories. Bolzano writes: Everything the author asserts of propositions in themselves in the rst section – with the exception of what he says at § 122, namely that they don’t exist – holds of thought propositions; likewise, in the second section, the “Di erences amongst propositions as regards their internal properties ” are all such that whoever admits of thought propositions can also admit of them. (Bolzano 1841: 50)13 13
See also (Bolzano 1841: 34 –35).
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But now the question arises whether someone who rejects the concept of propositions in themselves and accepts only that [for instance] of thought propositions could nonetheless admit of a connection amongst the latter more or less like the one Bolzano describes as objective. And this, we think, should be answered in the a rmative. (Bolzano 1841: 68)
On the face of it, the kind of semantic descriptivism Bolzano adopts seems incompatible with the claim that someone who rejects the notion of proposition could still admit his de nitions of logical notions. Indeed such statement contradicts what Př íhonsk ý, Bolzano’s close collaborator, seems to have assumed in the New Anti-Kant, namely that: All will be lost if they cannot grant us this concept [of a proposition], if they keep representing truths in terms of certain thoughts , appearances in the mind of a thinking being . . . (P ř íhonsk ý 1850: 5)
If Př íhonsk ý is right, Bolzano ’s move – the claim that denitions of logical notions are topic neutral – is at best a rhetorical concession made in order to win a reluctant public. The problem with this exegetical line is not that it is implausible. The problem is that it does not do justice to the coherence of Bolzano’s views. Notwithstanding what P ř íhonsk ý assumes (more on this below), and even granting that Bolzano as an internal logical realist has less of an axe to grind when it comes to defending the existence of propositions, it remains that the claim that de nitions of logical notions are topic neutral is not a mere rhetorical ploy. For Bolzano has the theoretical resources to make sense of this idea systematically. This is what I argue in Section 3. Nonetheless, if part of Bolzano ’s point is that the value of a logical theory does not reside in the nature of the entities that bear the properties it denes, but in the properties and relations they are meant to epitomize and that he would be willing to revise some of his ontological commitment as long as some other aspects of his theory are preserved, the onus is on him to show that his theory does present an advantage over that of his predecessors and contemporaries. Section 4 is dedicated to arguing that it does.
3.
Topic neutrality and implicit denition
Bolzano claims that everything he asserts of propositions in themselves, with the exception of their being non-actual, in the rst and second section of the second volume of the Theory of Science – what Bolzano calls “general characteristics” and “di e rences that arise from their internal
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constitution” – holds of thought propositions, the doxastic states in which such propositions are grasped.14 These characteristics and di erences include the following: (i) For all x, if x is a proposition, then x contains several ideas (§ 123) (ii) For all x, if x is a proposition, then x can be viewed as part of another proposition, even a mere idea (§ 124 ) (iii) For all x, if x is a proposition, then x is either true or false (for always and everywhere) (§125) (iv) For all x, if x is a proposition, then x is of the form ‘ A has b’ (§§127–128) (v) For all x, if x is a proposition, then the extension of x is identical with the extension of the subject-representation of x (§130) (vi) For all x, if x is a proposition, then x is either simple or complex (§132) (vii) For all x, if x is a proposition, then x is either conceptual or intuitional (§133) and so on. What is relevant here is the following observation: while (i)–(vii) take the form of descriptive statements, it is more accurate to think of Bolzano as resorting to what he calls “denition on the basis of use or context” ( 1837, §668: 547), that is, implicit de nitions. The idea is that in (i)–(vii), ‘proposition’ designates a primitive (simple) concept (i) –(vii) dene implicitly.15 Bolzano was aware from very early on of the bene t of this procedure when it comes to dening primitive notions. Though Bolzano’s paradigmatic examples come from mathematics, the procedure applies across the board, including in logic. It consists in: stating many propositions in which the concept that needs to be understood occurs in di erent combinations and which is designated by the word that is associated to it. By comparing these propositions, the reader himself will abstract exactly the concept designated by the unknown word. Thus for 14
The same holds for what he describes as the “objective connections” between propositions, including “formal properties”. More on this in the next section. 15 That ‘proposition’ is a simple concept is something Bolzano suggests at (1837, §128: 8) when he writes: “ From the mere fact that representations are the components of propositions we cannot infer that the concept of a representation must be simpler than that of a proposition. On the contrary, there is a lot to say for the idea that this mark which I use in § 48 merely as an explanation of the concept of a representation is the actual denition of the latter.” At (1837, §48: 216) Bolzano had written: “ Anything that can be part of a proposition in itself, without being itself a proposition, I wish to call a [representation] in itself . This will be the quickest and easiest way of conveying my meaning to those who have understood what I mean by a proposition in itself ”.
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What denes propositions, then, ultimately, is the system of all relevant implicit denitions. Implicit denitions (including (i)–(vii)) dene propositions as much as necessary for the purpose of Bolzano ’s logic. In (i)–(vii), ‘proposition’ occurs only as the name of what is in e ect being dened and, in this light, part of Bolzano ’s point is that substituting ‘thought (proposition)’ for ‘proposition (in itself )’ in (i) –(vii) has no bearing on the nature and structure of the properties and relations involved. Indeed, Bolzano has nothing to object to someone who would claim that the bearer of the properties involved in (i) –(vii) have further properties, e.g. the property of being types of mental processes, as long as she admits that mental processes have the properties involved in (i) –(vii). There is at least one other reason to take Bolzano ’s suggestion that his denitions are topic neutral seriously. Bolzano (1841, 68) claims that “objective connections” need not be predicated of propositions, that properties such as validity (§147), analyticity (§148), compatibility (§ 154 ) and deducibility (§155) – “formal” properties – could equally well be recognized by one who admits only thoughts. By a formal property, Bolzano means a property that is dened for “entire genera of propositions”, on the basis of the substitutional procedure.16 Beyond what I’ve argued above, I want to show that Bolzano ’s denitions of what counts as formal properties are topic neutral in the relevent sense. Formal properties are not properties of individual propositions but properties of what Bolzano calls ‘forms’, i.e. schematic expressions. 17 Bolzano makes copious use of schematic expressions – or their equivalent18 – when it 16
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Cf. Bolzano (1837, §12: 51) where he explains (my emphasis): “The clearest denitions say hardly more than that we consider the form of propositions and ideas when we keep an eye only on what they have in common with many others, that is, when we speak of entire species or genera of the latter. . . . one calls a species or genus of proposition formal if in order to determine it one only needs to specify certain parts that appear in these ideas or propositions while the rest of the parts which one calls the stu or matter remain arbitrary .” Cf. (Bolzano 1837, §9: 42 f ). Bolzano often speaks of propositions containing “variable representations” and he does not always revert to schemata to indicate variability. If [Caius] is taken to be variable in [Caius who has mortality, has humanity], the latter can in principle be designated by the schematic expression ‘ X who has humanity, has mortality ’; and [Caius is Caius] by ‘ A is A ’.
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comes dening formal notions. Take for instance Bolzano ’s claims that there are logically analytic or tautological propositions. Bolzano writes: The followi following ng are some some very very genera generall exampl examples es of analyti analyticc propos propositi itions ons which are also true: “ A is A ”, “ An A which is a B is an A ”, “ An A which is a B is a B ”, “Every object is either B or non-B ”, etc. Propositions of the rst kind, i.e., propositions cast in the form “ A is A ” or “ A has (the attribute) a ” are commonly called identical or tautological propositions. (Bolzano 1837, §148: 84 )
In this passage, Bolzano ascribes the property of being logically analytic to individual propositions, yet his examples – “ A is A ”, “ An A which is a B is an A ”, “ An A which is a B is a B”, “Every object is either B or non-B ”, etc. – are not examples of individual propositions at all. If we follow what Bolzano says in the Theory of Science , ‘ A A is A ’ does not stand for any “proposition” in particular. On Bolzano ’s account schematic expressions of the kind ‘ A A is A ’ represent classes of propositions that are de ned through a substitutional procedure. To say that a proposition “falls under a certain form” is to say that it belongs to a certain substitution class designated by this schematic expression.19 ‘ A A is A ’ represents the class of all propositions in themselves that correspond to the substitution instances of ‘ A A is A ’. If we use ‘[‘and’]’ to form designations for individual propositions (and their parts), we nd among the propositions designated by the substitution instances instances of ‘ A A is A ’ the following: [Caius is Caius] [Redness is Redness] [1 is 1]
and so on. It is certainly not incongruous for Bolzano to claim that ‘ A A is A ’ – the schematic expression – is logically analytic. Indeed, Indeed, it would seem that one need understand what it means for a proposition to belong to such a class – or to fall under such a “form” – in order to understand how it itself can be said to be analytic. The proposition: [Caius is Caius], for instance, falls under the form ‘ A A is A ’: ‘ A A is A ’ is a “determinate connection of words or signs” through which the class to which [Caius is Caius] belongs can be 20 To say that the individual proposition [Caius is Caius] is “represented”. 19
In one of his numerous historical digressions, Bolzano notes that the Latin word forma forma . . . . was . was in fact used as equivalent to the word species, i.e. the word class (Bolzano 1837, §81: 391 ). 20 See (Bolzano 1837, §81: 393 ). “
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logically analytic is to say that it is a member of a class of the latter kind: a class that can be represented by a determinate type of schematic expressions, namely one all of whose substitution instances designate propositions that have the same truth value. What’s interesting here is the fact that, on this interpretation, Bolzano is ultimately committed to the following view of logical analyticity: (LA) x is logically logically analytic analytic if x belongs to a class that can be represented represented by a schematic expression in which only logical terms occur essentially.
But (LA) is topic neutral. Nothing compels us to think of ‘x ’ in (LA) in terms of a proposition in itself. And as we have seen, once the logical work is done, Bolzano would not be disconcerted by such a move since “someone who rejects the concept of propositions in themselves and accepts only that [for instance] of thought propositions could nonetheless admit of a connection amongst the latter more or less like the one Bolzano describes as objective objective” (Bolzano 1841: 68).
4.
Bolzano s logic ’
Internal grounds to adhere to logical facts – or in Bolzano ’s case to fully desiderata or aims the edged semantic entities – are typically certain desiderata theory is meant to ful l. In Bolzano ’s case, one of the main purposes in introducing propositions in themselves is to achieve precise and satisfactory denitions. By way of consequence, on Bolzano ’s own account the success of the endeavour depends on whether his commitment to propositions allows him to deliver a “good” theory of logic, or at least one that is preferable to its rivals. To a large extent, Bolzano succeeds. It is not only that hat the Theo Theory ry of Scien Science ce is furn furnis ishe hed d with with rich rich and and remarka remarkably bly well-art well-articul iculated ated distinc distinctio tions ns and theoreti theoretical cal innovat innovation ionss but also that he set out to rede ne the very nature of a logical investigation in a way that is largely consistent with well-established contemporary endeavours. As Bolzano sees it, at its core, the purpose of logic is to tell us what it means for something to follow from something else, i.e. what it means for an inference to be valid or for a claim to be the consequence of some other claim(s). As an explanation of what it means for a truth to follow from others, Bolzano’s views on “deducibility ” ( Ableitbarkeit ( Ableitbarkeit ) are comparatively close to the ones that have become standard following Tarski in the twentieth century. Bolzano denes deducibility in the following terms:
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Hence I say that propositions M, N, O, . . . are deducible from propositions A, B, C, D, . . . with respect to variable parts i, j, . . . if every collection of representatio representations ns whose substitution substitution for i, j, . . . makes all of A, B, C, D, . . . true, also makes all of M, N, O, . . . true. Occasionally, since it is customary, I shall say that propositions M, N, O, . . . follow, or can be inferred or derived, from A, B, C, D, . . . Propositions A, B, C, D, . . . I shall call the premises, M, N, O, . . . the conclusions. (1837, §155: 114 )
The modern character of Bolzano’s denition, in itself, and especially the semantic machinery on which it rests is noteworthy enough. On Bolzano ’s account:21 0
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The propositions T, T , T . . . are ableitbar are ableitbar from from S, S , S with respect to representations i, j, . . . if and only if: 0
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i, j, . . . can be varied so as to yield at least one true variant of S, S , S , . . . and T, T , T , . . . (ii) (ii) when wheneve everr i, j, . . . are varied so as to yield true variants of S, S , S . . ., the corresponding variants of T, T , T , . . . are also true. 00
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To logicians and philosophers of logic today, the idea that the aim of logic is to dene validity via the elaboration of a theory of logical consequence consequence is unremarkable. Pointing to the similarities (and dissimilarities) between Bolzano’s deniti nition on of deduci deducibi bili lity ty and and Tarsk Tarskii’s deniti nition on of logi logical cal 22 consequence has become commonplace in the literature. This goes to show that at least some of the desiderata and aims that underlie Bolzano ’s logic rest on the kind of intuitions that have proven to be enduring. This should be emphasized for at least two reasons. First, when he published the Theory of Science in 1837, Bolzano’s views on deducibility were perfectly anachronistic. For one thing, by the end of the eighteenth century it had become usual for philosophers to think of logic as invested in the study of “reason” through an investigation of “thought” and to conceive of such an inve invest stig igat atio ion n to invo involv lvee the the study study of mind mind-d -depe epend nden entt oper operat atio ions ns and and product products. s. Though Though the methodo methodolog logies ies underlyi underlying ng these these invest investiga igatio tions ns varied widely – contrast Locke’s empirical approach in the Essay the Essay on Human Understanding with with Kant’s transcendental philosophy – they largely contributed to either discredit formal logic as a discipline 23 or, at best, to convey the opinion that it could not be improved on. 24 In this light, Bolzano’s e orts orts toward a new logic new logic based on an objective doctrine of 21 22 23 24
For a more detailed discussion of Bolzano’s theory of deducibility, see (Lapointe 2011: 72 –90). See, for instance, (van Benthem 1985; George 1986; Siebel 2002; Lapointe 2011). See (George 2003: 99 s). See Kant’s famous claim that logic is closed and complete ( 1781: Bviii).
