Wittgenstein's Apprenticeship with Russell

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Wittgenstein’s Apprenticeship with Russell

Wittgenstein’s Tractatus has generated many interpretations since its publication in 1921, but over the years a consensus has developed concerning its criticisms of Russell’s philosophy. In Wittgenstein’s Apprenticeship with Russell, Gregory Landini draws extensively from his work on Russell’s unpublished manuscripts to show that the consensus characterizes Russell with positions he did not hold. Using a careful analysis of Wittgenstein’s writings he traces the Doctrine of Showing and the ‘‘fundamental idea’’ of the Tractatus to Russell’s logical atomist research program which dissolves philosophical problems by employing variables with structure. He argues that Russell and Wittgenstein were allies in a research program that makes logical analysis and reconstruction the essence of philosophy. His sharp and controversial study will be essential reading for all who are interested in this rich period of the history of analytic philosophy. GREGORY LANDINI

is Professor of Philosophy at the University of Iowa.

Bertrand Russell and Ludwig Wittgenstein in conversation in 1922. From the archives of Dora Russell at McMaster University Library. Reproduced by permission of McMaster University Library.

Wittgenstein’s Apprenticeship with Russell Gregory Landini University of Iowa

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521870238 © Gregory Landini 2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007 eBook (EBL) ISBN-13 978-0-511-29510-2 ISBN-10 0-511-29510-3 eBook (EBL) ISBN-13 ISBN-10

hardback 978-0-521-87023-8 hardback 0-521-87023-2

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

For Frann to whom I am a grateful apprentice

Contents

Preface 1

Rereading Russell and Wittgenstein Two dogmas of Russellian interpretation Logical fictions of Russell

2

Logical atomism Logical atomism as a research program The logical independence of the facts that are truth-makers Acquaintance with logical objects Russell’s paralysis Retreat from Pythagoras

3

My fundamental idea Showing as radical Russellianism Ideal versus ordinary language Sub specie aeternitatis Russell’s rejection of Showing Kicking away the ladder

4

Logic as if tautologous Logic as if decidable Scaffolding Wittgenstein’s N-operator Quantification and the N-operator

5

Tractarian logicism Ramified types as scaffolding in Russell and Wittgenstein Equations versus tautologies Numbers as exponents of operations

6

Principia’s second edition Extensionality and neutral monism The oracle on Reducibility Slipshod notations? Ramsey’s extensional functions

page ix 1 5 11 24 25 40 53 65 72 77 79 90 94 100 103 107 112 118 125 134 147 150 158 170 189 192 201 208 214

vii

viii

Contents

7

Logic as the essence of philosophy Ontology as meaningless Ontology as structured variables Carnap versus Quine on ontology Return to Pythagoras

Appendix A: Exclusive quantifiers Appendix B: Modality in the Tractatus Bibliography Index

227 230 237 241 245 253 266 285 297

Preface

Wittgenstein’s Tractatus Logico-Philosophicus has generated many interpretations since its publication in 1921. Time has produced something of a consensus, however, concerning the nature of the Tractarian criticisms of Russell’s early philosophy. Challenging this consensus is the subject of the present book. Russell’s unpublished manuscripts have brought about a revolution in the understanding of his philosophy. The manuscripts reveal that the consensus about the Tractarian criticisms characterizes Russell with positions he did not hold. Rereading Russell requires rereading Wittgenstein. Russell’s logical atomism is not an empiricism couched in a logic of ramified and type-stratified entities (propositional functions) and based on a principle of acquaintance with sense-data. Logical atomism is a research program for dissolving philosophical conundrums by employing variables with structure – an ontologically austere structural realism. Wittgenstein transformed Russell’s method of structured variables into his Doctrine of Showing. The book identifies the Tractarian Grundgedanke (‘‘fundamental idea’’) with Showing and argues that Russell and his apprentice Wittgenstein were allies in a research program that makes logical analysis and reconstruction the essence of philosophy. In the Tractatus, Wittgenstein wrote, ‘‘The riddle does not exist. If a question can be framed at all, it is also possible to answer it’’ (TLP 6.5). Of course, paradoxes are all too easy to fall into. When my son Ansel was young, our bedtime ritual was for me to ask him ten questions. After a short time of repeating the questions every other night, he had memorized the answers. Finally, he demanded, ‘‘I want new hard questions.’’ I thought I’d be clever and give him a riddle. I fondly recalled walking with the logician Raymond Smullyan when I was a teaching assistant in graduate school. On one occasion we were off to his logic class and he turned to me and said, ‘‘Is the answer to this question, no?’’ I wanted something similar. I said to Ansel, ‘‘OK, here you go with the new hard questions: What is the first question?’’ Not surprisingly, he complained, saying that it doesn’t count as a question. I replied that it does count and is in fact easy to answer. The answer is, ‘‘What is the first question?’’ It seems that Ansel got ix

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Preface

the better of me in this exchange. He replied, ‘‘Well, if it is easy, then it is not among the new hard questions.’’ Now we have a puzzle. It would seem that my performative act of utterance assured that whatever question I asked is the first of the new questions if and only if it is hard. But is the question hard to answer? If it is hard, then it is the first question and so it is easy. If it is easy, then it is not the first question and that makes it a hard question to answer. What, after all, would then be the first question? Philosophical problems are, to my mind, puzzles whose solutions are very difficult because they require in some cases an almost complete understanding of logic, mathematics, physics, and metaphysics. Wittgenstein held that the task of the philosopher is to dissolve riddles (philosophical problems) by elucidating the misconceptions that are involved in their characterization. Russell clearly had a similar view. It was with just this attitude concerning the paradoxes plaguing logicism that he boldly entitled one of his papers ‘‘On ‘Insolubilia’ and Their Solution by Symbolic Logic.’’ From a historical perspective, this is perhaps the most important among Russell’s published papers for understanding the historical development of Principia Mathematica. It was originally published only in French with the title ‘‘Les paradoxes de la logique.’’ In the paper, Russell embraces a substitutional theory of propositional structure. He separates logical from semantic paradoxes, and shows how the structure of a type-theory of attributes (propositional functions) adequate to mathematics is to be emulated in a formal reconstruction of the first principles of logic that is type-free and makes no assumption that every formula comprehends a class (or an attribute). Based on Russell’s unpublished manuscripts on this theory, my book Russell’s Hidden Substitutional Theory set forth an entirely new account of the historical origins and nature of Principia. Russell’s substitutional theory, hidden among the voluminous collection of his manuscripts and work notes until the 1970s, shatters interpretations of Russell’s philosophy that have persisted for decades. Perhaps most startling among the discoveries I made is that Principia does not embrace a ramified type-hierarchy of entities. Principia offers a type-free ontology of universals, particulars, and facts. Its ‘‘type-theory’’ is a scaffolding built into its use of structured variables – a scaffolding which emulates the fundamental laws of a type- and order-regimented theory of attributes and thereby emulates a type-theory of classes and relations-in-extension. Principia does not embrace a hierarchy of orders of entities. It offers a recursive definition of ‘‘truth’’ which assumes that the facts that are truth-makers, and the universals (relating relations and properties) inhering in them, are logically independent of each other. It offers a ‘‘no-propositions theory’’ according to which logical connectives are not relation signs standing for entities. These discoveries, and more like them,

Preface

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require a new interpretation of Wittgenstein’s Tractatus. They demand a new account of logical atomism and the relationship between Russell and Wittgenstein. Earlier accounts are constructed with foundations in clay and washed away by Russell’s manuscripts. Early drafts of chapters of this work have greatly benefited from the comments of Pasquale Frascolla, Nicholas Griffin, Francesco Orilia, Russell Wahl, and Kenneth Williford. Each has made a mark on some chapter of the book. The manuscript was improved by being read in its entirety by Tuomas Manninen, who helped in translating parts of Mu¨ller’s Abriss of Schro¨der’s Algebra of Logik. Many thanks to the Bertrand Russell Research Center, McMaster University, Hamilton, Ontario, Canada, for permission to use scanned images.

1

Rereading Russell and Wittgenstein

Since its publication in 1921, Wittgenstein’s Tractatus Logico-Philosophicus has attracted a broad variety of interpretations. The work has been viewed as a revolution in the metaphysics of logic, with Wittgenstein inventing the truth-tables and ushering in modern logic and even modal logic. It has been viewed as the holy text of the antimetaphysical doctrines of logical empiricism – a work attempting to establish a foundational observation language grounding empirical (scientific) discourse in an effort to show that all philosophical or metaphysical propositions are pseudo-propositions.1 Wittgenstein’s attraction to the tragic lives of Schopenhauer, Weininger, and Kierkegaard has been a resource for irrationalist interpretations as well. Rejecting both logic and metaphysics as the focus of the Tractatus, they herald its entries on solipsism, value, religion, and mysticism as central to its message. Therapeutic interpretations attempt an even more radical break than do irrationalist interpretations. The therapeutic reading denies that there is any positive philosophical theory in the work. On this reading, the Tractatus is against philosophical theory and offers a treatment for the condition of thinking that there are riddles that must be solved by a philosophical theory. Thus, we find diametric opposition among even the most prominent philosophical interpretations of the Tractatus. We find those that take its central focus to be in epistemology, ontology, logic, semantics, ethics, religion, mysticism, or all of these together. We find interpretations of the text as realist, physicalist, phenomenalist, solipsist, idealist, existentialist, irrational, and therapeutic. It is no surprise, therefore, to find that interpretations differ significantly on what figures provide the best background orientation from which to understand the book. Is it to be Russell and Frege, or logical positivists such as Carnap, Ayer, and Popper? Is it to be Kant’s transcendental idealism or Schopenhauer’s mysticism? 1

Karl Popper, ‘‘Philosophy of Science: A Personal Report,’’ in C. A. Mace, ed., British Philosophy in the Mid-century: A Cambridge Symposium (London: Allen & Unwin, 1957), pp. 163–164.

