THE METAPHYSICS OF LOGIC

Featuring fourteen new essays from an international team ofrenowned contributors, this volume explores the key issues, debates,and questions in the metaphysics of logic. The book is structured inthree parts, looking first at the main positions in the nature of logic,such as realism, pluralism, relativism, objectivity, nihilism, conceptu-alism, and conventionalism, then focusing on historical topics such asthe medieval Aristotelian view of logic, the problem of universals, andBolzano’s logical realism. The final section tackles specific issues suchas glutty theories, contradiction, the metaphysical conception oflogical truth, and the possible revision of logic. The volume willprovide readers with a rich and wide-ranging survey, a valuable digestof the many views in this area, and a long overdue investigation oflogic’s relationship to us and the world. It will be of interest to a widerange of scholars and students of philosophy, logic, and mathematics.

penelope rush is Honorary Associate with the School ofPhilosophy and Online Lecturer for Student Learning at the Univer-sity of Tasmania. She has published articles in journals includingLogic and Logical Philosophy, Review of Symbolic Logic, South AfricanJournal of Philosophy, Studia Philosophica Estonica, and Logique etAnalyse. She is also the author of The Paradoxes of Mathematical,Logical, and Scientific Realism (forthcoming).

THE METAPHYSICS OF LOGIC

edited by

PENELOPE RUSHUniversity of Tasmania

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Library of Congress Cataloging in Publication dataThe metaphysics of logic / edited by Penelope Rush, University of Tasmania.

pages cmIncludes bibliographical references and index.

isbn 978-1-107-03964-3 (Hardback)1. Logic. 2. Metaphysics. I. Rush, Penelope , 1972– editor.

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isbn 978-1-107-03964-3 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this publication,and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

With thanks to Graham Priest forunstinting encouragement,

and to Annwen and Callum – never give up.

Contents

List of contributors page ix

Introduction 1Penelope Rush

part i the main positions 11

1 Logical realism 13Penelope Rush

2 A defense of logical conventionalism 32Jody Azzouni

3 Pluralism, relativism, and objectivity 49Stewart Shapiro

4 Logic, mathematics, and conceptual structuralism 72Solomon Feferman

5 A Second Philosophy of logic 93Penelope Maddy

6 Logical nihilism 109Curtis Franks

7 Wittgenstein and the covert Platonism of mathematical logic 128Mark Steiner

part ii history and authors 145

8 Logic and its objects: a medieval Aristotelian view 147Paul Thom

vii

9 The problem of universals and the subject matter of logic 160Gyula Klima

10 Logics and worlds 178Ermanno Bencivenga

11 Bolzano’s logical realism 189Sandra Lapointe

part iii specific issues 209

12 Revising logic 211Graham Priest

13 Glutty theories and the logic of antinomies 224Jc Beall, Michael Hughes, and Ross Vandegrift

14 The metaphysical interpretation of logical truth 233Tuomas E. Tahko

References 249Index 264

viii Contents

Contributors

jody azzouni, Professor, Department of Philosophy, Tufts University.

jc beall, Professor of Philosophy and Director of the UCONN LogicGroup, University of Connecticut, and Professorial Fellow at theNorthern Institute of Philosophy at the University of Aberdeen.

ermanno bencivenga, Professor of Philosophy and the Humanities,University of California, Irvine.

solomon feferman, Professor of Mathematics and Philosophy,Emeritus, and Patrick Suppes Professor of Humanities and Sciences,Emeritus, Stanford University.

curtis franks, Associate Professor, Department of Philosophy,University of Notre Dame.

michael hughes, Department of Philosophy and UCONN LogicGroup, University of Connecticut.

gyula klima, Professor, Department of Philosophy, Fordham University,New York.

sandra lapointe, Associate Professor, Department of Philosophy,McMaster University.

penelope maddy, Distinguished Professor, Department of Logic andPhilosophy of Science, University of California, Irvine.

graham priest, Distinguished Professor, Graduate Center, CUNY,and Boyce Gibson Professor Emeritus, University of Melbourne.

penelope rush, Honorary Associate, School of Philosophy, Universityof Tasmania.

ix

stewart shapiro, Professor, Department of Philosophy, The OhioState University.

mark steiner, Professor Emeritus of Philosophy, the Hebrew Universityof Jerusalem.

tuomas e. tahko, Finnish Academy Research Fellow, Department ofPhilosophy, History, Culture and Art Studies, University of Helsinki.

paul thom, Honorary Visiting Professor, Department of Philosophy,The University of Sydney.

ross vandegrift, Department of Philosophy and UCONN LogicGroup, University of Connecticut.

x List of contributors

IntroductionPenelope Rush

This book is a collection of new essays around the broad central theme ofthe nature of logic, or the question: ‘what is logic?’ It is a book about logicand philosophy equally. What makes it unusual as a book about logic isthat its central focus is on metaphysical rather than epistemological ormethodological concerns.By comparison, the question of the metaphysical status of mathematics

and mathematical objects has a long history. The foci of discussions in thephilosophy of mathematics vary greatly but one typical theme is that ofsituating the question in the context of wider metaphysical questions:comparing the metaphysics of mathematical reality with the metaphysicsof physical reality, for example. This theme includes investigations into: onexactly which particulars the two compare; how (if ) they relate to oneanother; and whether and how we can know anything about either ofthem. Other typical discussions in the field focus on what mathematicalformalisms mean; what they are about; where and why they apply; andwhether or not there is an independent mathematical realm. A variety ofpossible positions regarding all of these sorts of questions (and many more)are available for consideration in the literature on the philosophy ofmathematics, along with examinations of the specific problems and attrac-tions 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 themetaphysical status of logic and logical objects analogous to those thathave been asked about the metaphysical status of mathematical objects(or reality). Logic, as a formal endeavour has recently extended farbeyond Frege’s initial vision, describing an apparently ever more com-plex realm of interconnected formal structures. In this sense, it mayseem that logic is becoming more and more like mathematics. On theother hand, there are (also apparently ever more) sophisticated logics

1

describing empirical human structures: everything from natural lan-guage and reason, to knowledge and belief.

That there are metaphysical problems (and what they might be) for theformer structures analogous to those in the philosophy of mathematics isrelatively easily grasped. But there are also a multitude of metaphysicalquestions we can ask regarding the status of logics of natural language andthought. And, at the intersection of these (where one and the same logicalstructure is apparently both formal and mathematical as well as applicableto natural language and human reason), the number and complexity ofmetaphysical problems expands far beyond the thus far relatively small setof issues already broached in the philosophy of logic.

As just one example of the sorts of problems deserving a great deal moreattention, consider the relationship between mathematics and logic.Questions we might ask here include: whether mathematics and logicdescribe the same or similar in-kind realities and relatedly, whether thereis a line one can definitively draw between where mathematics stops andlogic 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 complexthan this?

Another central problem for the metaphysics of logic is that of pinningdown exactly what it is that logic is supposed to range over. Logic has beenconceived of in a wide variety of ways: e.g. as an abstraction of naturallanguage; as the laws of thought; and as normative for human reason. But,what is the ‘thought’ whose structure logic describes; how natural is thenatural language from which logic is abstracted?; and to what extent doesthe formal system actually capture the way humans ought to reason?

As touched on above, a key metaphysical issue is how to account for theapparent ‘double role’ – applying to both formal mathematical and naturalreasoning structures – that (at least the main) formal logical systems play.This apparent duality lines up along the two central, indeed canonicalapplications of logic: to mathematics and to human reason, (and/or humanthought, and/or human language). In many ways, the first applicationsuggests that logic may be objective – or at least as objective as mathemat-ics, in the sense that, as Stewart Shapiro puts it (in this volume) we mightsay something “is objective if it is part of the fabric of reality”. This in turnmight suggest an apparent human-independence of logic. The secondapplication, though, might suggest a certain subjectivity or inter-subjectivity; and so in turn an apparent human-dependence of logic,insofar as a logic of reason may appear dependent on actual humanthought or concepts in some essential way.

2 Penelope Rush

Both the apparent objectivity and the apparent subjectivity of logic needto be accounted for, but there are numerous stances one might take withinthis dichotomy, including a conception of objectivity that is nonethelesshuman-dependent. In Chapter 4, Solomon Feferman reviews one suchexample in his non-realist philosophy of mathematics, wherein “theobjects of mathematics exist only as mental conceptions [and] . . .the objectivity of mathematics lies in its stability and coherence underrepeated communication”. Others of the various positions one might takeup within this broad-brush conceptual field are admirably explored in bothStewart Shapiro’s and Graham Priest’s chapters, though from quite differ-ent stand points: Shapiro explores the nuances and possibilities in concep-tions of objectivity, relativity, and pluralism for logic, whereas Priest looksat these issues through the specific lens afforded by the question whether ornot 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; conceptualstructuralism; nihilism; realism; and anti-(or non-)realism, as well aspositions attempting to steer a path between the latter two. The followingessays cover all these positions and more, as defended by some of theforemost thinkers in the field.The first part of the book covers some of the main philosophical

positions one might adopt when considering the metaphysical nature oflogic. This section covers everything from an extreme realism wherein logicmay be supposed to be completely independent of humanity, to variousaccounts and various degrees in which logic is supposed to be in some wayhuman-dependent (e.g. conceptualism and conventionalism).In the first chapter I explore the feasibility of the notion that logic is

about a structure or structures existing independently of humans andhuman activity. The (typically realist) notion of independence itselfis scrutinised and the chapter gives some reasons to believe that there isnothing in principle standing in the way of attributing such independenceto logic. So any benefits of such a realism are as much within the reach ofthe 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 mightbe taken to represent the extreme opposite of realism. Azzouni argues thecase for logical conventionalism, the view that logical truths are true byconvention. For Azzouni, logic is a tool which we both impose by conven-tion on our own reasoning practices, and occasionally also to evaluate

Introduction 3

them. But Azzouni shows that although there seems to be a close relation-ship between conventionality and subjectivity, logic’s being conventionaldoes not rule out its also applying to the world.

Stewart Shapiro, in the third chapter, argues the case for logical relativ-ism or pluralism: the view that there is “nothing illegitimate” in structuresinvoking logics other than classical logic. Shapiro defends a particular sortof relativism whereby different mathematical structures “have differentlogics”, giving rise to logical pluralism – conceived of as “[the] view thatdifferent accounts of the subject are equally correct, or equally good, orequally legitimate, or perhaps even (equally) true”.Shapiro’s chapter looks in some depth at the relationship between

mathematics and logic, identified above as a central problem for ourtheme. But in particular, it investigates the extent to which logic canbe thought of as objective, given the foregoing philosophy. He offers athorough, precise, and immensely valuable analysis of the central concepts,and clarifies exactly what is and is not at stake in this particular debate.

In the fourth chapter, Solomon Feferman examines a variety of logicalnon-realism called conceptual structuralism. Feferman shares with Shapiroa focus on the relationship between mathematics and logic, extending thecase for conceptual structuralism in the philosophy of mathematics to logicvia a deliberation on the nature and role of logic in mathematics. He drawsa careful picture of logic as an intermediary between philosophy andmathematics, 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 todefinite conceptions. On this picture, when we speak of truth in aconception, that truth may be partial. Thus classical logic can be concep-tualised as the “logic of definite concepts and totalities”, but may itself bejustified on the basis of a semi-intuitionist logic “that is sensitive todistinctions that one might adopt between what is definite and what isnot”. Feferman shows how allowing that “different judgements may bemade as to what are clear/definite concepts”, affords the conceptualstructuralist a straightforward, sensible and clear understanding of the roleand nature of logic.

Penelope Maddy, in the fifth chapter, offers a determinedly second-philosophical account of the nature of logic, presenting another admirablyclear and sensible account, focusing in this case on the question why logicis true and its inferences reliable. ‘Second Philosophy’ is a close cousin ofnaturalism as well as a form of logical realism and involves persistentlybringing our philosophical theorising back down to earth.

4 Penelope Rush

In Maddy’s words: “The Second Philosopher’s ‘metaphysics naturalized’simply pursues ordinary science”. Thus Maddy investigates the questionfrom this ‘ordinary’ perspective, beginning with a consideration of rudi-mentary logic, and gradually building up (via idealisations) to classicallogic. On this account, logic turns out to be true and reliable in our actual(ordinary, middle-sized) world partly because that actual world shares theformal structure of logic (or at least rudimentary logic). Maddy gives anextensive account of some of the ways we might come to know of thisstructure, presenting recent research in cognitive science that supportsthe notion that we are wired to detect just such a structure. She thenoffers the (tentative) conclusion that classical logic (as opposed to any non-classical logic) is best suited to describe the physical world we live in,despite the fact that classical logic’s idealisations of rudimentary logic arebest described as ‘useful falsifications’.

In the final two chapters of the first part, Curtis Franks questions theassumption underpinning any metaphysics of logic at all: namely thatthere is “a logical subject matter unaffected by shifts in human interestand knowledge”; and Mark Steiner unpicks Wittgenstein’s idea that “Therules 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 logicalargument. Instead we [only] have regularities induced by linguistictraining”. So, Steiner argues, supposing that logic is grounded by anythingother than the regularities that ground rule following (say by some object-ive ‘fact’ according to which its rules are determined), is engaging in a kindof ‘covert Platonism’.Steiner identifies the key difference (for Wittgenstein) between math-

ematics and logic as the areas their respective rules govern: whereas bothmathematical and logical rules govern linguistic practices, (only) math-ematical rules also govern non-linguistic practices. Interestingly, whileSteiner argues that the line between mathematics and logic is thus moresubstantial than many may think, Franks argues that the line betweenmaths and logic is illusory, based on a need to differentiate the patterns ofreasoning we have come to associate with logic from other patterns ofreasoning, which itself is grounded on nothing more than a baselesspsychological or metaphysical preconception.Franks argues that logicians deal not with truth but with the “relation-

ships among phenomena and ideas” – and agrees with Steiner that lookingfor any further ‘ontological ground’ is misconceived (note, though, thatSteiner himself does not commit himself to the views he attributes to

Introduction 5

Wittgenstein. Rather he gives what he takes to be the best arguments inWittgenstein’s favour). As something of a side note, it is interesting tocompare Sandra Lapointe’s discussion of Bolzano’s notion of definition(in Part II) to that which Franks presents on behalf of Socrates. Lapointeargues that, for Bolzano, there is more to a definition than merely fixing itsextension, whereas Franks argues that Socrates was right to prioritise thefixing of an extension first before enquiring after the nature or essence of athing. Steiner’s discussion of the Wittgensteinian distinction betweenexplanation and description is also relevant here. This debate touches onanother important subtheme running throughout the book: the nature androle of intentional and extensional motivations of logical systems; and therelated tension (admirably illustrated by Franks’ discussion of the develop-ment of set theory) between appeals to form/formal considerations andappeals to our intuitions.

Both Steiner’s Wittgenstein and Franks agree that the image of logic as akind of ‘super-physics’ needs to be challenged, even eliminated; but eachtakes a different approach to just how this might be achieved, with Franksarguing for logical nihilism, and Steiner going to pains to show how, forWittgenstein, the rules of logic ought to be conceived as akin to those ofgrammar and as nothing more than this.

The next part of the book gives an historical overview of past investi-gations into the nature of logic as well as giving insights into specificauthors of historical import for our particular theme.

In the first chapter of this section Paul Thom discusses the thoughts ofAristotle and the tradition following him on logic. Thom focuses particu-larly on what sort of thing, metaphysically speaking, the objects of logicmight be. He traces a gradual shift (in Kilwardby’s work) from a concep-tion of logic as about only linguistic phenomena, through a conceptionwherein logic is also understood as also being about reason, to theinclusion of ‘the natures of things’ as a possible foundation of logic.Kilwardby considers a view whereby the principal objects of logic: ‘state-ables’, are not some thing at all (at least not in themselves), insofar as theydo not belong to any of Aristotle’s categories. Kilwardby opposes this viewon the basis of a sophisticated and complex argument to the effect thatthere may be objects of logic that are human dependent but also externalto ourselves, and can be considered both things of and things about natureitself. These insights are clearly relevant to the modern questions we askabout the metaphysics of logic and resonate strongly with the themesexplored in the first part. The range of possibilities considered offer afascinating and fruitful look into the historical precedents of the questions

6 Penelope Rush

about logic still open today: e.g. Thom notes that for Aristotle, the types ofthings 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 thepossible independence and objectivity of logic. The complexity of thatquestion is brought to the fore in Kilwardby’s detailed consideration ofthe various ‘aspects’ under which stateables can be considered, andaccording to which they may be assigned to different categories.Thom’s chapter goes on to offer a framework for understanding later

thinkers and traditions in logic, some of which (e.g. Bolzano in Lapointe’schapter) are also discussed in this part. His concluding section ablydemonstrates that understanding the history of our questions casts usefullight 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 itsapparent groundedness in the physical world. Klima focuses on the scho-lastics, comparing the semantic strategies of realists and nominalistsaround Ockham’s time. One of these was to characterise logic as the studyof ‘second intentions’ – concepts of concepts. Klima points out that whenlogic is conceived of in this way, the core-ontology of real mind-independent entities could in principle have been exactly the same for“realists” as for Ockhamist “nominalists”; therefore, what makes thedifference between them is not so much their ontologies as their differentconceptions of concepts, grounding their different semantics.Klima argues that extreme degrees of ontological and semantic diversity

and uniformity mark out either end of a “range of possible positionsconcerning the relationship between semantics and metaphysics, [from]extreme realism to thoroughgoing nominalism” and points out how theconceptualisation of the sorts of things semantic values might be variesaccording to where a given position sits within this framework. His chapterilluminates the metaphysical requirements of different historicalapproaches to semantics and the way in which the various possible meta-physical commitments we make come about via competing intuitionsregarding diversity: whether we locate diversity in the way things are orin 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 conceptionof logic appears to have lost touch with the relevant ways in which actualhuman reason can go wrong other than by not being valid. Offering aKantian view, Bencivenga suggests we adjust our conception of logic tothat of almost any structure we impose on language and experience, just so

Introduction 7

long as it is a holistic endeavour to uncover how our language acquiresmeaning. In this way almost all of philosophy is logic, but not all of whatwe commonly call logic makes the grade. For Bencivenga, logic shouldfocus on meaning: on the way language constructs our world. From thisperspective, the relationship of logic to reason is just one of many connec-tions between the world we create and the internal structure of any givenlogic. For example, while appeals to reason may motivate logic’s claims, sotoo do appeals to ethos and pathos.

Sandra Lapointe looks at the sorts of motivations and reasons we mighthave for adopting a realist philosophy of logic, pointing out that thesereasons may not themselves be logical and developing a framework withinwhich different instances of logical realism can be compared. Lapointeexamines Bolzano’s philosophy in particular and shows how his realismmay best be thought of as instrumental rather than inherent: adopted inorder to make sense of certain aspects of logic rather than as a result of anydeep metaphysical conviction.

Lapointe’s chapter shows how Bolzano’s works cast light on a wide arrayof issues falling under our theme, from his evocative analogy between thetruths of logic and the spaces of geometry to his critique of Aristotle’scriteria for validity. Lapointe’s discussion of the latter is worth drawingattention to as it deals with the topic mentioned earlier – of the tensionbetween external and intensional; and formal and non-formal motivationsfor logical systems. Lapointe compares the results of Bolzano’s motivationswith those of Aristotle for the definition of logical consequence and in sodoing, identifies some central considerations to help further our under-standing of this topic.

The final part of the book deals with the specific issues of the possiblerevision of logic, the presence of contradiction, and the metaphysicalconception of logical truth.

Graham Priest’s chapter deals with the question of the revisability oflogic and in so doing also offers a useful overview of much of what isdiscussed in earlier sections and indeed throughout this book. Priestoutlines three senses of the term ‘logic’ and asks of each whether it canbe revised, revised rationally, and (if so) how.

In some ways, Priest’s paper dovetails with Shapiro’s discussion of thepossible criteria used to judge the acceptability of a theory, and draws aconclusion similar to that of Shapiro’s ‘liberal Hilbertian’: i.e. “[that]There is no metaphysical, formal, or mathematical hoop that a proposedtheory must jump through. There are only pragmatic criteria of interestand usefulness” – which, for Priest, are judged against the requirements of

8 Penelope Rush

its application(s) and by “the standard criteria of rational theory choice”.And like Shapiro’s, Priest’s chapter is an immensely valuable overview ofthe key concepts informing any metaphysics of logic.In the next chapter, Jc Beall, Michael Hughes, and Ross Vandegrift look

at different repercussions of different attitudes toward “glutty predicates” –predicates which “in virtue of their meaning or the properties theyexpress . . . [are] both true and false”. Their chapter shows how our varioustheories and attitudes about such predicates may motivate different formalsystems. The formal systems in question here are Priest’s well-knownLP and the lesser-known LA advanced by Asenjo and Tamburino. Theupshot of the discussion is that the latter will suit someone metaphysically“commited to all predicates being essentially classical or glutty” andthe former someone for whom “all predicates [are] potentially classicalor glutty”.Thus, Beall et al. draw out some interesting consequences of the

relationships between our intuitions and theories regarding the metaphys-ical, the material, and the formal aspects of logic. They highlight both thepotential ramifications of the role we afford our metaphysical commit-ments and the ramifications of the particular type of commitments theymight be. So while Beall et al. look in particular at a variety of metaphysicaltheories about contradiction, and the impact of these on two formalsystems, their discussion also gives some general pointers to the way inwhich our metaphysical beliefs impact on other central factors in logic:crucially including the creation of the formal systems themselves and theevaluation of their differences.Tuomas Tahko finishes the book by examining a specific realist meta-

physical perspective and suggesting it as another approach we might taketo understand logic, especially to interpret logical truth. His case studyoffers an interpretation of paraconsistency which contrasts nicely with thatoffered in the penultimate chapter. Tahko’s approach is to judge logicallaws according to whether or not they count as genuine ways the actualworld is or could be. From this perspective, he argues, exceptions to thelaw of non-contradiction now appear more as descriptions of features ofour language than of reality. Thus he argues that the realist intuitiongrounding logic in how the world is (or could be) gives us good reasonto preserve the LNC. Tahko’s metaphysical interpretation of logical truthalso offers an interesting perspective on logical pluralism. From Tahko’smetaphysical perspective, pluralism may be understood as about subsets ofpossible worlds representing genuine possible configurations of the actualworld. Tahko’s chapter is a meticulous investigation into the links, both

Introduction 9

already in place and that (from this perspective) ought to be, between aninteresting set of metaphysical intuitions and those laws of logic we take tobe true.

In all, this book ranges over a vast terrain covering much of the ways inwhich our beliefs about the role and nature of logic and of the structures itdescribes both impact and depend on a wide array of metaphysical pos-itions. The work touches on and freshly illuminates almost every corner ofthe modern debate about logic; from pluralism and paraconsistency toreason and realism.

10 Penelope Rush

part i

The Main Positions

cha p t e r 1

Logical realismPenelope Rush

1. The problem

Logic might chart the rules of the world itself; the rules of rational humanthought; or both. The first of these possible roles suggests strong similaritiesbetween logic andmathematics: in accordance with this possibility, both logicand mathematics might be understood as applicable to a world (either thephysicalworldoranabstractworld) 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 ofhuman cognition in the way that mathematics might be; and relevant tothat cognition. The relevance of logic to cognition – or, at the very least,the human ability to think logically – seems indubitable. So any under-standing of the metaphysical nature of logic will need also to allow for aclear relationship between logic and thought.1

The broad aim of this chapter is to show that we can take logicalstructures to be akin to independent, real, mathematical structures; andthat doing so does not rule out their relevance and accessibility to humancognition, even to the possibility of cognition itself.Suppose that logical realism involves the belief that logical facts are

independent of anything human:2 that the facts would have been as they

1 Two things: note I do not claim we can or ought to show that logic underpins, describes, or arisesfrom cognition. In fact I think the relationship between thought and logic is almost exactly analogousto that between thought and mathematics (see Rush (2012)), and I disagree with the idea that there isany especially significant connection between logic and thought beyond this. Two: while this chapterdeals with the notion of ‘independence’ per se, it investigates this from the perspective of applyingthat notion especially to logic. That is, my main aim here is to indicate one way in which the realistconception of an independent logical realm might be considered a viable philosophical position butone primary way I hope to do this is by showing how attributing independence to logic need be nomore problematic than attributing independence to anything else (e.g. by arguing that the realistproblem applies across any ‘type’ of reality which is supposed to be independent).

2 See Lapointe’s characterisation as IND in this volume.

13

are regardless of whether or not humans comprehended them, or even hadexisted 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 cleardifference, or gap, between what the facts are and what we take them to be.(Folina 1994: 204)3

This sturdy objectivity is just one reason we might find logical realismappealing.4

There is, though, a well-known objection to the idea that we cancoherently posit the independence of facts (including logical facts) fromtheir human knowers (and human knowledge).

Wilfrid Sellars formulated a version of this objection in 1956. Sellarsargued that in order to preserve both the idea that there is somethingindependent of ourselves and epistemological processes, and the idea thatwe can access this something (e.g. know truths about it), we seem to haveeither to undermine the independent status of that thing (by attributing toit apparently human-dependent features) or to render utterly mysteriousthe way in which any knowledge-conferring relationship might arise fromthat access.

Sellars’ idea is that we cannot suppose that we encounter reality as it isindependently of us, unless we suppose something like a moment ofunmediated access. But, there can be no relevant relationship betweenindependent reality and us (e.g. we can make no justificatory or founda-tional use of such a moment) unless that unmediated encounter can betaken up within our own knowledge.

The obvious move is simply to say that this initial encounter is availableto knowledge. But this move undermines itself by casting what wasindependent as part of what is known: i.e. it attributes an alreadyin-principle knowability to a supposed fully independent reality (for moreon Sellars’ argument, see Fumerton (2010), and Sellars (1962)).

The broadly applicable Sellarisan objection bears comparison to Bena-cerraf ’s (1973) objection to mathematical realism, which extends, at least toa degree, to logical realism.5 Benacerraf argued that even our best theory of

3 Folina was talking about mathematical realism, but the sort of logical realism I want to examine hereis directly analogous to mathematical realism in this respect.

4 Lapointe (this volume) explores a variety of reasons that may play a role in holding some version oflogical realism, so I won’t go into these in depth here.

5 For more on the possible entities a logical realist might posit (e.g. meanings/propositions), seeLapointe (this volume). Regardless of which entities are selected and where these are situated on the

14 Penelope Rush

knowledge could not account for knowledge of mathematical reality just solong as that reality was conceived of in the usual mathematical realist way:as abstract, acausal, and atemporal. Part of the problem, as Benacerraf sawit, was that the stuff being posited as independently real is not sufficientlylike any stuff that we can know, and if it were, it would not be the sort ofthing intended by the mathematical realist in the first place.

Sellars’ objection can be understood as a generalisation of Benacerraf ’s:common to both is the idea that the fully independent reality posited bythe realist is not the type of thing we can know, or if it is, then it is not thetype of thing the realist says it is.Thus, even were the mathematical or logical realist to adjust his con-

ception of mathematical or logical reality by ruling out one or all of itsabstractness, atemporality, or acausality, the problem induced by its com-plete 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 whatindependently is and what we as humans grasp. The basic idea is that werethere no humans to experience or be conscious of it, logic would still be asit is. So it seems that being the type of thing which is experienced orknown can be no part of what it (essentially) is.6

The problem can be expressed this way: how can independent reality bepart of human consciousness and experience if our human consciousnessand experience of it can be no part of independent reality? A putativesolution, then, might show how independent reality could play a role inhuman consciousness, but such a solution would need also to affirm thenecessary 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 wherethere is no thought. . . . he speaks of thinking a world in itself andindependent of thought. But in saying this, does he not precisely speak ofa world to which thought is given, and thus of a world dependent on ourrelation-to-the-world? (Meillassoux 2011: 1)

abstract–physical scale of possible entities, just so long as the realist also posits IND (Lapointe, thisvolume), they’ll encounter some version of Benacerraf ’s or Sellars’ problem.

6 For more on the nuances of ‘independence’ available to the realist, see Jenkins (2005) – I takeessential independence to follow from modal independence, and I take modal independence ascharacteristic of the sort of realism I want to explore.

Logical realism 15

Husserl characterised the realist problem of independence (which he alsocalled ‘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 andwhat objectively is . . . [is] the source of the deepest and most difficultproblems. Taken collectively, they are the problem of the possibility ofcognition. (Husserl 1964: 10–15)

Each of the above characterisations of the realist’s situation turns on thecentral theme of how we can sensibly (and relevantly) conceptualisethe role that a reality independent of human consciousness could playin the realm of that consciousness.

Husserl’s characterisation of the problem already gives a clue as to hisoverall approach: rather than view the problem as bridging a gap of the sortFolina describes, Husserl suggests we view it as “the possibility ofcognition”.

2. The potential of phenomenology

I hope to show how Husserl’s approach potentially enables us to takeindependent 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 epistemol-ogy. That is, I hope to use his approach to see how we might accommodatethe idea that what is cognised, and what must (on a realist account) remainirreducibly external (or, in principle, separable) to what is cognised – canbe one and the same thing, or (perhaps) more accurately, a dual thing.7

At first glance, this might seem simply to concede the contradictionMeillassoux graphically outlines. I want to take a second glance – illustratinghow such a concession need be neither simple nor impotent but ratheroffer a way to conceptualise the elements underpinning the realist notionof independent reality and so begin, if not to resolve, then to make somesense of its intractability. That is, there are ways in which the Husserlianperspective can motivate us to find reasons and avenues by which we mightbegin to accommodate the independent reality the realist posits, even aspotentially contradictory – rather than to take its inherent instability asreason enough to brush it off as impossible and therefore irrelevant. Theseways 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, orcomponents, etc: ‘one that has two aspects/dimensions/components, etc’.

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perspective (admittedly most probably neither envisaged nor anywhereclaimed by Husserl himself ); namely that the realist predicament is itselfan 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 unbridgeabledichotomy between reality and our knowledge of it –Husserl characterisedas a product of the ‘natural’, ‘scientific’ attitude, which he saw as pervasiveall of philosophy (again, barring his own, e.g. Husserl 1964: 18–19).By contrast, phenomenology offers a picture of entangled cognition

wherein independent reality is inextricable from cognition itself. This sortof picture takes the first step toward accommodating both sides of thedivide insofar as it introduces the idea that our internal perspective itselfirreducibly incorporates the possibility, even the necessity, of there beingsomething 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 notattribute these possibilities to Husserl. As I understand him, for Husserl,experience is always experience of – and so cannot begin to be definedwithout allowing (at least) a place or a role for something external towardwhich it is directed at the outset. For me, the promising bit is this: that thissomething is both somehow outside or external to (‘constituting’) experi-ence and within it (‘being constituted’) at the same time.It is by examining and enlarging on this promising bit that I hope to

explore one way in which phenomenology (potentially) offers a role for therealist predicament itself as the (contradictory) structure of our relationshipto independent reality. I hope to sketch how accepting the predicament inthis way might enable us to make sense of reality, cognition, and experi-ence within a realist framework – to see the realist’s ‘predicament’ as acomplex and interesting structure that these elements share, as opposed toan impossible riddle or a problem in need of a solution.In what follows, I’ll briefly unpack just a couple of aspects of Husserl’s

account in order to show how we might use them to begin to open andexplore this possibility, specifically regarding the idea of a realisticallyimagined independent logical structure.9

8 Caveat: I’d like to argue that the predicament can play this role just so far as the basic idea of anindependent reality existing at all can. It is the latter that I see the framework in Husserl’s ideas asable to directly establish.

9 Or, again, to illustrate how conceiving logic as an independent objective structure akin tomathematics need not be considered an especially problematic instance of the general idea ofindependent reality itself, once that idea is effectively defended.

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3. Key aspects of phenomenology

3.1 The Platonic nature of logic

Husserl had a very broad concept of logic that embraces our usual modernidea of logic as well as something he called ‘pure logic’, which we canloosely characterise as something like ‘the fundamental forms of experi-ence’. For Husserl, logic as formal systems (and so too ‘modern logic’;incorporating classical, modal, and all the usual non-classical structures), isto be accounted for in much the same way as is mathematics: by itsrelationship to these fundamental forms. This relationship is roughly thatwhich holds between practice and theory – pure logic is the purelytheoretical structure (or, perhaps, structures – I don’t think it mattersmuch here) that accounts for logic as practised.

For Husserl, the fundamental forms of pure logic are an in-eliminablepart of experience: i.e. ‘experience’ encompasses direct apprehension ofthese inferential relationships. The apprehended structures are abstractand platonic; discovered, rather than constructed. Theory, empiricalobservation, and experience are in this sense fallible: they may or maynot ‘get it right’ and reveal the actual independent structure of logic.In Husserl’s words:

As numbers . . . do not arise and pass away with acts of counting, and as,therefore, the infinite number-series presents an objectively fixed totality ofgeneral objects . . . so the matter also stands with the ideal, pure-logicalunits, the concepts, propositions, and truths – in short, the significationsdealt with in logic . . . form an ideally closed totality of general objects towhich being thought and expressed is accidental. (1981: 149)

Thus both logic and mathematics, for Husserl, have a ‘pure’, ‘abstract’,‘theoretical’, ‘definite’, and ‘axiomatic’ foundation. Further, Husserlbelieved that:

one cannot describe the given phenomena like the natural number series orthe species of the tone series if one regards them as objectivities in any otherwords 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 asthey are characterised by Husserl, should encounter the realist problemof independence – neither are the sort of thing we can simply takeas 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 whateither mathematics or logic independently is.

3.2 Inextricability

As touched on above, one of Husserl’s most suggestive and promising ideasis that consciousness is not separable from consciousness of an object –intentionality is built into the structure of consciousness and experienceitself.The leading idea is consciousness as consciousness of: the very definition

of experience and consciousness as involving already what it is directedtoward, or what it is conscious of. Of course, this idea is also what a greatdeal of the controversy in Husserlian scholarship centres on. One reasonfor the controversy, I think, is the ambiguity in the prima facie simple ideaof an object (or realm, or reality) as an object of anything (including, forexample, consciousness, intention, act, or perception). Even on the mostsubjectivist reading, the notion is ambiguous between the idea of objects inexperience, and as experienceable. This ambiguity interplays in obviousways with the tension underpinning the realist’s problem: that between theobject as given to an epistemological human-dependent process, and theobject as independent. In turn (as we’ve seen) this ambiguity itself centreson a distinction between ‘internal’ (what we take the facts to be), and‘external’ (what the facts are).I suggest that the urge to disambiguate Husserl on this point should be

resisted,10 since to disambiguate here would be to miss a large part of thepotential of phenomenology. Indeed, Husserl himself seems at times todeliberately preserve ambiguity here (though whether he meant to or not istangential to the point). For example:

First fundamental statements: the cogito as consciousness of something . . .each object meant indicates presumptively its system. The essential related-ness of the ego to a manifold of meant objects thus designates an essentialstructure 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 ofconsciousness, passively or actively. On the other hand, the phenomen-ology of cognition has to do with these objects as presenting themselves inthis 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 inwhich they present’ give cognition its essential structure. It seems thatHusserl resists resolving the ambiguity in these phrases one way orthe other.

Husserl’s “phenomenology of cognition” is accomplished through aprior conceptual step called the ‘phenomenological reduction’. This‘reduction’ is related to Descartes’ method of doubt (e.g. in Husserl1964: 23. A useful elaboration can be found in Teiszen 2010: 80). Teiszenargues that for Husserl the crucial thing about the phenomenologicalreduction 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 uncoverin what remains (after Cartesian doubt) a lot more than an ‘I’ who isthinking. In particular, we can uncover direct apprehension of “the idealobjects of logic and mathematics” (Teiszen 2010: 9) whose pure formsextend far further than what Descartes ended up allowing as directlyknowable, and further than the knowable allowed for in Kant’sphilosophy.

Just as there is with what to make of the ‘consciousness as consciousnessof’ idea, so too there is much controversy surrounding exactly what thephenomenological reduction is and involves. To say that there is disagree-ment here among Husserl scholars is something of an understatement.Indeed: “there seem[s] to be as many phenomenologies as phenomenolo-gists” (Hintikka 2010: 91).

But the clarification of exactly what Husserl may have meant is notrelevant to my purpose here, which is to see if there are ideas we can drawfrom Husserl that might help a realist philosopher of logic.

I pause to note, though, that Teiszen’s interpretation of the reduction asa “‘suspension’ or ‘bracketing’ of the (natural) world and everything in it”(Teiszen 2010: 9) is standard; and the ‘ideal objects’ recovered in Teiszen’sconsequent ‘transcendental idealism’ (including their ‘constituted mind-independence’) are also standard for an established tradition of Husserlscholarship (adhered to by Føllesdal, among others). But these ‘ideal’objects are very far from the realist mind-independent realm that I wantto imagine has a place here (to hammer this point home, see Teiszen2010: 18).

Again, it is the (possibly resolute) ambiguity in Husserl’s accountthat allows for my alternate reading of phenomenology. Another casein point: “the description on essential lines of the nature ofconsciousness . . . leads us back to the corresponding description

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of the object consciously known” (Husserl 1983: 359). The phrase: “theobject consciously known” is ambiguous. It can be read differentlydepending on each term’s specific interpretation and on which termsare emphasized: e.g. the ‘consciously known’ can be read as ‘the object aswe know it’ (i.e. a strictly constituted – internal – object); or as ‘theobject that is known’. It is the latter interpretation that opens thepossibility of an ‘external element’ in the basic ingredients of the natureof 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 anindependent objective other, while nonetheless focusing on experienceand consciousness: my thought is that if we can argue that intendingreality as it appears (i.e. in the case of the realist conception of logic: asobjective and independent) is itself constitutive of cognition and evenof the possibility of cognition itself; then we can see a way in whichobjective independent reality is (complete with its attendant predica-ment) already there, structuring the essential nature of consciousnessand 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 tous to be (including being independent of us), in fact cannot be ruledout. Thus, we can see in the basic elements of the phenomenologicalanalysis how objective, independent reality enters the picture as objective,and independent – not only as an object of consciousness, but as consti-tuting consciousness itself. This is the case even if (or, as Husserl wouldhave it, especially if ) we try to focus only on ‘pure experience’ or ‘pureconsciousness’.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 beproblematic . . . This problem is resolved before hand with the idea ofthe intentionality of consciousness, since the presence of the subjectto transcendent things is the very definition of consciousness. (1998: 114,italics mine) [and]

the world is not only constituted but also constituting. The subject is nolonger pure subject; the object no longer pure object. The phenomenonis 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 totranscendent things defines consciousness,11 it is not a huge leap to see howthis initial subjective/transcendent relationship (even if it’s just one ofmutual ‘presence’) can incorporate the entire problematic outlined above:i.e. that the Sellars–Meillassoux contradiction is ‘built in’ just so far as itdescribes that relationship. Recall that Husserl equates that problematicwith the problem of the possibility of cognition (p. 16 above): it shouldnow be apparent how his equation can be understood as a means bywhich to understand (rather than resolve or dissolve) the ‘natural’,‘scientific’ perspective, complete with its consequent dilemma. That is,Husserl’s point:

‘The problem of the possibility of cognition is the traditional realist dilemma’

need not be interpreted thus: ‘the problem of the possibility of cognitionsupplants the traditional dilemma’. Rather, it may be interpreted thus: ‘thetraditional dilemma defines (in some way or other) the problem of thepossibility of cognition’.

Hintikka is another who seems to suggest that the contradictory rela-tionship between the subject and external reality is a part of Husserl’s(along with Aristotle’s) philosophy. He asks:

Is . . . the object that we intend by means of a noema12 out there in the real“objective” world? Or must we . . . say that the object “inexists” in the act?

He then points out:

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

My own point is that this characterisation of the relationship (one I agreeHusserl himself advocates) does not automatically eliminate or supplantthe traditional, ‘natural’ characterisation of the relationship, and so nordoes it eliminate the problem as it arises for that ‘natural’ characterisation.I suggest that the phenomenological perspective is best understood as are-conceptualisation of the same relationship that is characterised and

11 Note that this need not go the other way: we can retain the phenomenological insight without theinverse claim that the object itself depends on, or even is (either necessarily or always) present to,consciousness.

12 Husserl’s name for something akin to Fregean ‘sense’, but also apparently akin to (though more fine-grained 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 engagingdirectly with its key concepts (rather than as wholly re-interpreting,removing, or supplanting those concepts).

4. Overflow

I want now to discuss the idea of the “pregnant concept of evidence”(Husserl 1964: 46). Husserl says:

If we say: this phenomenon of judgement underlies this or that phenom-enon of imagination. This perceptual phenomenon contains this or thataspect, colour, content, etc., and even if, just for the sake of argument, wemake these assertions in the most exact conformity with the givenness ofthe cogitation, then the logical forms which we employ, and which arereflected in the linguistic expressions themselves, already go beyond themere cogitations. A “something more” is involved which does not at allconsist of a mere agglomeration of new cogitationes. (1964: 40–1)

Elsewhere, he notes:

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

The point I want to draw attention to is that Husserl takes both logical andphysical/perceptual ‘objects’ as the sort of thing that in one sense oranother ‘overflow’, or ‘go beyond’ what is given to cogitation.

The word ‘object’ must . . . be taken in a very broad sense. It denotes notonly physical things, but also, as we have seen, animals, and likewisepersons, events, actions, processes and changes, and sides, aspects andappearances 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 abstractlogical forms is not treated in any especially problematic way, all of what isgiven to experience can be explained in much the same fashion: “sensuousintuition means givenness of simple objects. Categorical intuition . . . meansgivenness of categorical formations, such as states of affairs, logical connectives,and essences” (Hartimo 2010b: 117). The structure underpinning logic – theform and structure of experience – is constituted and ‘given’ in experience. It is‘seen’13 analogously to the way physical objects are seen by perception.

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

Logical realism 23

So, the object of genuine perception and, by the extension I want tomake here, genuine categorical intuition, overflows what is given to the actof perception or comprehension itself. For this reason it is capable of beingveridical, and is opposed to Hyletic data, which is not.14

This is because genuine perception and intuition involve noema that areboth conceptual and objectual.15 It is because each noema is objectual thatour conceptual grasp can never fully contain the whole noema: i.e. that thisgrasp is always ‘pregnant’. Note that Husserl does not commit to therebeing two noemata for each act of perception or comprehension, butneither does he commit to the idea that the conceptual and the objectualare simply two aspects of the one noema.16 Rather, his claims regardingobjectual (or, to anticipate what’s to come: ‘non-conceptual’) phenomenaand conceptual phenomena are in tension with one another.

In every noema, Husserl says:

A fully dependable object is marked off . . . we acquire a definite system ofpredicates either form or material, determined in the positive form or left“indeterminate” – and these predicates in their modified conceptual sensedetermine the “content” of a core identity. (Husserl 1983: 364, italics mine)

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

14 Shim (2005) nicely characterizes hyletic data as the ‘sensual stuff ’ of experience. He gives thefollowing helpful example of the process of ‘precisification’ to contrast memory or fantasy withgenuine perception: “In remembering the house I used to live in, I can precisify an image of a redhouse in my head. The shape, the color and other physical details of that house must be ‘filled in’ byhyletic data. Now let’s say I used to live in a blue house and not a red house. There is, however, noveridical import to the precisifications of my memory until confronted by the correctiveperception . . . there is no sense in talking about the veridical import in the precisifications of [thememory or] fantasy” (pp. 219–220). In the latter cases, we may mistake merely hyletic data for non-conceptual (or objectual) phenomena (p. 220). An analogous situation might be said (by a logicalrealist) to occur for logical intuition when we encounter counter examples or engage directly withthe meaning of logical operators – in these situations we can see a genuine role for veridical inputcapable of correcting or ‘precisifying’ our intuition. On the other hand, perhaps analogously to whatoccurs in a fantasy or hallucination, we may mistake the mere manipulation of symbols for genuine(veridical) comprehension.

15 Shim gives a sophisticated argument for the idea that what provides perceptual noemata with‘overflow’ is that they have both conceptual and non-conceptual content. My idea is similar, but,as will be elaborated shortly, the duality I want to consider should not be rendered as (non-contradictory) aspects of one and the same object, but rather as a contradictory object; whereasI think that Shim means the duality he proposes to be interpreted in the former sense.

16 Thanks to Graham Priest for pressing this point.

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subject of its possible predicates”; “the pure X in abstraction from allpredicates”; “the determinable which lies concealed in every nucleus andis consciously grasped as self-identical”; “the object pole of intention”; and,best of all: “that which the predicates are inconceivable without and yetdistinguishable from”. This is conceptually located in a similar variety ofways, 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 ofthese characterisations are contradictory. What I hope to indicate, in whatfollows, is that this is as it should be.To review and sum up:The main points I get from Husserl are these: that independent abstract

‘reality’ is no more difficult to accommodate than is independent physicalreality; that conceptualising logical structures as similar to platonic math-ematical structures does not preclude conceptualising either as immedi-ately apprehendable objects of cognition; and thus that the idea ofindependent reality as (genuinely, problematically) independent finds aplace in phenomenology.

5. McDowell

It is useful to compare what has so far been drawn from Husserl to aspecific interpretation of McDowell.Neta and Pritchard in their (2007) article make a point that helps situate

Husserl’s programme: they argue that one way to understand attempts(specifically McDowell’s, but their ideas extend to Husserl’s) to reachbeyond our ‘inner’ world to an external realm is precisely by close examin-ation of the assumptions we bring to the Cartesian evil genius thoughtexperiment. The argument they present demonstrates links between aparticular (perhaps ‘natural’) way of conceiving the distinction between‘inner’ and ‘outer’, and the commonly held assumption that:

(R): The only facts that S can know by reflection alone are facts that wouldalso obtain in S’s recently envatted duplicate. (p. 383)

Neta and Pritchard argue that McDowell rejects R on the basis that there issomething about our actual, embodied experience of the world that cannotbe replicated by stimulus, no matter how sophisticated, experienced by abrain in a vat (compare this with Husserl’s differentiation between genuine‘pregnant’ perception and hyletic/sensuous data). The clue as to howMcDowell rejects (R) and to uncovering the similarities between his and

Logical realism 25

Husserl’s approaches is in the concept ‘experience of the world’. ForMcDowell, experience of (the world) is experience as (humans in theworld). The idea is that if indeed that is what we are talking about, thenwhen we talk of ‘experience in the world’, we cannot, as it were, ‘slice off ’the part that is us experiencing from the part that is being experienced.

Neta and Pritchard outline McDowell’s position as follows:

McDowell (1998a) allows . . . that one’s empirical reason for believing acertain external world proposition, p, might be that one sees that p is thecase. Seeing that is factive, however, in that seeing that p entails p. However,McDowell also holds that such factive reasons can be nevertheless reflect-ively accessible to the agent – indeed, he demands . . . that they be accessiblefor 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 tothe knower’s ‘space of reasons’. But her ‘satisfactory standing’ in the spaceof reasons in which p is so, involves ‘seeing that p’, which entails p itself.

McDowell’s ‘factive reasons’ are subtle things with clear similarities toHintikka’s characterisation of the Aristotelian/Husserlian ‘object of an act’:they are knowable by reflection alone, but also entail objective ‘external’states. I remember my then seven-year-old son once saying ‘I think thetrees have faces’, and thinking that this is a nice way of explaining some ofthe ideas in McDowell’s Mind and World (1994), which I take as anattempt to argue that what is external and objectively so is nonethelessalso accessible – available to us as conceptual content.

But I think that the McDowellian/Husserlian sort of manoeuvre canonly work if ‘what is experienced’ genuinely is the realist’s independentreality (at least as much as it is accessible content). To the extent thatany account re-casts or re-defines that independence, it is hard to seehow the specifically realist problem (which both McDowell and Husserlidentify in the ‘natural attitude’) is the problem their accounts actuallyaddress.

Put another way, if an account implicates the external in our human(reflective) experience simply by fiat (or by initial (re)design), then itbecomes difficult to see how such an account can help us understand theproblem that inspired it in the first place: i.e. the problem of the realist’sconception of independence as independence from human experience.McDowell’s and Husserl’s solution are of a kind, both answer the scepticalong the following general lines: you can’t take away reference to externalreality (as in the sceptical scenario) just because what we experience hasexternal reality somehow written into it. But if a position’s ‘inwritten’

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

6. Effectively defending ~R

The important word in the preceding paragraph is “somehow”. Expandingon the ‘somehow’, we can find a sense in which neither McDowell norHusserl escapes or resolves the traditional, ‘natural’ dilemma. Or rather, tothe extent that they can be said to, their solutions do not address thisoriginal dilemma. Conversely, I want to suggest it is just to the extent thatthey don’t escape the dilemma that they may (via expansion on the‘somehow’) be taken as having offered a sort of solution wherein whatwas unintelligible from the traditional/natural perspective, is made at leasta little intelligible. That is, their sort of insight might be taken as offering aperspective from which the contradiction inherent in speaking of a realityindependent of humans altogether need not automatically undermine thepossibility 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 thathas (R) at its core”.Neta and Pritchard outline McDowell’s challenge to R this way:

McDowell’s acceptance of reflectively accessible factive reasons . . . entailsthat the facts that one can know by reflection are not restricted to the“inner” in this way, and can instead, as it were, reach right out to theexternal world, to the “outer”. One has reflective access to facts that wouldnot obtain of one’s recently envatted duplicate, on McDowell’s picture. Ifthis is correct, it suggests that the popular epistemological distinctionbetween “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 morecomplex view of the original Cartesian experiment. To accept ~R, we needreasons to suppose that the thought experiment of ‘doubting everything’ isnot simply or not only constructible along lines drawn from our ‘natural’understanding of the ‘outer/inner’ distinction. Husserl offers the broadreason that consciousness per se is not possible – if we try to imagine sucha thing, we find a sense in which independent reality got there before us:consciousness itself incorporates ‘potentialities’ that, in turn, cannot bereduced to wholly ‘subjective’ or ‘internal’ phenomena.

Logical realism 27

Neta and Pritchard argue that the temptation to interpret McDowelleither wholly internally or wholly externally rests on believing R. BelievingR then, is very like holding fast to the possibility that in principle what isgiven to cognition and what cognition intends, can always be untangled.For Husserl, only a radically impoverished view of envattedness can deliverthe sceptical conclusion: a closer, careful look at cognition “in general,apart from any existential assumptions either of the empirical ego or of areal world” (Husserl 1981: 60) returns the world in all of its “modes ofgivenness” (Husserl 1981: 59), as constituted and constituting that cogni-tion. So I think it is reasonable to take Husserl similarly to McDowell onthe question of envattedness: i.e. to take Husserl as committed to therebeing a difference between envatted and non-envatted states.

But I want to take issue with Neta and Pritchard’s claim that: “Once (R)is rejected . . . these two aspects [internal and external] of the view are nolonger in conflict” (Neta and Pritchard 2007: 38b). And, for the samereasons, I take issue with similar claims Husserl makes regarding phenom-enology e.g.: “In . . . phenomenology . . . the old traditional ambiguousantitheses of the philosophical standpoint are resolved” (Husserl 1981: 34).

A genuine resolution of the ‘traditional antithesis’ could come about onlyvia an explicit defence of ~R in the original (‘traditional’) terms in whichR itself was conceptualised. In short, a ‘resolution’ of the problem gener-ated by the original dichotomy must directly address that dichotomy as agenuine dichotomy.

There are various ways ~R and an alternative conceptualisation of theinternal/external dichotomy might be defended, but only some of theseways can be said to address and so potentially resolve, the original realistdilemma. For example, ~R itself, or a set of key reasons offered tobelieve ~R, might be used as a sort of first principle, or established by fiat;then again, an approach might give a bunch of positive reasons or argu-ments for ~R (independent of the original reasons for R) in order toconvince us that ~R (along with any attendant, independent positivereasons offered for ~R) ought to replace or provide an alternative perspec-tive to the ‘traditional’ perspective. But neither of these cases can be said toresolve the original problem. They might be said to replace that problem,perhaps; or to render it irrelevant in the face of a potentially morecompelling 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 explicitlyengaging with that antithesis on its own terms (for a more detailed defenceof these ideas, see Rush 2005). That is, an explicit argument against

28 Penelope Rush

R (accommodating the terms and spirit in which it was intended) has todefend one of the following claims: an internal phenomenon is also not-internal (i.e. that a phenomenon able to act as internal in the R thoughtexperiment is also one able to act as not-internal in the thought experi-ment); an external phenomenon is also not-external (taking ‘internal’phenomena as ‘not-external’); or, for each case, there is no straightforwardeither/or dichotomy (i.e. it is not the case that such phenomena are ‘eitherexternal or not-external’, or ‘internal or not-internal’). That is, in R, theconcepts ‘external’ and ‘internal’ are explicitly (intended as) subject to boththe law of excluded middle (LEM): ~A v A; and the law of non-contradic-tion (LNC): ~(~A & A).17

So accounts that rest on or incorporate ~R in some way must alsodirectly challenge the applicability of these classical laws to the internal/external dichotomy. One such challenge might argue that the point of ~Ris that it gives us reason to doubt that the LEM should hold here. Therelationship between the phenomenological and ‘natural’ perspectivesmight then be seen as analogous to the relationship between the intuition-ist rendering of the continuum as viscous and the classical rendering of thecontinuum as discrete. From the intuitionist’s perspective, the continuumhas characteristics it does not have from the classical perspective. To see theformer, we need to allow the LEM to fail, in particular, for 8x8y((x<y) v(x=y) v (x>y)). In much the same way, we could argue that to see the morecomplex characteristics of our human experience in the world, we need toallow the LEM to fail for 8x(Ix v ~Ix) (where I is ‘internal’) and/or for 8x(Ex v ~Ex) (where E is ‘external’). (For more on the intuitionists’continuum, see Posy 2005, especially pp. 345–348.)Note that this means that the most effective defence of ~R challenges

the universal applicability of the laws of (classical) logic. So knowledge of(external, independent) logical truths is guaranteed only by an explicit,rather drastic instance of the corrigibility of that knowledge. Thus, theknowledge of logic that survives the phenomenological reduction is corri-gible knowledge – but this is perhaps what we should expect, given theindependence of logical truth: its fundamental role in cognition does notand cannot guarantee the infallibility of our own intuition.

17 That is, I think arguments for the claim that Husserl’s and McDowell’s accounts do not‘hypostatise’ ultimately fail (for examples of such arguments, see Hartimo 2010b, and Putnam2003, particularly p. 178). Or, to the extent that they succeed, the accounts themselves are renderedlargely irrelevant to the philosophical problem I am addressing here.

Logical realism 29

Another such challenge could argue that in the case of independent,external and dependent, internal phenomena, we have an explicit excep-tion to the LNC: there are occasions where each type of phenomenonboth is and is not that type (for more on this idea see Rush 2005 andPriest 2009).

Either way, these challenges undermine the notion that the LEM and/orthe LNC apply to internal and external phenomena. My own opinion isthat it makes more sense, for an account wishing to engage with thephilosophical problem, to mount the latter challenge – i.e. to argue thatthe LNC does not apply here, (given that it could be argued that LEMdefines the terms of the original thought experiment, R) – but the mainpoint is that only an explicit argument against (or recognising an implicitrejection of ) either or both of these classical rules can make such accountsas Husserl’s and McDowell’s relevant to the original ‘natural’ problem.

And I do think that Husserl was interested in addressing the original‘natural’ problem,18 but in a particular way:

one’s first awakening to the relatedness of the world to consciousness [i.e.the philosophical problem] gives no understanding of how the varied life ofconsciousness, . . . manages in its immanence that something which mani-fests itself can present itself as something existing in itself, and not only assomething meant but as something authenticated in concordant experience.(1981: 28) [and]

We will begin with a clarification of the true transcendental problem, whichin the initial obscure unsteadiness of its sense makes one so very prone . . .to shunt it off to a side track. (1981: 27)

In Husserl’s account then, there is a duality (akin to McDowell’s)‘within’ the constituted object itself, insofar as it is also ‘given’ asindependent. 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) asresiding strictly in the phenomenological attitude, and so Shim (2005) argues that thephenomenological ‘solution’ cannot neatly slot into a ‘natural’ answer to scepticism. But I thinkphenomenology is relevant to the natural answer to scepticism exactly insofar as it provides thisexplicit way of differentiating ‘being in the (real) world’ from ‘envattedness’. This differentiationdisrupts a neat holistic story, and so its lesson, carried through to science and the natural attitude, isperhaps not a ‘categorical mistake’ (Shim 2005: 225), but an alert as to the deficiencies of aphilosophy that disallows any perspective other than its own. What we know from thephenomenological attitude might resist reduction to naturalist/scientific knowledge, but itnonetheless can offer an insight into the items with which the scientific/philosophical attitude isconcerned: e.g. reality, experience, and knowledge. It is exactly what makes the phenomenologicalperspective “both tempt and frustrate . . . the very philosophical desire it should have satisfied”(Shim 2005: 225), that can make it relevant to that ‘desire’, and can potentially stop a too quick,neat, sealed holist answer from gaining complete purchase.