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inferences “in themselves” constitutes an important break from his immediate modern predecessors. Second, while Bolzano reaches back to Aristotle, his approach to the denition of validity also marks an important departure from Aristotle and most of his (early) traditional scholastic commentators. 25 Aristotle introduces the notion of a good “deduction” (i.e. (i.e. syllo syllogi gism sm)) in the the Prior Analytics . He writes: A deduction (syllogismos ) is speech (logos ) in which, certain things having been supposed, something di erent erent from those supposed results of necessity because of their being so. (Prior Analytics I. I. 2, 24 b18–20)
Let us call this the “intuitive intuitive Aristotelian Aristotelian notion of validity validity ”. Contemporary attempts at a de nition of logical consequence – one may think of Tarski-type model-theoretic denitions in particular – are generally understood to account for the intuitive intuitive Aristotelian Aristotelian notion notion of validity. validity. The same holds for Bolzano’s. What makes Bolzano’s account historically distinctive is the assumption that a good de nition of the intuitive Aristotelian notion of validity needs the support of a semantic theory. In this, his de nitional strategy ought to be contrasted with that of much of the Aristotelian tradi traditi tion on itse itself. lf. Ar Aris isto totl tlee and and his his early early medi mediev eval al succe success ssor orss are are most mostly ly known for their understanding of validity as epitomized in traditional syllogistic theories. But traditional syllogistic de nitions of validity are not not concer concerne ned d with with prov provid idin ingg a sema semant ntic ic acco account unt of valid validit ity. y.26 The standa standard rd and paradig paradigmat matic ic methodo methodology logy behind behind tradit tradition ional al syllogi syllogisti sticc theories of valid inference, and the one that is best known, is two-pronged. It rst consists in making a list of all possible forms of arguments (syllogisms) and then in identifying those forms whose instances e ectively ectively full the intuit intuitive ive Aristo Aristotel telian ian deniti nition on of vali validi dity ty.. In order order to dete determ rmin inee whether a particular inference is valid, one is thus required to determine whether it instantiates one of the forms identi ed as valid. There are at least three problems, from Bolzano ’s perspective, with this approach. First, traditional syllogistic de nitions of validity suppose that there is a nite (and implausibly small) number of possible forms of inference. Bolzano is right. If we follow the teachings of the Schoolmen, 25
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Here I am not concerned with comparing the Bolzanian and Aristotelian conception of the object of logic (see Thom, this volume, for such a discussion) but their views on validity. consequentia that Here, I exclude from what I call “syllogistic tradition” the theories of consequentia that emerged in the fourteenth century – those we nd in Occam and Buridan, for instance. The latter were attempts to generalize syllogistic and aimed at providing a new insight into the intuitive Aristotelian notion based on semantic considerations. On this topic, see (Novaes 2012). Something similar holds for Abelard. I am grateful to Julie Brumberg-Chaumont for this precision.
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ther theree are are exact exactly ly 256. This This numbe umberr come comess up as a resu result lt of vari variou ouss assumptions concerning the number of sentences involved in an argument – three! – and the form of such sentences. In particular, classical syllogistic theory assumes (i) that only categorical sentences (i.e. sentences of the form subject-copula-predicate ) are involved in arguments, (ii) that there are four variants of such forms (a, e, i, o) and (iii) that any given inference contains at most three di erent erent terms – subject, middle term and predicate – which yields four possible syllogistic “gures”. Second, (i) –(iii) mark out a syntax whose expressive resources are too limited to account for the richness of actual inferential practices. Hence, it cannot adequately model model (even some of the most basic forms forms of ) infere inference. nce. For instance, instance, it cannot model disjunctive and hypothetical syllogisms that require separate theories (at least if understood in its original sense, i.e. as a propositional logic). This is tributary to a third more general problem, namely the fact that that tradit tradition ional al syllogi syllogisti sticc denitions of validity are bound to a given syntax (namely the one de ned by (i)–(iii) above). But as is obvious from the relevant passage in Aristotle the intuitive notion of validity is not bound to any particular syntax – it is a “semantic” denition.27 Bolzano was aware of these three related problems. He writes: Aristotle began b egan with such a broad denition of the word syllogism that one is astonished that he could have subsequently restricted the concept of this kind of inference so severely. He writes (in Anal Pr. I, 1) “syllogism is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so ”. This denition obviously obviously ts every inference, not only with two, but also with three and more premises, and not only simple inferences but complex ones as well. (1837, §262: 535)
As Bolzano sees it, one need not suppose that the number of (valid) forms of inferences is nite or that it is linked to a determinate syntax, for instance that it can only be de ned for inferences that have only two categorical premises.28 Moreover, the three above problems concerning traditional syllogistic treatments of validity are linked to a fourth more general one. There are various ways of xing the extension of a concept, not all of which amount to denition. nition. The mere fact of knowing knowing which inferential inferential forms satisfy the 27
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This is even more obvious when one reads the beginning of the second book of the Prior Analytics , which was devoted to the relationship between premises and conclusion as regards their truth-value. ý 1850 1850: 115 f ). Some passages See (Př íhonsk íhonsk ý passages of the Prior the Prior Analytics suggest suggest that Aristotle was aware of the problem. See for instance (Prior (Prior Analyics Analyics I, 32). But Aristotle himself did not provide a systematic account of what it is for an inference that is not a syllogism to result of necessity.
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intuitive Aristotelian denition of validity does not, on Bolzano ’s account, amount to having a de nition of this notion. On Bolzano ’s account, my merely knowing what falls in the extension of a concept – say the class of all putatively valid syllogistic inferential forms – does not amount to my having a denition (Erklärung ) of that concept. De nition is a conceptual exercise: one that requires us to identify the components of a concept as well as the way in which they are connected. As Bolzano sees it, the theory of deducibility and the proposed denition above is what allows us to grasp the concept of validity. More importantly perhaps – though this might go beyond Bolzano ’s criticism – it seems that a good de nition of validity is one that is epistemically fruitful in the following sense: a good de nition of validity is one on the basis of which one can ascertain systematically for any newly encountered inference, whether or not it is valid. But the traditional syllogistic de nition of validity is not epistemically fruitful. There is no obvious reason to think that one could decide whether an as yet unknown argument form is valid when presented with it in any other way than by reverting to the intuitive notion. By contrast, Bolzano ’s de nition is epistemically fruitful: equipped with Bolzano’s denition, one can in principle determine for any new argument whether or not it instantiates the property in question. 5.
Conclusion
In light of what precedes, Bolzano’s internal realism is vindicated: Bolzano’s positing of propositions in themselves allows him to articulate a theory of deducibility that could do what the syllogistic theories of his predecessors could not: provide us with a general semantic theory of validity. Nonetheless, as those acquainted with recent scholarship know, there are problems with Bolzanian deducibility. (See, e.g. Siebel 2002.) For one, despite Bolzano’s claim to the contrary, his denition of deducibility fails to capture what is usually taken to be the modal insight that underlies the intuitive Aristotelian notion of validity, namely the idea that the conclusion of a good argument “results of necessity ”. Consequently, it overgenerates. Bolzanian deducibility systematically includes inferences that are merely materially valid. I say “systematically ” because if it is the case that all As are Bs, then ‘ X is B’ is invariably deducible from ‘ X is A ’. For instance, on Bolzano ’s account: X is no taller than three metres
is deducible from: X is a man
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with respect to X. This failure may strike as the result of a misunderstanding (coupled with contentious exegetical choices). Bolzano interprets the relevant passage of the Prior Analytics in the following terms: Since there can be no doubt that Aristotle assumed that the relation of deducibility can also hold between false propositions, the results of necessity can hardly be interpreted in any other way than this: that the conclusion becomes true whenever the premises are true. Now it is obvious that we cannot say of one and the same collection of propositions that one of them becomes true whenever the others are true, unless we envisage some of their parts as variable. For propositions none of whose parts change are not sometimes true and sometimes false; they are always one or the other. Hence when it was said of certain propositions that one of them becomes true as soon as the others do, the actual reference was not to these propositions themselves, but to a relation which holds between the in nitely many propositions which can be generated from them, if certain of their representations are replaced by arbitrarily chosen other representations. The desired formulation was this: as soon as the exchange of certain representations makes the premises true, the conclusion must also become true. ( 1837, §155: 129)
The main problem with Bolzano’s interpretation is that he assumes that “results of necessity ”, in this context, means the same as “ preserves truth from premises to conclusion”. Whatever the explanation for this confusion is – Bolzano does have a systematic account of necessity and one may wonder why he did not revert to it to interpret Aristotle on this occasion – it is unfortunate. Nonetheless one should not conclude from the fact that Bolzano’s denition of deducibility fails to grasp the modal insight that underlies the intuitive notion of validity that he achieved little toward a theory of logical consequence or that he missed the point entirely. This would not do justice to Bolzano’s accomplishment, both historical and philosophical. For one thing, while Bolzano’s own use of the substitutional method fails to do so, other philosophers have put a wager on a substitutional procedure of the type Bolzano was rst to introduce for the purpose of providing a satisfactory account of logical consequence. Tarskian-type model-theoretic approaches for instance can be seen as an extension of Bolzano ’s theory. Few would deny that Bolzano’s views on deductive knowledge were overall largely preferable to those of his predecessors and contemporaries. In particular, it is important to stress the fact that Bolzano did have views on epistemic modality – though unfortunately, there is no place for a discussion of the latter here. 29 At the very least, it ought to be mentioned 29
See (Lapointe 2014 ).
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that as an alternative to Kant s theory of pure intuition in arithmetic and geometry, Bolzano was rst to propose an account of epistemic necessity that rests on (i) the idea that truth by virtue of meaning can be de ned systematically (in a deductive system) and that (ii) a priori knowledge is accordingly always deductive. Regardless of the execution, (i) and (ii) are both manifestly valuable philosophical insights that deserve the attention of historians and philosophers alike. For one thing, one committed to (i) and (ii) cannot appeal to subjective justicatory devices such as certitude or evidence to warrant the truth of a priori claims. And, again, many even today would consider this to be an important lesson. What s relevant here is the fact that to the extent that Bolzano s views on a priori knowledge and deductive systems are parts and pieces of his theory of propositions in themselves, they are inseparable from his commitment to mind-independent logical facts. What this means is that logical realism also informs his views on a priori knowledge and nourishes insights that many of his successors, realist or not, will share. ’
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part iii
Speci c Issues
chapter 12
Revising logic Graham Priest
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What’ s at issue
Much ink has been spilled over the last few decades in disputes between advocates of “classical logic” – that is, the logic invented by Frege and Russell, and polished by Hilbert and others – and advocates of nonclassical logics – such as intuitionist and paraconsistent logics. One move that is commonly made in such debates is that logic cannot be revised. When the move is made, it is typically by defenders of classical logic. Possession, for them, is ten tenths of the law. The point of this chapter is not to enter into substantive debates about which logic is correct – though relevant methodological issues will transpire in due course. The point is to examine the question of whether logic can be revised. (And let me make it clear at the start that I am talking about deductive logic. I think that matters concerning non-deductive logic are much the same, but that is an issue for another occasion.) Three questions, then, will concern us: 1
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Can logic be revised? If so, can this be done rationally? If so, how is this done?
Unfortunately, debates about the answers to these questions are often vitiated by a failure to observe that the word ‘logic’ is ambiguous. Only confusion results from running the senses of the word together. Once the appropriate disambiguations are made, some of the answers to our questions are obvious; some are not. It pays, for a start, to be clear about which are which. 1
Thanks go to Hartry Field for many enjoyable and illuminating discussions on the matter. We taught a course on the topic together in New York in the Fall of 2012. Many of my views were claried in the process. 211
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We may distinguish between at least three senses of the word, which I will call: • • •
Logica docens Logica utens Logica ens
What each of these is will require further discussion and clarication. But as a rst cut, we may characterise them as follows. Logica docens (the logic that is taught) is what logicians claim about logic. It is what one nds in logic texts used for teaching. Logica utens (the logic which is used) is how people actually reason. The rst two phrases are familiar from medieval logic. The third, logica ens (logic itself ) is not. (I have had to make the phrase up.) This is what is actually valid: what really follows from what. Of course, there are important connections between these senses of ‘logic’, as we will see in due course. But the three are distinct, both intensionally and extensionally, as again we will see. I will proceed by discussing each of these senses of ‘logic’, and asking each of our three target questions about them. We have, then a nine-part investigation.
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Logica docens Can it be revised?
Let us start with logica docens . The discussion of this will form the longest part of the essay, since it informs the discussion with respect to the other two parts. The question of whether the logica docens can be revised is, however, the easiest to deal with. It can be revised because it has been revised. The history of logic in the West has three great periods. The rst was in Ancient Greece, when logic was founded by Aristotle, the Megarians, and the Stoics. The second was in the new universities of Medieval Europe, such as Oxford and Paris, where Ockham, Scotus, and Buridan ourished. The third starts in the late nineteenth century, with the rise of mathematical logic, and shows no signs yet of ending. Between these three periods were periods of, at best, mainly maintaining what was known, and at worst forgetting it. Much of Greek logic was forgotten in Europe, but fortunately 2
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The history of logic in the East has its own story to tell, but that will not be our concern here.
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preserved by the great Arabic scholars such as Al Farabi and Ibn Rushd. Most of medieval logic was simply wiped out by the rise of the Enlightenment, and the consequent obliteration of Scholasticism. It is only in the twentieth century that we have started to rediscover what was lost in this period. At any rate, one needs only a passing acquaintance with logic texts in the history of Western logic to see that the logica docens was quite di erent in the various periods. The di erences between the contents of Aristotle s Analytics , Paul of Venice s Logica Magna , the Port Royale Logic, or the Art of Thinking , Kant s Jäsche Logik , and Hilbert and Ackermann s Principle s of Mathematial Logic would strike even the most casual observer. It is sometimes suggested that, periods of oblivion aside, the development of logic was cumulative. That is: something once accepted, was never rejected. Like the corresponding view in science, this is just plain false. Let me give a couple of examples. One of the syllogisms that was, according to Aristotle, valid, was given the name Darapti by the Medievals, and is as follows: ’
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All As are B s All As are C s Some B s are C s
As anyone who has taken a rst course on modern rst-order logic will know, this inference is now taken to be invalid. 3 For another example: Classical logic is not paraconsistent; that is, the following inference (Explosion) is valid for all A and B : A, A ├ B . It is frequently assumed that this has always been taken to be valid. It has not. Aristotle was quite clear that, in syllogisms, contradictions may or may not entail a conclusion. Thus, consider the syllogism: :
No As are B s Some B s are As All As are As
This is not a valid syllogism, though the premises are contradictories. There are usually three distinct terms in a syllogism. The above has only two. But Aristotle is also quite explicit that two terms of a syllogism may be the same. So when did Explosion enter the history of Western logic? Matters are conjectural, but the best bet is that it entered with the ideas of the twelfth3
For further discussion of the matter, see (Priest 2006a: 10 .8).