1

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Wittgenstein’s Apprenticeship with Russell

It is difficult to avoid a pessimistic induction that reaches the conclusion that no satisfactory account of the Tractatus will be found. There seems to be no fundamental principle, no Archimedean point, which could unify the apparently diverse themes of the book. ‘‘The history of Tractatus interpretation,’’ writes Stern, ‘‘is for the most part a history of wishful thinking, each successive group of interpreters seizing on the passages they have found most interesting in order to reconstruct the doctrines they knew must be there.’’2 Stern believes that an important lesson can be exacted from this. No interpretation, he says, could be adequate to all the Tractarian doctrines because the work itself is in tension between diverse metaphysical, antimetaphysical, and antiphilosophical tendencies and motives. Russell’s philosophy had once served as an Archimedean point for viewing the Tractatus. Wittgenstein was, after all, Russell’s student. And Wittgenstein himself wrote that his book was an effort to address the problems he and Russell shared.3 But over the years interpreters have fought themselves free of interpreting Wittgenstein as a Russellian. Perhaps the breach began when Anscombe correctly pointed out that Tractarian ‘‘objects’’ cannot be identified with the sense-data Russell embraced in his Problems of Philosophy (1912), Our Knowledge of the External World (1914), and logical atomism lectures (1917). A consensus emerged among many interpreters that Russell’s logical atomism is not the proper starting point from which to understand Wittgenstein. In hope of illuminating the central questions that animated Wittgenstein, Anscombe abandoned Russell’s atomism and turned instead to Frege’s philosophy of language and arithmetic. No interpretation today finds Russell and Wittgenstein working in alliance on the same project in philosophy. The interpretations of Pears and Hacker, both very influential and important, acknowledge that the two started out that way with Russell imagining his ´ e.´ But they conclude that the Tractatus Austrian pupil to be his proteg ushers in an orientation that is in opposition to Russell’s philosophy. The marginalization of Russell is deeper in therapeutic readings of the Tractatus. For instance, Conant finds no precedent in Russell’s philosophy for the Tractarian notion that logical truths are meaningless (sinnlos). Conant does see a precedent in Frege’s philosophy. He likens the matter to Frege’s distinction between function and object. Functions, in Frege’s view, are incomplete or unsaturated (ungesa¨ttigt) while objects are complete. Since

2

3

David Stern, ‘‘The Methods of the Tractatus: Beyond Positivism and Metaphysics?’’ in Paolo Parrini, Wes Salmon, and Merrilee Salmon, eds., Logical Empiricism: Historical and Contemporary Perspectives (Pittsburgh: Pittsburgh University Press, 2003), p. 126. Bertrand Russell, The Autobiography of Bertrand Russell, vol. 2, 1914–1944 (Boston: Little, Brown & Co., 1968), p. 162.

Rereading Russell and Wittgenstein

3

function words represent unsaturated entities, Frege demands that they never occur in subject positions. Russell rejected Frege’s doctrine, noting that ‘‘inextricable difficulties’’ envelope the view.4 How are we to say of a function that it is unsaturated or even that it is a function without violating Frege’s doctrine? The very statement of Frege’s doctrine that functions are unsaturated violates its own proscriptions governing meaningfulness. This suggests to Conant an example of something literally meaningless and yet elucidatory.5 Diamond’s ‘‘resolute’’ reading of the Tractatus demands that its entries be regarded literally meaningless. In her work, the conceptual distance between Wittgenstein and Russell is maximized. Diamond writes: My way of talking about what is in the book is meant to reflect Wittgenstein’s ideas about his own authorship: there are lines of thought which he wanted a reader of his book to pursue for himself. In the case of the Tractatus, one can add that there are lines of thought which he wanted Russell, as reader – Russell in particular – to pursue.6

Far from an ally, Wittgenstein is now construed as antithetical to Russell’s philosophy. In Diamond’s view, parts of the Tractatus were intended as lessons of instruction for Russell. The viability of such interpretations may seem surprising given that in the preface of the Tractatus Wittgenstein disavows any ‘‘novelty in detail,’’ and mentions that he is ‘‘indebted to Frege’s great works and to the writings of my friend Mr. Bertrand Russell for much of the stimulation of my thoughts.’’ But in present debates over the message of the Tractatus the historical fact that the themes of the Tractatus were developed under Russell’s mentorship is regarded as little more than a platitude. The curious events surrounding Wittgenstein’s failed efforts to publish the Tractatus are often cited to buttress this attitude. Having been rejected by the publishers of Kraus, Weininger, and Frege, Wittgenstein beseeched Russell for help in bringing the work to press. Russell agreed to write an introduction, and on its basis Wittgenstein was able to negotiate with the Leipzig publishing house Reclam.7 Russell sent the introduction and quite naturally wrote that he would try to amend it if Wittgenstein found anything unsatisfactory in his remarks. Wittgenstein 4 5 6

7

Bertrand Russell, The Principles of Mathematics, 2nd ed. (New York: W. W. Norton & Co., 1937, 1964), p. 45. James Conant, ‘‘Elucidation and Nonsense in Frege and Early Wittgenstein,’’ in Alice Carey and Rupert Read, eds., The New Wittgenstein (London: Routledge, 2000), pp. 174–217. Cora Diamond, ‘‘Does Bismark Have a Beetle in His Box? The Private Language Argument in the Tractatus,’’ in Alice Carey and Rupert Read, eds., The New Wittgenstein (London: Routledge, 2000), pp. 262–292. Ludwig Wittgenstein, Cambridge Letters, ed. Brian McGuinness and G. H. von Wright (Oxford: Blackwell, 1995), p. 147.

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Wittgenstein’s Apprenticeship with Russell

responded by thanking Russell for the introduction, and included the following in his letter: There’s so much of it that I’m not in agreement with both where you are critical of me and also where you’re simply trying to elucidate my point of view. But that doesn’t matter. The future will pass judgment on us – or perhaps it won’t, and if it is silent that will be a judgment too . . . The introduction is in the course of being translated and will then go with the treatise to the publisher. I hope he will accept them!8

But when Wittgenstein sent the translated introduction to Reclam, he had a change of heart. He insisted that it was not to be published with the work but was only to help in the publisher’s own orientation with regard to the work’s significance. The note soured the deal with Reclam. Wittgenstein wrote a letter to Russell, taking responsibility and explaining that ‘‘when translated into German all the refinement of your English style was obviously lost in the translation and what remained was superficiality and misunderstanding.’’9 Wittgenstein’s rejection of Russell’s introduction suggests that the problems and motivations for the theses of the Tractatus were not shared by Russell. This bolsters the view that the Tractatus was not a work in alliance with Russell’s philosophy. Accordingly, interpreters of the work have not felt constrained by the context of Russell’s philosophical positions, goals, successes, and desiderata. Of course, all of this is set in the context of a fixed interpretation of Russell’s philosophy. But what were Russell’s philosophical positions? The answer is complicated. Russell’s philosophy evolved significantly in the years leading up to and after the publication of Principia Mathematica. Against which of Russell’s many philosophical theses was Wittgenstein allegedly rebelling? Hacker writes: Both Frege and Russell conceived of the logical connectives as names of logical entities . . . Russell construed them as naming functions from propositions to propositions. This conception was linked to their idea that propositions are names of truth-values (Frege) or complexes (Russell). But it is a dire error to think that ‘‘p v q’’ has the same logical form as ‘‘aRb.’’10

Hacker awards Wittgenstein the ‘‘achievement’’ of having ‘‘freed himself of many of Russell’s deep confusions about the role of logical expressions.’’11 But in the Principia, Russell had abandoned his early ontology of propositions and adopted the wedge (‘‘v’’) and the tilde (‘‘’’) as statement connectives 8 9 10 11

Ibid., p. 152. Ludwig Wittgenstein, Letters to Russell, Keynes and Moore, ed. G. H. von Wright with the assistance of B. F. McGuinness (Oxford: Blackwell, 1974), p. 86. P. M. S. Hacker, Wittgenstein’s Place in Twentieth-Century Analytic Philosophy (Oxford: Blackwell, 1996), p. 28. Ibid., p. 22.

Rereading Russell and Wittgenstein

5

in just the modern sense.12 Hacker should award Wittgenstein the achievement of having agreed with Principia on the logical connectives. A gap in one’s understanding of Russell’s positions on the nature of logic easily distorts one’s picture of Wittgenstein. Quite clearly, the degree of distance found between the Tractatus and Russell’s philosophy depends on what one takes Russell’s philosophy to have been. Though Wittgenstein published little in his life, he left a voluminous Nachlass of worksheets. The years since the publication of the Tractatus have produced a good many distinct interpretations of its central theses. Time has produced something of a consensus, however, concerning the Tractarian criticisms of Russell’s philosophy, and this is the subject of the present book. Russell also left voluminous worksheets, and these have shed an entirely new light on Russell. Much of the consensus as to the nature of the Tractatarian criticisms relies upon attributing to Russell positions he did not hold. In the last thirty years, there has been a significant rereading of Russell. Rereading Russell demands a rereading of Wittgenstein. Two dogmas of Russellian interpretation Two theses have dominated interpretations of Russell’s philosophy for many years. These two theses are so widely held that it is rare to find challenges to either in the vast literature on Russell. They are: (1) In Principia Mathematica, Russell advanced a ramified type-theory of entities. (2) Russell’s logical atomism is a form of reductive empiricism. Russell’s manuscripts and work-notes reveal that both are false. To borrow a colorful phrase from Kant, the manuscripts awaken us from a dogmatic slumber. Rejecting the two dogmas of Russellian interpretation has very important consequences for rereading Russell and Wittgenstein. Ray Monk’s recent biographies nicely illustrate how historical accounts of Russell and Wittgenstein are built upon the dogmas. The nature of the personal and intellectual relationship between Russell and Wittgenstein is invariably built around Russell’s letters to Ottoline Morrell. Much has been made, for instance, of Wittgenstein’s criticisms of Russell’s multiplerelation theory of judgment, a theory first espoused in Principia and later worked out in Russell’s 1913 manuscript for a book on the theory of knowledge. In the wake of a storm of protest from Wittgenstein, Russell abruptly abandoned his book project some 350 pages toward its completion. One can find Russell writing Ottoline that Wittgenstein’s criticisms 12

Sadly, the change in Russell’s view is often missed. There are many examples. Recently we find it in Thomas Ricketts, ‘‘Wittgenstein Against Frege and Russell,’’ in Richard Reck, ed., From Frege to Wittgenstein (Oxford: Oxford University Press, 2002), p. 228.

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were an event of first-rate importance in my life, and affected everything I have done since. I saw he was right, and I saw that I could not hope ever again to do fundamental work in philosophy. My impulse was shattered, like a wave dashed to pieces against a breakwater. I became filled with utter despair, and tried to turn to you for consolation.

Indeed, in the same letter Russell goes on to say that ‘‘Wittgenstein persuaded me that what wanted doing in logic was too difficult for me. So there was no really vital satisfaction of my philosophical impulse in that work, and philosophy lost its hold on me. That was due to Wittgenstein more than to the war.’’13 When Russell writes Ottoline of despair of ever doing fundamental work in philosophy, of suicidal depression over failed work – feelings which, he says, were caused by exasperating exchanges with Wittgenstein – Monk seizes upon what he takes to be evidence of Wittgenstein’s transformation from Russell’s pupil to Russell’s master. The exact nature of Wittgenstein’s criticism of Russell’s multiple-relation theory is, in fact, hard to pinpoint from the cryptic passages of the Tractatus and the remains of the exchanges between the two. On the interpretation advanced by Griffin and Sommerville – an interpretation that Monk assumes to be correct – Russell’s multiple-relation theory requires the assumption of Principia’s type-theory of entities. Wittgenstein allegedly rejected the theory of types of Principia, writing to Russell that ‘‘all theory of types must be done away with by a theory of symbolism showing that what seem to be different kinds of things are symbolized by different kinds of symbols which cannot possibly be substituted in one another’s places.’’14 Monk concludes that Wittgenstein was ‘‘jettisoning large parts of the logic that Russell had devised for Principia, in particular the theory of types.’’15 Wittgenstein’s idea, then, is supposed to have been that the theory of types must be rendered superfluous by a proper theory of symbolism. Monk offers what he takes to be a stinging blow: In the face of such a sweeping dismissal of his theory, Russell might have been expected to present a spirited defense of his position or at least some tough question as to how his logicist foundations of mathematics might avoid contradiction without a theory of types. But he had by this time abandoned logic almost entirely.16

13 14 15 16

Russell, Autobiography, vol. 2, p. 66. Ludwig Wittgenstein, Notebooks 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe, 2nd ed. (Oxford: Blackwell, 1979), p. 122. Ray Monk, Bertrand Russell: The Spirit of Solitude 1872–1921 (New York: The Free Press, 1996), p. 286. Ray Monk, Ludwig Wittgenstein: The Duty of Genius (New York: The Free Press, 1990), p. 71.