30 Penelope Rush

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 mostpromise. On the other hand the preservation of the LNC in this case callsfor resolution one way or the other and so renders an account open tobeing interpreted as wholly internal or wholly external, which I contend,would drastically impoverish it as an account of human experience. As itstands though, its own internal inconsistencies bear witness to the richnessof the very idea of phenomenology: of the inescapable, paradoxical, yetentirely natural thought that our human experience is irreducibly consti-tuted 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 forhelpful feedback on earlier drafts.

Logical realism 31

cha p t e r 2

A defense of logical conventionalismJody Azzouni

1. Introduction

Our logical practices, it seems, already exhibit “truth by convention.”A visible part of contemporary research in logic is the exploration ofnonclassical logical systems. Such systems have stipulated mathematicalproperties, and many are studied deeply enough to see how mathematics –analysis in particular – and even (some) empirical science, is reconfiguredwithin their nonclassical confines.1 What also contributes to the appear-ance 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 ofchoice will be replaced by one of these alternatives. If this happens, whyshouldn’t the result be the dethroning of one set of logical conventions foranother? One set of logical principles, it seems, is currently conventionallytrue; another set could be adopted later.

Quine, nevertheless, is widely regarded as having refuted the possibilityof logic being true by convention. Some see this refutation as the basis forhis later widely publicized views about the empirical nature of logic.Logical principles being empirical, in turn, invites a further claim thatlogical principles are empirically true (or false) because they reflect well (orbadly) aspects of the metaphysical structure of the world. Just as the truthor falsity of the ordinary empirical statement “There is a table in MinerHall 221B at Tufts University on July 3, 2012,” reflects well or badly how apart of the world is, so too, the Principle of Bivalence is true or false becauseit reflects correctly (or badly) the world’s structure. I’ll describe thisadditional metaphysical claim – one that I’m not attributing to Quine(by the way) – as taking logical principles to have representational content.Most philosophers think logical principles being conventional is

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

32

incompatible with those principles having representational content.2

I undermine the supposed opposition of these doctrines in what follows.That still leaves open the question whether logical principles do haverepresentational content; but I also undermine this suggestion. That mayseem a lot to do in under eight thousand words. Luckily for me (and foryou too), most of the important work is already done, and I can cite itrather 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 ofthe conventionality of logic is a dilemma. The famous Lewis Carrollinfinite regress assails only one horn of this dilemma, the horn thatpresupposes that the infinitely many needed conventions are all explicit.Quine (1936b: 105) writes, indicating the other horn:

It may still be held that the conventions [of logic] are observed from thestart, and that logic and mathematics thereby become conventional. It maybe held that we can adopt conventions through behavior, without firstannouncing them in words; and that we can return and formulate ourconventions verbally afterwards, if we choose, when a full language is at ourdisposal. It may be held that the verbal formulation of conventions is nomore a prerequisite of the adoption of conventions than the writing of agrammar is a prerequisite of speech; that explicit exposition of conventionsis merely one of many important uses of a completed language. So con-ceived, the conventions no longer involve us in vicious regress. Inferencefrom general conventions is no longer demanded initially, but remains tothe subsequent sophisticated stage where we frame general statements of theconventions and show how various specific conventional truths, used allalong, fit into the general conventions as thus formulated.

Quine agrees that this seems to describe our actual practices with manyconventions, but he complains that (Quine 1936b: 105–106):

it is not clear wherein an adoption of the conventions, antecedently to theirformulation, consists; such behavior is difficult to distinguish from that inwhich conventions are disregarded . . . In dropping the attributes of delib-erateness and explicitness from the notion of linguistic conventions we riskdepriving 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 oflogical conventionalism” as supporting the view that logical expressions “do not describe features ofthe world, but rather are mere conventional devices.”

A defense of logical conventionalism 33

We may wonder what one adds to the bare statement that the truths oflogic and mathematics are a priori, or to the still barer behavoristic state-ment that they are firmly accepted, when he characterizes them as true byconvention in such a sense.

These challenges aren’t specifically directed against the conventionalityof logic but against tacit “conventions” of any sort. One challenge isconcerned with making sense of when specific behaviors are in accordwith the proposed tacit conventions and when they’re not. One prob-lem, that is, is this: if the conventions are explicit, we know what theconventions are – because they’ve been stated explicitly – and thebehavior can be directly measured against them to determine deviations.But tacit conventions must be inferred from that very behavior, so thechallenge goes, and therefore a lot of unprincipled play becomes possiblebecause various conventions may be posited, these conventions differingin how far the practitioners’s behavior is taken to deviate from them.A second issue Quine raises is with the label “convention”; he wants toknow what’s distinctive about tacit conventions that makes them standapart from the simple “behavioristic” attribution that the population“firmly accepts them.”

So Quine’s two objections come apart neatly. There is, first, a challengeto the idea that a set of rules can be attributed to a population in theabsence of explicit indications like a set of official conventions. Even if thisfirst challenge can be circumvented, the second worry is why the set ofrules so attributed to a population should be called “conventions.”3

If we concede the requirement of explicitness to Quine, we’re forced tosomething like the Lewis account of convention:4

A regularity R, in action or in action and belief, is a convention in apopulation 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 con-formity by all but anyone.

3 See (Quine 1970b) for a reiteration of the first challenge with respect to linguistic rules.4 See (Lewis 1969: 78) – but I draw this characterization from (Burge 1975: 32–33).

34 Jody Azzouni

(5) There is at least one alternative R 0 to R such that the belief that theothers conformed to R 0 would give almost everyone a good anddecisive practical or epistemic reason to conform to R 0 likewise; suchthat there is a general preference for general conformity to R 0 ratherthan slightly-less-than-general conformity to R 0; and such that thereis normally no way of conforming to R and R 0 both.

(6) (1)–(5) are matters of common knowledge.

There are many problems with this approach – indeed, it’s no exaggerationto describe condition (6) as yielding the result that there are almost noconventions in any human population anywhere. But can Quine’s chal-lenges 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 tacitlinguistic conventions. Linguistic conventionality seem less complicatedthan logical conventionality if only because the intuitions that seem toaccompany 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 linguisticpractices are arbitrary: except for historical accident, they could have beenotherwise to roughly the same purpose.” He adds, “which linguistic andother social practices are arbitrary in this sense is a matter of dispute.” I’llshortly show that this matters to the empirical question whether language isconventional (and in what ways) – the thing Burge tells us we all agreeabout.But first, notice something important that Burge is explicit about

(although he doesn’t dwell on it): there are psychological mechanisms thatenable these regularities. Burge (1975: 35) writes, “the stability of conven-tions 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 forconventions: “Insofar as these latter play a role, they prevent the arbitrari-ness of conventional practices from being represented in the beliefs andpreferences of the participants.”Let’s focus on the important word “inertia.” This is an allusion to an –

ultimately neurophysiological – mechanism of imitation. The point ismade quite explicit some years later by Millikan when she characterizes

A defense of logical conventionalism 35

“natural conventionality” in terms of patterns that are “reproduced.”Crucial to the idea (Millikan 1998: 2) is that “these [conventional] patternsproliferate . . . due partly to weight of precedent, rather than due, forexample, to their intrinsically superior capacity to perform certain func-tions.” That is (Millikan 1998: 3), “had the model(s) been different . . . thecopy would have differed accordingly.”

Some may be worried about this characterization of conventional pat-terns.5 As I understand the characterization, for it to work we need tosharply distinguish between the patterns being conventional because theyare proliferating partly due to the weight of precedent, and the patternsinstead only being thought to be conventional because they’re thought toinvolve arbitrariness in our choice of a course of action. On the one hand,we can simply be wrong – thinking that arbitrariness is involved when itisn’t. On the other hand, there can be “arbitrariness” without our realizingit: there are other model-options we don’t know about, which, were theyin place, would have been imitated instead.

Consider the venerable practice of rubbing two sticks together to start afire.6 A tribal population may simply fail to realize that banging rockstogether will work instead. Their practice of rubbing sticks to start a fire isconventional despite their failing to realize this. Imagine, however, thatthey live where there are no such rocks, and where, presumably, there areavailable no other ways to start a fire. Then the practice isn’t conventional.Suppose (after many moons) the tribe migrates to an area where suitablerocks are located. Because of a change of location, a practice that wasn’tconventional has become conventional. (More generally, technologicaldevelopment can induce conventionality because it creates practical alter-natives that weren’t there before.) There is a lot of work to do here (muchof it empirical) detailing more fully the notion of “genuine practicalalternatives” – what sort of background factors should be seen as relevantand which not – but the need for hard empirical work isn’t problematicalfor this characterization of tacit convention.

Another worry. Many people believe (and some believe correctly) thatsome of their practices P are optimal. They engage (imitate) those practices(so they believe) precisely because they think these are optimal practicesand not because of the weight of precedent. Conventional or notconventional? Well, beliefs about optimality aren’t relevant; only the

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

6 I draw this example from (Epstein 2006: 4).

36 Jody Azzouni

efficacy of P ’s optimality to the spread of the practice through thepopulation is relevant. Suppose alternative suboptimal practices wouldnot have spread through the population, if instead they were the models,precisely because their suboptimality would have extinguished the prac-tices (or the population engaging in them). Then P isn’t conventional.Otherwise it is. Superior optimality, of course, can be why a practicetriumphs over alternatives. It’s an empirical question in what ways theoptimality of a practice relates to its popularity, but I’m betting thatsuperior optimality rarely counts for why a practice P spreads through apopulation.7 If a practice has enough optimality over other options tomake its superior optimality efficacious in its spread, then it isn’t conven-tional. On the other hand, some superior optimality clearly isn’t enough toerase conventionality. Therefore: How much superior optimality isrequired to erase conventionality is an empirical question, turning in parton how much damage a suboptimal practice will inflict on its population,how fast this will happen, how fast this will be noticed, and so on. Theseempirical complications, although of interest, don’t make the notion oftacit convention problematical.One point in the previous paragraph must be stressed further because

I seem to be definitively breaking with earlier philosophers on convention-ality on just this point. This is that roughly equivalent optimality is invari-ably built into the characterization of conventionality: the alternativepractices that render a practice conventional are ones that are reasonablyequivalent in their optimality – this is built into Lewis’ approach bycondition (3), that others conforming to such alternatives would givepeople “good and decisive” reasons for engaging in them as well – this isfalse if the alternative practices are suboptimal enough. It seems built intoMillikan’s approach – at least when conventional patterns serve functions –because alternatives should serve functions “about as well” (Millikan2005: 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 seriouslysuboptimal 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 thealternative suboptimal? There is controversy about this, but I think it is: I think this is why thealternative eventually died out among our progenitors (after thousands of years, that is). On the otherhand, 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 bytechnical developments in food processing. So the practice of eating all food raw needn’t be assuboptimal as it once was.

A defense of logical conventionalism 37

honest, a cold hard look at our own practices reveals exactly the samething). Evolution takes a really long view of things – even the extinction ofa population because it engages in a suboptimal practice may occur soslowly that the conventional fixation of that practice can occur for manygenerations, at least.8 How suboptimal a practice can be (in relation toalternatives) is completely empirical, of course, and turns very much on thedetails of the practices involved (and the background context they occur in);but optimality comparisons should play only a moderate role in an evalu-ation of what alternative candidates there are to a practice, and therefore inan 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 fromplaying a role in making conventional the logic we’ve adopted.)

One last additional point about conventionality that I’ve just touchedon in the last sentence. This is that it isn’t – so much – entire practices thatare conventional, but aspects of them that are. “Minor” variations in apractice are always possible, minor variations that we don’t normally treatas rendering the practice conventional because we don’t normally treatthose variations as rendering the practice a different one. There are manyvariations in how sticks can be rubbed together, for example. How wedescribe a practice or label it (how we individuate it) will invite ourrecognition of these variations as inducing conventionality or not. It’sconventional to rub two sticks together in such and such a way, but notconventional (say) to rub two sticks together instead of doing somethingelse that doesn’t involve sticks at all (in a context, say, where there are norocks). How we individuate “practices” correspondingly infects how and inwhat ways we recognize a practice to be conventional; but this is hardly anissue restricted to the notion of tacit convention, or a reason to think thenotion has problems.

4. Empirical evidence for tacit conventions

More than a serviceable notion of tacit convention is needed to respond toQuine. 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 doubtthe contemporary distribution of letters is suboptimal compared to alternatives; it’s clearly an inertialresult of the earlier arrangement of the keys on typewriters – which was probably also suboptimal inits time and relative to its context at that time. I’m not suggesting, of course, that keyboardconventions are contributing to a future extinction event – although I have no doubt that anumber of conventions that we currently use are doing precisely that.

38 Jody Azzouni

deliberateness and explicitness from the notion of linguistic convention werisk depriving the latter of any explanatory force and reducing it to an idlelabel.” As it turns out, and this is an empirical discovery, for conventionalpatterns to even be possible for a human population requires neurophysio-logical capacities and tendencies in those humans. These are currentlybeing intensively studied, and preliminary results reveal how humanchildren have a capacity to imitate that’s largely not shared with otheranimals.9 The recently discovered mirror-neuron system is crucial to thiscapacity (but is hardly the whole story). My point in alluding to thisempirical literature is to indicate how a systematic response to Quine’schallenges has emerged: Not only is a decent characterization of tacitconventionality – as noted above – now in place, but an explanation ofthe capacity for imitation that underwrites tacit conventions in this sense(and one that goes far beyond sheer behavioral facts about “firm accept-ance”) is also emerging due to intensive scientific study.10

Of course, Millikan (and Burge) seem to largely assume that language is“conventional” in the appropriately tacit sense. But this (on their ownviews) should be an entirely empirical question – patently so now that theneurophysiological mechanisms of imitation are being discovered. It’s anempirical question, for one thing, whether these mechanisms (mirrorneurons, etc.) are involved in language acquisition – more specifically,it’s an empirical question how they’re involved in language acquisition.Imagine (instead) that something like Chomsky’s principles and param-eters model is at work in language acquisition.11 Then the picture is this:the child starts language-acquisition with a massive prefixed cognitivelanguage-structure which is multiply triggered to a final state by specificthings the child hears. Imagine (what’s surely false, but will make theprinciple of the point clear) that there are (say) only three thousand andseventeen human languages that are possible, so that the child has only tohear a relatively small number of specific utterances for that child’s

9 See the introduction to (Hurley and Chater 2005a&b) for an overview of work as of that date. Seethe various articles in the volumes for details. The first sentence of the introduction (Hurley andChater 2005a: 1) begins, dramatically enough, with this sentence: “Imitation is often thought of as alow-level, cognitively undemanding, even childish form of behavior, but recent work across a varietyof sciences argues that imitation is a rare ability that is fundamentally linked to characteristicallyhuman forms of intelligence, in particular to language, culture, and the ability to understand otherminds.” It’s important to stress how recent these discoveries are – only within the last couple ofdecades.

10 One almost shocking development is that the study of these mechanisms is successfully taking placeat the neurophysiological level, and not at some more idealized (abstract) level – as is the case withmost language studies to date, specifically those of syntax.

11 See, e.g., (Chomsky and Lasnik 1995).

A defense of logical conventionalism 39

“language-organ” to fixate on a final language. A mechanism like this, evenif it helps itself to the neurophysiological imitation mechanisms to enablethe child to imitate the initial triggers, may leave very little of actuallanguage as conventional simply because the child’s final-state competencewould leave practically nothing for the child to subsequently learn.12

To respond to Quine, notice, what’s needed are both subpersonalmechanisms that allow alternative imitations (on the part of a population)as well as feasible alternative practices made available by the contextualbackground a population of humans is in. Without appropriate subperso-nal imitation mechanisms (as opposed to say, subpersonal mechanisms ofthe parameter/principles type), the apparent alternatives don’t render thecurrent practice conventional – because members of the population areactually incapable of imitating those alternatives. But if the feasible alter-native practices are absent from the contextual background then thepractice is rendered nonconventional because of this alone.

5. Three theories of logical capacity

I’ve just finished suggesting that the notion of tacit convention may(empirically) find almost no foothold in language, despite the appearanceof massive contingency, because the mechanisms of imitation – crucialto tacit convention – may play only a minimal role in language acquisi-tion.13 This is an empirical question, unresolved at the moment. Butwhat about logic?

Despite the subject matter of logic (in some sense) being so ancient, theactual principles of logic don’t become explicit until the very end ofthe nineteenth century. I now attempt to show that – possibly unlike

12 See (Chomsky 2003), specifically page 313. See (Millikan 2003), specifically pages 37–38. Thisempirical 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 localconventions to learn.” I don’t find convincing Millikan’s arguments against the empiricalpossibility of a (virtually final) steady-state for the language faculty: They seem to turn only onthe sheer impression that there’s always more language conventions for adults to acquire. But giventhat the empirical question is about what actual subpersonal mechanisms are involved in languageacquisition and also in the use of the language by adults who have acquired a language, it’s hard tosee why sheer impressions of conventionality deserve any weight at all.

13 One can always introduce the appearance of massive official conventionality by individuating thelanguage practices finely enough – e.g., minor sound-variations in the statistical norms of utterancesdetermining the individuation of utterance practices (recall the last paragraph of Section 3); but I’massuming this trivial vindication of the “conventionality” of language isn’t what either Burge orMillikan have in mind when they presume it as evident that natural language is full ofconventionality.

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the case of language, and rather surprisingly – tacit convention has agenuine 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 principlesenables the insight that these models of logic are, strictly speaking, generaltheories of the basis of human logical capacities, and not a priori character-izations of what logic must be. The earliest model, arguably, is thesubstitution one. Syllogistic reasoning especially, but also contemporaryreconstruals of logic in terms of schemata, invites the thought that logicalprinciples require an antecedent segregation of logical idioms. Logicaltruths are then characterized as all the sentences generated by the system-atic substitution of nonlogical vocabulary for nonlogical vocabulary withinwhat can be characterized as a recursive set of logical schemata or argumentforms. Such a characterization also allows the view that logical principlescan be recognized by their general applicability to any subject area: logicalprinciples 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: thecontent of an implication Im is contained in the content of the statementsIm is an implication of. An intensional version of this model is clearly atwork in Kant’s notion of analytic truth, and in notions of a numberof earlier thinkers as well. An extremely popular contemporary version ofthe content-containment approach externalizes the notion of content of astatement – taking it to be the possible situations, models, or worlds inwhich a statement is true. A deductive (intensional) construal of “content”understands the content of a statement to be all its deductiveconsequences.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 logicaldeduction is to be characterized in terms of a set of rules according towhich logical proofs must be constructed. Part of the reason this modelemerged so late for logic is that it required the extension of mathematicalaxiomatic methods to logic, something achieved definitively only byFrege.15

14 See (Sher 2001) for discussion and for citations of earlier proponents of this approach to logic.15 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-basedformat – how gappy those rules were. An early view was that a nonethymematic mathematicalproof was one without “missing steps” or gaps. But this view, based as it was on a picture of a

A defense of logical conventionalism 41

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

6. A case for the conventionality of logic

Let’s grant the suggestion that what logic is has finally stabilized (as of themiddle of the last century). The standard view is that an advantage of first-order logic over alternative logics is that all three models of logic can beapplied to it – and arguably, all three models converge as equivalent in thefirst-order context. The equivalence of the rule-governed model andthe substitution model is established by the existence of equivalent char-acterizations of first-order logical truths in terms either of sentence-axiomsor in terms of axiom-schemata. The equivalence of these characterizationsin turn with the content-containment model is enabled by Gödel’s com-pleteness theorem, subject to the model-theoretic characterization of thecontent-containment model via models (in a background set theory).

This sophisticated theoretical package of first-order classical logic isn’treflected in the psychological capacities of the humans who adhere (collect-ively) to this model of reasoning. In saying this, I’m not alluding to the richand developing literature on human irrationality17; I’m pointing out, rather,that as we become more sophisticated in our study of the neurophysiological

conceptual relationship between the steps in a mathematical proof, remained purely metaphorical(or, at best, promissory) until the notion of algorithm in the context of artificial languages emergedat the hands of Turing, Church, and others in the twentieth century.

16 See (Azzouni 2005: 18–19). It should be noted that this dramatic aspect of informal-rigorousmathematical proof is still with us despite the presence of formal systems that are apparently fullyadequate to contemporary mathematics. That is, informal-rigorous mathematical proof continues tooperate largely by conceptual implication – supplemented, of course, with substantialcomputational bits.

17 Nicely popularized by one of the major researchers in the area: See (Kahneman 2011).

42 Jody Azzouni

basis for our capacities for mathematics, and for reasoning generally, there isno echo in our neuropsychological capacities to reason, and to prove, of thesemantic/syntactic apparatus the contemporary view of logic (and even itscompetitors) provides.18 That apparatus is an all-purpose topic-neutral pieceof algorithmic machinery; how we actually reason, by contrast, involvesquite topic-specific, narrowly applied, highly componentalized,mental tools.This means that the role of formal logic can only be a normative one; it hasemerged as a reasoning tool that we officially impose on our ordinaryreasoning practices and that we (at times) can use to evaluate that reasoning.19

The foregoing, if right, makes the conventionality of logic quite plaus-ible even if it’s an optimal logic, compared to competitors.20 The fore-going, if right, also makes plausible the emergence of classical logic asexplicitly conventional in the twentieth century; and it makes plausible itsrole as tacitly conventional (at least in mathematical reasoning) for earliercenturies – before sets of rules for logic became explicit. I turn now todiscussing some of the reasons philosophers have for denying logic such aconventional status. The first kind of objection I’ll consider turns on howthe notion of truth is used in the characterization of validity; next I’llevaluate certain arguments that have been offered for why logical principleshave a (metaphysical) representational role.

7. Criticisms of the truth-preservation characterization of logic

We philosophers are all pretty familiar with the apparent truism – theapparent explanatory cliché, the apparently essential characterization – of

18 See (Carey 2009), especially chapter 4 – also see (Dehaene 1997) – for good introductions to thisremarkable and important empirical literature.

19 I’ve argued that this role of formal logic has emerged in the course of the twentieth century; it firstoccurred in mathematics but has spread throughout the sciences in large part because of themathematization 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 historicalcontingencies that seem involved in why the tacit employment of logical consequence inmathematical practice turned out to be in the neighborhood of a first-order and classical one: oneof those, I suggest (Azzouni 2005), is the psychological impression (on a case-by-case basis) that theintroduction and elimination rules for the logical idioms (“and,” “or,” “not,” etc.) are content-preserving, an impression that isn’t sustained for even quite short inference patterns, such as modusponens or syllogism.

20 Some philosophers argue that classical first-order logic isn’t optimal because of its representationaldrawbacks: proponents of higher-order logics (e.g., Shapiro), on the one hand, think that it can’trepresent mathematical concepts such as “finite,” proponents of one or another paraconsistentapproach (e.g., Priest) think it can’t represent certain global concepts, e.g., “all sentences.” AlthoughI’ve weighed in on these debates, they don’t matter for the issue of whether logic is conventionalprecisely because it’s been established in Section 3 of this chapter that suboptimality in relation tocompetitors doesn’t bar a practice from nevertheless being an alternative candidate.

A defense of logical conventionalism 43

deductive reasoning preserving truth: If the premises are true then theconclusion is true (must be true) as well. Many philosophers have takentruth-preservation to be a characterization of classical logical principles. Ifthe notion of truth, in turn, is a correspondence notion, then it wouldseem to follow that classical logic is semantically rooted in metaphysics, inwhat’s true about the world. And, it might be thought that what followsfrom this is that logic cannot be conventional. This argument-strategy failsfor a large number of reasons; for current purposes, I’ll focus on only threeof its failures. The first is that a characterization of deduction as “truth-preserving” fails to single out any particular set of logical rules – it fails toeven require that a set of logical principles be consistent! The second isthat, in any case, even if a characterization of logic in terms of truth-preservation singled out only classical logic (and not its alternatives), thatwouldn’t rule out the conventionality of classical logic: suboptimality ofalternatives is no bar to their rendering a practice conventional. The lastreason is that truth, in any case, is too frail an idiom to root logicsemantically in the world. This is because it functions perfectly adequatelyin discourses that bear no relationship to what exists.

The first claim is easy to prove. Relevant is that the truth idiom isgoverned by Tarski biconditionals: given a sentence S and a name of thatsentence “S,” “S ” is true iff S. Also relevant is that this condition can’t besupplemented by adding conditions to either wing of the Tarski bicondi-tionals that aren’t equivalent to the wings themselves.21 But these pointsare sufficient to make the truth idiom logically promiscuous: it’s compatiblewith 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 “truth-preserving” follows trivially, regardless of whether R is consistent ornot: If U ‘ V according to R, then, using T, we can show: U ‘ V iffT“U ” ‘ T“V ” holds in [R, T].

Notice that a characterization of a choice of logic being “legislated-true” is licensed by the foregoing: Start by choosing one’s logic, and thensupplement that choice with the T-schema. The resulting logic has been“legislated-true.” It might be thought that more substantial uses of thetruth idiom, in semantics and in model theory, can’t be executed in thecontext of a nonclassical logic. But this isn’t true either. In particular,

21 See (Azzouni 2010), 4.8 and 4.9.

44 Jody Azzouni

a model theory – characterized metalogically using intuitionistic con-nectives in the metalanguage – is homophonic to classical modeltheory.22

The second point has already been established: Imagine (contrary towhat has just been shown) that a population adopts a suboptimal set oflogical principles – ones strictly weaker than ours. (Intuitionist principles,for example.) Then one possible result would be a failure to know all sortsof things, both empirically and in pure mathematics, that we proponentsof classical logic know. Let’s say that this is suboptimal;23 but this is hardlyfatal. And so the conventionality of logic isn’t threatened by the presumedsuboptimality of other candidates.Lastly, a number of philosophers have thought that the Tarski bicondi-

tionals all by themselves characterize “truth” as a correspondence notion.There are many reasons to think they are wrong about this. Among them isthe fact that if a consistent practice of using nonreferring terms, such as“Hercules” or “Mickey Mouse” is established, such a practice remainsconsistent if it’s augmented with the T schema. Regardless of whetherthe truth idiom functions as a correspondence notion for certain dis-courses, it won’t function that way in this discourse. That shows that talkof truth has to be supplemented somehow to give it metaphysical traction.All by itself, it doesn’t do that job.The point generalizes, of course. In trying to determine whether logic is

conventional, some philosophers focus on specific statements like “Either itis raining or it is not raining,” and worry about whether this statement isabout the world or not; more dramatically, some philosophers worry aboutwhether the supposed conventionality of logic yields the result that we“legislate” the truth of a statement like this.24

But this misses the point. The claim that logic “is conventional” isorthogonal to the question of whether “logical truths” have content(worldly or otherwise), or (equivalently?) whether they are or aren’t “aboutthe world.” No doubt some philosophers have thought these claimslinked – especially philosophers (like the paradigmatic “positivists” influ-enced by Wittgensteinian Tractarian views) who are driven by epistemic

22 See (Azzouni 2008b), especially sections V and VI.23 Two issues drive my choice of the qualification-phrase: “let’s say.” First, mathematical possibilities

are richer in the intuitionist context than they are in the classical context – that could easily countagainst the supposed suboptimality of intuitionistic mathematics. Second, there are a lot of resultsthat show that the apparent restrictions of intuitionist mathematics – and constructivistmathematics, more generally – in applied mathematics can be circumvented.

24 See (Sider 2011: 203–204).

A defense of logical conventionalism 45

motives to deprive logical principles of “content.” The issue, to repeat, isn’twhether particular logical truths are or aren’t about the world, but insteadwhether our current set of logical principles lives in a space of viablecandidate alternatives. In addition, the claim that logical principles (ortruth) are about the world isn’t to be established by ruling out suchworldly content on the part of statements like “Either it is raining or itis not raining,” but by ruling in such worldly content on the part ofstatements like “Either unicorns have one horn or unicorns don’t haveone horn.”

I’d like to close out this section with a couple of remarks about thecurious project of trying to find individual representational contents forlogical idioms, such as disjunction, conjunction, and so on.25 Oneextremely natural way to try going about this is to give such notionscontent on an individual basis via introduction and elimination rules.We then understand the content of “and” (“&”) to be characterizedby 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. Anevident danger with this approach is that the holistic nature of “logicalcontent” emerges clearly when it’s recognized, for example, thatintuitionistic logic can be characterized by exactly the same introduc-tion and elimination rules, with the one exception of negation. Thatlogical truths not involving negation are nevertheless affected is an easytheorem.26

We can instead attempt to capture the individualized contents of theconnectives “semantically,” via truth tables for example. The problemhere is that truth tables are simply descriptions of truth conditions inneatly tabular form: e.g., A or B is true iff (A is true and B is not true) or(A is false and B is true) or (A is true and B is true). As noted earlier inthis section, such an approach simply amounts to a characterization oflogical principles (in a metalanguage) using those very same logicalprinciples plus the T-schema. The holism problem therefore is still withus. The appearance that we are semantically characterizing logical idiomson an individual basis, that is, is still the same illusion that we experiencewhen we approach the project directly by attempting to characterize thecontent of logical idioms individually, using natural deduction principles(for example).

25 Although the discussion is murky (or perhaps just metaphorical), this seems to be part of the projectundertaken 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.

46 Jody Azzouni

8. How Walker and Sider beg the questionagainst logical conventionalism

Much of the argument I’ve offered here has involved technical details thathave been deliberately kept off-stage. That was a necessity because technicaldetails in a paper restricted to eight thousand words must largely be kept inthe background for purely spatial reasons: to describe these technical detailsin even terse self-contained detail would expand the paper greatly – e.g.,details about the role of the truth idiom in metalanguages when character-izing a set of logical principles, or details about how the consequencerelation is holistically affected by how individual logical idioms synergistic-ally interact. But an important warning is in order. Discussion of theseissues – specifically, the issue of the conventionality of logic – often takesplace at an informal level that masks the fact that relevant technical pointsare 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 isnot 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 itsdisjuncts holds; to legislate-true a disjunction one would need to legislate-true one of its disjuncts. . . . It is open, of course, for the defender of truthby convention to supply a notion of legislating-true on which the argu-ment’s premises are false. The challenge, though, is that the premises seemcorrect 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 beggingthe question against logical conventionalism) is to implicitly adopt a loftymetalanguage stance, and then use the very words that are under conten-tion against the opponent. That doing this is so “intuitive” evidentlycontributes 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 non-contradiction, cannot be argued with. If they insist on their refusal there

A defense of logical conventionalism 47

is therefore nothing to be done about it, but for the same reason there is noneed to take them seriously.27

But this argument is just awful. Even if an opponent refuses to rely onmodus ponens as a law of logic, this doesn’t mean that opponent won’t beable engage in a debate using specific inferences that fall under classical modusponens. This is because all it means to deny modus ponens as a logicalprinciple is to claim that it has exceptions. That can nevertheless leaveenormous common ground for debate – that is, for arguments that bothdebaters take to be sound.28

Even if the reader who has gotten this far in the paper isn’t (or isn’tfully) convinced by the details of the intricate philosophical argument onoffer (both onstage and off ), I can at least hope the following take-awaymessage is convincing: This is that the issue of whether or not logic is“conventional” is a subtle and intricate (and interesting) philosophicalquestion that can’t be successfully adjudicated by merely superficiallyrehearsing Quine’s old arguments against “truth by convention,” andsupplementing that rehearsal with a semantic argument for the representa-tional content of logic that blatantly presupposes the very logical idiomsunder dispute. Also pertinent (or so I would have thought) is a discussionof the philosophical literature on tacit conventionality that has emergedsubsequent to Quine, including the relevant empirical results. I also think(and have tried to illustrate) that needed as well is a moderately deepdiscussion of whether and in what ways the attribution of “representa-tional” contents to logical idioms does or doesn’t contradict the supposedconventionality of logic.

27 See (Lewis 2005), where a similar refusal on similar grounds to debate the law of non-contradictionis expressed; see (van Inwagen 1981) for the same maneuver directed towards substitutionalquantification. As I said: it’s a popular maneuver with many illustrious practitioners.

28 Metalogical debates, in particular, are ones where proponents can easily debate one another oncommon ground, as many clearly do in the philosophical literature. See (Azzouni and Armour-Garb2005) for details.

48 Jody Azzouni

cha p t e r 3

Pluralism, relativism, and objectivityStewart Shapiro

I have been arguing of late for a kind of relativism or pluralism concerninglogic (e.g., Shapiro 2014). The main thesis is that there are different logicsfor different mathematical structures or, to put it otherwise, there isnothing illegitimate about structures that invoke non-classical logics, andare rendered inconsistent if excluded middle is imposed. The purpose ofthis chapter is to explore the consequences of this view concerning a coremetaphysical 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 debateon the issues appears to be bogged down because the participants do notuse the terms the same way. One group of philosophers uses the word“relativism” for what another group calls “contextualism”. So, in order toavoid getting lost in cross-purposes, we need a brief preliminary concern-ing 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 isno such thing as simply being Φ”. If Φ is relative, in this sense, then in orderto get a truth-value for a statement in the form “a is Φ”, one must implicitlyor explicitly indicate something else. A major discovery of the early twentiethcentury is that simultaneity and length are relative, in this sense. To get atruth-value for “a is simultaneous with b”, one needs to indicate a frame ofreference. 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 istasty”, one must indicate a judge, a taster, a standard, or something like that.This folk notion of “relativism” seems to be the one treated in Chris

Swoyer’s 2003 article in the Stanford Internet Encyclopedia of Philosophy.Swoyer suggests that discussions of relativism, and relativistic proposals,focus on instances of a “general relativistic schema”:

ðGRSÞ Y is relative to X :

49

In other words, in order to formulate a relativistic proposal, one firstspecifies what one is talking about, the “dependent variable” Y, and thenwhat that is alleged to be relative to, the “independent variable” X. So,according to special relativity, the dependent variable is for simultaneityand other temporal or geometric notions like “occurs before”, and phraseslike “has the same length as”. The independent variable is for a referenceframe. For predicates of personal taste, the independent variable is for agiven 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 folk-relativism concerning logical validity. They begin with what they call the“Generalised Tarski Thesis” (p. 29):

An argument is validx if and only if, in every casex in which the premises aretrue, so is the conclusion.

For Beall and Restall, the variable x ranges over types of “cases”. Classical logicresults from the Generalized Tarski Thesis if “cases” are Tarskian models;intuitionistic logic results if “cases” are constructions, or stages in construc-tions (i.e., nodes in Kripke structures); and various relevant and paraconsis-tent logics result if “cases” are situations. So Beall and Restall take logicalconsequence to be relative to a kind of case, and the General RelativisticSchema is apt. For them, the law of excluded middle is valid relative toTarskian models, invalid relative to construction stages (Kripke models).

Beall and Restall call their view “pluralism”, eschewing the term“relativism”:

we are not relativists about logical consequence, or about logic as such. Wedo not take logical consequence to be relative to languages, communities ofinquiry, contexts, or anything else. (p. 88, emphasis in original)

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

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

50 Stewart Shapiro

I propose below, and elsewhere, a particular kind of folk-relativism forlogic. The dependent variable Y is for validity or logical consequence, andthe independent variable X ranges over mathematical theories or, equiva-lently, structures or types of structures. The claim is that different theories/structures have different logics.Once it is agreed that a given word or phrase is relative, in the foregoing,

folk sense, then one might want a detailed semantic account that explainsthis. Are we going to be contextualists, saying that the content of the termshifts in different contexts? Or some sort of full-blown assessment-sensitiverelativist (aka MacFarlane (2005), (2009), (2014))? Questions of meaning,our present focus, thus come to the fore, and will be broached below. But,as construed here, folk-relativism, by itself, has no ramifications concern-ing semantics.Briefly, pluralism about a given subject, such as truth, logic, ethics, or

etiquette, is the view that different accounts of the subject are equallycorrect, or equally good, or equally legitimate, or perhaps even (equally)true (if that makes sense). Arguably, folk-relativism, as the term is usedhere, usually gives rise to a variety of pluralism, as that term is used here.All we need is that some instances of the “independent variable” in the(GRS) correspond to correct, or good, versions of the dependent variable.Define monism or logical monism to be the opposite of logical

relativism/pluralism. The monist holds that there is such a thing assimply being valid – full stop. The slogan of the monist is that there isOne True Logic.

1. Relativity to structure

Since the end of the nineteenth century, there has been a trend inmathematics that any consistent axiomatization characterizes a struc-ture, one at least potentially worthy of mathematical study. A keyelement in the development of that trend was the publication of DavidHilbert’s Grundlagen der Geometrie (1899). In that book, Hilbert pro-vided (relative) consistency proofs for his axiomatization, as well as anumber of independence proofs, showing that various combinations ofaxioms 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 aharmless kind that to use that word to promote it would dramatise the position too much.” Thepresent label “folk-relativism” is similarly meant to cut down on dramatic effect.

Pluralism, relativism, and objectivity 51

consistency of the axioms of geometry, since the axioms are all true(presumably of space).2 Hilbert replied:

As long as I have been thinking, writing and lecturing on these things,I have been saying the exact reverse: if the arbitrarily given axioms do notcontradict each other with all their consequences, then they are true and thethings defined by them exist. This is for me the criterion of truth andexistence.

The slogan, then, is that consistency implies existence.It seems clear, at least by now, that this Hilbertian approach applies, at

least approximately, to much of mathematics, if not all of it. Consistency,or some mathematical explication thereof, like satisfiability in set theory, isthe only formal criterion for legitimacy – for existence if you will. Ofcourse, one can legitimately dismiss a proposed area of mathematical studyas uninteresting, or unfruitful, or inelegant, but if it is consistent, orsatisfiable, then there is no further metaphysical, formal, or mathematicalhoop the proposed theory must jump through before being legitimatemathematics.

But what of consistency? The crucial observation is that consistency is amatter of logic. In a sense, consistency is (folk) relative to logic: a giventheory may be consistent with respect to one logic, and inconsistent withrespect to another.

There are a number of interesting and, I think, fruitful theories thatinvoke intuitionistic logic, and are rendered inconsistent if excludedmiddle is added. I’ll briefly present one such here, smooth infinitesimalanalysis, a sub-theory of its richer cousin, Kock–Lawvere’s synthetic differ-ential geometry (see, for example, John Bell 1998). This is a fascinatingtheory of infinitesimals, but very different from the standard Robinson-style non-standard analysis (which makes heavy use of classical logic).Smooth infinitesimal analysis is also very different from intuitionisticanalysis, both in the mathematics and in the philosophical underpinnings.

In the spirit of the Hilbertian perspective, Bell presents the theoryaxiomatically, albeit informally. Begin with the axioms for a field, andconsider the collection of “nilsquares”, numbers n such that n2 = 0.Of course, in both classical and intuitionistic analysis, it is easy toshow that 0 is the only nilsquare: if n2 = 0, then n = 0. But not here.Among the new axioms to be added, the most interesting is the principle

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

52 Stewart Shapiro

of micro-affineness, that every function is linear on the nilsquares.Its interesting consequence is this:

Let f be a function and x a number. Then there is a unique number d suchthat for any nilsquare α, f (x þ α) = f x þ dα.

This number d is the derivative of f at x. As Bell (1998) puts it, thenilsquares constitute an infinitesimal region that can have an orientation,but is too short to be bent.3

It follows from the principle of micro-affineness that 0 is not the onlynilsquare:

:ð8αÞðα2 ¼ 0 ! α ¼ 0Þ:Otherwise, the value d would not be unique, for any function. Recall,however, that in any field, every element distinct from zero has a multi-plicative inverse. It is easy to see that a nilsquare cannot have a multiplica-tive inverse, and so no nilsquare is distinct from zero. In other words, thereare no nilsquares other than 0:

ð8αÞ�α2 ¼ 0 ! ::ðα ¼ 0Þ�,which is just

ð8αÞ�α2 ¼ 0 ! :ðα 6¼ 0Þ�:So, to repeat, zero is not the only nilsquare and no nilsquare is distinctfrom zero. Of course, all of this would lead to a contradiction if we alsohad (8x)(x = 0_x 6¼ 0), and so smooth infinitesimal analysis is incon-sistent with classical logic. Indeed, :(8x)(x = 0_x 6¼ 0) is a theorem ofthe theory (but, since the logic is intuitionist, it does not follow that(9x):(x = 0_x 6¼ 0)).Smooth infinitesimal analysis is an elegant theory of infinitesimals,

showing that at least some of the prejudice against them can be traced tothe use of classical logic – Robinson’s non-standard analysis notwithstand-ing. Bell shows how smooth infinitesimal analysis captures a number ofintuitions about continuity, many of which are violated in the classicaltheory of the reals (and also in non-standard analysis). Some of theseintuitions have been articulated, and maintained throughout the historyof philosophy and science, but have been dropped in the main contem-porary account of continuity, due to Cantor and Dedekind. To take one

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

Pluralism, relativism, and objectivity 53

simple example, a number of historical mathematicians and philosophersfollowed Aristotle in holding that a continuous substance, such as a linesegment, cannot be divided cleanly into two parts, with nothing created orleft over. Continua have a sort of unity, or stickiness, or viscosity. Thisintuition is maintained in smooth infinitesimal analysis (and also inintuitionistic analysis), but not, of course, in classical analysis, which viewsa continuous substance as a set of points, which can be divided, cleanly,anywhere.

Smooth infinitesimal analysis is an interesting field with the look andfeel of mathematics. It has attracted the attention of mainstream mathem-aticians, people whose credentials cannot be questioned. One would thinkthat those folks would recognize their subject when they see it. The theoryalso seems to be useful in articulating and developing at least someconceptions of the continuum. So one would think smooth infinitesimalanalysis should count as mathematics, despite its reliance on intuitionisticlogic (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 thiswould depend on there being no theories that invoke a logic differentfrom those two. Admittedly, I know of no examples that are as compel-ling (at least to me) as the ones that invoke intuitionistic logic. Forexample, I do not know of any interesting mathematical theories thatare consistent with a quantum logic, but become inconsistent if thedistributive principle is added. Nevertheless, it does not seem wise tolegislate for future generations, telling them what logic they must use, atleast not without a compelling argument that only such and such a logicgives rise to legitimate structures. One hard lesson we have learned fromhistory is that it is dangerous to try to provide a priori, armchairarguments concerning what the future of science and mathematicsmust 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 classicaland intuitionistic logic, any inconsistent theory is trivial. A logic is calledparaconsistent if it does not sanction the ill-named inference of ex falsoquodlibet. Typical relevance logics are paraconsistent, but there are para-consistent logics that fail the strictures of relevance. The main observationhere is that with paraconsistent logics, there are inconsistent, but non-trivial theories.

54 Stewart Shapiro

If we are to countenance paraconsistent logics, then perhaps we shouldchange the Hilbertian slogan from “consistency implies existence” tosomething like “non-triviality implies existence”. To transpose the themes,on this view, non-triviality is the only formal criterion for mathematicallegitimacy. One might dismiss a proposed area of mathematical study asuninteresting, or unfruitful, or inelegant, but if it is non-trivial, then thereis no further metaphysical, formal, or mathematical hoop the proposedtheory 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, mighthold that there are no criteria for mathematical legitimacy. There is nometaphysical, formal, or mathematical hoop that a proposed theory mustjump 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 indus-try of developing and studying such theories.4 It is claimed that such theoriesmay even have applications, perhaps in computer science and psychology.I will not comment here on the viability of this project, nor on howinteresting and fruitful the systems may be, nor on their supposed applica-tions. I dowonder, however, what sort of argument onemight give to dismissthem 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 theoriginal version where consistency is the only formal, mathematicalrequirement on legitimate theories, or the liberal orientation where thereare no formal requirements on legitimacy at all. And let us assume that atleast some non-classical theories are legitimate, without specifying whichones those are. I propose to explore the ramifications for what I take to be alongstanding intuition that logic is objective. One would think logic has tobe objective, if anything is, since just about any attempt to get at the world,as it is, will depend on, and invoke, logic.

2. What is objectivity?

Intuitively, a stretch of discourse is objective if the propositions (orsentences) 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.

Pluralism, relativism, and objectivity 55

preferences, and the like. Many of the folk-relative predicates are charac-teristic of paradigm cases of non-objective discourses. Whether somethingis tasty, it seems, depends on the judge or standard in play at the time. Sotaste is not objective (or so it seems). Whether something is rude dependson the ambient location, culture, or the like. So etiquette is folk-relativeand, it seems, not objective. Etiquette may not be subjective, in the sensethat 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 humanjudgment, preferences, and the like.

One would be inclined to think that simultaneity and length areobjective, even though both are folk-relative, given relativity. As is thecase with much in philosophy (and everywhere else), it depends on whatone means by “objective”. We are told that whether two events aresimultaneous, and whether two rods are of the same length, depends onthe perspective of the observer. Does that undermine at least some of theobjectivity? But, vagueness and such aside, time and length do not seem todepend on anyone’s judgment or feelings, or preferences. A given observercan be wrong about whether events are simultaneous, even for eventsrelative to her own reference frame.

One might say that a folk-relative predicate P is objective if, for eachvalue n of the independent variable, the predicate P-relative-to-n does notdepend on anyone’s judgment or feelings. For example, if a given subjectcan be wrong about P-relative-to-n, then the relevant predicate is objective.However, even an established member of a given community can be wrongabout what is rude in that community. But one would not think thatetiquette is objective, even when restricted to a given community.

Clearly, to get any further on our issue, we do have to better articulatewhat objectivity is, at least for present purposes. Again, objectivity is tied toindependence from human judgment, preferences, and the like. There is atrend to think of objectivity in straightforward metaphysical terms. It mustbe admitted that this has something going for it. The idea is that some-thing, say a concept, is objective if it is part of the fabric of reality. Themetaphor 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 andrudeness 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?

56 Stewart Shapiro

One might argue that there can be at most one logic that is objective, inthis metaphysical sense. Sider does argue that at least parts of logic arefundamental. As it happens, the logic he discusses is classical, but, so far asI know, there is no argument supporting that choice of logic. It might becompatible with his overall program that, say, parts of intuitionistic logicor a relevant logic are fundamental instead. But perhaps two distinct logicscannot both be fundamental. Contraposing, if the present folk-relativismabout logic is correct, then logic is not objective, in the foregoingmetaphysical 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 arenot so fundamental can still be objective. Intuitively, the fact that theMiami Heat won the NBA title in 2012 is objective (like it or not), but (Ipresume) it is hardly fundamental. One can call a proposition objective if itsomehow supervenes on fundamental matters, but that requires one toaccept a contentious metaphysical framework, and to articulate the rele-vant notion of supervenience.More important, perhaps, several competing philosophical traditions

have it that there simply is no way to sharply separate the “human” andthe “world” contributions to our theorizing. Protagoras supposedly saidthat man is the measure of all things. On some versions of idealism, not tomention some postmodern views, the world itself has a human character.The world itself is shaped by our judgments, observations, etc. Perhapssuch views are too extreme to take seriously. A less extreme position isKant’s doctrine that the ding an sich is inaccessible to human inquiry. Weapproach the world through our own categories, concepts, and intuitions.We cannot get beyond those, to the world as it is, independently of saidcategories, 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 behad, no perspective from which we can compare our theories of the worldto the world itself, to figure out which are the “human” parts of oursuccessful theories and which are the “world” parts (see, for example,Burgess and Rosen 1997). On such views, the world, of course, is not ofour making, but any way we have of describing the world is in humanterms. As Friedrich Waismann once put it:

What rebels in us . . . is the feeling that the fact is there objectively nomatter in which way we render it. I perceive something that exists and put

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it into words. From this, it seems to follow that something exists inde-pendent of, and prior to language; language merely serves the end ofcommunication. What we are liable to overlook here is the way we see afact – 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 object-ivity to be had, or at least no objectivity that we can detect. Perhapsobjectivity is a flawed property, going the way of phlogiston and caloric, orwitchcraft. If this is right, then there simply is no answering the question ofthis paper – folk-relativism or no folk-relativism. Logic is not objective,since nothing is. Despite having sympathy with the Kant – Quine orien-tation, I would resist this rather pessimistic conclusion. There may not besuch a thing as complete objectivity – whatever that would be – but it stillseems that there is an interesting and important notion of objectivity tobe clarified and deployed. There seems to be an important difference – adifference in kind – between statements like “the atmosphere containsnitrogen” and statements like “the Yankees are disgusting”. The distinctionmay be vague and even context dependent, but it is still a distinction, and,I think, an important one. Our question concerns whether the presentfolk-relative logic falls on one side or the other of this divide (or perhaps onor near its borderline).

Crispin Wright’s Truth and objectivity (1992) contains an account ofobjectivity that is more comprehensive than any other that I know of,providing a wealth of detailed insight into the underlying concepts. Wrightdoes not approach the matter through metaphysical inquiry into the fabricof reality, wondering whether the world contains things like moral prop-erties, funniness, or numbers. He focuses instead on the nature of variousdiscourses, and the role that these play in our overall intellectual andsocial lives.

As Wright sees things, objectivity is not a univocal notion. There aredifferent notions or axes of objectivity, and a given chunk of discourse canexhibit some of these and not others. The axes are labeled “epistemicconstraint”, “cognitive command”, “the Euthyphro contrast”, and “thewidth of cosmological role”. In a previous paper, (Shapiro 2000), I arguethat logic easily passes all of the tests. The conclusion is (or was) that, oneach 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 object-ivity and non-objectivity, since most of the tests presuppose logic. That is,to figure out whether a given stretch of discourse is objective, on this orthat axis, one must do some logical reasoning or figure out what is

58 Stewart Shapiro

consistent, or the like. So it is hard to even apply the framework to mattersof 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 leaston the intuitive conception of objectivity. According to Resnik, to call anargument valid, or to call a theory consistent, is to manifest a certainattitude toward the theory.5

The present relativism/pluralism was not on the agenda then. The planhere is to return to the matter of objectivity with the present folk-relativism concerning logic in focus. Sometimes we will concentrate ongeneral logical matters, such as validity and consistency, as such, andsometimes we will deal with particular instances of the folk-relativism,such as classical validity, intuitionistic consistency, and the like. We willlimit the discussion to Wright’s axes of epistemic constraint and cognitivecommand.

3. Epistemic constraint

Epistemic constraint is an articulation of Michael Dummett’s (1991a)notion of anti-realism. According to one of Wright’s formulations, adiscourse is epistemically constrained if, for each sentence P in thediscourse,

P $ P may be known: ðp: 75ÞIn other words, a discourse exhibits epistemic constraint if it contains nounknowable truths.6

It seems to follow from the very meaning of the word “objective” that ifepistemic constraint fails for a given area of discourse – if there arepropositions in that area whose truth cannot become known – thenthat discourse can only have a realist, objective interpretation. AsWright puts it:

To conceive that our understanding of statements in a certain discourse isfixed . . . by assigning them conditions of potentially evidence-transcendent

5 It is perhaps ironic (or at least interesting) that Resnik argues against pluralism and relativism aboutlogic. 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 difference between the absence ofunknowable truths and the truth of the biconditional: P $ P may be known. That differencedoes 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 notpursue this further here.

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

In other words, if epistemic constraint fails for a given discourse, then it isobjective, and that is the end of the story. The other axes of objectivity –cognitive command, cosmological role, and the Euthyphro contrast – arethen irrelevant; they do not track a sense of objectivity (if the axis can beapplied at all). Or so Wright argues.

So what of logic? Are there, or could there be, unknowable truthsconcerning logical consequence, consistency, and the like? The presentfolk-relativism concerning logic pushes that question in a certain direction.Consider a given argument, or argument form Δ, and let P be a statementthat Δ is valid. Could something like P be an unknowable truth?

Not as it stands, but that is because, absent context, P is not a truth(or a falsehood) at all. According to the present folk-relativism, in orderto get a truth-value for P, one must specify something else, such as aparticular mathematical theory, a structure, or perhaps just a logic. Wehave to ask separately whether Δ is valid in classical logic, in intuitionisticlogic, in various relevant logics, etc. So it seems to me that in order toask whether logic is epistemically constrained, we have to considerstatements of validity and the like with the logic made explicit. We mustconsider statements in the form, Δ is valid in logic L, where L is one ofthe logics that can go in for the dependent variable in the generalrelativistic scheme.

To push the analogy, consider, again, relativity. Let p and q be twoevents. Say that p is a runner in baseball leaving third base, and q is anoutfielder catching a fly ball. Consider the statement S that p occurredbefore q (which an umpire sometimes has to adjudicate). According torelativity, we cannot get a truth-value for S without specifying a frame ofreference. So, a fortiori, we cannot even ask if there is an unknowable truthfor a statement about what happened before what without indicating areference frame. If a reference frame is specified (implicitly or explicitly),then, it seems, there can be unknowable truths in this area. For example, it

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may be unknowable whether the runner left base before the ball wascaught, from the perspective of the home plate umpire. For example, itmay be too dark or no human can see distinctions that fine, or whatever.So matters of temporal order, from a given reference frame, are notepistemically constrained. And, intuitively, matters of temporal order areobjective, vagueness aside.The point here is that with folk-relative discourses, we can only ask

about epistemic constraint for statements that have the relevant parametersfully specified, at least implicitly. So the central question is whether therecan be unknowable truths concerning whether the argument (form) Δ isvalid in a given logic L?In effect, this matter was dealt with in my earlier paper (Shapiro 2000),

and also in Shapiro (2007), which concerns mathematics. Classical logicwas in focus then, but to some extent, the argument generalizes. Whetherthere are unknowable truths in this area depends on what one means by“unknowable”. If we do not idealize on the knowers, then of course therecan be unknowable truths. Suppose that our argument Δ is an instance of&-elimination in which the premise and conclusion each have, say, 10100

characters. Then Δ is valid in, say, classical logic, but no one canknow that, since no one can live long enough to check that Δ is an instanceof &-elimination.So, to give epistemic constraint a chance of being fulfilled, we have to

idealize on the knowers. One sort of idealization is familiar. We assumethat our knowing subjects have unlimited (but still finite) time, attentionspan, and materials at their disposal, and that they do not make any simplecomputation errors. These idealizations are common throughout math-ematics, and we take them to be conceptually unproblematic (and thus weset aside issues concerning rule-following, as in, say Kripke (1982)). Then,if L is classical first-order logic, or intuitionistic logic, or most of therelevant logics, and Δ is an arbitrary argument form (with finitely manypremises and conclusions), then a statement that Δ is valid in logic L is trueif and only if that fact is knowable (by one of our ideal agents). That isbecause each of those logics has an effective and complete deductivesystem.Things are not so clear if the logic in question is classical second-order

logic, since its consequence relation is not effectively enumerable. Nor arethings so clear for statements that a given argument Δ is not valid in one ofthe aforementioned logics. Invalidity is not recursively enumerable, and sochecking invalidity is not a matter of running an algorithm. So if we are toinsist that all matters of logic are epistemically constrained, once the logic

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is fixed, we have to attribute to our knowers the abilities to decidemembership in non-recursive sets.

Things get vexed here. It is not at all clear what the relevant modality isfor the key phrase “knowable”. Moreover, as noted, the issues are essen-tially the same as with monism concerning logic. So I propose to just takeit as given, for the sake of argument, that the relevant discourse isepistemically constrained, in at least some relevant sense, so that we canmove on to another of Wright’s axes of objectivity.

4. Cognitive command

Assume that a given area of discourse serves to describe mind-independentfeatures of a mind-independent world. In other words, assume that thediscourse in question is objective, in an intuitive, or pre-theoretic sense.Suppose now that two people disagree about something in that area. Itfollows that at least one of them has misrepresented reality, and so some-thing went wrong in his or her appraisal of the matter. Suppose, forexample, that two people are arguing whether there are seven, as opposedto eight, spruce trees in a given yard. Assuming that there is no vaguenessconcerning what counts as a spruce tree and no indeterminacy concerningthe boundaries of the yard, or whether each tree is in the yard or not, itfollows that at least one of the disputants has made a mistake: either shedid not look carefully enough, her eyesight was faulty, she did not knowwhat a spruce tree is, she misidentified a tree, she counted wrong, orsomething else along those lines. The very fact that there is a disagreementsuggests that one of the disputants has what may be called a cognitiveshortcoming (even if it is not always easy to figure out which one of them itis that has the cognitive shortcoming).

In contrast, two people can disagree over the cuteness of a given baby orthe humor in a given story without either of them having a cognitiveshortcoming. One of them may have a warped or otherwise faulty sense oftaste or humor, or perhaps no sense of taste or humor, but there need benothing wrong with his cognitive faculties. He can perceive, reason, andcount as well as anybody.

The present axis of objectivity turns on this distinction, on whetherthere can be blameless disagreement:

A discourse exhibits Cognitive Command if and only if it is a priori thatdifferences of opinion arising within it can be satisfactorily explained onlyin terms of “divergent input”, that is, the disputants working on the basis of

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

Intuitively, cognitive command holds for discourse about spruce trees(vagueness and indeterminacy aside) and it fails for discourse about thecuteness of babies and the humor of stories.Later in the book, Wright (1992: 144) adds some qualifications to the

formulation of cognitive command, meant to deal with matters likevagueness. A discourse exerts cognitive command if and only if

It is a priori that differences of opinion formulated within the discourse,unless excusable as a result of vagueness in a disputed statement, or in thestandards of acceptability, or variation in personal evidence thresholds, so tospeak, will involve something which may properly be described as a cogni-tive shortcoming.