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century Paris logicians called the Parvipontinians , whose members included Adam of Balsaha and William of Soissans, who may well have developed the argument to Explosion using extensional connectives and the Disjunctive Syllogism. After that, the validity of Explosion was debated. But it certainly did not become entrenched in Western logic till the rise of classical logic.4 .
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Can it be revised rationally?
Logica docens , then, has been revised, and not in a cumulative fashion. The next question is whether revision can be rational. Arguably, not all the changes in the history of logic were rational (or perhaps better: occurred for reasons that were internal to the subject). Thus, logic fell into oblivion in the early Middle Ages in Western Christendom because the institutions for the transmission of philosophical texts collapsed. And later Medieval logic was written o on the coat-tails of the rejection of Scholasticism during the Enlightenment. 5 However, many changes that did arise were the result of novel ideas, reason, argument, debate. These are the things of which rational change are made. This should be pretty obvious with respect to the only change that most logicians are now familiar with: the rise of mathematical logic. In the mid nineteenth century, text book logic ( traditional logic ) was a highly degenerate form of medieval logic: essentially, Aristotelian syllogistic with a few medieval accretions, such as immediate inferences like modus ponens . But this was a period in which high standards of rigour in mathematics were developing. Mathematicians such as Weierstrass and Dedekind were setting the theory of numbers on a rm footing. And when it came to examining the reasoning required in the process, notably by Frege, it became clear that traditional logic did not seem to be up to the job. Hence Frege invented a logic that did much better: classical logic. The extra power of this logic made it much preferable rationally; and within 50 years it had replaced traditional logic as the received logica docens . I will come back to this in the next section. For the present, let us move on to our third question. “
”
“
”
4 5
For references and further discussion on all these matters, see (Priest 2007: sec. 2). Actually, my knowledge of the history of these periods is pretty sketchy; but I think that these claims are essentially correct.
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Logic as theory
So, what, exactly, is it in virtue of which one logica docens is rationally preferable to another, and so may replace it? To answer this question, we need to draw some new distinctions. Let us start with geometry. There are many pure geometries: Euclidean geometry, elliptical geometry, hyperbolic geometry, and so on. And as pieces of pure mathematics, all are equally good. They all have axiom systems, model theories; each speci es a perfectly ne class of mathematical structures. Rivalry between them can arise only when they are applied in some way. Then we may dispute which is the correct geometry for a particular application, such as mensurating the surface of the earth. Each applied geometry becomes, in e e ct, a theory of the way in which the subject of the application behaves. Geometry had what one might call a canonical application: the spatiotemporal structure of the physical cosmos. Indeed this application was coeval with the rise of Euclidean geometry. It was only the rise of nonEuclidean geometries which brought home the conceptual distinction between a pure and an applied geometry. And nowadays the standard scientic view is that Euclidean geometry is not the correct geometry for the canonical application. So much, I think, is relatively uncontestable. But exactly the same picture holds with respect to logic. There are many pure logics: classical logic, intuitionist logic, various paraconsistent logics, and so on. And as pieces of pure mathematics, all are equally good. They all have systems of proof, model theories, algebraicisations. Each is a perfectly good mathematical structure. But pure logics are applied for many purposes: to simplify electrical circuits (classical propositional logic), to parse grammatical structures (the Lambeck calculus), and it is only when di erent logics are taken to be applied for a particular domain that the question of which is right arises. Just as with geometries, each applied logic provides, in e ect, a theory about how the domain of application behaves. And just as with geometries, pure logics have a canonical application: (deductive) reasoning. A logic with its canonical application delivers an account of ordinary reasoning. One should note that ordinary reasoning, even in science and mathematics, is not carried out in a formal language, but in the vernacular; no doubt the vernacular augmented by many technical terms, but the vernacular none the less. (No one reasons à la Principia Mathematica .) And so applied, di erent pure logics may give di erent verdicts concerning an inference. If it is not the case that it is not
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the case that there is an in nitude of numbers, does it follow that there is an innitude of numbers? Classical logic says yes; intuitionist logic says no.6 In other words, a pure logic with its canonical application is a theory of the validity of ordinary arguments: what follows (deductively) from what. How to frame such a theory is not at all obvious. Many approaches have been proposed and explored. One approach is to take validity to be constituted modally, by necessary truth-preservation (suitably understood). Another is to de ne validity in terms of probabilistic constraints on rational belief. Perhaps the most common approach at present is to take a valid inference to be one which obtains in virtue of the meanings of (at least some of ) the words employed in it. This strategy has itself two ways in which it can be implemented. One takes these meanings to be spelled out in terms of truth conditions, giving us a model-theoretic account of validity; the other takes these meanings to be spelled out in inferential terms, giving us a proof-theoretic account of validity. It is clear that a theory of validity is no small undertaking. It requires an account of many other notions, such as negation and quantication. Moreover, depending on the theory in question, it will require an articulation of other important notions, such as truth, meaning, probability. No wonder it is hard to come up with plausible such theories! At any rate, it is crucial to distinguish between logic as a theory (logic docens , with its canonical application), and what it is a theory of ( logica ens ). In the same way we must clearly distinguish between dynamics as a theory (e.g., Newtonian dynamics) and dynamics as what this is a theory of (e.g., the dynamics of the Earth). This is enough to dispose of the Quinean charge (still all too frequently heard): change of logic means change of subject.7 If one changes one s theory of dynamics, one can still be reasoning about the same thing: the way the Earth moves. ’
2.4
What is the mechanism of rational revision?
With this substantial prolegomenon over, we can now address the question of the mechanism of rational change of logica docens . As we have seen, a pure logic with its canonical appication is essentially a theory of validity and its multitude of cognate notions. How do we determine which theory is better? By the standard criteria of rational theory choice. 6
Further on the above, see (Priest 2006a: chs. 10 , 12 ).
7
(Quine 1970a: p. 81 ).
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Given any theory, in science, metaphysics, ethics, logic, or anything else, we choose the theory which best meets those criteria which determine a good theory. Principal amongst these is adequacy to the data for which the theory is meant to account. In the present case, these are those particular inferences that strike us as correct or incorrect. This does not mean that a theory which is good in other respects cannot overturn aberrant data. As is well recognised in the philosophy of science, all things are fallible: both theory and data. Adequacy to the data is only one criterion, however. Others that are frequently invoked are: simplicity, non-(ad hocness), unifying power, fruitfulness. What exactly these criteria are, and why they should be respected, are important questions, which we do not need to go into here. One should note, however, that whatever they are, they are not all guaranteed to come down on the same side of the issue. Thus (the standard story goes), Copernican and Ptolemaic astronomy were about equal in terms of adequacy to the data; the Copernican system was simpler (since it eschewed the equant); but the Ptolemaic system cohered with the accepted (Aristotelian) dynamics. (The Copernican system could handle the motion of the Earth only in an ad hoc fashion.) In the end, the theory most rational to accept, if there is one, is the one that comes out best on balance. How to understand this is not, of course, obvious. But we do not need to pursue details here.8 I observe that this procedure does not prejudice the question of logical monism vs logical pluralism. If there is one true logic one s best appraisal of what this is is determined in the way I have indicated. If there are di erent logics for di erent topics, each of these is determined in the same way. Whether one single logic is better than many, is a meta-issue , and is itself to be determined by similar considerations of rational theory-choice. Let me nish this discussion by returning, by way of illustration, to the replacement of traditional logic by mathematical logic in the early years of the twentieth century. In the nineteenth century, much new data had turned up: specically, the microscope had been turned on mathematical reasoning, showing all sorts of inferences that did not t into traditional logic. Mathematical logic was much more adequate to this data. This is not to say that enterprising logicians could not try to stretch traditional logic to account for these inferences. But mathematical logic scored high on many of the other theoretical criteria: simplicity, unifying power, and so on. It was clearly the much better theory. “
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’
“
8
”
Matters are spelled out in detail on (Priest 2006a: ch. 8), and especially, (Priest to appear).
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A word of warning: it would be wrong to infer that classical logic did not have its problems. It had its own ad hoc hypotheses (to deal with the material conditional, for example). It had areas where it seemed to perform badly (for example, in dealing with vague language). And why should one expect a logic that arose from the analysis of mathematical reasoning to be applicable to all areas of reasoning? It was just these things which left the door open for the development of non-classical logics. That, however, is also a topic for another occasion. 9 We have seen, at least in outline, what the mechanism of rational change for a logica docens is. 3.
Logica utens
3.1
What is this?
So much for the discussion of logica docens . Let us now turn to the next disambiguation. Before we address our three questions, however, there is an important preliminary issue to be addressed. What exactly is logica utens ? I said that it is the way that people actually reason. This may make it sound like a matter of descriptive cognitive psychology; but it is not this, for the simple reason that we know that people often reason invalidly. Set aside slips due to tiredness, inebriation, or whatever. We know that people actually reason wrongly in systematic ways. 10 To take just one very well established example: the Wason Card Test. There is a pack of cards. Each card has a letter on one side and a positive integer on the other. Four cards are laid out on the table so that a subject can see the following: A
K
4
3
The subject is then given the following conditional concerning the displayed situation: If there is an A on one side of the card, there is an even number on the other. They are then asked which cards should be turned over (and only those) to check this hypothesis. The correct answer is: A and 3 . But a majority of people (even those who have done a rst course in logic!) tend to give one of the wrong answers: A, or A and 4 . Exactly what is going on here has occasioned an enormous literature, which we do not need to go into. The experiment, and ones like it, show that people can reason wrongly systematically. Of course, people are able 9
Some discussion can be found in (Priest 1989).
10
See (Wason and Johnson-Laird 1972).
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to appreciate the error of their reasoning when it is pointed out to them. But how to draw a principled distinction between correcting a standard performance error, and revising an actual practice is not at all obvious. Fortunately, we do not need to go into this here. I point these facts out only to bring home the point that logica utens is not a descriptive notion; it is a normative one. A logica utens is constituted by the norms of an inferential practice. Subjects in the Wason Card Test can see, when it is pointed out to them, that they have violated appropriate norms. How to understand the normativity involved here is a particularly hard question, which, fortunately, we also do not need to pursue. We have sucient understanding to turn to the rst of our three questions. Can a logica utens be revised? 3.2
Can it be revised?
Clearly, di erent reasoning practices come with di erent sets of norms. Thus, the norms that govern reasoning in classical mathematics are di erent from those that govern reasoning in intuitionist mathematics. I was trained as a classical mathematician, and have no di culty in reasoning in this way. But I have also studied intuitionist logic, and can reason (more falteringly) in this way too. Clearly, then, it is possible to move from one logica utens to another. I can reason like a classical mathematician on Mondays, Wednesdays, Fridays, and like a intuitionist on Tuesdays, Thursdays, and Saturdays. (And on Sundays ip a coin.) So practices can be changed. At this point one might wonder about the nature of inference sketched in Wittgenstein’s Philosophical Investigations . According to this, correct reasoning is simply how we feel compelled to go on after suitable training. If such is the case, then how can one change? The answer is that we must take the suitable training seriously. I can follow my training as a classical logician some days, and my training as an intuitionist on others – just as I can follow my training in cricket on some days, and my training in baseball on others. 3. 3
Can it be revised rationally?
So logica utens can change. Can it be changed rationally? Unless one is a complete relativist about inferential practices, the answer must be yes: some practices are better than others. And to move from one that is less good to one that is more good for principled reasons is clearly rational.
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Moreover, being a relativist about such practices is a hard pill to swallow. For we use reasoning to establish what is true, and what is not, about many things. A relativism about these practices therefore entails a relativism about truth. And such a relativism is problematic. To take an extreme example: suppose that reasoning in one way, we establish that the theory of evolution is correct, but that reasoning in another way, we establish that creationism is true and the theory of evolution false. Something, surely, must be wrong with one of these forms of reasoning. 11
3.4
How is it revised rationally?
Assuming, then, that rational change is possible, how is this to be done? The answer to that is easy. We determine what the best theory of reasoning is (the best docens ), and simply bring our practice (utens ) into line with that. How else could one be rational about the matter? 4. 4.1
Logica ens
Can it be revised?
We now turn to what I think is the hardest of the three disambiguations: logica ens . These are the facts of what follows from what – or better, to avoid any problems with talk of facts: the truths of the form ‘that so and so follows from that such and such’. Can these be revised? The matter is sensitive for a number of reasons. As we have seen, our logica docens , with its canonical application, is a theory about what claims of this form are true. Now, if one changes one ’s theory of dynamics, the dynamics of the Earth do not themselves change. Such realism about the physical world is simply common sense. But logic is not a natural science. It is a social science, and concerns human practices and cognition. When a theory changes in the social sciences, the object of the science may change as well. One has to look only at economics to see this. When free-market economics became dominant in the capitalist world in the 1980s, so did the way that the then deregulated economy functioned. So, in the social sciences one is not automatically entitled to the view that a change of theory does not entail a change of object. 11
It is quite compatible with this point that sometimes truth may be internal to a practice example, within classical and intuitionist pure mathematics. See (Priest 2013).