Rereading Russell and Wittgenstein

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Monk’s interpretation weds itself to the thesis that Russell embraced an ontology of types of entities. This is understandable, for this is part of the orthodoxy and appears in a great many works on Russell’s philosophy of mathematics. For instance, Hacker writes, ‘‘It is easy to suppose, Wittgenstein remarked in his first onslaught upon the Theory of Types . . . that ‘individual,’ ‘particular,’ ‘complex,’ etc., are primitive ideas (Urzeichen). But in so doing, we forget that these are not primitive ideas.’’17 Wittgenstein is often said to have pointed out that a type-theory of entities is not a solution but an ad hoc dodge of the paradoxes that confront logicism. He is said to have pointed out that Russell violates type-theoretical strictures in his effort to set out a theory of types of entities. Moreover, he is said to have revealed that ramified type-theory relies on contingent truths as if they could provide a foundation for logic. None of these interpretations can stand once it is discovered that Russell never embraced a theory of types of entities. This is just the discovery that faces us. The dogma that Russell advanced a ramified type-theory of entities clouds the proper understanding of Wittgenstein. Russell’s manuscripts and publications reveal that he had worked steadfastly since 1905 to formulate a theory of symbolism which made the type distinctions that block the paradoxes part of the formal grammar of a type-free calculus for logic. Russell’s work reached an apex with his ‘‘substitutional theory’’ of propositional structure. This theory attempts to solve the paradoxes plaguing logicism by showing how logic can reconstruct mathematics without the ontological assumption that every open formula comprehends an entity (attribute or ‘‘propositional function’’), and without the ontological assumption of classes. Russell’s manuscripts reveal that his work to build type and order distinctions into formal grammar evolved into the no-propositions, no-classes, and no-propositional function theory of Principia. Thus, the idea that grammar must supplant type distinctions among entities is a position Wittgenstein learned from Russell. It is not at all odd or perplexing, then, that Monk doesn’t find Russell worried about Wittgenstein’s alleged ‘‘sweeping dismissal of his [type-] theory.’’ There was no such dismissal because there was no type-theory of entities. The rejection of the first dogma requires an entirely new picture of the intellectual and philosophical relationship between Russell and Wittgenstein. When Wittgenstein wrote that ‘‘all theories of types must be done away with by a theory of symbolism,’’ he surely was not criticizing Russell’s ontology of types of entities. There was no theory of types of entities in Principia. The demand that types be built into grammar was a lesson Russell taught him. We shall

17

P. M. S. Hacker, Insight and Illusion (New York: Oxford University Press, 1972), p. 22.

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argue that his point was that Russell had not gone far enough in building type distinctions into formal grammar. Russell relies on a difference between a universal and a particular (another ‘‘type’’ distinction, as Wittgenstein saw it). In Wittgenstein’s view, this must also be built into logical grammar. Monk’s biographies attempt to explain issues pertaining to Russell’s life by tying them to his debates with Wittgenstein over philosophical logic. This is laudable, but it takes a serious risk. If one gets the philosophy wrong, the explanation collapses. In his efforts to demonstrate that Wittgenstein became Russell’s ‘‘master,’’ Monk relies on the dogmas of Russellian interpretation. But the dogmas are mistaken. Clarke’s 1975 biography of Russell started the ‘‘pupil becomes master’’ motif. He writes: From the early summer of 1912 Wittgenstein’s relationship with Russell, little more than six months old, began to change. On paper it might still be that of pupil and teacher, but the teacher was already eager for the pupil’s opinion of his work.18

Monk expanded the theme significantly. As characterized by Monk, Wittgenstein began as Russell’s student, but was soon to become his intellectual master where the philosophy of logic and mathematics were concerned. They first met in October 1911 during Cambridge’s Michaelmas term. Wittgenstein continued in January 1912 and over the next term. As Monk tells the story, he pursued his studies in mathematical logic with such vigor that, by the end of it, Russell was to say that he had learned all he had to teach, and indeed gone further.19 Monk takes Russell literally, and would have us believe that Wittgenstein learned everything important Russell knew about mathematical logic in less than one year. By January 1913, Monk proclaims that the cooperation between the two has come to an end. ‘‘In the field of logic, Wittgenstein, far from being Russell’s student, had become Russell’s teacher.’’20 Monk’s case is largely based on Russell’s self-deprecating letters to Ottoline. Yet there is a straightforward alternative explanation of the letters. Russell’s emotional life was a shambles during this period. His failed relationship with his first wife Alys was largely the cause. Lady Ottoline was his angel of mercy, his hope in new love for release from despair and suffering. Yet Ottoline was often aloof to Russell’s pouring sentimentality and took much of his letters as attempts to cajole her sympathies in hope of a level of intimacy and emotion she simply did not feel. Russell was then forty-one, and wanted a companion and children.21

18 19 21

Ronald Clark, The Life of Bertrand Russell (New York: Knopf, 1976), p. 176. Monk, Ludwig Wittgenstein (Penguin, 1990), p. 72. 20 Ibid. Bertrand Russell, Letter to Ottoline Morrell, 18 August 1913, in The Selected Letters of Bertrand Russell, ed. Nicholas Griffin (Boston: Houghton Mifflin Co., 1992), p. 469.

Rereading Russell and Wittgenstein

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Ottoline had no intention of abandoning her life with Philip Morrell and her daughter, and soon tired of Russell’s often maudlin declarations of suicidal feelings contrived to drive her away from them. Russell’s letters to Ottoline must be read in the context of his technical work, otherwise they lend support to a skewed picture of events. One oftenquoted letter concerns Wittgenstein’s ‘‘rewriting’’ of Principia. Russell once remarked to Ottoline that ‘‘Wittgenstein has persuaded me that the early parts of Principia are very inexact, but fortunately it is his business to put them right, not mine.’’22 August 1913 dates the following entry in Pinsent’s diary: ‘‘It is probable that the first volume of Principia will have to be re-written, and Wittgenstein may write himself the first eleven chapters. That is a splendid triumph for him!’’23 Monk takes this to provide evidence of Russell realizing his inferiority to Wittgenstein in mathematical logic and his bequeathing its foundations to his pupil. Monk writes: These remarks are revealing. They show how Russell was still inclined to look upon Wittgenstein’s work as a kind of ‘‘fine tuning’’ of his own. He talks as if the inexactitude of the early parts of Principia are a mere detail, but those early parts contain the very foundation upon which the whole of the rest was built. And Wittgenstein was not repairing it, as Russell continued to think, but was demolishing it altogether.24

The colorful image of the genius Austrian pupil who bested the most famous philosopher of the time now reaches a zenith. This has become part of the folklore of Wittgenstein, but it does not pass technical scrutiny. Sections A and B of Principia consist of eleven starred numbers: *1–*5 is the propositional system, *9, *10, and *11 concern quantification theory, *12 introduces Reducibility, *13 discuss identity, and *14 is Russell’s theory of definite descriptions. There were new technical developments concerning *1–*5 that came to Russell’s attention in 1913. Sheffer read a paper to the American Mathematical Society on 31 December 1912 maintaining all the quantifier-free formulas of Principia’s sentential calculus can be expressed via one logical connective.25 Russell received Sheffer’s paper on 15 April 1913 and was interested in the revisions to Principia that it enables. The Sheffer stroke appears in Wittgenstein’s ‘‘Notes on Logic 1913’’ which were composed in Norway.26 With the intercession of Moore, these notes were later offered on Wittgenstein’s behalf so 22 23 24 25 26

Russell, Letter to Ottoline Morrell, 23 February 1913, ibid., p. 446. Quoted from Brian McGuinness, Wittgenstein: A Life (Berkeley: University of California Press, 1988), p. 180. Monk, Bertrand Russell: The Spirit of Solitude, p. 290. H. Sheffer, ‘‘A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants,’’ Transactions of the American Mathematical Society 14 (1913): 481–488. Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 103.

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that he would fulfill the dissertation requirement for a Research Student for the B.A. degree.27 It would not in the least belittle Russell’s stature as mentor if he had agreed that Wittgenstein should make it part of his work for the degree to find a new sentential deductive system based on Sheffer’s new connective, reducing the number of primitive principles of Principia’s *1–*5. This is the likely explanation of the so-called ‘‘splendid triumph’’ that Pinsent mentions. But Wittgenstein did not find the reduction Nicod found in 1916. Nicod demonstrated that only one axiom, together with the rule of uniform substitution and one other inference rule governing Sheffer’s one logical connective, suffices to generate Principia’s sentential calculus.28 Nicod died tragically in 1924. In the 1925 introduction to the second edition of Principia, Russell recommended that Sheffer ‘‘rewrite’’ [foundational chapters of] the Principia in accordance with the new methods.29 There is no question that Wittgenstein had ideas for improving the philosophical foundations of Principia and that Russell was enticed by them. But this provides no basis for Monk’s conclusion that Russell abandoned Principia in favor of Wittgenstein’s work on logic. Russell was attracted to Wittgenstein’s suggestion that Reducibility would be obviated by the doctrine that ‘‘a [propositional function] can only occur through its values.’’30 In the first edition of volume 1 of Principia, Russell’s own remarks concerning the status of Reducibility are illuminating. He wrote: although it seems very improbable that the axiom should turn out to be false, it is by no means improbable that it should be found to be deducible from some other more fundamental and more evident axiom. It is possible that the vicious circle principle, as embodied in the above hierarchy of types, is more drastic than it need be, and that by a less drastic use the necessity for the axiom might be avoided. Such changes, however, would not render anything false which has been asserted on the basis of the principles explained above: they would merely provide easier proofs of the same theorems.31

In Principia, Russell offered a pragmatic justification of Reducibility. He recognized that it is ‘‘not the sort of axiom about which one can rest content.’’32 He maintained that some formulation embodying type structures must be correct, and expressed a hope that with further work in the

27 28

29 30 32

That Wittgenstein’s Notes on Logic were submitted as a dissertation to fulfill the requirements of the B.A. degree is argued by McGuinness. See McGuinness, Wittgenstein, p. 199. Jean Nicod, ‘‘A Reduction in the Number of the Primitive Propositions of Logic,’’ Proceedings of the Cambridge Philosophical Society 19 (1917): 32–41. The paper was read before the society on 30 October 1916. A. N. Whitehead and Bertrand Russell, Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925), p. xv. Ibid., p. xxix. 31 Ibid., p. 59 (quoted from the 2nd ed.). Ibid., p. xiv (quoted from the 2nd ed.).