So what of logic, again assuming the correctness of the foregoing folk-relativism? Let Δ be a given argument form, and consider two folks whodisagree – or seem to disagree – whether Δ is valid. One says it is and theother says it is not. Our question breaks into two, depending on whetherwe fix the logic. For our first type of case, let Δ be an instance of excludedmiddle 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. Forthe other sort of case, we fix the logic and ponder disputes concerning thatlogic. We imagine two folks who disagree – or seem to disagree – whetherΔ is valid in L, where L is, say, a particular relevant logic.We start with the second sort of case, disagreements that concern a fixed

logic. I would think that there is room for blameless disagreement con-cerning how a given argument, formulated in natural language, should berendered in a formal language. However, such issues would take us too farafield, broaching matters of the determinacy of meaning, the slippagebetween logical terms and their natural language counterparts, and theintentions of the arguer. It is not so clear whether a disagreement in howto render a natural language argument is “excusable as a result ofvagueness . . . or in the standards of acceptability, or variation in personalevidence thresholds” or the like.So let us set such matters aside, and just assume that our target

argument Δ is fully formalized. One of our characters says that Δ is valid

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in the given logic L and the other says that Δ is invalid in that logic L.Do we know (a priori) that at least one of them has a cognitiveshortcoming?

Suppose that the logic L is either defined in terms of a deductive systemor that there is a completeness theorem for it. So L can be classical first-order logic, intuitionistic logic, or one of the various relevant and para-consistent logics that are given axiomatically. So our disputants differ onwhether there is a deduction whose undischarged premises are among thepremises of Δ and whose last line is the conclusion of Δ. So, up toChurch’s thesis, our disputants differ over a Σ1-sentence in arithmetic,one in the form (9n)Φ, where Φ is a recursive predicate. So our questionconcerning cognitive command for this logic L reduces to whether cogni-tive command holds for these simple arithmetic sentences. I would thinkthat cognitive command does hold here. One of our disputants has made(what amounts to) a simple arithmetic error, and that surely counts as acognitive shortcoming. But I will rest content with the reduction. Cogni-tive command holds in this case if and only if it holds for 91-sentences (or,equivalently, Π1-sentences).

Now suppose that our fixed logic L is not complete. Say it is second-order logic, with standard, model-theoretic semantics. In that case, thequestion at hand reduces to set theory. Suppose, for example, that ourtarget argument Δ has no premises and that its conclusion is, in effect, thecontinuum hypothesis (see Shapiro 1991: 105). So Δ is valid if and only ifthe continuum hypothesis is true. So, in effect, our disputants differ overthe truth of the continuum hypothesis. Is that dispute cognitively blame-worthy? Surely, that would take us too far afield (but see Shapiro 2000,2007, 2011, 2012), and we will leave this case with the reduction.

Let us briefly consider the analogues of our question concerningcognitive command with our other examples of folk-relative predicates.Suppose that two judges differ on whether a certain event a occurredbefore another event b from the same frame of reference (putting aside thefact that this discourse is not epistemically constrained). Assume, forexample, that the two judges are in the same reference frame. Then,unless the disagreement is “excusable as a result of vagueness . . . or in thestandards of acceptability, or variation in personal evidence thresholds”,at least one of them exhibits a cognitive shortcoming. She did not lookcarefully enough, or did not time the events properly, or forgot some-thing. So cognitive command holds, and, of course, matters of temporalorder from a fixed frame of reference are intuitively objective. The samegoes for matters of length.

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Now consider two folks who disagree over whether a given food is tastyfor one and the same subject. Suppose, for example, that Tom and Dickdiffer over whether licorice is tasty to Harry. To keep things simple,assume that neither Tom nor Dick is Harry. Tom’s and Dick’s judgmentswould presumably be based on what Harry has told them and theirobservations of his reactions when eating licorice. We should assume thatTom and Dick have exactly the same body of such evidence (sinceotherwise one of them has the cognitive shortcoming of lacking relevantevidence). And we should set aside matters of “vagueness . . . standards ofacceptability, [and] variation in personal evidence thresholds”. Tom andDick may have come to opposite conclusions because they weighed certainpronouncements or reactions differently. In this case, perhaps, neither ofthem 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-objectivediscourse, but I am not sure whether discourse about Harry’s taste isobjective, intuitively speaking. Maybe we have a borderline case.Returning to matters logical, I’ve saved the hardest sort of situation for

last. That is on prima facie disagreements when the logic is not held fixed.To focus on a specific example, let Δ be an instance of the law ofexcluded middle (with no premises). Let h be a classicist who says thatΔ is valid and let b be an intuitionist who insists that Δ is not valid. Isthis a disagreement that is (cognitively) blameless? If so, then cognitivecommand fails here, and this aspect of logic falls on the non-objectiveside of this particular axis (assuming that cognitive command tracks a sortof 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 ofthem. So each is (cognitively) blameless, at least concerning this particularmatter. The only question remaining is whether they disagree. Here weencounter a matter that is treated extensively in the philosophical litera-ture, and I must report that the issues are particularly vexed. There doesnot seem to be much in the way of consensus as to what makes for adisagreement. John MacFarlane (2014), for example, articulates several

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

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different, and competing senses of “disagreement”. We will keep things ata more intuitive level, as far as possible.

One thesis, perhaps, is that a necessary condition on disagreement isthat the parties in question cannot both be correct. If so, and continuing toassume our folk-relativism concerning logic, we have that h and b do notdisagree. A fortiori, we do not have a case of blameless disagreement. Wecan still maintain that cognitive command holds when the logic is heldfixed, as above, and so logic passes this test for objectivity.

The thesis that in a disagreement both parties cannot be correct iscontroversial. It is sometimes taken as a criterion of being non-objectivethat parties can disagree and both be correct (see, for example, Barker2013). Suppose that Harry announces that licorice is tasty, and Jillresponds, “no it is not; licorice is disgusting”. That looks like a disagree-ment; Jill uses the language of disagreement, apparently denying Harry’sassertion. And yet, one might say, both are correct. Or at least one mightsay that both are correct.

To make any progress here, we have to get beyond the loose character-ization of folk-relativism and address matters of semantics. I’ll brieflysketch the framework proposed by John MacFarlane (2005), (2009),(2014) for interpreting expressions in a folk-relative discourse. The termsused by other philosophers and linguists can usually be translated into thisframework, though sometimes with a bit of loss.

Indexical contextualism about a given term is the view that the contentexpressed by the term is different in different contexts of use. The clearestinstances are the so-called “pure indexicals”, words like “I” and “now”. Thecontent expressed by the sentence “I am hungry”, when uttered by me on agiven day, is different from the content expressed by the same sentence,uttered by my wife at the same time. Intuitively, the first one says that I amhungry (then) and the second says that she is hungry (then). Clearly, theseare different propositions; they don’t say the same thing about the world –not to mention that one might be true and the other false.

Although very little is without controversy in this branch of philosophyof language, words like “enemy”, “left”, “right”, “ready”, and “local” seemapt 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 itis on the right). Intuitively, Jack and Jill do not disagree with each other,and the propositions they express are not contradictories. The reason is

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

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that the content of the word “left” is different in the two contexts. In thefirst, it means something like “to the left from Jill’s perspective” and inthe second it means “to the left from Jack’s perspective”. And they canboth be correct – intuitively one of them is correct just in case theother 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 extensioncan so vary according to a parameter determined by the context ofutterance.9 Suppose, for example, that a graduate student sincerely saysthat a local roller coaster is fun, and her Professor replies “No, that rollercoaster is not fun, it is lame”. According to a non-indexical contextualismabout “fun” (and “lame”), each of them utters a proposition that is thecontradictory of that uttered by the other – so they genuinely disagree. Yet,assuming both are accurately reporting their own tastes, each has uttered atruth, 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 theprofessor, 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 thatthe 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 studentand the professor each express a proposition contradictory to one expressedby the other. However, for the assessment-sensitive relativist, the term getsits extension from a context of assessment. Suppose, for example, that a thirdperson, a Dean, overhears the exchange between the graduate student andprofessor and, assume that the roller coaster is not fun-for-the-Dean.Then, from the context of the Dean’s assessment, the student uttered afalse proposition and the professor uttered a true one. And, from thegraduate student’s context of assessment, the Professor uttered a falseproposition, and from the Professor’s context of assessment, the studentuttered a false proposition.According to MacFarlane, the difference between non-indexical

contextualism and assessment-sensitive relativism is made manifest bythe phenomenon of retraction. That difference does not matter here, andwe can lump non-indexical contextualism and assessment-sensitive

9 Nearly all terms have different extensions in different possible worlds. That is not the sort ofcontextual variation envisioned here. For terms subject to non-indexical contextualism, therelevant 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 disputebetween our logicians h and b, then they do not disagree, and so cognitivecommand is saved. If we opt for a non-indexical contextualist or anassessment-sensitive interpretation, we do have a disagreement – in thesense that each of them accepts a content that is the contradictory of thataccepted by the other. As above, the disagreement is blameless (since bothare correct), and so cognitive command fails.

Recall that h says that our (fully formalized) argument Δ is valid and bsays that Δ is not valid. Recall that Δ is an instance of excluded middleΦ_:Φ, with no premises. There are two places to look here, but bothdeliver the same range of verdicts.

We can ask first about the content of the argument Δ. Do h and b meanthe same thing by the disjunction “_” and by negation “:”? We thusbroach the longstanding question of whether the classicist and the intu-itionist (or, indeed, advocates of any rival logics) are talking pasteach other.

Michael Dummett (1991a: 17) argues that the “disagreement” is merelyverbal:

The intuitionists held, and continue to hold, that certain methods ofreasoning actually employed by classical mathematicians in proving the-orems are invalid: the premisses do not justify the conclusion. The imme-diate effect of a challenge to fundamental accustomed modes of reasoning isperplexity: on what basis can we argue the matter, if we are not inagreement about what constitutes a valid argument? In any case how cansuch a basic principle of rational thought be rationally put in doubt?

The affront to which the challenge gives rise is quickly allayed by a resolveto take no notice. The challenger must mean something different by thelogical constants; so he is not really challenging the laws that we have alwaysaccepted and may therefore continue to accept.

Dummett goes on to argue that the classicist has no coherent meaning hecan assign to the connectives, but we can set that aside here (as inconsistentwith the foregoing folk-relativism).

From a very different perspective, W. V. O. Quine (1986: 81) also holdsthat the various connectives change their content in the different logicaltheories. Concerning the debate over paraconsistent logics, he wrote:

My view of this dialogue is that neither party knows what he is talkingabout. They think they are talking about negation, “�”, “not”; but surelythe notation ceased to be recognizable as negation when they took toregarding some conjunctions in the form “p.�p” as true, and stopped

68 Stewart Shapiro

regarding such sentences as implying all others. Here, evidently, is thedeviant logician’s predicament: when he tries to deny the doctrine he onlychanges 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, ifhe wishes to discuss it, he must state his methods clearly, and give syntac-tical rules instead of philosophical arguments.

Again, the key idea is that each logic is tied to a specific language.Presumably, the meaning of the logical terms differs in the differentlanguages.So the Dummett–Quine–Carnap perspective has it that we have a kind

of indexical contextualism here. The logical terms themselves have differ-ent contents for our characters h and b. Using a subscript-C to indicate aclassical connective and a subscript-I for the corresponding intuitionisticconnective, we have that h holds that Φ_C:CΦ is valid, while b holds thatΦ_I:IΦ is invalid. This is the same sort of situation as with Jack and Jilland the salt. There is no disagreement between h and b unless it be overwhether the other has a coherent meaning at all. If they are sufficientlyopen-minded, h and b might agree that Φ_C:CΦ is valid and that Φ_I:IΦis 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” concernsthe notion of validity for a single language, with a single batch oflogical terms. So there is not, for example, a separate “_C” and “_I”.There is just “_”. Restall (2002: 432) puts the difference with Dummett–Quine–Carnap well:

If accepting different logics commits one to accepting different languagesfor those logics, then my pluralism is primarily one of languages (whichcome with their logics in tow) instead of logics. To put it graphically, as apluralist, I wish to say that

A,:A ‘CB, but A,:A⊬RB

A and :A together, classically entail B, but A and :A together do notrelevantly entail B. On the other hand, Carnap wishes to say that

A,:CA ‘ B, but A,:RA⊬B

A together with its classical negation entails B, but A together with itsrelevant negation need not entail B.

Pluralism, relativism, and objectivity 69

So (Beall and) Restall reject an indexical contextualism concerning theconnectives (and quantifiers). Either there is no folk-relativism at all for theconnectives – each has a single, uniform content – or we have a non-indexical contextualism or an assessment-sensitive view.

Recall that h says that Δ is valid and b says that Δ is invalid. On theoption considered now, championed by Beall and Restall, we have that hand b mean the same thing by Δ. What about “valid”? Does that have thesame content in the two pronouncements?

Recall Beall and Restall’s (2006: 29) “Generalised Tarski Thesis”:

An argument is validx if and only if, in every casex in which the premises aretrue, so is the conclusion.

I presume that Beall and Restall did not intend to make a claim about thesemantics of an established term of philosophical English. However, thepresence of the subscript x in the statement of the thesis might indicate thatthe word “valid” has a sort of elided constituent, a slot where a logic can befilled 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 fromRestall [2002], when he is using his own voice. He says that, for him:

A,:A‘ CB, but A,:A⊬RB:

So the technical term “‘” seems to have an elided constituent, and thatsuggests 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 genuinedisagreement. They are in the analogous situation as Jack and Jill withthe salt. Beall and Restall insist that h and b give the same content to theargument Δ, but not to “valid”. For h, it is “classically valid”, “‘C”, and forb it is “intuitionistically valid”, “‘I”. So, once again, we do not have afailure of cognitive command.

To get cognitive command to fail, we have to assume that our logiciansh and b assign the same content to the terms in the argument Δ and wehave to assume that they assign the same content to the word “valid”.Given that Δ has the same content, “valid” must be folk-relative (sinceboth h and b are correct). The options for that term are thus non-indexicalcontextualism and assessment-sensitive relativism. I do not know ofanyone who explicitly defends that combination of views, and I won’tconsider 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

70 Stewart Shapiro

folk-relativism, statements of validity do not get truth-values unless onesomehow indicates a particular logic. If a particular logic is so indicated,then it depends on how much idealization goes into the notion of“knowable”.If we fix a particular logic, then either cognitive command holds

trivially, or, at worst, the question is reduced to one concerning math-ematics which is, I would think, almost a paradigm case of objectivity. Ifwe do not fix a particular logic, and consider statements of validitysimpliciter, then the question of cognitive command depends on somedelicate, and controversial semantic theses concerning both the logicalterminology 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 com-mand depends on the notion of disagreement and, as we saw, that doesturn 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 theor-izing (perhaps with some emphasis on “sharply”). So we might expectsome tough, borderline cases of objectivity. Add to the mix some widelyheld, 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 andsemantics is not so surprising.

Pluralism, relativism, and objectivity 71

cha p t e r 4

Logic, mathematics, and conceptual structuralismSolomon Feferman

1. The nature and role of logic in mathematics:three perspectives

Logic is integral to mathematics and, to the extent that that is the case, aphilosophy of logic should be integral to a philosophy of mathematics. Inthis, as you shall see, I am guided throughout by the simple view that whatlogic is to provide is all those forms of reasoning that lead invariably fromtruths to truths. The problematic part of this is how we take the notion oftruth to be given. My concerns here are almost entirely with the nature androle of logic in mathematics. In order to examine that we need to considerthree perspectives: that of the working mathematician, that of the math-ematical logician, and that of the philosopher of mathematics.

The aim of the mathematician working in the mainstream is to establishtruths about mathematical concepts by means of proofs as the principalinstrument. We have to look to practice to see what is accepted as amathematical concept and what is accepted as a proof; neither is deter-mined formally. As to concepts, among specific ones the integer and realnumber systems are taken for granted, and among general ones, notions offinite and infinite sequence, set and function are ubiquitous; all else issuccessively 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 inthem, and few of them have studied logic in any systematic way. Whenmathematicians consider axioms, instead it is for specific kinds of struc-tures: groups, rings, fields, linear spaces, topological spaces, metric spaces,Hilbert spaces, categories, etc., etc. Principles of a foundational characterare rarely mentioned, if at all, except on occasion for proof bycontradiction and proof by induction. The least upper bound principleon bounded sequences or sets of real numbers is routinely applied withoutmention. Some notice is paid to applications of the Axiom of Choice. To a

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side of the mainstream are those mathematicians such as constructivists orsemi-constructivists who reject one or another of commonly acceptedprinciples, but even for them the developments are largely informal withlittle explicit attention to logic. And, except for some far outliers, whatthey do is still recognizable as mathematics to the mathematician in themainstream.Turning now to the logicians’ perspective, one major aim is to model

mathematical practice – ranging from the local to the global – in order todraw conclusions about its potentialities and limits. In this respect, then,mathematical logicians have their own practice; here I shall sketch it andonly later take up the question how well it meets that aim. In brief:Concepts are tied down within formal languages and proofs within formalsystems, while truth, be it for the mainstream or for the outliers, isexplained in semantic terms. Some familiar formal systems for the main-stream are Peano Arithmetic (PA), Second-Order Arithmetic (PA2), andZermelo–Fraenkel set theory (ZF); Heyting Arithmetic (HA) is anexample of a formal system for the margin. In their intended or “standard”interpretations, PA and HA deal specifically with the natural numbers, PA2

deals with the natural numbers and arbitrary sets of natural numbers, whileZF deals with the sets in the cumulative hierarchy. Considering syntaxonly, in each case the well-formed formulas of each of these systems aregenerated from its atomic formulas (corresponding to the basic conceptsinvolved) by closing under some or all of the “logical” operations ofnegation, conjunction, disjunction, implication, universal and existentialquantification.The case of PA2 requires an aside; in that system the quantifiers are

applied to both the first-order and second-order variables. But we must becareful to distinguish the logic of quantification over the second-ordervariables as it is applied formally within PA2 from its role in second-orderlogic under the so-called standard interpretation. In order to distinguishsystematically between the two, I shall refer to the former as syntactic orformal second-order logic and the latter as semantic or interpreted second-orderlogic. In its pure form over any domain for the first-order variables,semantic second-order logic takes the domain of the second-order variablesto be the supposed totality of arbitrary subsets of that domain; in itsapplied form, the domain of first-order variables has some specified inter-pretation. As an applied second-order formal system, PA2 may equally wellbe considered to be a two-sorted first-order theory; the only thing thatacknowledges its intended second-order interpretation is the inclusion ofthe so-called Comprehension Axiom Scheme: that consists of all formulas

Logic, mathematics, and conceptual structuralism 73

of the form 9X8x[x 2 X$ A(x,. . .)] where A is an arbitrary formula of thelanguage of PA2 in which ‘X ’ does not occur as a free variable. Construingthings in that way, the formal logic of all of the above-mentioned systemsmay be taken to be first-order.

Now, it is a remarkable fact that all the formal systems that have been setup to model mathematical practice are in effect based on first-order logic,more specifically its classical system for mainstream mathematics and itsintuitionistic system for constructive mathematics. (While there are formalsystems that have been proposed involving extensions of first-order logicby, for example, modal operators, the purpose of such has been philosoph-ical. These operators are not used by mathematicians as basic or definedmathematical concepts or to reason about them.) One can say more aboutwhy 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 inmathematics is that of the philosopher of mathematics. Here there area multitude of positions to consider; the principal ones are logicism (andneo-logicism), “platonic” realism, constructivism, formalism, finitism, pre-dicativism, naturalism, and structuralism.1 Roughly speaking, in all of theseexcept for constructivism, finitism, and formalism, classical first-order logicis either implicitly taken for granted or explicitly accepted. In constructivism(of the three exceptions) the logic is intuitionistic, i.e. it differs from theclassical one by the exclusion of the Law of Excluded Middle (LEM).According to formalism, any logic may be chosen for a formal system. Infinitism, the logic is restricted to quantifier-free formulas for decidablepredicates; hence it is a fragment of both classical and intuitionistic logic.At the other extreme, classical second-order logic is accepted in set-theoreticrealism, and that underlies both scientific and mathematical naturalism; it isalso embraced in in re structuralism. Modal structuralism, on the otherhand, expands that via modal logic. The accord with mathematical practiceis perhaps greatest with mathematical naturalism, which simply takes prac-tice to be the given to which philosophical methodology must respond. Butthe structuralist philosophies take the most prominent conceptual feature ofmodern mathematics as their point of departure.

2. Conceptual structuralism

This is an ontologically non-realist philosophy of mathematics that I havelong 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).

74 Solomon Feferman

logic within it. I have summarized this philosophy in Feferman (2009) viathe following ten theses.2

1. The basic objects of mathematical thought exist only as mentalconceptions, though the source of these conceptions lies in everydayexperience in manifold ways, in the processes of counting, ordering,matching, combining, separating, and locating in space and time.

2. Theoretical mathematics has its source in the recognition that theseprocesses are independent of the materials or objects to which theyare applied and that they are potentially endlessly repeatable.

3. The basic conceptions of mathematics are of certain kinds of rela-tively simple ideal-world pictures that are not of objects in isolationbut of structures, i.e. coherently conceived groups of objects inter-connected by a few simple relations and operations. They are com-municated and understood prior to any axiomatics, indeed prior toany systematic logical development.

4. Some significant features of these structures are elicited directly fromthe world-pictures that describe them, while other features may beless certain. Mathematics needs little to get started and, once started,a little bit goes a long way.

5. Basic conceptions differ in their degree of clarity or definiteness. Onemay speak of what is true in a given conception, but that notion oftruth may be partial. Truth in full is applicable only to completelydefinite conceptions.

6. What is clear in a given conception is time dependent, both for theindividual and historically.

7. Pure (theoretical) mathematics is a body of thought developedsystematically by successive refinement and reflective expansion ofbasic structural conceptions.

8. The general ideas of order, succession, collection, relation, rule, andoperation are pre-mathematical; some implicit understanding ofthem is necessary to the understanding of mathematics.

9. The general idea of property is pre-logical; some implicit understand-ing of that and of the logical particles is also a prerequisite to theunderstanding of mathematics. The reasoning of mathematics is inprinciple logical, but in practice relies to a considerable extent onvarious forms of intuition in order to arrive at understanding andconviction.

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

Logic, mathematics, and conceptual structuralism 75

10. The objectivity of mathematics lies in its stability and coherenceunder repeated communication, critical scrutiny, and expansionby many individuals often working independently of each other.Incoherent concepts, or ones that fail to withstand critical examin-ation or lead to conflicting conclusions are eventually filtered outfrom mathematics. The objectivity of mathematics is a special case ofintersubjective objectivity that is ubiquitous in social reality.

3. Two basic structural conceptions

These theses are illustrated in Feferman (2009) by the conception of thestructure of the positive integers on the one hand and by several concep-tions of the continuum on the other. Since our main purpose here is toelaborate the nature and role of logic in such structural conceptions, it iseasiest to review here what I wrote there, except that I shall limit myself tothe set-theoretical conception of the continuum in the latter case.

The most primitive mathematical conception is that of the positiveinteger sequence as represented by the tallies: |, ||, |||, . . . From thestructural point of view, our conception is that of a structure (Nþ, 1,Sc, <), where Nþ is generated from the initial unit 1 by closure under thesuccessor operation Sc, and m < n if m precedes n in the generationprocedure. Certain facts about this structure (if one formulates themexplicitly at all), are evident: that < is a total ordering of Nþ for which1 is the least element, and that m < n implies Sc(m) < Sc(n). Reflecting ona given structure may lead us to elaborate it by adjoining further relationsand operations and to expand basic principles accordingly. For example, inthe case of Nþ, thinking of concatenation of tallies immediately leads us tothe operation of addition, m þ n, and that leads us to m � n as “m addedto itself n times.” The basic properties of the þ and � operations such ascommutativity, associativity, distributivity, and cancellation are initiallyrecognized only implicitly. We may then go on to introduce more dis-tinctively mathematical notions such as the relations of divisibility andcongruence and the property of being a prime number. In this language, awealth of interesting mathematical statements can already be formulatedand investigated as to their truth or falsity, for example, that there areinfinitely many twin prime numbers, that there are no odd perfectnumbers, Goldbach’s conjecture, and so on.

The conception of the structure (Nþ, 1, Sc, <, þ, �) is so intuitivelyclear that (again implicitly, at least) there is no question in the minds ofmathematicians as to the definite meaning of such statements and the

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assertion that they are true or false, independently of whether we canestablish them in one way or the other. (For example, it is an openproblem whether Goldbach’s conjecture is true.) In other words, realismin truth values is accepted for statements about this structure, and theapplication of classical logic in reasoning about such statements is auto-matically legitimized. Despite the “subjective” source of the positive inte-ger structure in the collective human understanding, it lies in the domainof objective concepts and there is no reason to restrict oneself to intuitio-nistic logic on subjectivist grounds. Further reflection on the structure ofpositive integers with the aim to simplify calculations and algebraic oper-ations and laws leads directly to its extension to the structure of naturalnumbers (N, 0, Sc, <, þ, �), and then the usual structures for the integersZ and the rational numbers Q. The latter are relatively refined conceptions,not basic ones, but we are no less clear in our dealings with them than forthe basic conceptions of Nþ.At a further stage of reflection we may recognize the least number

principle for the natural numbers, namely if P(n) is any well-definedproperty of members of N and there is some n such that P(n) holds thenthere is a least such n. More advanced reflection leads to general principlesof proof by induction and definition by recursion on N. Furthermore, thegeneral scheme of induction,

Pð0Þ ^ 8n½PðnÞ ! PðScðnÞÞ� ! 8nPðnÞ,is taken to be open-ended in the sense that it is accepted for any definiteproperty P of natural numbers that one meets in the process of doingmathematics, no matter what the subject matter and what the notions usedin the formulation of P. The question – What is a definite property? –requires in each instance the mathematician’s judgment. For example, theproperty, “n is an odd perfect number,” is definite, while “n is a feasiblycomputable number” is not, nor is “n is the number of grains of sandin a heap.”Turning now to the continuum, in Feferman (2009) I isolated several

conceptions of it ranging from the straight line in Euclidean geometrythrough the system of real numbers to the set of all subsets of the naturalnumbers. The reason that these are all commonly referred to as thecontinuum is that they have the same cardinal number; however, thatignores essential conceptual differences. For our purposes here, it is suffi-cient to concentrate on the last of these concepts. The general idea of set orcollection of objects is of course ancient, but it only emerged as an objectof mathematical study at the hands of Georg Cantor in the 1870s. Given

Logic, mathematics, and conceptual structuralism 77

the idea of an arbitrary set X of elements of any given set D, consideredindependently of how membership in X may be defined, we write S(D) forthe conception of the totality of all subsets X of D. Then the continuum inthe 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 isthe relation of membership of natural numbers to sets of natural numbers.Two principles are evident for this conception, using letters ‘X ’, ‘Y ’ torange over S(N) and ‘n’ to range over N.

I. Extensionality 8X 8 Y [8n(n 2 X $ n 2Y ) ! X = Y ]II. Comprehension For any definite property P(n) of members of N,

9X 8n½n 2 X $ PðnÞ�: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 tothe usual set-theoretical view, S(N) is a definite totality, so that quantifica-tion over it is well-determined and may be used to express definiteproperties P. But again that requires on the face of it a realist ontologyand in that respect goes beyond conceptual structuralism. So if we do notsubscribe to that, we may want to treat S(N) as indefinite in the sense thatit is open-ended. Of course this is not to deny that we recognize manyproperties P as definite such as – to begin with – all those given by first-order formulas in the language of the structure (N, 0, Sc, <, þ, �) (i.e.those that are ordinarily referred to as the arithmetical properties); thenceany sets defined by such properties are recognized to belong to S(N).

Incidentally, even from this perspective one can establish categoricity ofthe Extensionality and Comprehension principles for the structure(N, S(N), 2) relative to N in a straightforward way as follows. Supposegiven 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), letP(n) be the definite property, n 2 X. Using Comprehension for thestructure (N, S 0(N), 20), one obtains existence of an X 0 such that for alln in N, n 2 X iff n 2 X 0; then X 0 is unique by Extensionality. This gives a1–1map of S(N) into S 0(N) preserving N and the membership relation; it isseen to be an onto map by reversing the argument. This is to be comparedwith the standard set-theoretical view of categoricity results as exemplified,for example, in Shapiro (1997) and Isaacson (2011). According to that view,the subject matter of mathematics is structures, and the mère structures ofmathematics such as the natural numbers, the continuum (in one of itsvarious guises), and suitable initial segments of the cumulative hierarchy of

78 Solomon Feferman

sets are characterized by axioms in full second-order logic; that is, any twostructures satisfying the same such axioms are isomorphic.3 On thataccount, the proofs of categoricity in one way or another thenappeal prima facie to the presumed totality of arbitrary subsets of anygiven set.4

Even if the definiteness of S(N) is open to question as above, we cancertainly conceive of a world in which S(N) is a definite totality andquantification over it is well-determined; in that ideal world, one maytake for the property P in the above Comprehension Principle any formulaof full second-order logic over the language of arithmetic. Then a numberof theorems can be drawn as consequences in the corresponding systemPA2, including purely arithmetical theorems. Since the truth definition forarithmetic can be expressed within PA2 and transfinite induction can beproved in it for very large recursive well-orderings, PA2 goes in strength farbeyond PA even when that is enlarged by the successive adjunction ofconsistency statements transfinitely iterated over such well-orderings.What confidence are we to have in the resulting purely arithmeticaltheorems? There is hardly any reason to doubt the consistency of PA2

itself, even though by Gödel’s second incompleteness theorem, we cannotprove it by means that can be reduced to PA2. Indeed, the ideal worldpicture of (N, S(N), 2) that we have been countenancing would surely leadus to say more, since in it the natural numbers are taken in their standardconception. On this account, any arithmetical statement that we can provein PA2 ought simply to be accepted as true. But given that the assumptionof S(N) as a definite totality is a purely hypothetical and philosophicallyproblematic 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 differ on whetherthe 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 logicalconsequence.

4 In general, proofs of categoricity within formal systems of second-order logic can be analyzed to seejust what parts of the usual impredicative comprehension axiom scheme are needed for them. In thecase of the natural number structure, however, it may be shown that there is no essential dependenceat all, in contrast to standard proofs. Namely, Simpson and Yokoyama (2012) demonstrate thecategoricity 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 (PrimitiveRecursive Arithmetic). By comparison, it is sketched in Feferman (2013) how to establish categoricityof the natural numbers in its open-ended schematic formulation in a simpler way that is alsoconservative over PRA. For an informal discussion of the categoricity of initial segments of thecumulative hierarchy of sets in the spirit of open-ended axiom systems, see D. Martin (2001, sec. 3).

Logic, mathematics, and conceptual structuralism 79

Incidentally, all of this and more comes into question when we moveone type level up to the structure (N, S(N), S(S(N)), 21, 22) in whichCantor’s continuum hypothesis may be formulated. A more extensivediscussion of the conception of that structure and the question of itsdefiniteness in connection with the continuum problem is given inFeferman (2011). We shall also see below how taking N and S(N) to bedefinite but S(S(N)) to be open-ended can be treated in suitable formalsystems.

4. Where and why classical first-order logic?

Logic, as I affirmed at the outset, is supposed to provide us with all thoseforms of reasoning that lead invariably from truths to truths, i.e. it is givenby an essential combination of inferential and semantical notions. Butfrom the point of view of conceptual structuralism, the classical notion oftruth in a structure need not be applicable unless we are dealing with aconception (such as that of the structure of natural numbers) for which thebasic domains are definite totalities and the basic notions are definiteoperations, predicates, and relations. It is clear that at least the classicalfirst-order predicate calculus should be admitted both on semantical andinferential grounds, since we have Gödel’s completeness theorem to pro-vide us with a complete inferential system. But why not more? For example,model-theorists have introduced generalized quantifiers such as the cardin-ality quantifiers (Qκx)P(x) expressing that there are at least κ individuals xsatisfying the property P, where κ is any infinite cardinal; one couldcertainly consider adjoining those to the first-order formalism. A muchmore general class of quantifiers defined by set-theoretical means wasintroduced by Lindström (1966); each of those can be used to extendfirst-order logic with a model-theoretic semantics for arbitrary first-orderdomains. But for which such extensions do we have a completenesstheorem like that of Gödel’s for first-order logic? It is well known thatno such theorem is possible for the quantifier (Qωx)P(x) which expressesthat there are infinitely many x such that P(x). For, using that quantifierand thence its dual (“there are just finitely many x such that P(x)”) we cancharacterize the structure of natural numbers up to isomorphism, so all thetruths of that structure are valid sentences in the logic. But the set of suchtruths is not effectively enumerable, indeed far from it, so it is not given byan effectively specified formal system of reasoning.

Surprisingly, Keisler (1970) obtained a completeness theorem for thequantifier (Qκx)P(x) when κ is any uncountable cardinal; as it happens,

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that has the same set of valid formulas as for the case that κ is the firstuncountable cardinal. In view of the leap over the case κ = ω, one maysuspect that the requirement that the set of valid formulas be given bysome effective set of axioms and rules of inference is not sufficient toexpress completeness in the usual intended sense. We need to say some-thing more about how such axioms and rules of inference ought specific-ally to be complete for a given quantifier. The key is given by Gentzen’s(1935) system of natural deduction NK (or sequent calculus LK) whereeach connective and quantifier in the classical first-order predicate calculusis specified by Introduction and Elimination rules for that operation only.Moreover, for each pair of such rules, any two connectives or quantifierssatisfying them are equivalent, i.e. they implicitly determine the operatorin question. So a strengthened condition on a proposed addition by ageneralized quantifier Q to our first-order language is that it be given byaxioms and rules of inference for which there is at most one operatorsatisfying them. That was the proposal of Zucker (1978) in which he gave atheorem to the effect that any such quantifier is definable in the first-orderpredicate calculus. In particular, that would apply to the Lindströmquantifiers. However, there were some defects in Zucker’s statement ofhis theorem and its proof; I have given a corrected version of both inFeferman (to appear). To summarize: we have fully satisfactory semanticand inferential criteria for a logic to deal with structures whose domains arefirst-order and that are completely definite in the sense described above,and these limit us to the standard first-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 indicatesthat this augments an arithmetical structure onN such as (N, 0, Sc,<,þ,�).Again, if S(N) is considered as a definite totality, the classical notion oftruth is applicable and the semantics of second-order logic must beaccepted. But as is well known there is no complete inferential systemthat accompanies that, since again the arithmetical structure is categoricallyaxiomatized in this semantics and in consequence the set of its truths is noteffectively enumerable. In any case, as I have argued above, S(N) ought notto be considered as a definite totality; to claim otherwise, is to accept theproblematic 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 “nomin-alistic” system of plural quantification. This was incisively critiqued byResnik in his article “Second-order logic still wild”: “Boolos is involved in acircle: he uses second-order quantification to explain English plural

Logic, mathematics, and conceptual structuralism 81

quantification and uses this, in turn, to explain second-order quantifica-tion” (Resnik 1988, p. 83).

Though the Lindström quantifiers are restricted to apply to first-orderstructures and thus bind only individual variables they may well be definedusing higher-order notions in an essential way, in particular those neededfor the cardinality quantifiers. Another example where the syntax is first-order on the face of it but the semantics is decidedly second-order is IF(“Independence Friendly”) logic, due to Hintikka (1996). This uses for-mulas in whose prenex form the existentially quantified individual vari-ables are declared to depend on a subset of the universally quantifiedindividual variables that precede it in the prefix list. Explanation of thesemantics of this requires the use of quantified function variables; over anygiven first-order structure (D, . . .) those variables are interpreted to rangeover functions of various arities with arguments and values in D. Indeed,Väänänen (2001, p. 519) has proved that the general question of validity ofIF sentences is recursively isomorphic to that for validity in full second-order logic. Thus, as with the Lindström quantifiers, the formal syntax canbe deceptive. See Feferman (2006) for an extended critique of IF logic.

5. Where and why intuitionistic first-order logic?

Now let us turn to the question which logic is appropriate to structuralconceptions that are taken to lack some aspect of definiteness. Offhand,one might expect the answer in that case to be intuitionistic logic, but thematter is more delicate. The problem is that there is not one clear-cutsemantics for it; among others that have been considered, one has theso-called BHK interpretation, Kripke semantics, topological semantics,sheaf models, etc., etc. Of these, the first is the most principled one withrespect to the basic ideas of constructivity; it is that that leads one directly tointuitionistic logic but it does not determine it via a precise completenessresult. By contrast, as we shall see, not only does Kripke semantics take careof the latter but it relates more closely to the question of dealing withconceptions of structures involving possibly indefinite notions anddomains. For the details concerning both of these I refer to Troelstra andvan Dalen (1988), a comprehensive exposition of constructivism in math-ematics that includes treatments of the great variety of semantics and prooftheory that have been developed for intuitionistic systems.

The BHK (Brouwer–Heyting–Kolmogoroff ) constructive explanationof the connectives and quantifiers is described in Troelstra and van Dalen(1988, p. 9). It uses the informal notions of construction and constructive

82 Solomon Feferman

proof; for each form of compound statement C necessary and sufficientconditions are provided on what it is for a construction to be a proof ofC, in terms of proofs of its immediate sub-statements. Namely, a proofof A ^ B is a proof of A and a proof of B; a proof of A _ B is a proof of A ora proof of B; a proof of A ! B is a construction that transforms any proofof A into a proof of B; and a proof of :A is a construction that transformsany proof of A into a proof of a contradiction ⊥, i.e. is a proof of A ! ⊥.In the case of the quantifiers, where the variables range over a givendomain D, a proof of (8x)A(x) is a construction that transforms any d inD into a proof of A(d); finally, a proof of (9x)A(x) is given by a d inD and aproof of A(d). (D must be a constructively meaningful domain, so that itmakes sense to exhibit each individual element of D and for constructionsto be applicable to elements of D.)A statement A of the first-order predicate calculus is constructively valid

according to the BHK interpretation if there is a proof of A, independentlyof the interpretation of the domain D and the interpretation of thepredicate symbols of A in D. The axioms of intuitionistic logic in any ofits usual formulations are readily recognized to be constructively valid andthe rules of inference preserve constructive validity. But since there are noprecise notions of proof and construction at work here, we cannot state acompleteness result for the BHK interpretation. Instead, the literature uses“weak counterexamples” to show why it is plausible on that account that agiven classically valid form of statement is not constructively valid. Thus,for example, to show that A _ :A is not constructively valid as a generalprinciple one argues that otherwise one would have a general method forobtaining for any given statement A, either a proof of A or a proof thatturns any hypothetical proof of A into a contradiction. But if we had sucha universal method, we could apply it to any particular statement A thathas not yet been settled, such as the twin prime conjecture, to determineits truth or falsity. Similarly, the method of weak counterexamples is usedinformally to argue against the constructive validity of many other suchschemes, for example ::A ! A, though the converse is recognized to bevalid.5

Let us turn now to Kripke semantics for the language of first-orderpredicate logic (Troelstra and van Dalen 1988, Ch. 2.5–2.6). A Kripkemodel 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 preciseindependence results for such schemes, but are still not complete for intuitionistic logic. Cf. Troelstraand 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 non-empty set D(k) such that if k � k0 then D(k) � D(k 0), and (iii) v is afunction into f0, 1g at each k in K, each n-ary relation symbol R in thelanguage and n-ary sequence of elements of D(k), such that if k � k 0 andd1,. . .,dn 2 D(k) and v(k, R(d1,. . .,dn)) = 1 then v(k 0, R(d1,. . .,dn)) = 1. Onemotivating idea for this is that the elements of K represent stages ofknowledge, and that k � k 0 holds if everything known in stage k is knownin stage k 0. Also, v(k, R(d1,. . .,dn)) = 1 means that R(d1,. . .,dn) has beenrecognized to be true at stage k; once recognized, it stays true. The domainD(k) is the part of a potential domain that has been surveyed by stage k;the domains may increase indefinitely as k increases or may well bifurcatein a branching investigation so that one cannot speak of a “final” domainin that case.

The valuation function v is extended to a function v(k, A(d1,. . .,dn)) intof0, 1g for each formula A(x1,. . .,xn) with n free variables and assignment(d1,. . .,dn) to its variables in D(k); this is done in such a way that if k � k 0and d1,. . .,dn 2 D(k) and v(k, A(d1,. . .,dn)) = 1 then v(k0, A(d1,. . .,dn)) = 1.The clauses for conjunction, disjunction, and existential quantification arejust like those for ordinary satisfaction at k in D(k). The other clauses are(ignoring parameters): v(k, A! B) = 1 iff for all k 0 � k, v(k 0, A) = 1 impliesv(k 0, B) = 1; v(k, ⊥) = 0; and v(k, 8x A(x)) = 1 iff for all k 0 � k and d inD(k), v(k0, A(d)) = 1. As above, we identify:Awith A!⊥; thus v(k,:A) = 1iff for all k0 � k, v(k 0, A) = 0. We say that k forces A if v(k, A) = 1; i.e. A isrecognized to be true at stage k no matter what may turn out to be knownat later stages. A formula A(x1,. . .,xn) is said to be valid in a model (K, �,D, v) if for every k in K and assignment (d1,. . .,dn) to its free variables inD(k), v(k, A(d1,. . .,dn)) = 1. Then the completeness theorem for thissemantics is that a formula A is valid in all Kripke models iff it is provablein the first-order intuitionistic predicate calculus. We shall see in the nextsection how Kripke models can be generalized to take into accountdifferences as to definiteness of basic relations and domains.

Satisfying as this completeness theorem may be, there remains thequestion whether one might not add connectives or quantifiers to thoseof intuitionistic logic while retaining some form of its semantics. Thoughintuitionistic logic is part of classical logic, the semantical and inferentialcriterion above for classical logic doesn’t apply because of the differences inthe semantical notions. But just as for the classical case, on the inferentialside each of the connectives and quantifiers of the intuitionistic first-orderpredicate calculus is uniquely identified via Introduction and Eliminationrules in Gentzen’s natural deduction system NJ. Even more, Gentzen first

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formulated the idea that the meaning of each of the above operations isgiven by its characteristic inferences. Actually, Gentzen claimed more: hewrote that “the [Introduction rules] represent, as it were, the ‘definitions’of the symbols concerned” (Gentzen 1969, p. 80). Prawitz supported thisby means of his Inversion Principle (Prawitz 1965, p. 33): namely, it followsfrom the normalization theorem for NJ that each Elimination rule for agiven operation can be recovered from the appropriate one of its Introduc-tion rules when that is the last step in a normal derivation. Withoutsubscribing at all to this proposed reduction of semantics to inferentialroles, we may ask whether any further operators may be added via suitableIntroduction rules. The answer to that in the negative was provided by thework of Zucker and Tragesser (1978) in terms of the adequacy of what theycall inferential logic, i.e. of the logic of operators that can simply be markedout by Introduction rules. As they show, every such operator is defined interms of the connectives and quantifiers of the intuitionistic first-orderpredicate calculus. To be more precise, this is shown for Introduction rulesin the usual sense in the case of possible propositional operators, while inthe general case of possible operators on propositions and predicates – nowin accord with the BHK interpretation – “proof ” parameters and con-structions on them are incorporated in the Introduction rules, but thoseare eventually suppressed.6

6. Semi-intuitionism: the logic of partiallyopen-ended structures

An immediate generalization of Kripke structures is to allow many-sorteddomains, possibly infinite in number. Let I be a collection of sorts. Thenthe definition of Kripke structure is modified to have each of K, �, andD indexed by I, and the valuation function modified to accord with thedifferent sorts. Thus we deal with n-tuples k = (k1,. . .,kn) where km is ofspecified sort im; the � relation then holds between such n-tuples if itholds term-wise. Of course the basic predicates come with specified aritiesto show what sorts of objects they relate, and the variables in the first-orderlanguage over these predicates are always of a specified sort. Then thedefinition of the valuation function on arbitrary formulas for a many-sorted structure (K, I, �, D, v) proceeds in the same way as above. Now an

6 Incidentally, as Zucker and Tragesser show (p. 506), not every propositional operator given bysimple Introduction rules has an associated Elimination rule; a counterexample is provided by(A ! B) _ C.

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n-ary relation R may be considered to be definite if v(k, R(d1,. . .,dn)) = v(k0,R(d1,. . .,dn)) whenever k � k 0. A domain Di is definite if Di(k) = Di(k 0)for all k and k 0 in Ki, otherwise indefinite or open-ended. While theformulas valid in the structure obey intuitionistic logic in general, onemay apply classical logic systematically to formulas involving definiterelations as long as the quantified variables involved range only overdefinite domains.

This is illustrated by reasoning about the ordinary two-sorted struc-ture (N, S(N), 2, . . .) where (N, . . .) is conceived of as definite withdefinite relations, while S(N) is conceived of as open-ended. To treatthis as a two-sorted Kripke structure, take I = f0, 1g where N is of sort0 and S(N) is of sort 1. We may as well take K0 to consist of a singleelement, while K1 could be indexed by all collections k of subsets ofN, ordered by inclusion. Now the membership relation is definitebecause sets are taken to be definite objects, i.e. if X is in both thecollections k and k 0 then n 2 X holds in the same way whetherevaluated in k or in k 0. So classical logic applies to all formulas A thatcontain no bound set variables, though they may contain free setvariables, i.e. A is what is usually called a predicative formula. Butwhen dealing with formulas in general, only intuitionistic logic isjustified on this picture. This leads us to the consideration of semi-intuitionistic (or semi-constructive) theories in general, i.e. theories inwhich the basic underlying logic is intuitionistic, but classical logic istaken to apply to a class of formulas distinguished by containingdefinite predicates and quantified variables ranging over definitedomains. A number of such theories have been treated in the paperFeferman (2010), corresponding to different structural notions in whichcertain domains are taken to be definite and others indefinite. They fallinto three basic groups: (i) predicative theories, (ii) theories of countable(tree) ordinals, and (iii) theories of sets. The general pattern is that ineach case one has a semi-intuitionistic version of a correspondingclassical system, and they are shown to be proof-theoretically equivalentand to coincide on the classical part. Moreover, the same holds whenthe semi-intuitionistic system is augmented by various principles suchas the Axiom of Choice (AC) that would make the correspondingclassical system much stronger. It is not possible here to explain theresults in adequate detail, so only some of the ideas behind the formu-lations of the systems involved are sketched. The reader who prefers toavoid even the technicalities that remain can easily skim (or even skip)the rest of this section.

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6.1 Semi-intuitionistic predicative theories

Here the language of arithmetic is extended by variables for function(al)sin all finite types; following Gödel (1958, 1972) in his so-called Dialecticainterpretation, we also add primitive recursive functionals in all finitetypes. In many-sorted intuitionistic logic, the system obtained is denotedHAω. In the process of obtaining reduction to a quantifier-free system,Gödel showed that this system is of the same strength as Peano Arithmetic,PA; in fact the same holds for HAω þ AC. Now the latter is turned into asemi-intuitionistic system by adding the Law of Excluded Middle for allarithmetical formulas. For the proof-theoretical work on that, it proves tobe more convenient to add the least-number operator μ and an axiom (μ)that says that when the operator is applied to a function f : N ! N forwhich there exists an n with f(n) = 0, it yields the least such n. Under thisaxiom, all arithmetical formulas become equivalent to quantifier-free (QF)formulas, for which the LEM then holds. Thus one is led to considerHAω þ AC þ (μ), which turns out to be proof-theoretically equivalent toPAω þQF-ACþ (μ), and both are equivalent to ramified analysis throughall ordinals less than Cantor’s ordinal ε0. If one adds the Bar Rule forarithmetical orderings in both the semi-intuitionistic and the classicalsystems, we obtain systems of proof-theoretical strength full predicativeanalysis, i.e. ramified analysis up to the least impredicative ordinal Γ0. (TheBar Rule on an ordering allows us to infer transfinite induction w.r.t.arbitrary formulas from well-foundedness of the ordering.) On the otherhand, if in the basic system we restrict the primitive recursive functionalsto those with values in N and restrict induction to QF formulas, we obtaina semi-intuitionistic system Res-HAω þ AC þ (μ) that turns out to be ofexactly PA in strength.

6.2 Semi-intuitionistic theories of countable tree ordinals

By countable tree ordinals one means the members of the open-endedcollection O of countably branching well-founded trees. Add a sort forthe members of O to the preceding systems; extend the higher typevariables accordingly; add the operator of supremum that joins a sequenceof trees f : N ! O into a single tree sup( f ) in O; add the inverse operatorthat takes each sup( f ) in O and n in N and produces f (n); and, finally, addoperators for transfinite recursion on O. The resulting system is denotedSOO in intuitionistic logic and COO in classical logic; then SOO þ (μ) is asemi-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-theoreticalstrength: SOO þ AC þ (μ), COO þ QF-AC þ (μ), and ID1, the theory ofarbitrary arithmetical inductive definitions. It is known that the latterhas the same proof-theoretical strength in intuitionistic logic as inclassical logic.

6.3 Semi-intuitionistic theories of sets

We turn finally to the picture of the cumulative hierarchy structure, thestandard classical view of which leads us to the system ZFC, i.e. ZF þ AC.However, if we identify definite totalities with sets then by Russell’sparadox, the “universe V of all sets” must be considered to be an open-ended indefinite totality if we are to avoid contradiction. But in theSeparation Axiom scheme for ZF, 8a9b8x[x 2 b $ x 2 a ^ A(x)], oneallows the formula A to contain bound variables that range withoutrestriction over V, and hence in general do not represent definite proper-ties; the same criticism applies to the formulas A(x, y) in the ReplacementAxiom scheme. By a Δ0 formula is meant one in which all quantifiedvariables are restricted, i.e. take the form 8y(y 2 x ! . . .) or 9y(y 2 x^ . . .), written respectively (8y 2 x)(. . .) and (9y 2 x)(. . .). The system KPof Kripke–Platek set theory in classical logic has, like ZF, the axioms ofextensionality, ordered pair, union, infinity, and the scheme of transfiniteinduction on the membership relation. In place of the Separation Axiomscheme it takes Δ0-Separation, i.e. the Separation Axiom scheme restrictedto Δ0 formulas. And in place of the Replacement Axiom scheme, it takeswhat is called Δ0-Collection, i.e. the scheme that for each Δ0 formulaA, (8x 2 a)9yA(x, y) ! 9b(8x 2 a)(9y 2 b)A(x, y). This implies theReplacement Axiom scheme for Δ0 formulas. It is known that the systemKP is of the same strength as ID1.

The system IKP is taken to be the same as KP but restricted tointuitionistic logic. It turns out that we can strengthen it considerably byadding a bounded form ACS of the Axiom of Choice, namely (8x 2 a)9yA(x, y) ! 9f [Fun( f ) ^ (8x 2 a)A(x, f (x))], where Fun( f ) expresses thatthe set f is a function in the set-theoretical sense, and where now A is anarbitrary formula of the language of set theory. Under the assumption ACS

we can infer Collection for arbitrary formulas and hence Replacement forarbitrary formulas. Finally, since sets are considered to be definite totalities,we obtain a semi-intuitionistic system from IKP by adjoining the law ofexcluded middle for Δ0 formulas. The main result of Feferman (2010) isthat the semi-intuitionistic system IKP þ ACS þ Δ0-LEM is of the same

88 Solomon Feferman

proof-theoretical strength as KP and hence of ID1 in its classical andintuitionistic forms. Moreover, if we add the Power Set Axiom (Pow) weobtain a system that is of strength between that of KP þ Pow and that ofKP þ Pow þ (V = L).7,8

It is natural in the context of semi-intuitionistic theories T to say that asentence A in the language of T is definite (relative to T) if T proves LEMfor A, i.e. A _ :A. A question in set theory that has caused considerablediscussion in recent years is whether Cantor’s continuum hypothesis CH isa definite mathematical problem. One formulation of it is that every subsetof S(N) is either countable or in 1-1 correspondence with S(N). Of course,that is definite in the theory IKP þ Pow þ Δ0-LEM, because quantifica-tion 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 inFeferman (2011) – considering the weaker system T = IKP þ Pow(N)þACS þ Δ0-LEM, where Pow(N) simply asserts the existence of S(N) as aset. I conjectured there that CH is not definite relative to that system.9 Ofcourse, that would not show that CH is not a definite mathematicalproblem, but it might be considered as an interesting bit of evidence insupport of that.

7. Conceptual structuralism and mathematical practice

One criterion for a philosophy of mathematics that is often heard is that itshould accord with mathematical practice. It’s very hard to know just whatthat means since there are so many dimensions along which practice can beviewed. One particular interpretation of the criterion is that philosophershave no business telling mathematicians what does or doesn’t exist. Fam-ously, David Lewis wrote:

I’m moved to laughter at the thought of how presumptuous it would be toreject mathematics for philosophical reasons. How would you like the job oftelling the mathematicians that they must change their ways, and abjurecountless 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 setaxiom going back to the early 1970s. See Feferman (2010, sec. 7.2) for references to the relevantwork 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 usualargument for the relative consistency of (V = L) doesn’t work.

9 Michael Rathjen (2014) has recently verified this conjecture.10 Curiously, this quote is from Lewis’ book, Parts of Classes, which offers a revisionary theory of classes

that differs from the usual mathematical conception of such.

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But this is a caricature of what philosophy is after; philosophers take forgranted that mathematicians have settled problematic individual questionsof existence like zero, negative numbers, imaginary numbers, infinitesi-mals, points at infinity, probability of subsets of [0, 1], etc., etc., usingpurely mathematical criteria in the course of the development of theirsubject. The existence of some of these has been established by reductionto objects whose existence is unquestioned, some by qualified 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, ina way that cannot even be discussed within ordinary mathematical par-lance. Lewis could equally well have laughed at the idea that some generalprinciples accepted in the mathematical mainstream such as the Law ofExcluded Middle or the Axiom of Choice would be dismissed as false (orunjustified) for philosophical reasons. But again, the use of truth inordinary mathematical parlance is deflationary and the reasons foraccepting such and such principles as true has either been made withoutquestion or for mathematical reasons in the course of the development ofthe subject. The philosopher, by contrast, is concerned to explain in whatsense the notion of truth is applicable to mathematical statements, in a waythat cannot be considered in ordinary mathematical parlance. Whether themathematician should pay attention to either of these aims of the philoso-pher is another matter.

Conceptual structuralism addresses the question of existence and truthin mathematics in a way that accords with both the historical developmentof the subject and each individual’s intellectual development. It cruciallyidentifies mathematical concepts as being embedded in a social matrix thathas given rise, among other things, to social institutions and games; likethem, mathematics allows substantial intersubjective agreement, and likethem, its concepts are understood without assuming reification.11 Whatmakes mathematics unique compared to institutions and games is itsendless fecundity and remarkable elaboration of some basic numericaland geometrical structural conceptions. To begin with, mathematicalobjects exist only as conceived to be elements of such basic structures.The direct apprehension of these leads one to speak of truth in a structurein a way that may be accepted uncritically when the structure is such as theintegers but may be put into question when the conception of the structureis less definite as in the case of the geometrical plane or the continuum, and

11 For an interesting social institutional account of mathematics see Cole (2013); this differs fromconceptual structuralism in some essential respects while agreeing with it in others.

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should be put into question when it comes to the universe of sets. Onecriticism of conceptual structuralism that has been made is that it’s notclear/definite what mathematical concepts are clear/definite, and makingthat a feature of the philosophy brings essentially subjective elements intoplay.12 Actually, conceptual structuralism by itself, as presented in thetheses 1–10, takes no specific position in that respect and recognizes thatdifferent judgments (such as mine) may be made. Once such are con-sidered, however, logic has much to tell us in its role as an intermediarybetween philosophy and mathematics. As shown in the preceding section,one can obtain definitive results about formal models of different stand-points as to what is definite and what is not. Moreover, the results can besummarized as telling us that to a significant extent, the unlimited (defacto) application of classical logic in mainstream mathematics – i.e., thelogic of definite concepts and totalities – may be justified on the basis ofa more refined mixed logic that is sensitive to distinctions that one mightadopt between what is definite and what is not.13 In other words, oncemore they show that, at least to that extent, you can have your cake andeat it too.There are other dimensions of mathematical practice that reward meta-

mathematical study motivated by the philosophy of conceptualstructuralism. One, in particular, that I have emphasized over the yearsis the open-ended nature of certain principles such as that of induction forthe integers and comprehension for sets. This accords with the fact that inthe development of mathematics what concepts are recognized to bedefinite evolve with time. Thus one cannot fix in advance all applicationsof these open-ended schematic principles by restriction to those instancesdefinable in one or another formal language, as is currently done in thestudy of formal systems. This leads instead to the consideration of logicalmodels of practice from a novel point of view that yet is susceptible tometamathematical study. One such is via the notion of the unfolding ofopen-ended schematic axiom systems, that is used to tell us everything thatought to be accepted if one has accepted given notions and principles.Thus far, definitive results about the unfolding notion have been obtainedby Feferman and Strahm (2000, 2010) for schematic systems of non-finitistand finitist arithmetic, resp., and by Buchholtz (2013) for arithmetical

12 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.