–
for
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But the object of a social scienti c theory may not change when the theory does, for all that. (Many basic laws of psychology are, presumably, hard-wired in us by evolution.) Whether the truth of validity-claims can change will depend on what, exactly, constitutes validity. Let me illustrate. Suppose that one held a “divine command” theory of validity: something is valid just if God says so. Then, God being constant and immutable, what is valid could not change. On the other hand, suppose that one were to subscribe to the “dentist endorsement” view of validity: what is valid is what 90 per cent of dentists endorse. Clearly, that can change. These theories are, of course, rather silly. But they make the point: the truth of validity-claims may or may not change, depending on what validity actually is. An adequate answer to our question would therefore require us to settle the issue of what validity is, that is, to determine the best theory of validity. That is far too big an issue to take on here. 12 I shall restrict myself in what follows to some remarks concerning the model-theoretic and proof-theoretic accounts of validity. According to the rst, an inference is valid i every model of the premises is a model of the conclusion. But a model is a structured set, that is, an abstract object, the premises form a set, another abstract object, and the premises and conclusions themselves are normally taken to be sentence types, also abstract objects. According to the second, an inference is valid if there is a proof structure (sequence or tree), at every point of which there is a sentence related to the others in certain ways. But a proof structure is an abstract object, as, again, are the sentences. In other words, validity, on these accounts, is a realtionship between abstract objects. As usual, we may take these all to be sets. If this is so, then, at least if one is a standard platonist about these things, the truth of claims about validity cannot change.13 Claims about mathematical objects are not signi cantly tensed: if ever true true, always true. 4.2
Can meanings change?
That is not an end of the matter, though. The propositions about validity may not change their truth values. But we express these in language. It might be held that the words involved may change their meanings – and, moreover, do this in such a way that the truth values of the sentences 12 13
I have said what I think about the matter in (Priest 2006a: ch. 11 ). Certain kinds of constructivists may, of course, hold that the truth about numbers and other mathematical entities may change – for example, as the result of our acquiring new proofs.
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involved may change. If this is the case, then the sentences expressing validity claims can change their truth values. Can meanings change in such a way as to a e ct truth value? Of course they can. When Nietzsche wrote The Gay Science , it was a reference to the art of being a troubadour. Nowadays, one could hear it only as concerning a study of a certain sexual preference. In modern parlance, Nietzsche did not write a book about (the) gay science. Now, could there be such change of meaning in the case we are concerned with? Arguably, yes. In both a proof-theoretic and a modeltheoretic account of validity, part of the machinery is taken as giving an account of meanings – notably, of the logical connectives (introduction or elimination rules, truth conditions). If we change our theory, then our understanding of these meanings will change. This does not mean that the meanings of the vernacular words corresponding to their formal counterparts changes. You can change your view about the meaning of a word, without the word changing its meaning. However, if one revises one ’s theory, and then brings one ’s practice into line with it, in the way which we noted may happen, then the usage of the relevant words is liable to change. So, then, will their meanings – assuming that meaning supervenes on use (and some version of this view must surely be right). So the sentences used to express the validity claims, and maybe even which propositions the language is able to express, can change. 14 It might be thought that this makes such a change a somewhat trivial matter. Suppose we have some logical constant, c , which has di erent truth or proof conditions according to two di erent theories. Can we not just use two words, c 1 and c 2, which correspond to these two di erent senses? Perhaps we can sometimes; but certainly not always: for meanings can interact. Let me illustrate. Suppose that our logic is intuitionist. Then B ) A) A, is not logically valid. But suppose that “Peirce’s law ”, (( A we now decide to add a new negation sign to the language, which behaves as does classical negation. Then Peirce’s law becomes provable. The extension is not conservative. Another case: given many relevant logics, the rules for classical negation can be added conservatively, as can the natural introduction and elimination rules for a truth predicate. But the !
14
!
!
A pertinent question at this point is whether the meaning of ‘ follows deductively from’ – or however this is expressed – can itself change. Perhaps it can; and if it does, this adds a whole new dimension of complexity to our investigation. However, I see no evidence that the meaning of the phrase (as opposed to our theories of what follows from what) has changed over the course of Western philosophy. So I ignore this extra complexity here.
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addition of both (when appropriate self-reference is available) produces triviality. Meanings, then, are not always separable .15 “
4. 3
”
Can meanings change rationally?
So meanings can change, and not necessarily in a straightforward way. Can this happen rationally, and if so, how? The answers to these questions are implicit in the preceding discussion. Suppose we change our logica docens to a rationally preferable one. Suppose that we then change our logica utens rationally to bring it in line with this. Then the meanings of our logical constants, and so the language used to express the facts of validity, may also change. And the whole process is rational. 5.
Conclusion
Let me end by summarising the main conclusions we have reached, and making a nal observation. A logica docens may be revised rationally, and this happens by the standard mechanism of rational theory choice. A logica utens may be changed by bringing it into line with a logica docens ; and if the docens is chosen rationally, so is the utens . The answer to the question of whether or not the logica ens may change depends on one s best answer to the question of what validity is. However, under the model- or proof-theoretic accounts of validity, the answer appears to be: no. This does not mean, however, that the sentences used to express these facts may not change. And a rational change of logica utens may occasion such a change. Now the observation. The rational logica utens depends on the rational logica docens . The true logica docens depends on the facts of validity. And assuming a model- or proof-theoretic account of meaning, the language available to express these may depend on the logica utens . It is clear that we have a circle. If one were a foundationalist of some kind, one might see this circle as vicious: there is no privileged point where one can ground the entire enterprise, and from which one can build up everything else. However, I take it that all knowledge, about logic, as much as anything else, is situated.16 We are not, and could never be, tabulae rasae . We can start only from where we are. Rational revision of all kinds then has to proceed by an incremental and possibly (Hegel notwithstanding) neverending process. ’
15
On these matters, see (Priest 2006a: ch. 5).
16
See (Priest to appear).
chapter 13
Glutty theories and the logic of antinomies Jc Beall, Michael Hughes, and Ross Vandegrift
.
1
Introduction
There are a variety of reasons why we would want a paraconsistent account of logic, that is, an account of logic where an inconsistent theory does not have every sentence as a consequence. One relatively standard motivation is epistemic in nature. There is a high probability that we will come to hold inconsistent beliefs or inconsistent theories and we would like some account of how to reason from an inconsistent theory without everything crashing. Another motivation, rooted in the philosophy of logic or language, is that we want a proper account of entailment or relevant implication, where there is a natural sense in which inconsistent claims do not (relevantly) entail arbitrary propositions – where not every claim follows from arbitrary inconsistency. A third motivation, the one which will occupy our attention here, is metaphysical or semantic. One might, for various reasons, endorse that there are ‘true contradictions’, or as they are sometimes called, truthvalue gluts – true sentences of the form φ ^ : φ, claims which are both true and false. We shall say that a glut theorist is one who endorses glutty theories – theories that are negation-inconsistent – with the full knowledge that they are glutty. There are di erent kinds of metaphysical commitments that can lead one to be a glut theorist. One route towards glut theory arises from views about particular predicates of a language or the properties that those predicates express. Along these lines, a familiar route towards glut theory holds that certain predicates like ‘is true’, ‘is a member of ’ , or ‘exempli es’ are essentially inconsistent: they cannot be (properly) 1
2
1
2
For work in this tradition, see Rescher and Manor ( 1970); Schotch et al. (2009); Schotch and Jennings ( 1980). For work in this tradition, see Anderson and Belnap ( 1975); Anderson et al. ( 1992); Dunn and Restall (2002); Mares (2004); Slaney (2004).
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interpreted in a way that avoids there being objects of which these predicates are both true and false. Such essentially glutty predicates – everywhere glutty with respect to something if glutty anywhere with respect to anything – are antinomic , as we shall say. Of course, one need not hold that a predicate is essentially inconsistent to think that it can give rise to gluts: there may be only contingently glutty predicates. For some predicates, whether they are properly interpreted consistently or inconsistently may depend on facts about the world. Priest, for example, has suggested that predicates like ‘is legal’ and ‘has the right to vote’ are of this sort; see Priest ( 2006b). Acceptance of either sort of (essentially or contingently) inconsistent predicates is sucient for being a glut theorist – though not necessary. Another path one might take towards being a glut theorist is inevitable ignorance about the exact source of gluttiness. One might think that our best – and true – theory of the world will inevitably be inconsistent, even though we might, for all we know, remain ignorant of the source of the inevitable inconsistency. Indeed, one might have reason to be agnostic about the source of gluttiness: one is convinced that our best theory of the world (including truth, exemplication, sets, computability, modality, whatever) will be inconsistent, though also convinced that we will never be in good position to pinpoint the exact source of the inconsistency. Agnosticism about the particular predicates responsible for gluttiness remains an option for the glut theorist. The question that arises is: how do our metaphysical commitments inform our choice of logic? We cannot ask this question without attending to the di erence between formal and material consequence. Brie y, a logic takes a material approach to consequence when it builds in facts about the meaning of predicates, the properties they express, or the objects those predicates are about. A logic takes a formal approach to consequence when it abstracts away from all of these concerns. There are various ways a logic could be said to ‘build in’ such facts, and one of our aims below is to explore these in the context of metaphysical commitments to gluts. We carry out our discussion via a comparison of two paraconsistent logics, namely, the logic of paradox (LP) and the logic of antinomies (LA). The former is well-known in philosophy, discussed explicitly and widely by Priest (1979, 2006b);3 the latter is a closely related but far less familiar and 3
LP is the gap-free extension of FDE, the logic of tautological entailments; it is the dual of the familiar glut-free extension of FDE called ‘strong Kleene’ or ‘K 3’. See Dunn (1966, 1976), Anderson and Belnap (1975), and Anderson et al. ( 1992).
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equally less explored approach in philosophy, an approach advanced by Asenjo and Tamburino (1975).4 Below, we consider various philosophical motivations that could explain the logical di erences between LA and LP. We shall argue that LA re ects a fairly distinctive set of metaphysical and philosophical commitments, whereas LP, like any formal logic, is compatible with a broad set of metaphysical and philosophical commitments. We illustrate these points below. The discussion is structured as follows. §§ 2–3 present the target logics in terms of familiar model theory. § 4 discusses the main logical di erences in terms of di erences in philosophical focus and metaphysical commitment. §5 closes by discussing the issue of detachment.
.
2
The logic of antinomies
The logic of antinomies (LA) begins with a standard rst-order syntax. The logical vocabulary is _, : , 8. Constants c 0, c 1, . . . and variables x 0, x 1, . . . are the only terms. The set of predicate symbols is the union of two disjoint sets of standard predicate symbols: = f A0, A1, . . .g and = fB 0, B 1, . . .g. (Intuitively, contains the essentially classical predicates and the essentially non-classical, essentially glutty predicates.) The standard recursive treatment denes the set of sentences. 5 An LA interpretation I consists of a non-empty domain D , a denotation function d , and a variable assignment v , such that: • • •
for any constant c , d (c ) 2 D , for any variable x , v ( x ) 2 D , for any predicate P , d (P ) = P þ, P , where P þ [ P = D .
The only di erence from the standard LP treatment appears here, in the form of a restriction that captures the distinction between the antinomic (i.e., essentially glutty) and essentially classical predicates:
4
5
For purposes of accommodating glutty theories, the propositional logic LP was rst advanced in Asenjo (1966) under the name calculus of antinomies ; it was later advanced, for the same purpose, under the name ‘ logic of paradox ’ by Priest ( 1979), who also gave the rst-order logic under the same name (viz., LP). What we are calling ‘LA ’ is the rst-order (conditional-free) logic advanced by Asenjo and Tamburino (1975), which was intended by them to be a rst-order extension of Asenjo’s basic propositional logic. Due to what we call the LA Predicate Restriction (see page 227 ) LA isn’t a simple rst-order extension of Asenjo’s propositional LP – as will be apparent below (see §4 ). For simplicity, we focus entirely on unary predicates. Both LA and LP cover predicates of any arity, but focusing only on the unary case su ces for our purposes.
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LA Predicate Restriction . For any predicate P: • •
if P is in , then the intersection P þ \ P must be empty; if P is in , then the intersection P þ \ P must be non-empty .
As above, the Ai s are the essentially classical predicates, while the B i s are those which are antinomic.6 |φ|v is the semantic value of a sentence φ with respect to a variable assignment v , which is dened in the standard recursive fashion. (We leave the relevant interpretation implicit, as it will always be obvious.) For atomics: jPt jv
8> < ¼ >:
0 1 1 2
if I ðt Þ 2 = P þ and I ðt Þ 2 P if I ðt Þ 2 P þ and I ðt Þ 2 = P otherwise:
The inductive clauses are as follows:
|φ _ ψ |v = max f|φ|v |ψ |v g. 2. |:φ|v = 1 |φ|v . |∀ x φ|v = minf|φ|v : v 0 is an x -variant of v g. 3. 1.
0
Conjunction and existential quanti cation can be de ned from these in the normal way. LA consequence ‘LA is dened as preservation of designated value, where the designated values are 1 and 21 . Thus, Γ ‘LA φ holds (i.e., Γ implies/entails φ according to LA) if and only if no LA interpretation designates everything in Γ and fails to designate φ. 3.
The logic of paradox
We obtain the logic LP simply by dropping the LA predicate restriction, but leaving all else the same. Thus, for purposes of ‘semantics’ or model theory of LP, there’s no di erence between -predicates and -predicates: they ’re all treated the same. 4.
Contrast: LA and LP
We begin with formal contrast. While both logics are paraconsistent (just let jφjv ¼ 21 and jψ jv = 0, for at least some formulae φ and ψ ), there are some obvious but noteworthy formal di erences between the logics LA 6
The presentation in Asenjo and Tamburino ( 1975) is rather di erent; but we present their account in a way that a o rds clear comparison with LP.
