Reflexivity: From Paradox to Consciousness 9783110320183, 9783110319989

Table of contents :
CONTENTS
PREFACE
INTRODUCTION
Chapter 1: THE DYNAMIC STRUCTURE OF PARADOX
Chapter 2: THE LIMITATIVE THEOREMS
Chapter 3: NIHILISM, SKEPTICISM, AND THE COGITO
Chapter 4: THE ROLES OF REFERENCE
Chapter 5: THE MYSTERIES OF CONSCIOUSNESS
Chapter 6: MYSTERIES OF SEMANTICS AND FREE WILL
Chapter 7: CONCLUSION
APPENDICES
NAME INDEX

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Patrick Grim | Nicholas Rescher Reflexivity From Paradox to Consciousness

Dedicated to some of those whose work continues to surprise and inspire us: Johan van Benthem Graham Priest Neil Tennant

Patrick Grim | Nicholas Rescher

Reflexivity From Paradox to Consciousness

Bibliographic information published by Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

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2012 ontos verlag P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN 978-3-86838-135-1 2012 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work Printed on acid-free paper FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by CPI buch bücher gmbh

CONTENTS Preface Introduction

1

Chapter 1:

The Dynamic Structure of Paradox

9

Chapter 2:

The Limitative Theorems

37

Chapter 3:

Nihilism, Skepticism, and the Cogito

59

Chapter 4:

The Roles of Reference

95

Chapter 5:

The Mysteries of Consciousness

119

Chapter 6:

Mysteries of Semantics and Free Will

151

Chapter 7:

Conclusion

173

Appendices:

177

PREFACE

T

his book results from a collaboration inaugurated by Patrick in 2009. The project unfolded in ready dispatch and congenial harmony, though Nicholas insists on its being said that Patrick did the lion’s share of the work. Patrick wants to make it clear that the project would not have been as thorough or as wide ranging . . . and perhaps would not have been at all . . . without Nicholas’s critical reflection and major contributions. In any case the authors want to thank each other for the joys of collaboration: for open and encouraging exploration of ideas on each side and for critiques firmly but kindly rendered. Patrick also wishes to thank members of his graduate seminar for probing, testing, and sharpening major ideas: Adam Rosenfeld, Frances Bottenberg, Tim Hyde, Adam Kohler, Amir Jaima, and Jenny Strandberg. Patrick Grim Nicholas Rescher

INTRODUCTION

A

close-knit family of conceptual structures underlies the range of philosophical phenomena from Descartes’ Cogito through semantic and set-theoretical paradoxes to some of the major limitative results of twentieth-century logic. At issue are questions of indexicals, the nature of semantics, free will and determinism, and contemporary debates regarding the nature of consciousness. The conceptual structures that underlie all of these are variations on a single theme: the theme of reflexivity. Our attempt here is to characterize reflexive conceptual structures more thoroughly and more precisely than has been done before, making explicit the structure of paradox and the clear connections to major logical results. Our goal is to trace the structure of reflexivity in sentences, sets, and systems, but also as it appears in propositional attitudes, mental states, perspectives and processes. What an understanding of patterns of reflexivity offers is a deeper and de-mystified understanding of issues of semantics, free will, and the nature of consciousness. The main points of the book, roughly by chapter, are these: GRAPHING PARADOX

We can graph the conceptual structure of oscillational paradoxes. When we do, a central pattern emerges repeatedly through a range of variations: set-theoretical and semantic paradoxes share the same conceptual DNA. In terms of that common structure we can understand the crucial conditions for paradox, how oscillatory paradox works, and can do so in terms general enough to anticipate related conceptual structures elsewhere. Although this offers a new way of understanding paradox, it does not offer a ‘solution.’ For reflexive reasons of its own, the idea of an analysis sufficient to anticipate and avoid paradox in all applications itself proves vulnerable to paradox.

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THE LIMITS OF LOGIC The conceptual structure of paradox is not the DNA of paradox alone. It is also the DNA of a range of the most important twentiethcentury theorems in logic, foundations of mathematics, recursion theory and theory of computation. The affinity between paradox and some of these results has long been sensed—sensed, for example, in Gödel’s introduction to his own work. Here we make structural similarities explicit, extending techniques developed in graphing paradox to graphic representations of the basic arguments in Gödel’s incompleteness theorem and Turing’s Halting Problem. A more detailed graphical treatment of Gödel’s proof and of Tarski’s theorem is left to the Appendix. SELF-REFUTATION AND CARTESIAN CERTAINTY The study of reflexive structures makes it clear that those structures quite standardly bring with them automatic ‘ins’ and ‘outs’: sets that must be self-membered or cannot be, for example, sentences that must be in the extension of their predicates or must not, formulae that must appear as theorems or must not. Automatic ‘ins’ and ‘outs’ appear in richer reflexive structures as well—in reflexive structures within human practices of investigation, assertion, valuation, and debate. Written in the pragmatic contradictions of practice rather than the logical contradictions of formal systems, blanket positions of Nihilism, Skepticism, and Relativism appear as self-refuting ‘outs.’ The certainty of the Cartesian Cogito, in contrast, is traceable precisely to its status as an automatic ‘in.’ REFLEXIVE REFERENCE Reference is a crucial area in which reflexivity appears: it is reflexivity that characterizes indexicals ‘I,’ ‘here’, and ‘now,’ for example. In offering a structural analysis of the problem of the essential indexical, we focus on categories of reflexive reference as particular cases in a general phenomenon of non-equivalent referential forms. Here and elsewhere forms of reference prove irreducible, tied to contrasting

INTRODUCTION

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perspectives and practices, even where the referents themselves—the things referred to—may be the same. In conceptual development, we argue, it is quite common for ontologies of ‘things’ to be built from the materials of incompatible forms of reference. The result is apparently incompatible ontologies. Where one form of reference is essentially reflexive and another not, the result is apparently incompatible ontologies of subjectivity and objectivity—even if the original domain of reference is fundamentally the same. CONSCIOUSNESS One payoff from our studies of reflexivity is an outline of consciousness that renders it much less mysterious, a phenomenon understandable in terms of a clear structure and a clear place in evolutionary history. The characteristic feature of consciousness, we argue, is a reflexive structure within mental states much like that we have traced in various domains throughout. Conscious states are not states that ‘glow’ in a peculiar way but states of awareness of a particular reflexive form. States of phenomenal consciousness, in particular, are states of awareness that access reflexively their own ‘fast categories’ of information processing. The account of consciousness we develop allows us a deflationary analysis of Frank Jackson’s Black-and-White Mary, the explanatory gap, and Nagel’s ‘what is it like to be a bat?’ Considerations of reflexivity also allow us to extend our exploration to self-consciousness and the concept of a self. SEMANTICS AND FREE WILL The study of reflexivity carries further payoffs in an understanding of problems of semantics and the issue of free will and determinism. Quine’s ‘gavagai’ example is used to argue that there is no ‘truth of the matter’ regarding meaning. Searle’s Chinese Room is used to argue that semantics must be something beyond mere syntax. In different ways, we argue, both Quine and Searle force a view of language from without. Because meaning is accessible only reflexively within

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the practice of communication, both impose a perspective from which meaning is inevitably invisible. The concepts of choices, decisions, possibilities, alternatives, potential gains and losses, hazards and opportunities constitute elements of an ontology proprietary to the context of deliberation, built on reflexive reference within that context. Concepts of causality, in contrast, are essentially non-reflexive and non-deliberative. Here as elsewhere, however, a conceptual Dualism may be compatible with a fundamental Monism. Despite contrasting ontologies built on forms of reference tied to radically different forms of endeavor, the fundamental domain of reference may ultimately be the same. We have said that the structural understanding of paradox we offer does not constitute a ‘solution.’ In the same way, the structural understanding of problems of consciousness, semantics, and free will may fall short of what a ‘solution’ might seem to demand. Neither free will nor determinism emerges triumphant. Syntax and semantics remain inevitably conjoined and yet distinct. With regard to consciousness, a conceptual Dualism still leaves open the possibility of a fundamental Monism. In that sense our results may not offer ‘solutions’ to any of these. What we hope they offer instead is a new understanding of the real character of the conceptual issues at stake, of why they may be inevitable, and why they inevitably are as they are. As a whole, our effort is understand structures of reflexivity in greater depth and detail—explicitly enough to graph them—and to follow the trail of reflexivity that leads tantalizingly from semantic and set-theoretical paradox to indexicals and the Cogito, from limitative logical results to issues of consciousness, semantics, and free will. In all of these topics we are the beneficiaries of an extensive literature and a long philosophical history. Our focus is on a trail of ideas that links those various topics. For easier access, we have chosen to map that trail fairly uncluttered by footnotes and without extensive positioning in the detailed literature.1

INTRODUCTION

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NOTES 1

A useful outline of the larger context of reflexivity as well as extensive sufficiencies to the literature are given in the relevant article of the Enzyklopädie Philosophie und Wissenschaftstheorie, ed. by Jürgen Mittelstrass, Stuttgart: J. B. Metzler, 2004.

Chapter 1 THE DYNAMIC STRUCTURE OF PARADOX

O

ur aim in the following chapters is to trace a family of reflexive conceptual structures evident across a broad swath of philosophical territory. In this chapter we introduce those structures in what is perhaps their simplest and most familiar form: the conceptual structure of oscillation paradox. Why precisely do the paradoxes work as they do? Attempts to pin down the crucial mechanisms of paradox have often failed by being too vague and too general. There is indeed something paradigmatically self-referential about the paradoxes, for example, but ‘selfreferential’ is inadequate as an analysis of why the paradoxes do what they do. Tangled routes of reference can be indirect to the point that what is at issue is no longer literally self-reference. Self-reference alone is also clearly insufficient to produce paradox. It is selfreference that lies at the core of a range of powerful results in mathematics and computation theory. Much more immediately, it is clear that the fact that this sentence is self-referential causes no difficulty at all. Attempts to characterize the paradoxes have also sometimes failed by being too specific. The paradoxes were often carefully separated into the set-theoretical paradoxes on one side and the semantic paradoxes on the other, for example, with the implication that these differed importantly in kind. That taxonomy, we will argue, ignores the fact that both forms of paradox exhibit the same conceptual structure. That structure is the same despite being applied across different categories—to sets or collections on the one hand, to predicates or properties on the other. Here, starting with the simplest case, we want to do better. Our goal is to understand the conceptual mechanisms that are responsible for the magic of the paradoxes, avoiding both unjustified specificity and vague over-generality. We want to understand the core mecha-

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nisms well enough to graph them. That in turn will give us a better understanding of the reach of paradox, and the reach of related reflexive mechanisms across a range of philosophical concerns. 1. PARADOXICAL SETS Russell’s set is a set of all and only those sets that do not contain themselves. Does it contain itself or not? • If it does contain itself, since it is specified as a set of only those sets that do not contain themselves, it cannot contain itself. • If it does not contain itself, since it is defined as a set containing all those sets that do not contain themselves, it must contain itself. There are two ways in which this result might be characterized. The characterization we favor in this case, and in similar cases throughout, is one that emphasizes the logical dynamic at issue. The attempt to decide whether Russell’s set contains itself or not forces us to conceptual oscillation. The hypothesis that it does contain itself forces us to the opposite conclusion: that it does not. But that hypothesis—that it does not contain itself—forces us back to our starting point: we are forced to conclude that it does contain itself after all. But that leads again to the opposite conclusion . . . The result is an infinite oscillation between two opposing conclusions. That oscillation is characteristic of paradox, and is indeed the source of its perverse appeal. But it is far more standard to characterize Russell’s set statically. Does the Russell set contain itself as a member or not? Either hypothesis leads to contradiction: such a set can be a member of itself if and only if it is not. Contradiction. We conclude simply, and statically, that there can be no such set. A static picture in terms of contradiction does not ultimately conflict with the dynamic characterization, of course; the source of the contradiction is precisely what drives the oscillatory dynamics. But contradiction can arise in many ways:

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• It is green. It is not green. Contradiction need not be oscillational, and need not be inherently puzzling or intriguing. It is the dynamics of conceptual oscillation that is characteristic of the paradoxes, and it is that dynamics that constitutes a major part of their magic. To understand the paradoxes requires that we understand that dynamics. Why does Russell’s set produce an oscillation? Each of the following, even at first glance, seems a crucial component: • Sets can appear on either side of the relation of membership. Sets are things which have members, but are also things which can themselves be members. • The relation of membership can be instantiated reflexively: the same set can appear as a candidate for membership and as that of which membership is being considered. But the relation need not hold reflexively: a set may fail to stand in that relation to itself. • A well-placed negation plays a central role. It is sets that are not members of themselves that are at issue. • The Russell set is defined as containing all and only those sets that fit the negated reflexive condition. If sets could not be both members and have members, oscillational paradox would never get off the ground. If membership could not be reflexive—as hierarchical forms of set theory demand that it not be— there would be no oscillation. If the reflexive instantiation were not negated at a crucial point, we would get no oscillation. If our characterization of the Russell set were not an all and only characterization we would get some interesting results—as we will see—but we would not get conceptual oscillation. By graphing these elements we can get a better picture of how they work in oscillation, how they interact, and of how crucial the conditions listed really are.

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We will use an arrow to represent our core relation. Here that relation is one of membership, though in other cases we may use the arrow for a different core relation. That S is a member of S’ becomes:

We will use a cross bar for negation of the basic relation. That S is not a member of S’, then, becomes:

Any set that is not a member of itself is specified as a member of the Russell set. We represent our ‘any’ using blanks, with the specification that all blanks must be filled with a set specification, and that all blanks must be filled with the same set specification.

Our dotted arrow represents a pattern of inference, in this case laid down by specification: For any uniform filling of the set-name blanks, the fact that a set is not a member of itself entails that it is a member of the Russell set. The Russell set, however, is specified as containing all and only those sets which are not members of themselves. Our representation above captures ‘all’; to capture ‘only’ we have to indicate that any self-membered set is not a member of the Russell set:

The two conditions on the Russell set are established in these diagrams. But as long as blanks are filled by the same set specification, they can be filled by any set specification. We can therefore use R itself in those blanks, giving us the following:

THE DYNAMIC STRUCTURE OF PARADOX

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Together these give us an explicit picture of the dynamics of paradoxical oscillation:

We think of this as a ‘implicative dynamics diagram’, or ‘dynamics diagram’ for short:1 The diagram makes clear the necessity of each of the conditions we have listed. If sets could not appear on both sides of the relation, the basic elements on each side would not be formulable. If the relation at issue could not be instantiated reflexively, one of those basic elements would be impossible. Without the crucial placement of negation on one side, we would have no oscillation between conflicting conclusions. Were one of our ‘all’ or ‘only’ conditions missing, we would have the inference pattern of only half the graph—and again would have no oscillation. 2. STRUCTURE AND SOLUTION Our attempt above has been to grasp the structure of a first case of paradox. It should be noted that the attempt has not been to solve that paradox. Indeed we think that a more complete understanding of the reflexive structure at issue offers something more powerful than mere solution: an understanding of related conceptual structures across a range of philosophical issues. The structure that underlies oscillation in this case, however, also makes it clear what routes attempt at solution might take, and indeed illuminates a quite standard route that attempts have taken.

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Our initial observations regarding Russell’s set included the condition that the relation of membership can be instantiated reflexively: the fact that the same set can appear as a candidate for membership and as that of which membership is being considered. That condition appears graphically in the fact that the blanks in a representation like the following can be filled by any set specification, including R itself:

It is objection to precisely this condition that has been the primary route taken in the attempt at solution. The intuitive objection is that it is illegitimate to assume an item to exist specification of which makes reference to that item itself. In Principia Mathematica the intuitive objection takes the form of the Vicious Circle Principle: “An analysis of the paradoxes to be avoided shows that they all result from a kind of vicious circle. The vicious circles in question arise from supposing that a collection of objects may contain members which can only be defined by means of the collection as a whole. . . The principle which enables us to avoid illegitimate totalities may be stated as follows: ‘Whatever involves all of a collection must not be one of the collection’; or, conversely: ‘If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.’ We shall call this the ‘vicious-circle principle,’ because it enables us to avoid the vicious circles involved in the assumption of illegitimate totalities.”2

The formal mechanism Russell uses to enforce such a principle is the ramified theory of types. One of us has explored such a solution more generally in terms of a ‘Successful Introduction Principle.’ On such a principle, identificatory self-reference is prohibited: if X is by definition to be (ιx)Cx then mention of X cannot occur within C. The idea is that not merely circular arguments, circular explanations, but circular definitions and introductions must be rejected.3

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Such prohibitions, like prohibitions on self-reference generally, are open to the objection that they tar with too broad a brush. There appear to be legitimate cases of introduction in both mathematics and elsewhere that would violate such a principle, at least in its broadest form, just as in establishing the validity of a certain mode of argument we may have no choice but to use that mode of argument itself. What is clear from the implicative dynamics diagram in our first case above is that an attempt to block the paradox in this way, though sufficient, is not necessary. There are other structural elements crucial to the paradox—the specific role of negation, for example—and it is only in combination that all the elements together give rise to paradoxical oscillation. Were one to attempt to block paradox, and only paradox, in a surgically more precise way, it is all crucial elements of the graph that should be attended to. 3. THE STRUCTURE BEYOND SETS One of the benefits of seeing the dynamics of paradox in terms of explicit diagrams is that one can see the complexes of conditions involved in generating that dynamics. Another benefit is that one can see immediately the wider instantiability of the fundamental structure. Russell’s paradox concerns sets and set membership. But as Russell himself clearly saw, the deep structure of the paradox has nothing to do with sets and membership in particular. Russell offers the Barber’s paradox as a direct analogy. Here the relationship is not membership, applicable to sets, but the relationship of shaving. We assume a barber who shaves all and only those who do not shave themselves:

All that differs structurally from the case above is that we have B in relation to a blank on the right (the barber shaves . . .) as opposed to a blank in relation to R (. . . is a member of R). The direction of our arrow is a difference that makes no difference, since we could of course choose to express our relation as ‘is shaved by . . .’ The blanks in the

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Russell paradox are typed to sets, and can take any set as substitution. Here they are typed to people, and can take any person as substitution. Using B as the name of our barber in all blanks, we get an oscillation of precisely the structure above:

The Heterological paradox exhibits precisely the same form. Just as membership is nonreflexive with regard to sets, and shaving is nonreflexive with regard to people, ‘applies to’ is nonreflexive with regard to words: words may or may not apply to themselves. Consider then a term ‘autological’ that applies to all and only those words that apply to themselves:

‘Heterological,’ in contrast, applies to all and only those words that do not apply to themselves:

Does ‘autological’ apply to the term ‘autological,’ or not? Does ‘heterological’ apply to the term ‘heterological,’ or not? We can substitute ‘autological’ A for all word-typed blanks in the first case, and ‘heterological’ H for all blanks in the second. Only the second of these, however, gives us oscillational paradox. Here the crucial role of negation is clear: only the second case has conditions both of which link inferentially a positive application case with a negated application case, both with blanks open for substitution.

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It is clear that the same conceptual structure underlies the Russell paradox, the Barber, and the Heterological paradox. Such an analysis is confirmed by the fact that we can produce further paradoxes with that same structure at will. For a nonsymmetric and nonreflexive relation R, we specify an X as standing in relation R to all and only those of its type for which that relation does not apply reflexively. Does that X stand in that relation to itself, or not? • Consider thoughts, for example. Some thoughts are about themselves, while others are not. Consider moreover a thought about all and only thoughts that are not about themselves. Is such a thought about itself, or not? • Consider lists. Some lists contain themselves as items, while others do not. Consider a list of all and only those lists that do not contain themselves as items. Does that list contain itself, or not? • Consider photo albums. Some contain photos of themselves, while others do not. Consider then a photo album that contains photos of all and only those photo albums that do not contain photos of themselves. Does it contain a photo of itself, or not? The conceptual pattern at issue is in no way particular to sets and membership, to barbers and shaving, or to words and application. The classical categorization of set-theoretical as opposed to semantic paradoxes is therefore likely to mislead; the mechanism that lies beneath conceptual oscillation is both simpler and more abstract than specific issues of either sets or semantics. Hans Herzberger, following James Thomson, identified the structure in these terms:

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• Nothing stands in relation R to all and only those things that do not stand in relation R to themselves. 4 The proof will in all cases be by a reductio ad absurdum from oscillation on precisely the pattern above. Were there such a thing X, we could ask whether it stands in relation R to itself. If it does, it does not, because X has been specified as standing in relation R to only things that do not stand in relation R to themselves. If it does not stand in relation R to itself, on the other hand, it must: X has been specified as standing in relation R to all things that do not stand in relation R to themselves.

4. INS AND OUTS Closely related to oscillational paradox are automatic ‘ins’ and ‘outs,’ which we will track in various contexts throughout the book. The presence of both ‘all’ and ‘only’ conditions, we have seen, are crucial for full oscillation. If we weaken one of those conditions we get an automatic ‘in’. If we weaken the other we get an automatic ‘out.’ Consider a variation R’ of the Russell set that contains all sets that do not include themselves as members, though perhaps not only those. Here we are restricted to our ‘all’ condition:

Set-typed blanks, as before, can take any set as substitution, including R’:

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The hypothesis that R’ does not contain itself, therefore, leads to the conclusion that it does. But here we do not get full oscillation, because we are not assured of the ‘only’ condition that gave us an inference in the other direction. What we have instead is a classic reductio ad absurdum. The fact that a premise leads to contradiction forces us to reject that premise, in this case rejecting the premise that R’ does not contain itself and concluding instead that it must. But here, unlike in the oscillational case, that conclusion leads to neither oscillation nor inconsistency. The conclusion we are forced to is simply that R’ must be one of those members of R’ that do contain themselves as members. R’ must contain itself; unlike oscillational R, R’ is an automatic ‘in.’ Consider on the other hand a set R” that contains only sets that do not contain themselves as members, though perhaps not all of those.

or equivalently Does R” contain itself as a member or not? This time it is the hypothesis that it does contain itself that leads to contradiction.

By reductio, we are forced to the opposite conclusion: that R” does not contain itself as a member. That hypothesis is perfectly consistent, and in fact is the only consistent hypothesis: R” must be one of those sets that do not contain themselves as members but that happens to be excluded from R”. With regard to self-membership, R” is an automatic ‘out’. ‘Ins’ and ‘outs’, like oscillation, are a natural consequence of the central conceptual structure at issue. It is thus to be expected that we get similar ‘ins’ and ‘outs’ with other instantiations of that structure.

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‘Heterological’ is defined so as to include all and only those words that self-apply, and gives us an oscillation in its own case: is ‘heterological’ heterological or not? If we consider a term ‘heterological2,’ the extension of which is specified as applying to all those words that do not self-apply, but perhaps not only to these, we get a different conclusion. Is ‘heterological2’ heterological2 or not? If it is not—if it does not self-apply—it follows from our specification that it is heterological2. Contradiction.

‘Heterological2’ must therefore self-apply: it must be heterological2. Because we do not have an ‘only’ requirement for ‘heterological2,’ this hypothesis is entirely consistent and does not lead to oscillation. ‘Heterological2’ must be one of those things in its own extension that do self-apply. ‘Heterological2’ is an automatic ‘in’. Consider on the other hand ‘heterological3’, the extension of which is specified to include only terms that do not self-apply, but perhaps not all of these.

Is ‘heterological3’ heterological3 or not? Here it is the supposition that ‘heterological3’ is heterological3 that gives us a contradiction. ‘Heterological3’ is an automatic ‘out’; it cannot self-apply. It must therefore be one of the things that do not self-apply but are left out of the extension of ‘Heterological3’. Automatic ‘ins’ and ‘outs’ are natural accompaniments to the structure of conceptual oscillation. By weakening ‘all’ or ‘only’ conditions in the other cases listed above we can establish automatic ‘ins’ and ‘outs’ for these as well: lists, photo albums, and thoughts that are automatic ‘ins’ in that they must include themselves, for example; lists, photo albums, and thoughts that are automatic ‘outs’ in that they cannot.

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5. THE STRUCTURE OF THE LIAR • This sentence is not true The basic elements of conceptual oscillation outlined above appear again in the Liar paradox and in close relatives that we will track through later chapters. Instantiated in the Liar, however, the structure of oscillation is more complex in several ways. To this point we have considered sets of sets, words applicable to words, and lists of lists: X’s of X’s. What is at issue in the Liar is a sentence about a sentence, but it is crucial to understanding its structure that sentences—at least sentences of the form of the Liar—are composed of subjects and predicates. Sentences of at least this simple sort refer to something and say something about that thing. Reserving a left section for subject term and a right section for predicate, we can represent that simple structure as follows:

Here positions, it must be remembered, are typed: the right hand corner is reserved for predicates, the left hand corner for subjects. Consider now the predicate ‘is true’. ‘Is true’ is a predicate that applies to sentences. Hence an inevitable complication: in cases like the Liar we don’t have merely X’s of X’s—sets of sets, for example— but P’s that apply to dual compounds of P and S. ‘Is true’ is a predicate applicable to compounds of subject and predicate. ‘Is true’ is a predicate that takes as its potential range sentences in general, and thus may apply to subject-predicate compounds in which any predicate—including itself—may be a part. ‘Is true’, moreover, is a predicate that applies to sentences in virtue of something about predicates and something about application. ‘True’ is a predicate that holds of all and only those sentences the predicate of which holds of their subject. Using our relational arrow now as ‘applies to,’ we can represent the fact that ‘true’ holds of all sentences in which predicate holds of subject as follows:

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Here we specify that the subject place of the structure on the right must be occupied by a structure identical to the full structure on the left. If we fill in either, the other must match. ‘True’ applies to a compound of subject-predicate just in case the predicate of that compound applies to its subject. We can represent the fact that ‘true’ holds of only sentences in which predicate holds of subject as follows:

The ‘all and only’ condition that played a major role in our considerations above appears here in virtue of the nature of predication. A predicate ‘x’ is quite generally taken to apply to all and only those things that are x—that is, after all, what predication using ‘x’ is all about. The predicate ‘false’ thus applies to all and only those sentences in which predicate does not apply to subject:

It is a crucial and unavoidable fact that the subject of a sentence— what a sentence is about—can be the sentence itself. This is of course the case with the Liar, but is the case with each of the following as well: • This sentence is the subject of this sentence. • This sentence is in English. • This sentence appears on p. 16.

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The crucial fact in each case is that the sentence as a whole— composed of both subject and predicate—is taken as the subject of the sentence itself. The result is something like the Droste effect, in which a box of cocoa displays a picture of a woman holding the duplicate of that box, which of course displays a picture of the woman holding a duplicate of the box . . .5 Self-referential sentences of this sort might therefore be pictured in terms of an infinite regression of identical two-part subjects:

Fractals are defined as images self-similar at descending scales. This is precisely the character of the image envisioned above; the subject representation, or any series of embedded subject representations, can be substituted in subject position at any point without change in the whole. It is this possibility of substitution, in various forms, that will be important for our purposes throughout. A self-referential structure of the form exhibited by any of the sentences above entails intersubstitutable reference: reference to the subject of the sentence is interchangeable with reference to the sentence as a whole. Reference to a sentence may take any of various forms, which it will be important to disentangle at a later point. For now, however, we can represent the referential intersubstitutability characteristic of these sentences simply as follows:

We now have all the components needed to diagram the oscillational behavior of the liar. The ‘all and only’ conditions for the predicate ‘is false’ carry two types of blanks: those that can be filled consistently by any predicate to the right, and those that can be filled by any subject to the left:

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Because blanks on the right can be filled with any predicate, they can be filled with ‘false’ in particular. Note that the right hand corner of the embedded sentence form must be filled in precisely the same way:

Because blanks on the left can be filled with any subject, they can be filled with a subject that is itself a sentence. They can be filled with reference to a self-referential sentence, and can be filled in particular with reference to the self-referential sentence

With these substitutions, our all and only conditions for ‘false’ give us

In virtue of the referential intersubstitutability of S and can replace embedded appearances of with S, giving us:

, we

With these instantiations, our ‘all and only’ conditions for ‘is false’ give us the full dynamic oscillation of the Liar:

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In discussing Russell’s and other paradoxes we noted that a standard route to ‘solution’ was to block the possibility of a set taking itself as a member. The structure outlined for the Liar makes obvious the parallel approach here: to block the possibility of a sentence taking itself as subject. Parallel to the typed sets and objects of Russell’s theory of types are the typed sentences and predicates of Tarski’s and other meta-language approaches. The route to oscillation in the Liar involves two crucial elements that did not appear in the simpler case of Russell’s, the Heterological, and related paradoxes. The relations at issue there were between single elements: a relation of membership between X and X, for example. In the Liar the crucial relation is between a predicate—’is false’—and a two-part structure composed of both subject and predicate. Essential to the Liar is therefore a two-place structure in place of the one-place structure of the simpler paradoxes. Because of the two-part structure involved, a second element crucial to the Liar is the intersubstitutability of a two-place complex of subject and predicate with a term appropriate to subject position alone. Without that capability of self-reference—without the Droste effect of a two-part structure capable of including a representation of the whole as one of its parts—we would lack an essential structural component of the Liar. Given both of those elements, however, the Liar produces a conceptual oscillation in very much the way that the simpler paradoxes do—in particular, by way of ‘all and only’ conditions here built into the predicate ‘is false.’ 6. OSCILLATION BEYOND THE LIAR Once the structure of Russell’s paradox is clear enough to graph, other instantiations of that structure suggest themselves immediately. The same is true of the structure of the Liar.

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It is important for the Liar as we have outlined it that the subject of the sentence stand in for the sentence as a whole: a condition we have referred to as intersubstitutability. How that stand-in is accomplished, however, remains open. Reference may be direct, oblique, or turn on empirical facts. A sentence with any of the following subjects, in the right circumstances, can accomplish the requisite substitutability of sentence subject and sentence as a whole: • The last sentence on p. 8 . . . • The only sentence on the board in room 251 . . . • The sentence referred to by the second sentence on p. 9 . . . • The sentence Mike will next refer to . . . • Bill’s favorite sentence . . . Whether a sentence has a subject with the structural property required for oscillation may in fact be unknown. There may thus be sentences which may or may not be forms of the Liar: • The sentence Lauren last refers to in the year 2020 is false. It is also important for the Liar that the predicate that the sentence at issue applies to its subject be a predicate that applies to all and only those sentences in which the predicate does not apply to its subject. ‘Is false’ and ‘is not true’ fill that bill, but other predicates do as well. Consider for example: • . . . has a predicate which does not hold of its subject. The predicate at issue may also appear obliquely, in much the manner noted for the subject term. In the right conditions, each of the following might function as a predicate appropriate to oscillation:

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• . . . has the property attributed to another sentence by the first sentence on p. 12, where the property attributed by that sentence is ‘is false’ or ‘has a predicate which does not hold of its subject.’ • . . . has Bill’s favorite property of sentences, or • . . . is as true as the claim that p, where p is a claim that happens to be empirically false. In predicate position as in subject position we may have unknown forms of the Liar. Whether their truth-values must be oscillatory depends on the facts, but those facts may be beyond our reach: • The last sentence a 12-year-old writes in Ann Arbor in 2020 is as true as the claim that Goldbach’s Conjecture will be proven by 2040. Once we can see the structure of the Liar we can also offer a Thomson and Herzberger-like generalization for oscillational cases: • The supposition of a predicate that stands in relation R to all and only those sentences the predicate of which does not stand in relation R to their subject, leads to oscillation. Such a supposition leads to oscillation, of course, when we ask whether a sentence with that predicate, the subject of which stands in for the sentence itself, satisfies that predicate or not. If it does, it cannot; if it does not, it must, just as in our dynamics diagram above. Throughout, we have characterized the Liar in terms of sentences, subjects, and predicates. But talk of sentences rather than propositions is certainly not essential. A characterization in terms of propositions, subjects of those propositions, and properties attributed within them

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would exhibit precisely the same structure and do precisely the same work. We can in fact abstract the structure beneath the Liar still further. The fact that it is sentences or propositions that are at issue is inessential as well: a structure adequate for oscillation appears with any twopart structure that meets certain conditions. • Consider complexes Z with two parts: each Z has an X and a Y. Consider further complexes of such a sort which are ‘partially reflexive’ in that we have Z’s which are also their own X’s. Without specifying the complexes as significatory in the sense of either sentences or propositions, and without specifying X’s as referential expressions of any sort, we have laid down the conditions necessary for structures meeting the abstract conditions of

Given that abstract conception, consider further: • A specific Y* of type Y in complexes Z that bears relation R to all and only those Z’s such that their Y does not bear relation R to their X. Here neither Y* nor Y has been specified as anything like a predicate or property. Y* is merely a specific Y, and a Y is merely an element that appears in a two-part structure. The specification given for Y*, however, satisfies the abstract conditions of ‘all and only’ used above. With the solid arrow as our relation R we can again use the dotted arrow to represent inference:

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The supposition of a Y* meeting the conditions specified leads immediately to oscillation. Consider that complex Z* which (a) has Y* as its Y and (b) takes itself Z* as its X.