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area, Reducibility might yet be supplanted. Indeed, he acknowledged that there remained philosophical difficulties with Principia’s reliance on Reducibility and invited all readers to work on them.33 It is quite understandable that Russell would be agreeable to the interesting new ideas of Wittgenstein about how difficulties in the philosophical foundations of the Principia might be overcome. But to imagine possible amendments and improvements to the philosophical foundations of logic and mathematics offered in Principia is certainly not, as Monk thinks, to ‘‘replace’’ or ‘‘demolish it.’’ As we are beginning to see, an assessment of the Tractarian criticisms of Russell’s philosophy often depend essentially on an interpretation of some rather technical features of Russell’s work – his reconstruction of classes, ramified type-theory, and the like. Moreover, much of the literature on the relationship between Russell’s philosophy and Wittgenstein’s Tractatus is concerned to distinguish Tractarian doctrines from an empiricist reading of Russell’s theory of sense-data. This brings us to the second dogma of the Russellian interpretation – the doctrine that Russell’s atomism was a form of reductive empiricism. Again we ask: What becomes of such interpretations if it is discovered that the second dogma is mistaken? Again, this is precisely the situation that faces us. Indeed, the reasons for rejecting the first dogma of Russellian interpretation give rise to reasons for rejecting the second. We shall see that the lessons Russell distilled from his endeavors to build the structure of ramified types into logical grammar render an entirely new conception of Russell’s philosophy of logical atomism. The ‘‘logical constructions’’ or ‘‘logical fictions’’ of Russell’s philosophy of logical atomism cannot properly be understood in terms of empiricist reduction. Research coming from Russell’s manuscript has shed a flood of new light on this history, and a new picture is emerging. The new picture presents a fresh way to understand Russell’s philosophy, and with it comes a promising new approach to the interpretation of Wittgenstein’s Tractatus. Logical fictions of Russell Russell wrote in his autobiography that the theory of descriptions ‘‘was the first step toward solving the difficulties which had baffled me for so long.’’34 In a letter to Lucy Martin Donnelly of 13 June 1905, Russell wrote that the theory of descriptions ‘‘throws a flood of light on the 33 34

A. N. Whitehead and Bertrand Russell, Principia Mathematica to *56, 2nd ed. (Cambridge: Cambridge University Press, 1964), pp. vii, xiv, 60. Bertrand Russell, The Autobiography of Bertrand Russell, vol. 1, 1872–1914 (Boston: Little, Brown & Co., 1967), p. 79.

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foundations of mathematics.’’35 The ‘‘difficulties,’’ of course, pertained to the paradox of the class of all and only those classes that are not members of themselves. The floodlight (solution) from 1905 through 1907 was Russell’s substitutional theory. Definite descriptions of propositions are used to emulate a type-hierarchy of attributes in intension and thereby classes. In Principia (1910), the ontology of propositions is abandoned and Whitehead and Russell write that ‘‘the phrase that expresses a proposition is what we call an ‘incomplete’ symbol.’’36 The theory of incomplete symbols is involved in the ‘‘no-classes’’ theory according to which ‘‘classes, are in fact, like descriptions, logical fictions or (as we say) ‘incomplete symbols.’’’37 After Principia the notion of a ‘‘logical fiction,’’ ‘‘incomplete symbol,’’ or ‘‘false abstraction’’ came to pervade Russell’s philosophy. In the 1920s, Russell wrote that ‘‘I and the chair are both logical fictions, both being in fact a series of classes of particulars.’’38 Ramsey aptly described Russell’s theory of definite descriptions as ‘‘a paradigm of philosophy.’’ Wittgenstein proclaimed that ‘‘it was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one’’ (TLP 4.0031). But what is Russell’s conception of a ‘‘logical fiction’’? What connection, if any, does it have with empiricist reduction? Empiricist reductions in philosophy became popular with the success of Newton’s philosophy of science. The wide success of Newton’s science encouraged philosophers to investigate his scientific methods. Newton’s famous dictum, Hypotheses non fingo (I do not make hypotheses), aptly distinguished his natural philosophy from that of the mechanical philosophy that preceded it. Newton embraced a philosophy of science according to which crucial experiments are essential to the very nature of scientific discovery. The mechanical philosophy that preceded Newtonian science had sought to exclude appeals to Aristotelian metaphysical essences, entelechies, drives, animal spirits, life forces, and the like. The principle of demarcation of the mechanical philosophy was the requirement that only mechanical processes, governed by quantitative mathematical laws, are properly scientific. For example, mechanists regarded it to be a scientific hypothesis that the lodestone spews forth screw-shaped particles which permeate the iron, 35 36 37 38

Ibid., p. 152. A. N. Whitehead and Bertrand Russell, Principia Mathematica to *56 (Cambridge: Cambridge University Press, 1964), p. 44. Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1919), p. 182. Bertrand Russell, ‘‘The Philosophy of Logical Atomism,’’ in Logic and Knowledge: Essays 1901–1950, ed. Robert C. Marsh (London: Allen & Unwin, 1977), p. 241. This is a fourdimensional theory of time according to which the notion of a material continuant (enduring through time) obeying physical laws is to be abandoned and the laws of matter are to be reconstructed in terms of series of events.

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pulling the two together. The conjecture that it contains within it a force of magnetic attraction is not ‘‘scientific’’ because it is not mechanical. The hypothesis that air consists of many moving particles which exert a pressure when compressed is scientific. That nature abhors a vacuum is not. But strict adherence to mechanical philosophy threatens to exclude Galileo’s work on laws governing ‘forces’ and ‘accelerations.’ To the mechanical philosophy, forces are anathema; they are vestiges of Aristotelianism. Newton offered a grand synthesis of the mechanical and Galilean traditions by appeal to empiricism.39 Forces are legitimate in science only if precise mathematical laws govern them, laws that admit of empirical test. The criterion of a hypothesis’s being scientific is its amenability to empirical test. In the hands of philosophy, empiricism is the thesis that all nonanalytic knowledge is obtained by sense experience. This gives rise to a conceptual thesis of empiricism. The conceptual thesis has it that each meaningful nonanalytic statement is logically equivalent to some statement which involves only terms which refer to sense experience. The logical equivalence renders the empirical import of the statement. Every idea about the world must originate either in ideas of sense experience or be compounded of such ideas. For example, the meaning of a material object term such as ‘‘tomato’’ is wholly compounded from actual (and anticipated) sense experiences – tastes, smells, colors, feels, and the like. It soon became clear, however, that strict adherence to empiricism requires that notions of ‘‘matter’’ and ‘‘cause’’ also obtain their meanings from sense experience. Attempts to carry out such an empiricist meaning analysis of theoretical terms were greatly facilitated by the development of modern quantificational logic. But the new logic changed the emphasis of the project from terms to sentences. The project evolved into the plan of setting out the evidentiary relations between the laws in which theoretical terms are couched and sentences of sense experience (or observation). The derivations were to be piecemeal, sentence by sentence. Carnap illustrates this form of reductive empiricism in his book Der logische Aufbau der Welt (1928). Carnap’s valiant efforts conclusively showed that many commonplace theoretical terms (such as ‘‘temperature’’) can only be given conditional (partial) definitions in his observation language. Where T is the theoretical sentence in question and q contains only sentences pertaining to sense (or observation), we have: If p, then T if and only if q.

39

See Richard Westfall, The Construction of Modern Science (Cambridge: Cambridge University Press, 1977), p. 159. See also E. A. Burtt, The Metaphysical Foundations of Modern Science (Atlantic Highlands, N. J.: Humanities Press, 1952).

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For example, ‘‘If the mercury thermometer is in contact with x, then the temperature of x is c degrees if and only if the thermometer reads c.’’ When the condition p is not met, the theoretical sentence T is undefined. Be this as it may, Carnap did go some way in revealing how sentences in which theoretical terms are couched stand in evidentiary relationships to sentences in which only observation terms occur. Carnap’s partial definitions naturally give rise to the thesis that theoretical terms are not referential but serve simply as instruments to predict and control observations. Further investigation into scientific practice complicated matters even more. Duhem’s investigation of the experimental methods that provide evidence for (or against) a scientific theory resulted in the discovery that no sentence, taken in isolation from the empirical theories in which it is embedded, admits of confirmation of disconfirmation by experiment. This ‘‘holistic’’ relationship between theory and the conditions of its empirical evidence marginalizes the importance of the question as to whether a given theoretical term is referential. The successes of the experimental method in science were a main catalyst in favor of empiricism in philosophy. Holism is a fundamental feature of the empirical scientific method, and its discovery reverberated throughout empiricism. The unit of meaning reduction is not the term or the sentence, but the entire theory.40 Within reduction, two models emerge: Reductive Identity and Elimination. Both may be called ‘‘ontological reductions,’’ but the difference can be significant. Eliminations are exemplified by the historical development of physics and chemistry. Important eighteenth- and nineteenth-century theories offered a number of subtle fluid and aetheric objects that were highly successful at explaining a wide variety of phenomena. In the process of theory change, the research programs that gave rise to such theories were supplanted by atomistic physical theories couched within a new research program that abandons the aether. Empirical and conceptual problems pertaining to the aether (such as its elasticity) became mute, and an entirely new research program, with a new language and a new set of empirical and conceptual techniques, was inaugurated. Many successes of the earlier aether theories were retained by the theories of the new research program. Retention, however, is always partial. The confirmed predictions of an earlier theory in a rival research tradition do not always survive into the supplanting research tradition. Indeed, theoretical processes and mechanisms of earlier theories are at times treated as flotsam.41 The supplanting tradition may come to regard the 40 41

See Carl Hempel, ‘‘The Empiricist Criterion of Meaning,’’ in A. J. Ayer, ed., Logical Positivism (Glencoe, Ill.: The Free Press, 1959), pp. 109–129. Larry Laudan, Progress and Its Problems (Berkeley: University of California Press, 1977).

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terms of the earlier theories as nonreferential, or regard earlier ontologies as idle wheels that serve no explanatory purpose. It is tempting, given these examples of elimination in the history of science, to suggest the following simple way of distinguishing elimination from reductive identity. In an elimination, the ontology of the old theory is abandoned. The theoretical terms of the old theory don’t refer. Accordingly, one might expect that an elimination would regard the old theory’s statements as literally false. Reductive identity, in contrast, regards the terms of the old theory as referential – because they are suitably reinterpreted to be about entities countenanced in the new theory. In this way, the sentences of the old theory can be regarded as true. But this way of distinguishing the two conceptions of reduction is inadequate. An elimination may endeavor to reconceptualize the laws of the old theory without countenancing its objects in any way. It can regard certain of the laws of the old theory as approximately true, though not true of the objects the old theory countenanced. In 1914 Russell advocated a ‘‘scientific method in philosophy,’’ which puts logical constructions (the notion of a logical fiction) at center stage. Russell characterizes his program as one governed by a ‘‘supreme maxim of all scientific philosophizing,’’ namely this: ‘‘Wherever possible, logical constructions are to be substituted for inferred entities.’’42 In his paper ‘‘On the Relation of Sense-Data to Physics’’ he described the method thus: Given a set of propositions nominally dealing with the supposed inferred entities, we observe the properties which are required of the supposed entities in order to make these propositions true. By dint of a little logical ingenuity, we then construct some logical function of less hypothetical entities which has the requisite properties. This constructed function we substitute for the supposed inferred entities, and thereby obtain a new and less doubtful interpretation of the body of propositions in question.43

In his article ‘‘Logical Atomism’’ (1924), Russell stated the maxim in the form: ‘‘Wherever possible, substitute constructions out of known entities for inferences to unknown entities.’’44 Since Russell maintained in several works of the period that sense-data and universals are entities of immediate acquaintance, the maxim has naturally enough been interpreted as if it advocates an empiricist reduction. Indeed, even Russell’s ‘‘no-classes’’ theory has been shoehorned into this mold. For instance, Ayer writes:

42

43 44

Bertrand Russell, ‘‘On the Relation of Sense-Data to Physics,’’ in Mysticism and Logic and Other Essays (Totowa, N. J.: Barnes & Noble, 1976), p. 115 (first published in Scientia 4, 1914). Ibid., p. 116. Bertrand Russell, ‘‘Logical Atomism,’’ in Logic and Knowledge: Essays 1901–1950, ed. Robert C. Marsh (London: Allen & Unwin, 1977), p. 326.