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and LP. LP permits the existence of a maximally paradoxical object – an object of which every predicate is both true and false – whereas LA does not. Indeed, LA – but not LP – validates ‘explosion’ for certain contradictions; for example, for any A i in and any φ, Ai t ^ : Ai t ‘LA φ:
Similarly, LA validates the parallel instances of detachment (modus ponens): Ai t Ai t ⊃ φ ‘LA φ ,
where φ ⊃ ψ is dened as usual as :φ _ ψ . But LP is di erent: not even a restricted version of detachment is available; see Beall et al. ( 2013a). As a nal and nicely illustrative example, LA validates some existential claims that go beyond those involved in classical logic (e.g., 9 x (φ _ :φ), etc.), whereas LP does not. To see this, note that, for any Bi in , the following is a theorem of LA: 9 xB i x :
Since any predicate B i in must have at least some object in the intersection of its extension and anti-extension, it follows that something is in its extension. 4.1
Metaphysics, formal and material consequence
How are we to understand the logical di erences between LA and LP? For a rst pass, they might be naturally understood as arising from di erent notions of consequence: namely, material and formal consequence. The distinction may not be perfectly precise, but it is familiar enough. 7 Material consequence relies on the ‘matter ’ or ‘content’ of claims, while formal consequence abstracts away from such content. Example: there is no possibility in which ‘Max is a cat’ is true but ‘Max is an animal ’ is not true; the former entails the latter if we hold the meaning – the matter, the content – of the actual claims xed. But the given entailment fails if we abstract away from matter (content), and concentrate just on the standard rst-order form: Cm does not entail Am. The notion of formal consequence delivers conclusions based on logical form alone. Material consequence essentially requires use of the content of the claims or the meaning of things like predicates that appear in them. 7
See Read (1994 , Ch. 2) wherein Read provides a defense of material consequence as logical consequence, and also for further references.
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One way to understand Asenjo and Tamburino ’s proposal is that it gives a material consequence relation of a language arising from certain metaphysical commitments. It is clear that the logic re ects an assumption that certain predicates are essentially classical, and other predicates are essentially glutty – antinomic, as we have said. On this interpretation, the incorporation of essentially classical predicates reects a metaphysical commitment that gluts cannot arise absolutely anywhere. Similarly, the semantic restriction on the B i predicates reects a metaphysical commitment that certain predicates, in virtue of their meaning, or the properties they express, must give rise to gluts: there is bound to be at least some object of which B i is both true and false. Beall ( 2009) gives such a view: inconsistency unavoidably arises in the presence of semantic predicates like ‘is true’. The typical semantic paradoxes like the liar require an inconsistent interpretation of the truth predicate, but this is compatible with the commitment to the essential classicality of all predicates in the truth-free fragment of the language. But what if you wanted to give the formal consequence relation of a language that is motivated by Asenjo and Tamburino’s metaphysical commitments? LP, we suggest, provides the formal consequence relation of such a language – abstracting from the matter or content to mere form. LA ’s predicate restriction is not a purely formal matter: that a predicate is either antinomic or essentially classical depends on its meaning. If we ignore content, and focus just on purely formal features of sentences, LA ’s predicate restriction falls away as unmotivated. And that ’s precisely what happens in LP: if we abstract away to ‘pure form’ then the content of predicates doesn’t matter. We observe that one might be concerned with material consequence, and yet still be motivated to adopt LP rather than LA. Suppose that all predicates are on par with respect to (in-) consistency: each might be glutty with respect to something – or not glutty at all. If one held this commitment, then LA ’s predicate restriction is inappropriate, or at least unmotivated. Indeed, even one predicate which is either contingently consistent or contingently inconsistent arrests the motivation for LA ’s predicate restriction. And one might think that ordinary cases of such predicates are not hard to nd. Priest (2006b, Ch. 13), for example, discusses such cases arising from considerations of the law. Suppose that we had laws that all citizens have a right to vote and no felons have a right to vote . It is then a contingent matter whether or not there are any gluts about rights to vote; it depends on whether anyone commits any felonies, and whether or not anything is classied as a felony. And you might hold a view wherein all
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predicates are like that: potentially glutty, one and all, but none antinomic – none essentially glutty. There is another metaphysical route to LP. We might not start with any commitments about the nature of any predicates, their meaning, the properties they express, and whether or not they are essentially inconsistent. One might start with the commitment that one ’s theory is both true and inconsistent, while remaining agnostic about where to locate the origins of the inconsistency. There is no reason to think that this position excludes a material approach to consequence. It ’s just that such a view lacks any particular metaphysical commitments that would motivate a restriction on predicates like the LA predicate restriction. Of course, from the material point of view, LA and LP far from exhaust the possibilities. So far we’ve mentioned fairly strong, all-or-nothing approaches. On a material approach to consequence, the proponent of LA is committed to all predicates being essentially classical or glutty, while the proponent of LP is committed to all predicates being potentially classical or glutty. Mixed approaches are available. These are achieved by adding obvious combinations to the LA predicate restriction – for example, some antinomic, some essentially classical, some neither, etc. We leave these to the reader for exploration. We turn (briey) to an issue peculiar to the logics under discussion: detachment or modus ponens. 5.
Detachment
A salient problem for LP is that there is no detachable (no modus-ponenssatisfying) conditional de nable in the logic (Beall et al. ( 2013)); and thus, historically, LP has been viewed as unacceptably weak for just that reason. A lesson one might try to draw from the above observations is that LP can be improved by shifting focus to the material notion of consequence. But this is not quite right. Though one fragment of LA di ers from LP in that it satises detachment, LA is like LP in that detachment doesn ’t hold generally: arguments from φ and φ ⊃ ψ to ψ have counterexamples. On this score, Asenjo and Tamburino ( 1975), along with Priest ( 1979, 2006b), have a solution in mind. The remedy is to add logical resources to the base framework to overcome such non-detachment.8 But the remedy 8
Until very recently, Beall (2013), all LP-based glut theorists focused their e orts on the given task: adding logical resources to the base LP framework to overcome its non-detachment. Whether this is the appropriate response to the non-detachment of LP is something we leave open here.
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o e red by Asenjo and Tamburino doesn’t work, as we now briey indicate. Asenjo and Tamburino de ne a conditional ! that detaches (i.e., φ and φ ! ψ jointly imply ψ ). The conditional is intended to serve the ultimate purpose of the logic, namely, to accommodate paradoxes in non-trivial theories (e.g., theories of naïve sets), and is de ned thus:
jφ ! ψ jv
8> >< ¼ >>:
if jψ jv ¼ 0 and jφjv
0 1
if jψ jv ¼
2 1
1 2
n o 2 n o
and jφjv 2
1
,
1
2 1
,
1
2
otherwise !
The resulting logic, which we call LA , enjoys a detachable conditional. In particular, dening ‘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 satises ‘absorption’ in the form !
φ
! ð φ ! ψ Þ ‘ 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 f x : 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 dened from ! and ^ as per usual, we have c 2 c $ ð c 2 c !
Þ:
⊥
But, now, since the Asenjo –Tamburino arrow satises the given absorption rule, we quickly get c 2 c ! 9
⊥
Throughout, ⊥ is ‘explosive’ (i.e., implies all sentences).
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which, by unrestricted comprehension, is sucient 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 su cient for standard rstorder connectives, the ‘remedy ’ for non-detachment (viz., moving to LA ) is not viable: it leads to absurdity. 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. !
10
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 rst by Asenjo and Tamburino, and to highlight the salient di erences in the logics. We’ve argued that the di erences in logic reect a di erence 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
12
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 nding 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’) reects 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.
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 di erent uses, hence it is not surprising that it can be interpreted in di erent 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 rst. 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 di erence 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
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. 2 See Chateaubriand Filho (2001, 2005) for a version of the metaphysical interpretation of logical truth partly similar to mine. 3 ‘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. 4 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. 233
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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 di erence 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 intu- ition, 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 rst, 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 de discipline of metaphysics and I am interested in nding a use for logical truth within that discipline. I doubt there is enough initial common ground to fruitfully engage with Dummett. Let me briey 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 a airs’, 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 denition 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 < p> is true, it has a truthmaker. < p> corresponds to a state of a airs, namely the state of a airs which consists of a relation of coherence holding between < p> and the other members of a maximal set of propositions’. Consider the pragmatist. He may consistently say, ‘If < p> is true, it has a truthmaker. < p> corresponds to a state of a airs, namely the state of a a irs of < p>’s having the property of being useful to believe ’. It is controversial whether there exist states of a a irs. Let that pass. My point here is that the coherence theory and the pragmatic theory are each compatible with the admission of states of a airs. Furthermore, each of these theories is compatible with the admission of states of a a irs 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 di erent 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
6
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 denes 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 a a irs.8 Hence, we may dene 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 di erent theories of truth, then we can characterize the issue at hand as follows. There is an apparent and important di erence 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
Crivelli also denes 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 de ationism 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. 8 Admittedly, once we explicate isomorphism, reference to propositions, states of a a irs 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 rst 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 specication 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 specied the class of models, our de nition of truth in a model is guided by straightforward semantic intuitions, intuitions about the inuence 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 e ect 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 de nition 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 congurations of the world, and I am interested in the correct understanding of these possible con gurations (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 specied the class of genuinely possible congurations , we can dene 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. 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
‘
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 tting 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 de ne 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 conguration of the world.
ML leaves open the criteria for a ‘genuinely possible conguration 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 con gurations 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 conguration 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, de ned 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 di ers from Aristotle’s only with regard to the qualication regarding ‘the same domain’ – here the domain is the set of genuinely possible congurations 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). I will not 11
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 specically 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 briey below, but rst I wish to say something about the methodology of logical-cum-metaphysical inquiry. In terms of ML, demonstrating the falsity of LNC would rst require a genuinely possible conguration 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 con guration, as I will go on to explain. 13 In any case, if a possible world in which LNC is not true
12 13
a principle regarding structured exclusion relations (between properties, states of a airs, 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 nally 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 con gurations of the world and only them. Logical modality cannot do the job because it is not suciently 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 congurations 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 de nitions of concepts. Nomological modality is already too restrictive, as we sometimes need to consider congurations 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 congurations 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 de nitions 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 a air 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 reect the relations of individuals, properties, and states of a a irs rather than mere logical concepts. Indeed, this understanding e ectively 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 falsi ed 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 con gurations of the world. However, this approach seems biased towards historically prior logical principles, the ones that were formulated rst. It is not implausible that the reason why they were formulated rst 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 ‘rst’ principles. It would certainly be enough to challenge the metaphysical necessity of LNC, or other logical principles, if empirical evidence to the e ect 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 conguration of the world. Since I have not seen a convincing case to the e e ct that such a conguration 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 con guration of the world that contains macroscopic objects . Whether or not there are genuinely possible congurations that do not conform to PEP is an open question, but it seems unlikely that such a conguration 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 rst place; see Estrada-González (2013) for a case to the e ect 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 su cient 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 inuential 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 di erent, and irreconcilably so. Beall and Restall have formulated and defended this type of pluralism: Given the logical consequence relation dened on the class of cases , the logical truths are those that are true in all cases . If you like, they are the sentences that are x-consequences of the empty set of premises. The logical truths are those whose truth is yielded by the class of cases alone. Since we are pluralists about classes of cases, we are pluralists about logical truth. (Beall and Restall 2006: 100) x
x
x
x
x
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 de nition of ‘model’, this approach denes 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 denitions 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 de nition. 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 de nition 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 .) do say something about the matter. Speci cally, 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 truth- preservation 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 di e r 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 di erent 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 de nition (ML), which de ned a sentence as logically true if and only if it is true in every genuinely possible conguration of the world. Since in this view of logical pluralism there can be proper subsets of genuinely possible congurations with di erent 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 congurations 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 congurations, i.e., logical monism – in that case the relevant subset of possible worlds would not be a proper subset of the genuinely possible congurations. An alternative formulation of ML-P is possible, dismissing subsets altogether. We could understand logical pluralism by giving di erent interpretations to ‘genuinely possible congurations’.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 congurations of the world. Only metaphysically possible worlds are suciently 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 reect 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 congurations. 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 de ne a set of possible worlds in which the law of non-contraction holds, call it W LNC . The assumption is that W LNC includes the actual world. But since I have made no mention of any other laws of logic that hold (in the metaphysical sense) in W LNC , 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 W LNC may be dened. This is not quite as straightforward in other cases though. Very briey, 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 rst pass, causation might be o ered 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 di er er in spin (i.e., to have di erent erent 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 have spin-d spin-down own.. 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 logi logic, c, enabl enablin ingg the the sugg suggest ested ed inte interp rpre reta tati tion on of logic logical al plura plurali lism sm.. The resulting subset could be called W LNC þ MP . This hardly exhausts the debate about logical pluralism, but it appears that there are ways, perhaps several ways, to accommodate accommodate pluralism about logical truth within the metaphysical interpretation. ‘
5.