Does Y* bear relation R to that complex or not? • If it does, it does not—Y* is specified as bearing R to only those complexes that do not have a Y that bears R to their X. The Y of this complex would then not bear R to its X. But Y* is its Y, and X is the complex itself. Y* would then not bear R to the complex Z*. • If it does not, it does—Y* is specified as bearing R to all those complexes that do not have a Y that bears R to their X, and we have supposed in this case that Z* fits the bill and thus that Y bears R to Z*.

Just as the structure of the set-theoretical paradoxes shows that they are not essentially set-theoretical, the structure of a semantic paradox such as the Liar shows that it is not essentially semantic. Both are instances of a general reflexive pattern, and it is the structure of that pattern, rather than its instantiation in either sets or semantics, that is responsible for the dynamics of oscillation. 7. THE LIAR’S INS AND OUTS The oscillation of the Liar, like the oscillation of Russell’s and the Heterological paradox, requires full ‘all and only’ conditions: condi-

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tions implicit in predication in general in the case of ‘is false’, for example. Weakening those conditions in the case of Russell’s set and Heterological gave us automatic ‘ins’ and ‘outs.’ The same is true in the case of the Liar. Consider for example a predicate ‘troop’ that applies to all truths, but perhaps not only to truths. Consider moreover a troop Liar: • This sentence is not troop. Will this sentence be troop or not? If the sentence is not troop, it must be true, since that is precisely what it says. But if it is not troop, it must lie outside of that category of ‘troop’ specified as including all truths. It must therefore be false. Contradiction. We therefore have a reductio argument against the supposition that the sentence above is not troop. But suppose that the sentence is troop. It is then false, because the sentence explicitly denies that it is troop. Does that give us a contradiction? No. Since the application of ‘troop’ contains all truths but also perhaps a few falsehoods, it is perfectly consistent to assume the sentence above is ‘troop’. The only consistent assumption, in fact, is that the sentence is troop. Indeed the only consistent assumption is that the sentence is false but ‘troop.’ With regard to ‘troop,’ our troop Liar is an automatic ‘in’. Consider also a second predicate ‘troof’ that applies to only truths but perhaps not to all of them. And consider a troof Liar: • This sentence is not troof. Will this sentence be troof or not? Suppose it is troof. Since ‘troof’ applies to only truths, it is true. What it says, however, is that it is not troof; since it must be true, it is not troof. Contradiction. We have a reductio against the supposition that this sentence is troof. The contrary supposition, however, is entirely consistent. Suppose it is not troof. Because that is what it says, the sentence must be

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true. That does not give us a contradiction, however, since some truths may not be included as troofs. This must be one of them. With regard to troof, the troof Liar is an automatic ‘out’. 8. GRAPHING THE DUALIST Beyond the Liar, crucial elements of the pattern we have outlined are evident in more complicated tangles of reference. Consider for example the Dualist, consisting of not one sentence but two: S1. Sentence S2 is false. S2. Sentence S1 is true.6 Here we have two sentences at issue and two predicates: the Dualist uses both ‘true’ and ‘false’. We can use the ‘all and only’ conditions for ‘is false’ as before:

The relationship between ‘true’ and ‘false’ can be made explicit as follows:

(*) We instantiate each of our ‘all and only’ conditions for each of our sentences. For S1 we fill the open predicate slots uniformly with ‘F’ and subject spots with ‘S2’:

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For S2 we fill the open predicate slots with ‘T’ and subject spots with ‘S1’:

Finally, we note intersubstitutability:

Using these licenses for substitution, our instantiations of ‘all and only’ conditions for the two sentences above become:

(a)

(b)

(c)

(d)

With the relationships between truth and falsity outlined in * above, these give us a four-point oscillation between elements of the Dualist. Note that each element has a contradictory across the graph:

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9. THE PARADOX OF PARADOX ANALYSIS What we have offered by means of implicative dynamics diagrams is a new way of understanding the structure of paradox. But we have emphasized throughout that our goal is something other than ‘solution.’ Might not these techniques offer precisely that? Couldn’t we characterize the conceptual structures outlined as necessary and sufficient conditions for paradox, and couldn’t we avoid paradox precisely by avoiding those conceptual structures? The answer, it appears, is ‘no’—a result we think worthy of canonizing the Paradox of Paradox Analysis. Though dynamics diagrams and other techniques allow us to understand the structure of paradox, and to understand it quite generally, they cannot offer necessary and sufficient conditions. The paradoxical result is that there can be no successful analysis of paradox adequate to avoid it. For suppose any such analysis, successful for example in the case of sentences. Suppose we have a set of conditions C proposed as necessary and sufficient for paradox on the general model of the Liar. And then consider the following: • Either this sentence meets conditions C, or it is false. This sentence, like any other, must either meet conditions C or not. If it does meets conditions C, it will be simply true by virtue of its first disjunct—and thus not paradoxical at all. In that case our conditions C will not be sufficient for paradox. If it fails to meet conditions C, on the other hand, the first disjunct will be false and the truth of the sentence as a whole will turn on the second disjunct alone. Under those conditions the sentence will self-attribute falsehood precisely as the Liar does, and will be paradoxical for precisely the same reasons. It fails to meet C, but is paradoxical nonetheless: in that case our conditions C will not be necessary for paradox. No set of conditions C can be both necessary and sufficient for paradox. It is worth noting, however, that the structure of even the Paradox of Paradox Analysis is understandable in terms of the general con-

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cepts of reflexivity apparent in dynamics diagrams throughout. It is the structure of paradox itself that shows us there will be no specifiable necessary and sufficient for paradox. Paradox analysis, even in terms of reflexive conceptual structure, is doomed to be incomplete for reasons precisely of reflexive conceptual structure. 7 10. THE WIDER LESSONS OF REFLEXIVITY We have outlined conceptual structures within a range of traditional paradoxes explicitly enough to graph the dynamics of paradoxical oscillation. Implicative dynamics diagrams also allow us a glimpse of the formal extent of oscillation phenomena, characterized structurally, both within and beyond its instantiation in set-theoretical and semantic paradox. Our aim in the chapters that follow is to carry these first lessons of reflexivity farther. In the next chapter we turn to instances of related structure at the core of major 20th century limitative theorems in logic and computation. NOTES 1

Some of our students refer to these as either Grimgrams or DiaGrims. A diagrammatic anticipation appears in Graham Priest, Beyond the Limits of Thought. New York: Oxford University Press, 2002.

2

Alfred North Whitehead and Bertrand Russell, Principia Mathematica. Cambridge: Cambridge University Press, 1925, p. 37.

3

See Nicholas Rescher, Paradoxes: Their Roots, Range, and Resolution, Chicago: Open Court, 2001.

4

J. F. Thomson, “On Some Paradoxes,” in R. J. Butler, ed., Analytical Philosophy, Oxford: Blackwell, 1962; Hans Herzberger, “Paradoxes of Grounding in Semantics,” Journal of Philosophy 67 (1970), 145-167. Herzberger was clearly building on Russell's vicious circle principle. Graham Priest's Inclosure schema continues the tradition. See Priest, Beyond the Limits of Thought. New York: Oxford University Press, 2002.

5

The visual effect, which has often been repeated on cereal boxes, appears as early as 1320. The central panel of Giotto di Bondone’s Stefaneschi Triptych shows Cardinal Stefaneschi offering up the triptych itself to St. Peter, complete with a central panel showing Stefaneschi offering the triptych to St. Peter...

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NOTES 6

Analogous to the Dualist are situations of oscillatory reciprocal behavior in game theory. Consider for example two players in the Prisoner’s Dilemma. One is playing Tit for Tat, which starts with cooperation and then copies the opponent’s behavior on the previous round. The other is playing Suspicious Tit for Tat, different only in starting with a defection. In such a case both players will oscillate between cooperation and defection much as assignment to each sentence in the Dualist oscillates between true and false.

7

The conclusions to be drawn from the Paradox of Paradox Analysis bear a strong resemblance to both Gödel’s incompleteness theorem and Turing’s Halting Problem, further explored in chapter 2.

Chapter 2 THE LIMITATIVE THEOREMS

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n the preceding chapter we developed graphic techniques to capture the conceptual dynamics of traditional paradox. In this chapter we carry those techniques into two of the main limitative results of 20th century logic: Gödel’s first incompleteness theorem and Turing’s Halting Problem. With some important differences, a reflexive structure much like that evident in the ‘negative’ paradoxes of the preceding chapter underlies the ‘positive’ results at issue here. The conceptual structure of these major limitative results in 20th century logic— and of related results, left to the Appendix—can be graphed in much the same way. 1. GRAPHING THE LIAR It has not gone unnoticed that there are clear conceptual similarities between the traditional paradoxes and proofs for the limitative theorems. Gödel himself introduces his major work on incompleteness in terms of analogies with Richard’s paradox and the Liar. Let us start, then, with the structure of the Liar outlined in the previous chapter. Crucial to that structure is the fact that ‘false’ holds of all and only those compounds of subject and predicate in which the predicate does not hold of the subject. With left corner reserved for subject and right for predicate, we graphed that fact as follows:

Also crucial in construction of the Liar is the possibility of a sentence taking itself as subject, giving us a Droste effect image of retreating self-referential subjects, either in the fractal form

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or in a form that represents simply the substitutability at issue. With ‘false’ as our predicate, we get the Liar:

With F in all predicate positions in ‘all and only’ conditions for ‘false’ above, and using S and interchangeable in all subject positions, we have all elements required to graph the oscillation characteristic of the Liar:

As noted in the previous chapter, this structure can be generalized beyond sentences to any binary complex Z with two parts X and Y. Consider complexes of such a sort which are ‘partially reflexive’ in that Z’s can also appear as their own X’s, and consider a specific Y* of type Y that bears relation R to all and only those Z’s such that their Y does not bear relation R to their X. Whether or not Y* is a predicate, whether or not Z is a sentence, these conditions on a binary structure are sufficient to give us an oscillation of precisely the pattern of the Liar. 2. THE DYNAMICS OF GÖDEL’S PROOF Although the same crucial elements appear in the conceptual structure of Gödel’s proof, they appear in importantly different ways. In full development they also appear in a series of layered complications. Here we will sketch the dynamics of Gödel’s proof in a somewhat

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simplified version—that which appears in a number of popular presentations, closely related to a version that Gödel himself offers in the first section of “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.”1 A more technically complete version, with layered complications in place, is left to the Appendix. Gödel’s is a result concerning formal systems, which inevitably have a double nature. On the one hand, they are formal systems: welloiled syntactic machines that operate entirely by defined transformations of formulae in terms of their shape. On the other hand, they are always systems designed for a specific purpose, or with an intended interpretation in mind. Gödel’s explicit target is the system of Principia Mathematica, designed to capture mathematics. It is numerals and defined predicates within an axiomatic logic that are the grist of Russell and Whitehead’s formal mill. But that formal system is built with the end of mathematics—the telos of mathematical truth— in mind. Machines designed for interpretive purposes are now part of our daily lives. A laptop computer works in terms of electrical impulses through conductive webs embedded in silicon. The machine is constructed, however, for a purpose beyond that of electrical impulses. It is designed so as to make word-processing, tied to my communicative purposes, both possible and easy. In this analogy, the formal systems that Gödel is concerned with correspond to the physical device itself. Their intended interpretation is analogous rather to my word processing, or even my purposes in communicating—something for which the machine and the system are designed and used, but about which they themselves know nothing. In the case of formal systems, the mechanical properties of the system cash out in terms of demonstrations and theorems, both of which are definable entirely syntactically—in terms of the shape of symbols alone. The interpretational properties of the system are different: it is on interpretation that the system is about numbers, that its formulae represent claims regarding numbers, and that those formulae may or may not give us genuine truths regarding numbers. Theoremhood is a property within the syntax of the system. Truth is a property that lives in its interpretation.

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The ultimate purpose of formal systems, of course, is to systematize a body of truths: such is the semantic goal of syntactic systems. Ideally, what we want of a system is that it be sound—that all the formulae that appear as theorems are genuine truths on interpretation. Ideally, what we want of a system is that it be complete; that all the truths of the area we are interested in—all truths of number theory, for example, or at least all truths expressible in our system—appear as theorems. The desiderata of formal systems of this type—the accomplishment Russell and Whitehead implicitly claimed for Principia Mathematica—is that they be both sound and complete. What Gödel shows is that these desiderata cannot both be realized. Syntactic formalization has inherent limits, because of which it will always fall short of its full semantic telos. Gödel’s is the first of a string of theorems showing inherent limits in our hopes for formalization. In informal presentations the core sentence of Gödel’s proof is sometimes represented as a sentence of a formal system that says: • This is not a theorem or • ‘I am not a theorem.’2 This does indeed capture an important key to the Gödel construction, though we will also note some aspects that it obscures. What it offers is a clear view of how such a sentence would lead to a Liar-like oscillation. ‘I am not a theorem’ might be represented graphically as a selfembedding sentence of the Droste effect form outlined above. In this case:

(a) Unlike the case of the Liar, what are at issue here are theorems of a formal system. That formal system is defined by (1) syntactic criteria for admissible formulae, (2) specific formulae chosen as axioms, and

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(3) rules of inference. What it means for a formula to appear as a theorem of such a system is that there is a derivation of that formula within the system. What it means for there to be a derivation is that there is a series of formulae each of which is either an axiom of the system or a manipulation of earlier formulae in the series according to our syntactically specified rules of inference. For our purposes it again suffices to think of sentences and formulae as of subject-predicate form. We will represent the fact that a formula is derivable in a system—that the formula is a theorem—using a variation of our earlier arrow with an s for ‘system’:

The right corner is typed for open one-place predicates within the system, the left corner for ‘subjects’ to which those predicates apply. Here the graphical element as a whole represents the claim that the application of the one-place predicate to the subject term is demonstrable within the system. With a predicate for ‘theoremhood’ and this representation for demonstrability within a system, theorem status or the lack of it really appears in two ways. We can lay out the relationships explicitly as:

(b) ‘Applicability’, represented by our simple solid arrow, and theoremhood, represented by an arrow tagged by s, are both bivalent. Something is either a theorem of the system or it is not. A predicate actually applies to its subject or not. That bivalence gives us the following:

(c)

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With these preliminaries, we are in position to graph the basic dynamics of Gödel’s proof—not too far, we’ve noted, from Gödel’s own informal presentation. In the Liar, our entire argument was in terms of a single arrow of applicability between subject and predicate. Here we use two arrows: simple applicability as before, but also derivable applicability within a system. It is the relation between those two—or the lack of it—that is at the core of the incompleteness result. What Gödel shows is that no formal system of the sort at issue can be both sound and complete. A system is sound if everything demonstrable as a theorem in the system is true—if all theorems are true on interpretation. A system is complete if everything expressible in the system that is true on interpretation appears as a theorem. Within our graphic conventions, a system will be sound and complete if:

sound

complete For each of these modus tollens gives us

Beyond the self-embedding character of ‘I am not a theorem,’ all our assumptions up to soundness and completeness are either conventions of representation or assumptions as innocuous as bivalence for applicability and ‘is a theorem’. Once we add assumptions of soundness and completeness, however, we get oscillation. Here the pattern

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of oscillation is more complicated than that of the Liar because it travels through both applicability and system derivability, ‘is a theorem’ and ‘is not.’ The basic form, however, is much the same. At the top of the cycle we start by assuming that ‘I am not a theorem’ is demonstrable within the system:

Contradictions appear in our oscillation in representations directly across from each other, and appear within both claims of applicability and system derivability. Given the self-embedding character of ‘I am a theorem’ and the innocuousness of other assumptions, the only way to avoid oscillation—and contradiction—is to break the circle at either soundness or completeness. On pain of contradiction, then, no system of the sort at issue can be both sound and complete. It is important to note that Gödel gives us a positive result regarding the limits of formal systems rather than simply an oscillational paradox. The key, of course, is that in this instantiation our conceptual pattern involves two major concepts, rather than one: truth or applicability on the one side, and theoremhood or demonstrable applicability on the other. If the two are identified, as an assumption of soundness and completeness would demand, we get an oscillation with the essentials of the Liar. On pain of that oscillation and its incumbent contradiction, we get the positive result that the two therefore cannot be identified: formal derivability cannot be coextensive with truth.

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3. DETAILS AND EXTENSIONS Popular presentations in terms of ‘I am not a theorem’ capture something important and central in the Gödel result. But there are also important aspects they leave out. Formal systems, we have said, inevitably lead double lives. They are, on the one hand, well-oiled syntactic machines, designed to operate purely syntactically. The purposes for which they are designed and implemented, on the other hand, are semantic. They are intended and interpreted as systematizing and formalizing a body of truth— mathematical truths regarding numbers, for example. What Gödel’s proof relies on is a second layer of interpretation as well. Using the technique of Gödel numbering, the syntactic formulae of such a system can be shown to live not merely a double but a triple life. Syntactic formulae are interpretable, in precisely the sense of Principia Mathematica, as claims regarding numbers. But they are also interpretable, in an importantly reflexive way, as claims regarding formal properties of the system itself. These details of the Gödel result can be outlined in terms of several clever tricks. The first trick is that we can recoverably encode both formulae of our system and series of formulae as specific numbers. This can in fact be done in any of an infinite number of ways—there is not one Gödel numbering but infinitely many. The second trick of the Gödel result is to note that there will be genuine properties of numbers that correspond to syntactic properties of the formulae they encode by our chosen Gödel numbering. There will be a numerical property, for example, that holds of all and only those numbers that Gödel-number axioms of the system on our chosen encoding. There will be a numerical property of two numbers that will hold just in case the second encodes a formula that follows from the former by a syntactic rule of inference of the system. Because a derivation is simply a series of formulae of the system each of which is an axiom or follows from an earlier step by a rule of inference, there will ultimately be a numerical property that applies to pairs of numbers just in case the first numbers a derivation for the second. That a certain number is the second of a pair of numbers with that

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property will be a property of numbers N corresponding to derivability within the system of the formulae those numbers encode: a property N that will hold of all and only numbers that encode theorems of the system. The sentence which in fact plays the role of ‘I am not a theorem’ in the dynamics of Gödel’s proof is a sentence which applies the negation of this property N to a number that encodes that sentence itself. Representing that sentence as ‘I am not a theorem’ captures its role on one level of interpretation—that mediated by Gödel numbering—very well. What it tends to obscure is the more mundane level of interpretation on which the sentence, like others of the system, represents merely a specific claim regarding certain numbers. Given the two levels of interpretation, carefully constructed to operate in parallel, that claim regarding numbers is one which is true if and only if the system cannot capture it as a theorem. With regard to straight claims regarding numbers—the telos of systems of number theory—no formal system can be both sound and complete. Many presentations of Gödel’s result overlook its ubiquity. Crucial to the result is the assumption of a given Gödel numbering for the system. It is in terms of that assumed encoding that a formula has a particular number, and in terms of that encoding that a particular numerical property holds of a number just in case that number encodes a theorem of the system. As noted, however, there are infinite alternatives for Gödel-numbering any system. There are indeed alternatives for encoding that will never occur to us. For each of these there will be a Gödel sentence that plays the role of ‘I am not a theorem.’ What Gödel shows, in the dynamics of the structure we have outlined, is a disjunctive conclusion: that Principia Mathematica and similar systems must be either unsound or incomplete. The standard assumption is that the latter is the case: because assumed to be sound, standard systems of number theory must inevitably be incomplete, and the sentence at issue must therefore be a truth on interpretation that is not captured as a theorem. On the assumption of soundness, there will for each possible Gödel-numbering be a numerical truth that is not captured as a theorem. But whether or not a numerical claim is demonstrable is entirely independent of our encoding choices. Each possible encoding, therefore,

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reveals a mathematical truth incapable of demonstration within the system. Each possible encoding, indeed, reveals an infinite field of such claims, since any claim that is demonstrably equivalent to the target claim must have the same status. Were any of those demonstrable, so would be the target claim. These too must be truths our system fails to capture. Gödel’s proof is remarkable for being a constructive demonstration of incompleteness, offering a recipe for actually constructing a specific problematic formula for any system. But because of that the Gödel result is often presented as if it shows a small and constrained formal gap. What the result actually shows is a much more widespread limitation: for any formal mathematical system, there will be infinite fields of mathematical truth that will lie beyond formal grasp. Here we have indicated some of the details of the Gödel result as keys to its breadth and power. A more complete representation of the dynamics of the proof with full details is left to an Appendix, as is development of Tarski’s closely related but equally important theorem on the formal undefinability of truth. 4. TURING AND THE HALTING PROBLEM In Turing’s Halting Problem reflexivity reappears in the guise of computation theory. Here we deal not with formal axiomatic systems but with idealized symbol-manipulation machines. Turing machines are a standard model for computation—for any computation—unrestricted as to processing time or memory. That model of computation is so powerful that nearly everyone in the field accepts Church’s Thesis: that any algorithm can be instantiated as some Turing machine. Alternatively put, the thesis is that anything computable step-by-step by anything—machine, person, or angel— can be computed step-by-step by some Turing machine.3 Turing’s first achievement was to show how much could be computed by an abstract machine as simple as he envisaged. His second achievement was to show a limitative theorem corresponding to Gödel’s: that there are some things that nonetheless could never be computed by such a machine.

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We began in chapter 1 with categories of things that stand in certain relations to themselves: sets that take themselves as members, for example, or words that apply to themselves. We considered sentences that take themselves as subjects. In the Gödel result we have formulae within axiomatic systems which by Gödel numbering have a complex interpretational relationship to themselves. In the Turing result we envisage machines which apply to their own descriptions. Any Turing machine operates on input, canonically envisaged as set of symbols on an infinite tape. But it is also the case that the operation of any Turing machine can be finitely encoded, either as a flow diagram or equivalently as a machine table or linear series of quadruples or quintuples. We will call these ‘machine descriptions’: they are essentially sets of ‘if . . .then’ instructions of the form ‘If you are in state 25 and input the symbol 1, output the symbol 0, move one space to the left, and go into state 12.’ The important point is that the operating instructions for any Turing machine can be written out as a series of symbols—precisely the type of thing that a Turing machine is designed to take as input. A Turing machine can therefore take the operation description of a Turing machine as input, and can moreover take as a two-part input a machine description and an input which might be fed to that machine. This capacity is essential to what Turing envisaged as the Universal Turing Machine, which is in fact specifiable. The Universal Turing Machine takes as dual input (a) the specification of a Turing machine and (b) a secondary input, then simulating the operation of that machine on that input. Since a Turing machine can take a Turing machine description as input, or as part of its input, a Turing machine might also take its own description as input. Reflexivity of this type should by now be familiar. In formal systems theory it forms the core of the Gödel result. In computation theory it lies at the core of the Halting Problem. In the Gödel result we noted a crucial split between a systemproperty—theoremhood—and an interpretational desideratum—truth on interpretation. A similar split is important in the Halting Problem, though here that split is between what a machine signals as output and what is in fact true.

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The Halting Problem shows important limits to Turing machine computability: the fact that no Turing Machine can signal truthfully certain facts regarding the operation of any arbitrary Turing machine. As we will see, that theorem has a broad application with regard to a wide range of properties. It is usually introduced, however, in terms of the single property that Turing invoked: a proof that no Turing Machine can tell you, for arbitrary Turing machines, whether that Turing Machine will halt on a particular input or not. If Church’s thesis is right, the proof shows important limitations to computability in general. Here as before the argument proceeds as a reductio to contradictory oscillation. Suppose, contrary to what is to be proven, that there could be a Turing Machine that could tell you, for any arbitrary Turing machine, whether that machine would halt (or halt in standard configuration) when started on a given input. We will label that hypothetical machine M2, specified graphically as follows:

(h) Here we use our icons for something new. The space on the right is reserved for machines or machine specifications, the space on the left for input. Our corner brackets indicate a machine encoding—quadruples or quintuples representing the details of a specific machine’s operation, for example. That encoding is itself capable of being taken as part of input. In the diagram above we use corner brackets to indicate that the second part of the dual input to M2 represents a machine encoding. The diagram as a whole indicates how M2 behaves on that input relative to the behavior of the machine itself. It should be remembered, however, that inputs come purely as inputs— simply as strings of symbols on which the machine operates. They do not come labeled as machine codes as such.4 Our ‘h’ arrow above indicates that the machine in its right corner halts when started on the input in its left. ‘h1’ and ‘h2’ indicate that the machine at issue halts with a particular result on the relevant input: a signal ‘1’ for example, or a signal ‘2’. As a whole, then, the di-

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agrams specify that M2 halts with signal 1 when fed an input encoding a machine specification M and input I just in case the machine specified does in fact halt on I. M2 halts with signal 2 on such a input just in case the machine at issue does not halt on I. A machine that fit the specifications we’ve outlined for M2 would qualify as an infinite loop detector. One of the plagues of computer programming is the prospect of infinite loops, which lead a computer into infinite cyclic computation with no exit and to no effect. It would certainly be nice to have a detector for debugging which, when fed any program, would tell us whether that program contained any of the dreaded infinite loops. Given the fact that Turing machines are so incredibly flexible—flexible enough to underwrite near universal support for Church’s Thesis—it might seem initially plausible that a universal loop detector should be within reach. But what the Halting Problem shows, on pain of contradictory oscillation, is that a universal form of any such detector is flatly impossible. Much of the work and cleverness of the Gödel result lies in showing that a particular encoding of particular properties of the system is possible. Much of the work and cleverness of the Turing result lies in showing that, given M2, a further machine M3 would be possible as well. Here three simple facts about Turing machines are of importance: • The first simple fact is that we can build an ‘input duplicator’: in a very few states we can construct a machine which takes any input and ‘xeroxes’ it, leaving us with two copies instead of one. • Another simple fact about Turing Machines is that we can take any machine that signals ‘1’ or a ‘2’ or some other symbol at a point of output and change it so that the machine goes into an infinite loop at that point instead. • A third simple fact about Turing Machines is that we can compound them: we can connect an output of a machine A as input to a second machine B, effectively creating a hybrid machine which first does the work of A and then proceeds to do the work of B.

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With a hypothetical M2, Turing uses these simple facts to conceptualize a further machine M3. M2 itself might be envisaged as follows, taking a two part input of I and machine coding M, outputting a ‘1’ if that machine started on I halts and a ‘1’ otherwise:

By the first fact above we can use a duplicator to ‘double’ an input. By the second we can change the top output above to feed into an infinite loop. By the third fact we can string these together to create machine M3:

If M2 is possible, so is this M3. What M3 does is take an input on the left and feed it through a duplicator, resulting in two identical strings fed to the altered M2 as both target input I and machine specification M. If, so read, the machine at input would halt on input I, M3 goes into an infinite loop. If not, M3 halts, here with an output of a simple ‘2’. M3’s operation relative to M2 can thus be specified as follows:

In each case there are three empty slots, and the construction of M3 from M2 above entails that all three must be filled with the same entry. It is worthy of note in this specification, moreover, that the second part of the input for M2 is not specified in terms of machine coding. As indicated above, the two (identical) parts of the input in this speci-

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fication could be anything, though if the second encodes a machine it will be treated as such by M2. On the assumption of a machine M2, the diagram above would allow us to construct a genuine Turing machine M3. Every machine, we have said, can be encoded in a form suitable for input. M3 can therefore be encoded in such a way, and can take its own encoding as a possible input. Inserting M3’s encoding in the slots above gives us the following:

(i) Will M3 started on its own encoding as input halt, or not?

The hypothesis of a machine M3 started on its own code gives us an oscillation even more closely akin to the Liar than even the Gödel result. Here our complexes are not sentences composed of subject and predicate, however, but complexes of machines and inputs. Reflexivity arises not by the fact that a sentence can take itself as subject but by the fact that a machine can take its own encoding as input. The special element of such a complex that gives rise to oscillation is not a predicate that holds of all and only sentence complexes in which predicate does not hold of subject, but rather a machine that goes into a loop in

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the case of all and only those machine/input pairs in which the machine at issue does not go into an infinite loop on the input at issue. It is the structure of such elements, rather than their instantiation in terms of sentences or machines, that is the driving mechanism of oscillation. If a universal halting detector M2 is possible, we’ve said, so is M3. But were M3 possible, we could start M3 on its own code, and that hypothesis leads directly to the contradictory oscillations graphed above. On pain of oscillation, there can be no M3. On pain of oscillation, then, there can therefore be no universal halting detector. Like Gödel’s theorem, the Halting Problem is constructed in terms of a crucial split, and that split allows an alternative to contradictory oscillation—indeed forces us to that alternative. Here the split is between a machine’s verdict with regard to some machine and input— that it will halt, for example—and the truth of the matter: whether it will or not. What has been shown is that the assumption of a machine that will always give us a correct verdict leads to oscillation. The way out is to abandon the idea that any machine can do that: to abandon the notion that any machine could give us a correct verdict for machine-and-input pairs in all cases. Just as Russell’s paradox is taken as showing that there can be no set of all non-self-membered sets, the Halting Problem is taken as showing that there can be no universal loop detector of the sort envisaged. 5. THE LIMITS OF COMPUTATION Turing’s original presentation is phrased in terms of the impossibility of a universal loop detector, as is our representation above. What is often overlooked is how broadly the same conceptual structure applies. Universal detectors are problematic quite generally, independently of the special issue of halting. For a broad range of other properties, for precisely the same reasons, a universal mechanical detector will be impossible. If Church’s thesis is correct, there will be no detection algorithm possible for those properties at all. Some machines, started on some inputs, will give as a result the simple signal ‘1.’ Some will not, with a result either simpler or more complex. But there can be no machine which will infallibly tell us, for

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an arbitrary machine and input, whether that machine started on that input will give us a signal ‘1’ or not. If we suppose there were such a machine, we can simply duplicate the alterations made from M2 to M3, including the initial duplicator, except that we will ‘cross’ outputs differently. Instead of replacing the ‘1’ output of M2 with an infinite loop we replace the positive output of this machine with an operation that produces something more complex than ‘1’—’111’, perhaps, replacing the output corresponding to the negative ‘2’ of M2 with a simple ‘1’. Would that expanded machine, started on its own encoding, give us a simple ‘1’ as output or not? The hypothesis that it would leads us, in following the operation of the machine, to an output of ‘111’ and the conclusion that it would not give us a simple ‘1’. We must then conclude that it would not give us a simple ‘1’, but that forces us in turn to the conclusion that it would. The pattern of oscillation above has simply been instantiated for a different property in place of halting. There can be no Turing machine which tells us, for all Turing machines and inputs, whether our output would be an even number, or an English word, or greater than 12, or a string of 1’s. Contrary to widespread impression, the Halting Problem has nothing essentially to do with halting. What the proof shows is that machines capable of detecting all and only those machine-and-input pairs which will produce outputs with any of an unlimited range of formal properties are simply impossible. 6. ‘INS’ AND ‘OUTS’ In chapter 1 we saw that weakening ‘all and only’ conditions for Russell’s paradox, the Heterological paradox, and the Liar not only broke the cycle of oscillation but resulted in the phenomenon of automatic ‘ins’ and ‘outs’: sets that had to be self-membered or could not be, for example, sentences that had to be within the extension of their predicates or could not be. Given the similarity of structure, it is to be expected that we have automatic ‘ins’ and ‘outs’ in the realm of the limitative theorems as well. And we do. A given Gödel numbering ensures that the readings of a specific formula G as (a) a statement of number theory and as (b) a statement

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about itself and the system—‘I am not a theorem’—will be true or false together. That statement of number theory will be true if and only if it is not a theorem. But if our system is sound—if all theorems correspond to number-theoretical truths on interpretation—formula G cannot then be a theorem. Were it a theorem, it would be true on interpretation, and thus it would be true that it was not a theorem. On the assumption of soundness, then, ‘I am not a theorem’ is an automatic ‘out.’ But the Gödel result operates in terms of concepts of both theoremhood and truth on interpretation. The interesting point is that it is precisely because ‘I am not a theorem’ is an automatic ‘out’ with respect to theoremhood that it is an automatic ‘in’ with respect to truth. Precisely because it is not a theorem, and that is what it ‘says’ through levels of interpretation, it must be true. Automatic ‘ins’ and ‘outs’ give us another way of making the central point: If all theorems represent truths on interpretation, there will be a formula that is an automatic ‘out’ with respect to theoremhood but an automatic ‘in’ with respect to truth. Any formal system adequate for number theory, if sound, must be incomplete. The Halting Problem generates ‘ins’ and ‘outs’ as well. M2 was guaranteed to output a ‘1’ if a machine halted on a particular input, a ‘2’ if it did not. Consider a more limited relative M2' guaranteed to output a ‘1’ in all cases in which a machine halts on a particular input, but perhaps not only in those cases. If we build an M3' around this variant, like our M3 above in all other respects, the result is not full oscillation. What we get instead is M3' as an automatic ‘out’: M3' cannot halt when given its own specification as input. Assume the contrary: that M3' does halt on its own encoding. Were that the case, M2' fed a dual input of M3' encoding would guarantee an output of ‘1’. But the specification we have laid down for M3', constructed from M2', makes it clear that M3' would then actually go into an infinite loop, contradicting our assumption. The alternative assumption that M3' does not halt on its own encoding as input, however, is entirely consistent. On the assumption of an M2' that signals all but not only cases of halting, it is that conclusion we are forced to. If on the other hand we assume an M2" guaranteed to output a ‘1’ in only cases in which a machine halts on a particular input, but per-