13 These kinds of logical results can also be used to throw substantive light on philosophical discussionsas to the problem of quantification over everything (or over all ordinals, or all sets) such as are foundin Rayo and Uzquiano (2006).

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inductive definitions. As initiated in Feferman (1996), I am optimistic thatit can be used to elaborate Gödel’s program for new axioms in set theoryand in particular to draw a sharper line between which such axioms oughtto be accepted on intrinsic grounds and those to be argued for on extrinsicgrounds.

ACKNOWLEDGEMENTS

I would like to thank Gianluigi Bellin, Dagfinn Føllesdal, Peter Koellner,Grigori Mints, Penelope Rush, Stewart Shapiro, and Johan van Benthemfor their helpful comments on a draft of this essay.

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cha p t e r 5

A Second Philosophy of logicPenelope Maddy

What’s hidden in my hand is either an ordinary dime or a foreign coin of atype I’ve never seen. (I drew it blindfolded from a bin filled with just thesetwo 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 oflogical truth? – and common, also, to regard these questions as properlyphilosophical, to be answered by appeal to distinctively philosophicaltheories 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 toaddress the simple question afresh, without presumptions about whatconstitutes ‘logic’ or even ‘philosophy’. The thought is to treat inquiriesabout reliability of the coin inference and others like it as perfectly ordinaryquestions, in search of perfectly ordinary answers, and to see where thisinnocent 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 theworld and our place in it.1 She begins her investigations with everydayperceptions, gradually develops more sophisticated approaches to observa-tion and experimentation that expand her understanding and sometimesserve to correct her initial beliefs; eventually she begins to form and testhypotheses, and to engage in mature theory-formation and confirmation;along the way, she finds the need for, and pursues, first arithmetic andgeometry, then analysis and even pure mathematics;2 and in all this, sheoften pauses to reflect on the methods she’s using, to assess their

1 The Second Philosopher is introduced in (Maddy 2007), and her views on logic detailed in Part III ofthat book. The discussion here reworks and condenses the presentation there (see also (Maddy toappear)).

2 For more on the Second Philosopher’s approach to mathematics, see (Maddy 2011).

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effectiveness and improve them as she goes. When I propose to treat thequestion 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 itsimply as another of her straightforward questions about the world and herinvestigations of it.

The first thing she’s likely to notice is that neither the reliability of thecoin inference nor the truth of the corresponding if–then statement3

depends on any details of the physical composition of the item in herhand or the particular properties that characterize dimes as opposed toother coins. She quickly discerns that what’s relevant is entirely independ-ent of all but the most general structural features of the situation: an objectwith 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 – forany object a and any properties, P and Q, if Qa-or-Pa and not-Qa, thenPa – and from there to develop a broader theory of forms that yield suchhighly general forms of truth and reliable inference. In this way, she’s ledto consider any situation that consists of objects that enjoy or fail to enjoyvarious properties, that stand and don’t stand in various relations; sheexplores conjunctions and disjunctions of these, and their failures as well;she appreciates that one situation involving these objects and their inter-relations can depend on another; and eventually, following Frege, shehappens on the notion that a property or relation can hold for at leastone object, or even universally – suppose she dubs this sort of thing a‘formal structure’.4

Given her understanding of the real-world situations she’s out todescribe in these very general, formal terms, she sees no reason to supposethat every object has precise boundaries – is this particular loose hair partof the cat or not? – or that every property (or relation) must determinatelyhold or fail to hold of each object (or objects) – is this growing tadpole nowa frog or not? She appreciates that borderline cases are common and fullydeterminate properties (or relations) rare. Thinking along these lines, she’sled to something like a Kleene or Lukakasiewicz three-valued system: for agiven object (or objects), a property (or relation) might hold, fail, or beindeterminate; not-(. . .) obtains if (. . .) fails and is otherwise indetermin-ate; (. . .)-and-(__) obtains if both (. . .) and (__) obtain, fails if one ofthem fails, and is otherwise indeterminate; and so on through the obvious

3 I won’t distinguish between these, except in the vicinity of footnote 5.4 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 inferencepatterns – for example, the introduction and elimination rules for ‘not’,‘and’, ‘or’, ‘for all’, and ‘there exists’; the DeMorgan equivalences; and thedistributive laws – but the gaps produce failures of the laws of excludedmiddle and non-contradiction (if p is indeterminate, so are p-or-not-p andnot-( p-and-not-p).5 The subtleties of the Second Philosopher’s depend-ency relation undercut many of the familiar equivalences: not-(the rose isred)-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) andp obtain, q can’t fail or be indeterminate. Suppose the Second Philosophernow codifies these features of her formal structures into a collection ofinference patterns; coining a new term, she calls this ‘rudimentary logic’(though without any preconceptions about the term ‘logic’). She takesherself to have shown that this rudimentary logic is satisfied in anysituation 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 formalstructure. Common sense clearly suggests that our actual world doescontain objects with properties, standing in relations, with dependencies,but the Second Philosopher has learned from experience that commonsense is fallible and she routinely subjects its deliverances to carefulscrutiny. What she finds in this case is, for example, that the region ofspace occupied by what we take to be an ordinary physical object like thecoin does differ markedly from its surroundings: it contains a more denseand tightly organized collection of molecules; the atoms in those moleculesare of different elements; the contents of that collection are bound togetherby various forces that tend to keep it moving as a group; other forces makethe region relatively impenetrable; and so on. Similarly, she confirms thatobjects have properties, stand in relations, and that situations involvingthem exhibit dependencies.Now it must be admitted that there are those who would disagree, who

would question the existence of ordinary objects, beginning withEddington and his famous two tables:

One of them is familiar to me from my earliest years. . . . It has extention; itis comparatively permanent; it is coloured; above all it is substantial. By

5 Here, briefly, the distinction between logical truths and valid inferences matters, because the gapsundermine all of the former. Inferences often survive because gaps are ruled out when the premisesare taken to obtain.

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substantial I do not mean merely that it does not collapse when I lean uponit; I mean that it is constituted of ‘substance’. (Eddington 1928: ix)

Table No. 2 is my scientific table. . . . It . . . is mostly emptiness. Sparselyscattered in that emptiness are numerous electric charges rushing aboutwith great speed; but their combined bulk amounts to less than a billionthof the bulk of the table itself. Notwithstanding its strange construction itturns out to be an entirely efficient table. It supports my writing paper assatisfactorily as table No. 1; for when I lay the paper on it the little electricparticles with their headlong speed keep on hitting the underside, so thatthe paper is maintained in shuttlecock fashion at a nearly steady level. IfI lean upon this table I shall not go through; or, to be strictly accurate, thechance of my scientific elbow going through my scientific table is soexcessively small that it can be neglected in practical life. (Eddington1928: x)

So far, the Second Philosopher need have no quarrel; Eddington can beunderstood as putting poetically what she would put more prosaically:science has taught us some surprising things about the table, its propertiesand behaviors.

But this isn’t what Eddington believes:

Modern physics has by delicate test and remorseless logic assured me thatmy second scientific table is the only one which is really there. (Eddington1928: xii)

The Second Philosopher naturally wonders why this should be so, why theso-called ‘scientific table’ isn’t just a more accurate and complete descrip-tion of the ordinary table.6 In fact, it turns out that ‘substance’ inEddington’s description of table No. 1 is a loaded term:

It [is] the intrinsic nature of substance to occupy space to the exclusion ofother substance. (Eddington 1928: xii)

There is a vast difference between my scientific table with its substance (ifany) thinly scattered in specks in a region mostly empty and the table ofeveryday conception which we regard as the type of solid reality . . . Itmakes all the difference in the world whether the paper before me is poisedas it were on a swarm of flies . . . or whether it is supported because there issubstance below it. (Eddington 1928: xi–xii)

Here Eddington appears to think that being composed of something likecontinuous 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’veseen, 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 verydifferently than I might have at first imagined – that one couldn’t cometo realize this, that is, without also coming to realize that there is no suchthing as table No. 1. But why should this be so? Why should our initialconceptualization be binding in this way? For that matter, is it even clearthat our initial conceptualization includes any account at all of how andwhy the table supports paper or resists elbows? The Second Philosophersees no reason to retract her belief in ordinary macro-objects.7

So let’s grant the Second Philosopher her claim that formal structureas she understands it does turn up in our actual world. This means notonly that rudimentary logic applies in such cases, but that it does soregardless of the physical details of the objects’ composition, the precisenature of the properties and relations, any particular facts of spatiotem-poral location, and so on. This observation might serve as the first stepon a path toward the familiar idea, noted earlier, that questions likethese are peculiarly philosophical: the thought would be that if thecorrectness of rudimentary logic doesn’t depend on any of the physicaldetails of the situation, if it holds for any objects, any properties andrelations, etc., then it must be quite different in character from ourordinary information about the world; indeed, if none of the physicaldetails matter, if these truths hold no matter what the particular contin-gencies happen to be, then perhaps they’re true necessarily, in anypossible world at all – and if that’s right, then nothing particular toour 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 ideathat logical truths reflect the facts, not about our world, but about aplatonic world of propositions, or a crystalline structure that ourworld 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 scientific image’ with ‘themanifest image’. In fact, the manifest image includes much more than Eddington’s table No. 1 – ‘it isthe framework in terms of which, to use an existentialist turn of phrase, man first encounteredhimself ’ (Sellars 1962: 6) – but Sellars does come close to our concerns when he denies that ‘manifestobjects are identical with systems of imperceptible particles’ (Sellars 1962: 26). He illustrates with thecase of the pink ice cube: ‘the manifest ice cube presents itself to us as something which is pinkthrough and through, as a pink continuum, all the regions of which, however small, are pink’ (Sellars1962: 26), and of course the scientific ice cube isn’t at all like this. Here Sellars seems to think, withEddington, that science isn’t in a position to tell us surprising things about what it is for the ice cubeto be (look) pink; he seems to agree with Eddington that some apparent features of the manifest icecube can’t be sacrificed without losing the manifest ice cube itself. Indeed the essential features theycling to are similar: a kind of substantial continuity or homogeneity. The Second Philosopherremains 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 philoso-phy proper.

But suppose our Second Philosopher doesn’t set foot on this path.Suppose she simply notices that nothing about the chemical makeup ofthe coin is relevant, that nothing about where the coin is located isrelevant, – that only the formal structure matters to the reliability of therudimentary logic she’s isolated. From here she simply continues herinquiries, turning to other pursuits in geology, astronomy, linguistics,and so on. At some point in all this, she encounters cathode rays andblack body radiation, begins to theorize about discrete packets of energy,uses the quantum hypothesis to explain the photo-electric effect, andeventually goes on to the full development of quantum mechanics. Andnow she’s in for some surprises: the objects of the micro-world seem tomove from one place to another without following continuous trajectories;a situation with two similar particles A and B apparently isn’t differentfrom a situation with A and B switched; an object has some position andsome momentum, but it can’t have a particular position and a particularmomentum at the same time; there are dependencies between situationsthat violate all ordinary thinking about dependencies.8 Do the ‘objects’,‘properties’, ‘relations’, and ‘dependencies’ of the quantum-mechanicalmicro-world enjoy the formal structure that underlies rudimentary logic?The Second Philosopher might well wonder, and sure enough, her doubtsare soon realized. In a case analogous to, but simpler than position andmomentum, she finds an electron a with vertical spin up or vertical spindown, 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 lawof rudimentary logic doesn’t obtain!

We’re now forced to recognize that those very general features theSecond Philosopher isolated in her formal structures actually have somebite. Though it wasn’t made explicit, an object in a formal structure wasassumed to be an individual, fundamentally distinct from all others; havinga property – like location, for example – was assumed to involve having aparticular (though perhaps imprecise) property – a particular location, notjust some location or other. These features were so obvious as to gounremarked 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 ventureddown that path the Second Philosopher didn’t take were tempted to thinkthat her formal structure is to be found in every possible world, but it turnsout 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 onthe very general, but still not universal, features isolated in the SecondPhilosopher’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 wecome to know these things? If we followed the philosopher’s path andsucceeded in dismissing the vicissitudes of contingent world as irrelevantwell before the subsequent shocks dealt the Second Philosopher by quan-tum mechanics, then we might continue our reasoning along these lines: iflogic is necessary, true in all possible worlds, if the details of our contingentworld are beside the point, then how could coming to know its truthsrequire us to attend to our experience of this world?11 Again a range ofoptions flourish 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 fromthis observation – that the micro-world doesn’t seem to consist of individual objects – then go on toclassify the ordinary table, along with the botanist’s giant redwoods and the physical chemist’smolecules, as human constructs imposed for ‘epistemological book-keeping’ (p. 240) on an entirelyobjectless world. I suspect that this disagreement with the Second Philosopher traces at least in partto differing 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, thebotanist and the chemist that there are tables, trees and atoms, that trees are roughly constituted bybiological items like cells, cells by chemical items like molecules, molecules by atoms, and so on. Shedoesn’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 thisalone should undermine our belief in the objects of our ordinary world. In contrast, the ‘naturalizedmetaphysics’ of Ladyman and Ross is the work of ‘naturalistic philosophical under-labourers’(p. 242), designed to show ‘how two or more specific scientific hypotheses, at least one of whichis drawn from fundamental physics, jointly explain more than the sum of what is explained by thetwo hypotheses taken separately’ (Ladyman and Ross 2007: 37) – and it’s this project that deliversthe surprising result that ordinary objects are constructed by us. From their perspective, the SecondPhilosopher ‘metaphysics naturalized’ is just more science: the botanist and the physical chemistmake no contribution to ontology; metaphysics only begins when their hypotheses are unified withfundamental physics. From the Second Philosopher’s perspective, there’s no reason to suppose thatordinary objects are human projections or to insist that assessments of what there is must involveunification with fundamental physics. Indeed, from her perspective, given our current state ofunderstanding (see below), quantum mechanics is perhaps the last place we should look forontological guidance!

10 This incidentally removes another sort of skeptical challenge to the Second Philosopher’s belief inordinary 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 manyphilosophers recognize a posteriori necessities, but logical truth seems a poor candidate for this sortof 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 thuscan’t itself be confirmed. But let’s return to the Second Philosopher’s morenaïve inquiries, still well clear of the philosopher’s path, and ask how sheanswers the simple question: how do we come to know that rudimentarylogical inference is reliable?

In general, the Second Philosopher’s epistemological investigations takethe form of asking how human beings – as described in biology, physi-ology, psychology, linguistics, and so on – come to have reliable beliefsabout the world – as described in physics, chemistry, botany, astronomy,and so on.12 Work in psychology, cognitive science, and the like is primaryhere, but the Second Philosopher’s focus is somewhat broader; not onlydoes she study how people come to form beliefs about the world, she alsotakes it upon herself to match these beliefs up with what her otherinquiries have told her about how the world actually is, and to assesswhich types of belief-forming processes, in which circumstances, are reli-able. Though her epistemology is naturalized – that is, it takes placeroughly within science – it’s also normative.

In the case of rudimentary logic, the Second Philosopher’s focus is onformal structure: her other studies of the world have revealed the existenceof many objects, with properties, standing in relations, with dependencies,and she now asks how we come to be aware of these worldly features. Hereshe recapitulates the work of an impressive research community in con-temporary cognitive science.13 The modern study of our perception ofindividual objects reaches back at least to the 1930s, when Piaget usedexperiments based on manual search behavior14 to argue that a childreaches the adult conception of a permanent, external object by a seriesof stages ending at about age 2. Conflicting but inconclusive indicationsfrom visual tracking suggested that even younger children might have theobject concept, but it wasn’t until the 1980s that a new experimentalparadigm emerged for testing this possibility: habituation and preferentiallooking. In such an experiment, the infants are shown the same event overand over until they lose interest, as indicated by their decreased lookingtime (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 This is reminiscent of many of Quine’s descriptions of his ‘epistemology naturalized’, but Quinealso tends to fall back on more traditional philosophical formulations, asking how we manage toinfer our theory of the external world from sensory data (see Maddy 2007, §I.6; for more).

13 I can only give the smallest sampling of this work here. For more, with references, see (Maddy 2007,§III.5).

14 E.g., does the child lift a cloth to find a desirable object she’s seen hidden there?

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inconsistent with the adult understanding; if an infant is thinking likethe adult, the inconsistent display should draw a longer gaze (preferentiallooking).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 flat toward theviewer on the stage surface, and it can pivot through 180˚ arc to lie flataway from the viewer. The infant is habituated to seeing the screen movethrough this range of motion. Then the screen is positioned towardthe infant, a box is placed behind it, and the screen is rotated backwards.The consistent display shows the screen stopping when it comes to reston the now-hidden box; the inconsistent display shows it moving as beforeand coming to rest on the stage surface away from the viewer. If the infantthinks the box continues to exist even when it’s hidden by the screen, andthat the space it occupies can’t be penetrated by the screen, then theinconsistent display should draw the longest gaze. (Notice that the incon-sistent display is exactly the one the infant has been habituated to, so itsvery inconsistency would be sufficiently novel to overcome the habitu-ation.) In this early use of the new paradigm, this is exactly what wasobserved in infants around five months of age.

Obviously this is only the beginning of the story. For example, does theinfant understand the box as an individual object, as a unit, or just as anobstacle to the screen? Experiments of similar design soon indicated thatinfants as young as four months perceive a unit when presented with abounded and connected batch of stuff that moves together. Now imaginea display with two panels separated by a small space. An object appearsfrom stage left, travels behind one screen, after which an object emergesfrom behind the second screen, and vanishes stage right. One group offour-month-olds is habituated to seeing an object appear in the gapbetween the screens, as if it moved continuously throughout; anothergroup is habituated to seeing an object disappear behind the first screenand an object emerge from behind the second screen without anythingappearing in the gap. The test displays are then without panels, showingeither one or two objects. The result was that the infants habituated withthe apparently continuous motion looked longer at the two-object testdisplay than the infants habituated with the scene where no object wasseen in the gap. It seems an object is regarded as an individual if its motionis 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 concep-tualize 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 samespace, and they expect them to travel in continuous trajectories. In thewords of Elizabeth Spelke, a pioneer in this field, the infant’s objects are‘complete, connected, solid bodies that persist over occlusion and maintaintheir identity through time’ (Spelke 2000: 1233):

Putting together the findings from studies of perception of object boundar-ies and studies of perception of object identity, young infants appear toorganize visual arrays into bodies that move cohesively (preserving theirinternal connectedness and their external boundaries), that move togetherwith other objects if and only if the objects come into contact, and thatmove 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 tomove at constant speeds, for example; and they are more likely to maintaintheir connectedness over motion than they are to maintain a rigid shape.Infants’ perception appears to accord with the most reliable constraints onobjects. (Spelke et al. 1995: 319–320)

Partly because so much of this research depends on experiments conductedwith habituation/preferential-looking and closely related designs, partly forother reasons,15 these conclusions can’t be taken as irrevocably established,but then the fallibility of ongoing science is an occupational hazard forthe Second Philosopher. Let’s take the early emergence of a modesthuman ability to detect (some of ) the world’s individual objects as atentative datum.

As for properties and relations, the infants’ sensitivity to these plays arole in the habituation/preferential-looking studies mentioned earlier:habituating to green objects then preferentially looking at red ones mustinvolve noticing those colors, likewise the spatial relations of objects andthe screens. What’s surprising is that object properties aren’t initially usedto individuate them. Ten-month-old infants watched as a toy duckemerged from the left side of a single screen, followed by a ball emergingfrom the right side of the screen; one of the two test displays then showedthe duck and the ball, the other just the duck – and no significantdifference between their reactions was found! The same experiment run

15 E.g., Hatfield (2003) argues that the findings of Spelke and her collaborators only establish thatyoung infants perceive ‘bounded trackable volumes’ not ‘individual material objects’. Of course,Spelke (e.g., in Spelke et al. (1995), cited by Hatfield) does allow that the infant’s object conceptcontinues to develop in early childhood, so there is room here for clarification of levels or degrees ofobject perception. A question like this would prompt the Second Philosopher to get down to sortingthings 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 haveexpected: the test display with the duck alone drew greater attention.On reflection these results aren’t so bewildering. While infants begin

with simple but highly reliable spatiotemporal constraints on object iden-tity (as Spelke notes), property distinctions require more judicious appli-cation: a red ball can turn blue and still be the same object; a human canchange clothes and still be the same person. Some experience and learningmust be needed for the child to realize that ducks don’t generally turn intoballs, 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 nervoussystem were replaced. . . . Because we know that dogs but not cars havebehavioral and mental capacities supported by certain internal structures, weconsider certain transformations of dogs to be more radical than other,superficially similar transformations of cars. (Spelke et al. 1995: 302–303)

With this in mind, it’s less surprising that the beginnings of the child’sidentification of objects by their properties comes a couple of months laterthan their identification by the more straightforward spatiotemporalmeans, and perhaps even that this new development apparently coincideswith the acquisition of their first 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 conjunc-tions and disjunctions of object properties, the failure of properties orrelations, simple billiard-ball style causal dependencies, and so on. It’s alsonotable 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 spatio-temporally, to perceive their properties and relations, to notice dependen-cies, would have been as useful on the savanna as they are in modern life.All this leaves the Second Philosopher with two well-supported hypoth-

eses: the ability to detect (at least some of ) the formal structure present inthe world comes to humans at a very early age, perhaps largely due to ourevolutionary inheritance; whether by genetic endowment, normal matur-ation, or early experience, the primitive cognitive mechanisms underlyingthis ability are as they are primarily because humans (and their ancestors)interact almost exclusively with aspects of the world that display thisformal structure. From here it’s a short step to the suggestion that thepresence of these primitive cognitive mechanisms, all tuned to formal

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structure, is what makes the simpler inferences of rudimentary logic strikeus as so obvious. Assuming that the Second Philosopher has this right –that the formal structure is often present, that we are configured to detectit, and that this accounts for our rudimentary logical beliefs – then asufficiently externalist epistemologist might count this as a case of a prioriknowledge. An epistemologist of more internalist leanings might hold thatthe sort of a posteriori inquiry undertaken here would be required tosupport actual knowledge of rudimentary logic. The Second Philosopherisn’t confident that this disagreement has a determinate solution, isn’tconfident that the debate is backed by anything more substantial thanthe various handy uses of ‘know’, so she’s content to offer a fuller versionof the story sketched here, and to leave the decision about ‘knowledge’to others.

Notice, incidentally, that if this is right, if the Second Philosopher’sformal structure is so deeply involved in our most fundamental cognitivemechanisms, this explains why it’s so difficult for us to come up with aviable interpretation of quantum mechanics, where formal structure goesawry. But this observation raises another question: if formal structure andhence rudimentary logic are missing in the micro-world, and if these areso fundamental to our thought and reasoning, how do we manage tocarry out our study of quantum mechanics? Some suggest that we shouldadopt a special logic for quantum mechanics,16 but the question posedhere is how we manage to do quantum mechanics now, apparently usingour ordinary logic. I think the answer is fairly simple: what we actuallyhave in quantum mechanics isn’t a theory of particles with properties, inrelations, with dependencies, but a mathematical model, an abstractHilbert space with state vectors.17 This bit of mathematics displays allthe necessary formal structure – it consists of objects with properties, inrelations, with (logical) dependencies – so our familiar logic is entirelyreliable there.18 The deep problem for the interpretation of quantum

16 See the discussion of deviant logics below.17 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 ourcognition and our logic work, the Second Philosopher would predict that these authors shouldencounter some difficulty 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 thatdividing 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 ofmathematical relationships’ (Ladyman and Ross 2007: 299).

18 In fact, the mathematical world is in some ways more amenable to our logical ways (see footnote19 below).

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mechanics is to explain how and why the mathematical model works sowell, to figure out what worldly features it’s tracking, but in the math-ematics itself, our natural ways of thinking and reasoning are on impec-cable 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 eliminatesthe law of excluded middle, the principle of non-contradiction, and indeedall logical truths. An inference rule as central as reductio ad absurdum canbe seen to fail: that (q-and-not-q) follows from p only tells us that p iseither false or indeterminate. And the substantive requirements ondependency relations undercut most of our usual manipulations with theconditional. Though he’s speaking of a full Kleene system, with a truth-functional conditional, I think Feferman’s assessment applies to rudimen-tary logic as well: ‘nothing like sustained ordinary reasoning can be carriedon’ (Feferman 1984: 95).Under the circumstances, a stronger, more flexible 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 thatdescription is awkward or unworkable in various ways. To take oneexample, she can give a complete molecular description of water flowingin a pipe, but alas all practical calculation is impossible. In hope of makingprogress, she introduces a deliberate falsification – treating the water as acontinuous substance – that allows her to use the stronger and moreflexible mathematics of continuum mechanics. She has reason to thinkthis might work, because there should be a size-scale with volumes largeenough to include enough molecules to have relatively stable temperature,energy, density, etc., but not so large as to include wide local variations inproperties like these. This line of thought suggests that her deliberatefalsification might be both powerful enough to deliver concrete solutionsand benign enough to do so without introducing distortions that wouldundercut its effectiveness for real engineering decisions. She tests it out,and happily it does work! This is what we call an ‘idealization’, indeed asuccessful idealization for many purposes. (It would obviously be unaccept-ably distorting if we were interested in explaining the water’s behaviorunder electrolysis.) In similar ways, we ignore friction when its effects aresmall enough to be swamped by the phenomenon we’re out to describe; wetreat slightly irregular objects as perfectly geometrical when this does noharm; and so on.

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With the technique of idealization in mind, the Second Philosopherlooks for ways to simplify and streamline her theoretical account offormal structure, that is, her rudimentary logic, in ways that make itmore flexible, more workable, and to do so without seriously undermin-ing its reliability. To this end, she makes two key idealizations, introducestwo falsifying assumptions – that there is no indeterminacy, that anyparticular combination of objects and properties or relations either holdsor fails; and that dependencies behave as material conditionals19 – and ata stroke, she transforms her crude rudimentary theory into our modernclassical logic. There can be no doubt that full classical logic is anextraordinarily sophisticated and powerful instrument; the only openquestion is whether or not the required idealizations are benign. Andas in the other examples, this judgment can be expected to vary fromcase to case.

This is where some of the so-called ‘deviant logics’ come in. Proponentsof one or another of the various logics of vagueness, for example, may insistthat indeterminacy is a real phenomenon,20 may condemn ‘the lamentabletendency . . . to pretend that language is precise’ (J. A. Burgess 1990: 434).On the first point, the Second Philosopher agrees – indeterminacy is real –but she views the classical logician’s pretending otherwise as no different inprinciple than the engineer’s pretending that water is a continuous fluid;what determines the acceptability of either pretense isn’t the obvious factthat it is a pretense, but whether or not it is beneficial and benign in thesituation at hand. Most logics of vagueness begin from a picture not unlikethe Second Philosopher’s, in which, for example a property can hold of anobject, fail to hold, or be indeterminate for that object; there’s also theproblem of higher-order indeterminacy, that is, of borderline casesbetween holding and being indeterminate, between being indeterminateand failing. So far, I think it’s fair to say that there is no smooth andperspicuous logic of vagueness, no such logic that escapes Feferman’scritique. It is, of course, true that classical logic can lead us astray incontexts 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 Though these idealizations involve falsification in her description of the physical world, they aresatisfied in the world of classical mathematics: excluded middle holds and the dependencies arelogical. For more on the ontology of mathematics, see (Maddy 2011).

20 There is serious disagreement between various writers over the source of the indeterminacy: is itpurely linguistic or does the world itself include borderline cases and fuzzy objects? Here the SecondPhilosopher sides with the latter, but this shouldn’t affect the brief discussion here, despite theformulation in the quotation in the next clause above.

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classical logic with care,21 as one should any idealization, rather than switchto a less viable logic.22

Advocates of various conditional logics protest the Second Philosopher’sother bold idealization: replacement of real dependencies with the simplematerial conditional. There are many proposals for a more substantialconditional, far too many to consider here (even if my slender expertiseallowed it), but perhaps the conditional of relevance logic can be used as onerepresentative example. The motivation here speaks directly to the falsifica-tion in question: the antecedent of a conditional should be ‘relevant’ to theconsequent.23 To return to our earlier example, the redness of this rose isn’trelevant to the fact that 2 þ 2 = 4, despite the truth of the correspondingmaterial conditional (if the rose is red, then 2þ 2 = 4). Of course, as before,the Second Philosopher fully appreciates that the material conditional is afalsification, that the rose inference is an anomaly, but the pertinentquestions are whether or not the falsification is beneficial and benign, andwhether or not the relevance logician has something better to offer. AgainI 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 thendepends on the extent to which the falsifications introduced are beneficial andbenign, and on the systematic merits of the proposed alternative. But not alldeviant logics fit this profile; some concern not just the idealizations of classicallogic, but the fundamentals of rudimentary logic itself. Examples includeintuitionistic logic – which rejects double negation elimination – quantum

21 Sorensen (2012) credits this approach to H. G. Wells: ‘Every species is vague, every term goes cloudyat its edges, and so in my way of thinking, relentless logic is only another name for a stupidity – for asort of intellectual pigheadedness. If you push a philosophical or metaphysical enquiry through aseries of valid syllogisms – never committing any generally recognized fallacy – you neverthelessleave behind you at each step a certain rubbing and marginal loss of objective truth and you getdeflections that are difficult to trace, at each phase in the process’ (Wells 1908: 11).

22 Williamson (1994) also advocates retaining classical logic, but his reason is quite different: becausethere is no real vagueness, because apparent borderline cases really just illustrate our ignorance ofwhere 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 thatanything follows from a contradiction. For related reasons, full relevance logic rejects even somerudimentary logical inferences not involving the conditional, like disjunctive syllogism, but I leavethis 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 ofrudimentary logic: e.g., free logicians counsel us to reject the falsifying assumption that all namingexpressions refer. Here, too, our assessment depends on the effectiveness of the idealization and theviability of the alternative. In practical terms, leaving aside the various technical studies in the theoryof free logics, I’m not sure using a free logic is readily distinguishable from being careful about theuse of existential quantifier introduction in the context of classical logic. In any case, our concernhere is with worldly idealizations, not linguistic ones.

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logic – which rejects the distributive laws – and dialetheism – which holdsthat there are true contradictions. Given the connection of rudimentary logicwith the Second Philosopher’s formal structure, the challenge for each ofthese 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 metaphysicalpicture, but it’s suited to describing the world of constructive mathematics,not the physical world.26 Quantum logic at first set out to characterize thenon-formal-structure of themicro-world, but in practice it has not succeededin doing so;27 the problem of interpreting quantum mechanics remainsopen. And dialetheism faces perhaps the highest odds: as far as I know, itsdefenders have focused for the most part on the narrower goal of locating acompelling example of a true contradiction in the world, perhaps so farwithout conspicuous success.28 The Second Philosopher tentatively con-cludes that rudimentary logic currently has no viable rivals as the logic ofthe world, and that classical logic likewise stands above its rivals as anappropriate idealization of rudimentary logic for everyday use.

In sum, then, the Second Philosopher’s answer, an ordinary answer tothe question of why that coin must be foreign, is that the coin and itsproperties display formal structure and the inference in question is reliablein all such situations. This answer doesn’t deliver on the usual philosoph-ical expectations: the reliability of the inference is contingent, our know-ledge of it is only minimally a priori at best. The account itself results fromplain empirical inquiry, which may lead some to insist that it isn’tphilosophy at all. Perhaps not. Then again, if the original question –why is this inference reliable? – counts as philosophical – and it’s not clearhow else to classify it – then the answer, too, would seem to have someclaim to that honorific. But the Second Philosopher doesn’t care muchabout labels. After all, even ‘Second Philosophy’ and ‘Second Philosopher’aren’t her terms but mine, used to describe her and her behavior. In anycase, philosophy or not, I hope the Second Philosopher’s investigations dotell us something about the nature of that inference about the coin.29

25 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 Maddy2007: 293–296).

26 See the discussion of Creator Worlds in (Maddy 2007: 231–233, 296) and (Maddy to appear, §II).27 See (Maddy 2007: 276–279, 296). 28 See (Maddy 2007: 296–297).29 My thanks to Patricia Marino for helpful comments on an earlier draft and to Penelope Rush for

editorial improvements.

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cha p t e r 6

Logical nihilismCurtis Franks

1. Introduction

The idea that there may be more than one correct logic has recentlyattracted considerable interest. This cannot be explained by the mere factthat several distinct logical systems have their scientific uses, for no onedenies that the “logic” of classical mathematics differs from the “logics” ofrational decision, of resource conscious database theory, and of effectiveproblem solving. Those known as “logical monists” maintain that thepanoply of logical systems applicable in their various domains says nothingagainst their basic tenet that a single relation of logical consequence iseither violated by or manifest in each such system. “Logical pluralists” donot counter this by pointing again at the numerous logical systems, forthey agree that for all their interest many of these indeed fail to trace anyrelation of logical consequence. They claim, instead, that no one logicalconsequence relation is privileged over all others, that several such relationsabound.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 theirentire discussion pivots: the idea that one thing a logical investigationmight do is adhere to a relation of consequence that is “out there in theworld,” legislating norms of rational inference, or persisting some otherwise independently of our logical investigations themselves. My opinion isthat fixing our sights on such a relation saddles logic with a burden that itcannot comfortably bear, and that logic, in the vigor and profundity thatit displays nowadays, does and ought to command our interest preciselybecause of its disregard for norms of correctness.I shall not argue for the thesis that there are no correct logics. Although

I do find attempts from our history to paint a convincing picture of arelation of logical consequence that attains among propositions (or sen-tences, 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 reflection about what work the notion of a correctlogic is supposed to be doing, why the debate about the number of logicalconsequence relations is supposed to matter to a logician, and whether theactual details of logic as it has developed might be difficult to appreciate ifour attention is overburdened by questions about the correctness of logicalprinciples. Rather than issue any argumentative blows, I propose merely tolead 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 logicalcorrectness is the law of excluded middle, f _ :f. Is this law logicallyvalid, so that we know that its instances, “Shakespeare either wrote allthose plays or he didn’t,” “Either the continuum hypothesis is true or it isnot true,” “He’s either bald or he isn’t,” etc., each are true in advance ofany further information about the world?

One hardly needs to mention that hundreds of spirited disavowals anddefenses of lem have been issued in the last century. Many of these haveeven been authored by expert logicians. But let us turn our backs to theseideological matters and consider briefly some of what we have learnedabout lem quite independently of any question about its correctness.

The simplest setting for this is the propositional calculus.1 The intui-tionistic propositional calculus (IPC) differs from the classical propos-itional calculus (CPC) precisely in its rejection of lem. For a concreteand standard formalization of IPC one may take a typical Hilbert-styleaxiomatization of CPC and erase the single axiom for double negationelimination, ‘ ::f ⊃f, leaving the rules of inference as before.

In fact, one of the first things observed about lem is its equivalence withdne, which is most easily seen in the setting of natural deduction. Standardnatural deduction presentations of CPC have a rule allowing one to inferany formula f from the single premise ::f. It is easy to derive in such asystem the formula f _ :f. It is similarly easy to show that if we modify

1 I emphasize that this really is a matter of perspicuity. One should not think that the phenomenadescribed below are artifacts of peculiar features of propositional logic. They are nearly allconsequences 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 propositionalrules of Heyting Arithmetic (and of Heyting Arithmetic with Markov’s principle) are exactly those ofIPC (Visser 1999); the disjunction property holds for Heyting Arithmetic (Kleene 1945) and forintuitionistic Zermelo–Fraenkel set theory (Myhill 1973).

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this calculus by disallowing dne but now introduce a new rule that allowsone to write down any instance of lem in any context, f will be derivablein all contexts in which ::f is derivable.

In (1934–35), Gentzen observed to his own surprise2 that the sequentcalculus presentation of CPC admits an even more elegant modificationinto a presentation of IPC than the one just described. One simplydisallows multiple-clause succedents and leaves the calculus otherwiseunchanged. Thus lem and with it the entire distinction betweenintuitionistic and classical logic is subsumed into the background structureof 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 transform-ation. Thus it appears that a duly chosen logical calculus allows a preciseanalysis of what had been thought of as a radical disagreement about thenature of logic. When classicists and intuitionists are seen to admitprecisely the same inference rules, their disagreement appears in someways quite minor, if more global than first suspected.

Exactly how minor, on closer inspection, is the difference between thesesuperficially similar calculi CPC and IPC? Not very. Even beforeGentzen’s profound analysis, Gödel (1932) observed that IPC satisfiedthe “disjunction property”: formulas such as f _ ψ are provable only ifeither f or ψ is as well. At first sight this might appear to be no more thana restatement of the intuitionist’s rejection of lem. After all, that rejectionwas motivated by the idea that instances of lem are un-warranted whenneither of their disjuncts can be independently established. But, one mightthink, if any disjunction is warranted in the absence of independentverification of one of its disjuncts, those like f _ :f are, so rejectinglem should lead to something like the disjunction property. This reasoningstrikes me as worthy of further elaboration and attention, but it should beunconvincing as it stands. For one thing, the formal rejection of lem onlybars one from helping oneself to its instances whenever one wishes. Thegap between this modest restriction and the inability ever to infer anydisjunction of the form f _ :f at all, unless from a record of thatinference one could effectively construct a proof either of f or of :f, isa broad one. More, there are infinitely many formulas f unprovable in

2 Gentzen described the fact that lem prescribes uses of logical particles other than those given by theirintroduction and elimination rules as “troublesome.” The way that in the sequent calculus the logicalrules 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 firstformulating that calculus” (Gentzen 1938: 259).

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IPC, such that IPCþf is consistent but insufficient to prove lem.Examples like :p _ ::p may well fuel suspicions that even in the shadowof global distrust of lem, some disjunctions f _ ψ are more plausiblethan any instance of lem even in the absence of resources sufficient toderive f or ψ.

Thus the disjunction property is a non-trivial consequence of theinvalidity of lem. In fact, this situation exemplifies a recurring phenom-enon in logic, wherein from the assumption of a special case of somegeneral hypothesis, that hypothesis follows in its full generality. This oftenhappens even when, as in the present case, the general phenomenon doesnot appear, even in hindsight, to be a “logical consequence” of its instancein any absolute sense. The disjunction property has further consequencesof its own, however, to which we can profitably turn.

In the approach to semantics known as inferentialism, the meaning of alogical particle should be identified with the conditions under which one isjustified in reasoning one’s way to a statement governed by that particle.From this point of view, which is given expression already in some ofGentzen’s remarks, and owing to the separation in sequent calculus of lemfrom the logical rules, the meanings of the familiar logical particles mightbe said not to differ in intuitionistic and classical logic.

However, the disjunction property gives rise to an alternative interpret-ation of the logical particles of IPC in which each theorem refers back tothe notion of provability in IPC itself. For if f _ ψ is provable only if oneof f and ψ is as well, then a candidate and interesting reading of thesentence ‘ IPC f _ ψ is “Either ‘ IPC f or ‘ IPC ψ.” Expanding on thisidea, one might suggest that the provability of a conjunction “means”that each of its conjunctions is provable, that the provability of a condi-tional, f ⊃ ψ, “means” that given a proof of f one can construct a proofof ψ, and that the provability of :f “means” that a contradiction can beproved in the event that a proof of f is produced.

As we shall see shortly, this so far informal interpretation ofintuitionistic logic is riddled with ambiguities. All the same, somereflection should bring home the idea that some disambiguation of thisreading is a possible way to understand the theorems of IPC. When onecompares the situation with CPC, where conditional truth comes socheap and the disjunction property fails badly, one can only concludethat the difference between these calculi is in some sense great after all,greater even than the debate over the validity of lem alone first suggests.When one then recalls the earlier observation that these calculi can bepresented so that their formal differences are slight and their rules

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identical, the observation that radical differences in meaning result fromso 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 interpret-ation 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 inevery situation in which the set of theorems of IPC is closed under the rule“from f, infer ψ.” However, the disjunction property implies that theseexpectations 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) innatural deduction are proofs whose last inferences are instances of ⊃-elim(modus ponens), ^-elim, dne, or ⊃-intro. It is easy to see that a proofending 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 sameformula 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 proofnecessarily contains a subproof of ψ _ χ from the assumption :f (toallow for ⊃-intro). What might this subproof look like? Once more, dne isnot an option, so again by insisting that the proof is normalized (so that itdoesn’t end needlessly and awkwardly with ⊃-elim or ^-elim) we ensurethat its last step is an instance of _-elim. But this means that an initialsegment of this subproof is a derivation in IPC either of ψ or of χ from theassumption :f (it is here that the disjunction property rears its head), andin 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 theoremof 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 argumentestablished that the Kreisel–Putnam rule is admissible for IPC. Onemight expect that the rule is also “derivable,” that ‘IPC (:f ⊃ (ψ _ χ))⊃ ((:f ⊃ ψ) _ (:f ⊃ χ)). However, it is not (Harrop 1960). Thissituation is dis-analogous to that of classical logic, where all admissiblerules are derivable so that the distinction between admissibility andderivability vanishes. In the parlance, we say that CPC is “structurallycomplete” 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 intuitionisticallyvalid 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 finite basis of rules that generates them all (Rybakov 1997).

What ought one make of the structural incompleteness of IPC? Onething that can definitely 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” isproblematic for both the classicist and the intuitionist, but for differentreasons. This reading is wrong for the classicist, because the idea ofprocedurality simply does not enter into the conditions of classicalvalidity. By contrast, procedures of proof transformation are central forthe intuitionist. However, we now know that there are procedures fortransforming a IPC-proof of f into a IPC-proof of ψ in cases where f⊃ ψis not a theorem of IPC. So at best one could say that IPC is incompletewith respect to this semantics, and more plausibly one should say that thisreading of ‘IPC f ⊃ ψ is erroneous.

Thus we see a sense in which the phenomenon of structural complete-ness is related to a sort of semantic completeness: a structurally incompletelogic will be incomplete with respect to the most naive procedural readingof its connectives. It also happens that structural completeness bears aprecise relation to the phenomenon of Post-completeness, the situation inwhich any addition made to the set of theorems of some logic will trivializethe logic by making all formulas in its signature provable. To state thisrelationship, we refer to a notion of “saturation.” For a logical calculusL whose formulas form the set S, let Sb(X ) be the set of substitutioninstances of formulas in X � S and let CnL(X ) be the set of formulas fsuch 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 isstructurally complete if, and only if, it is saturated.

For these and perhaps other reasons many authors have felt that thepresence of non-derivable, admissible rules is a deficiency of systems likeIPC. The very term “structural incompleteness” suggests that something ismissing from IPC because correct inferences about provability in this logicare not represented as theorems in IPC. Rybakov (1997), for example,suggests that “there is a sense in which a derivation inside a [structurallyincomplete] logical system corresponds to conscious reasoning [and] aderivation using [its] admissible rules corresponds to subconsciousreasoning.” He faults such systems for having rules that are “valid inreality” yet “invalid from the viewpoint of the deductive system itself ”(10–11). Structurally complete systems, by contrast, are “self contained” inthe sense that they have the “very desirable property” of being conscious ofall 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 naiveprocedural understanding of the logical connectives. The situation oughtrather, I counter, lead one to appreciate the subtlety of proceduralityexhibited in intuitionism. For the logical lesson to be learned is thatin the absence of lem the context of inference takes on a new role. Thusf ⊃ ψ means that given any background of assumptions from which f isprovable, a proof of f can be transformed into a proof of ψ under thosesame assumptions, and this understanding does not reduce, as it does withlogics insensitive to context, to the idea that any proof of f can betransformed into a proof of ψ. This irreducibility strikes me as a “verydesirable property” for many purposes. I should like to know more aboutthe 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 classicallogic, are not sensitive in this way to context. Perhaps the constructivenature of IPC derives from its context-sensitivity. Surprisingly, Jankov’slogic, IPCþ:f _ : :f, appears to undermine any hope of establishing aconnection between these phenomena. Consider the Medvedev lattice ofdegrees of solvability. The setting is Baire space ωω (the set of functionsfrom ω to ω) and the problem of producing an element of a given subset ofthis space. By convention, such subsets are called mass problems, and theirelements are called solutions. One says that one mass problem reduces toanother if there is an effective procedure for transforming solutions of thesecond into solutions of the first. If one defines the lattice of degrees ofreducibility of mass problems, it happens that under a very naturalvaluation, the set of identities of corresponds to the set of theoremsof Jankov’s logic, so that the theory of mass problems provides aconstructive interpretation of this logic.3 However, Jankov’s logic is struc-turally complete (Prucnal 1976). Thus one sees that the so-called weak lawof 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 complexitytheory and because the special difficulty of representing within thesetheories their own consistency statements shed much light on the finedetails of arithmetization. We had a running gag, which is that formaltheories like PA are awfully weak, because with them one can’t draw verymany distinctions. The implicit punchline, of course, is that the bulk of

3 For the details of this interpretation, see (Terwijn 2004).

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the distinctions one can draw in theories of bounded arithmetic are amongstatements that are in fact equivalent. Lacking the resources to spot theseequivalences is no strength!

Something perfectly analogous happens in the case of substructurallogics. There are theorems of CPC that are unprovable in IPC, but notvice versa, so the latter logic is strictly weaker. Moreover, CPC proves allsorts of implications and equivalences that IPC misses. But if we stopbelieving for a moment, as the discipline of logic demands we do, and askabout the fine structure of inter-dependencies among the formulas ofpropositional logic, IPC delivers vastly more information. Consider justpropositional functions of a single variable p. In CPC there are exactlyfour equivalence classes of such formulas: those inter-derivable with p, :p,p _ :p, and p ^ :p. In IPC the equivalence classes of these same formulasexhibit a complicated pattern of implications, forming the infinite Rieger–Nishimura lattice.

One thought one may have is that IPC should be considered an expan-sion of CPC: every classical tautology can be discovered with IPC via thenegative translation of Gödel (1933) and Gentzen, so with IPC one gets allthe classical tautologies and a whole lot more. (Gödel at times suggestedsomething like this attitude.) But, of course, neither the negative transla-tion nor the very idea of a classical tautology arises within intuitionisticlogic. The thought that I encourage instead is this: The logician is loath tochoose between classical and intuitionistic logic because the phenomena ofgreatest interest are the relationships between these logical systems. Whowould have guessed that the rejection of a single logical principle wouldgenerate so much complexity – an r.e. set of admissible rules with no finitebasis, an infinite lattice of inter-derivability classes?

The intuitionist and the classicist have very fine systems. Perhaps withthem one gains some purchase on the norms of right reasoning or themodal structure of reality. The logician claims no such insight butobserves that one can hold fixed the rules of the logical particles and, bymerely tweaking the calculus between single conclusion and multi con-clusion, watch structural completeness come in and out of view. The sameswitch, he or she knows, dresses the logical connectives up in a construct-ive, context-sensitive interpretation in one position and divests them ofthis interpretation in the other. These connections between sequentcalculus, constructive proof transformation, structural completeness, andlem are fixtures from our logical knowledge store, but they cannotseriously be thought of as a network of consequences in some allegedlycorrect logic.

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3. Logic imposed and logic discovered

If I have conveyed my attitude successfully, then I will have inspired thefollowing objection:

You speak about an unwillingness to embrace any one or select few logicalsystems because of an interest in understanding all such systems and howtheir various properties relate to one another. But by making logical systemsinto objects of investigation, you inhabit an ambient space in which youconduct this investigation. It is legitimate to ask which logic is appropriatein this space. What is your “metalogic”?

I shall explain that this objection rests on various preconceptions that I donot share. I hope the explanation functions to aid the reader in seeing theseas 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 wemust hasten to adopt should be the one that assists us in detecting andunderstanding these relationships. Which relationships? Presumably, it hasbeen 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 statesof affairs and pledging allegiance to factual relations of ground and conse-quence? I have an intuitive sense of what a fact is; I have no such sense ofontological grounding. Nor have I seen any reason to expect that the studyof 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 indetecting and understanding whatever relationships we can find. Thenwe 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 thereseemed to have been only disorder. We might find that one hat helps timeand again, but we will be well-advised not to forget that we are wearing it.For if we never take it off, then we risk forever overlooking logicalrelationships of considerable interest. Worse, we risk coming to thinkof the relationships we can detect as “in the world,” “preconditions ofthought,” 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 properanalysis of the quantifiers that occur in natural language. Notably, quanti-fiers allow us to reason in succinct strokes about infinite collections. About

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a century ago, David Hilbert had the idea of analyzing quantifierswith consistency proofs, and he devised an intricate calculus with “trans-finite axioms” which allows proof figures involving quantifiers to betransformed (in principle) into figures without them. Hilbert’s idea wasthat reasoning about this transformation (which by its nature requiresbeing attentive to constructibility) would expose the quantifiers as innocu-ous parts of our mathematical language and also make perspicuous theircomplexity.4

Hilbert conducted this investigation in the shadow of a great ideologicalquarrel about the validity of various logical and mathematical principles. Inone quarter were dour skeptics who distrusted not only lem but otherforms of infinitary reasoning. Chief among them was Kronecker, againstwhom Hilbert (1922: 199–201) railed because he “despised . . . everythingthat did not seem to him to be an integer.” Less famously, but perhapsmore importantly, he faulted Kronecker also because “it was far from hispractice to think further” about what he did accept, “about the integeritself.” In a similar vein, Hilbert observed that Poincaré “was from the startconvinced of the impossibility of a proof of the axioms of arithmetic”because of his belief that mathematical “induction is a property of themind.” Thus Hilbert viewed these figures as short-sighted, not only intheir rejection of mathematical techniques that he wished to defend, butalso in their belief that the manifest validity of a principle precludes anyhope of our analyzing it. Both attitudes, he cautioned, “block the path toanalysis.”5

After only a decade of partial successes, it was discovered that the sort ofconsistency proofs Hilbert envisioned are not available. Specifically, Gödel(1931) demonstrated that no proof of the consistency of a reasonably strongand consistent mathematical system could be carried out within that samesystem. Typically it is the recursion needed to verify that the prooftransformation algorithm halts that cannot be so represented. This situ-ation raises the question whether proving with such principles that asystem 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 iscarried out in the relatively weak theory PRA together with the relativelystrong principle of transfinite induction up to the ordinal

4 For details, see section 2 of (Franks 2014).5 For more on Hilbert’s view, see Professor Shapiro’s contribution to this volume.

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�0 ¼ supn<∞

ω1ω2...ωn

Together with Gödel’s result, this proof demonstrates that transfiniteinduction through �0 is unprovable in PA. All of this is well known.Most people familiar with the history of logic are aware that Gentzenproved also that transfinite induction to any ordinal beneath �0, anyordinal ω1

ω2...ωn for n � N, is in fact provable in PA. But Gentzen pressed

even further. One can consider fragments of PA defined by restricting theinduction scheme to formulas with a maximum quantifier complexity (callthese the theory’s class of inductive formulas). Gentzen showed in (1943)that the height of the least ordinal sufficient for a proof of the consistencyof such a fragment corresponds with the quantifier complexity of thattheory’s class of inductive formulas, and that transfinite induction to anysmaller ordinal is provable in the fragment. So the number of quantifiersover which mathematical induction is permitted equals the number ofexponentials needed to express the ordinal that measures the theory’sconsistency strength. One quantifier equals one exponential. Thus thetheory of constructive proof transformations has turned up a precisemathematical analysis of the complexity of natural language quantifiers, aremarkable 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 theproof of PA’s consistency can really shore up our confidence in this theory.To anyone who has witnessed a talk about ordinal analysis sabotaged bythis debate, the scene will be familiar: someone reminds us that the proofuses a principle that extends the resources of PA. Someone else defends theprinciple despite this fact and points out that in every other way the proofis extremely elementary compared to the full strength of PA. Because thetwo theories PA and PRAþti�0

6 are in this basic way incomparable, thejury is out as to the gains made by reducing the consistency of one to thatof the other. From the point of view of logic, however, this is all adistraction from what Gentzen actually achieved: he showed that thequestion of the consistency even of elementary theories can be formulatedas a precise problem, and he showed that the solution to this problemrequires new perspectives and techniques and carries with it unexpectedinsights about logical complexity. If philosophers did not harbor skepti-cism about PA, then they would likely not be interested in Gentzen’s result

6 The theory of primitive recursive arithmetic extended with a principle of transfinite induction to �0.

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one way or the other. Their disinterest in the analogous results aboutfragments of PA is just evidence that they harbor no skepticism about thesetheories. We recognize Gentzen’s analysis of first-order quantifiers as oneof 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 thissection. Of course it is true that in the study of logical systems one mustengage in reasoning of some sort or another. This reasoning can possiblybe described by one or a few select logical systems. But why should anyoneassume that this amount of reasoning is anything more than ways ofthinking that have become habitual for us because of their proven utility?Further, why should anyone assume that there is any commonality amongthe principles of inference we deploy at this level over and above the factthat we do so deploy them?

To expand on the first of these points, it may be helpful to draw ananalogy between rudimentary logic and set theory. Often it is thought thatdecisions about which principles should govern the mathematical theory ofsets should be made by appealing to our intuitions about the set conceptand even about the cumulative hierarchy of sets. Doubtless such appealshave figured centrally in the development of set theory. But the history ofthe subject suggests that a complete inversion of this dynamic has also beenat play. Kanamori (2012: 1) explains:

[L]ike other fields of mathematics, [set theory’s] vitality and progress havedepended on a steadily growing core of mathematical proofs and methods,problems and results . . . from the beginning set theory actually developedthrough a progression of mathematical moves, whatever and sometimes inspite of what has been claimed on its behalf.

Whereas one can today find endless phenomenological and metaphysicaljustifications of, for example, the replacement axiom, Kanamori contendsthat set theory in fact evolved primarily by absorption of successfultechniques, like transfinite induction, devised to answer mathematicalquestions. “It was von Neumann’s formal incorporation of this methodinto set theory, as necessitated by his proofs, that brought in Replacement”(33). In similar fashion, the power set existence assumption, which origin-ally had many detractors, was not finally embraced in the wake of anyargument or philosophical insight. It merely happened that “iteratedcardinal exponentiation figured prominently” in Kurepa’s proofs in infini-tary combinatorics, “so that shedding deeper concerns the power setoperation became further domesticated” (46). The upshot? “Set theory is

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a particular case of a field of mathematics in which seminal proofs andpivotal problems actually shaped the basic concepts and forged axiomatiza-tions, 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 offormal set theory. The point is that those intuitions have evolved partly inresponse to our need to make sense of routinely counter-intuitive scientificdiscoveries. If our intuitions have been at least partially, perhaps largely,shaped by developments in logic, then the fact that we appeal to them onoccasion in order to refine our definitions and techniques or in order tochoose new axioms seems to lose its significance. Rather than develop atheory of sets that unpacks the fundamental truths we intuit, we havedeveloped an intuition about sets that makes sense of mathematicallyinteresting relations we have discovered.I need not argue that rudimentary logic has taken shape in a similar

fashion first because Professor Maddy has been persuasive on this samepoint in her contribution to this volume and second because it does notreally matter for my purposes whether, in the end, this is true. I amcontent simply to reveal the picture that holds us captive when we beginto think about logic. For the objection we solicited was that if our modernscience of logic thrives on an unprejudiced consideration of the full gamutof properties exhibited by logical systems and the relations among them, sothat issues of correctness do not arise, then it has only smuggled thoseissues in through the back door, in the metalogic that makes the sciencepossible. But it is a preconception that science is made possible byahistorical norms of right reasoning. Once one considers the possibilitythat logic may be studied with patterns of thought adapted to what welearn along the way, it becomes hard to understand what special status therudimentary principles we find ourselves reflexively appealing to are sup-posed 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 presum-ably have some features in common that led them there. Even if theseprinciples have been adopted over the course of time, the idea goes, theremust be some reason for their being adopted instead of other principles.Perhaps this reason can be repackaged as an explanation of their being thetrue 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 logicalvalidity 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 itmight be reasonable to look for it in whatever classes of inference rules wehappen to find collected together. But we have no such foreknowledgeand, in fact, the evidence suggests that the arrangement of our toolkit is ahighly contingent matter. Are we not better off shedding this vestigialbelief that among all the intricate and interesting consequence relations outthere, some have a special normative status? Can we not get by with theunderstanding that principles of inference with a rather wide range ofapplicability differ from those suited only to specific inference tasks only inhaving a wider range of applicability? Had this been the understandingthat our culture inherited, would anything we have learned from studyinglogic lead us to question it?