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when Russell spoke of an object as a logical fiction, he did not mean to imply that it was imaginary or nonexistent . . . Similarly, in the period during which Russell held that physical objects were logical constructions, he did not wish to suggest that they were unreal in the way that gorgons are unreal . . . Logical fictions do indeed exist, but only in virtue of the existence of the elements out of which they are constructed. As Russell put it, they are not part of the ultimate furniture of the world.45

Ayer has in mind reductive identity. Rocks and mountains do exist, according to reductive empiricism, their being orderings of sense experiences notwithstanding. Ayer interprets Principia similarly. He assumes that Principia is a theory of ramified types of entities (‘‘propositional functions’’), and that classes are logical fictions in the sense that they are identified with propositional functions. In this sense, classes exist. There is, however, a rival interpretation of Russell’s supreme maxim of all scientific philosophizing – an interpretation that separates Russell’s logical constructions from empiricism entirely. The rival interpretation is supported by the historical evolution of Russell’s attempts to solve certain paradoxes plaguing the logical theory of a class or attribute.The naive theory of classes assumes that for every condition Ax, there is a class of all and only those objects x satisfying the condition. In symbols the assumption can be expressed as follows: ðCP2 Þ ð9yÞðx1 Þ; . . . ; ðxn Þð5x1 ; . . . ; xn 4 2 y :: AÞ;

where y is not free in the formula A. Letting A be the condition x ¼ x, our naive assumption assures the existence of a universal class V, since every object is self-identical. Letting Ax be the condition x 6¼ x, our naive assumption assures that there is an empty class  because no object satisfies the condition of non-self-identity. Letting Ax be the condition x ¼ V, there is a class V whose only member is V. We now have V 2 V and V 2 V. Russell revealed that our naive assumption is contradictory. Letting Ax be x 2 = x, we get the Russell class r of all and only those classes that are not members of themselves. This yields a contradiction, namely, r 2 r .. r 2 = r. The naive theory of classes is little more than an extensional shadow of the logical conception of an attribute. On the naive view of attributes, every condition comprehends an attribute which is exemplified by all and only those objects that satisfy the condition ðCPÞ 45

ð9’Þðx1 Þ; . . . ; ðxn Þð’ðx1 ; . . . ; xn Þ :: AÞ;

A. J. Ayer, ‘‘Bertrand Russell as a Philosopher,’’ in The Meaning of Life (New York: Macmillan, 1990), p. 152.

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where ’ is not free in the formula A. There is an attribute ’V of being selfidentical, the attribute ’ of being non-self-identical, the attribute y of being identical to ’, and so on. On this view, attributes are intensional entities. That is, there are attributes y for which the coexemplification of attributes ’ and y, i.e., (x)(’x « yx), does not assure that y (’)  y (y). It does not assure it because coexemplifying attributes do not assure their identity. Classes, on the other hand, are extensional. If classes z and y have all the same members, i.e., (x)(x 2 y .. x 2 z) then they are identical. Hence, for any class w, we have y 2 w .. x 2 w. This difference can, however, be minimized. One can emulate extensionality within a theory of attributes. Russell showed the way. For any context y which is such that (x)(’x  yx) does not entail y (’) « y (y), we can appeal to (CP) to find an attribute , which is such that ðyÞððyÞ  ð9GÞðGz z yz : &: ðGÞÞÞ:

Hence we now have that (x)(’x  yx) does entail (’)  (y). In this way all theorems of the naive theory of classes simply shadow theorems of the logic of attributes. Since attributes in intension seem to be purely logical objects, the theory of classes (its extensional shadow) is a logical notion of a class. Of course, since the logical notion of a class simply shadows the logical theory of attributes, one should expect Russell’s paradox of classes to shadow a paradox of attributes. This is, in fact, what Russell discovered. The naive assumption of attributes is contradictory. We need only consider the attribute R that all and only those attributes have that do not exemplify themselves. That is, from (CP) one readily arrives at: ðyÞðRðyÞ  :RðRÞÞ:

In Russell’s view, the paradoxes show that the naive logical assumption of attributes in intension is on a par with the naive logical assumption of classes. Logic must get along without either assumption. The paradoxes require a genuine solution, and a solution can only be found by a reconceptualization of logic itself. It might be thought that mathematics does not rest on the ontology of logical objects such as attributes (or classes in the logical sense). This is hard to reconcile with the thesis that mathematics (or at least arithmetic) is necessary. Russell was prepared to accept synthetic necessities known a priori, but only if those necessities are logical necessities.46 In Russell’s view the only necessity is logical necessity. Modern set theories reject the logical notion of an attribute (or class) and instead postulate the existence 46

Russell, The Principles of Mathematics, p. 457.

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of nonlogical entities called ‘‘sets.’’ If arithmetic truth depends on sets, then arithmetic truths are either contingent truths about sets, or metaphysical (nonlogical) synthetic necessities governing such sets. Russell found both unacceptable. To see things right, one must find a solution of the paradoxes of attributes (and classes) by means of a deeper understanding of the nature of logic. The logical notion of an attribute (or class) seems impossible to maintain. Yet if one does not maintain it, mathematical necessity seems unintelligible.47 Frege’s Begriffsschrift (1879) had argued in favor of a new way of looking at logic – a way that made logic a genuine science. He rejected the long-held dogma famously recounted by Kant that logic can never be informative, that the conclusion of a deduction can never go beyond what is contained in the premises. Frege offered a new quantificational form for logic according to which functions of level n þ 1 ‘‘mutually saturate’’ with functions of level n. In Frege’s view, functions are unsaturated entities and this makes them distinct from objects. In Frege’s terminology, the first-level function f ‘‘falls within’’ (or ‘‘mutually saturates’’) the universal quantifier function (8x)’x. This is a new logical form, quite different from the notion of an object’s exemplification of an attribute. Frege embraces levels of mutual saturation. For instance, the quantifier function (8x)’x falls within the quantifier function (8’)M’. Frege’s notation demands that function signs always appear in function (predicational) positions. In this way, the notation respects his thesis that functions are entities, not objects. If Frege’s ontology of functions is part of logic, then logic is informative. The quantificational structures given in the premises of a deduction can yield entirely new information in the conclusion. In Frege’s view, arithmetic is founded upon these quantificational structures – arithmetic is a branch of logic. Frege’s new conception of logic as the science of functions is consistent. But Frege’s Grundgesetze der Arithmetic (1898) introduced a theory which collapsed the hierarchy of levels of functions. Each function of level n þ 1 is correlated with a function of level n, and in turn, each level-1 function is correlated with an object Frege calls its ‘‘course-of-values.’’ Frege’s philosophy of arithmetic was wedded to courses-of-values, and this was its Achilles heel. In a letter of 1901, Russell broke the bad news to Frege that he had found a paradox of classes which has an analog for the Grundgesetze theory of courses-of-values. 47

Russell’s paradoxes are quite different from the old conundrum about the barber of a village who shaves all and only those of the village that do not shave themselves. There is no ‘‘paradox’’ of such a barber because there is no reason whatever to embrace the existence of such a barber.

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Retreat to Frege’s hierarchy of levels of functions first appeared to Russell to be worth investigating. But Russell was not prepared to accept Frege’s doctrine of mutual saturation and the accompanying thesis that functions are entities, not objects. In Russell’s view, logic is a universal science whose laws apply to all objects alike, be they universals, functions, particulars, abstract, or concrete. Russell’s guiding principle is a doctrine that has come to be called his ‘‘doctrine of the unrestricted variables of logic’’: A formal calculus for the science of logic must adopt only one style of genuine variables – entity/individual/object variables.

The calculus for logic cannot embrace different variables for a hierarchy of types of objects. Frege and Russell agreed that logic cannot embrace a hierarchy of levels of objects. But Frege distinguished functions from objects and allowed levels of functions. Russell could not accept this. In Russell’s view, whatever is, is an object. Thus a retreat to Frege’s levels of functions is impossible for Russell. In his search for a deeper understanding of logic, Russell’s recognized that there is no way to find principles of pure logic which exclude the problematic cases such as Russell attribute R (or the class r) and yet retain attributes (or classes) as objects for the construction of mathematics. The lesson Russell drew from the paradoxes is that the calculus for logic must proceed without the ontological assumption that every open formula comprehends an attribute. It must also proceed without the ontological assumption that every open formula comprehends a class of just those entities that satisfy the formula. By December of 1905 Russell had used his new theory of definite descriptions to form a wonderfully unique reconstruction of the first principles of logic. The reconstruction enables one to emulate a type-theory of attributes without its ontology. It is a eliminativistic reconstruction, not reductive identity, that Russell had in mind when he said that classes are ‘‘logical fictions.’’ Contrary to Ayer, classes do go the way of the gorgon sisters. From 1903 until 1907, Russell regarded logic as the general science of propositional structure. No paradox plagues an ontology of propositions understood as mind- and language-independent states of affairs. Propositions are natural candidates as purely logical objects – candidates which could be used to emulate a logical theory of types (or levels) of attributes (and accordingly classes in the logical sense). Russell called this new reconstruction of logic his ‘‘substitutional theory.’’ The substitutional theory is type-free and makes no assumption that every condition determines an attribute (or a class). This does not mean that the substitutional theory denies that there are universals (attributes) which have both a predicable and an individual/objectual nature. Quite the contrary, the theory maintains that there is a universal (a logical relation) of ‘implication’ which

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holds between certain propositions. The point of the substitutional theory is to generate mathematics without comprehension principles such as (CP) and (CP2). The substitutional theory emulates a type-stratified theory of attributes (and classes in the logical sense) powerful enough to generate mathematics. This is what Russell meant by speaking of classes as ‘‘logical fictions.’’ By using ‘‘variables with structure’’ Russell emulates a type-theory of attributes in intension (and classes in the logical sense) without its ontology. In this way, Russell hoped to demonstrate that the necessity of mathematical truth is the synthetic logical necessity of the theory of propositional structure. Russell’s substitutional theory was a monumental achievement. It illustrates what Kuhn came later to call a ‘‘paradigm shift.’’48 It offers a genuine solution of the paradoxes plaguing the logical notion of an attribute (and class). Its formal calculus for logic is a genuinely universal calculus in the sense that its language is type-free and adopts only one style of variable (the entity/individual/object variable). Its laws hold of everything, be it a universal, a particular, or whatever. The substitutional language and ontology supplants the language and ontology of a type-stratified theory of attributes in intension (and/or a type-stratified theory of classes and relations-in-extension).The type distinctions that dismantle the paradoxes are built into the formal reconceptualization of logical first principles.49 Classes in the substitutional theory are not identified with any entities countenanced in the supplanting theory. The theory offers a logical analysis and reconstruction that is ontologically eliminative and structurally retentive. In this way, the major successes obtained by appeal to the existence of classes, the positive constructions of Cantor, Dedekind, Weierstrass, and Frege, are retained within substitution. Russell explained that ‘‘the principles of mathematics may be stated in conformity with the theory,’’ and the theory ‘‘avoids all known contradictions, while at the same time preserves nearly the whole of Cantor’s work on the infinite.’’50 The substitutional theory involves, as Russell put it, ‘‘an elaborate restatement of logical principles.’’ The results obtained by appeal to the existence of classes are conceptualized in an entirely new way within the research program of the substitutional theory. There will be some loss – some flotsam – such as Cantor’s transfinite ordinal number !!, the usual generative process for the series of ordinals, and the class of all ordinals. But this loss is to be measured

48 49 50

Thomas Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1962). See Appendix B for a brief sketch of how this is accomplished. Bertrand Russell, ‘‘On ‘Insolubilia’ and Their Solution by Symbolic Logic,’’ in Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 213.