’
Conc onclusi lusion on
In conclusion, I have demonstrated that there is a coherent metaphysical inte interp rpre reta tati tion on of logi logica call trut truth, h, and and that that this this inte interp rpret retat atio ion n has has some 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 Thanks to audiences audiences at the University University of Tampere Tampere Research Seminar and the First HelsinkiHelsinki-Tartu Tartu Workshop in Theoretical Philosophy, where earlier versions of the paper were presented. In partic particula ular, r, I d like like to than thankk Luis Luis Estr Estrad adaa-Go Gonz nzál ález ez for for exte extens nsiv ivee comm commen ents ts.. In addi additi tion on,, 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. ’
References
Anderson, A. R., and Belnap, N. D. 1975. Entailment: The logic of relevance and necessity, volume i . Princeton University Press Anderson, A. R., Belnap, N. D., and Dunn, J. M. 1992. Entailment: The logic of relevance relevance and necessity, necessity, volume i i . Princeton University Press Anon. 1967. Ars Burana, in L.M. de Rijk, Logica modernorum vol. i i : The origin and early development of the theory of supposition. Assen: Van Gorcum Aquinas, T. 2000. Opera omnia , edited by E. Alarcón. Pamplona: University of Navarre, www.corpusthomisticum.org/iopera.html Aristotle. 1963. Categories and de interpretatione , translated with notes by J. l. Ackrill. Oxford: Clarendon Press 1984 . Metaphysics , translated by W. D. Ross, revised by J. Barnes. Princeton University Press 1993. Metaphysics books gamma, delta and epsilon, translated with notes by C. Kirwan, second edition. Oxford: Clarendon Press 1994 . Posterior analytics , translated with a commentary by J. Barnes, second edition. edition. Oxford: Oxford: Clarendon Clarendon Press 2009. Prior analytics , translated and with a commentary by G. Striker. Oxford University Press Asenjo, F. G. 1966. “ A calculus of antinomies”, Notre Dame Journal of Formal Logic 7(1), 103–105 Asenjo, F. G., and Tamburino, J. 1975. “Logic of antinomies”, Notre Dame Journal of Formal Logic 16(1), 17–44 Azzouni, J. 2005. “Is ther here still till a sens ense in which hich mathe athema mati tics cs can can hav have foundations”, in G. Sica, (ed.) Essays on the foundations of mathematics and logic . Monza, Italy: Polimetrica International Scienti c Publisher 2006. Tracking reason: Proof, consequence, and truth . Oxford University Press 2008a. “ Why do informal proofs conform to formal norms?” Foundations of Science 14 , 9–26 2008b. “ Alternative logics and the role of truth in the interpretation of languages”, in D. Patt Patter erson son,, (ed.) (ed.) New essay essayss on Tars Tarski ki and and philo philosop sophy hy . Oxford University Press, 390–429 Oxford d 2010. Talki Talking ng about about nothing nothing:: number numbers, s, halluc hallucinat inations ions and ctions ctions . Oxfor University Press 249
250
References
2013. Semantic
perception: How the illusion of a common language arises and persists . Oxford University Press Azzouni, J. and Armour-Garb, B. 2005. “Standing on common ground ”, Journal of Philosophy Philosophy 102(10), 532–544 Barker, C. 2013. “Negotiating taste ”, Inquiry 56, 240–257 Batens, D., Mortensen, C., and Van Bendegem, J.P. (eds.) 2000. Frontiers of paraconsistent logic . Dordrecht: Kluwer Publishing Company Beall, Jc 2009. Spandrels of truth . Oxford University Press 2012. “ Why Priest’s reassurance is not reassuring ”, Analysis 72(3), 739–741 2013. “Free of detachment: logic, rationality, and gluts ”, Noûs 47(3), 1–14 Beall, Jc and Restall, G. 2006. Logical pluralism. Oxford University Press Beall, Jc, Forster, T., and Seligman, J. 2013. “ A note on freedom from detachment in the Logic of Paradox ”, Notre Dame Journal of Formal Logic 54 (1) Beebee, H. and Dodd, J. (eds.) 2005. Truthmakers: The contemporary debate . Oxford University Press. Bell, J. 1998. A primer of in nitesimal Cambridge University University Press nitesimal analysis . Cambridge Benacerraf, Benacerraf, P. 1973. “Mathematical truth ”, Journal of Philosophy 70, 661–79 Bencivenga, Bencivenga, E. 1987. Kant s Copernican Copernican revolution revolution. New York: Oxford University Press 2000. Hegel s dialectical logic . New York: Oxford University Press 2007. Ethics vindicated: Kant s transcendental legitimation of moral discourse . New York: Oxford University Press Benthem, J. van, 1985. “The variety variety of conseq consequenc uence, e, accordi according ng to Bolzan Bolzanoo”, ), 389–403 Studia Logica 44 (4 ), Berto, F. 2007. How to sell a contradiction: Studies in Logic 6 . London: College Publications 2008. “Άδύνατον and material exclusion ”, Australasian Journal of Philosophy 86(2), 165–90 Blackburn, Blackburn, S. 1984 . Spreading the word . Oxford: Clarendon Press Bolzano, B. 1810. Beyträge zu einer begründetereu Darstelluns der Mathematik . Prague: Widtmann 1837. Wissenschaftslehre . Sulzbach: Seidel 1839. Dr Bolzano und seine Gegner . Sulzbach: Seidel 1841. Wissenschaftsleh Wissenschaftslehre re (Logik) (Logik) und Religionswiss Religionswissenscha enschaft ft in einer beurtheilend beurtheilenden en Uebersicht . Sulzbach: Seidel 1985–2000. Wissenschaftslehre , critical edition by Jan Berg. Stuttgart: FrommanHolzboog 2004 . On the mathematical method and correspondence with Exner , translated by Rusnock, P., and George, R. Amsterdam-New York: Rodopi Boolos, G. 1975. “On second-order logic ”, Journal of Philosophy 72, 509–527. (Reprinted in Boolos 1998, pp. 37–53) 1984 . “To be is to be the value of some variable (or some values of some variables)”, Journal of Philosophy 81, 430–450. (Reprinted in Boolos 1998, pp. 54 –72) 1998. Logic, logic, and logic . Cambridge, MA: Harvard University Press ’ ’
’ ’
’ ’
References
251
Brady, R. 2006. Universal logic . Stanford, CA: Centre for the Study of Language and Information Brady, R. and Rush, P. 2009. “Four Basic Logical Issues ”, The Review Review of Symbolic Symbolic Logic 2(3), 488–508 Buchholtz, Buchholtz, U. 2013. The unfolding of inductive de nitions Dissertati ation, on, nitions , PhD Dissert Dept. of Mathematics, Mathematics, Stanford University University Burge, T. 1975. “On knowledge and convention convention”, in Foundations of mind ( ( 2007), 32–37. Oxford University Press 85 , Burgess, J. A. 1990. “The sorites paradox and higher-order vagueness”, Synthese 85 417–474 2010. “Review of Beall and Restall 2006,” Philosophy and Phenomenological Research 81, 519–522 Burg Burges ess, s, J. P., P., and and Rose Rosen, n, G. 1997. A subj subjec ectt with with no obje object ct:: Stra Strate tegi gies es for
nominalistic nominalistic interpretatio interpretationn of mathematics mathematics . Oxford University Press 1999. “ Which modal logic is the right one? ” Notre Dame Journal of Formal Logic 40, 81–93 Carey, S. 2009. The origin of concepts . Oxford University Press Carnap, R. 1934 . Logische Syntax der Sprache , Vienna, Springer, translated as The logical syntax of language , New York, Harcourt, 1937; another edition edited by Amethe Smeaton, Patterson, New Jersey, Littleton, 1959 Cesalli, L. 2007. Le réalisme propositionnel: sémantique et ontologie des propositions chez Jean Duns Scot, Gauthier Burley, Richard Brinkley et Jean Wyclif . Paris: Vrin Chatea Chateaubr ubriand iand Filho, Filho, O. 2001. Logica Logicall forms: forms: Part Part i – truth and description. Campinas: UNICAMP 2005. Logic Campinas: s: Logical al forms: forms: Part Part i i – – logic, logic, langua language, ge, and knowle knowledge dge . Campina UNICAMP Chomsky, N. 2003. “Reply to Millikan”, in L. M. Antony and N. Hornstein (eds.) Chomsky and his critics . Oxford: Oxford: Blackwell Blackwell Publishing, Publishing, 308–315 Chomsky, N. and Lasnik, H. 1995. “The theory of principles principles and parameters parameters”, in Noam Chomsky The The minimalist program. Cambridge, MA: The MIT Press, 13–127 Cole, J. 2013. “Towards an institutional account of the objectivity, necessity, and atemporality of mathematics ”, Philosophia Mathematica 21, 9–36 Condillac, E. 1780. La logique (Logic), (Logic), translated by W. R. Albury. New York:
Abaris Crivelli, P. 2004 . Aristotle on truth . Cambridge Cambridge University University Press. da Costa, N. 1974 . “On the theory of inconsistent formal systems ”, Notre Dame Journal of Formal Logic 15, 497–510 Daly Daly,, C. 2005. “So wher wheree’s the explanat explanation ion??” In Beeb Beebee ee and and Dodd Dodd 2005, 85–103 . Davidson, D. 1973. “In defense of convention T”, in H. Leblanc (ed.) Truth, syntax and modality modality . Amsterdam: North-Holland. Reprinted in his Inquiries into truth and interpretation. Oxford University Press. Dehaene, S. 1997. The number sense . Oxford University Press
252
References
Dummet Dummett, t, M. 1975. “The philos philosophi ophical cal basis basis of intuit intuitioni ionisti sticc logic logic”, Logic Colloquium 73, 5–40 1991a. The logical basis of metaphysics . Cambridge, MA: Harvard University Press 1991b. Frege: Philosophy of mathematics . Cambridge, MA: Harvard University Press Dunn, J. M. 1966. The algebra of intensional logics . PhD thesis, University of Pittsburgh 1976. “Intuitive semantics for rst-degree rst-degree entailments entailments and ‘coupled trees’”, Philosophical Studies 29, 149–168 Dunn, J. M., and Restall, G. 2002. “Relevance logic”, in D. M. Gabbay and F. Günt Günthn hner er edit editor orss Handbo (second edition), edition), Handbook ok of philos philosophi ophical cal logic logic (second volume v i . Dordrecht: D. Reidel Eddington, A. S. 1928. The nature of the physical world , New York: Macmillan Epstein, B. 2006. “Review of Ruth Millikan, Language: A biological model, ” Notre Dame Philosophica Philosophicall Reviews Reviews 5 Estrada-Gon Estrada-González, zález, L. 2013. “On two two argum argumen ents ts again against st reali realist st dial dialet ethei heism sm”, unpublished manuscript. Etchemendy, Etchemendy, J. 1988. “Models, Models, semanti semantics cs and logical logical truth truth”, Linguistics Linguistics and Philosophy 11, 91–106 1990. The concept of logical consequence . Cambridge, MA: Harvard University Press. Feferman, Feferman, S. 1984 . “Toward useful type-free theories I ”, Journal of Symbolic Logic 49, 75–111 1996. “Gödel’s program for new axioms. Why, where, how and what? ”, in P. Hájek (ed.) Gödel 96 ; Lecture Notes in Logic , 6 . Cambridge University Press, ’
’
3–22 2006. “ What kind of logic is ‘Independence Independence Friendly Friendly ’ logic”, in R. E. Auxier
and L. E. Hahn (eds.) The philosophy of Jaako Hintikka; Library of living vol. 30. Chicago: Open Court, 453–469 philosophers vol. 2009. “Conceptions of the continuum”, Intellectica 51, 169–189 Feferman and 2010. “On the strength of some semi-constructive theories ”, in S. Feferman W. Sieg (eds.) Proof, categories and computation: Essays in honor of Grigori Publications, 109–129; reprinted in U. Berger, et al. Mints . London: College Publications, (eds.) Logic, Frankfurt: rt: Ontos Ontos Verlag Verlag,, 2012, Logic, Construction, Construction, Computation Computation. Frankfu 201–225 2011. Is the
Continuum Hypothesis a de nite nite mathematical problem? lecture text, Harvard University, Dept. of Philosophy, EFI project. http://math. stanford.edu/ feferman/papers/IsCHdenite.pdf 2013 . “Categor Categoricit icityy of open-end open-ended ed axiom axiom system systemss”, lect lectur uree slid slides es for for the the conf confer eren ence ce,, “Intuition and Reason”, in honor of Charles Parsons, Tel-Aviv, Dec. 2, 2013; http://math.stanford.edu/ feferman/papers/CategoricityOpenEnded.pdf (to appear). “ Which quanti quantiers are logical logical?? A combined combined semant semantical ical and inferenti inferential al criterion”, http://math.stanford.edu/ feferman/papers/WhichQsLogical(text). pdf. “
” ”
~
~
~
References
253
Feferman, S. and Strahm, T. 2000. “The unfolding of non-nitist arithmetic ”, Annals of Pure and Applied Logic 104 , 75–96 2010. “The unfolding of nitist arithmetic”, Review of Symbolic Logic 3, 665–689 Feynman, Richard P. 1985. QED: the strange theory of light and matter . Princeton University Press Field, H. 2009. “Pluralism in logic”, The Review of Symbolic Logic 2(2), 342–359 Folina, J. 1994 . “Poincaré on objectivity ”, Philosophia Mathematica 2(3), 202–227 Føllesdal, D. and Bell, D. 1994 , “Objects and concepts”, in Proceedings of the Aristotelian Society, Supplementary Volumes , 68, 131–166 Franks, C. 2010. “ Cut as consequence”, History and Philosophy of Logic 31(4 ), 349–79 2013. “Logical completeness, form, and content: an archaeology ”, in J. Kennedy (ed.) Interpreting Gödel: Critical essays . New York: Cambridge University Press 2014 . “David Hilbert’s contributions to logical theory ”, in A. P. Malpass (ed.) The history of logic . New York: Continuum Frege, G. 1918, “Der Gedanke: Eine Logische Untersuchung ”, in Beiträge zur Philosophie des Deutschen Idealismus i . 58–77 1976. Wissenschaftlicher Briefwechsel , edited by G. Gabriel, H. Hermes, F. Kambartel, and C. Thiel. Hamburg: Felix Meiner 1980. Philosophical and mathematical correspondence . Oxford: Basil Blackwell Fumerton, R. 2010, “Foundationalist theories of epistemic justi cation”, The Stanford encyclopedia of philosophy , Edward N. Zalta (ed.), http://plato. stanford.edu/archives/sum2010/entries/justep-foundational/ Gentzen, G. 1934 –35. “Untersuchungen über das logische Schliessen”. Gentzen’s doctoral thesis at the University of Göttingen, translated as “Investigations into logical deduction” in (Gentzen 1969): 68–131 1936. “Die Widerspruchsfreiheit der reinen Zahlentheorie”, Mathematische Annalen 112, translated as “The consistency of elementary number theory ” in (Gentzen 1969): 132–213 1938. “Neue Fassung des Widerspruchefreiheitsbeweises für die reine Zahlentheorie”, Forschungen zur logik und zur Grundlegung der exackten Wissenschaften, New Series 4 , translated as “New version of the consistency proof for elementary number theory ”, in (Gentzen 1969): 252–86 1943. “Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der trans niten Induktion in der reinen Zahlentheorie ”. Gentzen’s Habilitationschrift at the University of Göttingen, translated as “Provability and nonprovability of restricted transnite induction in elementary number theory ”, in (Gentzen 1969): 287–308 1969. The collected papers of Gerhard Gentzen , M. E. Szabo (ed.). Amsterdam: NorthHolland George, R. 2003. “Bolzano and the problem of psychologism” in D. Fisette (ed.) Husserl s logical investigations reconsidered . Dordrecht: Kluwer, 95–108 1986. “Bolzano’s concept of consequence”, Journal of Philosophy 83, 558–563 Gödel, K. 1931. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I ”, Monatshefte für Mathematik und Physik 38, ’
173–98
254 1932,
References