At the end of Plato’s Phaedrus, Socrates explains that prior to investi-gating the essence of a thing, it is important to devise an extensionallyadequate definition of that thing so that we will be in agreement aboutwhat we are investigating. This attitude seems right to me, and it seems tome that the familiar debates about, for example, where logic leaves off andmathematics begins violate this principle unabashedly. Suppose it wereclear to everyone that some but not all patterns of reasoning are inescap-able and furthermore that it were easy to tell which these are. Then wewould have good reason to label this “logic” in distinction to patterns ofreasoning that we all recognize as the province of some special science orparticular application. It would be reasonable to wonder what accounts forthe privileged role that these principles play in our lives. As things stand,however, many of us seem instead to assume that there simply must bepatterns of reasoning that differ in kind from others. Typically, our mindsare already made up about the psychological or metaphysical circumstancesthat underwrite this difference. This is what Wittgenstein stressed with hisobservation that “the crystalline purity of logic was not the result of aninvestigation,” that instead “it was a requirement” (Wittgenstein 1953:§107). Driven by this assumption, we thrash about looking for someextensional definition that we can hang our ready-made distinction on.These definitions are simply unconvincing on their own. They can satisfyonly people who cannot tolerate the thought that there is no line tobe drawn.

When Gentzen began his study of logic, he parted ways with hispredecessors7 by not first defining logical validity and then seeking outlogical principles that accord with that definition. He simply observed that

7 There are historical precedents for Gentzen’s attitude: Aristotle, Condillac, Mill.

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some patterns of reasoning abound. Consider that to define “predicatelogic” he said only that it “comprises the types of inference that arecontinually used in all parts of mathematics” (1934–35: 68). This homelydefinition set Gentzen on a task of empirically tallying the techniques usedin mathematical proofs, ignoring those that are unique to geometry,arithmetic, and other specific branches of mathematics.8 Of course sucha survey is by no means guaranteed to be exhaustively executable, and itwas Gentzen’s good fortune that his subject matter happens to exhibit fewinstances.One should not, however, write off his success as purely a matter of

luck. Gentzen devised an ingenious argument to the effect that his tallywas in fact exhaustive. This involved constructing an innovative type oflogical calculus that is at once formal and patterned on the informalreasoning recorded in mathematical proofs. This scheme enabled him todo more than construct a serial tally of proof techniques, because theinference types identifiable with it are extremely few and systematicallyarranged 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 veryreason they fail to meet the criterion of ubiquity in mathematical practicethat 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’sdefinition of predicate logic does pick out a well-defined body of infer-ences,9 he did not concoct the definition in the service of a preconceivednotion of logical validity. He did not, for example, first stipulate a semanticnotion of logical consequence based on his own intuitions and then askwhether his calculi adequately capture this notion. Gentzen simply pro-posed that the intuitions guiding mathematicians in their research wouldbe worth isolating and studying, and he therefore modeled a logicalcalculus on the inferences mathematicians actually make.10

Of course mathematicians also deploy proof techniques that are lessuniversal, and the only observable difference between these and the onesthat meet Gentzen’s criterion of ubiquity is their relative infrequency.Mathematicians do not report a feeling that arithmetical reasoning is less

8 This is explicit in (Gentzen 1934–35) and even more vividly depicted in section 4 of (Gentzen 1936).9 Actually Gentzen vacillated over the inclusion of the principle of mathematical induction,ultimately deciding against it.

10 For details of this conception of completeness, see section 3 of (Franks 2010).

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valid or valid in some other way than general reasoning, and even if theydid, we should be inclined to ignore these reports if they did not reflect inmathematical practice. For these reasons Gentzen could never bring him-self to describe the distinction between inference rules that appear in thepredicate calculus and those that belong specifically to arithmetic as adistinction between the logical and the non-logical. He only thought thathe had designated a logical system, one that by design encodes some of theinferences he was bound to make when reasoning about it but whoselogical interest derives solely from what that reasoning brings to light.

Contrast this with one of the more famous attempts to demarcate thelogical: Quine’s defense of first-order quantification theory. Second-orderquantification, branching quantifiers, higher set theory, and such can eachbe dissociated from logic for failing to have sound and complete proofsystems, for violating the compactness and basic cardinality theorems, andother niceties. There is even a (1969) theorem, due to Lindström, to theeffect that any logic stronger than first-order quantification theory will failto exhibit either compactness or the downward Löwenheim–Skolem the-orem. Second-order logic, Quine concluded, is just mathematics “insheep’s clothing” because by using second-order quantifiers one is alreadycommitted to non-trivial cardinality claims (Quine 1986: 66).

How true will these remarks ring to someone who doesn’t know inadvance that they are expected to distinguish logical and mathematicalreasoning? Quine’s consolation is telling: “We can still condone the moreextravagant reaches of set theory,” he writes, “as a study merely of logicalrelations among hypotheses” (Quine 1991: 243). I should have thought thatthis accolade, especially in light of the intricate sorts of logical relations thatset-theoretical principles bear to one another and that set theory bears toother systems of hypotheses, would be used rather to enshrine a disciplinesquarely within the province of logic. For if we never suspected that amongthe plenitude of logical relations are a privileged few that capture the trueinter-dependencies of propositions, what else would we mean by “logic”than just the sort of study Quine described?

As to the properties that characterize first-order quantification theory, itshould now go without saying that from our perspective Lindström’stheorem, far from declaring certain formal investigations extra-logical,exemplifies logic. So too do results of Henkin (1949) and others to theeffect that second-order quantification theory and first-order axiomatic settheory each are complete with respect to validity over non-standardmodels. For a final example, I can think of none better than the recentresult of Fan Yang that Väänänen’s system of dependence logic (with

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branching quantifiers), extended with intuitionistic implication (the con-text sensitive and constructive operator that results from the denial of lem),is equivalent to full second-order quantification theory (Yang 2013).Wittgenstein (§108) expected his readers to recoil from the suggestion

that we shed our preconceived ideas by “turning our whole examinationaround.” Rather than impose our intuitions about logic on our investi-gations by asking which principles are truly logical, let us first ask if a closelook at the various inference principles we are familiar with suggests thatsome stand apart from others. If some do, then let us determine what it isthat sets them apart. But the question he puts in our mouths – “But in thatcase doesn’t logic altogether disappear?” – suggests that we know deepdown that our empirical investigation is bound to come up empty. Variouscriteria will allow us to demarcate different systems of inference rules tostudy, but when none of these indicate more than a formal or happen-stance association we will find ourselves hard pressed to explain why anyone 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 logicalis not that there is no such thing as logic. Let us agree instead that no onepart of what logicians study, contingent and evolving as this subject matteris, should be idolized at the expense of everything else. Logic outstrips ourpreconceptions both in its range and in its depth.

4. Conclusion

Traditional debates about the scope and nature of logic do not do justice tothe details of its maturation. In asking whether certain inferential practicesare properly logical or more aptly viewed as part of the special sciences, forexample, we ignore how modern logic has been shaped by developments inextra-logical culture. Similarly, questions about whether logic principallytraces the structure of discursive thought or the structure of an impersonalworld presuppose a logical subject matter unaffected by shifts in humaninterest and knowledge.I mean, by saying this, not just to suggest that the principles of

rudimentary logic are contingent, not different in kind from principlesthat we use only some of the time or very rarely and only for specific tasks.I do urge this attitude. But the caution against mistaking our default,multi-purpose habits of reasoning for something monumental is onlypreparation for a second, more valuable reaction. One should warm upto the trend of identifying logic with the specialized scientific 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 alogic’s correctness stems not only from an interest in exploring the proper-ties of possible logical systems in full generality but also from an appreci-ation, fostered by the study of logic, that no one such system can have allthe properties that might be useful and interesting.

In closing, let me re-emphasize that the idea of a true logic, one thattraces the actual inter-dependencies among propositions, is unscathed byall I have said. Part of the difficulty in questioning that idea is that it is amoving target: argue against it, you feel it again in that very argument;close the door, it will try the window. But this very circumstance onlyunderlines the fact that the idea is a presupposition, nothing thatemerges from any discovery made in the study of logic. For the samereason that we can marshal no evidence against it, we see that if we canmanage to forget it our future discoveries will not reveal to us that wehave erred.

This realization, coupled with the observation that a fixation on the truelogical relationships “out there” hinders the advancement of logic, certainlyrecommends nihilism on practical grounds. The question that remains iswhether we are capable of sustaining a point of view with no directargumentative support.

The proper antidote to our reflexive tendencies will surely extend ananalysis of modern logic and include a rehearsal of the subject’s history.I cannot offer that here.11 I can only mention that logic as a discipline hasevolved often in defiance of preconceived notions of what the true logicalrelations are. Logic has been repeatedly reconceived, not as a fallout fromour better acquaintance with its allegedly eternal nature, but in response tothe changing social space in which we reason. There is reason neither toexpect nor to hope that logic will not be continually reconceived. Suchreconceptions have been and likely will again be fundamental, so that whatmakes the moniker “logic” apt across these diverse conceptions is not aninvariable essence.

In these pages I have indicated instead logic’s modern contours, high-lighting the fact that the deepest observations logic has to offer come withno ties to preconceptions about its essence. The richness of logic comesinto view only when we stop looking for such an essence and focus insteadon the accumulation of applications and conceptual changes that have

11 For the details of the evolution of one central concept – that of logical completeness – in the pasttwo centuries, see (Franks 2013).

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made current logical investigations possible. The study of logic might bethe 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 thatlanguages would have been better made. It would, then, have requiredother philosophers than the ones we know” (237–8). Our interest in abetter made language is an interest in a language that traces a pre-existinglogical structure. Like Condillac, Wittgenstein warned that presupposingsuch a structure fosters dismissive attitudes about the languages we have:“When we believe that we must find that order, must find the ideal, in ouractual language, we become dissatisfied with what are ordinarily called‘propositions,’ ‘words,’ ‘signs’” (§105). When we stop believing for amoment, as the discipline of logic demands we do, the structures we findimmanent in our several, actual languages command our interest morethan anything we could have devised in the service of our ideal.

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cha p t e r 7

Wittgenstein and the covert Platonismof 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 ofmathematics, Cambridge, 1939 (Wittgenstein 1976), PG = Philosophicalgrammar (Wittgenstein and Rhees 1974).)

By the end of the 1930s, Wittgenstein’s thought on mathematics hadundergone a major, if often undetected, change.1 The change had to dowith the relationship between arithmetic, including elementary numbertheory and geometry,2 and empirical regularities, including behavioralregularities that are induced by training. During the first part of thedecade, Wittgenstein continued to regard mathematical theorems as akinto grammatical rules. As such, there was no need to seek a general theory ofmathematical applicability, as Frege did.3 “The applications,” he repeated,“take care of themselves.” (E.g. PG, III, 15: 308.) After all, grammaticalrules have no applications outside grammar itself, being norms, notdescriptions of nonlinguistic objects or processes. This does not imply,to be sure, that the environment in which language operates has no effecton which rules we use in language to describe the environment. In a1939 lecture at Cambridge (LFM XX: 194) Wittgenstein remarked “thereis, in all the languages we know, a word for ‘all’ but not for ‘all but one’.”

1 See (Steiner 2009).2 In this chapter, for the most part, I will not address the complicated question of to what extentWittgenstein’s ideas were intended to describe advanced mathematics, and to what extent he actuallysucceeded in describing advanced mathematics. Hence, we will focus upon arithmetic andelementary 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 quantified, where the predicateletters range over “concepts.” Then to apply an arithmetic proposition, all one needs to do is toperform universal instantiation, replacing each predicate variable with a constant predicate thatexpresses a particular concept. To apply arithmetic to empirical situations, for example, all weneed to do is instantiate empirical predicates for the universally quantified second-order variables.Note that mathematical applicability is in this account the same thing as logical applicability.

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He certainly meant to say that this is true because in our world it is mostconvenient to have a universal quantifier, not because logic is itself empirical.I believe that this part of what he says there, though not all of it, can beattributed to himwell before 1939.4 Even in 1939, Wittgenstein told the class:

To say “A reality corresponds to ‘2 þ 2 = 4’” is like saying “A realitycorresponds to ‘two’.” It is like saying a reality corresponds to a rule, whichwould come to saying: “It is a useful rule, most useful – we couldn’t dowithout 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 aregrammatical rules, the difference being which vocabulary the rules govern.This is not to say, with Frege and Russell, that mathematics is logic. Therules of logic are used to prove mathematical theorems, to be sure, but thisdoes 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 Philosophicalinvestigations. 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 ofthe time – when they are trained the same way. Rules label the deviationsfrom 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 butone” is the primitive idea. And this sort of thing would entirely change their outlook on logic.

This further idea, expressed, as I say, in 1939, I would not want to attribute to Wittgenstein in theearly 1930s (which is what I am discussing here) – and I will discuss it below, in the context of (whatI will call) his revolution of 1937. It is reminiscent of Nelson Goodman’s relativism concerningnatural kinds (“grue”).

5 The idea that the relationship between the empirical world and mathematical propositions is that theformer makes the latter useful is not replaced in 1939, but augmented by a much deeper connectionbetween mathematical propositions and empirical reality, which we will discuss later.

6 I don’t mean to say that the mathematical technique of perturbation theory is “normative.” I bringthe subject of perturbations in because Wittgenstein himself does:

Suppose we observed that all stars move in circles. Then “All stars move in circles” is anexperiential proposition, a proposition of physics – Suppose we later find out they are notquite circles. We might say then, “All stars move in circles with deviations” or “All stars movein 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 makethe description of it the description of a norm – we say, “This is what we are going tocompare things with.” It gives us a method of describing experiments, by saying that theydeviate from this by so much. (LFM X: 99)

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physics) and the like. There are a number of possible explanations of theutility of stigmatizing deviations in this way, at least in the areas of language,logic, and mathematics.7 Society has an interest in rendering certain prac-tices as uniform as possible, and adding negative and positive incentivesmay do the trick.8 This account is plausible in the areas of language andmathematics, which is our topic here.

The so-called “rule-following paradox,” as Wittgenstein himself labels itin 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 toexplain10 human practices like rule-following – i.e., almost all philosophersbesides Wittgenstein. The explanations emerge from diverse philosophies,from mentalism to physicalism, but all agree that there must be some factabout a person, beyond the regularities of his behavior, in virtue of whichwe can say he is following a specific rule at a specific time and not another.The explanation that follows here follows that of Saul Kripke, though,

as will become clear, it differs from his in some crucial details.11 Kripkeattributes a “skeptical argument” concerning rule-following to Wittgen-stein, and has drawn much criticism on this account. I agree with Kripkethat Wittgenstein did construct a skeptical argument, but I hold that theargument is supposed to be valid only for academic philosophy, as distinctfrom Wittgenstein’s own philosophy.

7 Wittgenstein never attempted to found an account of ethics on this basis – there are regularities inthe 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 perfectcertainty, and the lack of reasons does not trouble me.” (PI: 212) I believe that this passage reflectsactual occurrences in Wittgenstein’s life. After the publication of the Tractatus, Wittgenstein leftacademics and went into school teaching. Ray Monk (Monk 1991, pages 195–196, 232–233) reportsthat Wittgenstein used to inflict corporal punishment on his pupils if he thought they were notapplying themselves to the arithmetic lessons he was giving them. Not enough has been said aboutthe connection between the rule-following arguments in Philosophical investigations andWittgenstein’s short-lived experience as a schoolteacher, which came to an end, when one of hispupils 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 (sofar as I know) unpublished works. Felix Mühlhölzer draws my attention to the following passagefrom 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 beparticularly interested. . .

10 Wittgenstein condemns this kind of philosophy in one of the most famous passages in Philosophicalinvestigations: “We must do away with explanation and description must take its place.”(Philosophical investigations, 109; in the 4th edition this is translated: “All explanation mustdisappear. . .”)

11 Cf. (Kripke 1982).

130 Mark Steiner

Since it turns out that there is no such “fact” which can serve as anexplanatory criterion for rule-following, academic philosophers are facedwith a paradox since it now follows that there is no such thing as rule-following at all.12 This could be considered a skeptical argument, though anad hominem one.13

Let us now examine the arguments Kripke brings to support thecontention (on behalf of Wittgenstein) that there is no fact in virtue ofwhich somebody is following a rule. Kripke adduces two argumentsfor this, of which only one is actually in Wittgenstein. This is the“normative” argument: to ascribe rule-following to someone is to assertthat someone is acting according to a norm, i.e. following the rule“correctly” – as we say. The gap between “is” and “ought” then impliesthat no fact or state of the person at time t could be identified withfollowing the rule at t. The situation is different in other cases ofexplanation and reduction in science.14 Disposition terms (e.g. “solubil-ity”) can in principle be reduced to “state descriptions” of a substance,which actually replace the disposition term. This is the so-called “placeholder” theory of dispositional terms. Given the normative nature ofrule-following, i.e. its social nature, it applies to what interests society:behavior. Thus it cannot be reduced to, or identified with, but onlycorrelated with, an underlying state description either of the mind or ofthe brain; that is Wittgenstein’s argument.

Kripke has another argument, the so-called “infinity” argument,according to which any state of the brain,15 for example, is necessarilyfinite; while rule-following commits the trainee to infinitely many

12 The case is formally similar to Frege’s foundation of arithmetic upon “logic.” When Russell’sparadox showed that Frege’s “logic” is inconsistent, Frege overreacted by saying “arithmetictotters.”

13 Kripke compares Wittgenstein’s “skeptical argument” to that of David Hume, and the comparisonis just, but not in the way that Kripke imagines: both Wittgenstein and Hume use skepticalarguments 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 connectionsbetween events, which are used in rationalist explanations of causal reasoning. It is a skepticalargument only for them, because they hold that, without the necessary connections there is nocausal reasoning at all. See here (Steiner 2009: 26ff ).

14 I am here offering my own opinions, not those of Wittgenstein. In fact, I am not at all sure thatWittgenstein distinguished clearly between dispositions in science and abilities in humans, since inPhilosophical investigations, 193–194, he claims that academic philosophers make the same kind ofmistakes in discussing the abilities of humans with dispositions of machines. See also Philosophicalinvestigations, 182, where he compares “to fit” (said of bodies in holes) and “to be able,” “tounderstand,” said of humans.

15 As my colleague Oron Shagrir has cogently argued, Kripke seems to be thinking of a brain state asthe physical realization of a finite digital computer.

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applications of the rule – as in the rule “add two.” Not only is thisargument not in Wittgenstein, it couldn’t be, as I will argue below.16

The “rule-following paradox” is, then, a paradox only for academicphilosophy. For Wittgenstein himself there is no paradox to begin with.For paradox to loom, our ordinary discourse about some topic must beseen to lead to catastrophe. For example, Zeno’s paradoxes began withordinary conceptions of motion and showed that they lead to inconsist-ency, or to the conclusion that no motion is possible. Wittgenstein’saccount of rule-following involves the claim that all rules are supervenientupon regularities. In the case of the rule “add 2” which surfaces inPhilosophical investigations – how do we know that our trainee is followingthis rule if he manages to go 2, 4, 6, 8, . . ., 1000 – or another rule whichsays that after 1000 one starts “adding 4”? We don’t know, but ourexperience, both as students AND as teachers, is that almost all whoproduce this series go on the same17 way: to 1,002 and not to 1,004.(The claim is not that we reflect upon these regularities – that would bea misinterpretation – but that the regularities make our practice in thisregard possible and coherent.) This regularity allows us to attribute the rule“add two” (the rule which is “hardened” from just this regularity) withgreat confidence to our trainee, and to call his response “erroneous” orperhaps “provocative” if he says next 1,004. (We may filter out frivolousresponses by warning the trainee that he will be severely sanctioned if hedoesn’t give the right answer.) Wittgenstein expressed this idea quiteclearly in 1939:

Because in innumerable cases it is enough to give a picture or a section ofthe use, we are justified in using this as a criterion of understanding, notmaking further tests, etc. (LFM I: 21)

The position I am attributing to Wittgenstein is not that of Kripke. Kripkeseems to regard as a criterion that our trainee is following the rule þ2, if he

16 In fairness, however, I must add that my own presentation of Wittgenstein’s “normative” argumentthat there is no “fact” in virtue of which somebody is following a rule, also “improves” the argumentsomewhat. The distinction I draw between disposition terms – which are in principle reducible tostate descriptions – and rule-following ascriptions – which are not – is not in Wittgenstein. On thecontrary, Wittgenstein tends to see rule-following as precisely a disposition, but denies thatdisposition terms are reducible to underlying state descriptions. Since I think Wittgenstein ismistaken 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“infinity” argument, on the other hand, is one which in fact contradicts basic Wittgensteinianinsights and has no place even in an “improved” Wittgensteinian corpus.

17 I will deal below, page 133, with the objection that there is no objective meaning to “the same” andhence that the claim is circular.

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goes on to give the same responses that “we” would give, i.e. agreementwith society. On the contrary, society is itself predicated on empiricalregularities of its members. The criterion, then, is simply that our traineehas successfully followed the rule þ2 up to now. It is true that withoutbehavioral regularity upon training, this criterion would have no point.But to apply the everyday criterion, one does not have to reflect on thisregularity, 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 ourcommon training, so that there is no circularity in saying that a rule isfounded on regularity even though detecting a regularity requires an abilityof ours to follow rules. The same training teaches the concept of “thesame.” For this reason our previous statement that “People trained thesame act the same” is not susceptible to skeptical doubt.18

In the passages of Philosophical investigations we are discussing, thoughnot necessarily in RFM, Wittgenstein is employing a very simple conceptof “applying” a rule. We may understand Wittgenstein as saying thatapplying a rule is simply following it (correctly). Since rules are groundedin regularities, it is the ability to continue the series by doing what almosteverybody does when placed in the same situation, which grounds theability to apply the rule. In principle, there is no difference between rulesof logic and grammatical rules. The “hardness of the logical must” is a kindof projective superstition, much as the superstition that Hume thought hehad exposed in the idea there is “necessary connection” between causes andtheir effects. It is similar to the superstition of thinking that when one isreading he is having a characteristic experience of being “influenced” orbeing “guided” by the text. (PI, §§170ff.)Since rules are norms, there is no equivalence between saying that

somebody is following a rule and saying that his behavior falls under theunderlying regularity. Saying that somebody is following a rule is simplyevaluating his behavior, not describing it – even though the evaluationresults from observing his previous behavior and responding to it in lightof our own training in following rules, and the regularities that areinstilled by that training. In other words, we can describe the criteriathat a teacher is using to evaluate the student’s behavior as successfullyfollowing a rule, even though the teacher may not be aware of thesecriteria. In fact one of the purposes of Wittgenstein’s analysis in

18 One could say it is Wittgenstein’s version of a synthetic a priori proposition.

Wittgenstein and the covert Platonism of mathematical logic 133

Philosophical investigations is to unearth these criteria in support of hisview of philosophy as being purely descriptive.

To the question “How does the rule follower himself know that he isfollowing a rule”Wittgenstein answers that the exclamation “Now I can goon” 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 certainlyno form of self-knowledge, and given what Wittgenstein says in Oncertainty (1974), the conviction on the part of the rule follower whoexclaims “Now I can go on” actually rules out knowledge.20

In summation, logic consists of rules governing the use of logicalexpressions like “and,” “or,” “if. . .then,” “everything,” etc. As Wittgen-stein himself put it, even in the 1940s, “The rules of logical inference arerules of the language-game.” (RFM, VII: 401) There is nothing akin to“intuition,” “seeing,” and the like in following or producing a logicalargument. Instead we have regularities induced by linguistic training,which in hindsight are interpreted, or misinterpreted, by us as some kindof determination. Deviation from this regularity is labeled by society as“incorrect” reasoning. Wittgenstein’s aim is to demystify logic and logicalnecessity, just as Hume’s aim was to demystify causation by eliminatingthe alleged “necessary connection” between events. The image of logic as akind of “super-physics” is what needs to be debunked. Philosophical investi-gations contains a number of references to this mystification of logic:

89. With these considerations we find ourselves facing the problem: In whatway is logic something sublime?

For logic seemed to have a peculiar depth, a universal significance.

Logic lay, it seemed, at the foundation of all the sciences.

97. Thinking is surrounded by a nimbus. –Its essence, logic, presents anorder: namely, the a priori order of the world; that is, the order of possibil-ities, which the world and thinking must have in common.

108. . . .The preconception of crystalline purity [in logic] can only beremoved by turning our whole inquiry around. (One might say: the inquirymust 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 “noncontradic-tion”) he stated:

19 PI, §§151ff. I understand the Private Language Argument of Wittgenstein as saying that what iscalled “referring” to our mental states is more like expressing them than naming them.

20 Can one say “Where there is no doubt there is no knowledge either”?

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Let us go back to the law of contradiction. We saw last time that there is agreat temptation to regard the truth of the law of contradiction as some-thing which follows from the meaning of negation and of logical productand 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.”21 The term “follows fromthe meaning,” is incoherent, since meaning is tied to use, and it does notmake sense to speak of “following from” use. The “temptation” of whichWittgenstein speaks here, is the attraction of the academic philosopher (andthe earlyWittgenstein, of course) to a covert Platonism.Wittgenstein’s own“demystified” view is that logical laws are a special case of rules that are basedon regularities of speakers of language – i.e. rules of “grammar.” As in thegeneral 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 tonew cases.In another lecture, Wittgenstein said:

If we give a word one particular partial use, then we are inclined to go onusing it in one particular way and not in another. “Not” could be explainedby 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 wehave laid down part of the practice. Once this is done, we are inclinedwhen we go on to adopt one step and not another – for example, doublenegation being equivalent to affirmation. (LFM: 242–243)

We can say, then, that logical laws arise in a two step process. First, thechild is trained in the use of words like “not.” The training induces aregularity in this use, a regularity which society reinforces as “correct”usage. Within this regularity, however, there arises a subregularity, whenthe rules for using “not” are to be applied to special cases like doublenegation. Most trainees find themselves using double negation as theywould affirmation. This regularity is then “put in the archives” as a lawof logic.Something similar happens in arithmetic, according to Wittgenstein. In

applying the rules for division to 1/7, most proficient students find them-selves repeating the sequence 0.142857142857. . .22 In fact, most proficientstudents in dividing m by n always get a finite decimal or a repeatingdecimal. This subregularity is then converted into a rule in itself, a law or

21 This is a point that Quine also made during the very same period. See (Quine 1936a), reprinted in(Quine 1976).

22 See LFM, p. 123.

Wittgenstein and the covert Platonism of mathematical logic 135

proposition of arithmetic, as a result of a proof. This view is in accordancewith Wittgenstein’s arguments in PI (186ff.) that rule-following isgrounded, and grounded only, in the actual behavioral regularities ofindividuals and group. The covert Platonist wishes to say more: that, forexample, there is an objective fact of the matter by which the theoremabout repeating decimals is “determined,” in some further – perhapsmetaphysical – sense by the rules for division, once they are accepted.The search for a fact like that, however, Wittgenstein has argued, collapsesinto paradox.

It would be an error, however, to conclude that the only differencebetween arithmetic and logic is that they control different vocabularies:that arithmetic controls numerical terms and logic controls sententialconnectives and quantifiers. The year 1937 saw a revolution in Wittgen-stein’s view of arithmetic, and mathematics in general: arithmetic propos-itions remained rules as always – it was the nature of the rules thatchanged. Mathematical rules were to govern nonlinguistic practices as wellas linguistic ones.

Arithmetic propositions, in Wittgenstein’s post-1937 thought, are rulesthat govern our practice of counting. Geometrical propositions arerules that govern our practice of measuring.23 Not only does the applica-tion no longer “take care of itself,” it is the very heart of the mathematicalproposition. The canonical application is now precisely the regularity ofcounting or measuring which is “hardened” into a rule. The applications ofarithmetic and geometry are outside mathematics; they are empiricalapplications. The applications of logic remain, as before, within logic: anapplication of modus ponens, for example, is simply an inference of theform “If A, then B; A; therefore B.” To see the difference between thesetwo kinds of applications, consider an example Wittgenstein loves to use:the game of chess. When we apply the rules of chess, we are only playingchess. The rules can apply to infinitely many “chess sets,” which areunlimited in their physical composition and also shapes. However,although chess is essentially a war game between two “kingdoms,” thereare no applications of chess outside the game itself, even to war itself. Themore abstract theory of games, which is real mathematics, does have such“external” applications.

Wittgenstein even ventured the idea in 1939 that set theory is notmathematics at all, because it has only imaginary applications. In the

23 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 asof a mathematical concept, is determined by the application.24

In this scheme, proofs bring the mathematician to convert regularities torules. Wittgenstein came up with the idea that they do so by beingschematic “pictures” of these nonmathematical regularities, something likeflowcharts in computer science or schematic diagrams in electronics:

You might say that the relation between a proof and an experiment is thatthe proof is a picture of the experiment, and is as good as the experiment.(LFM, VII: 73)

What is interesting is that some of these new rules function as rulesthat determine whether previous rules have been followed. (Since, byWittgenstein’s rule-following considerations, there is no fact by whichthe previous rule has been followed, the idea as such does not harborany contradiction. For Wittgenstein, the “rule-following paradox” isnot only not a paradox, but it bolsters his account of mathematics.)Consider again the theorem that:

1/7 = 0.142857142857 and so on ad infinitum (a repeating decimal)

One might think that the infinite expansion of 1/7 is determined25 fromthe beginning by the rules for division that are learned in school (or wereonce learned in school). But the rules cannot outstrip the regularities thatare their basis, and the regularities, being regularities of human beingscannot go on forever, and in fact, at some finite point, the regularities willpeter out: the deviation will increase to the extent that no rule could befounded 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

24 Wittgenstein asserts:

It is the use outside mathematics, and so the meaning of the signs, that makes the sign-gameinto mathematics. (RFM, V: 2)

Here we have the extreme anti-formalist statement that the applications of mathematics givemeaning to its language.

In case the message has been missed, Wittgenstein relays it again at once:

What does it mean to obtain a new concept of the surface of a sphere? How is it then a conceptof the surface of a sphere? Only insofar as it can be applied to real spheres. (RFM, V: 4)

25 Wittgenstein recognizes a number of meanings for the concept of “determination” in mathematics,and some of them he might regard here as innocuous. See PI, 189. I thank Felix Mühlhölzer for thisreference. Compare also LFM, p. 28.

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first 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” anycalculation which does not lead to a repeating decimal; it overrules thenaïve use of the school rules. This kind of proof is characteristic ofmathematics:

It is just the same with 1:7 = 0.142857142. . . You say, “This must giveso-and-so.”

Suppose it doesn’t.

Suppose what doesn’t?

Here I am adopting a new criterion for seeing whether I divide thisproperly – and that is what is marked by the word “must”. But it is acriterion which I need not have adopted. For just as bricks measured withall exactness might give a curve (‘space is curved’), so 1 : 7 = 0. . . . lookedthrough 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’vemade up a new criterion for the correctness of the division. And I havemade it up because it has always worked. If different people got differentthings, I’d have adopted something different. (LFM XIII: 129)

In fact, Wittgenstein remarked in RFM that one should not regardcalculations with very large numbers as simple applications of the rulesfor the operations which we learned on small numbers:

We extend our ideas from calculations with small numbers to ones withlarge numbers in the same kind of way as we imagine that, if the distancefrom here to the sun could be measured with a footrule, then we should getthe very result that, as it is, we get in a quite different way. That is to say, weare inclined to take the measurement of length with a footrule as a modeleven for the measurement of the distance between two stars. (RFM,Part III: 147)

The reader should not be surprised to find Wittgenstein in a somewhatambivalent attitude towards “finitism”:26

If one were to justify a finitist position in mathematics, one should say justthat in mathematics “infinite” does not mean anything huge. To say“There’s nothing infinite” is in a sense nonsensical and ridiculous. But itdoes make sense to say we are not talking of anything huge here. (LFM: 255)

26 By “finitism” Wittgenstein always means what is now called “strict finitism,” according to which itis incorrect or false to assert “There are infinitely many natural numbers.”

138 Mark Steiner

We now can see where Kripke went wrong in attributing the “infinity”argument to Wittgenstein. The argument was supposed to defeat the ideathat “following a rule” is identifiable 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 infinitely many(correct) responses to the question, “What is . . . þ2”? But the brain, beingfinite, cannot produce infinitely many answers to questions of this kind.Kripke discusses a number of possible responses to this argument and findsfault with them all. He does not realize, however, that the major premise ofhis argument is in direct conflict with a basic feature of Wittgenstein’saccount of arithmetic: the idea that adopting an algorithm like “plus”determines in some physical, mental, or metaphysical way one’s responseto infinitely many exercises is nothing but covert Platonism, in many waysworse than the Platonism of objects.These reflections reflect on the application of logic to arithmetic. By the

application of logic to arithmetic I mean simply the substitution ofarithmetic propositions in the variables (or schematic letters, if you prefer)of logical rules or “truths.” Consider the law of the excluded middle, a lawof the “Propositional Calculus,” p_ � p. An application of this wouldbe: Either the Goldbach conjecture is true or its negation is true. TheGoldbach conjecture states that every even number greater than 2 is thesum of two primes (e.g. 8 = 5 þ 3). The conjecture has been shown to holdfor very large numbers, and there are corollaries of the conjecture whichhave been proved. But no proof of the full conjecture has been given,though most mathematicians are persuaded that it is “true.” (There arepseudo-probabilistic arguments for this, based on the fact that as thenumbers get larger, the “probability” that a given number can be parti-tioned into two primes rises monotonically, since the number of thepartitions 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 thatthe natural numbers form a closed totality, or what Aristotle called an“actual infinite,” so that we can say that either there is, or is not, acounterexample to the Goldbach conjecture in this closed totality. If wethink of the natural numbers through the metaphor of “becoming,”rather than “being,” then the present absence of a proof or of a refuta-tion of the Goldbach conjecture means only that the truth of theconjecture is not determined, and the law of the excluded middle cannotbe asserted. As an invalid rule of inference, it is thus banished fromclassical mathematics.

Wittgenstein and the covert Platonism of mathematical logic 139

Let us now apply Wittgenstein’s ideas to the Intuitionist program.Wittgenstein agrees entirely with the Intuitionist critique of the law ofexcluded middle. For the Goldbach conjecture to be “true” in the sense ofclassical mathematics, we have to say that the operations of arithmeticdetermine in advance that every even number, no matter how large, can bepartitioned into two primes. Wittgenstein agrees that this is not math-ematics, but metaphysics: a statement like this cannot be grounded on thebehavioral regularities inculcated in grade school. A statement true of allthe natural numbers can be based only upon a theorem – which lays downa new norm (on the basis of a proof ) which labels any deviation fromthe Goldbach conjecture a “mistake.” Hence, Wittgenstein agrees with theIntuitionists that one cannot regard the law of excluded middle for theGoldbach conjecture as a theorem of mathematics. It cannot be regardedas the “hardening” of a regularity.

How, then, are we to square this with Wittgenstein’s explicit disavowalof Intuitionism (“Intuitionism is all bosh,” he said, “entirely” (LFMXXIV: 237))?

There are two explanations available. The first has to do with theconnection of Intuitionism with . . . intuition. Brouwer writes as if thenumbers themselves are mental constructions, and the law of excludedmiddle does not apply to mental constructions, which can never produce aclosed totality. Wittgenstein is an implacable opponent of the concept ofmathematical intuition – he believes, among other things, that it has noexplanatory value, and hence its only rationale fails. From this point ofview, Intuitionism is a form of mentalism, the other side of the coin fromPlatonism. 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 haslittle or nothing to do with mathematical intuition. According to this pointof view, which is presumably heavily influenced by Wittgenstein’sthought, truth in general is associated with “assertibility.” And sincemathematical propositions are asssertible only when provable, Dummettthinks,27 one cannot assert an instance of the law of excluded middle attime t unless we can show at t that one of the two alternatives can beproved. Thus a proof of the following form is invalid at t, despite theacceptance of it by almost all mathematicians:

27 I actually deny this, and have given examples of mathematical propositions that were assertible evenwhen there was no proof of them in (Steiner 1975). But I will take for granted that Wittgensteinagrees with Dummett on this point, an agreement that has a solid basis in the corpus.

140 Mark Steiner

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 theIntuitionist attack on the law of excluded middle means: to revise math-ematics on the basis of a philosophical argument. Wittgenstein is quiteexplicit on this point in one of the most famous passages of Philosophicalinvestigations: no mathematical discovery is relevant to philosophy, and nophilosophical argument can revise accepted mathematical practice. Phil-osophy describes practice; and the only reason we need philosophy is thatwe 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. Atthe same time he refuses to revise mathematical practice on the basis ofDummett’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 mathemat-ical theorems are “hardenings” of regularities was never meant to be acharacterization of the “essence” of mathematics. The philosopher whoemphasized so strongly the idea that the referents of certain terms (andreally all terms) are related only by a “family resemblance” did not becomean “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 infinite case (i.e. eitherall natural numbers have property P or not all natural numbers haveproperty P) should be regarded as a “postulate” and was used as such inmathematics. Presumably the postulate should be judged by its usefulnessin mathematics, though Wittgenstein, ironically, rejected the most cele-brated attempt (Hilbert 1983) to “justify” the law of excluded middle –namely, by showing – without using the law of excluded middle – that thelaw of excluded middle does not lead to contradiction, when applied to“infinitary” statements: “Either all numbers have property P, or there is anumber that does not have property P.”A consistency proof can be compared to theorems to the effect that, in

chess, a forced checkmate is not possible from a certain position. And theattempt to find one is associated with David Hilbert’s programmatic “On

28 Mühlhölzer protests this translation, which has become entrenched in the philosophicalWittgenstein discourse, and insists that the right phrase is “multi-colored.”

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the Infinite” – the program which is almost universally thought to havebeen refuted by Gödel’s second theorem – which states that arithmeticcannot prove its own consistency even if the law of excluded middle isused, to say nothing of the kind of combinatorial, “metamathematical,”proof Hilbert had in mind. At the risk of digressing, I would now like todiscuss in a little more detail why a consistency proof for Wittgenstein isnot what we are seeking in showing the usefulness of the law of excludedmiddle.

Wittgenstein’s rejection of Hilbert’s program had nothing to do withGödel’s theorem, which he in any case regarded with suspicion. On thecontrary, he regarded Gödel’s theorems as part and parcel of what waswrong with the program to begin with – the concept of “metamathe-matics.” Nor did he regard the search for consistency proofs for math-ematics 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 piecewith his theory of rule-following in general.

How do we get convinced of the law of contradiction? – In this way: Welearn a certain practice, a technique of language; and then we are all inclinedto 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 meansthat if I give you orders I must do my best to avoid contradictions; though itmay be that what I wanted was to puzzle you or to make you lose time orsomething of that sort.

The Law of Contradiction is thus nothing but the hardening of a linguisticpractice into a rule.

From this it follows that the concept of a “hidden contradiction” does nothave a clear meaning.

There is always time to deal with a contradiction when we get to it. Whenwe get to it, shouldn’t we simply say, “This is no use – and we won’t drawany conclusions from it”? (LFM: 209)

At this point, Alan Turing (who attended Wittgenstein’s lectures atCambridge 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 casea 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 insidemathematics? Turing says, “Because something may go wrong with theapplication.” But nothing need go wrong. And if something does gowrong – if the bridge breaks down – then your mistake was of the kindof 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 othermathematical proposition, inconsistency is either a rule, or nothing. Aslong as it is not a rule, i.e. a proven theorem, physical applications go on asbefore.But let’s look at this a little closer. Wittgenstein discusses whether a

bridge could fall down because somebody divided by zero. This is certainlypossible; consider the equation x2 = x. 93 percent of precalculus students atCity College of the City University of New York, in a recent test, dividedby x and got the (only) answer x = 1.29 Not knowing about the solutionx = 0 could, in some scenarios, indeed cause a bridge to fall down. Muchmore sophisticated cases could be constructed in which somebody doesnot 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 ofteaching students an axiomatic number theory in which cancellation ofzero is possible, in other words an inconsistent system. The students mightnot 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 anactual case in which a hidden contradiction in a formal axiomatic systemcaused “bridges to fall down.”A good example of this quandary is the theory of quantum electrodynam-

ics (QED), pioneered by, among others, Schwinger and Feynman.30 Thecalculations afforded by this theory are remarkably accurate, but nobodyknows how to base the calculations in a consistent axiomatic mathematicalsystem. In fact, there are mathematical physicists who think it cannot bedone. One reason is as follows. In calculating the probability of events in

29 See for example www.nydailynews.com/new-york/education/cuny-math-problem-report-shows-freshmen-city-hs-fail-basic-algebra-article-1.418801.

30 I am grateful to Barry Simon and Shmuel Elitzur who helped me with the details.

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quantum mechanics, according to the usual rules, we come upon infiniteintegrals. To “renormalize” these integrals, which are intended to be (Feyn-man 1985) functions of the basic constants of physics, we change the rules:instead of theoretical values like e, the charge of the electron, we substitutethe 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 verywell for observed values at the low scales of energy with which we arefamiliar, as the energies get higher the observed electronic charge becomeslarger and larger, so that even the new rules are not valid. In fact, mathemat-ical physicists think that this charge may become infinite at some finite highenergy, so that QED is not defined at all as a universal “theory of light andmatter.”31

For Wittgenstein, this would just show what he was claiming all thetime: that the ideal of a formal system does not fit the reality of mathemat-ical physics.32 This would be a perfect example of “The disastrous invasionof mathematics by logic.” (RFM, V: 281) It is also plausible that theinconsistency which appears in the infinite integrals has its source in thephysics, not the mathematics, exactly as Wittgenstein says. But, further,the physicists managed to eliminate the troublesome integrals, albeit bytweaking the rules for calculations in QED by using, as we said, not the“naked” magnitudes that appear in the “Hamiltonian” of the system, suchas e, but the “dressed” magnitudes as measured in the laboratory. Thisartifice works, and no physicist worries that a possible inconsistency (whichis suspected though not proved) could somehow spoil the calculations wemake at familiar energies.

Coming back to the law of excluded middle, we see that the problem isnot that it has no formalist justification in terms of a combinatorialconsistency proof. It is rather that the law of excluded middle cannot beregarded as a hardened regularity in cases in which we are applying it to aputative infinite totality. But precisely because of this, there is no directcomparison possible between empirical observations and mathematicaltheorems in this type of proof. That is what Wittgenstein means by a“postulate.” The justification of such a postulate would be, in Quine’spithy words, “where rational, pragmatic.”33 It would seem that Wittgen-stein, in accommodating classical mathematics and rejecting the intuition-ist revisionism, ends up where Quine began: in holism.34

31 I am of course referring to the title of (Feynman 1985).32 Tim Chow made this point on the FOM list, on August 15, 2013. 33 (Quine 1953: 46).34 I am grateful to Felix Mühlhölzer and Penny Rush for their helpful comments.

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part ii

History and Authors

cha p t e r 8

Logic and its objects: a medieval Aristotelian viewPaul Thom

What is logic about? According to one familiar account logic tells uswhen arguments are valid; logic is thus about arguments. On anotheraccount logic tells us which propositions are (unconditionally) neces-sary; logic is thus about propositions (Smith 2012). Less familiar thaneither of these accounts is the Aristotelian tradition of thinking aboutlogic. Aristotelians have standardly thought of logic as being aboutterms, as well as propositions and arguments. Let us call propositionsand arguments, and whatever else logic has been supposed to beabout, the objects of logic. The general question that interests me is:What are the metaphysical types to which the objects of logic belong?More specifically, I will look at the way this question has beenaddressed in the Aristotelian tradition. I will not be dealing withanswers to our question proposed by Platonists or with the Stoicconcept 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 sympathisedwith his views. The latter include the ancient Greek commentators, amultitude of medieval logicians writing in Arabic or Latin, and a smallernumber of later thinkers (notably Bernard Bolzano). But my main focuswill be on just one of these, the thirteenth-century philosophical logicianRobert Kilwardby (d. 1279). Kilwardby dealt with our question at somelength, and his discussion is also useful in that it considers several viewsother than his own. Let us begin with Aristotle’s own ideas on ourquestion.

1. Aristotle

There is not much in Aristotle’s own writings that bears directly on ourquestion. Four passages are noteworthy.

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First, the Categories makes a remark about statements:

Statements and beliefs, on the other hand, themselves remain completelyunchangeable in every way; it is because the actual thing changes that thecontrary comes to belong to them. For the statement that somebody issitting remains the same; it is because of a change in the actual thing that itcomes 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 achange in itself – would be distinctive of substance, even if we were to grantthat beliefs and statements are able to receive contraries. However, this isnot true. For it is not because they themselves receive anything thatstatements and beliefs are said to be able to receive contraries, but becauseof what has happened to something else. For it is because the actual thingexists or does not exist that the statement is said to be true or false, notbecause it is able itself to receive contraries. No statement, in fact, or beliefis changed at all by anything. So, since nothing happens in them, they arenot able to receive contraries. (Aristotle 1963: 4a5)

Here Aristotle leaves two positions open: either statements do not changetruth-value at all, or else any change in their truth-value is due to a changein something external to them, namely the things which the statementsare about.

Second, in the De Interpretatione we find Aristotle apparently proposinga general semantic theory according to which the meaning of spoken andwritten utterances is to be found in the existence of mental items thatsomehow correspond to them:

Now spoken sounds are symbols of affections in the soul, and written markssymbols of spoken sounds. . . . Just as some thoughts in the soul are neithertrue nor false while some are necessarily one or the other, so also withspoken sounds. For falsity and truth have to do with combination andseparation. (Aristotle 1963: 16a2)

Here, the meaning of spoken and written language is derived from ‘affec-tions in the soul’, and truth and falsity are seen as residing primarily in thecombination or separation of mental items.

Third, there is a remark in the Posterior Analytics which, again, seems topoint to the soul as the locus of truth and demonstration.

By contrast, it is always possible to find fault with ‘external’ arguments (i.e.spoken or written ones): For demonstration is not addressed to externalargument – but to argument in the soul – since deduction is not either. Forone can always object to external argument, but not always to internalargument. (Aristotle 1994: 76b23)

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Finally, there is a remark in Metaphysics Book 6 which again locates truthand 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 ofbeing false, depends on combination and separation, and truth and false-hood together are concerned with the apportionment of a contradiction (fortruth has the affirmation in the case of what is compounded and thenegation in the case of what is divided, while falsity has the contradictoryof this apportionment – it is another question, how it happens that wethink things together or apart; by ‘together’ and ‘apart’ I mean thinkingthem so that there is no succession in the thoughts but they become aunity – ; for falsity and truth are not in things – it is not as if the good weretrue, and the bad were in itself false – but in thought; while with regard tosimple 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 isor is not in this sense; but since the combination and the separation are inthought 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 areeither unchanging or any change in them is due to a change insomething else;

2. the meaning of written and spoken language is to be explained byreference to what goes on in our minds;

3. truth and falsity belong in the first instance to combinations andseparations that occur in our minds.

These are scattered remarks. Aristotle doesn’t show how they could becombined in a coherent theory of terms, propositions and arguments. Wedo not find such a theory in Aristotle; we find only some materials thatseem 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 enlistelements drawn from Aristotle’s metaphysics or his account of scientificknowledge. Another would be to import non-Aristotelian ideas. We willsee that both approaches were used by later Aristotelians in their efforts toflesh 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 thecategories, our question becomes: Do the objects of logic belong to any ofthe 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 ofthinkers in the twelfth and thirteenth centuries.

2. Robert Kilwardby

The thirteenth-century English philosopher and churchman RobertKilwardby commented extensively on Aristotle’s logic, as well as compos-ing a treatise On the origin of the sciences and a set of questions on the fourbooks of Peter Lombard’s Sentences. Over the course of his career heshowed a continuing interest in the nature of the objects of logic, andindeed the nature of logic itself.

In his early question-commentary on the Prior Analytics Kilwardby takesthe view that logic is one of the language-related sciences along withgrammar and rhetoric. In the work’s first sentence he adopts Boethius’scharacterisation of logic as an art of discoursing (Kilwardby 1516: 2ra).1 Hegoes on to consider the meaning of the words ‘proposition’ [propositio,Aristotle’s protasis] and ‘syllogism’ [syllogismus] as they occur in Boethius’stranslation of Aristotle’s text, distinguishing propositions from statements[enuntiationes]. A statement is put forward on its own account, a propos-ition on account of the conclusion it is intended to support. A statementexpresses what is in the speaker’s soul, and accordingly is defined as thatwhich is either true or false since truth and falsity reside in the soul(Kilwardby 1516: 4rb).2 In his other writings Kilwardby will generallypreserve this distinction, reserving the term ‘proposition’ for the premiseof an argument.

He asks whether a syllogism should be defined as a kind of process,rather than a kind of discourse (following Aristotle’s definition). He agreesthat there is a sense in which a syllogism is a mental process, but says thatthis is a metaphorical sense (Kilwardby 1516: 4vb).3 And it must indeed beregarded as a transferred usage for someone whose starting-points areAristotle’s usage of ‘syllogism’ to mean a kind of discourse and Boethius’scharacterisation of logic as a science of language.

In his later work On the rise of the sciences logic is no longer characterisedpurely as a linguistic science, and the syllogism is no longer a purelylinguistic phenomenon. Logic is there presented under two guises. It is ascience of reason as well as being a language-related science:

1 Robert Kilwardby, Notule libri Priorum, Prologue.2 Kilwardby, Notule libri Priorum Lectio 2 dubium 5.3 Robert Kilwardby, Notule libri Priorum Lectio 4 dubium 1.

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It is called a science of reason not because it considers things belonging toreason as they occur in reason alone, since in that case it would not properlybe called a science of discourse, but because it teaches the method ofreasoning that applies not only within the mind but also in discourse,and because it considers the things belonging to reason as the reasonswhy things set forth in discourse can be reasoned about by the mind. . . ..It is, therefore, a ratiocinative science, or science of reason, because itteaches one how to use the process of reasoning systematically, and a scienceof discourse because it teaches one how to put it into discourse systematic-ally. (Kilwardby 1988: 265)

On this view, which places mental reasoning on a par with reasoning inwords, it cannot be right to dismiss as merely metaphorical a conception ofthe syllogism as a mental process.He raises the issue of the basis or foundation of logic, declaring that

there are three different kinds of basis on which a body of scientificknowledge can be founded. The science might be based on things thatactually exist. Or it might be based on potentialities, even when they areunactualised. Finally, a science might be based not on potentialities but onaptitudes of things. These are incomplete potentialities, such as the apti-tude for sight which exists even in a blind eye. Now, even though speechpasses away as soon as it is uttered, something remains, namely certainnatural principles wherein potentialities or aptitudes reside. Because ofthese, speech contains enough on which to found a science, even whenno-one is speaking (Kilwardby 1976: 429).

Later in On the rise of the sciences he adds that an art of reasoning has asufficient foundation in the natures of things through which they aresusceptible to a rational account. Among these natures he mentionsantecedents, consequents, incompatibles, universality, particularity,middles, extremes, figure 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 PeterLombard’s four books of the Sentences. Question 90 on Book One ofthe Sentences contains a detailed exposition of the metaphysical status ofthe objects of logic. It seems that Kilwardby himself attached someimportance to this exposition, for in the alphabetic index which hecompiled of the matters covered in his questions, he refers on fouroccasions to question 90 on Book One.4 It is therefore worth examininghis exposition in detail.

4 Kilwardby 1995: Entries under ‘Ens’, ‘Predicamentum’, ‘Ratio’ and ‘Res’.

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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 ofcomplex and incomplex things insofar as they concern reason. And the firstquestion about these is whether they are something, in such a way that theyare things in one or more of the categories. (Kilwardby 1986: 1, q.90: 1)

Although he refers here to ‘propositions’, he proceeds to discuss insteadwhat he calls ‘stateables’ (enuntiabilia). This is no doubt partly because ofthe distinction he had made earlier between propositions and statements;but this doesn’t explain why he talks about stateables rather than state-ments. Christopher Martin takes the expression enuntiabile in earlierauthors 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 fordoubting that this is Kilwardby’s meaning.

Concerning the nature of the objects of logic, Kilwardby mentions aview according to which stateables cannot be assigned to any of theAristotelian categories. Among the arguments he mentions in favour ofthis view, two rest on Aristotelian texts. First, there is the chapter of theCategories where the ten categories are presented as a classification of‘things said without any complexity’; stateables on the other hand, if theyare things at all, are things possessing complexity. The second Aristoteliantext is the one we noted above from Metaphysics book 6. Here, saysKilwardby, the ten categories are presented as being truly outside the mindor 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 andinfinite construction in Latin, thought that the question of what sort of thingsthese 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 thesentence its dictum (plural dicta). To the modern philosophical ear, Abelard’s dicta might sound likepropositions, abstract entities that are the timeless bearers of truth and falsity. But Abelard will havenothing to do with any such entities. He declares repeatedly and emphatically that despite beingmore than and different from the sentences that express them, dicta have no ontological standingwhatsoever. In the short space of a single paragraph he says that they are “no real things at all” andtwice calls them “absolutely nothing.” They underwrite sentences, but they aren’t real things. Foralthough a sentence says something, there is not some thing that it says. The semantic job ofsentences is to say something, which is not to be confused with naming or denoting some thing. It isinstead 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 anyof 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 theview 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 Englishtheologian Alexander of Hales (Hales 1951–1957: 1 d.39 n.1). Alexander heldthat the ontological type to which a statement belongs depends on whetherthe statement expresses an essential or an accidental predication. In theformer case the statement is nothing other than its subject, and thusbelongs 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 thecase of accidental predications, the statement can be reduced to theAristotelian categories in one of two ways: either it reduces to the categoryin which its accidental predicate is located, or partly to that category andpartly to the category of the subject. Thus ‘Fido is white’ turns out eitherto be a quality (and then it is the quality of whiteness) or partly a qualityand 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, state-

ables and the other objects of logic can be assigned to the Aristoteliancategories in their own right without having to be reduced to the categoriesto which their subjects and predicates belong. His view involves a complexreduction 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 fromGod by himself. Human things, in his parlance, do not include what issuesfrom humans solely in virtue of their existence as natural beings, but onlywhat comes about through human activity in the form of industry or skill.He classes the objects of logic, not among divine things, but amonghuman things in this narrow sense (Kilwardby 1986: 1 q.90: 102).

Among such things he distinguishes those that are internal to a humanand 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 besaid that the sayable [enuntiabile], like the predicable, is neither substance nor accident nor any kindof other category. For it has its own mode of existence [suum enim habet modum per se existendi]. Andit is said to be extracategorial, not, of course, in that it is not of any category, but in that it is not ofany of the ten categories identified by Aristotle. Such is the case with this category, which can becalled the category of the sayable [praedicamentum enuntiabile].’

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combinations, divisions, reasonings etc. The human things that are exter-nal include utterances, the making of works and the works made (e.g. themaking of a house, and the house that is made). This distinction betweenwhat is internal to the human and what is external appears to rest on adistinction between doing and making. While making can be consideredas a kind of doing, it can also be distinguished from other kinds of doinginsofar as it involves the production of something, or at least a processaimed at the production of something. Thus when we mentally combineor separate concepts, or when we reason ‘in our heads’, we do not therebyproduce anything external to ourselves: we have done something but wehaven’t made anything. But when we utter something, or build a house,we do produce something external, we make something. If this is whatKilwardby means, then the acts which he distinguishes from actions, andwhich he also considers to be internal, cannot be products of those actions.Being purely internal, they have no product. It is clear that the relationbetween acts and actions should be similar to the relation between worksand the making of works. But works stand to the making of works in morethan 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 noproducts. There is, however, another relation connecting works to theirmaking: the relation of completion. All actions, in principle, have comple-tions; and it is these completions, I believe, that Kilwardby refers to as actsor things-done. Thus the human things that are the objects of logic includecompleted acts of stating and reasoning, as well as the actions that havethose acts as their completions. According to Kilwardby, all of these arethings of reason. They are secondarily in a category, because they arefounded on things of nature in one of two ways. In the case of makingsand actions of reason, they are founded on things of nature in the sensethat the latter constitute their subject matter. In the case of things-done ormade by reason and art, they are founded on things of nature in the sensethat they are certain relations or accidental conditions of things of nature.Kilwardby takes both of these senses to indicate that the things of reasonand art have things of nature as their subjects; and he means here themetaphysical 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 thecategories, the things of reason and art can be assigned to the categories ina secondary sense, via the things of nature that are their metaphysicalsubjects (Kilwardby 1986: 1 q.90: 111).

Stateables and arguments, whether completed or incomplete, may existin 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 rulesof some art (he has in mind the arts of grammar and logic), and which hasthe purpose of communicating knowledge of something through visualperception. The characters are, let us say, written in ink; and theirmetaphysical subject is then the primary substances which these blobs ofink constitute. The written argument or the stateable is not these blobs ofink; it is constituted by certain relations and accidental features of the ink.Entities satisfying this complex description can be considered under morethan one aspect; and accordingly they will belong to different categories,depending on the aspect under which they are considered. Considered assigns they belong to the category of relatives. Considered as an ordering ofcharacters they could be assigned to the category of location or the category‘Where’. Considered as exhibiting a certain syntactic form they can beassigned to the fourth species of quality. In all these ways the relations orproperties which constitute the rational entity in question are accidentalfeatures of the underlying subject: it is not essential to the ink that it be asign, nor is it essential to the characters that they be so ordered as to makepropositions.Similar treatments can be given of spoken and mental statements and

arguments. Whether spoken or merely thought, these are signs and thusbelong to the category of relatives. As spoken they are qualities. As thoughtthey 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 relativesinsofar as they are signs of external things (Kilwardby 1986: 1 q.90: 163).