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haps not all of these, we get an automatic ‘in’. An M3" constructed on the basis of M2" cannot consistently be assumed not to halt on its own input. Were it to do so, M2" fed a dual input of M3" encoding would signal a ‘2’, leading directly to a halt. Given the limitations of M2", however, the assumption that M3" does halt on its own input is entirely consistent. On the assumption of an M2" that signals only but not all cases of halting, M3" must be an automatic ‘in’ with regard to halting on its own input. 7. THE UNIVERSAL PREDICTION ALGORITHM Given a structural understanding of the limitative results, we should be able to generate other results with that same structure. And indeed we can. We can show, for example, that there can be no universal prediction algorithm. For suppose there were: suppose a set of procedures which, on a given input of events before time t, would give us an accurate prediction of events after time t. In order to make things simple, suppose a set of procedures which would allow us, on the basis of data regarding events before 10:00 on Tuesday, to predict for all cases and with complete accuracy any event occurring at precisely 11:00 on that day. If our universe is one of deterministic causes, one might indeed think such a procedure possible. Given such an algorithm, we can imagine a written form of its projection for any case. Were there such an algorithm, there would be steps we could take to express its prediction for 11:00 on Tuesday in any chosen format. We can make the case even simpler. Consider a predictive algorithm for an extremely limited series of events: whether a particular machine M, operating on Tuesday, will turn on a green light at 11:00 or not. That is all that we demand that our prediction algorithm be capable of doing: of predicting, from events prior to 10:00, whether a machine will flash a green light at 11:00 or not. We word-process the prediction so that it ends with either the words ‘M will flash a green light’ or ‘M will not flash a green light.’ Now, however, let us complete the description of machine M. M is a machine which, fed a formatted description that ends ‘will flash a green light’, is set so that no light will go on. If fed a formatted de-

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scription that ends with the words ‘will not flash a green light,’ it turns on a flashing green light. You and I, tinkering in the garage with old computer parts, could create such a machine in a matter of hours. On Tuesday, at 11:00, we will feed the formatted version of whatever prediction our algorithm produces into the machine we have designed. Will that prediction specify that the machine will turn on the green light, or not? Our algorithm was conceived as one that was accurate for all cases. But it is clear that it cannot possibly be accurate in this case. Should our algorithm predict that the machine ‘will flash a green light,’ it will not. Should it predict that the machine ‘will not flash a green light,’ it will. There can therefore be no universally accurate predictive algorithm.5 For any algorithm we might consider, there will be a case for which its prediction is guaranteed to be wrong. What holds for algorithms holds for machines and holds for human predictors as well. There can be no machine capable of flawlessly predicting the future, for it could not accurately predict what a carefully designed machine would do when fed its prediction. There can be no gypsy fortune teller capable of flawlessly predicting the future, for she could not accurately predict what a carefully designed machine (or a perversely inclined person) would do on reading her prediction. The lesson is that the limitative theorems, though of a form perfected in the context of axiomatic systems and general models for computation, are not limited to those systems and models. Theirs is a conceptual structure that will reappear in other cases as well—cases, for example, of prediction. The structure of the argument against a universal predictive algorithm is precisely that outlined for the Halting Problem. In place of a machine intended to detect halting in all machines, we have an algorithm intended to predict accurately all future events. In the case of the Halting Problem we build a particular machine using the conjectured halting-detector which will give us a result directly opposite to what the detector determines for that machine. In the prediction case we construct a future light-blinking machine, operating at a designed time, which will give us a result directly opposite to what our prediction algorithm determines for that machine at that time. The concep-

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tual structure is the same in both cases: a structure involving reflexivity, application of a crucial predicate, and negation of that predicate. In this case, as in the Halting Problem and Gödel’s theorem, it is also crucial that there is a split: a split between a formula’s status as a theorem and its truth on interpretation, between the halting machine’s determination regarding a Turing machine and the machine’s actual operation, or between what the fortune teller predicts and what actually occurs. It is because of that split in each case that we get a limitative theorem—on systems, machines, or predictions—rather than the oscillation characteristic of the Liar. Here too it is perhaps worth noting ‘ins and outs’. The prediction result depends on demanding ‘all and only’: that our algorithms offer all and only true predictions. Here as elsewhere we get different results when that requirement is weakened. Consider for example an algorithm designed to give only true predictions, but perhaps not guaranteed to give predictions in all cases. Consider further a machine designed—like that envisaged above—to input a formatted prediction and output a result in contradiction to it. Here we would not get the conclusion that there could be no predictive algorithm of this limited ‘only but perhaps not all’ form. All that would follow is that such an algorithm could give no prediction in this particular case. 9. CONCLUSION In this chapter we have graphed the basic structure of two central 20th century theorems in metamathematics and computation theory, indicating their structural parallels with the paradoxes considered in chapter 1. The paradoxes, however, are phrased in terms of a single concept—a concept of truth, for example, or set membership. The limitative theorems operate in each case with a pair of concepts instead: machine projection of machine operation as opposed to true machine operation in the case of Turing’s Halting Problem, for example, and formal derivability as opposed to truth on interpretation in the case of Gödel’s theorem. In each case it is the attempt to take those concepts as coextensive that results in oscillation on the pattern of paradox: the force of the limitative theorems is precisely that on pain

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of oscillation the concepts in each pair must be treated as distinct. In precisely the same way, we’ve argued, such a structure shows that no algorithmic prediction can be identified with predictive truth. In chapter 3 we follow the trail of structural reflexivity into less formal but conceptually richer territory. There the topic is automatic ‘ins’ and ‘outs,’ both logical and pragmatic, within the context of philosophical practice. NOTES 1

“Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I,” Monatshefte für Mathematik und Physik 38 (1931) 173-98, trans. in Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1979-1931, Harvard University Press, Cambridge Mass., 1967, pp. 595616.

2

See Douglas Hofstadter, Gödel, Escher, Bach: The Eternal Golden Braid, New York: Basic Books, 1979, and Ernest Nagel & James R. Newman, Gödel’s Proof, New York: Routledge, 1958.

3

See however B. J. Copeland, “The Church-Turing Thesis,” in E. Zalta, ed., Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/church-turing/, 2002.

4

It should also be noted that because of our brackets the specification above is incomplete: it tells us conditions for M2 when its second input encodes a machine, but does not tell us what it does when the second portion of its input fails to do so. For present purposes that omission is harmless; in all cases at issue that second portion will in fact encode a machine. In general, M2 might be thought to do any of various things where the second part of input fails to encode a machine: offer a third signal ‘3’, for example, or fail to halt altogether.

5

Our argument here is similar to several that appear in chapter 12 of Nicholas Rescher, Predicting the Future: An Introduction to the Theory of Forecasting, Albany, NY: State University of New York Press, 1998, which in turn owe a debt to John L. Casti, Searching for Certainty, New York: Morris, 1990. Reflexivity is crucial across the entire range of examples: guaranteed failure cases for universal prediction machines, algorithms, or gypsy fortune tellers all rely crucially on ‘cross-circuited’ consequences of their own predictions, precisely in the manner of the Halting Problem.

Chapter 3 NIHILISM, SKEPTICISM, AND THE COGITO

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he graphical techniques developed to map the conceptual structure of oscillational paradox in Chapter 1 pointed up the associated phenomena of automatic ‘ins’ and ‘outs’. ‘True,’ for example, is a predicate applicable to all and only truths, and gives us the oscillational Liar: • This sentence is not true. The hypothesis that the sentence is true leads to the conclusion that it is not; the hypothesis that it is not true leads to the conclusion that it is. ‘Troop,’ however, was defined as applicable to all but perhaps not only truths. ‘Troof’ was defined as applicable to only but perhaps not all truths. The first of these gives us an automatic ‘in’: • This sentence is not troop. Here the hypothesis that the sentence is not troop cannot be maintained, forcing the alternative (and consistent) hypothesis that it is troop. We get the opposite conclusion with • This sentence is not troof. The hypothesis that this sentence is troof leads to contradiction, forcing the (consistent) conclusion that it is not troof. Here we have an automatic ‘out’. In Chapter 2 we extended the application of graphical techniques to major limitative results in 20th century logic. In this chapter we track core conceptual structures—particularly the phenomena of reflexive ‘ins’ and ‘outs’—into the important philosophical territory of self-

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refuting and self-validating positions: Nihilism, Skepticism, and the Cogito. 1. ALETHIC ‘OUTS’: THE CASE OF NIHILISM The Liar and its relatives are reflexive sentences in which it is only the sentence itself that is taken as subject: this sentence is not true. But what of cases in which the scope is wider, such as the following?: 1. Nothing is true. This is a form of Alethic Nihilism, and it gives us an automatic ‘out.’ This sentence cannot be true, since if it were true it would not be—contradiction. But there is no contradiction in assuming that the sentence is false. Given options of only ‘true’ and ‘false,’ therefore, this sentence must be false on pain of contradiction—an automatic ‘out’. Graphic techniques can be used to reveal conceptual structure here too, though they become more complex with the introduction of quantifiers. We will represent a universal quantification, to the effect that all things in a subject position have the property indicated in a predicate position, as on the left below. The claim that no s’s are P will be represented as on the right.

The s of ∀s effectively restricts our quantifier to s’s; within that scope the form to the left licenses the form to the right licenses

with any substitution in s position; with any s.

Our existential quantifiers are these, in each case requiring only that the form holds for some substitution in s position:

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With these complications, of course, contradictions take a different form. Consider the four quantificational representations above. In an arrangement familiar since Aristotle’s square of opposition, it is the formula on the upper left that contradicts that on the lower right, and that on the upper right that contradicts the formula on the lower left. In what follows we will be particularly concerned with quantified forms in which the ∀s is a quantifier over statements or claims, and thus is effectively a propositional quantifier. Allowable substitutions will include simple subject-predicate constructions, but will include quantified expressions as well. The graphics below, sketched for ‘is false’ F, outline a form of quantifier negation evident in such a case:

Alethic Nihilism of the sort represented by (1) is self-defeating: the position itself entails that it cannot be true. In terms of truth, Alethic Nihilism is an automatic ‘out’. There are actually two ways in which we can capture the reasoning involved. We can diagram the first as follows:

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This is the reasoning with which we began. We take ‘nothing is true’ as ‘everything is false,’ represented with a universal quantification over all statements s. This includes quantified statements, specifically that statement itself, giving us the second inferential step above. By the form of quantifier negation outlined above, we get our third step, the explicit contradiction outlined in square of opposition form. The second form of reasoning will be of further importance in what follows. If all statements are false, then this statement in particular is false: 2. This sentence is false. This, of course, is the Liar, a sentence S such that

But a hypothesis of falsehood for (2), as diagrammed in chapter 1, leads to oscillation. A second argument against Alethic Nihilism is therefore that it cannot be true on pain of oscillation in a particular instantiation. Graphically, the situation is this:

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It is only by way of the Liar, however, that the Alethic Nihilism of (1) leads to oscillation. The first argument above gives us a contradiction in quantificational form, without a full oscillation. Why is that? The difference is entirely one of scope. The Liar applies to a single case: itself. The Alethic Nihilism of (1) applies to all sentences, of which (1) is merely a single instance. Because the Liar’s reference is to itself alone, the supposition that it is false (given that it says it is false) forces us to assert that it is true—the crucial second half of the oscillation. Because (1)’s reference is not to itself alone, the conclusion that (1) is false forces us to assert that something is true—but since that does not entail that (1) in particular is true, we do not get the second half of the oscillation. Our principles do not therefore allow us to complete the second half of an oscillational cycle in the first large figure above. We get a simple ‘out’ instead. Indeed the dynamics of Alethic Nihilism’s self-defeat are very much the dynamics of ‘This sentence is not troof’ as an automatic ‘out’. It is crucial to the conceptual structure of Nihilism’s self-defeat that the statement applies to all statements, thereby including itself. It is interesting to note that the same structure would give us a similar selfdefeat—and a similar automatic out—even if the scope was not so wide as to include all sentences. Consider for example: 3. No universal generalizations are true. Because this is a universal generalization, (3) cannot be true: it is an automatic out. Were it the only universal generalization, it would give

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us an oscillation. But it is not: although it cannot be true, it can be false, given that some universal generalization is true. The point at issue is therefore purely one of scope. The following is an automatic ‘out’ for precisely the same reason: 4. No six word sentence is true. Consider now a philosophical claim that is a contrary of Alethic Nihilism, of equally wide scope: 5. Everything is true. (5) represents a position as self-defeating as that of Nihilism, though by a slightly more complicated mechanism. If everything is true, then the negation of that claim is true as well: the claim that not everything is true. Contradiction. (5) therefore cannot be true—it too is an automatic ‘out’. Other argumentative routes to the same conclusion can be taken through the patterns of arguments outlined for Alethic Nihilism above. The fact that both ‘Everything is true’ and ‘Nothing is true’ give us automatic ‘outs’ entails that another claim will be an automatic ‘in’: 6. Some claims are true and some claims are false. This claim must be true, since both disjuncts of its negation— 7. Either no claims are true or all claims are true. are automatic ‘outs’. 2. QUALIFICATIONS AND EXCLUSIONS Alethic Nihilism, we have argued, is an automatic ‘out.’ But what of a more qualified or mitigated Nihilism, such as the following? 8. Nothing is unqualifiedly true.

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9. Everything is only approximately true. 10. Nothing is obviously true. On the pattern above, each of these will apply to themselves. (8), for example, entails that (8) is true only with qualifications. (9) entails that (9) is only approximately true. (10) entails that (10) is not obviously true. Do these generate contradictions in the same way that Alethic Nihilism does? What this question points up, interestingly enough, is the issue of what constitutes a contradiction.1 Certainly the result in each case is not as direct as the ‘P and ~P’ of Alethic Nihilism. The result is, however: • P, and P only with qualifications. • P, and P only approximately. • P, and the truth of P is not obvious. The third case seems harmless; I can certainly hold that some P is true while recognizing that its truth is far from obvious. In the first two cases, though we do not have an explicit contradiction, one might argue that we have something similar. In those cases a claim is asserted simultaneously with a backpedaling qualification on precisely that claim. In each case the backpedaling makes it obscure what claim is being made. If it is only approximately true that everything is approximately true, does that mean that some things, however few, are precisely true? If so, of course, we have not merely qualification but explicit contradiction: if some things are precisely true, it is simply false everything is only approximately true. If it is not unqualifiedly true that nothing is unqualifiedly true, does it follow that some things, however few, are unqualifiedly true? If so, we would again get explicit contradiction: if some things are unqualifiedly true, it will simply be false that nothing is true without qualification.

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It is precisely because they flirt with contradiction that qualified claims like the first two above raise questions of meaning, of what putting forward a claim demands, and thus of pragmatic contradiction in the sense discussed in later sections. The case of partial exclusion is very different from that of blanket qualification. Consider for example: 11. Almost nothing is true. 12. Very few generalizations are true. 13. Almost no seven word sentence is true. (11) is a variant of (1) that escapes the self-contradiction of full Alethic Nihilism. Because it uses an ‘almost’ rather than a full universal quantifier, it need not entail that it itself is false, and thus avoids the pattern of contradiction graphed above. The same is true for (12) and (13). One might also try tightening exclusion: 14. Nothing is true, except this claim itself. Because a wide net of claims are entailed by any claim, however, an exclusion of this form will not work. For consider also: 15. Some claim is true. 16. Either (14) is true or Moscow is a suburb of Chicago. Neither (15) nor (16) is the same claim as (14). But if (14) is true, (15) and (16) must be true as well. (14) will therefore be self-defeating. Although a Nihilism qualified with the vague quantification of (11) escapes the pattern of contradiction outlined for (1), the wide net of entailment from any statement means that any attempt at a more precise self-exclusion on the model of (14) will fail. We have explored qualifications and exclusions here in the context of Nihilism. Although we will not go into detail, similar points will

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hold regarding qualifications and exclusions in the case of Skepticism and with regard to both Semantic Nihilism and Semantic Skepticism. 3. EPISTEMIC ‘OUTS’: THE CASE OF SKEPTICISM The patterns we have traced above work in terms of a concept of truth, as does the Liar and as do the limitative results considered in chapter 2. The patterns themselves, however, can be instantiated in terms of other concepts as well, with partially parallel and equally interesting conclusions. Consider not Nihilism but Skepticism: 17. Nothing can be known. Unlike the universal Nihilism of (1), the universal Skepticism of (17) is not self-contradictory. It is self-defeating in another way, however. Although there is no contradiction in asserting that (17) is true, there is a contradiction in asserting that (17) can be known to be true. Universal Skepticism is an automatic ‘out’ not in terms of truth but in terms of knowability. This is particularly important for social and linguistic practices in which propositions are put forward as known or at least as knowable. Academic debate is quite naturally characterized in these terms: opponents put forward alternative positions as items of knowledge, open to either challenge or auxiliary support through further argument or evidence. But knowledge, belief, and supportability are also common assumptions in everyday discourse. We assume that people who tell us things believe those things, and that they take themselves to be in a position to know those things. When Mike tells me that there has been an attack on the White House, the natural pragmatic assumption is that Mike takes himself to know that there has been an attack on the White House. With regard to any practice conceived or constructed in terms of knowledge in this way, the universal Skepticism of (17) is an automatic ‘out.’ In understanding and graphing such a case, the fact that we are dealing with a particular practice will be crucial. In other regards,

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however, we will find that the conceptual structure closely parallels those considered in the case of truth. In offering an implicative dynamics diagram for the universal Skepticism of (17) we introduce a new convention for the domain of a practice. In this case our domain is one of purported knowledge, with a circle used below to enclose claims taken as known. Given that convention, however, we will broaden our graphic representations to indicate two forms of implicature. One is the form we have marked throughout, tracking the straight logical entailment of content. What a claim says carries logical entailments we will now mark with an ‘e’. The other form of implicature that we now mark flows not from the content of a claim but from the fact that it plays a particular role in a practice—in this first representation, from the fact that the claim appears within a domain of purported knowledge.

All three conventions are illustrated in a simple case above. The major circle represents a domain of knowledge, or purported knowledge. A simple proposition within it carries a content entailment, represented by ‘e’: in this example we have simply indicated that the proposition entails itself. But the fact that such a proposition is within the domain of knowledge carries a domain implication or a practice entailment as well: the p entailment that such a claim is known. Consider now (17): the claim that nothing can be known. If this is hypothesized within the domain of knowledge, we get the following result:

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Claim (17) entails by content e that no propositions are known, and thus in particular that the proposition that no propositions are known is not known. If offered as an item of knowledge, however, the practice entailment p is that such a proposition is known. The two claims on the right are explicitly contradictory. This pattern is one that we will see repeatedly: a pattern of pragmatic contradiction. The skeptical claim is not itself inconsistent: it does not content-entail both p and ~p. If offered as an item of knowledge, however, it does practice-entail p while content-entailing ~p. However consistent it is on its own, it cannot be maintained as part of any practice that assumes knowledge. Because normal communication assumes that a speaker puts forward claims that they are in a position to know, a claim of universal Skepticism is pragmatically excluded. To the extent that argumentative defense presupposes a claim to knowledge—at least to knowability—a position of universal Skepticism is pragmatically self-defeating within any context of argumentative defense. Consider in a similar light: 18. This sentence is argumentatively indefensible. or 19. This sentence cannot be successfully argued for.

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If (18) is argumentatively defensible, it can be successfully argued that it is argumentatively indefensible. Here again we have a conceptual conflict, akin to but not the same as literal contradiction. With ‘untrue’ in place of ‘argumentatively indefensible’ we would get a strict contradiction: given basic principles regarding truth, the Liar has both ‘the Liar is untrue’ and ‘the Liar is true’ as strict entailments. The conflict in this case, in contrast, is between a strict entailment on the one hand (‘this sentence is argumentatively indefensible’) and a pragmatic entailment on the other (‘this sentence is argumentatively defended’, or ‘this sentence is argumentatively defensible’). Here again the feeling of contradiction turns on a practice; what is at issue is pragmatic contradiction. (18) and (19) are sentences for which one cannot attempt to give argumentative support without thereby defeating that very end. These are pragmatic ‘outs’ with regard to any practice of arguing for or against claims. Consider in the same light 20. No truth can be established by humans. If (20) is true, it cannot be established by humans. (20) cannot therefore be put forward in human discourse as an establishable claim. (20) offers a pragmatic contradiction within the human practice of establishing claims, and is an automatic ‘out’ in that domain. Here again it is clear that (20) is not explicitly contradictory. In that sense (20) might in fact be true, though of course unestablishable by us. The conflict that (20) represents is not one between claims that assert and deny some p—the realm of strict contradiction—but between a claim and the assumptions of a practice—in this case, the assumptions of a human practice of establishing claims. Within the practice of establishing claims, all of the forms of Skepticism we have considered are self-defeating: although on logical grounds alone they might be true, they could not possibly be established. In the light of our epistemic practices, they forever remain automatic ‘outs’. We get much the same result with: 21. It is not rational to believe anything.

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If (21) is true, it is not rational to believe it. Within any practice that aims at rational belief, (21) will constitute a pragmatic contradiction— an automatic ‘out’. 4. MOORE’S PARADOX All of the cases considered in the previous section are examples of pragmatic contradiction within broadly epistemic contexts. Here classical locus is Moore’s paradox: 22. I have boots on but I do not believe it. We can graph Moore’s paradox as follows:

The content of (22) is by no means inconsistent. It might be true of George that he has boots on and does not believe it. That might be true of anyone, and thus of course might be true of me. There is in that sense nothing contradictory about (22). The oddity of (22) is not explicit or content contradiction, but pragmatic contradiction—the oddity of asserting, proposing, or putting forward (22) in conversation. What (22) violates is not our logic but our fundamental conceptual and communicative practices. It is a general pragmatic assumption that people put forward only claims that they believe to be true. This is in fact enshrined in Grice’s pragmatic maxim of quality: ‘try to make your conversational contribution true.’2 In putting forward the first conjunct of (22), a practice entailment is therefore that I believe it, displayed in terms of the lower

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arrow above. But the second conjunct of (22) explicitly entails that I do not believe the first conjunct—a content entailment displayed in terms of the upper arrow. Practice entailment contradicts explicit entailment, and Moore’s paradox gives us a pragmatic contradiction. (22) has an ‘I’ in two positions: It is I who don’t believe it, but also I who have boots on. The graphic representation, however, makes clear that it is only the ‘I’ in belief position that is crucial. We get the same problem with (23): 23. The White House is being attacked but I don’t believe it. Both (22) and (23) include claims that contradict a pragmatic assumption of everyday discourse—the assumption that one puts forward only what one believes. Within any practice that carries such an assumption, these claims are pragmatically self-defeating: the claims at the core of Moore’s paradox are automatic ‘outs’. In the analysis above we have exhibited contradictory practice and explicit entailments outside the circle of belief. But because the inferences at issue are so simple, an agent with minimal powers of deducibility and minimal self-reflection regarding his or her beliefs can also be expected to instantiate the following variation on the pattern:

Here pragmatically contradictory beliefs appear within the domain of my own belief: with minimal reflection, a belief in (22) would commit me to inconsistent beliefs. This holds despite the fact that (22) is not itself inconsistent. Although it is something that with even min-

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imal self-reflection I cannot possibly believe, it might in fact be true that I have boots on but do not believe it. 5. PRAGMATIC OSCILLATION Just as there will be automatic ‘outs’ with regard to epistemic, argumentative, and investigatory practices, there will be automatic ‘ins’, considered below. There will also be conceptual oscillations, some of which route through pragmatic contradictions. Consider for example: 24. This sentence cannot be put forward as established. (24) can be represented as a sentence of the form encountered in earlier chapters: a Droste effect sentence in which progressive embeddings are substitutable:

‘Put forward as established,’ unlike ‘established as true,’ does not entail truth. The first moves of the argument regarding Skepticism therefore will not hold in this case. But there is a clear argument that (24) gives us a pragmatic contradiction nonetheless. We can graph the argument as follows:

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(24) carries as explicit entailment its own embedding, shown on the right above—the claim that (24) cannot be established. But to put forward (24) as established carries the practice entailment that (24) can be established, giving us a pragmatic contradiction. If we take the derivation of a pragmatic contradiction in this spirit as a form of reductio, the conclusion is that (24) cannot be put forward as established. On pain of pragmatic contradiction, then, (24) cannot be put forward as established. But that, of course, is what (24) itself maintains. What (24) claims is therefore true. What our reasoning to this point indicates is that there are truths that cannot be put forward as established: a reality beyond the reaches of our practice. In the diagram below, our representation for (24) therefore appears on the right as a truth in its own right, as the conclusion for our pragmatic reductio—not merely, as before, as a hypothesis in a conditional proof. 3

At this point the argument becomes even more complex. Oscillation appears with the realization that we have just established the truth of (24): appears within the circle of establishment on the left of the diagram with which the entire problematic began.

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On reflection, it becomes clear that our entire argument exhibits the kind of self-embedding characteristic of many of our sentences throughout. In this case, oscillation turns at an important point on reflection regarding pragmatic contradiction. This case points up a structural feature worthy of note in our examination of reflexivity with regard to ‘ins’, ‘outs’ and oscillations. The claims we have considered have all been of one of three forms. Most have been universal, like ‘No claims are true.’ Some have been particular, with a form like that of ‘some claims are true.’ In both cases it has been important that the claims at issue can take themselves as instances—as one instance, it should be noted, among others. Some claims crucial to our analyses, however, have been more closely targeted to themselves alone, with a solely indexical reference. ‘This sentence cannot be put forward as established’ is a clear example. Reflexive argument of the form we have emphasized can show universal sentences to be self-counter-exampling: automatic ‘outs’, either logically or pragmatically when put forward in a practice. The same pattern of argument cannot, at least directly, be used to establish universal claims as automatic ‘ins’. Reflexive argument can show particular claims, such as ‘some claims are true,’ to be self-verifying or automatic ‘ins’. But it cannot show them to be automatic ‘outs’. The sole form that can be either self-counter-exampling or self-verifying is that of sentences which index themselves and themselves alone. Most

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noteworthy is the fact that it is only these that we have found to be directly oscillational. Direct oscillation seems to demand a very precise focus: a claim’s reflexivity regarding itself and itself alone. 6. SEMANTIC ‘INS’ AND ‘OUTS’ ‘Ins’ and ‘outs’ appear not merely in alethic and epistemic cases but in semantic cases as well. (25), for example, is an automatic semantic ‘out’: 25. This statement does not mention rabbits. (26), in contrast, is an automatic semantic ‘in’: 26. This statement refers to one or more statements. Semantic reflexivity of such a sort is can appear for questions as well. The first question below is self-answering. The second carries a presupposition that it itself refutes: 27. Are there questions? 28. Why does no-one ever ask a question? These examples carry an amusing air of triviality. But semantic ‘ins’ and ‘outs’ will be crucial for forms of Semantic Nihilism and Skepticism as well. Consider for example: 29. No utterance has any meaning. (29) will be pragmatically self-defeating on precisely the pattern graphed for (17) above. Its explicit entailment is that (29) itself has no meaning. Putting forward (29) as even a statement for consideration, however, carries the practice entailment that it is meaningful. Semantic Nihilism will be pragmatically self-defeating. Much the same will hold for blanket Semantic Skepticism:

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30. We cannot know what any claim means. An explicit entailment of (30) is that we cannot know what (30) means.4 A practice entailment of putting forward claims at all, however, is that we can indeed know what they mean; proposing (30) in practice presupposes that we can know what (30) means. (30)’s explicit entailments and practice entailments constitute a classic pragmatic contradiction. Within any semantic practice, Semantic Nihilism and Semantic Skepticism must be self-defeating. 7. PRACTICES AND VALUES Pragmatic contradictions render certain claims self-defeating in the context of particular practices. We have emphasized epistemic and semantic contexts above, but the phenomenon of self-defeating pragmatic contradiction can be expected to appear more widely as well. Consider for example a case involving an order from a Sergeant to a Private in the army: 31. Don’t obey my orders. That is an order. This too will give us a pragmatic contradiction, despite the fact that it is entirely imperative in tone. Consider first the perspective of the Private. If he maintains both (a) that as a Private he must obey all his Sergeant’s orders and (b) that his Sergeant has given the order represented in (31), he will be forced into what might be considered a contradiction in action. Is he to do what his Sergeant has ordered in this case, or not? On the basis of (a), as a form of practice entailment, the answer is ‘yes’. On the basis of (b), in terms of explicit content entailment, the answer is ‘no’. The contradiction in action of ‘both yes and no’ offers a clear imperative parallel to ‘both p and ~p’ in the case of declaratives. The case can also be analyzed from outside the Private’s perspective. If military code specifies that a Private must obey all orders from his superior officer, the fact that the Sergeant gave an order carries a practice entailment that the Private ought to obey it. A clear content entailment of this particular order, however, is that the Private ought

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not to obey it. In these terms we have a pragmatic contradiction much like those above, here phrased in terms of ‘ought’; the patterns traced above will extend to questions of value. The general lesson is that practices carry value assumptions, and that those assumptions too can form the basis of pragmatic contradiction. Our communicative practices carry the value assumption that truths are preferable to falsehoods: that true sentences have a positive value that false sentences lack. Our argumentative and evidential practices carry the value assumption that sentences that are more defensible than their alternatives carry more weight: that a sentence more defensible than its alternatives has a value that a sentence that cannot be defended against its alternatives does not. Against the background of those value assumptions, however, consider the Value Nihilism of (32): 32. Nothing has any value. If (32) is true, an explicit entailment is that its contradictory— ‘something has value’—has no more or less value than it does. But if put forward in communication or argument, the practice entailment is that (32) does have more value than its alternatives. (32) therefore gives us a pragmatic contradiction once again, here in the realm of values. The context of putting forward and defending claims inevitably brings certain values with it: the preferential value of true claims over false, for example, and the preferential value of supported claims over those for which there is no evidence or argument. Because that context instantiates particular values of its own, Value Nihilism will inevitably be a pragmatic ‘out’—a position which cannot successfully be put forward or argued for. Much the same argument can be made with regard to Value Skepticism: 33. Nothing can be known to have any value. Argumentative contexts are ones in which positions are put forward as known, or at least as knowable, and in which arguable claims are to be preferred over their alternatives. Against those defining conven-

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tions of the practice of argument, (33) will be insupportable. Put forward in argument, its practice entailment will be that it is preferable to its alternatives. But that is precisely contrary to its content entailment. Because practices of assertion and argument inevitably carry specific values, (33) cannot successfully be put forward in assertion or argument. How strong are arguments from pragmatic contradiction against Value Nihilism and Value Skepticism? One can imagine a proponent of Value Nihilism responding ‘It is precisely because nothing has any value that your practices, complete with their value assumptions, must be rejected.’ ‘If the truth of Value Nihilism conflicts with basic assumptions of your practices, so much the worse for your practices.’ What gives this response some teeth is that we have to admit that neither (32) nor (33) is explicitly selfcontradictory: on the grounds of that logical standard, they could in fact be true. What the Nihilist cannot do, however, is engage in a value-defined practice without adopting its values in practice. It is not merely our practice of assertion that puts a preferential value on truth—it is the concept of assertion itself that carries such an assumption. Any practice without that basic value would simply not qualify as assertion. It is not merely our practice of argument that carries a preferential value on defensibility. It is argument that carries that value. If the Value Nihilist or Skeptic is to jettison those practices, he or she can do so only by renouncing the option of assertion and argument. The positions at issue are ‘outs’ with regard to those practices, self-excluding from the realm of that communicative discourse. It is here that universal forms of relativism belong as well: 34. Everything I believe is a mere product of my culture, and has no truth beyond that. If I believe (34), I must concede that this belief is also a mere product of my culture. Within any enterprise that aims at truths beyond mere cultural artifacts, any such claim will be an automatic ‘out’. It seems likely that cultural practices of claim-making and claimsupport, intended to establish claims as true or supportable in some

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culture-independent sense, are ubiquitous. Alethic Relativism of the form of (34) will offer pragmatic contradictions within any such practice: its content alone makes it an automatic ‘out’ within practices of claim support or defense. A proponent of (34) might see himself as deliberately stepping outside of such contexts. But if so, it is unclear what work something like (34) might be conceived of as doing. If put forward as a supportable or believable statement or claim, a context of claim-making and claim-support is immediate, and the claim at issue is immediately self-defeating. If not put forward as a supportable or believable statement or claim, it is far from clear in what sense the content of (34) is being put forward at all. It is clear that the widest claims of Value Relativism will be selfdefeating as well. Consider: 35. All values are mere artifacts of culture, with no reality beyond that. Because the context of putting forward and defending claims inevitably brings certain values with it, a blanket Value Relativism will again be pragmatically self-defeating. Is (35) something put forward as something true, arguable, or as a claim that should be believed or even seriously entertained? If put forward in any of these ways we invoke practices essentially conceived as instantiating values beyond ‘mere artifacts of culture.’ To put forward (35) in any of these ways is therefore to face pragmatic contradiction. Within any such practice, the Value Relativism of (35), like Nihilism and Skepticism before it, will be an automatic ‘out’. 8. AUTOMATIC ‘OUTS’, ‘INS’, AND THE COGITO We have traced the implications of reflexive conceptual patterns from alethic, epistemic, and semantic contexts to those of value, with an emphasis on pragmatic contradictions and the role of practices. Forms of linguistic practice in which claims play a part will quite generally have claims that are automatic ‘ins’, ‘outs’, and oscillations. As we have seen, even questions and commands—interrogations and instructions—will exhibit these reflexive patterns.