Zum intuitionistischen Aussagenkalkül”, Anzeiger der Akademie der Wissenschaftischen in Wien 69, 65–66 1933, “Zur intuitionistischen Arithmetik und Zahlentheorie ”, Ergebnisse eines mathematischen Kolloquiums 4 , 34 –38 1958. “Über eine bisher noch nicht benützte Erweiterung des niten Standpunktes”, Dialectica 12, 280–287; reprinted with facing English translation in Gödel (1990), pp. 240–251 1972. “On an extension of nitary mathematics which has not yet been used ”, in Gödel 1990, 271–280 1990. Collected works, vol. i i : Publications 1938 –1 974 (S. Feferman, et al., eds.), Oxford University Press Gómez-Torrente, M. 1999. “Logical truth and Tarskian logical truth ”, Synthese 117: 375–408 2009. “Rereading Tarski on logical consequence ”, The Review of Symbolic Logic 2(2), 249–297 Haddock, G. 2010. “Platonism, phenomenology, and interderivability ”, in Hartimo 2010a, 23–44 Hales, Alexander of, 1951–1957. Glossae in quatuor libros Sententiarum Magistri Petri Lombardi . Quaracchi: Florence Harrop, R. 1960. “Concerning formulas of the types A ! B _ C, A ! (Ex)B(x) in intuitionistic formal systems ”, Journal of Symbolic Logic 25, 27–32 Hartimo, M. 2010a. Phenomenology and mathematics . Dordrecht: Springer 2010b. “Development of mathematics ”, in Hartimo 2010a, 107–121 Hateld, G. 2003. “Getting objects for free (or not) ”, reprinted in his Perception and Cognition. Oxford University Press, 2009 212–255 Hegel, G. 1995. Lectures on the history of philosophy , vol. ii , translated by E. S. Haldane and F. H. Simson. Lincoln, NE and London: University of Nebraska Press Hellman, G. 2006. “Mathematical pluralism: the case of smooth in nitesimal analysis”, Journal of Philosophical Logic 35, 621–651 Henkin, L. 1949. “The completeness of the rst-order functional calculus”, Journal of Symbolic Logic 14 , 159–166 Hilbert, D. 1899. Grundlagen der Geometrie, Leipzig, Teubner ; Foundations of geometry , translated by E. Townsend. La Salle, IL: Open Court, 1959 1922. “Neubergründung der Mathematik. Erste Mitteilung ”, translated by William Ewald as “The new grounding of mathematics: rst report”, in P. Mancosu (ed.) From Brouwer to Hilbert: the debate on the foundations of mathematics in the 1920’ s . New York: Oxford, 1998, 198–214 1983. “On the innite”, in P. Benacerraf and H. Putnam (eds.) Philosophy of mathematics: Selected readings, second edition. Cambridge University Press, “
183–201 Hintikka, J. 1996. The
principles of mathematics revisited (with an appendix by
G. Sandu). Cambridge University Press 2010. “How can a phenomenologist have a philosophy of mathematics? ”, in Hartimo 2010a, 91–104
References
255
Horwich, P. 1998. Truth , second edition. Oxford University Press Hurley, P. J. 2008. A concise introduction to logic , eighth edition. Belmont, CA: Wadsworth/Thomson Learning Hurley, S. and Chater, N. 2005a. Perspectives on imitation: From neuroscience to social science, Volume i : Mechanisms of imitation and imitation in animals . Cambridge, MA: The MIT Press 2005b. Perspectives on imitation: From neuroscience to social science, Volume ii : Imitation, human development, and culture . Cambridge, MA: The MIT Press Husserl, E. 1964 . The idea of phenomenology , translated by W. P. Alston and G. Nakhnikian. The Hague: Martinus Nijho 1970. Logical investigations VII , translated by J. N Findlay. New York: Humanities Press 1981. Husserl shorter works , edited by P. McCormick and F. A. Elliston. University of Notre Dame Press 1983. Edmund Husserl: Ideas pertaining to a pure phenomenology and to a phenomenological philosophy ( rst book), translated by F. Kersten. Boston, MA: Nijho Isaacson, D. 2011. “The reality of mathematics and the case of set theory ”, in Z. Novak and A. Simonyi (eds.) Truth, reference and realism. Budapest: Central European University Press, 1–76 Jenkins, C. S. 2005. “Realism and independence”, American Philosophical Quarterly , 42(3) John of Salisbury 2009. The Metalogicon of John of Salisbury: A twelfth-century defense of the verbal and logical arts of the trivium, translated by D. D. McGarry, First Paul Dry Books, edition, 2009; originally published Berkeley, CA: University of California Press, 1955 Kahneman, D. 2011. Thinking, fast and slow . New York: Farrar, Straus and Giroux Kanamori, A. 2012. “Set theory from Cantor to Cohen ”, in D. M. Gabbay, A. Kanamori, and J. Woods (eds.) Handbook of the history of logic, vol. vi : sets and extensions in the twentieth century . Amsterdam: Elsevier Kant, I. 1781. Kritik der reinen Vernunft , second edition 1787(B), in Kants gesammelte Schriften, vol. iii and iv , Royal Prussian Academy of Science Berlin: de Gruyter, 1905 1996. Practical philosophy , translated by Mary J. Gregor, Cambridge University Press Karger, E. 2001. “ Adam Wodeham on the Intentionality of Cognitions”, in D. Perler (ed.) Ancient and Medieval Theories of Intentionality . Leiden, Boston, MA, Köln: Brill, 283–300 (forthcoming). “ Was Adam Wodeham an internalist or an externalist?” in Klima (2014 ) Keisler, H. J. 1970. “Logic with the quanti er ‘there exist uncountably many ’”, Annals of Mathematical Logic 1, 1–93 Kennedy, J. 2012. “Kurt Gödel”, The Stanford encyclopedia of philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL http://plato.stanford.edu/ archives/win2012/entries/goedel/.
256
References
Kilwardby, R. 1516. [Pseudo-] Aegidius Romanus. In libros priorum analeticorum Aristotelis Expositio . Venice 1976. De ortu scientiarum, edited by A. Judy. Oxford University Press for the British Academy 1986. Quaestiones in primum librum Sententiarum, herausgegeben von Johannes Schneider. Munich: Verlag. der Bayerischen Akademie der Wissenschaften 1988. “On the nature of logic”, in N. Kretzmann and E. Stump (eds.) Cambridge translations of medieval philosophical texts Volume i : logic and the philosophy of language . Cambridge University Press, 264 –276 1995. Quaestiones in quatuor libros Sententiarum, Appendix : Tabula ordine alphabeti contexta (Cod. Worcester F 43) Herausgegeben von Gerd Haverling. Munich: Verlag der Bayerischen Akademie der Wissenschaften King, P. 2010. ‘Peter Abelard ’, in The Stanford encyclopedia of philosophy (Winter 2010 Edition), Edward N. Zalta (ed.), URL http://plato.stanford.edu/ archives/win2010/entries/abelard/ Kleene, S. C. 1945. “On the interpretation of intuitionistic number theory ”, Journal of Symbolic Logic 10, 109–124 1971. Introduction to metamathematics . Amsterdam: North-Holland Klima, G. 1999. “Ockham’s semantics and ontology of the categories”, in P. V. Spade, (ed.) The Cambridge Companion to Ockham. Cambridge University Press, 118–142 2001a. “Existence and reference in medieval logic”, in A. Hieke and E. Morscher (eds.) New Essays in Free Logic . Dordrecht: Kluwer Academic Publishers 197–226 2001b. John Buridan’s Summulae de dialectica , translated by Gyula Klima, New Haven, CT and London, Yale University Press 2009. John Buridan, Great Medieval Thinkers, Oxford University Press 2011a. “Two summulae, two ways of doing logic: Peter of Spain ’s ‘realism’ and John Buridan’s ‘nominalism’”, in M. Cameron and J. Marenbon (eds.): Methods and methodologies: Aristotelian logic East and West, 500– 1500. Leiden and Boston, MA: Brill Academic Publishers, 109–126 2011b. “Being ”, Lagerlund, H. Encyclopedia of Medieval Philosophy, Springer: Dordrecht, 100–159 2012. “Being ”, in Marenbon, J. 2012, 403–420 2013a. “The medieval problem of universals”, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL http://plato. stanford.edu/archives/fall2013/entries/universals-medieval/ 2013b. “Being and cognition”, in D.D. Novotn ý and L. Novák (eds.) Neo- Aristotelian perspectives in metaphysics . To Be Published February 1st 2014 by Routledge: New York, 104 –115 2014 (ed.) Intentionality, cognition and mental representation in medieval philosophy . New York: Fordham University Press Klima, G. and Hall, A. 2011. (eds.) The demonic temptations of medieval nominalism, Proceedings of the Society for Medieval Logic and Metaphysics, Vol. 9, Cambridge Scholars Publishing: Newcastle upon Tyne Kripke, S. A. 1982. Wittgenstein on rules and private language . Cambridge, MA: Harvard University Press
References
257
Ladyman, J. and Ross, D. (with D. Spurrett and J. Collier). 2007. Every thing must go , Oxford University Press Lapointe, S. 2011. Bolzano s theoretical philosophy . Houndsmill: Palgrave 2014 . “Bolzano, Quine and logical truth ” in G. Harman, and E. Lepore (eds.) Blackwell companion to Quine . London: Blackwell Lepore, E. and Ludwig, K. 2006. “Ontology in the theory of meaning ”, International Journal of Philosophical Studies 14 , 325–335 Levinas, E. 1998. Discovering existence with Husserl . Illinois: Northwestern University Press Lewis, D. 1969. Convention: A philosophical study . Cambridge University Press 1991. Parts of classes . Oxford University Press 2005. Quotation, in G. Priest, Jc Beall, and B. Armour-Garb (eds.) The law of contradiction: New essays . New York: Oxford University Press, 1 Lindström, P. 1966. “First order predicate logic with generalized quanti ers”, Theoria 32, 186–195 1969. “On extensions of elementary logic ”, Theoria 35, 1–11 MacFarlane, J. 2005. “Making sense of relative truth”, Proceedings of the Aristotelian Society 105, 321–339 2009. “Nonindexical contextualism”, Synthese 166, 231–250 2014 . Assessment sensitivity: Relative truth and its applications . Oxford University Press Maddy, P. 2007. Second philosophy . Oxford University Press 2011. Defending the axioms . Oxford University Press (to appear) The logical must . Oxford University Press Marenbon, J. 2012. (ed.) The Oxford handbook of medieval philosophy . Oxford University Press Mares, E. D. 2004 . “Semantic dialetheism”, in G. Priest, Jc Beall, and B. Armour-Garb (eds.) The law of non-contradiction. Oxford: Clarendon Press, 264 –275 2007. Relevant logic: A philosophical interpretation. Cambridge University Press Martin, C. J. 2001. “Obligations and liars”, in M. Yrjönsuur (ed.), Medieval formal logic: Obligations, insolubles and consequences . Springer, 63–94 2010. “The development of logic in the twelfth century ”, in Pasnau 2010, ’
129–145 2012. “Logical consequence”, in Marenbon 2012, 289–311 Martin, D. A. 2001. “Multiple universes of sets and indeterminate truth values ”, Topoi 20, 5–16 Mathias, A. R. D. 2001. “The strength of Mac Lane set theory ”, Annals of Pure and Applied Logic 110, 107–234 McDowell, J. 1994 . Mind and world . Cambridge, MA: Harvard University Press Meillassoux, Q. 2011. “Contingency & the absolutization of the one ”, A lecture
delivered at the Sorbonne for a symposium. Source: http://speculativeheresy. wordpress.com/2011/03/24 . Meyer, Robert K., Routley, R. and Dunn, J. Michael. 1979. “Curry ’s paradox ”, Analysis 39, 124 –128
258
References
Millikan, R. G. 1984 . Language, thought, and other biological categories . Cambridge, MA: The MIT Press 1998. “Language conventions made simple”, The Journal of Philosophy 95(4 ), 61–180. Reprinted in Language: A biological model ( 2005) Oxford University Press, 1–23 2003. “In defense of public language”, in L. M. Antony and Norbert Hornstein (eds.) Chomsky and his critics . Oxford: Blackwell Publishing, 215–237. Reprinted in Language: A biological model (2005). Oxford University Press, 24 –52 2005. “Meaning, meaning, and meaning ”, in Language: A biological model . Oxford University Press, 53–76 Monk, R. 1991. Ludwig Wittgenstein: The duty of genius . Penguin Morris, M. 2005. “Realism beyond correspondence”, in Beebee and Dodd 2005, 49–66 Mortensen, C. 1995. Inconsistent mathematics , Dordrecht: Kluwer Academic
Publishers 2010. Inconsistent geometry, Studies in logic 27 . London: College Publications Mühlhölzer, F. 2001. “ Wittgenstein and the regular heptagon”. Grazer Philosophische Studien: Internationale Zeitschrift für Analytische Philosophie , 62, 215–247 Mulligan, K., Simons, P., and Smith, B. 1984 . “Truth-makers”, Philosophy and Phenomenological Research 44 , 287–321 Myhill, J. 1973. “Some properties of Intuitionistic Zermelo-Fraenkel set theory ”, Proceedings of the 1971 Cambridge summer school in mathematical logic (Lecture Notes in Mathematics 337): 206–231 Natalis, H. 2008. A treatise of Master Hervaeus Natalis: On second intentions . Vol. i – An English translation, and Vol. i i – A Latin edition, edited and translated, by John P. Doyle. Milwaukee: Marquette University Neta, R., and Pritchard, D. 2007. “McDowell and the new evil genius”, Philosophy and Phenomenological Research 74 (2), 381–396 Novaes, C. D. 2012. “Medieval theories of consequence”, The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta (ed.), URL http:// plato.stanford.edu/archives/sum2012/entries/consequence-medieval/ Ockham, W. 1974 . Summa logicae , opera philosophica i , edited by G. Gál and S. Brown, St. Bonaventure, NY: Franciscan Institute, St. Bonaventure University Panaccio, C. 2004 . Ockham on concepts . Aldershot: Ashgate Pasnau, R. 2010. (ed.) C. Van Dyke (assoc. ed.) The Cambridge history of medieval philosophy ( 2 vols.), Cambridge University Press Pirandello, L. 1990. Novelle per un anno , vol. i ii , edited by M. Costanzo, Milano, Mondadori. Plato. 1997. “Phaedrus”, translated by A. Nehamas and P. Woodru . In J. M. Cooper (ed.) Plato: Complete works . Indianapolis, IN: Hackett Popper, K. 1968. “Epistemology without a knowing subject”, in Logic, Methodology and Philosophy of Science iii . Elsevier, 333–373
References
259
Posy, C. 2005. “Intuitionism and philosophy ”, in Shapiro, S. (ed) The Oxford handbook of philosophy of mathematics and logic . Oxford University Press, pp. 318–356 Prawitz, D. 1965. Natural deduction: A proof-theoretical study ; second edition (2006), Mineola, NY: Dover Publications Priest, G. 1979. “The logic of paradox ”, Journal of Philosophical Logic 8, 219–241 1980. “Sense, entailment and modus ponens”, Journal of Philosophical Logic 9, 415–435 1987. In contradiction. The Hague: Martinus Nijho 1989. “Classical logic Aufgehoben”, in G. Priest, R. Routley, and J. Norman
(eds.) Paraconsistent logic: Essays on the inconsistent . Munich: Philosophia Verlag, ch. 4 2006a. Doubt truth to be a liar . Oxford University Press. 2006b. In contradiction, second edition. Oxford University Press 2007. “Paraconsistency and dialetheism”, in D. Gabbay and J. Woods (eds.) Handbook of the history of logic . Amsterdam: North-Holland, 194 –204 2009. “Beyond the limits of knowledge ”, in J. Salerno (ed.) New essays on the knowability paradox , Oxford University Press, 93–105 2013. “Mathematical Pluralism ”, Logic Journal of IGPL 21, 4 –20 (to appear). “Logical disputes and the a priori” Př íhonsk ý, F. 1850. Neuer Anti-Kant . Bautzen: Weller Prucnal, T. 1976. “Structural completeness of Medvedev ’s propositional calculus”, Reports on Mathematical Logic 6, 103–105 Putnam, H. 2000. “Meaning and reference”, in E. D. Klemke and Heimir Geirsson (eds.) Contemporary analytic and linguistic philosophies , second edition. Amherst, NY: Prometheus Books 2003. “McDowell’s mind and world ”, in Reading McDowell . London: Routledge, 174 –191 Quine, W. V. O. 1936a. “Truth by convention. In O. H. Lee (ed.), Philosophical essays for A. N. Whitehead . New York: Longmans, 90–124 1936b. “Truth by convention ”, in The ways of paradox , revised and enlarged edition (1976) Harvard University Press, 77–106 1953. Two dogmas of empiricism from a logical point of view . Cambridge, MA: Harvard University Press 1960. Word and object . Cambridge, MA: The MIT Press 1970a. Philosophy of logic . Prentice Hall 1970b. “Methodological reections on current linguistic theory ”, Synthese 21, 386–398 1976. The ways of paradox, and other essays (revised and enlarged edition).