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against the successes of the new program. Indeed, had the program yielded the conceptual successes that Russell had anticipated, present mathematics would regard the notion of a set (or class) as present physics regards phlogiston, caloric fluid, the aether, and other relics of past sciences. The substitutional theory sheds new light on the historical evolution of Principia and on Russell’s philosophy of logical atomism. It reveals the unity of Russell’s philosophical method. Unfortunately, the substitutional theory was largely unknown until the 1970s. The main reason is that Russell himself abandoned substitution sometime late in 1907 because of a paradox of propositions. It was not the semantic Liar Paradox (i.e., the proposition asserting of itself that it is false) that was the source of the difficulty, but a syntactic paradox that was unique to the substitutional theory itself.51 Indeed, the often-repeated interpretation that Russell never distinguished syntactic and semantic paradoxes is little more than myth.52 In 1906, Russell’s solution of the syntactic paradoxes (the paradoxes of classes and attributes) was to build type structure into the formal grammar of the theory of propositions.53 He offered an entirely distinct Tarski-style hierarchy of languages as a solution of the paradoxes (Richard, Ko¨nigDixon, Berry) which are now called ‘‘semantic.’’54 Russell’s propositions were intensional entities, not intentional ones involving semantic notions such as ‘‘truth,’’ ‘‘reference,’’ or ‘‘designation.’’ The paradox that spoiled the substitutional theory is syntactic in nature. Russell came to believe that to avoid the new paradoxes within the substitutional theory he would have to introduce a stratified language with variables restricted to orders. A theory of orders of entities, however, would undermine the legitimacy of the theory’s claim to be a universal science of logic. It is incompatible with Russell’s doctrine of the unrestricted variable. With the substitutional theory largely buried in unpublished manuscripts, Russell’s eliminativistic philosophical method has been lost. This has for years clouded the proper understanding of both Wittgenstein and Russell. In the Principia, Russell hoped to preserve his conception that the calculus for the science of logic must be type- and order-free. In this respect, Russell’s motive for the constructions of Principia are no different

51 52

53

54

Gregory Landini, New Evidence Concerning Russell’s Substitutional Theory (Oxford: Oxford University Press, 1998). See Gregory Landini, ‘‘Russell’s Separation of the Logical and Semantic Paradoxes,’’ in ´ Philippe de Rouilhan, ed., Russell en heritage, Revue internationale de philosophie 3 (2004): 257–294. Russell, ‘‘On ‘Insolubilia’ and Their Solution by Symbolic Logic,’’ p. 209. See also Bertrand Russell, ‘‘On the Substitutional Theory of Classes and Relations,’’ in Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 186. Landini, ‘‘Russell’s Separation of the Logical and Semantic Paradoxes.’’

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than for the constructions of the substitutional theory. In fact, Russell intended Principia to be more radically eliminative than the substitutional theory! In Principia, logic is to proceed without the blanket assumption of attributes, classes, or even propositions. Principia espouses a logical reconstruction which endeavors to philosophically explain and justify order indices on predicate variables. The explanation builds the structure of order into a nominalistic semantics which generates the truth-conditions for formulas – a recursive correspondence theory of truth based on Russell’s ‘‘multiple-relation theory of judgment.’’55 In short, type and order indices on the predicate variables of the language of Principia are explained away as ‘‘limitations built into the conditions of significance’’ of the use of the predicate variables. Principia’s order\types are given by an informal semantics for predicate variables. In Russell’s view, Principia is not a theory of types and orders of entities/objects. Russell’s ‘‘supreme maxim’’ for philosophy, his program for a new scientific philosophy based on logical form, evolved from this work in logic. In Russell’s view, the logical paradoxes (of classes and predication), and conundrums such as Kant’s antinomies and Zeno’s paradoxes of rational dynamics, are produced because ordinary grammar has not done justice to logical form. Russell’s work in logic is certainly not a form of empiricism. It is a method of supplanting one language and ontology by another, recovering the structure of the former without the ontological assumptions that generate philosophical paradoxes. Russell described his technique as ‘‘building structure into the variables.’’ That is, variables for classes, functions, and the like are not genuine variables but are syntactically structured so that the correct logical form is revealed. Ordinary language, of course, contains simple predicates for many concepts, and their grammar makes them appear as if they stand for (simple) properties like any other. Russell’s program challenged this. Once logical form is revealed, the conundrums, paradoxes, and speculative metaphysical doctrines surrounding these notions fall away. ´ e.´ He was enthralled by Wittgenstein was Russell’s student and proteg the ideas of Russell’s new eliminativistic approach toward finally solving the enduring conundrums of philosophy. Enthusiastic and brash, Wittgenstein hoped to extend Russell’s ideas widely – all philosophical problems would be solved, he thought, by means of eliminativistic analyses. Philosophical problems and riddles are generated by taking notions such as ‘‘identity,’’ ‘‘universal,’’ ‘‘particular,’’ ‘‘fact,’’ ‘‘truth,’’ ‘‘falsehood,’’ 55

The individual variables of Principia are treated objectually, however. Universals (properties and relations in intension) have both an individual and a predicable nature and are counted among individuals.

Rereading Russell and Wittgenstein

23

‘‘necessity,’’ ‘‘possibility,’’ ‘‘belief,’’ and nonextensional contexts generally, as if they were genuine. The proper role of philosophical analysis is to reveal logical form through eliminative conceptual analysis. The logical and semantic notions used in ordinary language have been misconstrued by philosophers, and their mistakes have led them to speculative metaphysics. Logical and semantic notions are, as Wittgenstein wrote in the Tractatus, ‘‘formal concepts and are represented in conceptual notation by variables’’ (TLP 4.1272). The suggestion that they be represented by ‘‘variables’’ is directly connected with Russell’s eliminativistic techniques for solving the problems facing logicism by ‘‘building structure into the variable.’’ Viewing Russell’s philosophy as eliminativistic creates a powerful new tool for a deeper understanding of Wittgenstein’s Tractatus. New and striking connections between Russell’s philosophy and the Tractatus are illuminated. With Russell’s nonempiricist methods revealed, we shall see how misleading it is to characterize his logical atomism as espousing a form of British empiricism which uses logic as a tool for its empiricist reductions. Attention to the historical evolution of Principia reveals that Russell worked to provide a genuine solution of the paradoxes by means of a theory of logical form. He distilled from this work a new research program which makes logical construction the essence of philosophy. In Russell’s program, philosophy involves the elucidation and analysis of logical form. It is quite misleading to characterize Wittgenstein’s Tractatus as being antithetical to this program. Wittgenstein was Russell’s ally. Russell knew that some of his own constructions for logicism, and certainly those for the foundation of the sciences, were incomplete. Wittgenstein was working to perfect and complete them.

2

Logical atomism

In 1918 Russell gave a series of lectures in London with the title ‘‘Logical Atomism.’’1 He glowingly praised Wittgenstein, writing that the lectures are largely concerned with explaining certain ideas which I learnt from my friend and former pupil Ludwig Wittgenstein. I have had no opportunity of knowing his views since August 1914, and I do not even know whether he is alive or dead. He has therefore no responsibility for what is said in these lectures beyond that of having originally supplied many of the theories contained in them.2

Russell was eager to launch his new program for philosophy as the science of logical form and to introduce Wittgenstein. But in his efforts to establish a reputation for Wittgenstein, he embellished matters greatly. Wittgenstein’s ideas were at a very immature stage during his conversations with Russell between 1912 and 1914, and his studies on logic in Norway were poorly developed. Wittgenstein’s ‘‘Notes Dictated to G. E. Moore in Norway’’ (dated April 1914) did not reach Russell until April 1915. The war intervened, and Russell lost contact with Wittgenstein. He did not hear of him again until June 1919. When Russell finally did get news, he found that Wittgenstein had managed to write a book while in the Austrian army. With the war at an end, Wittgenstein wrote a letter to Russell from an Italian prison camp in Cassino. In it, he announced that he had reached solutions to the problems he had been working on: I’ve written a book called Logisch-philosophische Abhandlung3 containing all my work of the past 6 years. I believe I’ve solved all our problems finally . . . This may 1 2 3

The lectures started on 22 January 1918 and ended on 12 March. They were published in The Monist 28 (Oct. 1918): 495–527; 29 (Jan., Apr., July 1919): 32–63, 190–222, 345–80. The Monist 29 (Apr. 1919): 205. Interestingly, the title is very similar to Johann Heinrich Lambert’s 1782 Logische und philosophische Abhandlungen. Wittgenstein later changed the title at the suggestion of Moore. Lambert wrote voluminously on logic, but his most important essays are in this volume which is composed of a series of studies with later ones referring to numbered results of the earlier. In C. I. Lewis’s A Survey of Symbolic Logic (Berkeley: University of California Press, 1918), p. 4, Lambert’s work is characterized as part of a tradition in logic and

24

Logical atomism

25

sound arrogant, but I can’t help believing it . . . But it upsets all our theory of truth, of classes of numbers and all the rest.4