In sum, Kilwardby holds that stateables and the other objects of logichave the following features:

1. They are human things.2. Some of them are spoken, some written, some mental.3. They are things of reason.4. They are grounded in things of nature.5. Considered as signs, they fit into the category of relatives.

Let us return to the meaning of ‘stateable’ in the light of this overallpicture. Whatever Kilwardby means by this word, it is evident that state-ables must satisfy the above five conditions. They must also satisfy theterms in which question 90 was framed: they have to be ‘in the humanreason and are brought about by reason’. Given these things, it is

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impossible to suppose that Kilwardby meant stateables to be propositionalcontents, in the modern sense of eternal abstract objects. If he meantstateables to be the contents of statements, he would have to have meant itin a sense that complies with the above constraints. Now, it might beproposed that a suitable notion of content can be devised, according towhich contents exist only when the things of which they are the contentsalso exist. Such a notion, it might be argued, complies with the aboveconstraints. Alternatively, using the Aristotelian notion of potentiality, wecould say that a stateable is just a potential statement. The second of theseapproaches, unlike the first, allows for the possibility that some stateablesare not (yet) actually stated.

How well does Kilwardby’s account of logic and its objects fit the sketchgiven by Aristotle? Aristotle envisaged two possible answers to the questionwhether statements are immutable. His first suggestion (that they are entirelyimmutable) does not figure in Kilwardby’s account. The objects of logic, onhis account, are human things and thus subject to change. And if stateables arepotential statements then they change when their potentiality is actualised.

However, Aristotle’s alternative suggestion, that statements might besuch that any change in them is really a change in other things, isconsistent with Kilwardby’s account. Mental compositions, considered assigns, are relative to that of which they are signs. Moreover, theirs is aspecial kind of relativity – a kind that gives rise to Cambridge change. I canchange from being on your left to being on your right simply because youwalk around to my other side while I remain stationary; and similarly thestateable that Socrates is sitting can change from being false to being truesimply because Socrates sits down.

The mentalistic semantics sketched in the De interpretatione is alsoconsistent with Kilwardby’s account of the objects of logic, as is hisaccount of composition and separation as located in the mind.

But only the second of the five points listed above is found explicitly inAristotle. Kilwardby’s specific conception of logic as an art – an art thatdeals with human things which are grounded in things of nature – is not tobe found in Aristotle. It is Kilwardby’s way of turning Aristotle’s sketchyaccount into a theory of the objects of logic.

Notwithstanding its departures from Aristotle’s own remarks on thenature of the objects of logic, Kilwardby’s account is wholly Aristotelian inits motivation. But the Aristotelian ideas on which he draws do not belongin logic itself; they belong in natural philosophy and metaphysics. Hisaccount is thus, to use the terminology of Sandra Lapointe (this volume),an external one.

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3. Later thinkers

Jakob Schmutz argues that scholastic ideas were transmitted to the earlymodern period along two paths. The first of these paths, which he calls ‘theidealistic main road’, took the subject-matter of logic to be the activity ofthe mind. The second path, ‘the realistic by-pass’, took logic to deal withindependent objects and structures (Schmutz 2012: 249). We have seen aversion of the first path in the writings of Kilwardby. Kilwardby was amoderate realist. But other versions of this path can be found in nominal-ists like William Ockham, for whom the objects of logic are individualwritten, spoken, or mental tokens.7 Walter Burley, who opposed Ock-ham’s views in most matters, appears to be working within the secondpath: for him, propositions are either complexes depending on mental actsof composition and separation, or intentional complexes existing in themind, or complexes existing outside the mind, which are signified by thosemental complexes. These extra-mental propositions [propositiones in re] arethe causes of truth of mental propositions (Cesalli 2007: 234).

The second path is taken up in the nineteenth century by BernardBolzano then by Frege. Bolzano believed in propositions in themselves(Sätze an sich), and held that it is the job of logicians to describe theseentities and their properties (Lapointe, this volume). He outlines hisnotion of a proposition as follows:

One will gather what I mean by proposition as soon as I remark that I donot call a proposition in itself or an objective proposition that which thegrammarians call a proposition, namely, the linguistic expression, but rathersimply the meaning of this expression, which must be exactly one of thetwo, true or false; and that accordingly I attribute existence to the graspingof a proposition, to thought propositions as well as to the judgments madein the mind of a thinking being (existence, namely, in the mind of the onewho thinks this proposition and who makes the judgment); but the mereproposition in itself (or the objective proposition) I count among the kindsof things that do not have any existence whatsoever, and never can attainexistence. (Bolzano 2004: 40)

The objects of logic, on Bolzano’s view, are not human things and are notgrounded in the things of nature. As Rusnock and George say, ‘It shouldbe possible, [Bolzano] thought, to characterize propositions, ideas, infer-ences, and the axiomatic organization of sciences without reference to athinking subject’ (Rusnock and George 2004: 177).

7 See (Panaccio 2004: 55).

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For Bolzano, propositions are not human things, they do not exist in themind 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 toa bare minimum so that propositions are understood simply as that whichis true or false, and arguments are understood as configurations ofpropositions.

4. Concluding remarks

In his essay in the present volume Graham Priest asks whether logic can berevised, whether this can be done rationally, and if so how. And hedistinguishes logic as something that is taught, logic as something that isused, 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 proposingto revise what hitherto had been accepted as the correct norms ofreasoning. Some of the great logicians – Abelard and Ockham along withthe well-known greats of the nineteenth century – saw themselves as notjust revising but reforming logic. Sometimes these reforms are motivatedby a sense that accepted logics are erroneous or in other ways inadequate toaccepted ideals of what logic should be. And sometimes what motivates areforming logician is a new vision of what logic should be. I think that themajor reformers of the nineteenth century had this sort of motivation.Looking at the ‘traditional’ logic of their day, which was a watered-downversion of medieval logic, usually along the lines of Schmutz’s ‘idealistroad’, they worked with a vision of logic as an objective science. We benefittoday from the fruits of that vision. But it can be salutary occasionally atleast to look back to the different aims of the ‘idealist’ logicians of the highMiddle Ages.

The reason why Kilwardby and other ‘idealist’ medieval logicians con-ceived of the objects of logic as human things is to be found in the aimswhich they thought logic should have. In treating logic as an art, they werecommitted to thinking that it should teach us how to construct gooddefinitions, divisions and arguments. So the objects of logic had to includehuman activities of defining, dividing and arguing.

Everyone agrees that an argument is faulty if it allows the conclusion tobe false while the premises are true; and accordingly any good logicaltheory has to include among its norms that one should not argue fromtruths to a falsehood. Faults and norms go together.

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But in the opinion of most medieval logicians arguments were subject toa variety of faults besides invalidity. Kilwardby is expressing a commonlyheld view when he says that Aristotle identifies two of these faults in hisdefinition of the syllogism: first when he specifies that the conclusion mustbe other than the premises, second when he requires that the conclusionfollow from the premises that are explicitly stated. According to Kilwardby,the first of these specifications rules out begging the question, and thesecond excludes the fallacy of stating as a reason what is not a reason [noncausa ut causa] (Kilwardby 1516: 4vb).8

In order to see why begging the question, and failing to state explicitlywhat premises on which the conclusion relies, are faults in reasoning, onehas to look at what the point of reasoning is. Many of the medievalsbelieved that it is the function [opus] of the activity of reasoning to makesomething known by proving it:

the function of the syllogism is to prove and make known. (Kilwardby1516, 12rb)9

If a particular argument is not suitable for making its conclusion known byproving it, then it is faulty in the way that a functional object is faultywhen it is incapable of performing its function. And if a form of reasoningis not suitable for making anything known, then it is faulty. Kilwardby’sidea here is that forms of reasoning which are intrinsically question-begging, or which include redundant material, cannot perform the func-tion that belongs to reasoning:

And it is to be said that there isn’t always a demonstration when theconclusion follows of necessity, but there has to be proof of the conclusionand it has to be made known, and further it is required that the premises areapt to prove the conclusion and to make it known. But this is lacking whenthe question is begged. (Kilwardby 1516: 72vb)10

In turning an art of human reasoning into an objective science, modernlogic has made enormous gains in comprehension and rigour. But it haslost its connection with a conception of reasoning as an activity whosepoint in human affairs makes it subject to other faults than invalidity.

8 Robert Kilwardby, Notule libri Priorum Lectio 4 dubium 1.9 Kilwardby, Notule libri Priorum Lectio 11 dubium 3.10 Kilwardby, Notile libri Priorum Lectio 67 dubium 3.

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cha p t e r 9

The problem of universals andthe subject matter of logic

Gyula Klima

1. Introduction: the subject matter of logicand the problem of universals

It might seem that the problem of universals should have little to do withthe issue of the subject matter of logic. After all, in (formal) logic we dealwith the deductive validity of arguments based on their formal structure,whereas the problem of universals, at least in one of its possible formula-tions, is the question of what corresponds to the universal terms of ourlanguage, which constitute precisely the “material” part of arguments, thepart we disregard or abstract from in formal logic. However, upon a closerlook, there is a certain connection. On the semantic conception of validity(which is also the intuitive motivation for syntactic rules of inference indeductive 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 spelledout in terms of the idea of compositionality, namely, the idea that thesemantic values of complex expressions are a function of the semanticvalues of their components. Given this idea of compositionality and therange of all possible evaluations of the components of the propositionsconstituting an argument, the semantic notion of validity can be spelledout by saying that an argument is valid just in case there is no possibleevaluation of the primitive components of its propositions that would,based on the composition of these components, render the premises trueand the conclusion false. Obviously, this notion of validity presupposesthat we have a pretty clear idea of what the range of all possible semanticvalues of the primitive components in question are and how those deter-mine the truth and falsity of propositions based on their compositionalstructure. But then, when we deal with predicate logic, some of thosepossible 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 inpredicate logic, being dependent on what the correlates of our universalterms are, demand at least a certain semantic clarification of the issue ofuniversals. Contemporary conventional wisdom that we can glean fromordinary 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 theyare possibly completely arbitrary collections of just any sorts of things, yetsomehow they are “abstract entities”. Clearly, ordinary logic text books canjust stop there. After all, they are not supposed to go into the metaphysicalproblems of “abstract entities”: qua logic texts, they are just supposed toprovide some validity-checking machinery, and need not worry about thepossible ontological qualms of metaphysicians these machineries involve,just like elementary math texts, as such, need not worry about theontological status of “mathematical entities” when they concern them-selves 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 testified byPorphyry’s famously raising the fundamental questions concerning univer-sals just in order to set them aside as pertaining to “deeper enquiries”, butnot to logic. (Spade 1994: 1) And of course it is one of the famous ironies ofthe history of ideas that it was precisely on account of these questions thatmedieval logicians got so much involved in these “deeper enquiries” thatJohn of Salisbury in his Metalogicon (John of Salisbury 2009: 111–116) hadto complain about how his contemporaries’ endless debates over theseissues 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’s questions determine to alarge extent the construction of logical semantics in general, and thus theunderstanding of the relationship between the subject matters of logic andmetaphysics in particular.

2. Realism, nominalism, conceptualism

Apparently, the primary issue concerning universals is ontological: arethere universal entities? After all, nobody in their right mind would doubtwhether we have universal words, i.e., words that on account of theirmeaning apply to a multitude, indeed, to a potential infinity 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 difference betweenuniversal and singular terms hinges on the ontological difference betweenthe kinds of entities these terms primarily name, rests on a relativelysimplistic understanding of the semantic relations of these terms: i.e. thenotion that their meaning consists in naming these different kinds ofentities in the same way. In fact, generalizing on this idea we might saythat on a realist conception semantic differences are accounted for in termsof the ontological differences of the semantic values of syntactical items ofdifferent categories, and not in terms of the differences in the semanticfunctions of these items themselves: on this approach, in realism we canhave semantic uniformity at the expense of ontological diversity.

By contrast, those medieval thinkers who were convinced by Aristotle’sand Boethius’s arguments against platonic universals (by John of Salis-bury’s time practically everybody (Klima 2013a: n. 27)) would account forthe semantic diversity of singular and common terms not on the basis ofthe ontological differences of the kinds of entities these terms denote, butrather in terms of how they denote the same kind of entities, namely,individuals, the only kind of real entities there are. Thus, on this under-standing of the Aristotelian view, we can have ontological uniformity on thebasis of semantic diversity. As we shall see, the two formulae just italicizedcan be regarded as the two extremes of a whole range of possible positionsconcerning the relationship between semantics and metaphysics, rangingfrom extreme realism to thoroughgoing nominalism. Indeed, let me callthe theoretical extreme of extreme realism the position that holds that allsemantic differences are ontological differences: different items in seman-tically different syntactical categories differ in what kinds of entities theirsemantic values are and not in what kinds of semantic functions relatethem to their semantic values. By contrast, on the other theoreticalextreme we have the position of extreme nominalism, which would holdthat all different items in semantically different syntactical categories differonly in the kinds of semantic functions that relate them to their semanticvalues, but all those semantic values are ontologically of the same kind, thesame, single kind of entities (or just the one single entity) there is. But inorder to see how actual historical positions can be arranged on thistheoretical scale, we should get into some further details concerning eachextreme.

On the platonic view, as we could see, the semantic relation betweencommon and singular terms and their semantic values would be of thesame kind: namely, denoting a single entity. What would make thedifference would be just the further ontological relation of the entity

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denoted by the common term, the universal, to its singulars as theirexemplar. It is only on account of this ontological relation that we canuse these terms to denote secondarily the singulars imitating or participat-ing in their exemplar, but what the terms truly and primarily denote is theexemplar itself. So, on this platonic understanding, the semantic functionof universal terms would be the same as that of singular terms, namely,denoting a single entity, just like the representative function of a portrait isto represent a single individual. However, just as the portrait of a monarchcan stand for a whole nation and thus can identify someone as a memberof that nation (say, in a passport), so the name of the universal can standfor a whole kind and thus identify any individual participating in it as amember of that kind.On the Aristotelian view, on the other hand, universal terms are

universal precisely because they apply to a multitude of singular entities,the same singular entities we can denote by their proper names, butdifferently, namely, in a universal fashion, in abstraction from their individ-ual differences. So, on this conception, what accounts for universality isabstraction, a mental activity, the activity of the Aristotelian agent intellect(nous poietikos, intellectus agens), which by this activity produces the firstuniversal representations, the so-called intelligible species out of the singularrepresentations of sensible singulars stored in sensory memory, the so-called phantasms. The intelligible species, however, although they areuniversally representing mental acts, generally were not regarded as theuniversals Porphyry meant to consider in his work. An intelligible specieson this conception is rather an acquired disposition enabling the receptiveintellect (nous pathetikos, intellectus possibilis) to form a universal concept inactual use. For example, once I acquire the intelligible species of circles,that enables me to form actual thoughts about circles in general, but thatdoes not mean that I am thinking of circles all the time. Thus, inpossession of the intelligible species my mind still needs to form timeand again another mental act, the so-called formal concept, to form anactual thought, as when I actually think that all circles touch a straight linein one point. However, this mental act is still not the universal. It is auniversally representing singular act of a singular human mind; so, myuniversal concept of circles is not the same item as your universal conceptof circles, even if those concepts are exactly alike in their representationalcontent, just like my dance moves I perform with my body are not thesame items you perform with yours, even if we are making exactly the samekinds of moves, say, in a chorus line. What is the universal in the intendedsense 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 differences of the mental acts whereby we have it,just like we can be said to make the same dance moves, despite theindividual differences of our bodies whereby we make them. Therefore,this commonly intended object, the universal representational content ofboth of our individual mental acts, was rightly called by later scholasticthinkers the objective concept or intention, both because it is the universalrepresentation of the ultimately intended objects, namely, all singulars ofthe same kind from some of which the intelligible species giving rise to thisconcept was abstracted in the first place, and because it is the commonobjective content of the formal concepts of all those individual humanminds that are capable of thinking this objective concept at all.

Now, even if this notion of a universal (as the objective representationalcontent of individual mental acts representing a natural kind of singulars inan abstract fashion) may seem to be rather contrived from a contemporaryperspective, it should be clear that the conception that treats universals asobjective concepts, the universality of which is the result of the intellectualactivity of abstraction, does not allow in its “core ontology” the sort of“abstract objects” Plato entertained. On this view, the intellect can formuniversal objects of thought, but those objects of thought are not objects orthings absolutely speaking. Since they are the results of a mental activity,they are ontologically posterior to that activity. (Although Scotus and hisfollowers would insist that among individuals of a certain kind there is acertain less-than-numerical unity that is ontologically prior even to thisactivity, and even Aquinas would admit a certain formal unity amongindividuals of the same kind prior to any activity of the intellect (Klima2013a: n. 39)). As Averroes was often quoted by medieval authors: intellec-tus facit universalitatem in rebus – it is the understanding that generatesuniversality among things.

3. Scholastic “conceptualisms”

To see this issue in a little more detail, we should see exactly how the piecesof the theory presented so far fit together in this tradition of medieval logic,which I like to call “via antiqua semantics”, in contrast to a radicallydifferent medieval logical tradition that emerged from the works of Wil-liam Ockham, John Buridan, and their fellow nominalists, which I refer toas “via moderna logic” (Klima 2011a, 2013a). As we shall see, both of theseapproaches to logical semantics are basically variations on what may still becalled conceptualism; however, they are based on radically different

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conceptions of what concepts are and how they are related to theirobjects, and accordingly give rise to very different constructions of logicalsemantics.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 ananimal”. This is an affirmative, universal categorical proposition (in themedieval sense of ‘proposition’, meaning sentence-token), both terms ofwhich are common or universal terms, joined by a copula and determinedby a universal sign of quantity (a universal quantifier, as we would say). Onthe common via antiqua analysis, the subject and predicate terms of thisproposition, its categorematic terms, have their semantic property ofsignifying human and animal natures, respectively, on account of beingsubordinated to the respective concepts our minds abstracted from theirindividuating conditions in the humans and animals we have been exposedto. Thus, although whatever it is on account of which I am a man (i.e., ahuman being, regardless of gender) is a numerically distinct item fromwhatever it is on account of which you are a man, the concept weabstracted from humans we have been exposed to in forming our conceptof man abstracts from any individual differences (“individuating condi-tions”). This is precisely the reason why this concept will represent notonly the humans we have been exposed to, but any past, present, futureand merely possible humans, that is to say, whatever it is that did, does,will or can satisfy the condition of being human, whatever this conditionis, and whatever means we have (or don’t have) for verifying the satisfac-tion of this condition (which would be a question of epistemology and notof 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 theAnglicized form of the scholastic technical term commonly used in thesecondary literature) supposits for all human beings that presently exist.(For an overview of scholastic theories of “properties of terms”, includingsupposition, see Read 2011) The reason why this term supposits only forpresently existing humans is the present tense of the copula, which restrictsthe supposition (reference) of the term to present individuals that actuallysatisfy the condition of its signification, namely, those individuals thatactually have human nature signified in general by this term. By contrast,with different tenses or modalities, or when construed with verbs and theirderivatives that signify acts of the cognitive soul (i.e., sensitive or intellect-ive, as opposed to the purely vegetative, soul) that are capable of targetingobjects beyond the presently existing ones (such as memory, imagination,

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anticipation, abstract thought, etc.), the supposition of this term would beextended, or ampliated, to use the Anglicized form of the scholastic term,to past, future, or merely possible humans. (Klima 2001a, 2014) Sincemedieval philosophers did not equate ontological commitment with quan-tification à la Quine, they did not find any special ontological difficulty intalking about “non-existent objects”, that is, objects of our cognitivefaculties beyond the objects directly perceived in our present environment.In fact, even the ontologically most squeamish nominalists would nothesitate to quantify over mere possibilia, simply because the flexibility oftheir theory of quantification and reference, namely, the theory of suppos-ition coupled with the theory of ampliation, allowed them to contend thatthese mere objects of thought (and of other cognitive acts) are simplynothing, and so to inquire into their nature and ontology would be just awild 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 first place,and perhaps most fundamentally, the medieval “realists” (practically any-body before Ockham), even if they did not buy into Plato’s “stratifiedontology” of universals vs. singulars, and had a much more sophisticatedsemantic theory than the uniform naming relation between differentkinds of words and correspondingly different kinds of things, they didpreserve some sort of semantic uniformity at the expense of some sort ofontological diversity.

As we have seen, the signification of common terms, based on the ideaof words being subordinated to concepts to inherit their natural semanticfeatures, coupled with the Aristotelian theory of abstraction, led to apeculiar theory of predication within this framework, often referred to inthe literature as the inherence theory of predication. The theory is simpleenough: the predication ‘x is F’ is true, just in case the F-ness of x actuallyexists, or equivalently, just in case F-ness, the form or property signified bythe predicate F in the individual x actually inheres in x. The problems startwhen we consider all sorts of substitution instances of F. For then we startrealizing that, apparently, by the lights of via antiqua semantics, asOckham put it “a column is to the right by to-the-rightness, God iscreating by creation, is good by goodness, just by justice, mighty by might,an accident inheres by inherence, a subject is subjected by subjection, theapt is apt by aptitude, a chimera is nothing by nothingness, someone blindis 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) generatemetaphysical problems where there shouldn’t be any, simply on account ofa misconception of semantics, because their conception would “multiplybeings according to the multiplicity of terms” (Ockham 1974: I, 51).

To be sure, the “realists” did make a number of metaphysical distinc-tions between the types and modes of being depending on the substitutioninstances of F to avoid apparent metaphysical absurdities (such as a thingundergoing change without losing or acquiring a property, action at adistance, non-beings undergoing change, etc. – cf. Klima 1999), but forOckham and his ilk, that is precisely the problem: to maintain a certaintype of semantic uniformity, the realists introduce ontological diversitywhere there shouldn’t be any, since the difference is not in the thingssignified by our different terms, but in the different concepts signifying thesame things in different ways (Klima 2011a).To illustrate the sort of semantic uniformity and the requisite onto-

logical diversity in the via antiqua approach to semantics, let us brieflyreturn 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 sametype of significative function, namely, signifying the individualized naturesof individuals represented by their corresponding concepts in an abstract,universal fashion. The subject term, in turn, has the function of standingfor those individuals that actually have this nature at the time connoted bythe tense of the copula, whereas the predicate has the function of attribut-ing the nature it signifies 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 thenature signified by the predicate in actuality at the time connoted bythe copula, the propositional complex, variously called dictum, enuntiabile,or complexe significabile, resulting from the combination of subject andpredicate by the copula as further determined by the universal sign, will beactual just in case all individuals supposited for by the subject actually havethe nature signified by the predicate. As can be seen, the via antiquaanalysis of this single proposition apparently requires an extremely com-plex, multilayered ontology; however, the payoff in the end is the simple,uniform semantic criterion of truth originally proposed by Aristotle: aproposition is true just in case what it signifies exists. But it is instructiveto take a closer look at the ontological status of the items required by thisanalysis.In the first 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 essentiallyrational animals, their existence also requires the existence of theirrationality and animality. Furthermore, on this conception, if somethingexists in actuality, its existence also has to be in actuality, and it is thisactual existence that is supposed to be signified by the copula. But thecopula also co-signifies the union of what is signified by the predicate andby the subject, thereby indicating that the existence of the thing signifiedby the predicate is also the existence (whether substantial or accidentalexistence, but in the case of the proposition at hand, it is the substantialexistence) of the thing supposited for by the subject, which is actual at thetime connoted by the tense of the copula.

Now, these are just the real, mind-independently existing items requiredfor the truth of this proposition. However, as we could see, these items canbe picked out by the relevant syntactical items from reality only on accountof these syntactical items being subordinated to their respective conceptsthat renders them meaningful in the first place. So, on this analysis,all propositions require a further “ontological layer”, as it were, the layerof 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 theseindividual mental acts, some of which represent extra-mental individuals ina universal manner, but without representing the sorts of universal thingsimagined by Plato. Thus, these objective concepts form another onto-logical layer, the layer of beings of reason, which in the strict sense are mereobjects of thought (the representational contents of formal concepts), butwith a more or less remote “foundation in reality” (as opposed to merefigments). In the case of universal concepts, this “foundation in reality”consists in the individualized natures of the things from which theseobjective concepts derive in the process of abstraction and concept forma-tion (through the generation of intelligible species).

But the objective concepts do not occur to the mind in isolation. Theyenter into the composition of complex thoughts, which are formed bymeans of syncategorematic concepts, such as the copula, which, as wecould see, besides its syncategorematic function of joining the concepts ofsubject and predicate also has the categorematic function of signifying theexistence of what is signified by the predicate in the relevant supposita(referents) of the subject, the relevant supposita being determined by thesyncategorematic concept of the sign of quantity, in the present case the

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universal sign (quantifier). The propositional complex thus formed isanother 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 affairs (dictum, enuntiabile, etc.) that every man is ananimal actually obtains just in case there are humans and each of themactually has its animality. But then, providing the rules of composition forall types of thought, based on the syntactical structure of the propositionexpressing it, one can provide the uniform Aristotelian criterion of truthfor all types of propositions, and based on that, the uniform criterion forthe formal validity of an inference or consequence. In fact, since in thisframework truth is defined in terms of the content of propositions, astronger entailment relation is also definable, in terms of the more fine-grained notion of content-containment, as was proposed by some authorsin this tradition (Martin 2010, 2012; Read 2010; for comparison, aninteresting contemporary development of the idea of entailment basedon content- or meaning-containment can be found, for instance, in Bradyand Rush 2009). However, as we could see, this could be obtained only interms of the multi-layered ontology that provoked Ockham’s and hisfellow-nominalists’ charges.Nevertheless, we should also emphasize that Ockham’s and his follow-

ers’ charges were not entirely justified, and, accordingly, the ultimatedifference between late-medieval realists and nominalists did not lie simplyin their different ontologies or simply in their different semantics thatallowed them to handle their ontological problems in rather different ways,but rather in their different conceptions of concepts underlying even theirsemantic differences.The Ockhamist charge of multiplying entities with the multiplicity of

terms was unjustified for several reasons. In the first place, even “realists”had at their disposal at least two different kinds of strategies to reduce theontological commitment of their semantics: (1) the identification of thesemantic values of terms belonging to different categories, and (2) attrib-uting a “reduced” form of existence to some of the semantic values of someterms in some categories.The first strategy could rely already on the authority of Aristotle, who in

his Physics identified action and passion with the same motion, but severaloriginal considerations allowed scholastic thinkers to identify relationswith their foundations (i.e., with entities in the “absolute” categories ofsubstance, quantity and quality), and in general entities in the remainingsix categories with those in the first three.

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The second strategy, as we could already see, was based on the idea thatthe mind’s different ways of conceiving of mind-independent entities ofexternal reality produces certain mind-dependent, intentional objects, theobjective concepts, the information contents of our mental acts, by means ofwhich we variously conceive ultimately those mind-independent objectsthat satisfy the criteria of applicability set by these objective concepts, orintentions. This is precisely why in this tradition the subject matter of logicwas generally characterized as the study of second intentions, that is, ofconcepts of concepts (such as the concepts of subject, predicate, propos-ition, negation, or the ultimately targeted notion of valid consequence).So, the core-ontology of real mind-independent entities could in principlehave 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 different conception of concepts and their role inlogic, semantics, and epistemology. In this connection, it is informative tocompare Ockham’s earlier, fictum-theory of universals with that of the viaantiqua conception discussed so far. For the important difference betweenthe two is that even if Ockham’s ficta are ontologically on the same footingas the objective concepts of the realists (they are beings of reason), and theywould be best characterized in the same way, namely, as the objectiveinformation content of individual mental acts, they do not have the samerole in Ockham’s theory.

In fact, as prompted by the arguments of his confrere, Walter Chatton,Ockham came to realize that ficta did not play any significant role in hislogic at all, and so, grabbing his famous razor, he painlessly cut them outfrom his ontology. The reason why Ockham could do so is that for himthe universality of universal representations (whether ficta or universallyrepresenting mental acts) consists merely in their indifferent representationof a number of individuals (in the case of a natural kind, all past, present,future, and merely possible individuals of the same kind). However, thisindifferent representation is due not to some abstracted condition of havinga certain nature that individuals of a given kind satisfy, but, as a matter ofbrute fact, to the indifference of the causal impact of one individual oranother of the same natural kind on the human mind.

Accordingly, for Ockham, there is no question whether there is a realdistinction between the nature of an individual represented by a universalconcept and the individual itself (as this emerged as a metaphysicalquestion in the via antiqua), because what these concepts indifferentlyrepresent are just the individuals themselves. Therefore, for him, the

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supposita of the terms subordinated to these concepts are not the individ-uals that actually have these natures relative to the time connoted by thecopula (as was conceived in the via antiqua), but simply the individualsrepresented by the concept that are actual at that time. As a result, terms inthe predicate position do not signify inherent natures either, so Ockhamand his followers endorse the identity-theory of predication, as opposed tothe inherence-theory. According to the identity-theory, an affirmative predi-cation is true, just in case the terms of the proposition supposit for thesame thing or things. But this is obviously not a general “definitionof truth”. In order to achieve a truth-definition on this approach, oneshould provide similar “satisfaction clauses” for all logically differentproposition-types, such as negatives, universals, particulars, not to mentionthe propositional complexes, such as conjunctions, disjunctions, etc.(Klima 2009: c. 10).As can be seen, on this nominalist approach, just as terms do not figure

into the calculation of truth-values with their intensions, but their exten-sions, so too, the truth of propositions themselves is not determined interms of their intension or signification, but solely by the extensions (sets ofsupposita) of their categorematic terms. Accordingly, nominalist semanticsas such has no use for enuntiabilia or complexe significabilia, as is brilliantlyillustrated by the logic of John Buridan.On Buridan’s theory, propositional signification is simply the set of all

significata (and connotata) of a proposition’s categorematic terms, which ofcourse yields a very coarse-grained conception of propositional significa-tion. In fact, on this conception, contradictory propositions must signifythe same, although differently, on account of the concept of negationincluded in the one, but not in the other of the contradictory pair ofpropositions (Klima 2009: c. 9). However, Buridan does not have to caremuch. On his account, truth is not a function of signification, so, twopropositions of the same signification can have opposite truth-values.Thus, when he needs a more fine-grained semantics of propositionalsignification (as in intentional contexts) he can always refer to the diversityof the corresponding propositions on the mental level, where, of course, inline with his nominalist ontology, the mental propositions in question arejust inherent qualities, individual acts of individual human minds, just asare the concepts entering into their semantic make-up (Klima 2009: c. 8).So, nominalist semantics can afford to be based on an entirely homoge-

neous, parsimonious ontology (containing only two or three distinctcategories of entities, namely, substances, quantities – sometimes identifiedwith substances or qualities, as by Ockham – and qualities). However, this

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parsimonious homogeneity is achieved at the “expense” of massive seman-tic diversity, assigning some of these entities various, distinctive semanticfunctions, especially on the mental level. However these semantic func-tions are always defined in terms of the extensions of these mental items,the formal concepts inherent in individual human minds, to the exclusionof items in the “ontological limbo” of the objective concepts of theolder model.

Still, even the nominalist version of scholastic conceptualism couldmaintain that logic is the study of second intentions without lapsing intosubjectivism, conventionalism, or skepticism, let alone psychologism – featuresthat in a modern context are so often associated with conceptualism. Well,how come? Actually, answering this question will allow us to draw somegeneral conclusions concerning both major versions of scholastic concep-tualism sketched out here, and some general lessons we can learn fromthese scholastic theories concerning the subject matter of logic andmetaphysics.

4. Conclusion: the lessons we can learn from the scholastics

In the first place, it should be quite obvious that the objective concepts ofthe 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 inindividual human minds as their individualized forms (the formal con-cepts), but also in the modern sense of being intersubjectively accessibleand the same for all. For an objective concept is the common, abstractinformation content of any formal concept that carries this information,and any formal concept that does not carry the same information is justnot a formal concept of the same objective concept. Thus, in this frame-work it simply cannot happen that you and I have different concepts of thesame kind of entities as such, or the same concept of entities of differentkinds. If I have the concept of H2O and you have the concept of XYZ,then we are just not talking about the same thing, no matter that we usethe same word in our miraculously matching English idioms of Putnam’sTwin Earth scenario. (Putnam 2000: 422) Since I use the term ‘water’ assubordinated to my concept and you use it as subordinated to yours, weuse our words equivocally, no matter how phenomenally similar the twokinds of things are, and how similarly we would describe their phenomenalproperties. Therefore, if I say ‘This is water’ pointing to a glass of H2O andyou say, pointing to the same, ‘No, that is not water’, we actually do notcontradict each other, although, of course, it can take a while until we

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figure out just why. But that is an issue of epistemology, not semantics. Asfar as meaning is concerned, on this conception you can have the sameconcept as I do only if our individual mental acts latch onto the same kindof objects in the same way, carrying exactly the same information. To besure, one of us may have a better understanding of the nature of the thingor things thus conceived, on account of being more aware of the relation-ships among this concept and others, picking out the same nature differ-ently, on account of other, more specific or more generic information, aswhen one of us knows the genuine quidditative definition of the kind ofthing in question. But regardless of whether either of us has this definitionin mind or knows what it would be, we can be said to have a concept ofthis kind of thing as such, only if we managed to form the objectiveconcept of its essence, which must be the same for both of us, or we justdo 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 non-

conventionally objective. For what determines the information content ofour abstracted, simple concepts is what kinds of things they are abstractedfrom, that is to say, the nature or essence of those things themselves. To besure, we can construct complex concepts out of these simple ones as wewish and agree to express them by words we wish (ad placitum, as thescholastics said), but whether the concept we both abstracted from samplesof H2O will apply to all and only samples of that kind of thing (even if wecannot infallibly identify all such samples in all possible scenarios) is clearlynot 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 withtheir own individual, subjective mental acts, their formal concepts; thelogical relations among these mental acts are not a matter of the causal orother psychological relations among them, but a matter of the relations oftheir objective semantic contents, the relations among their objectiveconcepts. So, no wonder scholastic thinkers working in this traditionwould identify the subject matter of logic as those second intentions orobjective concepts of our objective concepts that express precisely theseobjective 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 different from the laws of psychology.For while the former are the laws of the logical relations among objectiveconcepts, the latter are the laws of the causal relations among formalconcepts. Thus, whereas logic can be normative, prescribing the laws of

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valid inference, cognitive psychology can only be descriptive, describingand perhaps explaining those psychological mechanisms that can, forinstance, 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, especiallyinstructive modifications. As for the issue of objectivity, the nominalistauthors, after Ockham had dropped his ficta as being ontologically both-ersome and theoretically unnecessary, still insisted on our simple mentalconcepts being externally determined and not just subjectively made up byus, grounding this belief in the generally reliable natural mechanisms ofsense perception and universal concept formation (Panaccio 2004; Klima2009: c. 4).

So, for the nominalists there are no longer quasi-entities in mereobjective being: our concepts are anchored in extramental reality throughthe objective (in the modern sense, meaning mind-independent) laws ofnatural causality. However, there is a slight, but very significant shift in theway nominalists conceived of this “anchoring”, as opposed to their realistcounterparts. For realists, what did the “anchoring” was a certain “formalunity”, the sameness of the information content that was encoded in themental acts carrying this information and that was realized in the verynature of the things these concepts represented to the subjects havingthem. This conception of formal unity provided a much stronger“anchoring” for the via antiqua conception than what is available in thenominalist via moderna.

The via antiqua conception, as we could see, builds the identity of thenature of the represented objects into the identity-conditions of a conceptitself, hence tying the identity of its objects by logical necessity to theidentity of the concept in question. By contrast, the via moderna concep-tion ties the identity of the concept by mere natural, causal necessity to theidentity of its objects. In a medieval theological context, however, thisdifference amounts to the difference of what could and could not be doneby divine absolute power. Therefore, it should come as no surprise thatanticipations of Descartes’ famous “Demon argument” crop up precisely inthis context, once the nominalist conception opened up at least the logicalpossibility of a cognitive subjects’ having exactly the same concepts plantedin his mind by a deceptive God without the mediation of these concepts’adequate objects; i.e., the subject having exactly the same phenomenalconsciousness, regardless of whether any items of it are veridical, faithfulrepresentations of reality or not (Klima and Hall 2011; Kargerforthcoming).

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As for conventionalism, the nominalists usually laid even more emphasison the conventionality of spoken and written languages than their realistcounterparts; however, they equally strongly emphasized the non-conventional, natural character of the language of thought or mentallanguage, based on the fixed laws of nature. So, even for the nominalists,our simple concepts, anchored in natural kinds by causality, are not madeup 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 ofdifferent reactions to this possibility, but perhaps most typically offering a“reliabilist” solution, dismissing the overblown certainty-criteria of theskeptic in terms of different sorts of reliability-criteria for our variouscognitive powers and mechanisms utilized differently in different cognitivescenarios (Aristotle providing again a good authority by his remark thatone should not expect mathematical certainty in all fields of inquiry)(Klima 2009: c. 12). In any case, in deductive logic, nominalists stillrequired 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 nominalistsdid not take logic to collapse into psychology. Perhaps, the best illustrationof this fact comes from Adam Wodeham’s “thought experiment” concern-ing the presumed “perfect telepathy” of human minds uncorrupted byoriginal sin (of which now we have no actual example) and of good angels(the ones that did not fall with Lucifer). These minds, according toWodeham, are perfectly capable of intuiting each other’s thoughts, how-ever, this would still not amount to communication, because they wouldnot be able, simply on account of this intuition, to decode the contents ofthose thoughts (pretty much like brain scans can give us some informationabout some sort of brain activity, but not about what that activity isabout). Accordingly, based on these observations, these minds could comeup with a natural science describing the regularities of these mentalactivities, 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 different science, namely, logic. (Karger2001: 295–6)So, what conclusions can we draw for ourselves from this however

sketchy, general comparison of the two main scholastic approaches tothe 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 differentiated not bytheir different ontologies, but rather by their different conceptions ofconcepts, determining different kinds of constructions of logical semantictheories.

These different theories can be arranged on a “theoretical scale”, rangingfrom extreme realism to extreme nominalism, meaning maximal semanticuniformity along with maximal ontological diversity on the realist end(every linguistic item has the same type of semantic function, say, namingsome entity, while these items differ semantically on the basis of what typeof entity they name), and maximal ontological uniformity with maximalsemantic diversity on the nominalist end (having just one ontological typeof entities, while all semantic differences consist in the different semanticfunctions of some of these entities of the same ontological type). Even if,perhaps, no actual historical theory can be placed on either extreme(although, if Parmenides had had one, it would have probably been closeto the extreme nominalist end, whereas if Plato had had an articulatedsemantic 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 variouslyremoved from either extreme on account of various elements of varietyintroduced either on the side of ontology, by multiplying the semanticallyrelevant distinct categories of entities, or on the side of semantics, bymultiplying the different types of semantic relations that map the syntac-tical categories of language onto the ontological categories distinguished bythe theory.

In the scholastic theories discussed here, these different types ofsemantic relations were understood in terms of how our different kindsof concepts relate our words to things in our ontology. Here, in the viaantiqua framework “things” can be understood rather loosely for anyobject or quasi-object of our thought, whereas in the via modernaframework, they would be restricted to really existing entities in thecategory of substance and quality (or for Buridan and his followers alsoin the category of quantity). It is very telling, however, that the “coreontology” (i.e., the categories of real entities to the exclusion of beings ofreason) of the via antiqua framework could be just as parsimonious asthe nominalist “core ontology” was (as is illustrated, for instance, by theontology of the late-scholastic Domingo Soto). Furthermore, even the viamoderna framework could have in principle reduced its ontology to onehomogeneous 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 would avoid, in their own ways, contemporary worries aboutconceptualism in general, we can conclude that such comparisons can beespecially useful for refining our understanding of the implications of thevarious theories that can be arranged on the theoretical scale set up in thischapter. Such refinements in the end will allow us to overcome certainmodern theoretical reflexes (nominalism entails skepticism, conceptualismleads to psychologism, etc.) by shining a new light on the historical originsof these reflexes themselves.

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cha p t e r 1 0

Logics and worldsErmanno Bencivenga

There are no specifically logical objects. A logic is a theory of the logos, ofmeaningful discourse: a theory of how discourse acquires the meaning itdoes. Traditionally, logics have investigated the behavior of syncategore-matic words like “and,” which contribute to the meanings of their contextswhile having no meaning in isolation, and hence have studied the contrast,say, between “and” as it occurs in sentences like

(1) They are married and have a child(2) They got married and had a child.

But there is no reason why they should not also study the relation, say,between (1) and

(3) They are married and have a pet;

in particular, why they should not inquire into whether (1) and (3) areinterchangeable. A logic that studied the contrast between (1) and (2) wouldbe (among other things) a logic of “and” (a theory of the 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 thangreater generality, and attendant lesser detail, is gained by concentrating onlogics of conjunctions, adverbs, and pronouns; and at no level of generalitydoes it make any sense to grace a word (or maybe a diacritical sign, like aparenthesis) with the label “logical constant.” If you ever get sidetrackedinto a Quinean fatuous search for the nonexistent Eldorado of “purelogic,” I recommend a refreshing immersion into Buridan’s subtle, percep-tive study of the infinite nuances of signification and supposition.1

But, if there is no specific ontological realm for logic, there definitely areontological questions pertaining to it. Two questions, primarily: Do logicallaws (the laws bringing out the meaningful behavior of various words) have

1 See Klima 2001b.

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an objective status? And, if so, how do they acquire it? No answers to suchquestions 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 beforemoving on.2 (Though I must note that, here, the matter dealt with is notinnocent: an ontology is a logic of being, hence what ontological status alogic 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 thereis a God or humans are free, but to address semantical issues like what themeaning of “God” or “freedom” is. The reason why the formidable epithet“transcendental” is attached to it is precisely the misunderstandingI alluded to above: if you think that logic only deals with (some) conjunc-tions, adverbs, and pronouns, then you are forced to qualify this narrowconcern as “general logic” and to conjure up some other name for the fullline of business.Within the semantical space where the (transcendental) logical enter-

prise is located, one can take different words as primitives and establish anetwork of semantical relations and dependencies based on those primi-tives and involving other words, each time resulting in (the beginning of ) adifferent transcendental philosophy/logic; as more such structure isexposed, the meanings of the words involved will become correspondinglybetter established and clearer. If we want, we can even talk about “con-cepts”: clusters of largely interchangeable words resonating with a commontheme, not necessarily spoken but suggestively intimated by the resonance.A transcendental realism (TR) is a transcendental philosophy/logic

that takes a cluster of largely interchangeable words including “object,”“substance,” “thing,” and “existence” as primitives, and then turns tothe (hopeless) task of defining words like “experience” or “knowledge”on that basis. A transcendental idealism (TI) – my chosen course – is atranscendental philosophy/logic that takes its cue from a differentcluster including “experience,” “representation,” and “consciousness,”and then moves to defining “object” and “existence.” Not surprisingly,a TI has a lot more to say about objectivity – what makes an object anobject – than a TR: of primitives we will forever be dumb and, thoughoccasionally that incapacity is depicted as mystical depth, the bottomline is that no interesting account of what primitives mean is forthcom-ing. In a TI, however, objectivity belongs to a derived cluster; hence its

2 For additional details, see my 1987 and 2007.

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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 some-thing: their so-called intentional objects, which despite their name are notobjects yet, indeed never will be. And neither representations nor theirintentional objects can be objective in isolation: they can only be object-ive to the extent that they are categorially connected – that they aremutually consistent; that there are relations of mutual determinationamong them; that there is a definite fact of the matter of how many ofthem there are, hence how they are identical with, or distinct from, oneanother. The graduality signaled by the locution “to the extent that”would only be redeemed at the limit: by a system of representations towhich nothing further could be added and where each member were fullydetermined to be what it is by its relations with all others. Within thatsystem (suddenly, as soon as completeness were reached), all representa-tions would be objective and all their intentional objects would beobjects, period: existent objects. The limit cannot be experienced, inthe strongest sense of “cannot”: it would be contradictory (antinomical)to suppose otherwise, hence all intentional objects will forever stay thatway – remain appearances. But this conclusion is only going to worrythose who reduce a TI to a series of empirical claims about what takesplace (or can take place) in a mind. As none of that is implied here, forwhat is in question is rather the semantics of “objectivity,” we have all wecould ask for: a regulative idea that orients our everyday, always falliblevicissitudes, signaling the direction in which we are likely to find moreobjectivity and the standards we must enforce to maximize it (coherence,agreement, inclusiveness, mathematical structure), inevitably staying clearof such a complete realization of the idea as would make ( per impossibile)the whole project fall apart.

What the system of representations envisioned at the limit wouldrepresent is (as one might expect) a system of objects to which nothingfurther could be added and where each member were fully determined tobe what it is by its relations with all others. I call (and Kant calls) thissystem a world. No one ever experiences a world, though most everyone(everyone but severely disturbed people) ordinarily presumes herself to beexperiencing part of one, and sometimes goes through the catastrophe ofseeing what she took to be part of a world explode into incoherence anddisconnectedness. When such unfortunate events take place, we try hardto blame them on contingent occurrences (on misreadings of data) whilekeeping 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 isnot always that easy; for, what about the semantical necessity, until circa1905, that a wave is not a particle? And about the logical clash that ensuedwhen we were forced to deny that necessity: a clash whose logical characterwould be missed by more parochial characterizations of logic? If, on theother hand, you want to insist on a parochial characterization and attributethat clash to the empirical realm, I urge you to consider what happened afew years earlier, when the very logic of sets blew up in people’s face. Asvery unfortunate happenings of this kind can never finally be ruled out,logics align themselves with worlds, in the following way:A logic cannot be a theory of meaningless 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 setsconsistent, connected relations among them – once again, it is an all-or-nothing affair. Only a logic associated with this kind of dictionary wouldbe objective in the sense of possibly describing a world of objects (would bea real, not an apparent, logic), independently of the data that gave empir-ical content to its entries. As no such dictionary can ever be at hand, we arenever in possession of a logic but only of something we presume to be afragment of one, and which is always at risk of dissolving into the stuffdreams are made of. A logic (like a world) is worse than a territoryconstantly under threat of being conquered by enemies: it is constantlyunder 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 acomplete system of representations or a complete world, we are to maxi-mize the consistency, connectedness, and inclusiveness of what systems ofrepresentations or of intentional objects we do have, in the absence of alogic we are to maximize our closeness to one, walking away from thedepopulated citadels of the propositional and the predicate calculi toward afiner and finer 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 graduatefrom a tenseless doctrine into a concrete practice ready to uncover seman-tical treasures under any rock, and carry semantical threads around anycorners. 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 shunnedin 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 field.

The two most obvious examples of this search for an objective logic, inour tradition, come from Aristotle and Hegel3 – or, I should say, fromHegel and from Aristotle as interpreted by Hegel. For the official storyabout Aristotle is that logic for him is an organon, a neutral tool to beprefixed to research proper, indeed to be done with exhaustively beforeembarking in any research. If that were true, Aristotle’s would no more bea logic than quantification theory or S4 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 segmentof Aristotle’s philosophy (and science) deepens and widens his logicalanalysis4 – his logic as analysis, his analytic logic. Whether he is talkingabout the challenge sea-anemones bring to the logical distinction betweenanimals and plants, or he is illuminating through a careful examination ofcourage or friendship the relation between focal and extended/analogicalmeanings, Aristotle is reshaping his dictionary (including what it is to be adictionary) 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 aTR. 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; moreprecisely, 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 presenceinside Kant) then the issue is simple: as soon as we face a contradictionbetween two representations, or their intentional objects, a distinctionmust be made – there must be at least two things. The Aristotelian worldis structured by contraries: by what cannot be true together and invokes asplitting. If waves can cause interference phenomena and particles cannot,then waves are not particles, and for a definition of light we have only twochoices: we can either have particles or waves but not both, or give up on

3 For a systematic account of the contrast between Aristotelian and Hegelian logic, see my 2000.4 See the following passages from Hegel’s 1995: “in his metaphysics, physics, psychology, etc., Aristotlehas not formed conclusions, but thought the concept in and for itself ” (p. 217; translation modified);“it must not be thought that it is in accordance with . . . syllogisms that Aristotle has thought. IfAristotle 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 hisdeterminations should be brought into a necessary systematic whole” (p. 223). While he thusacknowledged the comprehensive character of Aristotle’s logic, however, Hegel did not see it as analternative 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 objectiveidentity be spatiotemporal shows him committed to this Aristotelian route:one and the same thing cannot be at two different locations at the sametime. But, when it comes to the semantics of regulative ideas, including theones that determine the criteria of objectivity, his inclination seems to beproto-Hegelian, witness his “derivation” of positive from negative free-dom, or of reciprocal action from simultaneity.5

With Hegel, on the other hand, contradiction is not a threat: it is anopportunity. When the semantics of a word faces a bifurcation betweencontradictory options, its fate is to take both, and its job is to evolve insuch a way that both options be present in a dialectical overcoming of theircontrast. Light is both particles and waves: the two are complementarydescriptions of one and the same complex reality, indeed belong to thevery substance of that reality, which is nourished (adds to its “concrete-ness,” Hegel would say) by their antagonism. Therefore the world that noone will ever experience but of which everyone takes herself to be experi-encing a portion is a monistic one: as not even contradictions can divide,no two things are radically divided; all divisions are but chapters of onestory. And the very unfortunate events that might bring this logic to a crisiswill not be the surfacing of contradictions, as is the case with its analyticcounterpart. It will rather be the confronting of occurrences (the Holo-caust, say) that simply cannot be integrated within one and the samecomprehending, 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 tobe found where one would be most likely to look for them. As I pointedout already, individual calculi cannot be regarded as logics, unless they arepart of an ambitious program that extends over a substantial area ofexperience, indeed potentially all of it. But, whereas most of what fallsunder the academic discipline of logic does not qualify as logic for me, a lotof traditional philosophy does. Transcendental philosophy is not a newway of doing philosophy initiated by Kant: it is a new way of looking atwhat philosophy has always done, without much awareness and hencewith considerable self-deception. Of course pre-Kantian, and many post-Kantian, philosophers typically took themselves to be establishing factualclaims like the existence of God or human freedom, but the way they did

5 For the former, see Groundwork of the Metaphysics of Morals, in Kant 1996, p. 94: “The precedingdefinition of freedom is negative and therefore unfruitful for insight into its essence; but there flowsfrom it a positive concept of freedom. . .”. For the latter, see my 1987, p. 149.

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this was by launching apodeictic demonstrations, that is: by trying toprove that the existence of God or human freedom were more than facts –that they were necessities. Therefore, it was not facts about God orfreedom they were directly addressing: it was the meanings of “God” andof “freedom,” and facts only insofar as they were inescapable consequencesof 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 werePlato or Hobbes; the logic of art, if they were Plotinus or Schiller; the logicof economic exchanges, if they were Ricardo or Marx. Far from being justan organon of philosophy, logic constitutes the very body of it: all philo-sophical theses, arguments, and theories are but logical matters – pages ofan ideal dictionary by which we try to make sense of experience. And, asI showed earlier with Aristotle, most of these theses, arguments, andtheories, though often grown on TR soil, can be put to profitable use indeveloping a TI, specifically in defining objectivity for various TI philoso-phies/logics.

If each logic (within a TI framework) develops its own definition ofobjectivity, one important feature of logic as ordinarily understood turnsout to be mistaken – and one more way emerges in which traditionalphilosophers were deluded about their own work. The feature I refer to isthat logic, one often believes, allows us to conclusively refute an opponentby proving him conceptually confused, or to conclusively establish ourposition by proving it sustained by necessary argument; which is just whatphilosophers traditionally took themselves to be doing when they offeredthe apodeictic demonstrations I mentioned. Here too it might help to getto 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 currencywithin an analytic framework. Hegelian, dialectical logic has other conver-sational and confrontational modes – which explains the repeated failure ofattempts at coming up with a dialectical theory of inference. But mutatismutandis what I say could be extended to the Hegelian camp.)

According to Kant, ethics is rationality,6 and never mind at the momentthat rationality, like objectivity, cannot be definitively established. Supposewe pronounce an ethical judgment that, on as solid grounds as we canmanage, stigmatizes a certain behavior as immoral. Assuming the groundsto 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 toempirical laws, so only empirical factors like temperament, education, oremotions can have motivating force there. Reason has none. One can onlyhope (if one takes reason’s side) that those empirical factors will promotewhat reason would want to see done; that moral feeling, say, will ally itselfwith rational judgment and make the agent move the “right” way – in thiscase, take his distance from the stigmatized behavior. If the agent decidesotherwise, 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 beimpressed by any of this. In fact, he can appropriate words like “reason”and “rational” and provide them with his own semantics; and there will beno forcing him to recognize that as an error. Reason (whatever that is) isplaying its own game and, however consistent and connected the gamemight 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 rulescurrent in a particular game, I prove that, than which nothing greater canbe thought, necessarily to exist. If I am a believer, I rejoice in thus seeingmy faith confirmed and I generously broadcast my proof to all others, sothat they can see the light also. And I am puzzled when many of thoseothers, instead of coming to a harmonious, reasonable agreement with me,use their disbelief as the premise of a modus tollens and start looking forwhat is wrong with my proof. Eventually they might focus on somethingI took to be included in the semantics of “greater”: that existing, say, isgreater than not existing. And they might deny it: adopt an alternativeaccount of “greater.” What can I do then? Clearly, they are playing adifferent game; and, no matter how loud I protest, there is no convincingthem that my game is the one they should be playing. This last judgment isinternal to my game, and of course from that internal perspective it looksirrefutable. From the outside, it just looks like something else onecould say.In and by itself (more about this qualification shortly), logic has no

persuasive force. Despite the metaphors of constraint that are invariablybrought up in its wake, it can constrain no one. If anything, the practice oflogic (as opposed to the often deceptive theory of it) has a liberating effect.You felt constrained to making a certain inferential step (say, from some-thing being necessary to its being necessarily so); but, when you bringlogical acuity and attention to bear upon it, you realize that it was a matterof 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 eitherway, you have a choice in the matter that you had missed at first. Think ofthe story long told about Girolamo Saccheri:7 of how he wanted to firmup, once and for all, the necessity of Euclid’s fifth postulate (to withdrawthe 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 objectivityever going to pay off? Its value judgments, I said, are internal to the gamethe logic is playing; it is only from within that game that certain principlesappear secure, certain inferential steps apodeictical, certain objectionsuntenable. 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 thedevelopment of an organism: a realization of its own potential and afunctional interaction of all its components. Repeatedly, I will come upontheoretical options, and the game I am playing (I decided to play,I committed myself to playing) will sometimes determine my choice ofone of them, in which case I will “naturally” accept that choice, andsometimes not, in which case I will reflect on what else I want to add tothe rules of the game in order to have it cover more ground, to make itmore 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 myinitial commitments: I will regard those commitments as confirmed to theextent that I approve of my destination; I will correct them to the extentthat I find it unwelcome. There will even be surprises along the way:locations I never thought I would reach but my rules irresistibly take meto, either to be more powerfully reassured that I am on the right track, ormore 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 scaffolding of a world and, if given enough data,even a large part of its actual construction. (So, as anticipated earlier, alogic includes its own ontology.) Luigi Pirandello called it a “corrosive”,infernal “little machine”8 because from any imagined variation in theexisting circumstances it could engineer, one step after the other, themost 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 A Jesuit priest and professor of mathematics at Pavia, who published in 1733 Euclides ab omni naevovindicatus, a presumed reductio proof of Euclid’s fifth postulate from his other assumptions, longregarded as the first (unintended) development of a non-Euclidean geometry.