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With regard to speaking as a way of making claims, (36) will be an automatic ‘out’: 36. I am not speaking. Whenever uttered, (36) is guaranteed to be false—a warning, among other things, against attempting to use ‘true whenever uttered’ as definitional of any deeper metaphysical or logical necessity. With regard to simply the practice of thinking, both (37) and (38) are automatic ‘outs’: 37. No-one is thinking about this. 38. No-one is thinking. There will also be automatic ‘ins.’ With regard to the practice of speaking, the following will be true whenever uttered: 39. I am speaking. Within the practice of thinking, the following will be true whenever thought: 40. Someone is thinking about this. 41. Someone is thinking. Automatic ‘ins’ within the practice of thinking delineate the region of ‘the Cogito’—Descartes’ ‘I think therefore I am.’ In Descartes, of course, the Cogito is intended as the firm and indubitable foundation for a reconstruction of all human knowledge. Whenever thought, (41) must be true. Within the practice of thinking, (41) is thus the opposite of a pragmatic contradiction: what we might term a pragmatic tautology. Interestingly, a pragmatic tautology is no more a logical truth or necessity than a pragmatic contradiction is a strict contradiction. It is no more a logical truth that someone is thinking than it is logically impossible that George has boots on and

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does not believe it. The realm of the Cogito appears as the flip side of Moore’s paradox—a pragmatic necessity within the realm of thinking much as Moore’s paradox is a pragmatic contradiction within the realm of making claims. Sometimes a practice will have automatic ‘ins’ as a direct result of automatic ‘outs.’ Consider the practice of doubting. Can I doubt that I am doubting? If so, I am doubting. Within the practice of doubting, therefore, given at least a minimal measure of self-reflection, there is something that I cannot doubt: that I am doubting. A search for something that cannot be doubted within a general methodology of skeptical doubt, and the discovery of the indubitable in the claim that one is doubting, is very much Descartes’ route to the Cogito.5 Thinking that and doubting that are propositional attitudes: psychological states or conceptual practices that deal in claim-like ‘that’s. If the analysis offered above is correct, we should expect similar automatic ‘ins’ and ‘outs’ in the case of other propositional attitudes as well. That is precisely what we find. With regard to hope, ‘that no hopes will ever be fulfilled’ is an automatic ‘out’: One cannot hope that no hopes will ever be fulfilled. A hope that some hope will at some point be fulfilled is correspondingly an automatic ‘in’ with regard to the practice: one cannot hope without hoping that some hope will at some point be fulfilled. I cannot, at least with minimal reflection, believe that there are no beliefs, and cannot believe that no beliefs are true. Both of these give us automatic ‘outs’. I cannot expect that no expectations will be fulfilled. I cannot be afraid that there is never anything to be afraid of; I cannot fear that there is never anything to fear. As a reflection of the discussion of Skepticism above, I cannot know that knowledge is impossible. I cannot predict that no predictions will come true. All of these share precisely the mechanism of Descartes’ Cogito. Descartes himself, and many of his followers, have treated the Cogito as if it were something more: as if it could play the role of a logical or metaphysical necessity rather than merely a pragmatic necessity. That it cannot do, any more than the oddity of my claiming I have boots on and don’t believe it entails anything about my boots.

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In later chapters, however, we will argue that the reflexive mechanism that drives the Cogito does have significant things to show us about the concept of consciousness. The mechanism of the Cogito involves reference within a practice—the practice of doubting, or of thinking—to aspects of that practice itself. It is a major clue to consciousness and related phenomena that communicative practice can focus reflexively on aspects of communication itself, that a decision can focus on decision procedure, and that one focus of attention can be aspects of attention itself. 9. REALITY, REFLEXIVITY, AND THE LIMITS OF KNOWLEDGE In Descartes, reflexivity generates a pragmatic ‘in’ taken as a kernal of subjective certainty. But reflexivity also gives us a certainty of a reality beyond the subjective. In both formal instances and in broader philosophical reflections, reflexivity gives us a concept of a reality beyond us. My belief set cannot possibly coincide with truth: my beliefs must either contain some falsehood or omit some truth. For consider: 42. Eli does not believe this. If true, there is a truth that Eli does not believe. If false, then Eli does believe this—and thus believes a falsehood. Because the pattern will hold for any named subject, the belief set of no subject can contain all and only truths. It is interesting to note that this reasoning can be clear to Eli as well: an assumption of either truth or falsity for (42) leads to the inevitable conclusion that Eli’s belief set has inherent limitations. That holds even though any attempt on Eli’s part to decide whether (42) is true or false leads to oscillation. If Eli believes (42) is true, he is forced to conclude it is false, and so not to believe it. If he doesn’t believe (42) is true, he is forced to admit that what (42) states is true, and so to believe it. What holds for subjects individually will hold for us collectively as well. Consider for example (43):

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43. This sentence cannot be established to be true by humans. The supposition that (43) is false leads to contradiction along either of two lines of reasoning: • If (43) is false, it certainly cannot be established to be true. (43) therefore cannot be established to be true by humans. But that is just what (43) says. It must therefore be true, contradicting our assumption of falsity. • If (43) is false, it can be established to be true by humans. But what can then be established, and is hence true, is that it cannot be established to be true by humans. Contradiction. Either route gives us a reductio argument. (43) cannot be false, and must therefore be true. But if (43) is true, since it says it cannot be established to be true by humans, (43) must represent a humanly unestablishable truth. In this first step, reflexivity forces us to envisage a reality independent of our epistemic abilities. We noted with regard to (42) that the assumption of either truth or falsity leads to the conclusion that Eli’s beliefs must be out of alignment with truth, even though Eli’s attempt to decide whether (42) is true or false will lead to oscillation. Something similarly peculiar holds with regard to (43). The argument above leads by reductio to the conclusion that there are truths beyond the ability of humans to establish. As human reasoners, however, recognition of that fact leads us into oscillation as well. The assumption that (43) is false leads us to contradiction, establishing that (43) must be true. (43) therefore represents a truth that cannot be established by humans. But by framing and following such an argument, it seems clear that we as human reasoners have established that (43) is true. Because (43) claims it cannot be established by humans, (43) must be false . . . and we are thrown again to the other side of the oscillation. Sentence (42) concerns belief regarding a specific belief: Eli does not believe this. But consider also those partially reflexive cases in

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which reference is to a larger inclusive class. A paragraph might include one sentence among many, for example, which maintains that some sentence in that paragraph is false. In much the same way, we standardly conceive of ourselves as imperfect believers—as individuals who sometimes believe falsely. This might be represented by (44): 44. At least one of my beliefs is false. One might think that if I believe (44) I cannot be wrong. The argument would go as follows: If some belief of mine other than (44) is false, (44) is unproblematically true. If all of my other beliefs are true, what of (44)? In that case, were (44) true, it would be false . . . contradiction. (44) must therefore be false. Since I believe (44), I believe a falsehood. I believe at least one falsehood, and thus I am right to believe (44). In the case in which all my other beliefs are true, however, what (44) gives us is not a guaranteed truth but an oscillation. Up to the last line, the reasoning above is flawless. But if (44) is false, and I therefore believe at least one falsehood, (44) is true. But if true, and all my other beliefs are true, it must be false . . . There are facts that others can establish, but that particular individuals clearly cannot. Consider facts of the form: 45. F is a (particular) fact that Eric does not know. Establishing (45) may be altogether unproblematic for you—you need only know the truth of F and a fact regarding Eric’s ignorance. But (45) is inaccessible to Eric: for no instance of the schema at issue can he claim knowledge. From Eric’s perspective: the contention ‘F is a fact that I do not know’ makes no sense. For in claiming F to be a fact one already claims to know it. Eric need have no difficulty, however, in accepting the general 46. There is some fact f that Eric does not know. The general thesis is unproblematic, for Eric as for anyone else. Concrete instantiations, while unproblematic for others, are unrealiza-

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ble for Eric. Using Ke for ‘Eric knows,’ the critical difference here reflects that that between Ke(p)~Kep and (p)Ke~Kep. It is not the fact of my ignorance but the details of my ignorance that are inevitably hidden from me in obscurity. Perhaps the most fundamental thing that we know about our knowledge is that it is incomplete. We humans—all of us—are finite knowers: there are facts that we just do not know.6 Can we even know how much we do not know? 10. COGNITIVE REFLEXIVITY One of the earliest philosophical injunctions arises in relation to cognitive reflexivity, encapsulated in the Socratic dictum: know thyself (gnôtni seauton). Here the issue of knowledge about our knowledge comes to the fore. Cognitive reflexivity is a matter of what an individual knows about his own knowledge. Throughout the discussion we will use Kxp to symbolize ‘x knows that p.’ We will let x will range over knowers, giving us (x)(p)Kxp. Every individual knower certainly knows that he himself knows something. One can regard the thesis: (x)(p)KxKxp as an epistemic postulate. Something like this is assured by the Cogito, for example. Given minimal rationality, this means that all knowers will know that someone knows something: (x)Kx(p)(y)Kyp And this in turn means that there is a proposition that all knowers know in common, namely

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(p)(y)Kyp Since every knower knows that somebody knows something, there a truth known to every knower: an item of universally shared knowledge. Going beyond this, it seems plausible to stipulate that every finite knower knows that he is finite, i.e., that there is something he does not know: (x)(p)(p & ~Kxp) It seems plausible, moreover, to stipulate that every finite knower knows this. If so, we will have it that: (x)((p) ~Kxp  Kx(p~Kxp)) But of course no finite knower knows specifically what it is that he does not know. A knower cannot know that he does not know a truth p to be the case, for in knowing it to be a truth he would inescapably know p. But do we have it that when x knows p, he thereby knows this very fact to be so?: Kxp  KxKxp The KK-thesis, as it is called, has occasioned considerable controversy. This seems to carry cognitive self-awareness one step too far. We would be extremely reluctant in general to allow any significant conclusion to be drawn from sheer ignorance: from a person’s not knowing that he doesn’t know something. That is, one should surely look with suspicion at any proposed thesis of the format: ~Kx~p  p But once we adopt

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Kxp  KxKxp We are saddled with ~KxKxp  ~Kxp And thereby with ~Kx~[~Kxp]  [~Kxp] which, of course, has the suspect format. To be sure, it x were an omniscient being g we would have p  Kgp And this would automatically assure ubiquitous cognitive reflexivity Kgp  KgKgp But to stipulate this for finite beings is carrying things too far. For such beings not only knowledge but cognitive self-knowledge is essentially limited. One important aspect of the incompleteness of our knowledge inheres in the fact that present cognition cannot speak for future knowledge. We can say nothing about the substance of future discoveries—for if we could they would not, after all, have to await the future. The discoveries for future sciences are—virtually by definition— unavailable to the scientists of the present. And not only cannot the substances of future knowledge be presently discussed but neither can its scope. With geographic explanation the size of the end could be determined in advance and thereby the scale and scope of unexplained terra incognita could be assessed. Nothing like this is possible with cognitive as opposed to geographic exploration. The knowledge of the present cannot speak for the scale and scope of future knowledge: the quantity and extent of its reach cannot be foretold.

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C. S. Peirce viewed scientific discovery on an analogy with global exploration. As he saw it, later is always lesser. Geographic discovery moved from the large to the small. The first great explorers discovered hemispheres, their successors discovered continents, later explorers discovered islands, but finally the targets of exploration came down to climbing mountains and discovering stream-sources. Peirce thought that scientific discovery was much like this, adding further discrete parts to further refine predetermined magnitudes. But this is just not how things work. Small-scale findings can have massive reverberations. Small discrepancies in the perihelion of Mercury can shake the entire ground edifice of classical Newtonian physics. Small anomalies can upset big theories and effect radical innovations. Later will often be different but not necessarily lesser. The advance of science does not indicate a clear pattern toward the progressively small or minor. In our ignorance of the future of science we are ignorant not merely of particulars but of the scope of our own ignorance. Just as we cannot foresee the scope of what we do not know, we cannot foresee the significance of what we do not know. Not only must we recognize the fact of our ignorance, not only must we recognize our ignorance of the scope of our own ignorance, but we must recognize an ignorance of how significant our ignorance might be. We have no way of knowing how important the facts that we know not are in the overall scheme of things. A peculiar and interesting aspect of our ignorance is at work in the mode of reference that occurs when an item is referred to obliquely in such a way that its specific identification is flat-out precluded as a matter of principle. This phenomenon is illustrated by claims to the existence of • a thing whose identity will never be known. • an idea that has never occurred to anybody. • an occurrence that no-one has ever mentioned. • an integer that is never individually specified.

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There are certainly bound to be such things, but we obviously cannot identify them.7 In these cases, those particular items that render ‘Some x has F’ true are referentially inaccessible: to indicate them individually and specifically as instances of the predicate at issue is ipso facto to unravel them as so-characterized items.8 In examples (45) and (46) we considered the case of a knowledge of general ignorance incapable of being cashed out in specifics. Here we have the corresponding concept of a predicate that is somehow applicable but accessibly noninstantiable. While it is clearly true in abstracto that such a property is exemplifiedthat (u)Fu will be truenevertheless the very manner of its specification makes it impossible to identify any particular individual u0 such that Fu0 obtains. Such predicates are ‘vagrant’ in the sense of having no known address or fixed abode. Though they indeed have applications, these cannot be specifically instanced—they cannot be pinned down and located in a particular spot. Accordingly we may define: • F is a vagrant predicate iff (x)Fx is true while nevertheless Fx0 is false for each and every specifically identified u0. Predicates of this sort will be such that on the basis of general principles one can show that while there must indeed be items to which they apply, nevertheless it can be shown that no such items can ever be concretely and specifically identified.9 While the predicates indeed have application, we are destined to be ignorant about where they apply.10 The following predicates represent properties that are clearly noninstantiable in this way: • being an ever-unstated proposition (or theory, contention, etc.). • being a never-mentioned topic (or idea, object, etc.). • being a truth (a fact) no one has ever realized (or learned, stated). • being someone whom everyone has forgotten.

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• being a never-identified culprit. • being an issue no-one has thought about since the 16th century. Noninstantiability itself is certainly not something that is noninstantiable: many instances of it are readily adduced. We can clearly demonstrate in many cases that a vagrant predicate applies— demonstrating at the same time, of course, that it will be impossible to specify any specific instance. Consider, for example, • being an integer that is never individually specified. There are infinitely many positive integers. But the earth has a beginning and end in time. Its overall history has room for only a finite number of intelligent earthlings, each of whom can only make specific mention of a finite number of integers. (They can, of course, refer to the set of integers at large, but they can only specifically take note of some finite number of them.) There will accordingly be some everunmentioned, ever unconsidered integers that are individually and explicitly never taken into considerationindeed an infinite number of them. But clearly no-one can give a specific example of this. The substitutional interpretation of quantifiers will not work with these vagrant predicates. Or again consider • being an unstated proposition Since in the history of the species there can only be a finite number of specifically stated propositions, while actual truths must be infinite in number, we know that there will be some such unstateable.11 But to say specifically of a particular proposition that it is unstated will be impossible. We can allude to such items generically but cannot actually identify them specifically. Vagrant predicates indicate another area where reflexivity— reflexivity regarding our own conceptual abilities—shows us both the existence of a reality beyond us and our inevitable ignorance regard-

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ing that reality. We can know that such predicates apply but not where they do so. Ignorance is inevitable here. Once we begin to add to the description of things such cognitive inaccessibility qualifiers as ‘never specifically contemplated,’ ‘never identified,’ ‘ever-overlooked,’ and the like, we push them into a cognitive black hole. Although vagrant predicates are by nature noninstantiable, it is important to note that we can nevertheless use them to individuate items that we can never identify. Consider for example • the oldest unknown (i.e., never-to-be identified) victim of the eruption of Krakatoa We can clearly make various true claims about the so-individuated person—for example that he or she was alive at the time of Krakatoa’s eruption. Reference of some sort is no problem here, though identificatory specific reference is impossible. Predicative vagrancy thus reinforces a distinction between mere individuation and actual identification. It also points to the fact of importantly different and nonequivalent forms of reference. It is the role of different forms of reference of this kind, differently accessible from within different perspectives and practices, that we will track in the next chapter. NOTES 1

See Patrick Grim, “What is a Contradiction?,” in Graham Priest, J. C. Beall, and B. Armour-Garb, The Law of Non-Contradiction: New Philosophical Essays, Oxford Univ. Press 2005, pp. 49-72.

2

H. P. Grice, Studies in the Way of Words. Cambridge, MA: Harvard University Press, 1998.

3

There is some similarity in the reflective argument here—but only some—to a perennial argument that Gödel’s proof shows not only the limitations of formal systems but the superiority of mind (see J. R. Lucas, “Minds, Machines, and Gödel,” Philosophy, 36, 1961, 112-127, and Roger Penrose, The Emperor’s New Mind, Oxford Univ. Press, 1989). For any formal system there will be a sentence undemonstrable in that system—moreover, a sentence that the argument shows will in fact be true. Reflection on the structure of the argument, it is held, affords us a grasp of truths beyond those capturable in any formal system or in its mechanical instantiation. Here we merely point out the similarity of argument structure, turning at an important point on reflection on argument structure. For replies, see Paul Benacer-

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NOTES

raf, “God, the Devil, and Gödel,” Journal of Symbolic Logic (1968), 613-615, David Lewis, “Lucas Against Mechanism”, Philosophy 44, 1969, pp. 231-233 and “Lucas Against Mechanism II”, Canadian Journal of Philosophy 9, 1979, pp. 373376, and Solomon Feferman, “Penrose’s Gödelian Argument”, Psyche 2 (1995). 4

It can be argued that Quine’s arguments for indeterminacy of translation entail precisely this type of Semantic Skepticism, and thus that a reductio regarding Semantic Skepticism is at the same time a reductio regarding Quine.

5

The limits of even such a pragmatic necessity are clear in Russell’s critique. Descartes concludes that a methodology of doubt cannot extend to ‘I am doubting’ or ‘I am thinking,’ and from that attempts to derive ‘I exist’: cogito ergo sum. Russell objects that the true kernal of certainty is merely ‘Doubting is,’ or ‘Thinking is,’ which carries no fully fleshed-out concept of an ‘I’. See Bertrand Russell, A History of Western Philosophy. New York: Simon and Schuster, 1945, p. 567.

6

It is often said that knowledge is a matter of agreement with reality (adequatio ad rem). But this contention has its problems. For knowledge can be imprecise. I can know that there are roughly 20 people in the room. But reality has to be exact (at any rate above the quantum level). Reality cannot put merely roughly twenty people into that room; reality has to ‘make up its mind’ about precisely how many. So among the things that I know about my knowledge is not only that it is incomplete but also that it is inexact and thus fails to do justice to reality in this way as well.

7

We have the true K(x)Fx but certainly not the false (x)KFx. So the former obtains despite ~(x)KFx.

8

We can, of course, refer to such individuals and even to some extent describe them. But what we cannot do is to identify them ass specific particular.

9

A uniquely characterizing description on the order of ‘the tallest person in the room’ will descriptively single out a particular individual de dicto, as it were, without specifically identifying him de re.

10

The classic Paradox of the Heap (Sorites) affords an illustration. One might propose that we know abstractly there is some number n as of which collected sand grains come to constitute a heap, but there is no specifiable number n of which we can say that this begins to be so.

11

Indeed truths must be more than infinite. See Rescher and Grim, “PlenumTheory,” Noûs 42 (2008), 422-439, and Beyond Sets: A Venture in Collection-Theoretic Revisionism, Ontos-Verlag, 2011.

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n important part of our analysis of self-defeating forms of nihilism and skepticism in the previous chapter, of Moore’s paradox and the Cogito, was an emphasis on domains of practice. Skepticism proves self-defeating within philosophical argument or any practice tied to knowledge, for example, because of a conflict between the position’s content entailments and the practice entailments of putting it forth within that practice domain. ‘I am doubting’ is an automatic ‘out’ within a methodological practice of doubt. ‘I am thinking’ is an automatic ‘in’ within the propositional context of thinking, and it is in these that the certainty of the Cogito exists. In this chapter we again emphasize the role of practice and propositional-attitude contexts, here with a focus on the roles of reference within such contexts. There are conceptual puzzles that turn on varieties of reference—on the ways in which things are referred to as well as the fact that they are referred to. In the chapters to come, both the form and fact of reflexive reference will prove important as we turn to issues of consciousness, semantics, and free will. 1. THE PUZZLE OF THE ESSENTIAL INDEXICAL John Perry tells the following story, borrowed in a number of essentials from Hector-Neri Castañeda and with a history that traces back to A. N. Prior: I am walking the aisles of the supermarket when I notice a trail of sugar in front of me. ‘Someone must have a broken bag of sugar in their cart,’ I think. ‘That person is making a mess.’ I glance at a figure in a fish-eye mirror at the end of an aisle, noticing the broken bag of sugar in the reflected cart. ‘Aha,’ I think, ‘That’s the guy who is making a mess.’

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It takes me a minute to realize that the reflection that I see is mine, and to look down to the broken bag of sugar in my own cart. ‘Oh,’ I gasp, ‘I’m the one who is making the mess.’ Factual knowledge is classically characterized as the grasping of a proposition.1 Consider then the proposition that I come to know when I realize that I am making a mess. That proposition, it appears, cannot be the same as this one: 1. Patrick Grim is making a mess. The two propositions cannot be the same because I might know (1) without knowing that I am making a mess—I might not know, after all, that I am Patrick Grim. If propositions are the objects of knowledge—the things that are known—these cannot be the same proposition, for I can know one and not the other. We can, moreover, explain certain things in terms of my knowing one proposition that we cannot explain in terms of knowing the other. The fact that I realize that I am making a mess fully explains the fact that I quickly gather up the bag of sugar and go in search of a broom. My realizing that Patrick Grim is making a mess would not fully explain that behavior. . .unless, of course, we add the additional element that I know I am Patrick Grim. The same argument can be used to show that what I come to realize when I realize I am making a mess is not what someone might know in knowing that: 2. He is making a mess, even when the ‘he’ in question is me. In the story, I know the proposition expressed by (2) when I identify the figure in the fish-eye mirror as the culprit. But at that point I don’t yet know that it is me who is making the mess. My recognition that I am making the mess fully explains my attempts to tidy up. My recognition that it is he who is making the mess would not fully explain my behavior—not, that is, unless we add that I recognize that he is me. What I know when I know that I am making a mess appears to have an essentially indexical element—the crucial ‘I.’ Replacing that ‘I’

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with a co-referential ‘Patrick Grim’ or even ‘he’ does not give us the same proposition or the same thing known. The conclusion that seems to follow is that there are things that I can know that no-one else can know. Whatever anyone else knows about me and my mess, it cannot be equivalent to what I know when I know that I am making a mess. Much the same problem, driving much the same conclusion, can be posed for other indexicals as well. What I know when I know the meeting is starting now is not what someone knows when they know the meeting starts at 12:00 on January 16th. One can know the latter without knowing the former, and knowing the former offers a sufficient explanation for certain behaviors—my running down the hall toward the conference room, for example—that knowing the latter does not. My positional knowledge that ground zero is here will work in similar ways. The problem of the essential indexical has been used to challenge standard concepts of propositions, to argue for a radically different understanding of knowledge, and as proof that there is a basic epistemic gap between a realm of indexically accessible propositions and those accessible non-indexically.2 Our approach here is different: to take the puzzle of the essential indexical as an introduction to a range of important lessons regarding the roles of reference in perspectival and practice contexts. We concentrate on those lessons in the next few sections, returning to the puzzle itself in section 5. 2. OPACITY The problem of the essential indexical is a problem regarding knowledge: whether what I know when I know that I am making a mess is what others know when they know, for example, that Patrick Grim is making a mess. In most contexts, if two things are identical, referential terms for them are intersubstitutable salva veritate. If Samuel Clemens is the same person as Mark Twain, then these propositions must be true or false together: 3. Samuel Clemens is buried in Hannibal, Missouri.

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4. Mark Twain is buried in Hannibal, Missouri. It has long been recognized, however, that knowledge contexts—like propositional attitude contexts generally—are different. The examples above are extensional contexts, preserving truth with co-referential substitution. Propositional attitude contexts, in contrast, are intensional or ‘opaque.’ The issue of opacity lies at the core—and resolution—of the classical paradox of the Hooded Man. Diogenes Laertius credits seven paradoxes to Eubulides of Megara. The first of these, appropriately, is the Liar. The second is the Hooded Man: “You say that you know your brother. But the man who came in just now with his head covered was your brother, and you did not know him.”3 Because of the knowledge operator, these two propositions are not guaranteed to be true or false together: 5. Bill knows that Mark Twain wrote Huckleberry Finn. 6. Bill knows that Samuel Clemens wrote Huckleberry Finn. If Bill has just taken an exam in American literature, the propositional attitudes in the following will make sense. Again it is clear that the pairs are not guaranteed to be true or false together: 7. Bill hopes that Mark Twain wrote Huckleberry Finn. 8. Bill hopes that Samuel Clemens wrote Huckleberry Finn. 9. Bill is afraid that Mark Twain wrote Huckleberry Finn. 10. Bill is afraid that that Samuel Clemens wrote Huckleberry Finn. In propositional attitude contexts, the truth of the claim that something is hoped for, feared, believed, known, suspected and the like can depend not only on what is referred to but on the way it is referred to.

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Claims are generally true or false of things independently of the ways in which those things are referred to. But it is true or false that things are hoped for, feared, believed, known, suspected and the like only in terms of particular modes of reference. We can graph the fact of referential opacity using the tools of the previous chapter:

The diagram on the left represents the fact that in terms of simple truth, the fact that Mark Twain wrote Huckleberry Finn and that Mark Twain is Samuel Clemens do entail that Samuel Clemens wrote Huckleberry Finn. In the second diagram, because ‘Bill knows . . .’ is a knowledge context and thus veridical, the fact that Bill knows that Mark Twain wrote Huckleberry Finn, together with the fact that Mark Twain is Samuel Clemens, entail outside the knowledge context—in the realm of fact—that Samuel Clemens wrote Huckleberry Finn.4 What the third diagram shows is that the two propositions do not carry that same entailment inside the context of ‘Bill knows . . .’ Knowledge contexts are opaque: identity of referents, though sufficient to guarantee substitutability salva veritate outside of such a context, is not sufficient to guarantee substitutability within it. In order to get a similar entailment within the knowledge context, both initial elements would have to appear within that specific domain:

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Even here, it is interesting to note, the entailment isn’t strictly guaranteed. We can guarantee that Bill’s knowledge that Mark Twain wrote Huckleberry Finn together with his knowledge that Mark Twain is Samuel Clemens will give Bill the knowledge that Samuel Clemens wrote Huckleberry Finn only on the hypothesis that Bill notes the two premises and carries through the inference. In that sense even the significance of our graphed our lines of entailment is contextually indexed. Although it is usually illustrated in terms of referential terms, opacity applies to predicates as well. The claim that ether can put people to sleep and that ether has a soporific capacity must be true or false together, simply because what is predicated is the same in each case. But it does not follow that: 11. Bill knows that ether can put people to sleep. and 12. Bill knows that ether has a soporific capacity. must be true or false together. Things are hoped for, feared, believed, known, and suspected in terms of particular ways of predicating just as they are in terms of particular ways of referring. It should not be surprising that such the phenomenon of opacity extends not merely to the truth and falsity of claims involving knowledge but to the ontology of ‘things people know.’ What is known has as part of its identity conditions not merely the thing referred to but the way in which reference is made, and not merely what is predicated but the form that predication takes. What Bill knows when he knows that Mark Twain is the author of Huckleberry Finn is not what Mike knows when he knows that Samuel Clemens is the author of Huckleberry Finn, because one might know one of these things without knowing the other. For two different questions on the test, Bill may get question #1 right and question #3 wrong because he knows one of these things and not the other. Pete may get one question right on his medical exam and one question wrong because he knows that ether can put people to sleep but does not know that it has a soporific

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capacity. The ontology of what is hoped for, feared, believed, and suspected carries the same feature: each of these is sensitive not merely to the fact but to the form of reference and predication. Opacity is a feature of reference and predication within certain contexts, among which the contexts of propositional attitudes are primary. It is because the concept of what is known, believed, or hoped for inherits aspects of reference and predication that issues of opacity will carry into issues of ontology as well. 3. UNMEDIATED REFERENCE Opacity is a phenomenon in which reference is contextualized: the role of referential terms within a knowledge context is unlike that outside that context. In the realm of simple truth, co-referential terms are substitutable. Within a knowledge context they are not. Reference can be contextualized in other ways as well. Much of our reference is mediated by description. I can refer to the tallest man in Chicago using that very description, for example. I can refer to the first postmaster general as such. One of the intriguing facts about reference is that I can refer to someone without knowing almost anything about him. As it happens, I know that the first postmaster general was Benjamin Franklin. But I can also refer to the second and third postmaster generals, about whom I know almost nothing other than there must have been people who fit those descriptions. I can in fact refer to someone while being quite mistaken about him or her. I may think that the second postmaster general was a woman, living alone in Philadelphia in 1800 and rich from taking in laundry. I will be wrong across the board in thinking that the second postmaster general fits that description—I may in fact be wrong in thinking that anyone fits that description—but the person I fail to describe accurately will nonetheless be the person I do fully succeed in referring to: the second postmaster general. In all these cases reference is mediated by description. But given specific contexts we can also use forms of reference that are unmediated. In referring to that person, to that table, or simply to that, my reference need not be mediated by any particular description, or indeed by any description at all. In referring to this place, or that place, here or there I am not referring by means of description. In referring

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to this point in time or that, then or now I am not referring by means of description. In referring to him or her, to us or them, to it or you or me I am not referring by means of description. Unmediated reference appears in ostensive reference and in the use of indexicals: this, that, here, there, now, then, him, her, it, them, us, you and me. Unmediated reference of any of these forms does, however, require a context of ostension. That usually demands that speaker and hearer—the producer and consumer of reference—share the same time or locale. Such a context was in fact merely pretended between we as authors and you as reader in the previous paragraph. In order to genuinely refer to that book, that person, or him, we would have to be in the same spatial location—a context within we could point to the book or person, for example. In order to genuinely refer to what is happening right now, we would have to share the same immediate time context. Anaphoric pronouns are those the reference of which must be set by a previous context—at the end of a historical passage regarding John Stuart Mill, we can use ‘he’ without ambiguity. There it is the earlier passage that sets a quasi-ostensive context. Reference by description is often referred to as reference de dicto. Unmediated reference is often referred to as reference de re. ‘That guy. . .the one with the pink martini. . .’ uses both unmediated and description-mediated reference. ‘That guy. . .the famous author. . .’ does as well. When two forms of reference are used, however, reference breakdown can arise from their conflict. It may be that the man I am pointing to is not a famous author—my descriptive reference conflicts with my indexical reference. A similar conflict can arise purely descriptively, as in the case of ‘the fifth signer of the Declaration of Independence and first Postmaster General.’ If James Monroe is the fifth signer of the Declaration of Independence and Benjamin Franklin is the first Postmaster General, who does my phrase refer to? We will have more to say about referential failure at a later point. For now suffice it to say that there is no hard and fast answer for all such cases: in cases of reference failure, reference and lack of reference appear to be open to negotiation on a case-by-case basis.5 A further warning is in order as well. Although descriptive terms are characteristic of reference de dicto and indexicals are characteristic of reference de re, the real distinction at issue is one between particu-

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lar uses of referential terms rather than specific syntactic forms. This is particularly clear in the ambiguity of second-hand reports regarding propositional knowledge. We have noted the opacity of knowledge contexts: that Bill may know that Mark Twain wrote Huckleberry Finn and not know that the man in the photograph wrote Huckleberry Finn, despite the fact that Mark Twain is the man in the photograph. In some cases, however, my report that Bill knows that the man in the photograph wrote Huckleberry Finn may be merely a report that Bill knows of the man in the photograph that he wrote Huckleberry Finn. ‘The man in the photograph,’ in other words, may function as my referential phrase as reporter rather than as a form of reference I am attributing to Bill. Understood in such a way, my claim that Bill knows that the man in the photograph wrote Huckleberry Finn may be true despite the fact that Bill has never seen the photograph. Any attribution of propositional knowledge to animals—the claim that kitty knows that Mike is away, for example—demands a similar referential use, as do cases in which what someone knows is reported in a language that they do not speak. On uttering “Mark Twain a écrit Huckleberry Finn” it may be true to say that Antoine said that Mark Twain wrote Huckleberry Finn despite the fact that Antoine speaks no English. 4. CONTEXTUAL POINTS OF ORIGIN Within the context of ostensive reference are a special sub-category of unmediated referential terms that target basic elements crucial to ostension itself. It is directed attention that is essential to ostension, and that attention must be directed from a particular point. The point of origin from which ostension is made is therefore crucial in defining the context. Not merely one of the things that can be pointed to, that point of origin is assumed in all action within the context—in any act of pointing or directed attention. Reference to the origin point within the context therefore has a character more basic and fundamental than other reference within the context, even where that other reference is equally unmediated by description. Reference to the point of origin is reference to the point from which reference is made, and is therefore not merely unmediated but transparently immediate. In the case of

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points of origin, the act of reference itself fixes reference without possibility of error. The point of origin in ostension is that targeted by ‘I’, with its immediate spatial and temporal coordinates targeted by ‘here’ and ‘now’. To the extent that any such terms can be defined at all, ‘here’ might in fact be defined as the point from which spatial ostension is made, ‘now’ as the point from which temporal ostension is made, and ‘I’ as the central actor in ostension—the pointer himself. Because these represent the contextual origin, reference to ‘I’, ‘here’, and ‘now’ have a status reflexively immediate in a sense in which reference to ‘you’, ‘him’, ‘there’ and ‘then’ do not. Of these it is ‘I’ that can claim to be most fundamental: one can define ‘here’ as the place I am, ‘now’ as the time I find myself in, but one cannot define ‘I’ uniquely as a person here and now. Given the multiple referential routes represented by maps, temporal recordings, and fish-eye mirrors, we might refer ostensively to two times, places, or people without knowing that these are in fact the same times, places, or people. By some confusion of mode of viewing or representation, for example, I might not realize that that place is the same as this one. I might not realize that the person I point to through a window is in fact that same person I point to on a television screen. By the same token, I might under certain conditions refer de re to a time, place, or person without knowing that time is now, that place is here, or that person is me. Because ‘I’ targets the origin of ostension in general, however, I cannot in the act of ostension be under some mistaken impression that it is some other person rather than me that is ‘I’. I cannot be under some mistaken impression that it is some earlier or later time that is actually ‘now’ or that it is some distant place that is really ‘here’. ‘I’, ‘here’ and ‘now’ play a uniquely immediate and transparent referential role, keyed to the context origin of the practice of ostension itself. Consider propositions of the form: 13. I am X where X ranges over individual identifiers such as:

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• the person who made a mess • the person who caused the explosion • the person who earned less money than anyone else in the room. For what X would it be effectively impossible for a claim of the form of (13) to be false when affirmed with declarative attempt?6 This will hold in a trivial sense with characterizations which reflect the fact of declarative intent itself, which only an intelligent being can have. In the spirit of Descartes, (13) will be guaranteed, for example, where we substitute for X • the person who is making this statement • an existing being • an assertion-capable being • an intelligent agent. Can we go further? Where X uses an anaphorical pronoun like ‘my,’ linked to ‘I,’ we will again get claims about which even Descartes’ evil demon would have difficulty deceiving us. I am, for example, • the person who lives my life • the person who has my memories • the person who thinks my thoughts. Once we leave reference tied to points of origin in those ways, however, the evil demon might again get a grip. Descartes’ demon can again get a grip, for example, even with • the person whose finger is wiggling here.