Cambridge, MA: Harvard University Press 1986. Philosophy of logic , second edition. Cambridge, MA: Harvard University Press 1991. “Immanence and validity ”, Dialectica 45, reprinted with emendations in W. V. O. Quine, Selected logic papers . Cambridge, MA: Harvard, 1995, 242–250
260
References
Rathjen, M. 2014 . “Indeniteness in semiintuitionistic set theories: On a conjecture of Feferman”, arXiv: 1405448 Rayo, A. and Uzquiano, G. (eds.) 2006. Absolute generality . Oxford: Clarendon Press Read, S. 1994 . Thinking about logic: An introduction to the philosophy of logic . Oxford University Press 2010. “Inferences”, in Pasnau 2010 173–184 . 2011. “Medieval theories: Properties of terms ”, The Stanford encyclopedia of philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), URL = http:// plato.stanford.edu/archives/spr 2011/entries/medieval-terms/ Rescher, N. and Manor, R. 1970. “On inference from inconsistent premises”, Theory and Decision, 1(2), 179–217 Resnik, M. 1988. “Second-order logic still wild ”, Journal of Philosophy , 85, 75 –87 1996. “Ought there to be but one logic? ”, in B. J. Copeland (ed.) Logic
and reality: Essays on the legacy of Arthur Prior . Oxford University Press, 489–517 1997. Mathematics as a science of patterns . Oxford University Press Resnik, Michael, 2000. “ Against logical realism”, History and Philosophy of Logic 20, 181–194 Restall, G. 2002. “Carnap’s tolerance, meaning, and logical pluralism”, Journal of Philosophy 99, 426–443 Rush, P. 2005. PhD thesis: The philosophy of mathematics and the independent
other ”, School of History, Philosophy, Religion and Classics, The University of Queensland. http://espace.library.uq.edu.au 2012. “Logic or reason? ”, Logic and Logical Philosophy 21, 127–163 Rusnock, P. and George, R. 2004 . “Bolzano as logician”, in D. M. Gabbay and J. Woods (eds.) The rise of modern logic: from Leibniz to Frege. Handbook of the history of logic , vol. iii . Springer, 177–205 Russell, G. 2008. “One true logic?” Journal of Philosophical Logic 37, 593–611 Rybakov, V. 1997. Admissibility of logical inference rules . Amsterdam: Elsevier Sainsbury, R. M. 2009. Paradoxes , third edition Cambridge University Press Schmidt, R. W. 1966. The domain of logic according to Saint Thomas Aquinas . The Hague: Martinus Nijho Schmutz, J. 2012. “Medieval philosophy after the Middle Ages ”, in Marenbon 2012, 245–270 Schotch, P., Brown, B. and Jennings, R. (eds.) 2009. On preserving: Essays on preservationism and paraconsistent Logic . University of Toronto Press Schotch, P. and Jennings, R. 1980. “Inference and necessity ”, Journal of Philosophical Logic 9, 327–340 Sellars, W. 1956. “Empiricism and the philosophy of mind ”, in H. Feigl and M. Scriven (eds.) Minnesota Studies in the Philosophy of Science , Vol i , Minneapolis: University of Minnesota Press, 253–329 1962. “Philosophy and the scienti c image of man ”, reprinted in his Science, perception and reality . London: Routledge & Kegan Paul, 1–40 “
References
261
Shalkowski, S. 2004 . “Logic and absolute necessity ”, The Journal of Philosophy 101(2), 55–82 Shapiro, S. 1991. Foundations without foundationalism: A case for second-order logic . Oxford University Press 1997. Philosophy of mathematics. Structure and ontology . Oxford University Press 2000. “The status of logic”, in P. Boghossian and C. Peacocke (eds.) New essays on the a priori . Oxford University Press, 333 –366; reprinted in part as “Quine on Logic”, in T. Childers (ed.) Logica Yearbook 1999 Prague: Czech Academy Publishing House, 11–21 (ed.) 2005. The Oxford handbook of the philosophy of mathematics and logic . Oxford University Press 2007. “The objectivity of mathematics”, Synthese 156, 337–381 2011. “Mathematics and objectivity ”, in J. Polkinghorne (ed.) Meaning in mathematics . Oxford University Press, 97–108 2012. “Objectivity, explanation, and cognitive shortfall”, in A. Coliva (ed.) Mind, meaning, and knowledge: themes from the philosophy of Crispin Wright . Oxford University Press, 211–237 2014 . Varieties of logic , forthcoming Sher, G. 2001. “The formal-structural view of logical consequence”, The Philosophical Review 110(2), 241–261 2011. “Is logic in the mind or in the world? ” Synthese 181, 353–365 Shim, M. 2005 “The duality of non-conceptual content in Husserl’s phenomenology of perception”, Phenomenology and the Cognitive Sciences 4 , 209–229 Sider, T. 2011. Writing the book of the world . Oxford University Press Siebel, M. 2002. “Bolzano’s concept of consequence”, The Monist 85(4 ), 580–599 Simpson, S. G., and Yokoyama, K. 2012. “Reverse mathematics and Peano categoricity ”, Annals of Pure and Applied Logic 164 , 229–243 Slaney, J. 2004 . “Relevant logic and paraconsistency ”, in L. Bertossi, A. Hunter, and T. Schaub (eds.) Inconsistency tolerance . Heidelberg: Springer-Verlag, 270–293
Smith, Nicholas J. J. 2012. Logic: The laws of truth . Princeton University Press Sorensen, R. 2012. ‘Vagueness’, The Stanford encyclopedia of philosophy (Summer 2012 Edition), Edward N. Zalta (ed.), URL http://plato.stanford.edu/ archives/sum2012/entries/vagueness/ Spade, P. V. 1994 . (trans.) Five texts on the mediaeval problem of universals: Porphyry, Boethius, Abelard, Duns Scotus, Ockham. Indianapolis, Cambridge: Hackett Spelke, E. 2000. “Core knowledge”, American Psychologist 55, 1233–1243 Spelke, E., Grant, G. and Van de Walle, G. 1995. “The development of object perception”, in S. Kosslyn and D. Osherson (eds.) Visual Cognition. Cambridge, MA: MIT Press, 297–330 Steiner, M. 1975. Mathematical knowledge . Ithaca and London: Cornell University Press 2009. “Empirical regularities in Wittgenstein ’s philosophy of mathematics ”, Philosophia Mathematica 17, 1–34
262
References
Swoyer, C. 2003. “Relativism”, Stanford internet encyclopedia of philosophy , http:// plato.stanford.edu/entries/relativism Tahko, T. E. 2009. “The law of non-contradiction as a metaphysical principle ”, The Australasian Journal of Logic 7, 32–47. URL: www.philosophy.unimelb. edu.au/ajl/2009/ 2012. “Boundaries in reality ”, Ratio 25 ( 4 ), 405–24 Tarski, A. 1944 . “The semantic conception of truth”, Philosophy and Phenomenological Research 4 , 13–47 Teiszen, R. 2010. “Mathematical realism and transcendental phenomenological idealism”, in Hartimo 2010a, 1–22 Terwijn, S. A. 2004 . “Intuitionistic logic and computation”, Vriendenboek ofwel Liber Amicorum ter gelegenheid van het afscheid van Dick de Jongh , Institute for Logic, Language, and Computation. Amsterdam Tokarz, M. 1973. “Connections between some notions of completeness of structural propositional calculi”, Studia Logica 32(1), 77–91 Troelstra, A. S., and van Dalen, D. 1988. Constructivism in mathematics. An introduction, Vol. i . Amsterdam: North-Holland Väänänen, J. 2001. “Second-order logic and foundations of mathematics”, Bulletin of Symbolic Logic 7, 504 –520 van Inwagen, P. 1981. “ Why I don’t understand substitutional quantication”, in Ontology, identity, and modality: Essays in metaphysics (2001). Cambridge University Press, 32–6 Visser, A. 1999. “Rules and arithmetics”, Notre Dame Journal of Formal Logic 40, 116–140
Waismann, F. 1945. “Veriability ”, Proceedings of the Aristotelian Society , Supplementary Volume 19, 119–150; reprinted in A. Flew (ed.) Logic and language . Oxford: Basil Blackwell, 1968, 117–144 Walker, Ralph C.S. 1999. “Induction and transcendental argument”, in R. Stern (ed.) Transcendental arguments . Oxford University Press, 13–29 Wason, P. C., and Johnson-Laird, P. 1972. Psychology of reasoning: Structure and content . Cambridge, MA: Harvard University Press Weber, Z. 2009. “Inconsistent mathematics”, Internet encyclopedia of philosophy , www.iep.utm.edu/math-inc/ Wells, H. G. 1908. First and last things . Teddington Echo, 2011 Willard, D. 1981. Introduction to “ A reply to a critic”, in Husserl 1981, 148–151 Williamson, T. 1994 . Vagueness . London: Routledge Wilson, M. 2006. Wandering signi cance . Oxford University Press Wittgenstein, L. 1953. Philosophical investigations , second edition, translated by G. E. M. Anscombe. Oxford: Blackwell. 2001 1974 . On certainty , edited by G.E.M. Anscombe and G.H. von Wright. Translated by D. Paul, and G.E.M. Anscombe. Oxford: Blackwell 1976. Wittgenstein s lectures on the foundations of mathematics , Cambridge, 1939. Ithaca: Cornell University Press 1978. Remarks on the foundations of mathematics (revised edition). Cambridge, MA: MIT Press ’
References
263
2009. Philosophical investigations , translated by P. M. S. Hacker, G. E. M.
Anscombe, and J. Schulte, fourth revised edition, edited by P. M. S. Hacker and J. Schulte. Wiley-Blackwell Wittgenstein, L. and Rhees, R. 1974 . Philosophical grammar . Oxford: Basil Blackwell Wright, C. 1992. Truth and objectivity . Cambridge, MA: Harvard University Press 2008. “Relativism about truth itself: haphazard thoughts about the very idea ”, in M. García-Carpintero and M. Kölbel (eds.) Relative truth . Oxford University Press Yang, F. 2013. “Expressing second-order sentences in intuitionistic dependence logic”, Studia Logica 101(2), 323–342 Zucker, J. I. 1978. “The adequacy problem for classical logic”, Journal of Philosophical Logic 7, 517–535 Zucker, J. I., and Tragesser, R. S. 1978. “The adequacy problem for inferential logic”, Journal of Philosophical Logic 7, 501–516
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
264
Index Field, H, 211 , 244 –245 rst-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 Hateld, 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
265
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
266
Index
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 denition 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