8 Pirandello 1990, pp. 1108–1109.

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Whatever the outcome, whether horrid or even benevolent, logic isrevealing of our powers: of the creative process by which we shape ourworld and hence of the responsibility that follows from it. Decades ofexistentialist thought (starting with Kant!) have made it clear that none ofthat 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 causerespect), pathos (an appeal to the public’s emotions), and logos (an appeal toreasoning and argument). On the face of it, this taxonomy would seem tocontradict my (and Kant’s) claim that reason has no motivational force,indeed it might look like the premise of an edifying call to exercisingpersonal and emotional restraint and letting the austere business of logostake control of public exchanges and safely guide the community toperfectly reasonable outcomes. In light of what I said so far, this attitudewould be delusional: if in fact a speaker were to convince an interlocutor toswitch to her side by the use of logical argument, it would only be bycleverly hiding the optional character of her principles and the controver-sial nature of her inferential steps, and that itself would happen, mostlikely, because of the competence, hence the authority, the interlocutorattributes to her, hence ultimately because of her implicit use of ethos, herimplicit appeal to her superior ability (and honesty) in dealing with thesematters.9 Rhetoric aficionados are fond of making some such point, and ofcollapsing logos into a fraudulent mannerism, which will succeed (when itdoes) by couching in impressive, authoritarian pseudorational garb sub-jective (and often repressive) opinions and policies. This extreme, unwar-ranted stance issues from a gut reaction to the equally extreme, and equallyunwarranted, claim that logic, in and by itself, can persuade anyone; andwe have now prepared the ground for a more plausible and balancedposture – and for finally explaining what I have meant by the qualification“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 retortsthat evolution is a fact are only signs of bad faith. But that evolution (andcreation 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 significant list of parameters:they are more detailed, more discriminating, more resourceful, moreingenious. And, when compared with creation, evolution is all that. Onecan 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 descriptionof a world; as that is one more limit that cannot be reached, we will makedo with what approximations we can reach, and develop theories thatweave a finer and finer texture of a presumed reality. Then we will throwour theories onto the marketplace of ideas and defend them as best we can,hoping that others will buy them. Some theories (creationism, forexample) will have a clear advantage in terms of pathos: they will line upthe support of strong feelings – the fear of death, the already mentionedfear of freedom. Others will have more modest, though no less genuineemotions on their side: intellectual curiosity, the fascination of complexsolutions, the aesthetic satisfaction of seeing things fall into place. And theywill stir such emotions more, the more detailed, discriminating, resource-ful, and ingenious they are, that is: the more reason structures theminternally – the reason that is forever asking pointed questions andexpecting relevant answers, the reason Socrates taught us how to use.10

A logic is a highly ambitious theory: one that attempts to construct auniversal language. In and by itself, this theory will be found persuasive onlyby those who are already committed to the particular view it expresses andarticulates. But, themore the view is articulated, themorematerial it includesand makes fit in a well-organized, thoroughly sensible structure, the more itwill look to others like the groundwork for a majestic cathedral, and themorethey might find it attractive. Despite the attraction, they might never leavethe hovels they are used to, since those give them more comfort andreassurance; still, however slim a chance logos has of winning over fear, byeliciting the waner passions germane to itself, this is a chance, that in happy(safe, relaxed, sociable) circumstances might well come to fruition. Forcefullyasserting our axioms and proudly marching to the tune of our proofs willnever get us but a reputation for arrogance; patiently working out a thing ofbeauty and making it a paradigm (an example, that is) of internal richnessand consistency might make a few others want to play with us. Not becausethey have to; but because that internal richness and consistency – the logosthat 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 Similar points could be made about ethos. The character and competence of a speaker, in and bythemselves, will have no power to persuade an audience unless the latter feels respect for them. So, asappropriate to a discussion of persuasion – that is, of how an audience can be manipulated – , it isalways pathos (the emotions the speaker is able to instigate in the audience) that works if anythingdoes; and the real distinction is among the emotions that are in play. Both logos and ethos will onlybe successful if the speaker can raise the emotions akin to them, and if these emotions, under thecircumstances, prevail over conflicting ones.

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cha p t e r 1 1

Bolzano’s logical realismSandra Lapointe

1. Framework

The term ‘logical realism’, as it is commonly understood, picks out afamily of views that are committed to at least two theses.1 The first, letus call it ‘(LF)’, is that there are “logical facts”. Here (LF) is construed inthe widest possible sense to include any theory that assumes that there is afact 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 theputatively true claim that modus ponens is a valid principle of inference.The realist may be committed to there being “something” – whatever thisturns 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 interpretationsof (LF), and every other one in between, raise a number of questions thatgo 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 thefollowing: whatever she understands “logical facts” to be, what makes theadherent to (LF) a realist about logic is a further assumption (IND), that“logical facts” are independent of our cognitive and linguistic make-up andpractices; 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 offers a number of theoretical alternatives to logical realism.

What’s common to nihilism, pragmatism and pluralism,2 for instance, isthe fact that they deny (LF). By contrast, the proponents of naturalism(of which there are many variants, including logical psychologism) and

My thanks to Matt Carlson, Nicholas F. Stang, Penny Rush, David Sanson, Ben Caplan and PeterHanks, Julie Brumberg and Teresa Kouric for their input on previous versions of this chapter.1 This characterization of logical realism draws on Resnik’s 2000: 181.2 I revert to the definition of “pluralism” given by Stewart Shapiro in the chapter included in thiscollection.

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conventionalism need not reject (LF). A psychologistic logician – think ofJohn Stuart Mill – need not disagree with the idea that there is a fact of thematter as to whether or not ‘Modus ponens is valid’ is true. Rather, hemight be denying that the truth of this claim can be established independ-ently of psychological knowledge and therefore independently of certainfacts concerning our mind. Likewise, the conventionalist may assume thatthere are determinate facts concerning our (linguistic) practices that deter-mine whether, for instance, the claim that modus ponens is valid ought tocount as true or false. What logical psychologism and conventionalismshare is the fact that they reject (IND).

As I’ve characterized it so far, logical realism is compatible with certainkinds of relativism. In the chapter included in this volume, Shapirodescribes the view he calls “logical folk-relativism”. While one who holdsthis view assumes (i) that the truth-value of ‘y is a consequence of x’, forinstance, 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 agiven 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 narrowersense that does not include relativism of this sort; the type of logicalrealism I will be discussing below is “monistic”.

The ontological questions that underlie logical realism – e.g. what kindsof “facts”, if any, ground the truth or falsity of logical claims? – are to bestrictly separated from the types of concerns that arise when explaininghow we come to know the truth of a claim about logic. The distinctionbetween questions about the epistemology of logic and questions about itsmetaphysics is important, among other reasons, for assessing the consist-ency of some theories. Take Edmund Husserl, for instance. At least in thefirst edition of the Logical Investigations (1900–1901), he adopts a form oflogical realism of the more robust kind. What makes claims about, say,validity, true according to the Logical Investigations are certain features ofabstract entities that exist independently of us: “Bedeutungen”. Nonethe-less, Husserl believed that the only way to know the truth-value of logicalclaims is to engage in certain (admittedly rather esoteric types of ) psycho-logical analyses. Whatever its other merits, Husserl’s theory is notinconsistent. The ontological position according to which there aremind-independent “logical facts” need not be at odds with the epistemo-logical position according to which we can only discover the truths of logicthrough an investigation of the mind. More generally, logical realists, whilethey hold that a claim about logic, if it is true, is true independently ofwhat 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 ormind works and/or reflect upon our cognitive abilities, psychologicaldispositions, linguistic conventions or other uses and practices. The alter-native would presumably be to assume that we come to recognize thetruths of logic through some sort of immediate logical “grasp”. And whilethis cannot be excluded a priori, it is an assumption that might seemdubious to anyone who has ever taught introductory logic to collegestudents.3

Logical realism raises a number of interesting metaontological questions.Consider two simple toy semantic theories. Let us assume that the contentof both theories is defined as the set of true instances of:

(1) s means p

where ‘s’ is taken to stand for a sentence of natural language and ‘p’ for themeaning of this sentence, say the “proposition” that p. To the extent thatone holds that at least some instances of (1) are true, both theories commitone to there being sentences and, more controversially, to there beingpropositions. What makes the two theories different theories may be avariety of things: they may diverge on which instances of (1) are true, theymay rest on different accounts of what a sentence is, or have different viewson what propositions consist of (e.g. structured entities, sets of possibleworlds). Or they may agree on all this and still not be identical.When it comes to comparing types of logical realism, differences that

reside in the metatheory, and in particular in the kinds of “grounds” thatunderlie commitment to the existence of proposition-like entities, can beespecially 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 herholding x to be true has explanatory value when it comes to accounting forA’s belief that y. The notion of explanation used here is to include the casein which y follows from x (in a sense of ‘follows’ to be specified) as well as arange of other cases I will discuss below. What’s peculiar about all thesecases is the fact that the relation between y and x is to be construed inepistemic terms. As I use the terms ‘ground’ and ‘explain’ here, whether y“objectively” follows from x is not ultimately what matters when it comes

3 The theory of “eidetic variation” and “Wesensschau” Husserl eventually committed to is an instanceof this kind of epistemology, and this explains in good part why, in many circles, his theorieseventually 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 acommon objection to the doctrine as a whole. In her chapter in this volume Penny Rush argues forthe 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 beliefthat x. It is sufficient 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’sbelief that y that A effectively believes that y is a consequence of x. Thisqualification is important if we are to account for the fact that grounds thatare unclear, implausible or otherwise mistaken nonetheless have explana-tory value when it comes to understanding an agent’s motivation forcertain claims. If it makes sense to say that A holds the belief that y becauseA holds the belief that x then A’s belief that x – and the correspondingclaim – 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 logicalrealism may have “external” grounds to assume that there are putativelogical facts, even if these grounds are implicit, unconvincing or otherwiseflawed. In the context of logical realism, what I mean by “externalgrounds” are grounds that arise out of a concern that is not itself forlogic. While it might be difficult to define precisely what counts as a logicalconcern, the idea that some concerns pertain to logic while others don’t isuncontroversial enough. On the contemporary understanding the defin-ition of validity and logical consequence belongs to logic – construedwidely enough to include semantics. The investigation of what is involvedin perception and cognition, what moral principle(s) we should abide byand what there is in the world, by contrast, do not.

The (more or less well defined) boundaries between the various philo-sophical subdisciplines are not hermetic and indeed are often such that thegrounds we have to hold a belief in one, are effectively driven by another.For instance, there’s nothing that forbids that a logical realist’s grounds tocommit to (LF) and (IND) be external to the extent that they are driven bymetaphysical concerns i.e. concern for what there is in the world in additionto 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 thekinds of grounds that underlie realist commitments, when they are external,are typically not metaphysical. The realist’s grounds for positing mind andlanguage-independent logical facts, when they are external, are typicallydriven 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 whichdoes 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 commitsLeibniz to truths that are independent of human minds (and language).And taken together, these two assumptions explain how human knowledgeis possible on Leibniz’s view. Leibniz’s primary concern in introducingpropositio is not for what there is, but for how we acquire knowledge.Leibniz’s grounds to commit to the existence of proposition-like entitiesare thus (in part) that the supposition of such entities – and the furtherassumption that a benevolent God exists! – is required to provide acoherent theory of knowledge. Similarly, Popper’s grounds for thinkingthat there is “objective knowledge” – in (Popper 1968) for instance –whatever their merit, is that this allegedly explains certain features of thesciences, such as the relatively autonomous character of scientific theoriesand problems. Whether they are epistemological or otherwise, as long asthe logical realist’s grounds for believing in the existence of logical facts arenot 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. Whatoften underlies one’s commitment to logical facts may correspond to(implicit) theoretical desiderata or aims. Desiderata and aims are types ofgrounds in the relevant sense: they have explanatory value when it comesto accounting for the ontological commitments that come with a logicaltheory. Let us call the kind of logical realism that would underlie such atheory ‘internal’. Historically, many instances of realism in logic have beeninternal. The exact nature of the grounds that underlie the internal realist’scommitment to logical facts vary. It may be that the logician desires to seecertain “intuitions” satisfied or certain epistemic “purposes” fulfilled by thelogical theory. Why precisely these intuitions and purposes ought to besatisfied by the theory is bound to be a matter of contention, but there’s acase to be made to the effect that they pervade logic and its philosophy.6

What I call “intuitions” here correspond to certain claims that seemmore 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 ofthe real foundation of this certainty of eternal truths. This question brings us at last to the ultimatefoundation of truth, namely to that Supreme and Universal Mind who cannot fail to exist and whoseunderstanding is indeed the domain of eternal truths. . .That is where I find the pattern for the ideasand 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 benefit from a closerinvestigation, 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 theyreally are). The logical realist may be convinced, for instance, that truth,whatever it is, is “immutable”, in the sense that it cannot be changed ordestroyed and she won’t regard the theory as adequate unless the immut-ability of truth is a consequence of it. Since she is also likely to hold thebelief that individual sentence- and thought-tokens do not persist indefin-itely (for they don’t) and thus cannot be the fundamental bearers of truth(and falsity), she might deem it necessary to introduce ontologically robustabstract entities, precisely in order to satisfy this intuition. If that is thecase, then A’s (desire to satisfy this) intuition has explanatory value when itcomes to accounting for her commitment to proposition-like entities:there is a definite sense in which A believes that there are propositionsbecause she believes that truth is immutable.

Quantifying over meanings may also serve certain more or less clandes-tine “purposes” within the theory. The logical realist may, for instance, beguided by the fact that systematically including instances of (1), above, in asemantic theory (surreptitiously) introduces a paraphrastic procedurethat can be used to “clarify” natural language sentences or make themmore “exact”.7 If one’s motive, be it explicitly or not, in introducing thesemantic operator ‘means’ and in quantifying over propositions are the(stealthy) epistemic gains that come from translations of this type, one’sgrounds to commit to propositions are subservient to the semantic theoryand the type of realism they embrace is internal.

Admittedly, in certain cases, it could be unclear whether one’s groundsare internal or external. Take the case in which A’s belief that there arelogical facts is the consequence of certain assumptions concerningthe relation between language and the world. A may believe that thereare objective logical facts because A believes (TM):

(TM) The truth of a claim implies its correspondence to something thatmakes it true (or the existence of a “truth-making” relation), whatever thisturns out to be.

Assuming that some claims about logic are true, (TM) implies the exist-ence of entities that fulfil this truth-making role in logic. Commitment to(TM) explains the commitment to logical facts. Indeed, (TM) epitomizes

7 See (Lepore and Ludwig 2006). They write: “The assignment of entities to expressions, which was tobe the key to a theory of meaning, turns out to have been merely a way of matching object-languageexpressions with metalanguage expressions thought of as used (in referring to their own meaning), sothat we are given an object-language expression and a matched metalanguage expression weunderstand, 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 Tractatuswas meant to put into question.8 But is (TM) an “external” ground tosubscribe to logical facts? After all (TM) is a claim that belongs to themetaphysics of logic and one could argue that logic does include its ownmetaphysics. 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 sufficient for our purposes that this question has an answer in principleeven if it is a difficult one.

2. Bolzano’s internal realism in logic

Bolzano’s Theory of Science (1837) presents the first explicit and methodicalespousal of internal logical realism. It also contains a formidable number oftheoretical innovations. They include (i) the first account of the distinctionbetween “sense” (Sinn, Bedeutung) and “reference” (or “objectuality”:Gegenständlichkeit), (ii) definitions of analyticity and consequence, i.e.“deducibility” (Ableitbarkeit) based on a new substitutional procedure thatanticipates Quine’s and Tarski’s, respectively, and (iii) an account of math-ematical knowledge that excludes, contra Kant, recourse to extraconceptualinferential steps and that is rooted in one of the earliest systematic reflectionson the nature of deductive knowledge. (i)–(iii) all assume the existence ofmind- and language-independent entities Bolzano calls “propositions andideas in themselves” (Sätze an sich). Take (i) for instance. Appeal to propos-itions in themselves in this context serves Bolzano’s antipsychologism inlogic: according to Bolzano, the sense (Sinn) of a sentence – the propositionit expresses – is to be distinguished from the mental act in which it isgrasped. Just like what is the case in Frege, a sentence has the semanticproperties it has (e.g. truth) on Bolzano’s account derivatively, by virtue ofits relation to mind-independent entities: the primary bearers of semanticproperties are the propositions that constitute their Sinne.

Bolzano’s version of logical realism is among the more robust. It yields aunique form of “semantic descriptivism”: there are objective,9 immutable10

entities, “propositions” (for short), that bear certain properties and rela-tions, which it is the task of logicians to describe. The first two books of theTheory of Science (together they make up the Theory of Propositions and

8 On this, see (Mulligan, Simons and Smith 1984: 289). 9 Cf. (Bolzano 1837, §21: 84).10 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 hiscommitment to logical realism.

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Representations in Themselves) are divided into chapters whose headersinclude reference to propositions’ and representations’ “general character-istics”, “properties”, “relations” (among themselves, to objects) and“internal constitution”. Bolzano even devotes entire sections of the bookto the analysis of claims in which such properties and relations are ascribedto representations and propositions.11 As Bolzano sees it, truths of logic –including definitions of such notions as meaning, analyticity and apriority –amount to descriptions in which (often multifaceted) properties or relationsare 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 ofcertain kinds of non-natural entities which, because it is explicit andindeed unequivocal, is perhaps somewhat perplexing: one could be leftwith the impression that Bolzano’s ontology of logic is a more direct targetfor standard naturalistic objections than some other varieties of logicalrealism.12 There are at least two grounds why this impression is misleading.First, to the extent that ontological commitments cannot be measured on ascale and that all logical realists subscribe to (LF) and (IND), all variants ofrealism are equally ontologically “candid” from a naturalistic standpoint.There is in principle nothing more ontologically damning about Bolzano’ssemantic descriptivism than about the kind of realism Frege will eventuallyput 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 thenaturalist than others. The naturalistic criticism of logical realism is typicallymotivated by a concern for metaphysical economy (Is it consistent topostulate entities that do not exist in the causal realm?) or by relatedepistemological reservations (What does it mean to “grasp” or cognize orbe epistemically related to something that does not exist causally?). For thisreason, one who has independent (external) metaphysical or epistemologicalgrounds to subscribe to (LF) and (IND) is a more direct target for naturalisticcriticism. But Bolzano’s commitment to the existence of non-naturalentities, while it is uncompromising, is also clearly motivated by the kindof internal grounds that makes him least susceptible to the naturalisticconcern.

11 See for instance (Bolzano 1837, §§164–168).12 There are other types of objection to realism. Rush (in this volume) for instance discusses Sellars’

objection. I won’t be discussing this point.

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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 purposeof reaching satisfactory definition of logical notions (understood broadly):

The usefulness of the distinction [between propositions in themselves andthought propositions] manifests itself in tens of places and in the mostsurprising way in that it allows the author to determine objectively anumber of concepts that had not been explained before or that wereexplained 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 consideredto have bearing outside of logic. As Bolzano sees it, logicians should beallowed to appeal to entities that may reveal themselves to be inconsistentwith paradigmatic metaphysical and/or epistemological theories:

Thus, to give another example, the logician must have the same right tospeak of truths in themselves as the geometrician who speaks of spaces inthemselves (i.e., of mere possibilities of certain locations) without thinkingof them as filled with matter, although it is perhaps possible to givemetaphysical 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 isthe fact that, while he argues that he needs to posit what he calls ‘propos-itions’ to arrive at satisfactory definitions, and while he assumes that thebearers of the logical properties and relations he defines in fact bear thisname: Bolzanian definitions of logical notions are in principle compatiblewith a number of different ontologies. This idea – it amounts to claimingthat logic is “topic neutral” – emerges from a series of remarks Bolzanomakes in a text he published some years after the Theory of Science, theWissenschaftslehre (Logik) und Religionswissenschaft in einer beurtheilendenUebersicht (1841) whose (failed) purpose was to arouse the public’s interestfor Bolzano’s theories. Bolzano writes:

Everything the author asserts of propositions in themselves in the firstsection – with the exception of what he says at §122, namely that theydon’t exist – holds of thought propositions; likewise, in the second section,the “Differences amongst propositions as regards their internal properties”are all such that whoever admits of thought propositions can also admitof 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 ofpropositions in themselves and accepts only that [for instance] of thoughtpropositions could nonetheless admit of a connection amongst the lattermore or less like the one Bolzano describes as objective. And this, we think,should be answered in the affirmative. (Bolzano 1841: 68)

On the face of it, the kind of semantic descriptivism Bolzano adopts seemsincompatible with the claim that someone who rejects the notion ofproposition could still admit his definitions of logical notions. Indeed suchstatement contradicts what Příhonský, Bolzano’s close collaborator, seemsto have assumed in the New Anti-Kant, namely that:

All will be lost if they cannot grant us this concept [of a proposition], if theykeep representing truths in terms of certain thoughts, appearances in themind of a thinking being . . . (Příhonský 1850: 5)

If Příhonský is right, Bolzano’s move – the claim that definitions of logicalnotions are topic neutral – is at best a rhetorical concession made in orderto win a reluctant public. The problem with this exegetical line is not thatit is implausible. The problem is that it does not do justice to thecoherence of Bolzano’s views. Notwithstanding what Příhonský assumes(more on this below), and even granting that Bolzano as an internal logicalrealist has less of an axe to grind when it comes to defending the existenceof propositions, it remains that the claim that definitions of logical notionsare topic neutral is not a mere rhetorical ploy. For Bolzano has thetheoretical resources to make sense of this idea systematically. This is whatI argue in Section 3. Nonetheless, if part of Bolzano’s point is that the valueof a logical theory does not reside in the nature of the entities that bear theproperties it defines, but in the properties and relations they are meant toepitomize and that he would be willing to revise some of his ontologicalcommitment as long as some other aspects of his theory are preserved, theonus is on him to show that his theory does present an advantage over thatof his predecessors and contemporaries. Section 4 is dedicated to arguingthat it does.

3. Topic neutrality and implicit definition

Bolzano claims that everything he asserts of propositions in themselves,with the exception of their being non-actual, in the first and secondsection of the second volume of the Theory of Science – what Bolzano calls“general characteristics” and “differences that arise from their internal

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constitution” – holds of thought propositions, the doxastic states in whichsuch propositions are grasped.14 These characteristics and differencesinclude 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 tothink of Bolzano as resorting to what he calls “definition on the basis of useor context” (1837, §668: 547), that is, implicit definitions. The idea is thatin (i)–(vii), ‘proposition’ designates a primitive (simple) concept (i)–(vii)define implicitly.15 Bolzano was aware from very early on of the benefit ofthis procedure when it comes to defining primitive notions. ThoughBolzano’s paradigmatic examples come from mathematics, the procedureapplies across the board, including in logic. It consists in:

stating many propositions in which the concept that needs to be under-stood occurs in different combinations and which is designated by the wordthat is associated to it. By comparing these propositions, the reader himselfwill 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 hewrites: “From the mere fact that representations are the components of propositions we cannot inferthat 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 theconcept of a representation is the actual definition of the latter.” At (1837, §48: 216) Bolzano hadwritten: “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 mymeaning to those who have understood what I mean by a proposition in itself ”.

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example, every one can grasp which concept is designated by the word pointon the basis of the following propositions: the point is the simple object inspace, it is the limit of the line without being part of the line, it is extendedneither lengthwise nor according to width, nor according to depth, etc. Asis well known, this is the means through which we all learn the firstsignifications of our mother tongue. (1810, II, §8: 54–55)

What defines propositions, then, ultimately, is the system of all relevantimplicit definitions. Implicit definitions (including (i)–(vii)) definepropositions as much as necessary for the purpose of Bolzano’s logic. In(i)–(vii), ‘proposition’ occurs only as the name of what is in effect beingdefined and, in this light, part of Bolzano’s point is that substituting‘thought (proposition)’ for ‘proposition (in itself )’ in (i)–(vii) has nobearing on the nature and structure of the properties and relationsinvolved. Indeed, Bolzano has nothing to object to someone who wouldclaim that the bearer of the properties involved in (i)–(vii) have furtherproperties, e.g. the property of being types of mental processes, as long asshe admits that mental processes have the properties involved in (i)–(vii).

There is at least one other reason to take Bolzano’s suggestion that hisdefinitions are topic neutral seriously. Bolzano (1841, 68) claims that“objective connections” need not be predicated of propositions, thatproperties such as validity (§147), analyticity (§148), compatibility (§154)and deducibility (§155) – “formal” properties – could equally well berecognized by one who admits only thoughts. By a formal property,Bolzano means a property that is defined for “entire genera of propos-itions”, on the basis of the substitutional procedure.16 Beyond what I’veargued above, I want to show that Bolzano’s definitions of what counts asformal properties are topic neutral in the relevent sense.

Formal properties are not properties of individual propositions but prop-erties of what Bolzano calls ‘forms’, i.e. schematic expressions.17 Bolzanomakes copious use of schematic expressions – or their equivalent18 – when it

16 Cf. Bolzano (1837, §12: 51) where he explains (my emphasis): “The clearest definitions say hardlymore than that we consider the form of propositions and ideas when we keep an eye only on whatthey have in common with many others, that is, when we speak of entire species or genera of thelatter. . . . one calls a species or genus of proposition formal if in order to determine it one only needs tospecify certain parts that appear in these ideas or propositions while the rest of the parts which one calls thestuff or matter remain arbitrary.”

17 Cf. (Bolzano 1837, §9: 42f ).18 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 hasmortality, 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 defining formal notions. Take for instance Bolzano’s claims that thereare logically analytic or tautological propositions. Bolzano writes:

The following are some very general examples of analytic propositionswhich are also true: “A is A”, “An A which is a B is an A”, “An A whichis a B is a B”, “Every object is either B or non-B”, etc. Propositions of thefirst kind, i.e., propositions cast in the form “A is A” or “A has (theattribute) a” are commonly called identical or tautological propositions.(Bolzano 1837, §148: 84)

In this passage, Bolzano ascribes the property of being logically analytic toindividual propositions, yet his examples – “A is A”, “An A which is a B isan 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 whatBolzano says in the Theory of Science, ‘A is A’ does not stand for any“proposition” in particular. On Bolzano’s account schematic expressions ofthe kind ‘A is A’ represent classes of propositions that are defined through asubstitutional procedure. To say that a proposition “falls under a certainform” is to say that it belongs to a certain substitution class designated bythis schematic expression.19 ‘A is A’ represents the class of all propositionsin themselves that correspond to the substitution instances of ‘A is A’. Ifwe use ‘[‘and’]’ to form designations for individual propositions (and theirparts), we find among the propositions designated by the substitutioninstances of ‘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 is A’ – the

schematic expression – is logically analytic. Indeed, it would seem that oneneed 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 besaid to be analytic. The proposition: [Caius is Caius], for instance, fallsunder the form ‘A is A’: ‘A is A’ is a “determinate connection of words orsigns” through which the class to which [Caius is Caius] belongs can be“represented”.20 To say that the individual proposition [Caius is Caius] is

19 In one of his numerous historical digressions, Bolzano notes that “the Latin word forma . . . was infact 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: aclass that can be represented by a determinate type of schematic expres-sions, namely one all of whose substitution instances designate propos-itions that have the same truth value.

What’s interesting here is the fact that, on this interpretation, Bolzano isultimately committed to the following view of logical analyticity:

(LA) x is logically analytic if x belongs to a class that can be represented by aschematic expression in which only logical terms occur essentially.

But (LA) is topic neutral. Nothing compels us to think of ‘x’ in (LA) interms of a proposition in itself. And as we have seen, once the logical workis done, Bolzano would not be disconcerted by such a move since “some-one who rejects the concept of propositions in themselves and accepts onlythat [for instance] of thought propositions could nonetheless admit of aconnection amongst the latter more or less like the one Bolzano describesas objective” (Bolzano 1841: 68).

4. Bolzano’s logic

Internal grounds to adhere to logical facts – or in Bolzano’s case to fullyfledged semantic entities – are typically certain desiderata or aims thetheory is meant to fulfil. In Bolzano’s case, one of the main purposes inintroducing propositions in themselves is to achieve precise and satis-factory definitions. By way of consequence, on Bolzano’s own accountthe success of the endeavour depends on whether his commitment topropositions allows him to deliver a “good” theory of logic, or at leastone that is preferable to its rivals. To a large extent, Bolzano succeeds. Itis not only that the Theory of Science is furnished with rich andremarkably well-articulated distinctions and theoretical innovationsbut also that he set out to redefine the very nature of a logical investi-gation in a way that is largely consistent with well-established contem-porary endeavours.

As Bolzano sees it, at its core, the purpose of logic is to tell us what itmeans for something to follow from something else, i.e. what it means foran inference to be valid or for a claim to be the consequence of some otherclaim(s). As an explanation of what it means for a truth to follow fromothers, Bolzano’s views on “deducibility” (Ableitbarkeit) are comparativelyclose to the ones that have become standard following Tarski in thetwentieth century. Bolzano defines deducibility in the following terms:

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Hence I say that propositions M, N, O, . . . are deducible from propositionsA, B, C, D, . . . with respect to variable parts i, j, . . . if every collection ofrepresentations whose substitution for i, j, . . . makes all of A, B, C, D, . . .true, also makes all of M, N, O, . . . true. Occasionally, since it is custom-ary, I shall say that propositions M, N, O, . . . follow, or can be inferred orderived, from A, B, C, D, . . . Propositions A, B, C, D, . . . I shall call thepremises, M, N, O, . . . the conclusions. (1837, §155: 114)

The modern character of Bolzano’s definition, in itself, and especially thesemantic machinery on which it rests is noteworthy enough. On Bolzano’saccount:21

The propositions T, T0, T00 . . . are ableitbar from S, S0, S00 with respect torepresentations i, j, . . . if and only if:

(i) i, j, . . . can be varied so as to yield at least one true variant of S, S0,S00, . . . and T, T0, T00, . . .

(ii) whenever i, j, . . . are varied so as to yield true variants of S, S0, S00 . . .,the corresponding variants of T, T0, T00, . . . are also true.

To logicians and philosophers of logic today, the idea that the aim of logicis to define validity via the elaboration of a theory of logical consequence isunremarkable. Pointing to the similarities (and dissimilarities) betweenBolzano’s definition of deducibility and Tarski’s definition of logicalconsequence has become commonplace in the literature.22 This goes toshow that at least some of the desiderata and aims that underlie Bolzano’slogic rest on the kind of intuitions that have proven to be enduring. Thisshould be emphasized for at least two reasons. First, when he published theTheory of Science in 1837, Bolzano’s views on deducibility were perfectlyanachronistic. For one thing, by the end of the eighteenth century it hadbecome usual for philosophers to think of logic as invested in the study of“reason” through an investigation of “thought” and to conceive of such aninvestigation to involve the study of mind-dependent operations andproducts. Though the methodologies underlying these investigationsvaried widely – contrast Locke’s empirical approach in the Essay on HumanUnderstanding with Kant’s transcendental philosophy – they largely con-tributed to either discredit formal logic as a discipline23 or, at best, toconvey the opinion that it could not be improved on.24 In this light,Bolzano’s efforts toward a new logic based on an objective doctrine of

21 For a more detailed discussion of Bolzano’s theory of deducibility, see (Lapointe 2011: 72–90).22 See, for instance, (van Benthem 1985; George 1986; Siebel 2002; Lapointe 2011).23 See (George 2003: 99s).24 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 imme-diate modern predecessors.

Second, while Bolzano reaches back to Aristotle, his approach to thedefinition of validity also marks an important departure from Aristotle andmost of his (early) traditional scholastic commentators.25 Aristotle intro-duces the notion of a good “deduction” (i.e. syllogism) in the PriorAnalytics. He writes:

A deduction (syllogismos) is speech (logos) in which, certain things havingbeen supposed, something different from those supposed results of necessitybecause of their being so. (Prior Analytics I.2, 24b18–20)

Let us call this the “intuitive Aristotelian notion of validity”. Contempor-ary attempts at a definition of logical consequence – one may think ofTarski-type model-theoretic definitions in particular – are generally under-stood to account for the intuitive Aristotelian notion of validity. The sameholds for Bolzano’s. What makes Bolzano’s account historically distinctiveis the assumption that a good definition of the intuitive Aristotelian notionof validity needs the support of a semantic theory. In this, his definitionalstrategy ought to be contrasted with that of much of the Aristoteliantradition itself. Aristotle and his early medieval successors are mostlyknown for their understanding of validity as epitomized in traditionalsyllogistic theories. But traditional syllogistic definitions of validity arenot concerned with providing a semantic account of validity.26 Thestandard and paradigmatic methodology behind traditional syllogistictheories of valid inference, and the one that is best known, is two-pronged.It first consists in making a list of all possible forms of arguments (syllo-gisms) and then in identifying those forms whose instances effectively fulfilthe intuitive Aristotelian definition of validity. In order to determinewhether a particular inference is valid, one is thus required to determinewhether it instantiates one of the forms identified as valid.

There are at least three problems, from Bolzano’s perspective, with thisapproach. First, traditional syllogistic definitions of validity suppose thatthere is a finite (and implausibly small) number of possible forms ofinference. Bolzano is right. If we follow the teachings of the Schoolmen,

25 Here I am not concerned with comparing the Bolzanian and Aristotelian conception of the object oflogic (see Thom, this volume, for such a discussion) but their views on validity.

26 Here, I exclude from what I call “syllogistic tradition” the theories of consequentia that emerged inthe fourteenth century – those we find in Occam and Buridan, for instance. The latter wereattempts to generalize syllogistic and aimed at providing a new insight into the intuitive Aristoteliannotion based on semantic considerations. On this topic, see (Novaes 2012). Something similar holdsfor Abelard. I am grateful to Julie Brumberg-Chaumont for this precision.

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there are exactly 256. This number comes up as a result of variousassumptions concerning the number of sentences involved in an argu-ment – three! – and the form of such sentences. In particular, classicalsyllogistic theory assumes (i) that only categorical sentences (i.e. sentencesof the form subject-copula-predicate) are involved in arguments, (ii) thatthere are four variants of such forms (a, e, i, o) and (iii) that any giveninference contains at most three different terms – subject, middle term andpredicate – which yields four possible syllogistic “figures”. Second, (i)–(iii)mark out a syntax whose expressive resources are too limited to account forthe richness of actual inferential practices. Hence, it cannot adequatelymodel (even some of the most basic forms of ) inference. For instance, itcannot model disjunctive and hypothetical syllogisms that require separatetheories (at least if understood in its original sense, i.e. as a propositionallogic). This is tributary to a third more general problem, namely the factthat traditional syllogistic definitions of validity are bound to a givensyntax (namely the one defined by (i)–(iii) above). But as is obvious fromthe relevant passage in Aristotle the intuitive notion of validity is notbound to any particular syntax – it is a “semantic” definition.27

Bolzano was aware of these three related problems. He writes:

Aristotle began with such a broad definition of the word syllogism that oneis astonished that he could have subsequently restricted the concept of thiskind of inference so severely. He writes (in Anal Pr. I, 1) “syllogism is adiscourse in which, certain things being stated, something other than what isstated follows of necessity from their being so”. This definition obviously fitsevery 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) formsof inferences is finite or that it is linked to a determinate syntax, forinstance that it can only be defined for inferences that have only twocategorical premises.28

Moreover, the three above problems concerning traditional syllogistictreatments of validity are linked to a fourth more general one. There arevarious ways of fixing the extension of a concept, not all of which amountto definition. The mere fact of knowing which inferential forms satisfy the

27 This is even more obvious when one reads the beginning of the second book of the PriorAnalytics, which was devoted to the relationship between premises and conclusion as regardstheir truth-value.

28 See (Příhonský 1850: 115f ). Some passages of the Prior Analytics suggest that Aristotle was aware ofthe problem. See for instance (Prior Analyics I, 32). But Aristotle himself did not provide asystematic account of what it is for an inference that is not a syllogism to result of necessity.

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intuitive Aristotelian definition of validity does not, on Bolzano’s account,amount to having a definition of this notion. On Bolzano’s account, mymerely knowing what falls in the extension of a concept – say the class ofall putatively valid syllogistic inferential forms – does not amount to myhaving a definition (Erklärung) of that concept. Definition is a conceptualexercise: one that requires us to identify the components of a concept aswell as the way in which they are connected. As Bolzano sees it, the theoryof deducibility and the proposed definition above is what allows us to graspthe concept of validity.

More importantly perhaps – though this might go beyond Bolzano’scriticism – it seems that a good definition of validity is one that is epistemic-ally fruitful in the following sense: a good definition of validity is one on thebasis of which one can ascertain systematically for any newly encounteredinference, whether or not it is valid. But the traditional syllogistic definitionof validity is not epistemically fruitful. There is no obvious reason to thinkthat one could decide whether an as yet unknown argument form is validwhen presented with it in any other way than by reverting to the intuitivenotion. By contrast, Bolzano’s definition is epistemically fruitful: equippedwith Bolzano’s definition, one can in principle determine for any newargument whether or not it instantiates the property in question.

5. Conclusion

In light of what precedes, Bolzano’s internal realism is vindicated: Bolzano’spositing of propositions in themselves allows him to articulate a theory ofdeducibility that could do what the syllogistic theories of his predecessorscould not: provide us with a general semantic theory of validity. Nonetheless,as those acquainted with recent scholarship know, there are problems withBolzanian deducibility. (See, e.g. Siebel 2002.) For one, despite Bolzano’sclaim to the contrary, his definition of deducibility fails to capture what isusually taken to be the modal insight that underlies the intuitive Aristoteliannotion of validity, namely the idea that the conclusion of a good argument“results of necessity”. Consequently, it overgenerates. Bolzanian deducibilitysystematically 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’ isinvariably 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 misunderstand-ing (coupled with contentious exegetical choices). Bolzano interprets therelevant passage of the Prior Analytics in the following terms:

Since there can be no doubt that Aristotle assumed that the relation ofdeducibility can also hold between false propositions, the results of necessitycan hardly be interpreted in any other way than this: that the conclusionbecomes true whenever the premises are true. Now it is obvious that wecannot say of one and the same collection of propositions that one of thembecomes true whenever the others are true, unless we envisage some of theirparts as variable. For propositions none of whose parts change are notsometimes true and sometimes false; they are always one or the other.Hence when it was said of certain propositions that one of them becomestrue as soon as the others do, the actual reference was not to these propos-itions themselves, but to a relation which holds between the infinitely manypropositions which can be generated from them, if certain of their repre-sentations are replaced by arbitrarily chosen other representations. Thedesired formulation was this: as soon as the exchange of certain representa-tions 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 frompremises to conclusion”. Whatever the explanation for this confusion is –Bolzano does have a systematic account of necessity and onemaywonder whyhe 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 definitionof deducibility fails to grasp the modal insight that underlies the intuitivenotion of validity that he achieved little toward a theory of logical conse-quence or that he missed the point entirely. This would not do justice toBolzano’s accomplishment, both historical and philosophical. For one thing,while Bolzano’s own use of the substitutional method fails to do so, otherphilosophers have put a wager on a substitutional procedure of the typeBolzano was first to introduce for the purpose of providing a satisfactoryaccount of logical consequence. Tarskian-type model-theoretic approachesfor instance can be seen as an extension of Bolzano’s theory.

Few would deny that Bolzano’s views on deductive knowledge wereoverall largely preferable to those of his predecessors and contemporaries.In particular, it is important to stress the fact that Bolzano did have viewson epistemic modality – though unfortunately, there is no place for adiscussion 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 andgeometry, Bolzano was first to propose an account of epistemic necessitythat rests on (i) the idea that truth by virtue of meaning can be definedsystematically (in a deductive system) and that (ii) a priori knowledge isaccordingly always deductive. Regardless of the execution, (i) and (ii) areboth manifestly valuable philosophical insights that deserve the attentionof historians and philosophers alike. For one thing, one committed to (i)and (ii) cannot appeal to subjective justificatory devices such as “certitude”or “evidence” to warrant the truth of a priori claims. And, again, manyeven today would consider this to be an important lesson. What’s relevanthere is the fact that to the extent that Bolzano’s views on a prioriknowledge and deductive systems are parts and pieces of his theory ofpropositions in themselves, they are inseparable from his commitment tomind-independent logical facts. What this means is that logical realism alsoinforms his views on a priori knowledge and nourishes insights that manyof his successors, realist or not, will share.

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part iii

Specific Issues

cha p t e r 1 2

Revising logicGraham Priest

1. What’s at issue

Much ink has been spilled over the last few decades in disputes betweenadvocates of “classical logic” – that is, the logic invented by Frege andRussell, and polished by Hilbert and others – and advocates of non-classical logics – such as intuitionist and paraconsistent logics. One movethat 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 tran-spire in due course. The point is to examine the question of whether logiccan be revised.1 (And let me make it clear at the start that I am talkingabout deductive logic. I think that matters concerning non-deductive logicare much the same, but that is an issue for another occasion.) Threequestions, then, will concern us:

• 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 oftenvitiated by a failure to observe that the word ‘logic’ is ambiguous. Onlyconfusion results from running the senses of the word together. Once theappropriate disambiguations are made, some of the answers to our ques-tions are obvious; some are not. It pays, for a start, to be clear about whichare which.

1 Thanks go to Hartry Field for many enjoyable and illuminating discussions on the matter. We taughta course on the topic together in New York in the Fall of 2012. Many of my views were clarified in theprocess.

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We may distinguish between at least three senses of the word, whichI will call:

• Logica docens• Logica utens• Logica ens

What each of these is will require further discussion and clarification. Butas a first cut, we may characterise them as follows.

Logica docens (the logic that is taught) is what logicians claim aboutlogic. It is what one finds in logic texts used for teaching. Logica utens (thelogic which is used) is how people actually reason. The first two phrases arefamiliar from medieval logic. The third, logica ens (logic itself ) is not. (Ihave had to make the phrase up.) This is what is actually valid: what reallyfollows 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, bothintensionally and extensionally, as again we will see.

I will proceed by discussing each of these senses of ‘logic’, and askingeach of our three target questions about them. We have, then a nine-partinvestigation.

2. Logica docens

2.1 Can it be revised?

Let us start with logica docens. The discussion of this will form the longestpart of the essay, since it informs the discussion with respect to the othertwo 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 beenrevised.

The history of logic in the West has three great periods.2 The first was inAncient Greece, when logic was founded by Aristotle, the Megarians, andthe Stoics. The second was in the new universities of Medieval Europe,such as Oxford and Paris, where Ockham, Scotus, and Buridan flourished.The third starts in the late nineteenth century, with the rise of mathemat-ical logic, and shows no signs yet of ending. Between these three periodswere periods of, at best, mainly maintaining what was known, and at worstforgetting it. Much of Greek logic was forgotten in Europe, but fortunately

2 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 Enlighten-ment, and the consequent obliteration of Scholasticism. It is only inthe twentieth century that we have started to rediscover what was lostin 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 different inthe various periods. The differences between the contents of Aristotle’sAnalytics, Paul of Venice’s Logica Magna, the Port Royale Logic, or theArt of Thinking, Kant’s Jäsche Logik, and Hilbert and Ackermann’sPrinciple’s of Mathematial Logic would strike even the most casual observer.

It is sometimes suggested that, periods of oblivion aside, the develop-ment of logic was cumulative. That is: something once accepted, was neverrejected. Like the corresponding view in science, this is just plain false. Letme 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:

All As are Bs

All As are C sSome Bs are C s

As anyone who has taken a first course on modern first-order logic willknow, this inference is now taken to be invalid.3

For another example: Classical logic is not paraconsistent; that is, thefollowing inference (Explosion) is valid for all A and B: A, :A ├ B. It isfrequently assumed that this has always been taken to be valid. It has not.Aristotle was quite clear that, in syllogisms, contradictions may or may notentail a conclusion. Thus, consider the syllogism:

No As are BsSome Bs are AsAll 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 onlytwo. But Aristotle is also quite explicit that two terms of a syllogism may bethe 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 twelfth-

3 For further discussion of the matter, see (Priest 2006a: 10.8).

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century Paris logicians called the Parvipontinians, whose membersincluded Adam of Balsaha and William of Soissans, who may well havedeveloped the argument to Explosion using extensional connectives andthe Disjunctive Syllogism. After that, the validity of Explosion wasdebated. But it certainly did not become entrenched in Western logic tillthe rise of classical logic.4

2.2 Can it be revised rationally?

Logica docens, then, has been revised, and not in a cumulative fashion. Thenext question is whether revision can be rational.

Arguably, not all the changes in the history of logic were rational (orperhaps better: occurred for reasons that were internal to the subject).Thus, logic fell into oblivion in the early Middle Ages in Western Chris-tendom because the institutions for the transmission of philosophical textscollapsed. And later Medieval logic was written off on the coat-tails of therejection 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 changeare made. This should be pretty obvious with respect to the only changethat most logicians are now familiar with: the rise of mathematical logic.In the mid nineteenth century, text book logic (“traditional logic”) wasa highly degenerate form of medieval logic: essentially, Aristoteliansyllogistic with a few medieval accretions, such as “immediate infer-ences” like modus ponens. But this was a period in which high standardsof rigour in mathematics were developing. Mathematicians such asWeierstrass and Dedekind were setting the theory of numbers on afirm footing. And when it came to examining the reasoning required inthe process, notably by Frege, it became clear that traditional logic didnot seem to be up to the job. Hence Frege invented a logic that didmuch better: classical logic. The extra power of this logic made it muchpreferable rationally; and within 50 years it had replaced traditionallogic as the received logica docens.

I will come back to this in the next section. For the present, let us moveon to our third question.

4 For references and further discussion on all these matters, see (Priest 2007: sec. 2).5 Actually, my knowledge of the history of these periods is pretty sketchy; but I think that these claimsare essentially correct.

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2.3 Logic as theory

So, what, exactly, is it in virtue of which one logica docens is rationallypreferable to another, and so may replace it? To answer this question, weneed 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 aspieces of pure mathematics, all are equally good. They all have axiomsystems, model theories; each specifies a perfectly fine class of mathemat-ical structures. Rivalry between them can arise only when they are appliedin some way. Then we may dispute which is the correct geometry for aparticular application, such as mensurating the surface of the earth. Eachapplied geometry becomes, in effect, a theory of the way in which thesubject of the application behaves.Geometry had what one might call a canonical application: the

spatiotemporal structure of the physical cosmos. Indeed this applicationwas coeval with the rise of Euclidean geometry. It was only the rise of non-Euclidean geometries which brought home the conceptual distinctionbetween a pure and an applied geometry. And nowadays the standardscientific view is that Euclidean geometry is not the correct geometry forthe canonical application.So much, I think, is relatively uncontestable. But exactly the same

picture holds with respect to logic. There are many pure logics: classicallogic, intuitionist logic, various paraconsistent logics, and so on. And aspieces of pure mathematics, all are equally good. They all have systems ofproof, model theories, algebraicisations. Each is a perfectly good math-ematical structure. But pure logics are applied for many purposes: tosimplify electrical circuits (classical propositional logic), to parse grammat-ical structures (the Lambeck calculus), and it is only when different logicsare taken to be applied for a particular domain that the question of whichis right arises. Just as with geometries, each applied logic provides, in effect,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 anaccount 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 manytechnical terms, but the vernacular none the less. (No one reasons à laPrincipia Mathematica.) And so applied, different pure logics may givedifferent verdicts concerning an inference. If it is not the case that it is not

Revising logic 215

the case that there is an infinitude of numbers, does it follow that thereis an infinitude of numbers? Classical logic says yes; intuitionist logicsays no.6

In other words, a pure logic with its canonical application is a theory ofthe validity of ordinary arguments: what follows (deductively) from what.How to frame such a theory is not at all obvious. Many approaches havebeen proposed and explored. One approach is to take validity to beconstituted modally, by necessary truth-preservation (suitably under-stood). Another is to define validity in terms of probabilistic constraintson rational belief. Perhaps the most common approach at present is to takea valid inference to be one which obtains in virtue of the meanings of (atleast some of ) the words employed in it. This strategy has itself two waysin which it can be implemented. One takes these meanings to be spelledout in terms of truth conditions, giving us a model-theoretic account ofvalidity; the other takes these meanings to be spelled out in inferentialterms, giving us a proof-theoretic account of validity.

It is clear that a theory of validity is no small undertaking. It requires anaccount of many other notions, such as negation and quantification.Moreover, depending on the theory in question, it will require an articula-tion of other important notions, such as truth, meaning, probability. Nowonder it is hard to come up with plausible such theories!

At any rate, it is crucial to distinguish between logic as a theory (logicdocens, with its canonical application), and what it is a theory of (logicaens). In the same way we must clearly distinguish between dynamics as atheory (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 Quineancharge (still all too frequently heard): change of logic means change ofsubject.7 If one changes one’s theory of dynamics, one can still bereasoning 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 questionof the mechanism of rational change of logica docens. As we have seen, apure logic with its canonical appication is essentially a theory of validityand its multitude of cognate notions. How do we determine which theoryis 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).

216 Graham Priest

Given any theory, in science, metaphysics, ethics, logic, or anythingelse, we choose the theory which best meets those criteria which determinea good theory. Principal amongst these is adequacy to the data for whichthe theory is meant to account. In the present case, these are thoseparticular inferences that strike us as correct or incorrect. This does notmean that a theory which is good in other respects cannot overturnaberrant data. As is well recognised in the philosophy of science, all thingsare 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 berespected, are important questions, which we do not need to go into here.One should note, however, that whatever they are, they are not allguaranteed to come down on the same side of the issue. Thus (thestandard story goes), Copernican and Ptolemaic astronomy were aboutequal in terms of adequacy to the data; the Copernican system was simpler(since it eschewed the equant); but the Ptolemaic system cohered with theaccepted (Aristotelian) dynamics. (The Copernican system could handlethe motion of the Earth only in an ad hoc fashion.) In the end, the theorymost rational to accept, if there is one, is the one that comes out best onbalance. How to understand this is not, of course, obvious. But we do notneed to pursue details here.8

I observe that this procedure does not prejudice the question of logicalmonism vs logical pluralism. If there is “one true logic” one’s best appraisalof what this is is determined in the way I have indicated. If there aredifferent logics for different topics, each of these is determined in the sameway. Whether one single logic is better than many, is a “meta-issue”, and isitself to be determined by similar considerations of rational theory-choice.Let me finish this discussion by returning, by way of illustration, to the

replacement of traditional logic by mathematical logic in the early years ofthe twentieth century. In the nineteenth century, much new data hadturned up: specifically, the microscope had been turned on mathematicalreasoning, showing all sorts of inferences that did not fit into traditionallogic. Mathematical logic was much more adequate to this data. This is notto say that enterprising logicians could not try to stretch traditional logic toaccount for these inferences. But mathematical logic scored high on manyof the other theoretical criteria: simplicity, unifying power, and so on. Itwas clearly the much better theory.

8 Matters are spelled out in detail on (Priest 2006a: ch. 8), and especially, (Priest to appear).

Revising logic 217

A word of warning: it would be wrong to infer that classical logic did nothave its problems. It had its own ad hoc hypotheses (to deal with thematerial conditional, for example). It had areas where it seemed to performbadly (for example, in dealing with vague language). And why should oneexpect a logic that arose from the analysis of mathematical reasoning to beapplicable to all areas of reasoning? It was just these things which left thedoor open for the development of non-classical logics. That, however, isalso a topic for another occasion.9 We have seen, at least in outline, whatthe 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 nextdisambiguation. Before we address our three questions, however, there isan important preliminary issue to be addressed. What exactly is logicautens?

I said that it is the way that people actually reason. This may make itsound like a matter of descriptive cognitive psychology; but it is not this,for the simple reason that we know that people often reason invalidly. Setaside slips due to tiredness, inebriation, or whatever. We know that peopleactually 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 positiveinteger on the other. Four cards are laid out on the table so that a subjectcan see the following:

A K 4 3

The subject is then given the following conditional concerning the dis-played situation: If there is an A on one side of the card, there is an evennumber on the other. They are then asked which cards should be turnedover (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 first coursein 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, showthat 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).

218 Graham Priest

to appreciate the error of their reasoning when it is pointed out to them.But how to draw a principled distinction between correcting a standardperformance 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; itis a normative one. A logica utens is constituted by the norms of aninferential practice. Subjects in the Wason Card Test can see, when it ispointed out to them, that they have violated appropriate norms. How tounderstand the normativity involved here is a particularly hard question,which, fortunately, we also do not need to pursue. We have sufficientunderstanding to turn to the first of our three questions. Can a logica utensbe revised?

3.2 Can it be revised?

Clearly, different reasoning practices come with different sets of norms.Thus, the norms that govern reasoning in classical mathematics are differ-ent from those that govern reasoning in intuitionist mathematics. I wastrained as a classical mathematician, and have no difficulty in reasoning inthis way. But I have also studied intuitionist logic, and can reason (morefalteringly) in this way too. Clearly, then, it is possible to move from onelogica utens to another. I can reason like a classical mathematician onMondays, Wednesdays, Fridays, and like a intuitionist on Tuesdays,Thursdays, and Saturdays. (And on Sundays flip a coin.) So practicescan be changed.At this point one might wonder about the nature of inference sketched

in Wittgenstein’s Philosophical Investigations. According to this, correctreasoning 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 musttake the suitable training seriously. I can follow my training as a classicallogician some days, and my training as an intuitionist on others – just asI can follow my training in cricket on some days, and my training inbaseball on others.

3.3 Can it be revised rationally?

So logica utens can change. Can it be changed rationally? Unless one is acomplete relativist about inferential practices, the answer must be yes:some practices are better than others. And to move from one that is lessgood to one that is more good for principled reasons is clearly rational.

Revising logic 219

Moreover, being a relativist about such practices is a hard pill toswallow. For we use reasoning to establish what is true, and what is not,about many things. A relativism about these practices therefore entails arelativism about truth. And such a relativism is problematic. To take anextreme example: suppose that reasoning in one way, we establish that thetheory of evolution is correct, but that reasoning in another way, weestablish that creationism is true and the theory of evolution false. Some-thing, 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 reasoningis (the best docens), and simply bring our practice (utens) into line withthat. How else could one be rational about the matter?

4. Logica ens

4.1 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, toavoid any problems with talk of facts: the truths of the form ‘that so and sofollows from that such and such’. Can these be revised? The matter issensitive for a number of reasons.

As we have seen, our logica docens, with its canonical application, is atheory about what claims of this form are true. Now, if one changes one’stheory of dynamics, the dynamics of the Earth do not themselves change.Such realism about the physical world is simply common sense. But logicis not a natural science. It is a social science, and concerns human practicesand cognition. When a theory changes in the social sciences, the object ofthe science may change as well. One has to look only at economics to seethis. When free-market economics became dominant in the capitalistworld in the 1980s, so did the way that the then deregulated economyfunctioned. So, in the social sciences one is not automatically entitled tothe 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 – forexample, within classical and intuitionist pure mathematics. See (Priest 2013).

220 Graham Priest

But the object of a social scientific theory may not change when thetheory does, for all that. (Many basic laws of psychology are, presumably,hard-wired in us by evolution.) Whether the truth of validity-claims canchange will depend on what, exactly, constitutes validity. Let me illustrate.Suppose that one held a “divine command” theory of validity: something isvalid just if God says so. Then, God being constant and immutable, whatis valid could not change. On the other hand, suppose that one were tosubscribe to the “dentist endorsement” view of validity: what is valid iswhat 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 whatvalidity actually is. An adequate answer to our question would thereforerequire us to settle the issue of what validity is, that is, to determine thebest 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 themodel-theoretic and proof-theoretic accounts of validity. According tothe first, an inference is valid iff every model of the premises is a modelof the conclusion. But a model is a structured set, that is, an abstractobject, the premises form a set, another abstract object, and the premisesand conclusions themselves are normally taken to be sentence types, alsoabstract objects. According to the second, an inference is valid if there is aproof structure (sequence or tree), at every point of which there is asentence related to the others in certain ways. But a proof structure is anabstract 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 ofclaims about validity cannot change.13 Claims about mathematical objectsare not significantly tensed: if ever true true, always true.

4.2 Can meanings change?

That is not an end of the matter, though. The propositions about validitymay not change their truth values. But we express these in language. Itmight 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 I have said what I think about the matter in (Priest 2006a: ch. 11).13 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.

Revising logic 221

involved may change. If this is the case, then the sentences expressingvalidity claims can change their truth values.

Can meanings change in such a way as to affect truth value? Of coursethey can. When Nietzsche wrote The Gay Science, it was a reference to theart of being a troubadour. Nowadays, one could hear it only as concerninga study of a certain sexual preference. In modern parlance, Nietzsche didnot write a book about (the) gay science.

Now, could there be such change of meaning in the case we areconcerned with? Arguably, yes. In both a proof-theoretic and a model-theoretic account of validity, part of the machinery is taken as giving anaccount of meanings – notably, of the logical connectives (introduction orelimination rules, truth conditions). If we change our theory, then ourunderstanding of these meanings will change. This does not mean that themeanings of the vernacular words corresponding to their formal counter-parts changes. You can change your view about the meaning of a word,without the word changing its meaning. However, if one revises one’stheory, and then brings one’s practice into line with it, in the way whichwe noted may happen, then the usage of the relevant words is liable tochange. So, then, will their meanings – assuming that meaning superveneson use (and some version of this view must surely be right). So thesentences used to express the validity claims, and maybe even whichpropositions the language is able to express, can change.14

It might be thought that this makes such a change a somewhat trivialmatter. Suppose we have some logical constant, c, which has different truthor proof conditions according to two different theories. Can we not justuse two words, c1 and c2, which correspond to these two different senses?Perhaps we can sometimes; but certainly not always: for meanings caninteract. Let me illustrate. Suppose that our logic is intuitionist. Then“Peirce’s law”, ((A! B)! A)! A, is not logically valid. But suppose thatwe now decide to add a new negation sign to the language, which behavesas does classical negation. Then Peirce’s law becomes provable. Theextension is not conservative. Another case: given many relevant logics,the rules for classical negation can be added conservatively, as can thenatural 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 howeverthis is expressed – can itself change. Perhaps it can; and if it does, this adds a whole new dimensionof complexity to our investigation. However, I see no evidence that the meaning of the phrase (asopposed to our theories of what follows from what) has changed over the course of Westernphilosophy. So I ignore this extra complexity here.