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We can graph the position of different forms of reference as shown below. Forms of reference external to ostension—in the wider public realm, as it were, shown outside the circle—include both reference mediated by description and common names.7 Reference is possible unmediated by description within ostension, indicated by ‘he,’ ‘that,’ ‘then,’ and ‘there.’ It is of interest that these referential terms cannot escape to the uncontextualized public realm at large; they can reference a place, time, and person only relative to the indicated context of ostension. Deep within the context of ostension we graph an inner sanctum of those terms tied to contextual origin and therefore immediate within the practice.

The boundaries in the diagram above are boundaries of nonequivalence. No referential term within the uncontextualized realm of public description can do precisely the work of an ostensive term, because of the simple fact that ostensive reference is tied to a specific context. In a particular case, an ostensive use of ‘he’ and the descriptive phrase ‘the tallest man in the room’ may in fact be co-referential. But no purely descriptive term can be precisely equivalent to ‘he’: there is no non-ostensive equivalent for ostensive forms of reference. By the same token, no ostensive term outside of the inner circle of immediate origin terms can be precisely equivalent to any of those within the inner circle. Although ‘that time’ and ‘now’ may in some context be co-referential, there is no equivalent to ‘now’ phrased purely in non-immediate terms: there is no ‘that time . . .’, without a smuggled origin term, that is equivalent to ‘now’. There is no ‘that place . . .’ without a smuggled origin term that is equivalent to ‘here’.

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There is no ‘he . . .’ without a smuggled origin term that is equivalent to ‘I’. What the graph represents is ways of referring or forms of reference isolated from one another in the sense that they have no precise equivalents across borders. Within the context of ostension, we have noted, are a category of immediate and transparent referential terms that target points of origin: ‘I’, ‘here’, and ‘now’. There is also a further class of referential terms that have something of the same character: those that reflexively target a context as a whole. In the process of ostension I can refer to the act of ostension itself: ‘pointing’ or ‘what I am doing now.’ Propositional attitudes can all be referenced in this way, as can mental states, and can often be referenced within the context of the propositional attitude or mental state itself. In a state of fear I can refer not only to myself, to what I am afraid of, but to my fear: a reference to the general context of the state itself. I can in fact be afraid of being afraid; ‘we have nothing to fear but fear itself’ instantiates precisely such a reflexivity. I can similarly refer to hope quite generally, or to my current hoping, and can hope that my hopes are not in vain. I can call attention to particular things, but can also choose to focus on the act of attention itself. The same sort of reflexivity that holds for attitudes and mental states can also obtain for perspectives, processes, and practices. I approach things from a particular conceptual perspective, and I can refer to that perspective itself. In some cases a perspective can itself be viewed from that perspective—reflexively from within, as it were. Among the things open to logical or scientific investigation are the processes of logical and scientific investigation themselves. Among the topics open to ethical deliberation in a particular case is the extent to which lengthy ethical deliberation is justified in that case. The fact that propositional attitudes, mental states, perspectives, and practices can reflexively target themselves transparently and immediately as points of origin will play a major role in the work of the following chapters. It should also be noted, however, that all of these can target themselves in part as well as whole. A state of awareness can focus, among other things, on particular aspects of that state. I can fear particular aspects of my fear, and can hope that some of my hopes in particular—or particular aspects of my hope—are not in vain. As an

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essential part of an ongoing investigation I can attend to methodological principles within that investigation itself. Philosophy is of course that discipline obsessively reflexive par excellence, in both part and whole. 5. ANALYZING THE ESSENTIAL INDEXICAL The puzzle of the essential indexical is a problem of knowledge and thus of referential opacity. It is a problem of non-equivalent reference within and outside of ostensive contexts. The ontology of the problem—an ontology of propositions and what is known—is an ontology of things instantiating these complex patterns of opaque and ostensive reference. Within knowledge contexts, reference is opaque. Though coreferentiality of two expressions does often guarantee substitutability salva veritate, it does not guarantee substitutability within knowledge or propositional attitude contexts generally. With referential terms fixed by ostension, the problem of the essential indexical invokes an ostensive context like that graphed above, but further enclosed within the opacity of a knowledge context as well. The fact that I know that the youngest man in the room is the tallest man in the room need not entail that I know that he is the tallest man in the room, precisely because I may not know that he is the youngest man in the room. The fact that I know that he is the tallest man in the room, indicating someone whose reflection I see in the mirror, need not entail that I know that I am the tallest man in the room, even where he and I are the same. I may, after all, not realize that it is me in the mirror. We have noted that referential terms often cannot cross the borders of ostensive contexts. Ostensive terms may have co-referentials but will have no precise equivalents in descriptive terms: there is no nonostensive phrase guaranteed to have the same reference as ‘this.’ The same holds for terms within and outside of the inner sanctum of origin terms within ostension: no other ostensive phrase de re is guaranteed to have the same reference as ‘I’. Thus my knowledge that someone falling under some non-ostensive description has a certain property will never be equivalent to my knowledge that he has that property, no

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matter what non-ostensive description might be chosen. By the same token, my knowledge that he has a certain property, where he is indicated by some non-origin route of ostension, will never be equivalent to my knowledge that I have that property, no matter what non-origin route of ostension is chosen. Given the isolation of referential terms inside and outside the borders of ostensive contexts, knowledge of these types will be essentially non-equivalent. The last structural element of the puzzle is the move to ontology, from whether someone knows . . . to what someone knows. An ontology of ‘what is known’ inevitably inherits the referential opacity of knowledge contexts, sensitive to ways of referring. Where knowing x is not equivalent to knowing y, x and y must be distinct. Given opacity, it is inevitable that what I know in knowing that Patrick is making a mess and what I know in knowing that he is making a mess must be different things, just as it is inevitable that what is known in knowing that Mark Twain is the author of Huckleberry Finn and what is known in knowing that Samuel Clemens is the author of Huckleberry Finn must be distinct. Given essential boundaries of non-equivalence between referential types in contexts of extension, moreover, it is inevitable that what I know in knowing that Patrick is making a mess, in knowing that he in the mirror is making a mess, and in knowing that I am making a mess must be essentially and irreducibly distinct. Given an ontology of things known constructed from the materials of opaque and non-equivalent referential contexts, it is because only I can use ‘I’ to index the origin for my ostensive context that no-one can know precisely what I know in knowing that I am making a mess. If this analysis in terms of opacity and ostension is correct, we should get parallels to the problem of essential indexical for other propositional attitudes as well. And indeed we do. There are clearly ontological issues traceable to opacity across the spectrum of propositional attitudes. What I fear when I fear that 14. Samuel Clemens has risen from the dead is not what I fear when I fear that 15. Mark Twain has risen from the dead,

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for I could fear one without fearing the other. Wrapped in a nightmare, the fact that I moan ‘Oh no . . . Samuel Clemens is going to get me . . .’ can be fully explained by pointing out that I fear the first, but cannot be fully explained by saying that I fear the second—or at least cannot without adding that I conceive of Samuel Clemens and Mark Twain as one and the same. Interestingly, in this case and others, it is not full knowledge of identity that is required. For it to follow from the fact that Bill fears that (14) that he also fears that (15), given at least minimal powers of inference on his part, it is not necessary that he knows Samuel Clemens is Mark Twain. It is sufficient merely that he fears Samuel Clemens is Mark Twain. Given the assumption of minimal powers of inference, for it to follow from the fact that someone believes that (14) that they believe (15) it is sufficient merely that they believe Samuel Clemens to be Mark Twain. For it to follow from the fact that someone hopes (14) to be true that they also hope (15) to be true what is required is merely that they hope Samuel Clemens is Mark Twain. As in the discussion of knowledge context in section 2, what is required for inferences of this type is the inclusion of an identity claim within the propositional attitude context and thus with its attitudinal character, whether than context be one of fear, belief, or hope. Parallels to the fully ostensive puzzle hold as well. What I believe in believing that Patrick is making a mess is not what I believe in believing that he is making a mess, precisely because I might not believe that he is Patrick. Because I can believe that he is making a mess without believing that I am, even where the person indicated is in fact me, the beliefs at issue must themselves be distinct. What I may be afraid of is not merely that he is making a mess but that I am. In another case, what I may hope is not merely that he will act honorably or courageously but that I will. By an argument precisely parallel to Perry’s, whatever someone else believes, fears, or hopes in such cases, it cannot be precisely what I do. The ontological individuation of beliefs, hopes, and fears will be fine-tuned to forms of reference, including forms that turn on ostension, just as in the case of knowledge. Does the analysis we have offered solve the problem of the essential indexical? No more than the implicative dynamics diagrams we

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have offered solve the paradoxes. If a ‘solution’ for the problem of the essential indexical requires showing that everything is ultimately accessible from some general public realm—that what I know when I know that I am making a mess is after all something accessible to anyone at any time—our analysis offers no such solution. If what we are after is an explanation of why and how such a puzzle arises, on the other hand—‘How can it be that what I know isn’t the same as what you know?’—such an analysis offers precisely what we are after. Reference comes in importantly different forms—mediated by description, unmediated in ostension, and fully immediate and transparent in contextual points of origin. Knowledge contexts are opaque, sensitive to those differences in reference, and the ontology of ‘what is known’ inherits that opacity over ostensive reference. Such is the conceptual structure beneath the puzzle. Once we understand that, we know all there is to know. There is also a lesson to be drawn from this discussion regarding a major piece of philosophical terminology. ‘Proposition’ is a term of philosophical art; the closest one gets in ordinary speech is perhaps ‘statement’ or ‘what someone said.’ But propositions are taken to serve at least two different philosophical roles. Propositions are the bearers of truth—they are what is true or false. Propositions are also the objects of propositional attitudes (hence the term ‘propositional attitudes,’ of course): what is known, believed, feared and hoped for. What the discussion above indicates is that identity conditions for propositions will be radically different in the two cases. As bearers of truth or falsity, independent of propositional contexts, propositions allow co-referential substitution salva veritate. As what is known, believed, feared or hoped for, in contrast, propositions will be opaque with respect to co-referential substitution. The fact that identity conditions differ so significantly for propositions as (a) truth bearers and (b) objects of propositional attitudes may indicate that we are asking a term of art to satisfy not merely different but conflicting ends—a sign of deep inconsistencies in the category itself.

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6. DIFFERENCES IN REFERENCE VERSUS DIFFERENCES IN REFERENTS One of the interesting aspects of Perry’s presentation is that there are in fact not two but three importantly different referential forms involved: Patrick Grim, he (indicating the figure in the mirror de re), and I. We could also add reference by description as a fourth: the person who is dropping sugar from his cart. We have noted that these distinct forms of reference generate distinct propositions in the sense of objects of propositional attitudes: different things known, believed, hoped for, or feared, for example. Those propositions are about the same person, and attribute the same property to that person, and yet remain distinct: differences in ways of referring or form of reference distinguish both different acts of knowing, hoping, and fearing, and the things that are known, hoped, and feared. Does it follow that what is referred to using ‘I’ is somehow different from what is indicated by ‘Patrick Grim,’ ‘the man in the mirror’, or ‘the person who is dropping sugar from his cart’? Certainly not. Although the referential forms are distinct, the things referenced need not be: I am Patrick Grim, after all, I am the man in the mirror, and I am the person who is dropping sugar from his cart. Differences in forms of reference can fool us into thinking that what is referred to is distinct. In some cases reference is ostensive and unmediated, perhaps rich with experiential associations. When I put my hand on a friend’s shoulder and refer to this man, de re, I do so in a context which may carry emotional resonance and in which there is much about him that I can sense and see. When I refer flatly to ‘the current resident of 204 Berkeley Street’ I may have no other experiential content than that of the description. In using the ostensive referential form we may dwell on the experiential richness of the ostensive context. In using a referential form by residential address we may dwell on the thinness of that bare description. Can this man, we ask, dwelling on the richness of experiential content in ostensive reference, be merely the current resident of 204 Berkeley Street? That ‘merely’ aside, the answer is ‘yes’: they can in fact be the same man. It is all too easy to be misled by the character and experiential richness of dif-

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ferent referential forms and contexts into thinking that we are dealing with different referents, themselves distinct in character and experiential richness. We may not be. No purely descriptive reference, we have noted, will be precisely equivalent to any ostensive referential term. No non-transparent ostensive term of the form of ‘he’, ‘then’ or ‘there’ will be precisely equivalent to transparent referential points of origin: ‘I’, ‘now’, and ‘here’. ‘The man in the corner with the brown hat is drunk’ is never equivalent to ‘He is drunk.’ But despite non-equivalence of referential forms, referents may in fact be the same: it may well be the case that he is the man in the corner with the brown hat. ‘He is drunk,’ where ‘he’ is used to indicate de re someone seen in the mirror, never entails ‘I am drunk.’ But despite non-equivalence of referential forms, referents may be the same: the man in the mirror may in fact be me. It will be important in the work of later chapters that there are contextual borders of non-equivalence between forms of reference that may nonetheless target the same referents. Just as the first postmaster general was the inventor of the bifocals, though there is nothing in either description that entails mutual reference, I am the man in the mirror, though there is nothing in either form of reference that entails coreference. It will also be important in later work that we can be tempted to see ontological differences—differences in referents—when what is fundamentally at issue are merely contextual differences in ways of referring. 7. CLASHING CONTEXTS OF REFERENCE There is a further fact regarding forms of reference that will also be of importance in following chapters. Forms of reference may be appropriate to, embedded in, and proprietary to contexts of either theory or practice. One family of reference contexts are perspectival, closely related to indexicals in general. I can refer to a particular individual using the simple phrase ‘my sister.’ You cannot refer to that same individual using that same simple phrase; your use of such a phrase, if it refers at all, will tag a different individual entirely.

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Another family of reference contexts, more hotly debated but perhaps ultimately less well understood, are theoretical contexts. Were J. J. Thomson, Ernest Rutherford, and Niels Bohr talking about the same things when they were talking about ‘electrons’? An argument can be made, and has been made, that the answer is ‘no’; that Thomson’s, Rutherford’s, and Bohr’s theories are different enough that the term ‘electron’ simply means something different in the three theories. But if Thomson, Rutherford, and Bohr meant different things by ‘electron,’ how could their models contradict one another? If moreover this kind of difference of meaning is a general characteristic of successive theories, how can we say that a previous theory is refuted, or that replacing one theory by another should be marked as progress? 8 These questions form the core of an extensive debate regarding the history of science, the meaning of theoretical terms, the character of theoretical conflict and the nature of scientific progress. Part of the recalcitrance of the issue, however, may be due to the fact that it has so often been approached in terms of the meaning of theoretical terms. A promising alternative is to analyze what is at issue in terms of reference. The term ‘electron’ may indeed mean something different in three theories. But ‘electron’ is a referential term, and it may nonetheless be the case that Thomson, Rutherford, and Bohr are all referring to the same thing as ‘electrons.’ We’ve seen that very different descriptive terms can nonetheless be co-referential. We should expect it to be the case that certain scientific terms, though embedded within radically different and even conflicting webs of conceptual connections and empirical claims, might nonetheless reference the same things. The fact that Thomson, Rutherford, and Bohr’s ‘electrons’ are nonequivalent in meaning need not therefore entail that they are different in reference. Despite their radical differences, all three theories may be talking about precisely the same things. The point that will be important for the work that follows is that there may also be terms that can reference only from a particular theoretical context. The phrase ‘electron shells’ may be appropriate to, embedded in, and proprietary to a specific theory, with the result that it cannot have referential traction outside of that context. The same may be true for cases of ‘possession by the devil’ and ‘psychotic epi-

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sodes,’ each of which is embedded in a particular theoretical context, with reference in those terms appropriate only from that context. When referential terms are restricted to a theoretical context, characterization from a conflicting theoretical perspective itself becomes difficult in interesting ways. There are in fact several ways in which radically cross-theoretical characterization is attempted. One can imagine a contemporary psychologist saying “‘Possessions by the devil’ were in fact psychotic episodes.” Here the phrase in single quotes plays the role of a reference within the earlier theory, but the single quotes signal a mock reference: the speaker is acting as if one could refer using such a phrase. The work of those single quotes can also be done by saying “So-called ‘possessions by the devil’ were in fact psychotic episodes.” In each case the referential forms of the disputed theory are used, rather than merely mentioned, but are used obliquely. On the other hand, earlier referential forms may simply be renounced. One can imagine a psychologist, on precisely the same basis and with precisely the same justification, saying “There are not and never were any possessions by the devil. The poor wretches were merely suffering from psychotic episodes.” Contexts of practice and purpose may embed proprietary forms of reference just as theoretical contexts may. ‘All Privates must defer to legal orders from their Sergeants’ uses forms of reference—’Privates’ and ‘Sergeants’—appropriate to the hierarchical organization of military service. ‘All those with symptoms of the flu are to report to sick bay’ uses medical rather than military categories, though the domain of individuals at issue in the two cases may be the same. The clash between the two referential contexts is indicated by the fact that the military order could not be phrased in medical terms, nor the medical injunction in those of military hierarchy. One could not specify who is to report to sick bay in terms of rank, nor who should defer to whom in terms of flu symptoms. Despite the potential for co-referentiality, we have forms of reference keyed to and only appropriate in the context of very different purposes and practices. There are projects aimed at particular ends, and forms of reference appropriate to those: forms of reference that embody the context of a particular kind of activity, deliberation, or mode of access. The ‘strikes’ and ‘errors’ of baseball and the ‘fumbles’ and ‘interceptions’

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of football, for example, have a referential context keyed not only to the rules of the game but to the goals of the teams. To attempt to find precisely co-referential terms within physics would be a mistake. ‘Counter-evidence,’ ‘margins of error,’ and ‘statistical significance’ have a referential context within scientific exploration, and cannot, in a very real sense, be exported from the context of that project. ‘Precedents,’ ‘standing’, and ‘jurisdiction’ are appropriate to, embedded in, and proprietary to contexts of legal practice. The attempt to find equivalents from without a practice often indicates a mistake in understanding the character of reference within that practice. Much the same, we will argue, will be true for decisional contexts and the concept of choice.9,10

NOTES 1

There are two basic forms of knowledge: factual knowledge and performative knowledge. Performative knowledge is a matter of knowing how to do certain things—how to eat with chopsticks or how to serve at tennis, for example. Factual knowledge is a matter of knowing that some-thing-or-other is the case. For a critique of that standard distinction, however, see Jason Stanley and Timothy Williamson, “Knowing How,” The Journal of Philosophy 98 (2001), 411-44.

2

See Perry, op. cit, David Lewis, “Attitudes De Dicto and De Se,” The Philosophical Review 88 (1979), 513–543, and Patrick Grim, “Against Omniscience: The Case from Essential Indexicals,” Noûs 19 (1985), 151-180.

3

Eubulides’ paradox seems to be phrased in terms of knowledge by acquaintance rather than propositional knowledge of the form ‘he is my brother,’ but the same general point will apply. There is, however, a remaining difficulty. Consider an updated form of Eubulides: ‘A man entered the room. He wore a monkey suit. He was in fact my brother. But I felt certain than I had never seen him before.’ The difficulty is in interpreting the final anaphoric pronoun ‘him.’ Does this refer extensionally to my brother or by way of some non-extensional characterization instead? We leave this further issue as an exercise for the reader.

4

Were the context at issue one of mere belief or hope, which are not veridical, even this would not hold.

5

See Peter Ludlow, “Cheap Contextualism,” Philosophical Issues 18 (2009) 104129. Related issues appear somewhat more peripherally in Augustin Rayo, “Vague Representation,” Mind 117 (2008), 329-373.

6

We exclude the case of a machine uttering those words or an actor reciting them without understanding.

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NOTES 7

Our claim is not, of course, that these are the same. There will be forms of nonequivalent co-reference in the public realm as well.

8

See N. R. Hanson, Patterns of Discovery, Cambridge: Cambridge University Press, 1958; Thomas Kuhn, The Structure of Scientific Revolutions, Chicago: University of Chicago Press 1962, 1970; Paul Feyerabend, Against Method: Outline of an Anarchistic Theory of Knowledge, London: Verso, 1975; Imre Lakatos and Alan Musgrave, eds., Criticism and the Growth of Knowledge, London: Cambridge University Press, 1970.

9

Of course reflexivity can function in the context of justification as well as one of reference. Ever since Aristotle theorists have argued that in order to achieve cogent validation we cannot always justly validate propositions deductively and discursively by means of others. Some must be justified immediately in view of their own constitution. These propositions must be evident in view of their own nature. However this sort of epistemological reflexivity leads beyond the range of present concern.

10

John Perry, “The Problem of the Essential Indexical,” Noûs 13(1979) 3-21; HectorNeri Castañeda, “On the Logic of Attributions of Self- Knowledge to Others,” The Journal of Philosophy 65 (1968) 439-456 and “He: A Study in the Logic of SelfConsciousness,” Ratio 8 (1966) 130-157.

Chapter 5 THE MYSTERIES OF CONSCIOUSNESS

T

he traditional mind-body problem is a question of the relation of the mental and the physical realm. It constitutes a mine field of philosophical positions with problematic triggers. If the mental and the physical are as essentially distinct as forms of Dualism insist, how could they interact in the way that they do? How, on the other hand, could this—the realm of subjective experience, indicated with outstretched arms—be merely a matter of electro-chemical activities in three pounds of matter? The ‘hard problem of consciousness’ is the apparent explanatory gap between subjective experience and physical accounts—not merely any physical accounts that we have at present, but any that we can imagine having in the future.1 The inevitable and forbidding character of the explanatory gap plays a large part in the contemporary arguments of Neo-dualists, Mysterians, and Panpsychists.2 Given the gap, consciousness seems to become mysterious—immediately obvious in our experience and yet forever just beyond our understanding. An analysis of consciousness in terms of reflexive processes, we will argue, does much to dispel this atmosphere of mystery. 1. THE PROBLEM If asked to name major landmarks in the development of cognitive processing over the course of evolution, we might offer something like the following list: • passive reactivity • active reactivity • consciousness

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• consciousness of a self Passive reactivity would mark a minimal accomplishment, indicating merely that an organism reacts to given environmental conditions. The category is admittedly vague. Just as it may be debatable in some cases whether what is at issue qualifies as an ‘organism,’ it may be debatable in some cases whether causal consequence amounts to a ‘reaction.’ It may also be unclear whether certain forms of passive reactivity qualify as cognitive or not. Active reactivity invokes something more; an ability to act toward favorable conditions or to avoid unfavorable ones. A paramecium selectively following chemical gradients might offer an example of active reactivity at the simple end of the spectrum. The flight reactions of higher organisms offer a clearer case of active reactivity on the other end. But here again the concept is admittedly vague; the line between passive and active reactivity at the lower end seems particularly indeterminate. Consciousness stands out as a clear landmark in the development of cognitive processing, though that landmark proves extremely difficult to characterize without resort to the term ‘conscious’ itself or to some close synonym. What is at issue is active reactivity not merely by means of internal states but by means of conscious states or processes—conscious in the sense that creatures like you and I are aware of conscious states or processes. What consciousness in this sense actually entails is the topic of the chapter as a whole. The last entry on the list is intended to mark something beyond momentary conscious processing. Here what is at issue is an organism’s consciousness of a self-organizing and instantiating a history of consciousness over time. Just as it proves difficult to define ‘consciousness’ in the sense at issue without resort to the term ‘conscious’ itself, it proves difficult to define ‘self-consciousness’ without redundant resort to the term ‘self.’ This rough exercise in outlining vague cognitive landmarks is clearly insufficient to define any of the terms at issue, let alone to settle any major issues. But it does point up the central problem we are after. There is one clear leap that appears in the list above—a leap between steps two and three. It is at that transition, the K-T boundary of

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cognitive development, that the mysteries of consciousness seem to arise. We understand how processes of inputting and updating environmental information can operate, and can envisage an evolutionary history that would produce informational processes of that kind— environmental reactivity, both passive and active. Animals are aware of their environments: they interact in real time with environmental changes and events by inputting information regarding their environments, tracking changes over time, and behaving in ways that reflect and exploit that information. No-one regards that facility of awareness as a philosophically problematic issue. We can envisage building robots with awareness in that sense. Awareness as reactivity is not the ‘hard problem of consciousness’ but one of the ‘easy problems,’ and does not saddle us with an explanatory gap. Consciousness, on the other hand, seems to be something essentially different in kind. With the case of phenomenal consciousness foremost in mind, the third landmark appears to demand something above and beyond mere awareness. Environmental awareness in the sense of active reactivity seems possible without the wealth of subjective experience with which we are familiar: without the ‘raw feels’ of sound, color, texture, and smell. What is this thing that consciousness adds, above and beyond awareness, and how are we to understand it? 2. CONSCIOUSNESS: THE REFLEXIVITY OF AWARENESS Whatever the distinctions between awareness and full consciousness, it should be noted, it is within the facility of awareness that richer subjective consciousness has its home. There are no states of consciousness that are not states of awareness in some sense. What then is it about some and not other states of awareness that qualifies them as conscious states? What is it about some psychological processes and not others that qualifies them as conscious? The tools we have developed in studying reflexivity offer an answer. In a nutshell, the answer is this: that conscious states are states of awareness of a particular reflexive form. Consciousness is not some additional element added to awareness. Consciousness, including phenomenal consciousness, is awareness with a specifically reflexive vec-

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tor, rather than awareness with a mysterious subjective add-on. A conscious state is a state of awareness in which the focus of awareness includes aspects of that state itself. A conscious process is one that includes an awareness of aspects of that process itself. In the previous chapter we considered contexts of ostension, and noted that a small sub-category of referential terms play a special role in such contexts. Ostension involves directed attention. That attention is within a particular act, directed from a particular point—a point of origin for the act itself. Reference to points of origin within ostension is thus not merely unmediated by definition but transparent and immediate within the act itself. Such is the character of ‘I’, ‘here’, and ‘now.’ Such is also, we noted, reference to the act of ostension itself— ’this pointing’ or ‘this act of ostension’. The structure of conscious awareness matches that outlined for ostension in several ways. Awareness, like ostension, involves directed attention. Just as ostensive reference can reflexively reference the act of ostension itself, awareness can include awareness of the act of awareness itself. Conscious awareness is awareness with that character: a state of awareness that includes reflexive awareness of the state itself. The reflexive awareness required for consciousness, however, demands reflexivity of a particular character. It is not enough that my state of awareness focuses on such a state under a description—’the attentiveness characteristic of contestants in the game,’ for example. It is not enough that my state of awareness is of something I am aware of de re in the form of ‘that state . . .’ In line with forms of reference transparent and immediate in that they target points of origin, the reflexive awareness required for consciousness must be the immediate internal self-reference of ‘this act of awareness,’ correlate to ‘this pointing’ in the case of ostension. What is consciousness, then? A state of awareness with a specific reflexive character: a state of awareness that includes itself within its scope, and does so in terms of transparent and immediate reflexivity. Consciousness is a state of awareness that is transparently and immediately self-aware. Conscious processes are those that involve such an awareness.