The letter makes it clear that Wittgenstein took the Tractatus to be addressing ‘‘our’’ problems – problems shared with Russell. Unfortunately, this point has been forgotten, and in the absence of an understanding of the complex historical development of Russell’s philosophy, Wittgenstein’s views have come to be interpreted as offering a unique form of logical atomism opposed to Russell. As we shall see, there was only Russell’s logical atomism and Wittgenstein’s apprenticeship. Logical atomism as a research program One of the earliest appearances of the phrase ‘‘logical atomism’’ occurs in Russell’s 1914 paper ‘‘On Scientific Method in Philosophy.’’ Russell calls his philosophical approach ‘‘logical atomism’’ to contrast it with those philosophical systems, exemplified by philosophers such as Hegel, who were (as Russell sees it) inspired by religious and ethical desires to build a metaphysics tailored to human nature. ‘‘The essence of philosophy,’’ writes Russell, is analysis, not synthesis. To build up systems of the world, like Heine’s German professor who knit together fragments of life and made an intelligible system out of them, is not, I believe, any more feasible that the discovery of the philosopher’s stone. What is feasible is the understanding of general forms, and the division of traditional problems into separate and less baffling questions.5

To emphasize this piecemeal approach, Russell proclaims that there are no propositions of which the ‘universe’ is the subject. He writes: ‘‘The philosophy which I wish to advocate may be called logical atomism or absolute pluralism, because, while maintaining that there are many things, it denies that there is a whole composed of those things.’’6 Russell focuses on logical analysis, rejecting metaphysical speculations inspired by religious, moral, or self-oriented conceptions of what the world must be like. The following passage nicely captures this attitude: Intellectually, the effect of mistaken moral considerations upon philosophy has been to impede progress to an extraordinary extent. I do not myself believe that philosophy can either prove or disprove the truth of religious dogmas, but ever

4 5 6

arithmetic leading to Cantor, Dedekind, Frege, and Russell – a tradition distinguished from the algebraic logic of Boole which reached its perfection in the work of Peirce and especially in Ernst Schro¨der’s Vorlesungen u¨ber die Algebra der Logik (1890). Bertrand Russell, The Autobiography of Bertrand Russell, vol. 2, 1914–1944 (Boston: Little, Brown & Co., 1968), p. 162. Bertrand Russell, ‘‘On Scientific Method in Philosophy,’’ in Mysticism and Logic (Totowa, N.J.: Barnes & Noble, 1976), p. 84. Ibid.

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Wittgenstein’s Apprenticeship with Russell

since Plato most philosophers have considered it part of their business to produce ‘‘proofs’’ of immortality and the existence of God . . . In order to make their proofs seem valid, they have had to falsify logic, to make mathematics mystical, and to pretend that deep-seated prejudices were heaven-sent intuitions. All this is rejected by the philosophers who make logical analysis the main business of philosophy . . . For this renunciation, they have been rewarded by the discovery that many questions formerly obscured by the fog of metaphysics, can be answered with precision, and by objective methods . . . Take such questions as What is number? What are space and time? What is mind, and what is matter? I do not say that we can here and now give definitive answers to all these ancient questions, but I do say that a method has been discovered by which, as in science, we can make successive approximations to the truth.7

In his logical atomism lectures, Russell used the metaphor of a zoological taxonomy of logical forms to describe his atomism. ‘‘I think one might describe philosophical logic,’’ he wrote, ‘‘as an inventory, or if you like a more humble word, a ‘zoo’ containing all the different forms that facts may have.’’8 Russell characterized logic as having two parts: The first part investigates what propositions are and what forms they may have; this part enumerates the different kinds of atomic propositions, of molecular propositions, of general propositions, and so on. The second part consists of certain supremely general propositions which assert the truth of all propositions of a certain form. This second part merges into pure mathematics, whose propositions all turn out, on analysis, to be such general formal truths. The first part, which merely enumerates forms, is the more difficult, and philosophically the more important; and it is recent progress in this first part . . . that has rendered a truly scientific discussion of many philosophical problems possible. The problem of the nature of judgment or belief may be taken as an example of a problem whose solution depends upon an adequate inventory of logical forms. We have already seen how the supposed universality of the subject–predicate form made it impossible to give a right analysis of serial order, and therefore made space and time unintelligible. But in this case it was only necessary to admit relations of two terms. The case of judgment demands the admission of more complicated forms.9

Russell calls his new philosophy a logical atomism because it ‘‘is concerned with the analysis and enumeration of logical forms.’’ Moreover, because it essentially involves the analysis of logical form, Russell concludes that philosophy becomes indistinguishable from logic. In Our Knowledge of the External World (1914) he writes that ‘‘every philosophical problem, when it is subjected to the necessary analysis and purification, is found either to 7 8 9

Bertrand Russell, A History of Western Philosophy (New York: Simon & Schuster, 1945), p. 835. Russell, ‘‘The Philosophy of Logical Atomism,’’ in Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. Robert C. Marsh (London: Allen & Unwin, 1924), p. 216. Bertrand Russell, Our Knowledge of the External World (London: Allen & Unwin, 1926), p. 67. (First published 1914.)

Logical atomism

27

be not really philosophical at all, or else to be, in the sense in which we are using the word, logical.’’10 Logic is the essence of Russell’s philosophy. In his paper ‘‘Logical Atomism’’ (1924) Russell makes no mention of Wittgenstein’s Tractarian ideas, but instead traces the origin of his philosophy to Principia. Russell writes: One very important heuristic maxim which Dr. Whitehead and I found, by experience, to be applicable in mathematical logic, and have since applied in various other fields, is a form of Ockham’s razor. When some set of supposed entities has neat logical properties, it turns out, in a great many instances, that the supposed entities can be replaced by purely logical structures without altering any of the detail of the body of propositions in question. This is an economy, because the entities with neat logical properties are always inferred, and if the propositions in which they occur can be interpreted without making this inference, the ground for the inference fails, and our body of propositions is secured against the need for a doubtful step.11

In Our Knowledge of the External World, Russell heralds Frege’s analysis of the notion of cardinal number as ‘‘the first complete example’’ of ‘‘the logical-analytic method in philosophy.’’12 But he also discusses Principia on classes, rational numbers, and real numbers (construed as lower sections of Dedekind cuts), and Whitehead on points and instances in physics. He greatly admires Cantor on infinity and continuity, and Weierstrass on the notion of the ‘‘limit’’ of a function.13 Their studies eventuated in new logical analyses of these notions. With Cantor, the notion of continuity which seemed impossible to render by any notion of magnitude, depends only on the notion of order. The derivative and the integral became, through the new definitions of ‘‘number’’ and ‘‘limit,’’ not quantitative but ordinal concepts. Continuity lies in the fact that some sets of discrete units form a dense compact set. ‘‘Quantity,’’ wrote Russell, ‘‘has lost the mathematical importance which it used to possess, owing to the fact that most theorems concerning it can be generalized so as to become theorems concerning order.’’14 Weierstrass had banished appeals to infinitesimals in the calculus. He showed that the notion of the ‘‘limit’’ of a function, which used to be understood in terms of quantity, as a number to which other numbers in a series generated by the function approximate as nearly as one pleases, should be replaced by a quite different ordinal notion. In Russell’s view, Cantor’s work on the transfinite put to rest centuries of speculative 10 11 12 13

14

Russell, Our Knowledge of the External World, p. 42. Bertrand Russell, ‘‘Logical Atomism,’’ in Logic and Knowledge, p. 326. Russell, Our Knowledge of the External World, p. 7. Bertrand Russell, ‘‘Mathematics and the Metaphysicians,’’ in Mysticism and Logic and Other Essays (Totowa, N. J.: Barnes & Noble, 1976), pp. 59–74. (First published with the title ‘‘Recent Work in the Philosophy of Mathematics’’ in The International Monthly, 1901.) Ibid.

28

Wittgenstein’s Apprenticeship with Russell

metaphysics surrounding the ‘‘infinite’’ and the notion ‘‘continuity.’’ Russell writes: ‘‘Continuity had been, until he [Cantor] defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics . . . By this means a great deal of mysticism, such as that of Bergson, was rendered inadequate.’’15 In his 1924 paper ‘‘Logical Atomism,’’ Russell added yet more examples of logical analyses. He includes his neutral monism which constructs both physical continuants and conscious minds as series of events, and he includes his ‘‘perdurance’’ (or four-dimensional) theory of time according to which ordinary enduring physical objects are logical constructions of phases. In A History of Western Philosophy (1945), Russell even includes Einstein on space-time and the new theory of quantum mechanics as an examples of the analytic methods of logical atomism. ‘‘Physics,’’ Russell tells us, ‘‘as well as mathematics, has supplied material for the philosophy of philosophical analysis . . . What is important to the philosopher in the theory of relativity is the substitution of space-time for space and time.’’ Turning to quantum theory, Russell writes: ‘‘I suspect that it will demand even more radical departures from the traditional doctrine of space and time than those demanded by the theory of relativity.’’16 Conspicuous by its absence from Russell’s many examples of his program in philosophy is any mention of empiricism. An empiricist epistemology involving a ‘‘meaning analysis’’ or reduction of theoretical terms and sentences to their evidentiary conditions in sense experience (sensedata) is certainly not the paradigm. Nonetheless, a longstanding interpretation defines Russell’s logical atomism as a form of empiricistic reduction, with acquaintance as its fundamental feature. Pears writes: ‘‘Logical atomism’’ is Russell’s name for the theory that there is a limit to the analysis of factual language, a limit at which all sentences will consist of words designating simple things . . . His theory of knowledge led him to claim that the only simple particulars that we know are sense-data, and that the only simple qualities and relations we know are certain qualities and relations of sense-data. Their simple qualities and relations are those with which we have to achieve acquaintance in order to understand the words designating them. This fixes the character of his logical atomism. It is a version of empiricism and it uses a criterion of simplicity based on the exigencies of learning meanings . . . The doctrine of forced acquaintance is the foundation of Russell’s logical atomism.17

Russell allegedly operates with a criterion of simplicity that allows him to identify logical atoms as things with which we are familiar, namely 15 16 17

Bertrand Russell, A History of Western Philosophy (Simon & Schuster, 1945), p. 829. Ibid., p. 833. David Pears, The False Prison, vol. 1 (Oxford: Oxford University Press, 1987), p. 63.

Logical atomism

29

sense-data and their properties.18 The foundation of Russell’s atomism, on this view, is an empiricist principle of acquaintance. It takes sense-data as its logical atoms, and offers an empiricistic construction of factual language. On this view, acquaintance also plays a role in Russell’s taxonomy or inventory of logical forms. For example, Pears characterizes Russell’s ‘‘inventory’’ metaphor as the thesis that philosophy comprises a body of knowledge that is gained by exploring a ‘‘second world,’’ a Platonic heaven of logical entities.19 Pears writes: A short answer to the question about the relation between Wittgenstein’s view of logic in the Tractatus and Russell’s . . . is that they were really opposed to one another. Russell believed that the logician’s task is to carry out a survey of ‘‘logical objects,’’ some of which are forms while others are the real counterparts of the logical connectives. Wittgenstein’s view was that there are no logical objects, . . . and logical connectives do not stand for anything in the world.20

On Pears’s reading, Russell arrives at his inventory of logical forms by appeal to acquaintance with other-worldly logical entities. In contrast, Pears characterizes Wittgenstein’s atomism as unconcerned with epistemology and sense-data. Wittgenstein endeavors to set ‘‘limits of thought’’ (from within) and show the ‘‘logical scaffolding’’ of thought, language, and the world. Anscombe made the point earlier: Russell was thoroughly imbued with the traditions of British empiricism. Wittgenstein’s admirers have generally been like Russell in this, and have assumed that Wittgenstein was too; therefore they have had assumptions about what is fundamental in philosophical analysis that were quite out of tune with the Tractatus.21

Anscombe is quite correct that it is questionable history to assume that the ‘‘objects’’ of Wittgenstein’s Tractatus are Russellian sense-data.22 An identification of the elementary (atomic) propositions of the Tractatus with sense-data statements (or with phenomenalistic observation statements) is not well supported by the textual evidence. But her point does not at all reveal a difference between Russell’s logical atomism and the Tractatus. For it is every bit as questionable to import empiricist (and phenomenalistic) epistemology into Russell’s conception of logical atomism. For a time, Russell’s epistemology adopted sense-data, but he was not an empiricist. Russell had very strong nonempiricist leanings because of his logicism. In fact, in The Problems of Philosophy (1912) he thought that 18 21 22

Ibid., p. 68. 19 Ibid., p. 22. 20 Ibid., p. 25. G. E. M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 2nd ed. (New York: Harper & Row, 1959), p. 14. There has been a recent revival of this early interpretation of the Tractatus. See Merrill Hintikka and Jaakko Hintikka, Investigating Wittgenstein (Oxford: Basil Blackwell, 1986).