222 Graham Priest

addition of both (when appropriate self-reference is available) producestriviality. Meanings, then, are not always “separable”.15

4.3 Can meanings change rationally?

So meanings can change, and not necessarily in a straightforward way. Canthis happen rationally, and if so, how? The answers to these questions areimplicit in the preceding discussion. Suppose we change our logica docensto a rationally preferable one. Suppose that we then change our logica utensrationally to bring it in line with this. Then the meanings of our logicalconstants, and so the language used to express the facts of validity, mayalso change. And the whole process is rational.

5. Conclusion

Let me end by summarising the main conclusions we have reached, andmaking a final observation.A logica docens may be revised rationally, and this happens by the

standard mechanism of rational theory choice. A logica utens may bechanged by bringing it into line with a logica docens; and if the docens ischosen rationally, so is the utens. The answer to the question of whether ornot the logica ensmay change depends on one’s best answer to the questionof what validity is. However, under the model- or proof-theoretic accountsof validity, the answer appears to be: no. This does not mean, however,that the sentences used to express these facts may not change. And arational 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. Andassuming a model- or proof-theoretic account of meaning, the languageavailable to express these may depend on the logica utens. It is clear that wehave a circle. If one were a foundationalist of some kind, one might see thiscircle as vicious: there is no privileged point where one can ground theentire enterprise, and from which one can build up everything else.However, I take it that all knowledge, about logic, as much as anythingelse, is situated.16 We are not, and could never be, tabulae rasae. We canstart only from where we are. Rational revision of all kinds then has toproceed by an incremental and possibly (Hegel notwithstanding) never-ending process.

15 On these matters, see (Priest 2006a: ch. 5). 16 See (Priest to appear).

Revising logic 223

cha p t e r 1 3

Glutty theories and the logic of antinomiesJc Beall, Michael Hughes, and Ross Vandegrift

1. Introduction

There are a variety of reasons why we would want a paraconsistentaccount of logic, that is, an account of logic where an inconsistenttheory does not have every sentence as a consequence. One relativelystandard motivation is epistemic in nature.1 There is a high probabilitythat we will come to hold inconsistent beliefs or inconsistent theoriesand we would like some account of how to reason from an inconsistenttheory without everything crashing. Another motivation, rooted in thephilosophy of logic or language, is that we want a proper account ofentailment or relevant implication, where there is a natural sense inwhich inconsistent claims do not (relevantly) entail arbitrary propos-itions – where not every claim follows from arbitrary inconsistency.2

A third motivation, the one which will occupy our attention here, ismetaphysical or semantic. One might, for various reasons, endorse thatthere are ‘true contradictions’, or as they are sometimes called, truth-value gluts – true sentences of the form φ ^ : φ, claims which are bothtrue and false. We shall say that a glut theorist is one who endorsesglutty theories – theories that are negation-inconsistent – with the fullknowledge that they are glutty.

There are different kinds of metaphysical commitments that can leadone to be a glut theorist. One route towards glut theory arises fromviews about particular predicates of a language or the properties thatthose predicates express. Along these lines, a familiar route towards gluttheory holds that certain predicates like ‘is true’, ‘is a member of ’, or‘exemplifies’ are essentially inconsistent: they cannot be (properly)

1 For work in this tradition, see Rescher and Manor (1970); Schotch et al. (2009); Schotch andJennings (1980).

2 For work in this tradition, see Anderson and Belnap (1975); Anderson et al. (1992); Dunn and Restall(2002); Mares (2004); Slaney (2004).

224

interpreted in a way that avoids there being objects of which thesepredicates are both true and false. Such essentially glutty predicates –everywhere glutty with respect to something if glutty anywhere withrespect to anything – are antinomic, as we shall say. Of course, one neednot hold that a predicate is essentially inconsistent to think that it cangive rise to gluts: there may be only contingently glutty predicates. Forsome predicates, whether they are properly interpreted consistentlyor inconsistently may depend on facts about the world. Priest, forexample, has suggested that predicates like ‘is legal’ and ‘has the rightto vote’ are of this sort; see Priest (2006b). Acceptance of either sort of(essentially or contingently) inconsistent predicates is sufficient forbeing 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 ourbest – and true – theory of the world will inevitably be inconsistent, eventhough we might, for all we know, remain ignorant of the source of theinevitable inconsistency. Indeed, one might have reason to be agnosticabout the source of gluttiness: one is convinced that our best theory of theworld (including truth, exemplification, sets, computability, modality,whatever) will be inconsistent, though also convinced that we will neverbe in good position to pinpoint the exact source of the inconsistency.Agnosticism about the particular predicates responsible for gluttinessremains 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 attendingto the difference between formal and material consequence. Briefly, a logictakes a material approach to consequence when it builds in facts about themeaning of predicates, the properties they express, or the objects thosepredicates are about. A logic takes a formal approach to consequence whenit abstracts away from all of these concerns. There are various ways a logiccould be said to ‘build in’ such facts, and one of our aims below is toexplore these in the context of metaphysical commitments to gluts. Wecarry out our discussion via a comparison of two paraconsistent logics,namely, the logic of paradox (LP) and the logic of antinomies (LA). Theformer is well-known in philosophy, discussed explicitly and widely byPriest (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 familiarglut-free extension of FDE called ‘strong Kleene’ or ‘K3’. See Dunn (1966, 1976), Anderson andBelnap (1975), and Anderson et al. (1992).

Glutty theories and the logic of antinomies 225

equally less explored approach in philosophy, an approach advanced byAsenjo and Tamburino (1975).4

Below, we consider various philosophical motivations that could explainthe logical differences between LA and LP. We shall argue that LA reflectsa fairly distinctive set of metaphysical and philosophical commitments,whereas LP, like any formal logic, is compatible with a broad set ofmetaphysical and philosophical commitments. We illustrate thesepoints below.

The discussion is structured as follows. §§2–3 present the target logics interms of familiar model theory. §4 discusses the main logical differences interms of differences 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 first-order syntax. Thelogical vocabulary is _, :, 8. Constants c0, c1, . . . and variables x0, x1, . . .are the only terms. The set ℙ of predicate symbols is the union oftwo disjoint sets of standard predicate symbols: = fA0, A1, . . .g and = fB0, B1, . . .g. (Intuitively, contains the essentially classical predi-cates and the essentially non-classical, essentially glutty predicates.) Thestandard recursive treatment defines the set of sentences.5

An LA interpretation I consists of a non-empty domain D, a denotationfunction 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) = 〈Pþ, P�〉, where Pþ [ P� = D.

The only difference from the standard LP treatment appears here, in theform of a restriction that captures the distinction between the antinomic(i.e., essentially glutty) and essentially classical predicates:

4 For purposes of accommodating glutty theories, the propositional logic LP was first advanced inAsenjo (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 first-order logic under the samename (viz., LP). What we are calling ‘LA’ is the first-order (conditional-free) logic advanced byAsenjo and Tamburino (1975), which was intended by them to be a first-order extension of Asenjo’sbasic propositional logic. Due to what we call the LA Predicate Restriction (see page 227) LA isn’t asimple first-order extension of Asenjo’s propositional LP – as will be apparent below (see §4).

5 For simplicity, we focus entirely on unary predicates. Both LA and LP cover predicates of any arity,but focusing only on the unary case suffices for our purposes.

226 Jc Beall, Michael Hughes, and Ross Vandegrift

LA Predicate Restriction. For any predicate P:

• if P is in , then the intersection Pþ \ P� must be empty;• if P is in , then the intersection Pþ \ P� must be non-empty.

As above, the Ais are the essentially classical predicates, while the Bis arethose which are antinomic.6

|φ|v is the semantic value of a sentence φ with respect to a variableassignment v, which is defined in the standard recursive fashion. (We leavethe relevant interpretation implicit, as it will always be obvious.) For atomics:

jPt jv ¼0 if I ðtÞ =2 Pþand I ðtÞ 2 P�

1 if I ðtÞ 2 Pþand I ðtÞ =2 P�1

2otherwise:

8><

>:

The inductive clauses are as follows:

1. |φ _ ψ|v = maxf|φ|v |ψ|vg.2. |:φ|v = 1 � |φ|v.3. |∀xφ|v = minf|φ|v0: v0 is an x-variant of vg.Conjunction and existential quantification can be defined from these inthe normal way.LA consequence ‘LA is defined as preservation of designated value,

where the designated values are 1 and 12. Thus, Γ ‘LA φ holds (i.e., Γ

implies/entails φ according to LA) if and only if no LA interpretationdesignates 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 modeltheory of LP, there’s no difference 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 (justlet jφjv ¼ 1

2 and jψjv = 0, for at least some formulae φ and ψ), there aresome obvious but noteworthy formal differences between the logics LA

6 The presentation in Asenjo and Tamburino (1975) is rather different; but we present their account ina way that affords clear comparison with LP.

Glutty theories and the logic of antinomies 227

and LP. LP permits the existence of a maximally paradoxical object – anobject of which every predicate is both true and false – whereas LA doesnot. Indeed, LA – but not LP – validates ‘explosion’ for certaincontradictions; for example, for any Ai in and any φ,

Ait ^ :Ait ‘LA φ:

Similarly, LA validates the parallel instances of detachment (modusponens):

Ait ,Ait ⊃ φ‘LAφ

where φ ⊃ ψ is defined as usual as :φ _ ψ. But LP is different: not even arestricted version of detachment is available; see Beall et al. (2013a).As a final and nicely illustrative example, LA validates some existential

claims that go beyond those involved in classical logic (e.g., 9x(φ _ :φ),etc.), whereas LP does not. To see this, note that, for any Bi in , thefollowing is a theorem of LA:

9xBix:

Since any predicate Bi in must have at least some object in the intersec-tion of its extension and anti-extension, it follows that something is in itsextension.

4.1 Metaphysics, formal and material consequence

How are we to understand the logical differences between LA and LP? Fora first pass, they might be naturally understood as arising from differentnotions of consequence: namely, material and formal consequence. Thedistinction may not be perfectly precise, but it is familiar enough.7

Material consequence relies on the ‘matter’ or ‘content’ of claims, whileformal consequence abstracts away from such content. Example: there isno possibility in which ‘Max is a cat’ is true but ‘Max is an animal’ is nottrue; the former entails the latter if we hold the meaning – the matter, thecontent – of the actual claims fixed. But the given entailment fails if weabstract away from matter (content), and concentrate just on the standardfirst-order form: Cm does not entail Am.

The notion of formal consequence delivers conclusions based on logicalform alone. Material consequence essentially requires use of the content ofthe 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 logicalconsequence, and also for further references.

228 Jc Beall, Michael Hughes, and Ross Vandegrift

One way to understand Asenjo and Tamburino’s proposal is that it givesa material consequence relation of a language arising from certain meta-physical commitments. It is clear that the logic reflects an assumption thatcertain predicates are essentially classical, and other predicates are essen-tially glutty – antinomic, as we have said. On this interpretation, theincorporation of essentially classical predicates reflects a metaphysicalcommitment that gluts cannot arise absolutely anywhere. Similarly, thesemantic restriction on the Bi predicates reflects a metaphysical commit-ment that certain predicates, in virtue of their meaning, or the propertiesthey express, must give rise to gluts: there is bound to be at least someobject of which Bi 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 inconsistentinterpretation of the truth predicate, but this is compatible with thecommitment to the essential classicality of all predicates in the truth-freefragment 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 metaphysicalcommitments? LP, we suggest, provides the formal consequence relationof 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 iseither antinomic or essentially classical depends on its meaning. If weignore content, and focus just on purely formal features of sentences,LA’s predicate restriction falls away as unmotivated. And that’s preciselywhat happens in LP: if we abstract away to ‘pure form’ then the content ofpredicates 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 allpredicates are on par with respect to (in-) consistency: each might be gluttywith respect to something – or not glutty at all. If one held this commit-ment, then LA’s predicate restriction is inappropriate, or at least unmoti-vated. Indeed, even one predicate which is either contingently consistentor contingently inconsistent arrests the motivation for LA’s predicaterestriction. And one might think that ordinary cases of such predicatesare not hard to find. Priest (2006b, Ch. 13), for example, discusses suchcases arising from considerations of the law. Suppose that we had laws thatall citizens have a right to vote and no felons have a right to vote. It is then acontingent matter whether or not there are any gluts about rights to vote;it depends on whether anyone commits any felonies, and whether or notanything is classified as a felony. And you might hold a view wherein all

Glutty theories and the logic of antinomies 229

predicates are like that: potentially glutty, one and all, but none anti-nomic – none essentially glutty.

There is another metaphysical route to LP. We might not start with anycommitments about the nature of any predicates, their meaning, theproperties they express, and whether or not they are essentially inconsist-ent. One might start with the commitment that one’s theory is both trueand inconsistent, while remaining agnostic about where to locate theorigins of the inconsistency. There is no reason to think that this positionexcludes a material approach to consequence. It’s just that such a viewlacks any particular metaphysical commitments that would motivate arestriction on predicates like the LA predicate restriction.

Of course, from the material point of view, LA and LP far from exhaustthe possibilities. So far we’ve mentioned fairly strong, all-or-nothingapproaches. On a material approach to consequence, the proponent ofLA is committed to all predicates being essentially classical or glutty, whilethe proponent of LP is committed to all predicates being potentiallyclassical or glutty. Mixed approaches are available. These are achieved byadding obvious combinations to the LA predicate restriction – forexample, some antinomic, some essentially classical, some neither, etc.We leave these to the reader for exploration.

We turn (briefly) 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-ponens-satisfying) conditional definable 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 canbe improved by shifting focus to the material notion of consequence. Butthis is not quite right. Though one fragment of LA differs from LP in thatit satisfies detachment, LA is like LP in that detachment doesn’t holdgenerally: 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 tothe base framework to overcome such non-detachment.8 But the remedy

8 Until very recently, Beall (2013), all LP-based glut theorists focused their efforts on the given task:adding logical resources to the base LP framework to overcome its non-detachment. Whether this isthe appropriate response to the non-detachment of LP is something we leave open here.

230 Jc Beall, Michael Hughes, and Ross Vandegrift

offered by Asenjo and Tamburino doesn’t work, as we now brieflyindicate.Asenjo and Tamburino define a conditional! that detaches (i.e., φ and

φ ! ψ jointly imply ψ). The conditional is intended to serve the ultimatepurpose of the logic, namely, to accommodate paradoxes in non-trivialtheories (e.g., theories of naïve sets), and is defined thus:

jφ ! ψjv ¼0 if jψjv ¼ 0 and jφjv 2

1

2, 1

n o

1

2if jψjv ¼

1

2and jφjv 2

1

2, 1

n o

1 otherwise

8>>><

>>>:

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

φ, φ ! ψ ‘LA!ψ:

The trouble, however, comes from Curry’s paradox. Focusing on the set-theoretic version (though the truth-theoretic version is the same), Meyeret al. (1979) showed that, assuming standard structural rules (which are inplace in LP and LA! and many other logics under discussion), if aconditional detaches and also satisfies ‘absorption’ in the form

φ ! ðφ ! ψÞ ‘ φ ! ψ

then the given conditional is not suitable for underwriting naïve founda-tional principles. In particular, in the set-theory case, consider the set

c ¼ fx : x 2 x ! ⊥gwhich is supposed to be allowed in the Asenjo and Tamburino (andvirtually all other) paraconsistent set theories.9 By unrestricted comprehen-sion (using the new conditional, which is brought in for just that job),where $ is defined from ! and ^ as per usual, we have

c 2 c $ ðc 2 c ! ⊥Þ:But, now, since the Asenjo–Tamburino arrow satisfies the given absorp-tion rule, we quickly get

c 2 c ! ⊥

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

Glutty theories and the logic of antinomies 231

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

c 2 c:

But the Asenjo–Tamburino arrow detaches: we get ⊥, utter absurdity.The upshot is that while LA may well be sufficient for standard first-

order connectives, the ‘remedy’ for non-detachment (viz., moving to LA!)is not viable: it leads to absurdity.10 Other LP-based theorists, notablyPriest (1980) and subsequently Beall (2009), have responded to the non-detachability of LP by invoking ‘intensional’ or ‘worlds’ or otherwise ‘non-value-functional’ approaches to suitable (detachable) conditionals. Weleave the fate of these approaches for future debate.11

6. Closing remarks

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

10 We note that Asenjo himself noticed this, though he left the above details implicit. We have notbelabored the details here, but it is important to have the problem explicitly sketched.

11 We note, however, that Beall has recently rejected the program of finding detachable conditionalsfor LP, and instead defends the viability of a fully non-detachable approach (Beall (2013)), but weleave this for other discussion.

12 We note that Priest’s ultimate rejection of LP in favor of his non-monotonic LPm (elsewhere called‘MiLP’) reflects a move ‘back’ in the direction of the original Asenjo–Tamburino approach, whereone has ‘restricted detachment’ and the like, though the latter logic (viz., LA) is monotonic. Weleave further comparison for future debate. For some background discussion, see Priest (2006,Ch. 16) and Beall (2012) for discussion.

232 Jc Beall, Michael Hughes, and Ross Vandegrift

cha p t e r 1 4

The metaphysical interpretation of logical truthTuomas E. Tahko

1. Two senses of logical truth

The notion of logical truth has a wide variety of different uses, hence it is notsurprising that it can be interpreted in different ways. In this chapter I willfocus on one of them – what I call the metaphysical interpretation. A moreprecise formulation of this interpretation will be put forward in what follows,but I wish to say something about my motivation first. Part of my interestconcerns the origin or ground of logic and logical truth, i.e., whether logic isgrounded in how the world is or how we (or our minds) see the world.1

However, this is notmy topic here. Rather, I will assume that logic is groundedin how the world is – a type of realism about logic – and examine the status oflogical truth from the point of view of logical realism. The upshot is aninterpretation of logical truth that is of special interest to metaphysicians.2

My starting point is the apparent difference between what we might callabsolute truth and truth in a model, following Davidson (1973). The notionof absolute truth is familiar from Tarski’s T-schema: ‘Snow is white’ is trueif and only if snow is white – in the world and absolutely. Instead of beinga 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 sentencesand models.3 Davidson suggested that philosophy of language shouldbe interested in absolute truth exactly because relative truth does not yieldT-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 groundedeither in the mind or in the world, and defends that it is grounded in both – hence logic has a dualnature. See also the opening chapter of this volume.

2 See Chateaubriand Filho (2001, 2005) for a version of the metaphysical interpretation of logical truthpartly similar to mine.

3 ‘Models’ are to be interpreted in a wide sense: they may for instance be interpretations, possibleworlds, 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 isnot directly relevant for my purposes. However, see Shapiro (2000) for an interesting discussion ofQuine on logical truth.

233

To clarify, relative truth is an understanding of logical truth in terms oftruth 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 isrealist about the models, as the models can be understood interpretationallyor representationally, along the lines suggested by John Etchemendy (1990).We will discuss the difference between these views in the next section, butultimately none of these alternatives are expressive of the metaphysicalinterpretation of logical truth. Instead, we need a way to express absolutetruth, 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 MichaelDummett’s view would be discussed, or at least used as a foil, butI prefer not to dwell on Dummett. The primary reason for this is thatDummett’s methodology is entirely opposite to the one that I use. Here isa 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 ourlanguage. We must not try to resolve the metaphysical questions first, andthen construct a meaning-theory in the light of the answers. We shouldinvestigate how our language actually functions, and how we can constructa workable systematic description of how it functions; the answers to thosequestions will then determine the answers to the metaphysical ones.(Dummett 1991a: 338)

Since I am analyzing logical truth from a realist, metaphysical point ofview, Dummett’s methodology is obviously not going to do the trick. Inmy view, there is a bona fide discipline of metaphysics and I am interestedin finding a use for logical truth within that discipline. I doubt there isenough initial common ground to fruitfully engage with Dummett.

Let me briefly return to Davidson and Tarski before proceeding. Whenconsidering the distinction between absolute and relative truth, an initialpoint of interest is absolute truth’s characterization by the T-schema. Onequestion that emerges is the connection between the T-schema andmetaphysics. A likely approach is to explicate this connection in terms ofcorrespondence. However, at least according to one reading, Tarski (1944)considered truth understood as a semantic concept to be independent ofany 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 correspond-ence theories of truth, despite the appearance of a correspondence relation

234 Tuomas E. Tahko

between sentences and the world.5 Yet, Tarski’s (1944: 342–343) initialconsiderations on the meaning of the term ‘true’ explicitly take intoaccount an ‘Aristotelian’ conception of truth, where correspondence withthe world is central. Davidson (1973: 70) as well seems to have somesympathy for the idea that an absolute theory of truth is, in some sense,a ‘correspondence theory’ of truth, although he insists that the entities thatwould act as truthmakers here are ‘nothing like facts or states of affairs’, butsequences (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 anunderstanding of the correspondence relation which is neutral in terms ofour theory of truth. It is this type of weak correspondence intuition thatI believe central to the metaphysical interpretation of logical truth. But itshould be stressed that the correspondence intuition itself is not necessarilyexpressive of realism (Daly 2005: 96–97). For instance, Chris Daly’ssuggested definition of the intuition is simply that a proposition is true ifand only if things are as the proposition says they are. Daly explains theneutrality of (his version of ) the correspondence intuition as follows:6

Consider the coherence theorist. He may consistently say ‘If <p> is true, ithas a truthmaker. <p> corresponds to a state of affairs, namely the state ofaffairs which consists of a relation of coherence holding between <p> andthe other members of a maximal set of propositions’. Consider the pragma-tist. He may consistently say, ‘If <p> is true, it has a truthmaker. <p>corresponds to a state of affairs, namely the state of affairs of <p>’s havingthe property of being useful to believe’. It is controversial whether thereexist states of affairs. Let that pass. My point here is that the coherencetheory and the pragmatic theory are each compatible with the admission ofstates of affairs. Furthermore, each of these theories is compatible with theadmission of states of affairs standing in a correspondence relation to truths.(Daly 2005: 97)

A neutral version of the correspondence intuition is desirable because I donot want to rule out the possibility of different approaches to truth, despiteassuming realism in the present context. A central appeal of the corres-pondence intuition is, I suggest, its wide applicability. However, a slightly

5 Furthermore, the idea that the T-schema or the correspondence theory are somehow expressive ofrealism has been forcefully disputed. See for instance Morris (2005) for a case against the connectionbetween realism and correspondence; in fact Morris argues that correspondence theorists should beidealists. See also Gómez-Torrente (2009) for a discussion about Tarski’s ideas on logicalconsequence as well as on Etchemendy’s critique of Tarski’s model-theoretic account.

6 The angled brackets describe a proposition, following Horwich (1998).

The metaphysical interpretation of logical truth 235

better formulation than Daly’s can be found by following Paolo Crivelli(2004), who interprets Aristotle as an early proponent of the correspond-ence theory. Crivelli defines correspondence-as-isomorphism as follows: ‘atheory of truth is a correspondence theory of truth just in case it takes thetruth 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 ofpropositions, or indeed states of affairs.8 Hence, we may define thecorrespondence intuition as follows:

(CI) A belief, or an assertion, is true if and only if its content is isomorphicwith reality.

This formulation preserves Daly’s idea. ‘Reality’ in CI may consist, say, ofwhat it is useful to believe, as the pragmatist would have it, so neutrality ispreserved. If we accept that CI is neutral in terms of different theories oftruth, then we can characterize the issue at hand as follows. There is anapparent and important difference between truth understood along thelines of CI, and truth understood as a relation between sentences andmodels. I take this to be at the core of Davidson’s original puzzle concern-ing absolute and relative truth. We ought to inquire into these two sensesof truth before we give a full account of logical truth. This is exactly whatI propose to do, arguing that the metaphysical interpretation of logicaltruth must respect CI.

Tarski and the model-theoretic approach may have made it possibleto talk about logical truth in a manner seemingly independent ofmetaphysical considerations, but important questions about the meta-physical status of logical truth and the interpretation of models remain.One thing that makes this problem topical is the recent interest inlogical pluralism, or pluralism about logical truth (e.g., Beall and Restall2006). In the second section I will assess the metaphysical status of thenotion of logical truth with regard to the two senses of truth familiarfrom Davidson. The third section takes up the issue of interpretinglogical truth in terms of possible worlds and contains a case study of the

7 Crivelli also defines a stricter sense of correspondence, which can be found in Aristotle. Butsometimes Aristotle’s view on truth is also considered as a precursor to deflationism about truth,so we shouldn’t put too much weight on the historical case. For a more historically inclineddiscussion, see Paul Thom’s chapter in this volume.

8 Admittedly, once we explicate isomorphism, reference to propositions, states of affairs or somethingof the sort could easily re-emerge. This shouldn’t worry us too much, because it is likely that we wanta structured mapping from something to reality. The reason to opt for isomorphism here is merely tokeep the door open for one’s preferred (structured) ontology.

236 Tuomas E. Tahko

law of non-contradiction. A brief discussion of logical pluralism willtake 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, theabsolute and the relative? As Etchemendy (1990: 13) notes, the obvious wayto attempt this would be in terms of generalization: if absolute truth is amonadic predicate of the form ‘x is true’, then it may be helpful to analyzeit in terms of a relational predicate of the form ‘x is true in y’, for instance‘x is a brother’ could be analyzed by first analyzing ‘x is a brother of y’, thususing the generalized concept of brotherhood. However, this does notapply to truth: ‘[C]learly the monadic concept of truth, the concept weordinarily employ, is no generalization of any of the various relationalconcepts. A sentence can be true in some model, yet not be true; asentence can be true, yet not be true in all models’ (1990: 14). Accordingly,generalization will not help in reconciling the two senses of truth.Another alternative that Etchemendy considers is to interpret absolute

truth as a specification of truth in a model, namely, absolute truth could beconsidered equivalent to truth in the right model, the model that corres-ponds with the world. This maintains the correspondence intuitionexpressed by CI above, but note that ‘correspondence with the world’already suggests a realist theory of truth, so the neutrality of the formula-tion is in question.9

However, there are good reasons to think that the notion of ‘model’ isnot entirely appropriate when discussing absolute truth, as it is closelyassociated with relative truth. Hence, interesting as Etchemendy’s charac-terization may be, it is unlikely to result in a metaphysical account of logicaltruth. Still, Etchemendy’s account may help pinpoint the issue; considerthe following passage:

Once we have specified the class of models, our definition of truth in amodel is guided by straightforward semantic intuitions, intuitions about theinfluence of the world on truth values of sentences in our language. Ourcriterion here is simple: a sentence is to be true in a model if and only if itwould have been true had the model been accurate – that is, had the worldactually 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 thelogician. For instance, as Burgess (1990: 82) notes, it is the metaphysician’s task to determine thecorrect 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.

The metaphysical interpretation of logical truth 237

There is an important requirement in the passage above, namely, it mustbe the case that the model could have been true. How do we interpret themodality in effect here? If we understand it as saying that it must be thecase that the world could have turned out to be like the model depicts,then this supports the case for a metaphysical interpretation of logicaltruth, for it introduces as a requirement for the notion of ‘model’ that it isa possible representation of the world. This representational approach, or‘representational semantics’ can be contrasted with ‘interpretationalsemantics’, which Etchemendy discusses later on:

[I]n an interpretational semantics, our class of models is determined by thechosen satisfaction domains; our definition of truth in a model is a simplevariant of satisfaction. (Etchemendy 1990: 50)

Etchemendy claims that the Tarskian conception of model-theoreticsemantics is of the ‘interpretational’ kind, although his interpretation ofTarski can certainly be questioned (e.g. Gómez-Torrente 1999). But I donot wish to enter the debate about Tarski or interpretational semantics.According to Etchemendy, in the representational approach models mustrepresent ‘genuinely possible configurations’ of the world, and I aminterested in the correct understanding of these possible configurations(cf. Etchemendy 1990: 60). However, instead of developing Etchemendy’srepresentational account, I will propose a pre-theoretic account of absolutetruth, which aligns nicely with Etchemendy’s analysis. The biggest compli-cation 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 specifiedthe class of ‘genuinely possible configurations’, we can define relative truthaccording to Etchemendy’s suggestion. In this regard, my analysis will notfollow that of Etchemendy’s, as the case for absolute truth will come beforeEtchemendy’s account. Etchemendy’s representational approach notwith-standing, the notion of ‘model’ is not ideal for this task, as it is stronglyreminiscent of relative truth.10

Instead of ‘models’, I propose to resort to talk of ‘possible worlds’. WhatI have in mind is interpreting possible worlds as metaphysical possibilities.

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

238 Tuomas E. Tahko

This is, of course, somewhat controversial, but as we will see, there arereasons to think that only metaphysical modality is fitting for the task. Inany case, more needs to be said about how the space of metaphysicalpossibilities is restricted. We will return to this in the next section.We are now in the position to define a provisional sense of logical truth

which I propose to call metaphysical:

(ML) A sentence is logically true if and only if it is true in every genuinelypossible configuration of the world.

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

3. Genuinely possible configurations and the caseof the law of non-contradiction

The puzzle can now be expressed in the following form: What sort ofcriteria can be established to evaluate whether a given possible world is agenuinely possible configuration of the world, i.e., could have turned outto correspond with the actual world? Let me approach the problem with acase study. Take, arguably, one of the most fundamental laws of logic, thelaw of non-contradiction (LNC). When I say that the law of non-contradiction is true in the ‘metaphysical sense’, I mean that LNC is truein the sense of absolute truth, i.e., it is a genuine constraint on the structureof reality. The metaphysical formulation of LNC takes a form familiarfrom Aristotle (Metaphysics 1005b19–20), although my proposed formula-tion is somewhat weaker, defined as follows:

(LNC) The same attribute cannot at the same time belong and not belongto the same subject in the same respect and in the same domain.

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

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

The metaphysical interpretation of logical truth 239

repeat my arguments here, but it may be noted that this is not strictly aquestion for logic. For instance, Priest’s most celebrated arguments in favorof true contradictions (in the metaphysical sense) concern the nature ofchange and specifically motion, the paradoxical nature of which is sup-posedly demonstrated by Zeno’s well-known paradoxes. Although theseparadoxes can quite easily be tackled by mathematical means, the relevantquestion is whether change indeed is paraconsistent.12 The answer to thisquestion requires both metaphysical and empirical inquiry. I will return tothis point briefly below, but first I wish to say something about themethodology of logical-cum-metaphysical inquiry.

In terms of ML, demonstrating the falsity of LNC would first require agenuinely possible configuration of the world where LNC fails. That is, itis not enough that we have a model where LNC is not true, such asparaconsistent logic, but we would also need to have some good reasons tothink that the world could have been arranged in such a way that theimplications of the metaphysical interpretation of LNC do not follow. Thispoint deserves to be emphasized, for it would be much easier to show thata paraconsistent model can be useful in modelling certain phenomena, orinterpreted 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 metaphysicalsense, can be shown to fail.

Note that we may also ask whether LNC is necessary, i.e., are there anypossible worlds in which LNC does not hold – even if we did have a goodcase for its truth in the actual world? In fact, this is the question we shouldbegin with, since if LNC is necessary, then it could not fail in the actualworld either. However, it is not clear how we could settle this questionconclusively, given that we are dealing with the metaphysical interpret-ation of LNC. Moreover, I do think that there could (in an epistemicsense) be possible worlds in which LNC fails, and hence I take the debateabout LNC seriously. Yet, I am uncertain about whether such a paracon-sistent possible world is in fact a genuinely possible configuration, as I willgo on to explain.13 In any case, if a possible world in which LNC is not true

a principle regarding structured exclusion relations (between properties, states of affairs, etc.), andthe world is determinate insofar as it conforms to this principle.

12 For discussion regarding Zeno’s paradoxes, see for instance Sainsbury (2009: Ch. 1).13 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 FrancescoBerto, is that the world cannot be a model, because it contains everything, and there’s no domain ofeverything, on pain of Cantor’s paradox. The result is that something can be a logical truth in thesense of being true in all models, without being true in the absolute sense, for the world is not amodel. My proposed treatment of this issue proceeds by understanding absolute truth in terms of

240 Tuomas E. Tahko

were genuinely possible, then LNC would obviously not be necessary.This should be relatively uncontroversial, but I should finally say some-thing more about ‘genuine possibility’.

As was mentioned in the previous section, there are reasons to under-stand genuine possibility in terms of metaphysical possibility, as onlymetaphysical modality could secure the correspondence between a possibleworld and the structure of reality – this is also what CI requires. Therelevant modal space must consist of all possible configurations of theworld and only them. Logical modality cannot do the job because it is notsufficiently restrictive. This can be demonstrated with any traditionalexample of a metaphysical, a posteriori necessity, such as gold being theelement with atomic number 79. Assuming that it is indeed metaphysicallynecessary that gold is the element with atomic number 79, we must be ableto accommodate the fact that gold failing to be the element with atomicnumber 79 is nevertheless logically possible. But since we are interested ingenuinely possible configurations of the world, we ought to rule outmetaphysically impossible worlds, such as the world in which gold failsto be the element with atomic number 79. The upshot is that if we acceptthe familiar story about metaphysical a posteriori necessities of this type,then there are necessary constraints for the structure of reality which logicalnecessity does not capture.14

The only other viable alternative in addition to metaphysical and logicalmodality is conceptual modality, i.e., necessity in virtue of the definitions ofconcepts. Nomological modality is already too restrictive, as we sometimesneed to consider configurations of the world that are nomologically impos-sible but at least may be genuinely possible (e.g., superluminal travel).However, conceptual modality is too liberal, quite like logical modality, asit also accommodates configurations of the world which are not genuinelypossible, such as violations of the familiar examples of metaphysical a poster-iori necessities. If we accept these examples, then neither definitions ofconcepts nor laws of logic rule out things like gold failing to be the elementwith atomic number 79. Accordingly, if one accepts that there are metaphys-ical necessities that are not also conceptually and logically necessary – some-thing that most metaphysicians would accept – the only availableinterpretation of genuine possibility is in terms of metaphysical possibility.

metaphysical modality, but the dialetheist could, in principle, endorse paraconsistent set theory andposit that absolute truth is just truth in the world-model – the model whose domain is the world.

14 I should add that cashing out these constraints is, I think, a much more complicated affair than thetraditional Kripke–Putnam approach to metaphysical a posteriori necessities suggests.

The metaphysical interpretation of logical truth 241

There is, however, a way to understand logical modality which maydo a better job in capturing the relevant sense of logical truth. This typeof understanding has been proposed by Scott Shalkowski, who suggeststhat ‘logical necessities might be explained as those propositions true invirtue of the natures of every situation or every object and property, thuspreserving the idea that logic is the most general science’ (Shalkowski2004: 79). On the face of it, this suggestion respects the criteria forgenuine possibility. According to this approach, logical modality con-cerns the most general (metaphysical) truths, such as the law of non-contradiction when it is considered as a metaphysical principle (as inTahko 2009). In this view, logical relations reflect the relations ofindividuals, properties, and states of affairs rather than mere logicalconcepts. Indeed, this understanding effectively equates metaphysicaland logical modality. The idea is that the purpose of logic is to describethe structure of reality and so it is ‘the most general science’. AsShalkowski (2004: 81) notes, denying the truth of LNC would, in termsof this understanding, amount to a genuine metaphysical attitudeinstead of, say, the fairly trivial point that a model in which the lawdoes not hold can be constructed.

Do we have any means to settle the status of LNC in the suggestedsense? 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 chal-lenged and sometimes falsified even by empirical means, but merely thefact that we can formulate models in which they do not hold is notenough to challenge their truth; it will also have to be demonstrated thatthere are possible worlds which constitute genuinely possible configur-ations of the world. However, this approach seems biased towardshistorically prior logical principles, the ones that were formulated first.It is not implausible that the reason why they were formulated first isbecause they are indeed the best candidates for metaphysically necessaryprinciples: for Aristotle, the law of non-contradiction is ‘the most certainof all principles’ (Metaphysics 1005b22). But this is admittedly quitespeculative – we ought to be allowed to question even the ‘first’principles.

It would certainly be enough to challenge the metaphysical necessity ofLNC, or other logical principles, if empirical evidence to the effect that theprinciple 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 ofchange and Zeno’s paradoxes, but I remain unconvinced. As I have argued(Tahko 2009), Priest’s examples can all be accounted for in terms ofsemantic rather than metaphysical dialetheism – a distinction developedby Edwin Mares (2004). The idea is that there may be indeterminacy insemantics, but this does not imply that there is indeterminacy in the world.Only the latter type of indeterminacy would corroborate the existence of agenuinely possible paraconsistent configuration of the world. Since I havenot seen a convincing case to the effect that such a configuration isgenuinely possible, I take it that LNC is a good candidate for a metaphys-ically necessary principle. If I am right, this means that a paraconsistentpossible world could not have turned out to accurately represent the actualworld. The fact that there are paraconsistent models has no direct bearingon this question. I do not claim to have settled the status of LNC once andfor all, but I think that a strong empirical case for the truth of LNC can bemade, on the basis of the necessary constraints for the forming of a stablemacrophysical 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 (Tahko2009). For instance, as PEP states, it is impossible for two electrons (orother fermions) in a closed system to occupy the same quantum state at thesame time. This is an important constraint, as it is responsible for keepingatoms from collapsing. It is sometimes said that PEP is responsible for thespace-occupying behavior of matter – electrons must occupy successivelyhigher orbitals to prevent a shared quantum state, hence not all electronscan collapse to the lowest orbital. Here we have a principle which capturesa crucial constraint for any genuinely possible configuration of the worldthat contains macroscopic objects. Whether or not there are genuinelypossible configurations that do not conform to PEP is an open question,but it seems unlikely that such a configuration could include stablemacroscopic 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 aclosed system) at the same time. Compare this with Aristotle’s formulationof LNC: ‘the same attribute cannot at the same time belong and notbelong to the same subject in the same respect’ (Metaphysics 1005b19–20).LNC is of course a much more general criterion than PEP – it concerns

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

The metaphysical interpretation of logical truth 243

one thing rather than things of a certain kind – but its underlying role isevident: if any fermion were able to both be and not be in a certainquantum state at the same time, then PEP would be violated and macro-scopic objects would collapse. If LNC is needed to undergird PEP, then wehave a strong case in favor of the metaphysical interpretation of LNC inworlds that contain macrophysical objects, given the necessity of PEP forthe forming of macrophysical objects. This is of course not sufficient toestablish the metaphysical necessity of either principle, but it is an interest-ing result in its own regard.

4. Pluralism about logical truth

Now that we have a rough idea about the metaphysical interpretation oflogical truth, we can consider the implications of this interpretation ina wider context. Here I would like to focus on the topic of logicalpluralism, which has lately received an increasing amount of attention.Perhaps the most influential form of logical pluralism derives frompluralism about logical consequence, i.e., the view that there are modelsin which the logical consequence relation is different, and irreconcilablyso. Beall and Restall have formulated and defended this type ofpluralism:

Given the logical consequence relation defined on the class of casesx, thelogicalx truths are those that are true in all casesx. If you like, they are thesentences that are x-consequences of the empty set of premises. The logicalxtruths are those whose truth is yielded by the class of casesx alone. Since weare pluralists about classes of cases, we are pluralists about logical truth.(Beall and Restall 2006: 100)

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

244 Tuomas E. Tahko

Restall have in mind. Moreover, Field suggests two reasons why model-theoretic accounts are irrelevant to logical pluralism:

One of these reasons is that by varying the definition of ‘model’, thisapproach defines a large family of notions, ‘classically valid’, ‘intuitionisti-cally valid’, and so on; one needn’t accept the logic to accept the notion ofvalidity. A classical logician and an intuitionist can agree on the model-theoretic definitions of classical validity and of intuitionist validity; whatthey disagree on is the question of which one coincides with genuinevalidity. For this question to be intelligible, they must have a handle onthe idea of genuine validity independent of the model-theoretic definition.Of course, a pluralist will contest the idea of a single notion of genuinevalidity, and perhaps contend that the classical logician and the intuitionistshouldn’t be arguing. But logical pluralism is certainly not an entirely trivialthesis, whereas it would be trivial to point out that by varying the definitionof model one can get classical validity, intuitionist validity, and a wholevariety 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 casecorresponding to the actual world. There is no obvious reason why asentence 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 instancesof excluded middle as true in all classical models, while doubting that theyare true in the real world. (Field 2009: 348; italics original)

Field goes on to suggest that Beall and Restall must have meant that thereis an implicit requirement for interpreting ‘truth in a case’, namely, thattruth in all cases implies truth. Field then argues that this will not producean interesting sort of logical pluralism as the pluralist notion of logicalconsequence suggested by Beall and Restall does not capture the normalmeaning of ‘logical consequence’. But it should be noted that Beall andRestall (2006: 36 ff.) do say something about the matter. Specifically, theysuggest that on one reading of ‘case’ (the TM account), Tarskian modelsare 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 anargument entails the existence of an actual (abstract) model that invalidatesthe 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 theTM account it is a Tarskian model and on the NTP account it is a possibleworld. The latter is of immediate interest to us, given that the metaphysical

The metaphysical interpretation of logical truth 245

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 NTPaccount 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 possibleworlds, distinguished in terms of logical truths that are true in each set ofpossible worlds. Only one possible world is actual, but the logical truthsthat are true in the actual world will also be true in all worlds which are inthe same set of possible worlds, i.e., these worlds may differ in otherregards, but they are close to the actual world in the sense that all thelogical truths are shared.

Accordingly, pluralists about logical truth, in the metaphysical sense,hold that there are distinct sets of possible worlds in which different logicaltruths hold. The metaphysical interpretation of logical truth can accom-modate this sense of logical pluralism, provided that possible worlds areinterpreted appropriately – this also enables us to preserve CI.16 However,accommodating pluralism in the metaphysical interpretation of logicaltruth does require a revision in our original definition (ML), which defineda sentence as logically true if and only if it is true in every genuinelypossible configuration of the world. Since in this view of logical pluralismthere can be proper subsets of genuinely possible configurations withdifferent 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 possibleworld of a given subset of possible worlds representing genuinely possibleconfigurations of the world.

ML-P can of course also accommodate the situation where the laws oflogic are the same across all subsets of genuinely possible configurations,i.e., logical monism – in that case the relevant subset of possible worldswould not be a proper subset of the genuinely possible configurations.

An alternative formulation of ML-P is possible, dismissing subsetsaltogether. We could understand logical pluralism by giving differentinterpretations to ‘genuinely possible configurations’.17 This formulation

16 Why is interpreting logical truth on the basis of metaphysical possibility the only way to preserveCI? Because we’ve seen that only by restricting our attention to metaphysically possible worlds canwe preserve a sense of correspondence between logical truth and genuinely possible configurations ofthe world. Only metaphysically possible worlds are sufficiently constrained to take into account allthe governing principles such as metaphysical a posteriori necessities.

17 Thanks to Jesse Mulder for suggesting this type of formulation.

246 Tuomas E. Tahko

could be developed by adopting a line of thought from Gillian Russell(2008). Russell suggests that we can distill a sense of pluralism by under-standing logical validity as the idea that in every possible situation in whichall the premises are true, the conclusion is true (2008: 594), where possibility isambiguous between logical, conceptual, nomological, metaphysical, orother senses of modality, hence producing a similar ambiguity concerningvalidity. A friend of the metaphysical interpretation of logical truth couldaccept this idea, but only provided that we prioritize the reading wherepossible situations reflect metaphysical possibility, as CI is preserved onlyin this reading. Nevertheless, there may still be room for a type ofpluralism concerning metaphysical possibility and hence genuinely pos-sible configurations. Unfortunately I have no space to develop thisapproach 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 ofpluralism is limited. Given that I consider there to be strong reasons tothink that LNC holds in the actual world, we can define a set of possibleworlds in which the law of non-contraction holds, call it WLNC. Theassumption is that WLNC includes the actual world. But since I have madeno mention of any other laws of logic that hold (in the metaphysical sense)in WLNC, the sense in which we can talk of a logic may be questioned. Inother words, it may be wondered if the resulting sense of logical pluralismis 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, wecan extend the metaphysical interpretation to other laws of logic as well insuch a way that a subset of WLNC may be defined. This is not quite asstraightforward in other cases though.Very briefly, consider modus ponens (A ^ (A! B))! B. If thought of

as a rule, it is not obvious that modus ponens can be applied to the worldin the sense that I have suggested with regard to LNC. Yet, there are clearcases of physical phenomena that feature a modus ponens type structure.As a first pass, causation might be offered as a candidate of ‘real worldmodus ponens’, but there are obvious complications with this suggestion,as it depends on one’s theory of causation. However, there are bettercandidates. 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 ExclusionPrinciple. In particular, since the electrons cannot be in the same quantumstate at the same time, we know that the only way for them to occupy thesame orbital (i.e., having the same orbital quantum numbers) is for them

The metaphysical interpretation of logical truth 247

to differ in spin (i.e., to have different spin quantum numbers). Accord-ingly, when we observe electron A having spin-up, we immediately knowthat any electron, B, in the same orbital as A must have spin-down.Moreover, there can be only two electrons in the same orbital and theymust 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 alogic, enabling the suggested interpretation of logical pluralism. Theresulting subset could be called WLNCþMP.This hardly exhausts the debate about logical pluralism, but it appears

that there are ways, perhaps several ways, to accommodate pluralism aboutlogical truth within the metaphysical interpretation.

5. Conclusion

In conclusion, I have demonstrated that there is a coherent metaphysicalinterpretation of logical truth, and that this interpretation has someinteresting uses, such as applications regarding logical pluralism. It hasnot been my aim to establish that this interpretation of logical truth is thecorrect one, but only that it is of special interest to metaphysicians. I haveassumed rather than argued for a type of realism about logic for thepurposes of this investigation, but I contend that for realists about logic,one interesting interpretation of logical truth is the one sketched here.18

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

248 Tuomas E. Tahko

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Index

admissible inference rule, 110, 113–114, 116analyticity, 195–196, 200, 202Aristotelian, 147, 204

categories, 149–150, 152–153Aristotle, 6–8, 22, 54, 117, 139, 147–150, 205, 207,

212–213, 239, 242–243Crivelli interpretation of, 236notion of validity, 204, 206–207, See Bolzano

Axiom of Choice, 72, 86, 88, 90

Beall, Jc, 229–230, 232Beall and Restall, 245Beall and Restall’s pluralism, 50, 69–70, 236,

244–246, See Restall, GregBeall, Hughes and Vandegrift, 9Priest and Beall, 232

bounded arithmetic, 116Burgess, J.A., 50–51, 57, 237

Carnap, R, 69, 233Dummett-Quine-Carnap, 69–70

classical validity, 59, 114, 245cognitive command, 58–60, 62–63, 65–66,

68–71completeness, 80–83, 114, 116, 123, 180–181

theorem, 42, 64, 80, 84, 181, See GödelCondillac, 122, 127conditional

logic, 107material, 106–107, 218the, 105, 107, 231

consistency, 51–52, 55, 58–60, 79, 115, 118–119,141–142, 144, 188

constructive, 82–83, 115–116, 119, 125mathematics, 74, 108semi-constructive, 86

constructivism, 74, 82contextualism, 49, 66–67, 69–70, 257continuum, 29, 54, 64, 76–80, 90,

97, 105Continuum 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, 115Lewis account of, 34–35tacit, 34, 36–38, 40–41truth by, 32, 34, 47–48

conventionalismlinguistic, 35logical. See logical conventionalism

conventions, 33–35, 45, 191explicit, 33–35logical, 32–33optimality of, 36–38, 44–45tacit, 34–35

correspondence, 44–45, 195, 234–237, 241, 2461-1, 89

criteria forvalidity, 8

criterion fora philosophy of mathematics, 89legitimacy, 52mathematical legitimacy, 55rule-following, 131validity, 169

Darapti, 213Davidson, D, 57, 233–234, 236, 244dialetheism, 108

metaphysical, 243semantic, 243

disjunction property, 110–113Dummett, M, 59–60, 68–70, 128, 140–141, 234

Eddington, A.S., 95–97, 99epistemic constraint, 58–61, 70Etchemendy, J, 234–235, 237–238Explosion, 107, 213–214, 228

Feferman, S, 3–4, 75–77, 79–82, 86, 88–89,91–92, 105–106

264

Field, H, 211, 244–245first-order logic, 42–43, 61, 64, 74, 80, 82, 213,

226Føllesdal, D, 20, 92formalism, 74Frege, G, 1, 41, 94, 128, 131, 157, 195, 214

and Russell, 129, 211correspondence with Hilbert, 51–52Fregean, 22realism, 196

Gentzen, G, 111–112, 116, 122–124meaning, 85proof, 118–120system of natural deduction, 81, 84

geometry, 8, 52, 93, 123, 128, 136, 208application, 136, 215axioms, 52Euclidean, 41, 77, 128, 215non-Euclidean, 186, 215

Gödel, K, 87, 92, 111, 116, 118–119, 142, 191completeness theorem, 42, 80incompleteness theorem, 79

Goldbach conjecture, 76, 139–141

Hatfield, G, 102Hilbert

Hilbertian, 8Hilbert, D, 51–52, 110, 118–119, 211

Hilbertian, 52, 55Hilbert’s program, 142space, 72, 104

Husserl, E, 17, 25–28, 30cognition, 22concept of evidence, 23–24conception of logic, 18–19Husserlian, 26logical realism, 190phenomenological reduction, 19–21transcendence, 17

idealization, 71, 106–107classical logic, 106–108, See logic, classicalof rudimentary logic, 5, See logic, rudimentarytechnique of, 105–106throughout mathematics, 61

implication, 41, 73intuitionistic, 125

incompleteness, 113–114theorem, 79, See Gödel

independence, 3, 13conceptions of, 26essential and modal, 15human-, 2, 15IF Independence Friendly, 82

mind-, 20, 56of facts, 14of logic, 7of logical truth, 29proofs, 51realist, 3, 15–18results, 83

intuitionism, 115, 140intuitionist validity, 245intuitionistic, 69, 74, 112analysis, 52, 54consistency, 59intuitionistic logic, 116intuitionistically, 45, 70, 113logic, 46, 50, 52, 54, 57, 60–61, 63–64, 74, 77,

82–83, 108, 111–112predicate calculus, 84–86propositional calculus, 110semi-, 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-, 198ethics, 184–185Kantian, 7, 179, 181, 183Kant-Quine, 58, 71

KF-structure, 94

Ladyman, J, 99Ladyman, J and Ross, D, 99, 104language acquisition, 40, 103law of excluded middle (LEM), 29, 50, 65, 74,

87–88, 90, 139–142, 144weak, 115

law of non-contradiction (LNC), 9, 29, 48, 239,242

Lewis Carroll regress, 33logicapplied, 215canonical application, 2, 215–216, 220classical, 4–5, 42–45, 50, 52–53, 77, 81, 84,

86–88, 107, 111–113, 115, 211, 214, 216, 218,228

application to mathematics, 91idealization, 106–108rise of, 214valid in, 60–61, 63

conditional, 107content-containment model of, 42deviant, 69, 104, 106–107intuitionist, 215–216, 219mathematical, 72–73, 212, 214, 217

Index 265

logic (cont.)medieval, 147, 158–159, 161, 164, 212–214Megarian, 212non-classical logic, 5, 49, 211, 218paraconsistent, 50, 54–55, 64, 68, 108, 211, 215,

225, 240, See paraconsistencyPort Royale, 213pure, 18, 178, 215–216relevance, 54, 107rudimentary, 5, 95, 97–100, 104–108, 120–121, 125rule-governed model of, 41–42semi-intuitionist, 4substitution model of, 42traditional, 214, 217

logica docens, 212–216, 218, 220, 223logica ens, 212, 216, 220, 223logica utens, 212, 218–219, 223logical

connectives, 23, 115–116, 222consequence, 8, 51, 59–60, 79, 109–110, 112,

123, 235Beall and Restall’s, 50, 244–245Bolzano, 203–204, 207in mathematical practice, 43material approach to, 232Read’s defense of material, 228traditional definition of, 192

conventionalism, 3, 33, 47, 190inference, 5, 41, 93, 100, 134pluralism, 4, 9, 217, 237, 244–248realism, 4, 8, 13–15, 189–192, 195–197, 208, 233schemata, 41

logical validity, 50, 56, 121–123, 161, 237, 247logicism, 74

MacFarlane, J, 51, 65–67Maddy, P, 5, 121mathematical

objects, 1, 90, 221proof, 42, 120, 123realism, 14reality, 1, 15

McDowell, J, 25–30meaning

of a mathematical proposition, 137of ‘all’, 78of logical operations, 24, 85of logical particles, 112of logical predicates, 225, 228of logical terms, 69, 135of ‘proposition’, 150of spoken and written utterances, 148–149of ‘stateable’, 155

Medvedev lattice, 115metalogical, 45

metalogical debates, 48mirror neuron, 39model theory, 44–45, 226–227

model-theoretic, 42, 64, 80, 204, 207, 235–236,238, 245

account of validity, 221–223modus ponens, 43, 48, 95, 113, 136, 189–190, 214,

228, 230, 247–248monism, 51, 54, 62, 217, 246

naturalism, 3–4, 74, 189necessity, 42, 159, 186, 204–207, 241,

See possibilitycausal, 174epistemic, 208follows of, 205logical, 35, 134, 174, 241metaphysical, 242, 244natural, 175semantical, 181

non-realist, 3–4, 74norms of reasoning, 158

objectivity, 3, 14, 56–60, 65–66, 71, 174, 179–180,184, 186

axes of, 62criteria of, 183of logic, 7of mathematics, 76

open-texture, 71

paraconsistency, 9–10, See logic, paraconsistentPeano Arithmetic, 73, 87Piaget, 100Plato, 18, 122, 162, 164, 166, 168, 176, 184platonic, 18, 25, 74, 97, 162platonism, 5, 135, 139–140platonist, 136, 147, 221pluralism, 9, 49, 51, 69, 189, 247, See logical:

pluralismpossibility, 70, 197, 228, 247, See necessity

genuine, 241logical, 174metaphysical, 241, 247of cognition, 13, 16–17, 21–22of logic, 32

Priest, G, 3, 9, 158, 225–226, 229–230, 232arguments against LNC, 240, 243

principle of bivalence, 32principles and parameters model, 39

quantum mechanics, 98–99, 104, 108, 144Quine, W.V.O.

on second-order logic, 124substitutional procedure, 195

266 Index

Quine, W.V.O., 32, 81, 100, 124, 135, 166challenge, 32–35, 38–40, 48Dummett-Quine-Carnap, 68–69holism, 144Kant-Quine, 58, 71Putnam and Davidson, 57

Quinean, 178, 181, 216

rationality, 168, 184relativism, 49, 51, 129, 190, 220, 238

folk-, 49–51, 57–60, 63, 65–66, 68, 70–71, 190logical, 49, 51proper, 67–68, 70

rule-following, 61, 129–132, 137, 142

second-order logic, 61, 64, 73, 79, 81, 128classical, 74full, 79, 82Quine on, 81, 124semantics, 81

Sellars, W, 22, 97, 99, 196objection, 14–15

set theory, 6, 64, 81, 88, 92, 124, 136, 181, 231background, 42development of, 120–121Kripke-Platek, 88paraconsistent, 241satisfiability in, 52Zermelo-Fraenkel (ZF), 73

Shapiro, S, 2–4, 9, 43, 61, 78–79, 118, 189, 233smooth infinitesimal analysis, 52–54Spelke, E, 102–103structuralism, 3, 74conceptual, 3–4, 78, 80, 90–91in-re, 74modal, 74

Tarski, A, 195, 202, 234–236, 238biconditionals, 44–45definition of logical consequence, 203Generalised Tarski Thesis, 50, 70T-schema, 233

-type, 204Tarskian conception, 238Tarskian model, 50, 207, 245theory choice, 9, 216, 223truthabsolute, 233–234, 237–239, 241by convention, 32, 47–48in a model, 233logical, 3, 9, 29, 41, 46, 95, 99, 233, 240, 242

all, 105first-order, 42ground of, 93interpretation, 233–234, 248metaphysical, 8–9, 233, 235–239, 244–247pluralism about, 244–245Quine on, 233realist, 238reflecting facts, 97

preservation, 44, 160, 222, 245preserving, 44, 160relative, 234, 236–238, 244

truth tables, 46T-schema, 44, 46, 233–235

vagueness, 56, 61–65logics of, 106real, 107worldly, 96

Waismann, F, 57, 71Wason Card Test, 218–219Wittgenstein, L, 5, 125, 144, 195, 219and physics, 144means by ‘postulate’, 144on mathematics, 128–129on rule-following, 129–132, 143rejection of Hilbert’s program, 142Steiner on, 6

Wittgensteinian, 45, 132, 141Wright, C, 49, 58–60, 62–63, 70

Zermelo-Fraenkel set theory, 73

Index 267