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Essential elements of David Rosenthal’s ‘higher-order thought’ theory of consciousness are incorporated in this approach.3 There are two regards, however, in which the current account departs from his— in both of which, we would argue, it represents an improvement. Rosenthal insists that conscious mental states are not mental states with a peculiar subjective character—a specific glow—but merely mental states that are the object or target of higher-order states. His account faces difficulties, however, in plausibly pegging the ‘right kind’ of targeting, since I may well think of some of my states under a description or in an oblique manner that fails to make them conscious. Rosenthal’s account also invites an ascending hierarchy of higherorder states. If conscious states are those that are immediately and transparently reflexive, we face neither difficulty. It is the point of origin form of self-reference, familiar from cases of ostension, that is the targeting crucial for consciousness. Because the state itself is reflexive, it forms a loop rather than an ascending ladder—precisely as states of consciousness seem to. ‘Consciousness’ is an extremely ambiguous term in common usage. When the doctor wants to know whether the patient is conscious, he simply wants to know whether he or she is responsive—something which those asleep or in coma are not, but which even those who are operating on automatic pilot are. Here it appears that any state of awareness counts as conscious. Rosenthal’s is a richer sense of ‘consciousness’, and his account demands more—that a mental state be an object of awareness in some sense. But there are in fact three phenomena: states of awareness, states of which one is aware, and states which are reflexively self-aware. It is the second that Rosenthal emphasizes. It is the third that is crucial to the full sense of consciousness at issue here. A major test of any account of consciousness is whether something could satisfy that account and yet not count as conscious in the sense at issue. Any account of consciousness as mere awareness fails such a test: an organism might well be responsive to environmental changes in real time without what we think of as a conscious ‘inner life.’ Higher order thought accounts such as Rosenthal’s, though they do capture the fact that consciousness is a structure of state awareness of state, also fail such a test: I might be aware of some of my mental states by an

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indirect referential route, as it were—such as ‘the kind of mental state that often causes me to wake up’ or ‘the mental state that explains my nervous finger-drumming’—that does not guarantee that what are at issue are conscious states. As Shaun Gallagher and Dan Zahavi note, I might be aware that I am processing spatial information by means of blindsight without that processing becoming conscious and thereby ceasing to be blind.4 The account of consciousness as a state of awareness that is transparently and immediately reflexively self-aware, however, does seem to pass such a test. It is impossible, we would argue, to think of transparent and immediately reflexive awareness within a state of attention that does not count as a conscious state. Consciousness is a form of awareness, but awareness of a particular type. It involves a state of attention taking a mental state as its object—reflexively, in fact, taking itself as at least part of its object. But it also involves that state taking itself as an object in a particular way: with the immediate and transparent ‘point of origin’ focus of ‘this particular state.’ Any state of awareness with that character, we hold, will of necessity be a conscious state. On such an account consciousness does mark mental states with a particular character, but not the character of a subjective add-on. What consciousness marks is not states with a magic ‘glow’ as a content nor with content that ‘glows’ but states of awareness with a particular structure. Consciousness marks those states of awareness that in addition to any other focus take themselves as a focus, accessed immediately in the form of ‘this awareness . . . .’ There are several complications we should add. It is quite generally true that conscious states of awareness involve awareness of something beyond themselves. Indeed a ‘pure’ state of awareness solely of that awareness may be as impossible as it would appear to be empty. It is also true that a state’s reflexive self-awareness may be limited to certain aspects of that state. Just as one may point to the importance of one’s ostension without pointing to its intention, and just as one may attend to the emotional character of one’s attention without attending to its duration, one may be aware of certain aspects of one’s act of awareness but not others. Reflexive awareness in general will qualify

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as conscious, but may be reflexively focused on particular aspects of that awareness just as awareness in general may be focused. Is consciousness necessarily intentional? Brentano and much of the phenomenological tradition insist that it must be: that consciousness is always consciousness of.5 Our account not only confirms such a claim, but explains it. It is ultimately awareness that is necessarily intentional. A state of awareness that is an awareness of nothing makes no sense; awareness is always awareness of. The intentionality of consciousness is therefore neither a unique nor defining characteristic; consciousness is inevitably intentional simply because it is a species of awareness. 3. QUALIA AND CATEGORICAL FAST PROCESSORS If consciousness is what we have said it is, why does it have the particular experiential character that it has? If consciousness is reflexive awareness in the sense outlined, why does it feel the way it feels? The examples of consciousness that appear most insistently within the philosophical literature, and that in many ways seem the most perplexing, are examples of conscious perception. It is in dwelling on one’s conscious perception that one is aware of the subjective qualia, raw feels, or felt qualities of consciousness. What are these felt qualities, and what place can they possibly have in an objective or physical universe? Here it helps to remember what perception is for. Perception represents information input from a changing environment in ways that serve the needs, aims, and self-preservation of an organism. It also helps to remember the characteristics of information input required for perception to serve those ends. The information input at issue has to be fast, because the organism may need to respond quickly to changes in the environment. It has to be indexed to the organism at issue and its place in the environment, because it is information relevant to the organism’s action and reaction that is required. If perception is to serve the needs of an organism—responding to cases like those in the past in the ways that have been most productive, for example—it also has to be categorical. Aspects of what is perceived must come either immediately tagged or very quickly taggable in terms of those features

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that are of past and potentially future relevance for the organism. Environmental input must be immediately indexed and served up in terms of immediate informational categories.6 Consciousness, we’ve said, is a transparently reflexive state of awareness. When that state of awareness is perceptual, it will include the features crucial for perception, including the fast categorical processing outlined above. We have noted that reflexive awareness can focus on specific aspects of that awareness. Within a conscious perceptual state, reflexive awareness focuses on the activation of specific fast categories. It is the activation of those specific categories, reflexively attended to as they are triggered, that are the subjective qualia, raw feels, and felt qualities of consciousness. In conscious perception of this type, we are reflexively aware of the information processing as it happens, and aware moreover of the mechanisms by which it is happening. It is the triggering of specific fast categories in perceptual processing, accessed reflexively in the course of that processing, that are what fear feels like, what lemons taste like, and what red looks like. It is those that are the subjective qualia, raw feels, or felt qualities of phenomenal consciousness. The first illustration below is a schematic representation of awareness, directed toward some object beyond it. The second represents conscious awareness, in which the object of awareness includes the process itself, or some aspect that process. The third illustration represents the specific form of reflexivity in which qualia appear: fast categories of processing, in operation, themselves made a focus of awareness.

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In chapter 1 we tracked sets and categories defined in terms of selfmembership, extending that study to sentences that take themselves as their own subjects. In chapter 2 we extended the study to formulae that can encode themselves and aspects of formal systems in which they themselves appear. In chapters 3 and 4 we extended our study of reflexivity to positions, practices, propositional attitudes and mental states that self-apply, and to distinct forms of reference within these. A similar reflexivity appears in stages of informational processing that transparently access themselves and their processing mechanisms. A state’s awareness of its own processing mechanisms—in particular, of the fast processing categories in operation—gives us the subjective character of phenomenal consciousness. What then are subjective experiences? In the case of perception, they are aspects of perceptual processing, accessed reflexively in the course of that processing. What are the subjective qualities of experience—the subjective feel of velvet and the taste of pineapple? They are information processing categories in operation, accessed within the context of that information processing itself.

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4. BLACK-AND-WHITE MARY AND THE EXPLANATORY GAP The distinctions noted help us to make sense of some of the standard puzzles regarding consciousness. One of these is Frank Jackson’s Black-and-White Mary. Mary is raised in an exclusively black and white world, but becomes the world’s scientific expert on color vision. She reads—in black and white, of course—all that is known regarding the physiology of color vision. She knows, Jackson assures us, all that any scientifically objective or physiological theories can tell us regarding color vision. On Tuesday she leaves her black-and-white room and enters a world with color for the first time. “So that’s what red things look like,” she exclaims.7 On Tuesday, Jackson argues, Mary has learned something new: she has learned through first-person subjective experience what red things look like. That is something she did not know before, though she knew before all that scientifically objective or physiological theory could tell us with regard to color. The conclusion Jackson draws is that a certain forms of reductionism must be false: there must be important things on the ‘subjective’ side of the universe that science and physiology cannot reach. As presented, the argument is incomplete— there seems, for example, no particular reason to believe that all that can be scientifically or physiologically known can be conveyed in black and white texts. Jackson himself has apparently changed his mind several times as to how effective the argument is and what it actually shows.8 The importance of Black-and-White Mary for our purposes, however, is as a clear and prominent example invoking phenomenal consciousness, subjective experience, qualitative raw feels, or the view ‘from the inside.’ There appears to be something Mary has access to for the first time on Tuesday, and something new that she comes to know. What are these? On Tuesday, exposed to a world of colored objects for the first time, Mary will employ fast categories in perceptual processing that she has never had occasion to use before: the fast categories appropri-

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ate to color perception. She is doing things she has never done before—perceiving colored things—and her information processing in that task makes use of fast categories unused in a black and white world. If her perception is conscious—if it is reflexive in terms of those categories—she will also be aware of something of which she was never aware before, because it was never active and so available for conscious awareness before: the fast categories now in use for the first time. Those fast categories, referenced in awareness, are what qualia, raw feels, and subjective qualities are. Frank Jackson’s ‘scientific’ or ‘objective’ view, written out in black and white, is best conceived as referring to what is at issue in descriptive terms. It cannot refer to what is at issue within the context of perceptual processing de se, and thus cannot reference what is at issue in the essentially experiential form of qualia, raw feels, and subjective qualities. It is this access to what is at issue that has been denied Mary until Tuesday. Does Mary now have access to something new—something inaccessible from a scientific, objective, or physicalist perspective? We might approach the issue in two ways. Because the same thing may be referenced in different ways, from different contexts, we might propose that what Mary references or has access to from two different perspectives are nonetheless ultimately the same things. The descriptive claims she has read and memorized regarding units of perceptual processing may be true of precisely the processing units she is accessing now, de se within the processing itself. Differences in forms of reference need not entail differences in referents. On the other hand, it is clear that we construct an ontology of entities in terms of the reflexive referential access that comes with conscious perception. Because that form of referential access is built into the very concept of ‘raw feels’ and ‘qualia,’ there is a set of entities that cannot be referenced from the very different context of objective scientific description in black and white. It is those things—the way red things look, for example, or what it is like to see red—that are open to Mary for the first time on Tuesday. We have outlined this sketch in terms of new ‘items’ of acquaintance. But the same story can be told in terms of properties. Mary comes to know ‘that red things look like that. . .’ or that they ‘have

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this qualitative appearance.’ These are properties with which Mary comes in contact for the first time on Tuesday. Red things have the property of being processed in terms of a particular fast category. It is reflexive access to that processing and thus that—access with the immediacy and transparency of ‘this processing’—that she has for the first time on Tuesday. ‘What is it like to be a bat?’, Thomas Nagel asks, and later makes it clear that he is asking what is like for the bat to be a bat. But ‘what it is like’ references something accessed in a particular way. Just as ‘what one knows’ may build in a particular form of reference, ‘what it is like’ builds in a particular form of access. It is clear that we are not bats, and that we therefore cannot consciously or reflexively access, as it if were our own, the informational processing of bats. When form of access is built into the ontology of ‘what it is like,’ it should then be no surprise that we are faced with a realm of things—bat qualia and bat experiences—that are inevitably inaccessible to us. Jackson’s Black-and-White Mary, Nagel’s bat, and the concept of the ‘explanatory gap’ are all of a piece.9 When we ask ‘How could all this. . .’—indicating the scope of our current visual experience—’. . .be merely that . . .’ —pointing to an electro-chemical process in three pounds of gray matter—we are mixing two radically different forms of reference. One is a mode of transparent self-reference within a form of awareness. One is objective reference de re, demanding neither transparency nor reflexivity within any ongoing mode of access. At base, it is our forms of reference that are radically different. The explanation for the explanatory gap is the fact that we read differences in referents off what are fundamentally differences in referential form. 5. CONSCIOUS STATES AND BRAIN STATES There is a conceptual pattern explored in the previous chapter that reappears in questions of how the mental is related to the physical— how states of consciousness, for example, are related to brain states. Here it helps to review that pattern.

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In the previous chapter we explored the pattern at issue in terms of the essential indexical. ‘I’, ‘here’, and ‘now’ involve an immediate and transparent reflexive reference to points of origin within any act of ostension. ‘He’,’ ‘that’, ‘there,’ and ‘then’ are equally unmediated by description, but do not play the same central role. For that reason the de se form of reference that is ‘I’ can never be given an equivalent in terms of a de re ‘he’, nor can a de se ‘now’ be given an equivalent referential phrase purely in terms of ‘then’. In such a case we have essentially non-equivalent forms of reference. Because each of these are unmediated by description, moreover, we have at least three forms of non-equivalent reference, since neither ‘I’ nor ‘he’ can be given nonindexical equivalents in terms of description alone. The pattern includes a second movement as well. The construction of an ontology—a realm of things—often incorporates forms of reference. In the case of indexicals this occurs in considering what is believed, known, or feared: the objects of propositional attitudes. Because propositional attitudes constitute opaque contexts, whether it can truly be said that something is believed, known, or feared may depend on an embedded form of reference. Non-equivalent forms of reference, such as the non-equivalent forms ‘I’, ‘he’, and ‘the man spilling sugar in the supermarket’ give us non-equivalent cases of potential belief, knowledge, or fear. Once we construct an ontology of the things believed, known, or feared, it should then not be surprising that what is known in knowing that I am making a mess, for example, cannot be merely what I knew previously in knowing that he—the man in the fish-eye mirror—was making a mess. ‘Propositions’ is simply another term for the items of such an ontology.

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Note that at base what is at issue in this pattern is merely alternative forms of reference, and that distinct forms of reference do not guarantee distinct referents. In this case, for example, although ‘I’, ‘he’, and ‘the man spilling the sugar in the supermarket’ are non-equivalent forms of reference, what is referred to in each case may in fact be the same. Whether I realize it or not, I am in fact the man spilling the sugar in the supermarket and am in fact him—the figure I see in the fisheye mirror. The fact that we have non-equivalent forms of reference at the base, therefore, does not guarantee that at that level we are not referring to precisely the same things—that in the case of indexicals we are not referring to the same person, for example. There are not three people at issue in the story—’I’, ‘he’, and ‘the man spilling sugar in the supermarket’. There is only one, referred to in non-equivalent ways. The same cannot be said, however, for the items of the ontology constructed from that base. Our conception of the propositional attitudes and of ‘what is known’ is sensitive not merely to what is referred to but to how it is referred to. Despite the fact that ‘I’, ‘he’, and ‘the man in the supermarket’ may be co-referential at base, the propositional or attitudinal objects constructed from these materials—what I know in knowing that I am making a mess, what I know in knowing that he is making a mess—are not and cannot be co-referential. At the bottom end of our ontology we have non-equivalent forms of reference (‘I’, ‘he’) that may nonetheless be co-referential. At the top end of an ontology built from that base we have forms of reference (the proposition that I . . ., the proposition that he . . .) that cannot be co-referential precisely because embedded forms of reference are nonequivalent. This conceptual pattern gives us a route into the question of the relation between consciousness and the brain. A conscious process of awareness, we have said, is one that references or accesses, in the immediate and transparent manner of points of origin, aspects of that process of awareness itself. In order to qualify as conscious, a state must self-reference in such a way. We have seen that certain forms of reference are built into our concept of propositions and into their individuation—what makes something the same proposition or a different one. In much the same way, form of reference—reflexive selfreference—is built into the very concept of states of consciousness

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and into their individuation: what makes something ‘this state’ rather than that in our ontology of mental states. What is a brain state? Despite decades of spilled philosophical ink, it has never been very clear what a ‘brain state’ is supposed to be—a snapshot of a particular individual’s brain at a particular time? Of a particular part of that individual’s brain? A composite snapshot of multiple brains? A motion picture version of each of these? What is clear is that the notion of a brain state has always been of something that can be publicly referenced and accessed in any of various ways, by any of various investigators. What are we asking, then, when we ask whether consciousness might be a brain state, or whether a particular conscious state might be a specific brain state? At least many of the difficulties and conundrums inherent in the mine field of controversy that is the mind-body problem—many of the mysteries of consciousness—can be traced to a radical difference in forms of reference. There is an inherent impetus toward Dualism in the fact that forms of reference (a) characteristic of the mental realm, and (b) characteristic of the public and objective order of brain states, are essentially non-equivalent and irreducible. The reflexively de se ‘this state’ characteristic of consciousness has no non-reflexively de re equivalent in ‘that state’ nor any merely descriptive equivalent of the form ‘neural pattern n.’ In thinking of my states of consciousness I use a form of reflexive reference proprietary to those states of consciousness and inherently non-exportable. When we ask ‘how could this . . .,’ indicating a present state of conscious awareness, ‘be merely that . . .,’ indicating a pattern observed in an fMRI, we switch mid-sentence from one to another radically irreducible form of reference. Dualism is fueled by the fact that major portions of our mental ontology are built in such a way as to incorporate these specific forms of reference. ‘My experiences’ and ‘my sensations’ build in precisely the aspect of self-reflexive consciousness we have outlined. ‘The feel of velvet,’ ‘the taste of pineapple,’ ‘the warmth of day-glo red’ and the ontology of qualia in general concretizes these forms of reference in the individuation of objects of consciousness. ‘What it is like . . .’ is the phrase that Thomas Nagel uses to define what it is for an organism to have conscious mental states: “the fact that an organism has con-

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scious experience at all means, basically, that there is something it is like to be that organism . . . something it is like for the organism,” which builds in referential reflexivity in precisely the same way.10 An ontology that builds in forms of reference impossible within another perspective will include things that can find no home in that other perspective. It is for that reason that mental states and mental phenomena will inevitably escape the net of the purely physical and objective perspective characteristic of the study of brain states. To this extent, and at this upper end of an ontology constructed on incompatible forms of reference, Dualism appears largely inevitable. It should be noted, however, that this is a distinctly different Dualism from many that have preceded it. The root source of duality here lies in ourselves rather than in the universe at large: in the forms of reference proprietary to the alternative perspectives and projects open to us. We are creatures that can observe and describe, using the public and objective forms of reference characteristic of ‘brain states’. We are also creatures that can self-reference our own states of awareness, leading to an ontology of experiences, sensations, qualia and raw feels. It is because the forms of reference proprietary to each project are irreducible that we end up with classes of radically different entities—the inherently ‘subjective’ objects of a ‘mental’ and the inherently ‘objective’ entities of a ‘physical’ world. Despite the fact that it is so easy to read the result as an inherent duality discovered in the substance of the universe, that result is ultimately the product of proprietary forms of reference within different but both very human forms of endeavor. A root difference in forms of reference, while both underwriting Dualism and offering an explanation for its inevitability, opens other perspectives as well. It is with respect to experiences, sensations, qualia and ‘what it is like’ that the strongest case can and has been made for contemporary Dualism. But it should be noted that the terminology of the ‘mental’ is not all of a piece, and it may be that full reflexive reference is not uniform across the board. ‘Reactivity,’ ‘receptivity,’ and ‘responsiveness’ might well be classed as ‘mental’ terms, but there is not the same pull to Dualism for these. In asking whether an organism is responsive or reactive rather than comatose it seems that we might well be referring

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to a characteristic capability of a nervous system. ‘Surprise,’ ‘alarm,’ ‘fear’, and ‘desire’ are all mental terms, but notably are not those put forward as proof that mental terms cannot be understood in purely physical terms. Essential self-reflexivity and the impetus to Dualism may thus be clearer in the case of some aspects of the mental than others. Indeed this may even be true even for the terminology of consciousness. In common usage, we have noted, ‘consciousness’ has a wide range of applications. In asking whether a patient is conscious, the medical profession is often asking merely whether the patient is responsive and awake. One can infer that one was conscious of an approaching vehicle from the fact that one reacted appropriately, despite the fact that one was operating on ‘automatic pilot’ and reflex. Philosophers have in fact had to struggle to isolate terminologically the specific target of ‘phenomenal consciousness.’ Consider moreover two terms as close as ‘state of consciousness’ and ‘conscious state.’ The idea that a state of consciousness is to be identified with some physical property within three pounds of brain matter faces all the objections outlined above regarding cross-circuited forms of reference. Having outlined conscious states as those that are inherently self-reflexive, however, it might not seem beyond the pale to imagine looking at a brain during surgery, seeing a recognized pattern of electro-chemical activity across a network of neurons, and saying of that state that it is in fact a conscious state—meaning not that we are accessing it as such, but merely that it is being accessed as such. Because mental terminology is not all of a piece, the question of whether a specific ‘mental’ term is inherently reflexive and whether it might be considered co-referential with some physical correlate or process must be considered on a case by case basis. It nonetheless remains true that for a large and important class of such terms the ontological instantiation of radically incompatible forms of reference makes aspects of Dualism inevitable. As noted, this is a different Dualism. Explanation in terms of forms of reference makes it tempting to consider the possibility of an underlying Monism as well. Because we will not generally be able to treat specific mental terms as co-referential with specific physical terms, such a Monism appears most plausible when expressed in a way that abstracts from particular referential terms on each side.

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As emphasized throughout, the fact that two forms of reference are essentially non-equivalent does not prove that that what is referred to in each case might not be precisely the same. Characteristic of ‘the mental’, most clearly in the terminology of phenomenal consciousness, are de se forms of immediate and transparent reflexive reference. Characteristic of ‘the physical’ are de re and descriptive forms which lack these characteristics. That alone does not prove, however, that the field of reference—that part of the universe within which each takes its referents—is not precisely the same. Consistent with the Dualism of constructed ontologies outlined, in other words, is the possibility that fundamental domains of reference might be the same despite radically different forms of reference. Consistent with such a Dualism will be an underlying Monism. ‘The mental and the physical are ultimately one substance.’ ‘The basic reality is a psycho-physical complex, with both mental and physical aspects.’ ‘Reality is one. It is our ways of accessing it that are multiple.’ It is unclear how to characterize the underlying unity of such a Monistic vision, and there are inherent difficulties in any attempt to do so. Were the language of the mental and of the physical uniformly and irrevocably opposed, such a Monism could not be formulated at all. Our situation is certainly not that hopeless, but because the terminology characteristic of ‘the mental’ and of ‘the physical’ face quite general issues of non-equivalence, any jointly underlying reality cannot be described solely or even primarily in the terms of one of these alone. The Monism with which this different Dualism is consistent can be neither standard Materialism nor standard Idealism, each of which insists on the priority of one form of characterization over the other. Despite the difficulties of formulation, the impetus for such a Monism is strong. Interestingly, it is a very different impetus from that behind Dualism. The motivation to Dualism, traced to alternative forms of reference, is entirely conceptual. The motivation to Monism includes major empirical elements. Despite conceptual irreducibility, the correlational and causal connection is clear between those events for which we use a mental terminology and those for which we use a physical terminology. Things describable in purely physical terms are the clear causal consequences of events of conscious awareness, deci-

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sions, and deliberative action. Things describable in purely mental terms are the clear consequents of physical events. When events of a particular class on one side occur, events of a particular class on the other side predictably occur in concert. The empirical facts require at the very least a recognition of causal interaction between elements of our mental and physical ontologies. Historically, resistance to the possibility of such interaction has been based in the myth that like can act only on like. For the Greeks, the idea that factors as conceptually distinct as night and day could nonetheless interact causally was anathema. That ancient idea of causal homogeneity continued to exert influence as late as Descartes, with his Great Wall separation of mind from matter. But the reality of it stands otherwise. Motion creates heat via friction. Pressure on the piston regulates the temperature in the chamber. Sounds engender salivation by Pavlovian conditioning. Since Hume, the idea of crosscategory causation no longer seems all that odd to us. We do not, or should not, see any inherent impossibility in the idea that physical processes can causally engender mental responses, and that mental processes can engender physical responses. The dentist-hypnotist’s instruction can shut off bleeding in the gums. The phase of the moon can influence the suicide rate. The psychology of investors can change the production or extraction of commodities. Rumors can cause stockmarket crashes; radio broadcasts can cause panics; mental stress can cause stomach aches. The scraping of cat-gut can produce music. The idea that only like can act on like is falsified all around us. And the idea that thought can control bodily activity, just as the mental can react to physical input, is simply an elemental fact in everyday experience. The possibilities opened up for a Monism consistent with referential Dualism go deeper than causal interaction, however. It could be that the portion of the universe accessed through the referential mechanisms of ‘the mental’ and that portion of the universe accessed through the very different referential mechanisms of ‘the physical’ is nonetheless one and the same portion of the universe. Despite nonequivalent forms of reference and essentially irreducible ontologies built on those forms of reference, it might be that the general territory in which reference is made is the same. One way of expressing such a

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possibility is to say that the mental and the physical may be coconstitutive: not that there are two kinds of things that causally interact, but that ultimately, beneath differences in form of reference, there is fundamentally just one kind of thing. The Monistic vision is that there is a single psycho-physical reality which appears in different aspects simply because it is accessed in different ways. Though consistent with the different Dualism sketched here, such a Monism is interestingly inconsistent with both the Monism of Materialism and the Monism of Idealism in their familiarly exclusive forms. The psychophysical reality at issue could not be said to be ultimately ‘physical’ in vindication of the Materialist: on the outline given, the referential structure of the ‘physical’ instantiates just one form of access and one associated ontology. For precisely the same reason, the common field of reference could not be said to be ultimately ‘mental’ in the sense of traditional Idealism. Though Monism is opened as a possibility, motivated by empirical facts of correlation, it must be admitted that proper formulation remains a difficulty. On the outline given, the referential and conceptual structures characteristic of talk of ‘the mental’ and those characteristic of talk of ‘the physical’ are embedded in very different human projects and perspectives. Talk of the link between them has to be within a philosophical context of reflection wider than either but with some form of access, however general and oblique, to each. 6. WHO NEEDS CONSCIOUSNESS? An important part of our examination began with the question of what perception was good for. That leads, we think, to a better understanding of why perceptual consciousness has the qualitative character that it does. A correlate question, however, is what perceptual consciousness is good for. Who needs consciousness? Awareness not merely of an environment but of the mechanisms used to access information from that environment allows for higherorder learning and generalization. It also allows an organism to correct or override modes of information access. Awareness of mechanisms used to access information thus allows for cognitive processing that is not possible without that higher-order or reflexive awareness.

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Consider the case of illusion. As long as action is guided uncritically by immediate fast categories of perceptual awareness, those cases that trigger a fast category of ‘bent stick’ will trigger the same actions, whether the stick is really bent or appears to be so because of refraction in a glass of water. Cases may trigger the same fast category whether the sinuous shape in shadow is that of a live snake or a limp rope. At the most immediate stimulus-response level of learning, cases that prompt the same fast categories will be impossible to distinguish, and thus differences will be impossible to learn. One will be at the mercy of one’s immediate perceptual processing. Intelligent beings need self-consciousness in order to do all the sorts of things they characteristically do. Take self-reproach, which in intelligent beings is a significant mode of reflexive action. When we (properly) reproach ourselves we do so either for performing wrongly or for failing to. _______________________________________________________ OCCASIONS OF SELF-REPROACH THEMATIC MODE

ACTIVE

INACTIVE

Prudential

incur a negativity

fail to realize a positivity

Ethical/moral

do something wicked

fail to do something good

Social

make a faux pas, lose face

fail to shine socially

Intellectual

fall into error, make a mistake

fail to grasp truth

_______________________________________________________ With awareness of the categories of processing, and coordinate awareness of context and memory, one can learn that some perceptual categories are misleading. One can guide one’s actions accordingly, avoiding both real snakes and overreaction to limp ropes, betting on the bentness of sticks outside of glasses of water with different odds than those within. There may indeed be cases of learning that are impossible without conscious awareness. It seems certain that there are cases of learning—and correlatively, of sophisticated reaction to envi-

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ronment—that are facilitated by conscious awareness. The awareness of fast categories that is perceptual consciousness can allow one to override what would otherwise be automatic reactions to fast categories. It can also allow one to construct complex categories of reaction from experience which, although not so fast (at least initially), are better suited to appropriate action in one’s environment. Were one to look through an optical microscope without ever considering the nature of the instrument itself, one would be misled by a range of optical effects—false color and rainbow effects are only the beginning. It is attention to the device itself, and the development of theories of optics regarding its function, that allow one to correct for such effects and thereby to correct for misleading aspects of information input. It is this same form of reflective awareness, with the same sorts of benefits, that is the function of conscious perception. There is at least anecdotal empirical evidence that supports such a view. In times of stress from unfamiliarity—in wartime, and on the battlefield—critical processing of incoming information, with suspicious and careful review of processing categories, can mean the difference between life and death. Is that a shadow, or a sniper? Is he a friend, or an infiltrator? Is this message genuinely from the claimed source? In such cases fast categories from other contexts can mislead, and it is thus particularly important that they be subjected to critical awareness. Interestingly, it is in reflecting on such times of stress in their lives that people often report being ‘most alive’—’attuned to every moment because every moment might be my last.’ Those reflections support precisely the kind of sketch we have outlined for the nature and function of consciousness. Fast categories are not all innate, though some may be. Many are learned, and in the process of learning one must learn how information is best grouped for fast, efficient, but accurate decisions. Those decisions require attention to categories within which the information is already processed, with an eye to required distinctions, shortcut combinations, and what immediate inferences might be justified—all of which requires attention to the categories and process of information processing itself. That seems to explain why the learning of a new task—a new language, or a new instrument—is so often an intensely conscious process. In the process of learning one is conscious of as-

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pects of the task of which one is not conscious later. Indeed that seems to be the general purpose of learning: to distil fast and efficient categories from the wide field of possibilities open for consideration at the beginning. Over time, attention to the process of information processing is no longer required—the processing itself can move without a hitch, and consciousness of the same intensity is no longer required. It is clear that higher-order cognitive processing has important benefits quite generally, whatever its field of application. It is therefore to be expected that higher-order cognitive processing with regard to the fast categories of information input—the lower levels of perceptual processing—will have benefits as well. It is those benefits that explain why we are conscious at all. Consciousness is wider than perceptual consciousness, but its basic character is the same throughout: the character of transparently reflexive states of awareness. Here a mental state is about a mental state, just as we had sets of sets and formulae regarding systems of formulae. That mental state, moreover, takes itself or an aspect of itself immediately and transparently as its own referent; a reflexive structure that runs from paradox to the phenomena of consciousness. 7. CONSCIOUSNESS OF SELF We began this chapter by attempting a sketch of landmarks in the evolution of cognitive processing: • passive reactivity • active reactivity • consciousness • consciousness of a self Our concentration has been on understanding the character of conscious states and conscious processes characteristic of the third stage. But beyond that stage lies a fourth, in which consciousness is not merely reflexive awareness within a state or process, but within a con-

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cept of a larger ‘self,’ persisting and binding a history of conscious states over time. What is a self? ‘Self’ in general is a marker of anaphoric retroreference: • He placed one block on top of another which itself rested on yet another block. A book can refer to itself, a map can contain an image of itself, a list of items can list itself. We can also signal reflexive incapacities using the anaphoric retro-reference of ‘self’: a book cannot contain itself in its entirety as one chapter among many, a map cannot include itself completely, a list of items cannot contain itself completely as one part among others. We have noted that ‘consciousness’ is ambiguous in ordinary use, signaling anything from mere responsiveness to mental states reflexively self-aware in the sense centrally at issue here. Given the additional ambiguity of ‘self’, ‘self-consciousness’ proves more ambiguous still. Ostension, we have noted, is always from a particular point—what we have characterized as the point of origin. Awareness too is always from a particular point, and in that limited sense may always involve an origin or an ‘I’. Consciousness as a form of awareness will share that characteristic, though in neither case is it clear that attention need be directed to the point of origin per se. A pivotal problem of first-person reflexivity was set by Hume’s unsuccessful quest for the ego. Franz Brentano took the line that all consciousness involved self-consciousness, insisting that ‘It is I who am aware’ is operative in every instance of consciousness. The German school of phenomenology, including Alexander Pfänder and Theodor Litt, stressed this idea of ownership: that our conscious mental acts always involve an implicit awareness that they belong to us: the work of an ego possessed of a sense of self (Selbst gefühl).11 On such an approach, all consciousness qualifies as self-consciousness. Such a claim can be maintained, we think, only for a particularly thin sense of ‘self’. Just as the ‘I’ for which Descartes’ ‘I think’ will hold with certainty must be an ‘I’ thin to the point of having no further

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characteristics, an implicit ‘I’ that constitutes only the implicit point of origin in awareness or consciousness would have to be a sense of ‘self’ attenuated in the extreme. What we are after is consciousness of self in a richer sense. In principle, a conscious state might reflexively reference any broader pattern of which it is a part. It might include an awareness of itself as similar to another state, for example, incorporating an awareness of that state as well. It might include an awareness of itself as one of a series of states equally self-aware—as part of a history or stream of consciousness. It might include an awareness of itself as part of a continuing process or goal-directed project. Richer concepts of a ‘self’ represent a still larger reflexive focus: an individual’s reflexive awareness of a substantive ‘I’ to which states, series of states, processes and projects all belong. It is perhaps understandable that the precise target of such self-reflection can seem variable or vague. For precisely that reason, it may sometimes be appropriate to speak of a being self-conscious in some regards but not others. It is one thing to be conscious, and another to be conscious of oneself as an organism with a history and a future, perhaps successes and failures, persisting through time. It is one thing to be aware of something, another to be aware of this awareness, and another to incorporate a sense of a wider self in my awareness of this awareness as mine. It is one thing to be a cognizing being who grasps certain facts, and another to be an intelligent being with a sense of self, capable of taking himself to grasp certain facts. In this richer sense of self, consciousness clearly does not entail self-consciousness. Animals at early stages of cognitive development may be conscious and yet not have a developed sense of self. Indeed there is a small industry in trying to develop adequate tests for a ‘sense of self’ across species in precisely this sense.12 The distinction in cognition between being aware of something and being aware that one is aware, between consciousness and selfconsciousness in both weaker and richer senses of ‘self’, carries over to agency as well. Self-consciousness can operate in two dimensions: the purely cognitive and the agency/practical. I can realize that I apprehend something, and I can realize that I can do something.

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One can manage to do various things without being aware of one’s ability to do them. A creature can believe X to be so without being aware that X’s being so is a belief of his. Analogously one can do X without any awareness of this being something one has done—i.e., without relating it to oneself. People can make claims without any awareness that they have done so—let alone that they are speaking English. ‘Who needs consciousness?’ we asked above. Here we might also ask ‘Who needs self-consciousness?’ Just as awareness of aspects of awareness seem to afford operational gains—gains in possibilities for detecting, correcting, and compensating for biases and defects in the ‘fast categories’ operational in certain contexts, for example—so selfconsciousness incorporating a wider sense of self can be expected to afford operational gains. At even the simple end of the evolutionary spectrum, it is crucial that an organism’s behavior differentiate between itself, its predators, and its prey. The success of more complex organisms in more complex environments depends further on the balancing and coordination of specific actions within a sustained pattern of input and goaldirected action over time. In a relatively unchanging environment, activity of such a sort might be immediate and instinctual: there may just be something that feels good about burying nuts in the late summer. For organisms that must deal with a changeable and unpredictable environment, memory of the organism’s past experiences of attack and injury, intentional attempts to avoid their repetition, and activity targeted to future retrieval of foodstuffs all seem to be required. These suggest organization in terms of something more like a concept of self, though what is at issue may still fall short of a full ‘consciousness of self’ both in terms of the sense of self and the sense of consciousness at issue. Social interaction within a species seems to require something more. Cooperation and competition within group action, deliberate transfer of information, and deliberate deception would all seem impossible without awareness of the social processes of coordination and information themselves. That awareness in turn seems to demand a consciousness of oneself as a social participant among others. The self-consciousness evident here, interestingly enough, seems to coor-

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dinate with a perspective on meaning from within the human practice of linguistic communication that plays an important role in the next chapter. The ‘theory of mind’ that forms the focus of a wide range of psychological experimentation is just one step away.13 The prerequisites for ethics also lie just one step away. The fact that we acknowledge our actions as ours, and as reflective of the person we take ourselves to be, is crucial for moral credit and discredit—for moral responsibility both negative and positive. The self-respect of pride and the self-reproach of shame are crucial factors here. Reflexivity in both a consciousness of self and in self-conception form the very crux of morality. The reciprocity of personal recognition that enables me to see in you a person much like myself seems crucial alike for our selfdefinition as individuals and for communal survival. Many higher organisms have a rudimentary sense of self based on possession and ownership: my children, my food, my turf. But it seems to be only humans who have a sophisticated sense of self and social bond in which selfhood is taken to deserve recognition, carrying both rights and claims vis-à-vis others and obligations to them. Without the reflexivity of self- and social consciousness as an integral element in our thinking, we could not be the sort of creatures we are. 8. THE REFLEXIVITY OF PERSONHOOD This is not the place to retrace the history of German Idealism, but a few words on the topic seem appropriate nonetheless. At the core of German Idealism is Kant’s distinction between the ‘empirical self’ we encounter in our everyday conscious experience and the ‘transcendental self’ (or ego) that the human mind postulates and presupposes to conjoin the varied totality of our experience into the unity of possession by a single individual. All this was a starting point for Fichte—the preeminent devotee of the reflexive I. In his early Wissenschaftslehre, as with Kant, the Fichtean self is not discovered but postulated and presupposed from the outset, and thereby ‘self-constructed.’ But in recognizing its own limitations it acknowledges a wider realm to which it stands in contrast and opposition. This idea was then elaborated by Fichte into the

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concept of a self-consciousness which, in positing itself, also posits a greater world—a non-I which includes other self-conscious beings as well. This larger domain is then seen as a lawful order—the theater of operations of natural laws, recalling Kant’s dictum that “Man is the law-giver of nature.” The point of this historical excursus is simply to note that much of the speculative anthropology of the German Idealists can be recast in more contemporary terms—specifically in the idea of a reflexive conceptual stance that we employ to conduct the business of thinking. In the recasting of that tradition, the idea of a person would come to the fore. To be a person, in a sense deeper and more far-reaching than being a member of Homo sapiens, is to be a rational agent, capable of acting on the basis of self-acknowledged reasons. Such a conception has been thought of as carrying certain definitive features: • A rational agent must see himself as such—as able to act on the basis of reason. • A rational agent must prize and value this standing: few if any of his possessions or abilities will matter to him as much as his reason. •

A rational agent must be prepared to acknowledge duly qualified others as fellow rational agents and must interact with them in ways that take account of this.

•

A rational agent must prize and respect that rationality of others even as he prizes and values his own.

•

A rational agent must acknowledge the limited range of rationality: he must acknowledge that he is part of a manifold of existence that also includes things quite different from rational agents.

All of these come down to saying that a person as rational agent must, in view of his nature as such, have a certain sort of self-image. The concept we use to describe what we are—to characterize our role in

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the scheme of things—carries a heavy load of reflexive conditions. We are, by nature, the sort of being for which reflexivity is central. We are made the sort of beings we are through the act of thinking of ourselves in a certain sort of way. 9. COGNITIVE REFLEXIVITY There is some surface plausibility to the idea of a special status for cognitive reflexivity with its insistence that we humans have a privileged access to our own thoughts. For it looks rather paradoxical to say that there are facts about one’s own thoughts and beliefs which are better known to others than to oneself—let alone that others can know facts about my thought which I myself cannot possibly know. But although this may seem decidedly strange on first sight, it is nevertheless quite correct. After all, you can know that I have forgotten that my cousin lives on Main Street, but I cannot. To be sure, I can know that I have forgotten my cousin’s street. But I cannot know that I have forgotten the fact of its being on Main Street. For to know that this particular fact has dropped from my memory, I must know that it is a fact, so that by hypothesis I cannot have forgotten it. The facts regarding the deficiencies to my cognitive condition are doubtless accessible to others, but they are not accessible to me. You can certainly know that I mistakenly believe that Thomas Jefferson was the second president of the USA, but I cannot. For my realization of the mistake at issue would eradicate that belief. The reality of it is that I cannot know the specifics of my own cognitive limitations. Such facts as are at issue in my: • being incorrectly convinced • mistakenly believing • erroneously accepting • ill-advisedly thinking that something is so are inaccessible to me.

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Such characterizations of one’s beliefs which in effect indicate their falsity are simply unavailable to oneself. Reflexive beliefdecertification is semantically impracticable because it affirms something that one cannot possibly know. The exact detail of my ignorance and error is cognitively inscrutable to me. I cannot possibly identify the detail of my cognitive gaps and errors. No one else can know in greater accuracy and detail what a person’s feelings are. Nor can another know more extensively and accurately what a person’s beliefs and convictions are. But the quality of a person’s beliefs and convictions—their truth or accuracy or clarity— are not matters of access to him—let alone privileged access. They are matters about which others be far better informed. So what we have here is a bit of a paradox. We incline to think that a person has access—and indeed privileged access—to his own thoughts and convictions. But this idea calls for recognizing a very important distinction. For while this privileged access may hold as regards content (substance) of our belief it is emphatically not true as regards their condition and quality. One can certainly criticize one’s own beliefs and convictions. (It is not senseless to say “I am under the insufficiently evidentiated conviction that he will come today.”) But one cannot meaningfully reject them. (It makes no sense to say “I have the erroneous conviction that he will come today—the retrospection of I had would be OK.) The actuality of it is that there are two decidedly different aspects to one’s thought: the purely phenomenological that characterize thought solely with respect to the descriptive content purported by that thought itself and the substantively ontological which relates that thought to the actual condition of its projected objects. These two divisions deal with very different sorts of facts regarding one’s thoughts. And while it may be true that we have privileged access to the former—the substance of our thought—it is decidedly false that we have privileged access to the latter, its relationship to the actuality of things. And with this distinction in view it becomes possible to come to terms with the seemingly paradoxical circumstances that there are facts about one’s own thoughts that are knowable to others but cannot possibly be known by oneself. On this basis the idea of a privileged access to information about one’s own thought must be qualified. It re-

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quires limitation to the phenomenological side of out thought and its objectively ontological side must be excluded from its purview.

NOTES 1

See for example Thomas Nagel, “What is it Like to be a Bat?,” Philosophical Review 83, 1974, 435-504.

2

See for example Galen Strawson, Mental Reality, MIT Press 1994; David Chalmers, The Conscious Mind: In Search of a Fundamental Theory, Oxford University Press 1996; Colin McGinn, The Mysterious Flame: Conscious Minds in a Material World, Basic Books 1999 and Consciousness and its Objects, Oxford 2004.

3

David Rosenthal, Consciousness and Mind, Oxford: Clarendon Press, 2005.

4

Shaun Gallagher and Dan Zahavi, The Phenomenological Mind. New York: Routledge, 2008, pp. 58-60.

5

Brentano took intentionality to be the mark of the mental, but identified mental states with conscious states. ‘Consciousness is consciousness of’ comes from Sartre (Being and Nothingness, trans. H. Barnes. New York Philosophical Library, 1943). See also Alphonso Lingis, “The Signs of Consciousness,” SubStance 13 (1984), 314.

6

The indexing requirements of information input are termed ‘de se’ by Tyler Burge in “Memory and Persons,” Philosophical Review 112 (2003), 289-337. Close in the background of the points made here is also the analysis of perception in terms of ‘affordances’ by J. J. Gibson. See The Ecological Approach to Visual Perception, London: Lawrence Erlbaum 1986.

7

We follow philosophical precedent in ignoring the complications that are involved in any real approximation of such a case. On these complications, see Marius von Senden, Space and Sight: The Perception of Space and Shape in the Congenitally Blind Before and After Operation, London: Methuen, 1960; Oliver Sacks, “To See or not See,” in An Anthropologist on Mars, Vintage 1995, 108-153, and Richard Held, et. al., “Revisiting the Molyneux Question,” Journal of Vision 8 (2008), 523.

8

See Jackson, “Epiphenomenal Qualia,” Philosophical Quarterly 23 (1982), 127136; “What Mary Didn’t Know,” Journal of Philosophy 83 (1986) 291-295, and Peter Ludlow, Yujin Nagasawa, and Daniel Stoljar, eds, There’s Something about Mary: Essays on Phenomenal Consciousness and Frank Jackson’s Knowledge Argument, Cambridge: MIT Press, 2004.

9

The concept is introduced in J. Levine, “Materialism and qualia: The explanatory gap,” Pacific Philosophical Quarterly 64:354-61, but is pursued in many later authors. See for example Galen Strawson, Mental Reality, MIT Press 1994, and Da-

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NOTES

vid Chalmers, The Conscious Mind: In Search of a Fundamental Theory, Oxford University Press 1996. 10

Thomas Nagel, “What Is it Like to Be a Bat?” Philosophical Review 83, 1974, 435504, p. 436.

11

Alexander Pfänder, Phänomenologie des Wollens: Eine Psychologische Analyse. Leipzig: Barth, 1900; Theodor Litt, Geschichte und Leben. Leipzig/Berlin 1918.

12

See Marc Hauser, Wild Minds: What Animals Really Think, New York: Henry Holt, 2000; G. G. Gallup, Jr., “Self-awareness,” in J. R. Erwin and G. Mitchell, eds., Comparative Primitive Biology, Volume 2B: Behavior Cognition, and Motivation, New York: Alan Liss, 1987, pp. 3-16; D. J. Povinelli, “Monkeys, apes, mirrors and minds: The evolution of self-awareness in primates,” Human Evolution 2, 1987, 493-507.

13

H. Wimmer & J. Perner, “Beliefs about beliefs: Representation and constraining function of wrong beliefs in young children’s understanding of deception,” Cognition 13, 1983, 103-128; S. Baron-Cohen, A. M. Leslie, & U. Frith, “Does the autistic child have a ‘theory of mind’?” Cognition 21, 1985, 37-46; S. Baron-Cohen, “Precursors to a theory of mind: Understanding attention in others,” in A. Whiten, ed., Natural Theories of Mind: Evolution, Development, and Simulation of Everyday Mindreading, Oxford: Basil Blackwell, 1991, pp. 233-151; A. N. Meltzoff, “Imitation as a mechanism of social cognition: Origins of empathy, theory of mind, and the representation of action,” in U. Goswami, ed., Handbook of Childhood Cognitive Development, Oxford: Blackwell, 2002, pp. 6-25.

Chapter 6 MYSTERIES OF SEMANTICS AND FREE WILL

W

hen I hear another person speak, what I hear is a proposal, a claim, a promise, a challenge, or a question. It is the semantics that is immediate and obvious to me; I pay virtually no conscious attention to the particulars or idiosyncrasies of sound patterns. A physical transcription of such an event, however, registers only sound patterns. How can mere production of a series of sounds amount to something as significant as a promise? How is it that an event at the level of sound patterns, or even of mere syntax, can also take on the magic character of meaning? I make free decisions and choose from alternative options, guiding my future activity by my own deliberation and choice. I am, on the other hand, a creature in a universe that seems to operate by chains of causality. How could both of these claims be true? Must it not be that I am either wrong in thinking that the universe is closed under causal relations between events, or that my decisions are not free in the sense I take them to be? This is the ancient perplexity of free will and determinism. In the previous chapter we considered reflexivity within states of awareness, applying some of the lessons of alternative reference developed in chapter 4. Here we use the same tools to approach reflexivity within human practices. Attention to patterns of reflexive reference contextualized within forms of human activity, we will argue, dispels many of the mysteries of both semantics and free will. 1. SEMANTICS In the previous chapter we focused primarily on information input at the level of the individual: on an organism’s perceptual relation to its environment. But like many organisms, human beings are also social creatures. None of us goes it alone; each of us owes much of what

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we know to communication from others. Much of our information is socially instantiated, socially conveyed, and socially accessed. We are such social creatures, in fact, so tied to social communication, that we tend to speak to ourselves in precisely the terms appropriate in speaking to others. Conventions of inter-agent communication come to serve purposes of intra-agent conception as well; it is this phenomenon that explains the fact that thought—at least our thought—seems to be so thoroughly linguistic in form. Our agent-to-agent information transfer is by means of conventions of communication, the full flowering of which is evident in human languages. Spoken language has a phonology; both spoken and written language have a syntax. But the purpose of language is neither phonology nor syntax: conventions of phonology and syntax are fully in the service of meaning and information, the semantics of spoken and written speak. What is this extra ingredient that is the semantics of a language? What is meaning? Suppose an alien engineer were to record a specific communicative event—my making a promise to you on Tuesday next, for example. The recording is complete and detailed, in both analog and digital form, computationally analyzable in terms of amplitude, azimuth, sound energy flux, acoustic impedance, auditory masking and the like. The alien spends hours scrutinizing the recordings in great detail, using advanced sound-pattern algorithms, searching with overlaid audio filters for that elusive aspect of the event that is the meaning. No matter what advanced technologies are put to work, the alien won’t find what he is looking for. There is no aspect of the sound pattern—no timbre, no tone, no graphable pattern of amplitude—that constitutes the meaning. From solely the perspective of a physical examination of sound patterns, meaning is something that will forever be invisible. Part of the alien engineer’s problem is the isolation of a single event. Because linguistic conventions are crucial to language use, language use is understandable only on the time-scale, or event-scale, on which conventions are understandable in general. Just as an instantaneous event has the historical significance it does only against the background of a history far broader than the time-span of the event it-

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self, so an instance of language use has the linguistic significance it does only against the background of a pattern of use far broader than the time-span of that particular instance. Suppose then that our alien were given a far broader class of linguistic events. Were he to use the same physicalist perspective, even that broader class of events would be insufficient to make meaning visible. Suppose he had tapes of all my linguistic exchanges, from birth to the present, and suppose he were to scrutinize all of those for the aspects of sound patterns that constitute the meaning. Within a certain tolerance or approximation, he would certainly be able to find repeatable sound patterns. But no matter how carefully he attended to those patterns, he would be unable to filter out that sound characteristic that is the meaning. Meaning is simply not a physical sound characteristic, of either a particular exchange or an extended series of exchanges. Meaning is an aspect of communication, not merely sounding—of information transfer, not merely of regularity in the production of sounds. Because of that, meaning can be understood only at the level that communicative acts of claiming, acknowledging, warning, and promising are understood. That is a level of understanding, or a perspective on linguistic phenomenon, that is perfectly familiar to any participant in conventionalized communication. But it is a level of understanding impossible without such a context. Semantics demands a perspective internal to a mode of communication in ways in which phonology and syntax do not; a perspective that incorporates the informational and communicative purposes of the human practice that language represents. Given our internal familiarity with conventions in our own communication, we can approach the sound patterns of other peoples and ask—analogously—about the semantics operative in their exchanges. We hear the unfamiliar tonal rises and falls of the Xhosa language for the first time, or the distinctive clicks of Hadza, and wonder what the semantics of those sound characteristics might be. Here as elsewhere our reasoning is firmly grounded in presumption; a presumption that another people’s sound exchange is like that we know reflexively from our own case—a communicative exchange with a distinct and regular semantics. We operate with a presumptive default assumption that their sound patterns are like ours; an economical hypothesis of

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similarity that is defeasible, of course, but only overridden given strong counter-indications. In many cases the default assumption works effectively, offering retrospective validation as well.1 We also rely on reflexivity in our own case when wondering about the sound patterns of creatures further removed from us. We wonder whether dolphins and whales are merely singing or actually communicating. We wonder what they might be communicating, and how. Even asking such questions, however, involves extrapolating from our own experience as communicative participants. The general concept of meaning, like the identification of particular meanings, demands a background context of participatory involvement in communicative practice. Talk of specifically what we mean, of what we are trying to convey, and of what our words and phrases mean is possible only because language can itself operate reflexively. One of the things that can be conveyed through a mode of communication is information regarding the mode of communication itself. We employ conventions of communication so ubiquitous that they become invisible to us, but one of the things that can be addressed by means of those conventions are issues regarding the conventions themselves. When I ask you to tell me what ‘egregious’ means, I am asking you a reflexive question regarding aspects of our mode of communication within that mode of communication itself—in particular, the communicative conventions for ‘egregious’. As noted, we are such social creatures that modes of social communication are internalized as modes of conceptualization itself. Even alone, my attempts to remember what ‘egregious’ means are essentially reflections within a mode of social communication on conventional aspects of that communication. An ontology of our own meanings is possible only because of linguistic reflexivity. A projected ontology of meanings in languages not yet understood is possible only with an operative presumption that sound patterns between other people are like those we know reflexively from our own case. Even the possibility of raising the question of whether very different animals have a language is possible only on the analogy of our reflexive understanding of meanings within our own communicative practice.

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Semantics and meaning remain a mystery if we look for them as some ingredient parallel but additional to the sound patterns, phonology, or syntax of a particular exchange. Part of the mystery is dispelled once we realize that semantic conventions, like all conventions, demand a perspective wider than that of an isolated event. Much of the mystery is dispelled once we realize that the character of semantic conventions as semantic conventions is tied crucially to the purposes of linguistic exchange—informational purposes of claiming, questioning, informing, challenging, promising, and the like. An understanding of those purposes demands the perspective of a participant in communicative exchange, and thus will forever be invisible from a perspective that is divorced from communicative participation. Meaning is a phenomenon obvious from the perspective of participation in a practice, but invisible without it. 2. THE ‘GAVAGAI’ MYSTERIES Despite its obsession with language, much of twentieth-century philosophy failed to incorporate the lesson that meaning and semantics are aspects visible only reflexively within linguistic practice. On the approach we have outlined, several persistent philosophical puzzles seem to dissolve. Quine envisages a linguist attempting to translate a native language: A rabbit scurries by, the native says ‘Gavagai’, and the linguist notes down the sentence ‘Rabbit’ (or ‘Lo, a rabbit’) as tentative translation, subject to testing in further cases . . . . . .consider ‘gavagai’. Who knows but what the objects to which this term applies are not rabbits after all, but mere stages, or brief temporal segments, of rabbits? . . . Or perhaps the objects to which ‘gavagai’ applies are all and sundry undetached parts of rabbits . . . A further alternative likewise compatible with the same old stimulus meaning is to take ‘gavagai’ as a singular term naming the fusion, in Goodman’s sense, of all rabbits: that single though discontinuous portion of the spatiotemporal world that consists of rabbits .

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. . And a still further alternative in the case of ‘gavagai’ is to take it as a singular term naming a recurring universal, rabbithood.2 The result, Quine concludes, is radical indeterminacy of translation: any of various translation manuals will be consistent with any body of behavioral evidence. Despite a number of attempts to claim otherwise, that central point is a particular instance of the quite general phenomenon of underdetermination of theory by available evidence. The original context of Quine’s discussion is as an objection to the distinction between analytic and synthetic in Carnap’s program for Logical Empiricism. But both Quine and later Quineans have been willing to draw much broader conclusions. The existence of alternative translations for ‘gavagai,’ it has been held, shows that there simply is no fact of the matter as to meaning: no fact of the matter, for example, as to whether ‘gavagai’ means rabbit, rabbit stages, undetatched rabbit parts, rabbit fusions, or rabbithood. The lesson drawn is not merely the epistemic lesson that we may forever remain uncertain as to what the native term means, but the ontological lesson that there is no fact of the matter as to what ‘gavagai’ means. Donald Davidson embraces the Quinean conclusion straightforwardly as indeterminacy of meaning.3 There is nothing in the argument, however, that limits it to translation of a foreign language. Since I have only a limited basis of behavioral data, I am in the same situation with respect to your use of ‘tomorrow’ as I would be with respect to the native’s use of ‘gavagai’. On the same grounds, it appears that we could claim not merely that I may forever remain uncertain as to what you mean by ‘tomorrow’, but that there is no fact of the matter as to what you mean. From you it is a short step to me, and the conclusion that I do not know what I mean by ‘tomorrow’ or that there is no fact of the matter as to what I mean. The conclusions that have been drawn are more sweeping still when the example of ‘gavagai’ is combined with a principle of charity in translation: a principle that all things being equal we should translate utterances of another people or another individual so as to maximize truth. Davidson uses these materials to construct an argument against the very idea of radically different conceptual schemes.4 In Putnam’s hands, they form the basis for an apriori argument that a

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majority of our beliefs, and of those of a brain in a vat, must be true.5 There are passages in which Quine uses a similar argument, entertaining Pythagoreanism on the grounds of the Lowenheim-Skolem theorem.6 All English utterances of all speakers over a finite time might be given a ‘translation’ in which all are claims solely about numbers and all turn out to be true. Why not then think that the meaning of ‘I will be there tomorrow’ is actually a truth of number theory? There are several maneuvers that are as questionable here as they are central to the line of argument. Operating just below the surface is an amalgamation of behaviorism and verificationism that is constantly used to slide from a claim that there can be no behavioral test for particular meaning claims to the conclusion that ‘there is no fact of the matter’ regarding such claims. That assumption, of course, leaves out two possibilities: (a) that purely behavioral evidence is not all that is available, and (b) that a meaning claim may be true despite lacking conclusive behavioral proof. A more explicit assumption in the extended argument is the principle of charity in translation. But that principle has the bite it does only because of the behavioral verificationist assumptions already in play. Like the body of behavioral evidence for ‘gavagai’, the grounds and application for charity are standardly given an extremely limited representation.7 For some purposes, the artificial portrayal of the linguist’s predicament in Quine and others is harmless, or even informative: there is a sense in which we may operate with linguistic hypotheses as with others. But there is also a clear sense in which that artificiality offers a radically distorted picture of our practice, either with speakers of a different language or with other speakers of our own. In both cases we operate with clear default presumptions of similarity. We operate with a presumption that sound patterns represent the kind of communication that we know reflexively from our own case. We operate with a presumption that linguistic communication will employ a semantics similar to that we know from our own case, tied to communicative goals and contextual concerns. Given those goals and concerns, the presumption is immeasurably on the side of ‘rabbit’ rather than some function in number theory. For any alternative of the ilk of ‘rabbitstages’ that is genuinely distinct from ‘rabbit’, the strong presumption is in favor of ‘rabbit’ simpliciter. One group with whom the ‘gavagai’

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example has failed to make any strong impression are practicing linguists and ethnographers. One reason it has failed to make an impression there is that their scientific practice is simply not as Quine’s example portrays it. Field linguistics is not an armchair balancing of arbitrary hypotheses in search of theoretical guarantees, but the extension in practice of a cognitively economical presumption of similarity, including semantic similarity, from what we know reflexively in our own case. Our success in dealing with people speaking very different languages, of course, serves as retrospective validation for precisely these kinds of semantic presumptions. If meaning is a phenomenon visible only from within communicative practice, or in presumptive extension of such practice, it is unsurprising that purely behavioral characterization from the limited perspective of Quine’s linguist will fail to capture it. It is certainly true that we are not always sure of meanings, others’ or our own. But it is certainly not true that we are never sure of meanings—as sure as we are of our own purposes in communication. A perspective that does not incorporate those purposes—as behavioral observation from without does not—will fail to see them. The considerations of ‘gavagai’ do not therefore entail that meaning is either inscrutable or non-existent. All they entail is that meaning is to be understood from a perspective different from that which Quine sets up as an artificial epistemic criterion and, given verificationism, as an ontological standard as well. The mysteries of ‘gavagai’ are merely the mysteries of a misplaced perspective. What Quine’s, Davidson’s, and Putnam’s conclusions show is not that meaning is essentially problematic but that it is a phenomenon that is not appropriately approached from the perspective that defines Quine’s approach from the start. 3. THE MYSTERY OF THE CHINESE ROOM Another philosophical mystery that dissipates with this understanding of semantics is that of the Chinese Room. John Searle imagines himself in a closed room, taking ‘squiggles’ on paper as input through one slot, following a rulebook that links marks to marks, and putting other marks on paper through another slot. With the right rulebook, he imagines, the input marks and the

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output marks might qualify as questions and answers in a language foreign to him—Chinese, perhaps. But genuinely understanding Chinese, Searle argues, must require more than that. Since any computational procedure can be modeled by an individual in a room blindly following a rulebook, genuinely understanding Chinese must require something beyond any computational procedure. The Chinese Room is often used to suggest that there is a missing element in any programmable or computational parallel to the kind of information processing characteristic of language users. That suggestion is buttressed by Searle’s insistence that only something with the ‘causal powers’ of an organic brain will be able to supply the missing element. In his later work, Searle gives a more general characterization of the problem: that mechanical operations on mere syntax will inevitably leave out semantics. Because it is not visible from an external behaviorist perspective, Quine’s ‘gavagai’ is used to impugn the concept of meaning. Because it is not visible in a characterization of input and output according to rules, Searle’s ‘Chinese room’ is used to suggest that meaning is a mysterious something beyond. Both miss the fact that visibility can be a matter of perspective. What Searle’s Chinese Room does is to force a perspective in which the flow of information is taken out of the context of communicative practice. Any context of communication—of questions asked and answered, of claims made, of warnings leveled—is larger than the room and beyond it. As the man inside, working rote from a rulebook, Searle incorporates none of the purposes of communication, and in that sense is no longer a participant. Nor is he able to project a presumption of communication from reflexive consideration of his own case. Trapped in the room, he cannot even see the larger context in which any communicative agents would be distinguishable. Small wonder that the dimension of semantics is invisible within the Chinese Room. Searle’s example also employs the misdirection characteristic of sleight of hand. Our attention is focused on the individual in the room. Because he is ignorant of Chinese but fluent in English, he is a potential participant in other communicative exchanges. That focus dis-

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tracts our attention from any communicative exchange operative outside the room. Searle’s argument is anticipated several hundred years earlier. In the Monadology, Leibniz offers the following argument: . . .Perception, and all that which depends on it, are inexplicable by mechanical causes . . . Supposing that there were a machine whose structure produced thought, sensation, and perception, we could conceive of it as increased in size with the same proportions until one was able to enter into its interior, as he would into a mill. Now, on going into it he would find only pieces working upon one another, but never would 8 he find anything to explain Perception . . .

Leibniz’s argument, if successful, would work not only for a machine but for a brain. Were we to conceive of a brain ‘increased in size with the same proportions until one was able to enter its interior,’ processes of thought, sensation and perception would be equally inscrutable. But that does not establish is that they do not occur within the mechanism of the brain, visible from a very different perspective. Leibniz’s argument similarly fails to establish that they could not occur—visible from a very different perspective—within a machine. If taken as an argument for a mysterious additional semantic something, Searle’s argument fails precisely as Leibniz’s does. Searle rightly emphasizes something very important about our current use of computing devices currently available, however. It is we who incorporate those machines into our practices, epistemic as well as communicative. It is we who build them or read them as taking ‘input’ at a particular point in operation, and we who read certain of their behaviors as ‘output.’ Our machines do therefore have a context of information processing and communication, but the context is entirely derivative: the context is entirely ours. It is in the context of our practices that computers have a role even as calculators—it is we who design them in such a way that we can interpret their activity as such. Meaning and semantics demand the perspective of participation in a process of communication; it is that larger context that is crucially omitted in the description of the Chinese room.

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The Chinese room portrays a linguistic event in terms of rulefollowing manipulation at the level of syntax. Consider another portrayal of a linguistic event: portrayal of a conversation in terms of a causal chain of verbal events within a nexus of agents. A utters sounds X which cause B to utter sounds Y which cause A to utter sounds Z . . . In both of these portrayals, the referential terms used—Searle’s ‘squiggles’ coming in and going out of the room, and ‘sounds X and Y’—are external terms of reference non-reflective of the practice of communication itself. Those terms of reference cannot tag aspects of the practice that can be referenced only reflexively from within it. They cannot, in particular, tag aspects of meaning. There can therefore be no reduction of meaning to mere ‘scribbles’ or ‘sounds’, no reduction of semantics to syntax, any more than there can be a reduction of strikes in baseball to physical velocities or spins. At issue are two very different contexts of practice, and both forms of reference and the ontologies built on them are proprietary to those two very different contexts. It will nonetheless be true that language does have a syntax, and that syntax does capture and constrain certain aspects of semantic practice. It will also be true that patterns of communicative interchange between agents in a community will take the form of chains of causal verbal events, and that those exchanges cannot occur without those chains of events. There is much that can be learned regarding the structure and dynamics of language from those perspectives; much that can be learned even regarding conditions sufficient for the emergence of language.9 An understanding of semantics, however, demands an understanding of semantics in its own terms—in terms of an ontology of meanings, for example, built from reflexive reference within the practice of human language. 4. FREE WILL Practices of communication afford proprietary forms of reflexive reference, with the possibility of an ontology built on those forms. Practices of decision do as well. It is within a mode of decision that I transparently self-reference ‘my free will,’ or my power of choice. The practice of human deliber-

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ation is one that relies crucially on possibilities for action and alternative courses of possible action—those things between which choices are made. Deliberation involves envisaging prospects and possible consequences, potential gains and losses, hazards and opportunities. Each of these forms part of the conceptual ontology of the practice of deliberation, real within and because of that practice. Free choices are aspects of an ontology proprietary to the human practice of choice. My concept of ‘your free will’ or of free will in general is an extension of reflexivity within my own experience of decision. It is not an inductive generalization in the sense of proceeding from present experience to future cases. What is at issue, here as in the case of communication, is a default presumption of similarity between your case and that which I know reflexively from my own—an empathetic presumption of the same sort of reflexivity within your mode of decision. Much as I operate with a reflexive presumption of meaning like mine in contexts of language use, I use reflexive presumption to generalize a notion of free will applicable in cases other than my own. Here as in the case of meaning, our success with such a presumption offers retrospective validation as well. But consider now a specific decision—my decision to commence this paragraph with this particular sentence, for example. What precisely is the referent of ‘my decision’? Given our knowledge of my physical limits and my neurological constitution, there seems reason to say that such a decision must be an event in my brain. What else could it be? But we know, or think we know, that events in the universe operate in terms of sufficient causal chains. There are no magical interventions. Given earlier events, later events are causally determined in accord with natural law. With that we have all the materials necessary for the problem of free will and determinism. My decisions, if free, must be undetermined. My decisions, if events in the universe, must be determined. All attacks on the problem of free will and determinism are attempts to choose one side or the other or to evade the apparent contradiction between them. Determinists of one form or another choose the second horn and deny the possibility of genuinely free choice. Libertarians of one stripe or another insist on free choice and thereby deny a universe

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closed under causal law. Compatibilists of one form or another attempt to show the apparent contradiction to be merely apparent. There are multiple options here. One of us has explored the possibility of backwards convergence in causation.10 Contrary to the implication standard to the problem, it does not follow from the fact that every event has a preceding cause—even a sufficient cause—that such an event is either predictable or determined at a given time t1 in the past. What standard presentations overlook is the possibility, graphed below, that there may be an infinite series of sufficient causes for an event E2 at time t2 that proceed Zeno-like into the past, converging toward but not extending beyond some previous event E1 at t1. Given such a possibility, universal determinism—the claim that every event has a cause—need not entail determination of events from earlier points in time. E1

E2

t1

t2

It is a different form of compatibilism that that is suggested in the outline we have given. The central question is clearly this one: • Is my decision an event in my brain? If the answer is ‘no’, we seem to have a mystery—what else could my decision be? If the answer is ‘yes’, putting aside the possibility of convergent causation for the moment, the threat of causal determinism to the very concept of free choice seems a short step away. The lesson of considering reflexive perspectives, however, is this: that in asking such a question one must realize the kind of question one is asking. For this is a question that crosses perspectives, phrased in terms of referential terms that clash precisely because they are appropriate to the radically different contexts of radically different human practices. In previous sections and previous chapters we have encountered similar questions of identity across rival perspectives— questions of whether aspects of semantics might reduce to syntax, of whether consciousness could be a brain state, of whether references de

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se might have equivalents in references de re, of whether propositions embedding indexicals can be the same as those without. In each case mystery dissipates when we realize what kind of question we are asking. As we have emphasized throughout, difference in reference need not entail difference in referents. To that extent the fundamental referential domain for ‘decisions’ and ‘brain events’ might in fact be the same. Scientific data, tallying brain scans with self-reports of decision, for example, offers correlation tight enough that it seems targets of reference must in some way overlap. As we have also emphasized throughout, however, ontologies within practices can be built on forms of reference proprietary to those practices. That certainly seems to be the case with propositions, distinct because of embedded modes of reference (‘I’ as opposed to ‘the man in the mirror’) despite a base identity of referents. In the case of consciousness we have argued that this is the case with respect to subjective states and qualia as well. The issue here is similar. What we have to recognize regarding the apparently innocent question of identity is that the terms on one side—the side of decisions—are referentially propriety to the context of deliberation and choice. Could it be that what I am referencing as ‘my choice’ or ‘my decision’ is in actuality a fully determined causal pattern? Here we trip over two radically different forms of reference. One, built into the concept of free will, involves immediate and transparent reflexivity within a mode of deliberation and decision. The other form of reference is non-transparent, appropriate to external rather than reflexive description. Could what I am referencing as ‘my choice’ or ‘my decision’ be a fully determined causal pattern? The real answer in such cases begins ‘You must understand the kind of question you are asking . . .’ Given an ontology built on forms of reference appropriate to deliberative practice, there will be a radical difference between concepts appropriate to choice and those appropriate to causal description. Choices and decisions, as well as the material on which they operate—prospects and possible consequences, potential gains and losses, hazards and opportunities—are all aspects of a mode of decision internally self-referenced. Options and alternatives live within the prac-

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tice of decision-making: decision-making is possibility’s home. Once these are ontologized as ‘things’, options, alternatives, and possibilities have no place in a mode of approach that deals exclusively with actuality—with actual casual chains between states of a physical brain, for example. This divergence between perspectives is what one of has dealt with in terms of ‘events’ and ‘eventuations.’11 In the sense that we think of the universe as operating in terms of causal chains of successive events, there seems to be no sense in which the options I consider, the alternatives I seek, the dangers I fear, or the hazards I broach are literally events. Choices and decisions are parts of the same ontology and live in the same referential space as these do; in that regard choices and decisions are not events, either. They are ‘eventuations’, properly thought of not as events in chains of causes but as the termini of chains of deliberation. The difficulty of a question as apparently innocent as ‘are choices events in my brain?’ is that it crosses contexts of reference as distinct as the human practices in which they are embedded. The first step in approaching such a question is to recognize that it is a question of that kind. If the answer to the question is ultimately ‘no, choices are not literally brain events,’ it is not a ‘no’ that suggests a shadow realm of spiritual events beside the physical. It is a ‘no’ reflecting simply the radical difference between referential frames. Could fundamental referents nonetheless be the same, despite radical differences in ways of referring? Perhaps. Here an analogy from robotics is suggestive. One can envisage constructing a robotic decision procedure that involves generating, say, three descriptions. In the normal functioning of the robot, on the basis of input from its sensors, one of those descriptions becomes active. Those three descriptions play the role of considered ‘possibilities’ in the program. Viewed from without they are actual existent states of the machine. But their role in the decision process is a role as descriptions of non-actual possibilities. Which are they, actual or non-actual? The truth is that we have two forms of reference for those descriptions: the ‘external’ mechanical and the ‘internal’ teleological, conceived in the terms of a decision-making process. Hence the illusion that we have two different things with essentially different properties, when what is at issue is

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one thing referenced in radically different ways. It is because distinct ontologies may be built on differences in forms of reference that ontologies may prove irreducible despite a crucial co-referentiality at the base. In chapter 5, in the case of consciousness, we characterized a conceptual divergence of this type in terms of a new form of Dualism, while also acknowledging a clear motivation toward Monism. There the pressure toward monism was strongly empirical; the fact of ubiquitous connection and correlation between those things we reference from different perspectives. The same will be true here. Our decisions have causal consequences ‘downstream’: decision would lose its sense were we not able to put decisions to effect. Our decisions often trigger chains of events beyond them, and part of deliberation involves anticipating those chains. By the same token, our decisions are constrained by actual events ‘upstream.’ It is chains of events in the past that put us in positions that demand decision, and in light of which particular decisions are made or even forced. Both ‘upstream’ and ‘downstream,’ then, decisions are embedded in a vision of the causal world in which those decisions play a causal part. Even if decisions are not to be identified with events, it is a crucial assumption of reflective deliberation that decisions are linked to patterns of causality in a single unified world. Our reflective deliberations thus carry an informal concept of causally connected events. It was because of the situation that I decided to take the course of action I did, we say. It was his intransigence that caused me to act as I did. It was because I decided to go to Montauk that weekend that the leak went undetected, a decision that caused a great deal of inconvenience and later regret. In these terms, it should be noted, we do not speak of causality as operative in merely one direction. It is not merely that external events cause certain decisions to be made; decisions result in external chains of events as well. The causal initiative can be on either side.12 The world we perceive in ordinary deliberation strongly suggests a Choice-Causality Monism. Such a view is reinforced by the fact that particular events in the brain are correlated with particular aspects of the deliberative or decisional process. But empirical results regarding the brain cannot, and need not, overthrow the presumptive weight of

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genuine choice, embedded in fundamental human practice of decision. What is called for is a more sophisticated vision embracing both cause and choice, much as both are incorporated in ordinary deliberation. A fundamental Monism is clearly not entailed by a conceptual Dualism that distinguishes between aspects of decision accessible reflexively within a context of deliberation and the events and causes external to such a perspective. But it need not be inconsistent with such a Dualism. Such a Monism would not reduce either side to the other— for reasons given, it could not simply identify choices, decisions, options and possibilities with simple events, for example. A Monism consistent with conceptual Dualism would on the contrary recognize an underlying reality that can be accessed from different perspectives in terms of either causes or choices. Here as in the case of consciousness it must be admitted that proper formulation would remain a difficulty. There are different human projects and perspectives in play, and talk of links between them demands a philosophical context that somehow incorporates both. But it is clear that our familiar and informal talk of constrained choices and the causal consequences of decisions is an everyday instantiation of a wider perspective of precisely this Monistic form. The philosophical challenge is to work toward a more sophisticated articulation of such a view. 5. A PARADOX OF FREE WILL Our investigation of reflexivity began in chapter 1 with an examination of paradox. Here paradox returns. There is a paradox that purports to show that choice and cause cannot be made compatible in the way suggested; in particular, that free choice forecloses any possibility of causal determinism; My choices cannot map onto any pattern of causal events—any pattern of brain transitions, for example, however sophisticated. Were such a mapping possible, it would be possible to predict what my future choices were going to be. That cannot be: given any such prediction, I could read it and falsify it by choosing to do exactly the opposite of what was predicted.

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There are a number of lacunae in the argument. It demands, for example, that any causal pattern of events will in principle be predictable. But let us put those aside. For our purposes it is enough to note that the paradox buried in the argument is simply a paradox of prediction. The paradox is very much that of the Universal Prediction Algorithm considered in chapter 2. One can see that it nothing essentially to do with free will from the fact that it can be rewritten as a paradox of prediction for a mere machine: Consider a machine, completely understood in terms of mechanical causes, that flashes a green light in certain contingencies and a red light in others—entirely deterministically and in terms of antecedent causes. We have built it to turn on either a green or a red light when given any message on a slip of paper. We have built it to turn on a single green light on scanning a message that ends ‘will turn on a red light.’ We have built it to turn on a single red light on scanning a message that does not end with that phrase. We want a prediction for what the machine will do at 1:00 on Tuesday, after scanning a particular piece of paper. That prediction will either be ‘it will turn on a red light’ or ‘it will turn on a green light’. Any such prediction can be formatted so that the quoted passage is its last sentence. Now suppose that the piece of paper that the machine will scan just before 1:00 on Tuesday is our prediction itself. If our prediction is ‘the machine will turn on a red light,’ the machine will turn on a green one instead. If our prediction is ‘the machine will turn on a green light,’ the machine will turn on a red one. No matter what our prediction is, it cannot be correct. Whatever we predict, the machine will do the opposite.13 The result in this case is perfectly parallel to that in the argument above, but has nothing to do with the free will of the machine. This is a reflexive paradox regarding predictions of events that turn on those predictions themselves. Because it has nothing essentially to do with free will, the paradox does not support any radical incompatibilism regarding free will, nor any radically separatist ontology of free choice. There are analogous difficulties for prediction that are much more real, including self-fulfilling predictions. How in general are we to

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predict events at time t when we know that our prediction itself will be one of the conditions relevant to what occurs at time t? In some cases we can include self-referential knowledge: together with other forces, our prediction that p may help bring it about that p. Such is the case of a (partially) self-fulfilling prophecy. In other cases, however, we may face a situation analogous to that of the machine above: a prediction that p will produce or contribute to ~p; a prediction that ~p will produce or contribute to p. This is a self-defeating prophecy, and a case in which either prediction made will be self-defeating. In the terms developed in earlier chapters, a self-fulfilling prophecy is something like an ‘automatic in’. A self-defeating prophecy is an ‘automatic out’. The machine prophecy is an oscillator which drives a wedge between the content of prophecy and the actual events prophesized: precisely the kind of split between theorem and truth, or machine-assertion regarding halting and halting itself, that we found in Gödel’s theorem and the Halting Problem. 6. PURPOSIVE AND PRACTICE-BOUND CONTEXTS OF REFERENCE Semantics and free will are specific cases of a quite general phenomenon of purposive and practice-bound contexts of reference. It is within the context of communication that aspects of meaning can be referenced as such, and that an ontology of meanings is possible. It is within the context of decision that possibilities and choices can be referenced; it is there that an ontology of choices, decisions, options, alternatives, and possibilities is possible at all. Once the phenomenon of ontologies built on practice-bound and purpose-relative reference is understood, it becomes visible everywhere. It is for example within the practice of evaluation and adjudication, of one’s own actions and of others’, that rights and obligations live. Reference to rights, legal or ethical, is proprietary to the normative practice of ethical evaluation—any attempt to find them in a nonnormative account of human action would be radically misguided. From the perspective of a purely descriptive psychology, neurophysiology, or anthropology—were any purely descriptive form of these possible—ethical rights and obligations would simply be invisible.

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The attempt to find something that could be referenced in purely descriptive language and yet would play the full role of rights is simply futile: nothing in the wide field of beliefs, mores, social rules, roles, or ostracization practices, even if these terms could be used purely descriptively, will give us a full concept of rights. There is nothing purely descriptive that rights are, because they form part of the ontology of, and can only be referenced within, the normative practice of ethical evaluation or decision. The is/ought gap, or better the normative/descriptive gap, is a gap not between different aspects of the universe but between radically different human practices—as different as description and evaluation. The same pattern of contrasting contexts of reference is evident in a number of easy cases. National borders are not a part of a purely geological description, and limestone formations are not part of a purely political description. To ask for necessary and sufficient geological characteristics for a ‘border’ is to miss the point that very different purposes, practices, and concerns are at issue. To ask for a purely political equivalent for a ‘limestone formation’ is to make the same mistake. Grand slams, errors, and foul tips are part of the ontology of baseball, referenced from within that practice. To ask for a physics specific to grand slams or errors would be to confuse the very different forms of reference and derivative ontologies appropriate to radically different practices and purposes. It is often the case that the same process can be characterized in radically different terms, for very different purposes and within different practices. The same process may be characterized, for example, in mechanical or functional terms. There was one extended process over time that was the evolution of the moth’s wing. The history of that evolution might be described (in Laplacian principle, if not in practice) in terms of statistical mechanics with reference to the physics of particular molecules. The evolutionary process can also be described, much more practically and much more usefully, in terms of a different set of entities—moths rather than molecules—and in terms of the function of the moth’s wing and considerations of environmental ‘fit’. It is nonetheless one process and one history that we are describing— here as elsewhere differences of reference within different purposive contexts do not prohibit co-reference. Although it may be impossible

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to state necessary and sufficient conditions in one vocabulary specific to entities recognized in the other, the two need not conflict. Different practices, purposes, perspectives and modes of approach embody different forms of reference and give rise to different proprietary ontologies. When we ask questions of identity that cross those, we often act as if there will be a simple ‘yes or no’ answer. Are decisions a particular kind of brain state? Is a right merely a matter of social approval? But that is to act as if there were only one human practice, only one human purpose, and only one context of reference. The appropriate analysis in such cases demands a recognition that these are questions that cross contexts of practice and purpose. The real answer in such cases begins ‘You must understand the kind of question you are asking. . . .’ When we do, we can come to appreciate the sense in which these things can be the same—the sense in which there is only one history of life on earth, for example. We also come to appreciate the sense in which they will not and cannot be the same, ultimately as different as the different forms of reference appropriate to our different human practices.

NOTES 1

See Nicholas Rescher, Presumption and the Practices of Tentative Cognition, Cambridge Univ. Press, 2006.

2

W. V. O. Quine, Word and Object, Cambridge, MA: MIT Press, 1960, pp. 26, 5152.

3

Donald Davidson, “Coherence Theory of Truth and Knowledge,” in E. Lepore, ed., Truth and Interpretation, Oxford: Blackwell, 1986.

4

Donald Davidson, “On the Very Idea of a Conceptual Scheme,” Proceedings and Addresses of the American Philosophical Association, 47 (1973-4), 5-20.

5

Hilary Putnam, Reason Truth and History, Cambridge University Press, 1982, and “Brains in a Vat”, in K. DeRose and T. A. Warfield (eds.), Skepticism: a Contemporary Reader (Oxford: Oxford University Press).

6

See W. V. Quine, “Ontological Relativity,” Ontological Relativity and Other Essays, New York: Columbia University Press, 1969, pp. 60-62.

7

A principle of charity properly applied would certainly not merely maximize truth of utterances. It would rather maximize credibility of utterances given available ev-

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NOTES

idence, maximize appropriateness of utterances in context, and maximize the rationality of patterns of inference, all with a strong qualification of ‘all things being equal.’ 8

Monadology 17, trans. George Montgomery. From The Rationalists. Garden City, N.Y.: Dolphin Books, 1960.

9

For work along these lines see Grim, Paul St. Denis, and Trina Kokalis, “Learning to Communicate: The Emergence of Signaling in Spatialized Arrays of Neural Nets,” Adaptive Behavior 10 (2003), 45-70, and Grim, Trina Kokalis, Ali AlaiTafti, Nick Kilb, and Paul St. Denis, “Making Meaning Happen,” Journal for Experimental and Theoretical Artificial Intelligence 16 (2004), 209-244.

10

Nicholas Rescher, Free Will: A Philosophical Reappraisal, New Brunswick: Transaction, 2009, p. 59ff.

11

Ibid. p. 63ff.

12

Ibid. chapter 8

13

As indicated in chapter 2, the argument is a close relative of several that appear in chapter 12 of Rescher, Predicting the Future: An Introduction to the Theory of Forecasting, Albany, N.Y.: State University of New York Press, 1998.

Chapter 7 CONCLUSION

T

his has been a complex investigation, focusing on a single family of conceptual structures, but tracking those structures across a wide swath of philosophical terrain. A summary of major points is perhaps in order. With a particular use of negation and an ‘all and only’ condition, a specific reflexive structure generates oscillational paradox for a wide range of categories and relations. Where sets and membership are at issue, that structure instantiates as Russell’s paradox. Where men and shaving are at issue, we get the Barber’s paradox. Where the category is that of words and the relation is that of applicability, the result is the Heterological paradox. When single categories are replaced with the two-part structures of subject and predicate, we get the Liar; with further complications, the same structure gives us the oscillation of the Dualist. That structure appears again and with reflexive vengeance in the paradox of paradox analysis. Across the territory of paradox, we’ve noted, weakening ‘all and only’ to either ‘all’ or ‘only’ gives us not oscillation but automatic ‘ins’ and ‘outs’: sets that must be members of themselves or cannot be, for example, sentences that must apply to themselves or cannot. Closely related structures are evident in major results of 20th century logic, though here it is important that the dynamics is played out across pairs of concepts: truth and theoremhood in the case of Gödel’s result, machine prediction and machine operation in Turing’s Halting problem. In each case identification of the two concepts proves impossible short of oscillation of precisely the form outlined for classical paradox. Within no sound and complete formal system can theoremhood and truth on interpretation coincide precisely. Given the simple elements required for Turing’s abstract model of computation, machine prediction and machine operation cannot coincide precisely. There can, for example, be no machine capable of detecting infinite loops in all machines.

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Reflexive ‘ins’ and ‘outs’ appear in pragmatic as well as logical form in the context of investigatory and philosophical practice. Here Nihilism of various forms proves logically self-defeating for reflexive reasons akin to those evident in the study of paradox. Skepticism is self-defeating in another way: as a pragmatic ‘out,’ with Moore’s paradox as the clear paradigm of pragmatic contradiction. The certainty of Descartes’ Cogito, on the other hand, is the certainty of a pragmatic ‘in’ within the practice of reflective thinking. Beyond simple categories and formal systems, then, reflexivity and the reflexive structures we have traced play a crucial role in the context of perspectives and practices. Reflexivity also plays a crucial role with regard to reference in the context of propositional attitudes and mental states. The familiar opacity of knowledge contexts is one case among many of forms of reference proprietary to propositional attitude, perspective or practice. Description-mediated reference de dicto and unmediated ostensive reference de re will be non-equivalent across the boundary between them, with reference de se as a further non-equivalent and still more intimate category. Within ostension, the reflexive indexicals ‘I’, ‘here’, and ‘now’ mark the point of ostensive origin—these reference the point from which reference is made. As such, indexical reference must be immediate and transparent in ways no other form of reference can be. Ostensive reference to the act of ostension itself will carry that same character. Consideration of reflexive reference within propositional attitudes and mental states offers an analysis of the puzzle of the Essential Indexical. That analysis carries further lessons as well. Different forms of reference need not entail distinct referents. But forms of reference will often be proprietary to specific practices and perspectives, and we often build our ontologies in terms of those forms of reference. The result is ontologies that will be essentially irreducible despite a fundamental domain of common reference. The act of ostension, we have seen, can reflexively reference itself immediately and transparently. The fact that this is true of states of attention quite generally offers a direct analysis of consciousness and its place in evolutionary history. Conscious states are not states with a mysterious ‘glow’ but simply states of awareness which involve the

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reflexive awareness of themselves that we have outlined for indexical points of origin in general. Awareness quite generally employs fast categories of information processing. The qualia of phenomenal consciousness are precisely those fast categories, as they are triggered, accessed reflexively in awareness. The analysis of consciousness in terms of forms of reference casts light on a range of contemporary issues posed as mysteries—what it’s like to be a bat, black-and-white Mary, and the explanatory gap generally. In each case we have irreducible ontologies built on radically different forms of reference. As we’ve noted, however, differences in forms of reference need not entail different referents. Here as elsewhere, a conceptual Dualism seems inevitable. But here as elsewhere this is a new form of Dualism, consistent with the possibility of a fundamental underlying Monism. And the mysteries of semantics and of free will, we have proposed, have much the same structure. Meaning is part of an onotology built on reflexive reference within the human practice of communication. It is for precisely that reason that meaning is invisible outside of active involvement in the communicative context; it is for precisely that reason that meaning remains invisible within the Chinese Room or in the ‘gavagai’ example. Decisions, choices, options, alternatives, and possibilities live reflexively within the human practice of deliberation, invisible from a purely descriptive perspective limited to causal links and actuality alone. In the case of both semantics and free will, irreducible ontologies result from essential differences between reflexive and other forms of reference. Here as elsewhere, however, differences in forms of reference need not entail different referents. Here as elsewhere, the possibility of a fundamental Monism beneath an inevitable conceptual Dualism remains. Our attempt has been to track a family of reflexive conceptual structures across philosophical territory from paradox to consciousness. Reflexivity can indeed prove problematic: in ways specific enough to graph, for example, it forms the core mechanism of paradoxical oscillation. But reflexivity also instantiates a conceptual structure we could not possibly do without. In the end, reflexivity in selfawareness and self-conception is a critical aspect of the kinds of beings we are.

APPENDICES

I

n chapter 2 we sketched the dynamics of Gödel’s Incompleteness Theorem, alluding to the complications of Gödel numbering and levels of interpretation necessary in order to appreciate its true generality. Here we offer a more complete sketch that incorporates those complications. We offer a corresponding sketch of a further landmark in 20th century logic: Tarski’s Theorem on the Formal Indefinability of Truth.

1. GÖDEL IN MORE DETAIL Popular presentations in terms of ‘I am not a theorem,’ we have said, capture something important and central in the Gödel results. But there are also important aspects they leave out. The telos of a system like that of Principia Mathematica is mathematical truth: the applicability of number-theoretical properties to numbers. ‘Truth on interpretation’ for such a system concerns whether certain numbers really have certain properties: whether 144 is divisible by 3, for example, or 37 is really a prime. In our discussion of the Liar we used a quite general graphical representation for applicability:

One form of that applicability (but only one) will involve numbers as subjects in the left corner position and properties of numbers in the right. Numbers themselves are abstract objects. Properties of numbers are abstract properties. But formal systems deal with fully abstract numbers no more than your computer deals with abstract concepts. Formal systems are syntactic devices just as computers are electrical devices, and what formal systems are forced to deal with are not numbers but syntactic numerals. In Principia Mathematica, for example, what are taken as numbers 1, 2, 3 and 4 ‘on interpretation’ appear within the system as numerals constructed from 0 and a successor function ‘—

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numerals 0', 0'', 0''', 0''''. In place of properties of numbers are complexes of symbols which function as syntactic predicates of numerals. The graphic icon above represents an aspect of mathematical truth: the fact that a real number-theoretical property holds of a real number. The closest we can get within a formal system is a shadow of that relation: the demonstrable applicability of a predicate to a numeral. This we will represent as:

Our ‘s’ arrow marks derivable applicability within the system— alternatively expressed, the fact that such a complex appears as a theorem within the system. Our dots serve as a reminder on the left that we are dealing with a numeral standing in for the number at issue, on the right that we are dealing with a predicate standing in for a property. Here as before, but incorporating the more sophisticated recognition of a level of numerals, the fact that something is a theorem within the system appears in two ways in our graphical representation:

(b') With a similar sophistication, bivalence for ‘is a theorem’ and our solid arrow of applicability become:

(c') From this point the Gödel result can be outlined in terms of two clever tricks. The first trick is that we can recoverably encode both formulae of our system and series of formulae as specific numbers. This can in fact be done in any of an infinite number of ways.

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We will use brackets to indicate an assumed (and specific) Gödel numbering for a formula of our system. The graphical icon on the left represents a number—a number encoding the displayed syntactic formula. The graphical icon on the right represents the applicability of a full-blooded property to that number: the property, perhaps, that this is the Gödel number of an axiom of our formal system:

The second trick of the Gödel result is to note that there will be genuine properties of numbers that correspond to syntactic properties of their numerals. There will be a numerical property, for example, that holds of all and only those numbers that Gödel-number axioms of the system on our chosen encoding. There will be a numerical property of two numbers that will hold just in case the second encodes a formula that follows from the former by a syntactic rule of inference of the system. Because a derivation is simply a series of formulae of the system each of which is an axiom or follows from an earlier step by a rule of inference, there will ultimately be a numerical property that applies to pairs of numbers just in case the first numbers a derivation for the second. That a certain number is the second of a pair of numbers with that property will be a numerical property of numbers corresponding to derivability within the system. That is a numerical property that will hold of all and only those numbers that encode theorems of the system. As outlined above, we graphically represent theoremhood—or derivability within the system—in two interchangeable ways:

(b') Given the encoding of theoremhood as a numerical property—the second trick of the Gödel result—we add a further relationship. The

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conditions in each case above will hold if and only if a particular numerical property holds of a particular number: only if the numerical property corresponding to theoremhood holds of the Gödel number for the formula in question. Using to represent that numerical property,

Just as there will be a numerical property corresponding to ‘theoremhood’, there will be its converse: a property corresponding to nontheoremhood:

(b") The simple introduction to Gödel in chapter 2 was written in terms of a sentence ‘I am not a theorem’: a sentence much like the Liar, selfreferential in precisely the sense that the Liar is. Here we can admit that ‘I am a theorem’ functioned there merely as a stand-in. The true result is constructed in terms of a merely analogous sentence, with a reflexivity merely analogous to self-reference. For any one-place predicate of the system there will be a sentence S for which the following is demonstrable in the system:

This is the Diagonal Lemma: for any one-place predicate of the system there will be a sentence which is provably equivalent to one in which that predicate is applied to the Gödel number of the sentence itself. Although the proof for the Diagonal Lemma proceeds by constructing a clever formula around what is ultimately its own Gödel number, what is at issue is not literal self-reference. For one thing, the

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numeral on the right encodes only under a Gödel numbering which is itself external to the system—about which the system itself, in effect, knows nothing. For another, the relation actually used in Gödel’s argument is provable equivalence rather than identity. It is however this systematic reflexivity under encoding, analogous to self-reference, that gives us the same kind of substitutable embeddability used in both the Liar and the informal Gödel of chapter 2. Where ‘=’ is used merely to indicate intersubstitability, each of these is intersubstitutable in our representations above:

The numerical properties corresponding to ‘is a theorem’ and to ‘is not a theorem’ are expressible in our system, and thus substitutable for F. This, finally, is the genuine article that functions as did ‘I am not a theorem’ in the simple version of the result offered above:

(d') Up to this point all aspects of our structure are reflections of clever Gödel numbering, conventions of representation, or theorems such as the Diagonal Lemma demonstrable in the system. These cause difficulties—difficulties at the core of Gödel’s proof—only when we add assumptions of soundness and completeness. In our richer representation, those assumptions amount to the following:

soundness

completeness

Given bivalence for our application arrows and their slashed negations, by modus tollens we have

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It is the addition of these final assumptions of soundness and completeness that gives us full oscillation and contradiction. We start at the top with the assumption that our sentence in (d’) is demonstrable as a theorem:

On pain of oscillation, no system with the structural features outlined can be both sound and complete. In order to avoid oscillation, one of these must go: any such system must either unsound or incomplete. This is the structure of Gödel’s first incompleteness theorem, this time in more complete detail. What prevents Gödel’s proof from being simply an instantiation of the Liar is the distinction throughout between genuine application, represented by

, and derivable application, represented by

.

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Were these either identified or inferentially bound in the sense that soundness and completeness would demand, we would be condemned to the oscillation and contradiction of the Liar. The fact that we can break that oscillation by denying their identification—by denying either soundness or completeness—converts the paradoxical structure of the Liar into Gödel’s positive demonstration that number-theoretical truth and formal demonstrability cannot be coextensive. 2. TARSKI AND THE FORMAL UNDEFINABILITY OF TRUTH Tarski’s Theorem builds on Gödel’s, with much the same structure, but brings the role of truth even more clearly to the fore. Raymond Smullyan has argued that Tarski’s Theorem captures what is most genuinely of philosophical interest in the cluster of results.1 Gödel’s theorem applies to formal languages and axiomatized formal systems, with the last fact crucial for the expressability of ‘is a theorem’ within the system. But consider the same formal language for number theory within the somewhat looser concept of a ‘theory,’ an extension of the system in which some larger set of sentences of the language are taken as theorems. ‘Arithmetic’ is that extension of our formal system in which the theorems correspond to the arithmetical truths; that theory in which theorems correspond to precisely those sentences of our language that are true on interpretation.2 This concept takes us a step beyond formal systems in the sense of the Gödel result. Arithmetic, precisely because its theorems are taken to correspond to all and only those sentences of the language true on interpretation, is by definition both consistent and complete. It can then be shown using precisely the proof above that Arithmetic so conceived cannot be axiomatizable: that additional assumption would lead directly to oscillation. For even Arithmetic, however, there is a crucial limitative theorem. There will be those formulae that are theorems of the system, and which are moreover true on interpretation. Because we are dealing with a formal language of precisely the same type as before, all formulae of the system will have Gödel numbers on a chosen encoding. The notion of truth on interpretation for such a system, however, will not be formally definable in the system: There will be no formula of

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the system that applies as a theorem to all and only the Gödel numbers of those formulae true on interpretation. In Tarski’s apt phrase, Arithmetical truth will not be Arithmetically definable. Here the proof is again by reductio to oscillation. Arithmetic as defined above is guaranteed to be both sound and complete: all and only the theorems of Arithmetic are formulae true on interpretation. Using restricted to formulae of the system and T to mean that a formula is true on interpretation:

(e) ‘True on interpretation’ and our solid applicational arrow are again bivalent. This we can represent as:

(f) The assumption of formal definability of arithmetical truth is the assumption of a predicate

in the language of the system for which it

is a theorem that applies to the numerical representation of the Gödel number of a formula just in case that formula is true on interpretation. Our specific interest is in the correlate predicate

such that

(g) Arithmetic is an extension of our earlier systems so as to include theorems corresponding to all truths on interpretation. Because it is an extension of that earlier system, however, the system is still one for which the Diagonal Lemma will be demonstrable. Where ‘=’ is used

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merely to indicate intersubstitability, then, and given our assumed predicate tations:

, each of these will be intersubstitutable in our represen-

(h) The assumption of a property corresponding to ‘true on interpretation’ and formally definable within the system once again generates oscillation. We start at the top with the assumption that S will appear as a theorem of Arithmetic:

On pain of oscillation, Arithmetical truth is not Arithmetically definable. NOTES 1

For details see G. Boolos, J. Burgess, and R. Jeffrey, Computability and Logic, Cambridge: Cambridge University Press, 1974, pp. 106-107, p. 176.

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NOTES 2

Raymond Smullyan, Godel’s Incompleteness Theorems, New York: Oxford University Press, 1992.

NAME INDEX Alai-Tafti, Ali, 172n9 Aristotle, 61, 117n9 Armour-Garb, Bradley P., 92n1 Baron-Cohen, Simon, 150n13 Beall, J. C., 92n1 Benacerraf, Paul, 92-93n3 Bohr, Niels, 114 Boolos, George, 185n1 Bottenberg, Frances, 1 Brentano, Franz, 125, 142, 149n5 Burge, Tyler, 149n6 Burgess, John, 185n1 Carnap, Rudolf, 156 Castañeda, Hector-Neri, 95, 117n10 Casti, John L., 58n5 Chalmers, David, 149-50n9 Church, Alonzo, 46, 48-49. 52 Copeland, B. Jack, 58n3 de Bondone, Giotto, 34n5 Davidson, Donald, 156, 158, 171n3, 171n4 Descartes, René, 3, 81-83, 93n5, 105, 137, 174 Diogenes Laertius, 98 Eubulides of Megara, 98, 116n3 Feferman, Solomon, 93n3 Feyerabend, Paul, 117n8 Fichte, Johann Gottlieb, 145 Frith, Uta, 150n13 Gallagher, Shaun, 124, 149n4

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Gallup, Gordon G., Jr., 150n12 Gibson, James J., 149n6 Gödel, Kurt, 4, 35n7, 37-47, 49, 51-54, 57, 92n3, 169, 173, 177-184 Goldbach, Christian, 27 Goodman, Nelson, 155 Grice, Herbert P., 92n2 Grim, Patrick, 1, 92n1, 93n11, 116n2, 172n9 Hanson, Norwood R., 117n8 Hauser, Marc, 150n12 Held, Richard, 149n7 Herzberger, Hans, 17, 27, 34n4 Hofstadter, Douglas, 58n2 Hume, David, 137 Hyde, Timothy, 1 Jackson, Frank, 5, 128-130. 149n8 Jaima, Amir, 1 Jeffrey, Richard, 185n1 Kant, Immanuel, 145-146 Kilb, Nicholas, 172n9 Kohler, Adam, 1 Kokalis, Trina, 172n9 Kuhn, Thomas, 117n8 Leibniz, G. W., 160 Leslie, Alan M., 150n13 Levine, Joseph, 149n9 Lewis, David, 93n3, 116n2 Lingis, Alphonso, 149n5 Litt, Theodor, 142 Lucas, J. R., 92n3 Ludlow, Peter, 116n5 Lowenheim, Leopold, 157 McGinn, Colin, 149n2

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189

Meltzoff, Andrew N., 150n13 Mill, John Stuart, 102 Moore, G. E., 71-72, 82, 95, 174 Nagel, Ernest, 58n2 Nagel, Thomas, 5, 130, 133, 149n1, 150n10 Newman, James R., 58n2 Peirce, C. S., 89 Penrose, Roger, 92n3 Perner, Josef, 150n13 Perry, John, 95, 110, 112, 116n2, 117n10 Pfänder, Alexander, 142, 150n11 Povinelli, D. J., 150n12 Priest, Graham, 34n1, 34n4, 92n1 Prior, A. N., 95 Putnam, Hilary, 156, 158 Quine, W. V. O., 5, 93n4, 155, 156-159, 171n2, 171n6 Rayo, Augustin, 116n5 Rescher, Nicholas, 1, 34n3, 58n5, 93n11, 171n1, 172n10, 172n13 Rosenfeld, Adam, 1 Rosenthal, David, 123, 149n3 Russell, Bertrand, 10-12, 14-18, 25, 29-30, 34n2, 34n3, 39-40, 52-53, 93n5, 173 Rutherford, Ernest, 114 Sacks, Oliver, 149n7 Sartre, Jean-Paul ,149n5 Searle, John, 5, 158-161 Skolem, Thoralf, 157 Smullyan, Raymond, 183, 186n2 St. Denis, Paul, 172n9 Stanley, Jason, 116n1 Stefaneschi, Cardinal, 34n5 Strandberg, Jenny, 1

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Strawson, Galen, 149n2, 149n9 Tarski, Alfred, 4, 25, 46, 177, 183-185 Thomas, J. J., 114 Thomson, James, 17, 27, 34n4 Turing, Alan, 4, 35n7, 46-55, 57, 173 von Senden, Marius, 149n7 Whitehead, Alfred North, 34n2, 39-40 Williamson, Timothy, 116n1 Wimmer, Heinz, 150n13 Zahavi, Dan, 124, 149n4