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Wittgenstein’s Apprenticeship with Russell

he had achieved a new compromise between empiricism and rationalism. He allows for intuitive (synthetic and a priori) knowledge of relations between universals.23 This, he hopes, provides the ground not only for knowledge of logic and mathematics, but also for synthetic a priori knowledge of the principle of induction, and even some ethical principles.24 Logical atomism is not a form of empiricism. Russell’s Our Knowledge of the External World bears a resemblance to the constructions of empiricists such as Berkeley, Hume, and the phenomenalism of C. I. Lewis’s Mind and World Order.25 But the resemblance is superficial. In stark contrast to traditional empiricism and especially phenomenalism, Russell’s analysis begins with what is physical, not with what is mental. Indeed, it seems likely that Russell always intended his sense-data to be physical entities, though not bits of ‘‘matter’’ (physical continuants that persist through time). In ‘‘The Relation of Sense-Data to Physics’’ and in Our Knowledge of the External World, Russell was explicit that sense-data are physical.26 In The Problems of Philosophy, however, matters are more difficult. Russell then accepted a more traditional distinction between the mental and the physical. He hints that sense-data are not mental but rather physical appearances, which, because they depend both upon the sensory apparatus of an observer’s body and her position in space, are likely to be private and ephemeral.27 Sense-data are the objects of sensations (which are mental), and they serve as the physical evidentiary data for inference concerning ‘‘physical objects’’ (i.e., continuants which persist through space-time). In Our Knowledge of the External World, Russell explains that Whitehead’s work on points and instances in space prompted him to avoid the view espoused in Problems that inferences from the data of sense grounds knowledge of the existence of material continuants.28 Instead of inference, Russell offers an eliminativistic reconstruction, supplanting the language and ontology of material continuants (matter) by a new theory of relations among physical appearances (sense-data) and recovering the structure of physical theory of matter (continuants) within it. By 1915, Russell is beginning to chip away at the traditional distinction of mental and physical. Let me quote at some length from Russell’s letter to the editor of The Journal of Philosophy, Psychology and Scientific Methods: 23 24 25 26 27 28

Bertrand Russell, The Problems of Philosophy (London: Oxford University Press, 1912), pp. 73–74, 103. Ibid., p. 112. C. I. Lewis, Mind and World Order (New York: Charles Scribner’s Sons, 1929). Bertrand Russell, ‘‘On the Relation of Sense-Data to Physics,’’ in Mysticism and Logic and Other Essays, p. 111. (First Published in Scientia, January 1914.) Russell, The Problems of Philosophy, p. 41. See Russell, Our Knowledge of the External World, p. 8.

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In a quotation from the Athenaeum printed in this Journal, I am represented as having said, ‘‘there may be perspectives where there are no minds; but we cannot know anything of what sort of perspectives they may be, for the sense-datum is mental.’’ I did not see the Athenaeum, and do not remember what I said, but it cannot have been what I am reported as having said, for I hold strongly that the sense-datum is not mental – indeed my whole philosophy of physics rests on the view that the sense-datum is purely physical . . . A particular which is a datum is not logically dependent upon being a datum. A particular which is a datum does, however, appear to be causally dependent upon sense-organs and nerves and brain. Since we carry those about with us, we cannot discover what sensibilia, if any, belong to perspectives from places where there is no brain. And since a particular of which we are aware is a sense-datum, we cannot be aware of particulars which are not sense-data, and can, therefore, have no empirical evidence as to their nature.29

Russell says explicitly that sense-data are physical. He then goes on in his letter to imagine how one might go about defining ‘‘mental.’’ Noting that a world without mind contains no relations such as perceiving, remembering, desiring, enjoying, and believing, he writes: ‘‘This suggests that no particulars of which we have experience are to be called ‘mental,’ but that certain facts, involving certain relations, constitute what is essentially mental in the world of experiences.’’30 In Russell’s view, the traditional notion of what is mental is questionable. Russell soon adopted neutral monism and abandoned his distinction between the physical sense-datum and the mental sensation. He endeavored to transcend the traditional distinction between mind and matter, constructing both as continuants in time built out of orderings of the physical events that are their phases. Acquaintance is not the foundation of Russell’s logical atomism. In the logical atomism lectures, Russell embraces acquaintance with complexes (facts) that have universals and particulars among their constituents. Thus, being acquainted with an object is not sufficient for that object’s being a logical simple (atom). Neither is acquaintance a necessary condition for logical simplicity. Russell allows that there might be logical atoms with which we are not acquainted. Consider the following passage from 1914: Sense-data at the times when they are data are all that we directly and primitively know of the external world; hence in epistemology the fact that they are data is allimportant. But the fact that they are all we directly know gives, of course, no presumption that they are all that there is. If we could construct an impersonal metaphysics independent of the accidents of our knowledge and ignorance, the privileged position of the actual data [sense-data] would probably disappear, and 29

30

Bertrand Russell, letter to the editor published in The Journal of Philosophy, Psychology, and Scientific Methods 12 (1915): 391–392. Reprinted in The Collected Papers of Bertrand Russell, vol. 8, The Philosophy of Logical Atomism and Other Essays, ed. John G. Slater (London: Allen & Unwin, 1986), p. 87. Ibid.

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Wittgenstein’s Apprenticeship with Russell

they would probably appear as a rather haphazard selection from a mass of objects more or less like them. In saying this, I assume only that it is probable that there are particulars with which we are not acquainted. Thus the special importance of sensedata is in relation to epistemology, not to metaphysics. In this respect, physics is to be reckoned as metaphysics: it is impersonal, and nominally pays no special attention to sense-data. It is only when we ask how physics can be known that the importance of sense-data re-emerges.31

Russell’s logical atomism leaves the question as to what are to be the logical atoms open for investigation. Logical atoms, Russell tells us, are what he calls ‘‘particulars,’’ and these are terms of relations (or properties) in atomic facts. Russell writes: ‘‘It remains to be investigated what particulars you can find in the world, if any. The whole question of what particulars you actually find in the real world is a purely empirical one which does not interest the logician as such.’’32 Russell makes no dogmatic claim about what must be the logical atoms found at the end point of logical analysis. His reply to Urmson in My Philosophical Development (1959) expresses the point emphatically: As regards simples, I can see no reason either to assert or to deny that they may be reached by analysis. Wittgenstein in the Tractatus and I on occasion spoke of ‘‘atomic facts’’ as the final residue in analysis, but it was never an essential part of the analytic philosophy which Mr. Urmson is criticizing to suppose that such facts were attainable.33

In the atomism lectures, Russell entertained the possibility that simples (or logical atoms) might never be reached. ‘‘I think it is perfectly possible,’’ he wrote, ‘‘to suppose that complex things are capable of analysis ad infinitum, and that you never reach the simple. I do not think it is true, but it is a thing that one might argue certainly.’’34 Russell did think that the endpoint of analysis was epistemically attainable, but not that it must be. In Human Knowledge: Its Scope and Limits (1948), he wrote: there is nothing erroneous in an account of structure which starts from units that are afterwards found to be themselves complex. For example, points may be defined as classes of events, but that does not falsify anything in traditional geometry, which treated points as simples. Every account of structure is relative to certain units which are, for the time being, treated as if they were devoid of structure, but it must never be assumed that these units will not, in another context, have a structure which it is important to recognize.35

31 32 33 34 35

Russell, ‘‘On the Relation of Sense-Data to Physics,’’ p. 110. Russell, ‘‘The Philosophy of Logical Atomism,’’ in Logic and Knowledge, p. 199. Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 221. ‘‘The Philosophy of Logical Atomism,’’ p. 202. Bertrand Russell, Human Knowledge: Its Scope and Limits (New York: Simon & Schuster, 1948), p. 252.

Logical atomism

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In My Philosophical Development he explained the point further: ‘‘I believed originally with Leibniz that everything complex is composed of simples, and that it is important in considering analysis to regard simples as our goal. I have come to think, however, that although many things can be known to be complex, nothing can be known to be simple.’’36 Acquaintance does not determine what must be the simples of logical atomism. In truth, Russell’s notion of acquaintance underwent rather dramatic changes as his logical analyses expanded. The significance of the changes is not always manifest in Russell’s writings. In My Philosophical Development, he wrote: I have maintained a principle, which still seems to me completely valid, to the effect that, if we can understand what a sentence means, it must be composed entirely of words denoting things with which we are acquainted or definable in terms of such words. It is perhaps necessary to place some limitation upon this principle as regards logical words – e.g., or, not, some, all. We can eliminate the need of this limitation by confining our principle to sentences containing no variables and containing no parts that are sentences. In that case, we may say that, if our sentence attributes a predicate to a subject or asserts a relation between two or more terms, the words for the subject or for the terms of the relation must be proper names in the narrowest sense.37

Russell formulates the principle to exempt words for logical particles and words for properties and relations. Understanding, in those cases, is a much more complicated matter. Complicated indeed. Radical changes occurred in Russell’s conception of ‘‘acquaintance.’’ In ‘‘On Denoting’’ (1905), the principle of acquaintance is stated as follows: In every proposition that we can apprehend (i.e., not only in those whose truth or falsehood we can judge of, but in all that we can think about), all the constituents are really entities with which we have immediate acquaintance.38

In his ‘‘Knowledge by Acquaintance and Knowledge by Description’’ (1910–1911) and in Problems (1912), Russell renders the principle as follows: Every proposition which we can understand must be composed wholly of constituents with which we are acquainted.39

The word ‘‘proposition’’ wants clarification in these passages. In ‘‘On Denoting,’’ Russell embraced an ontology of atomic, molecular, and 36 38 39

My Philosophical Development, p. 165. 37 Ibid., p. 169. ‘‘On Denoting,’’ in Essays in Analysis, ed. D. Lackey (London: Allen & Unwin, 1973), p. 119. ‘‘Knowledge by Acquaintance and Knowledge by Description,’’ in Mysticism and Logic, p. 159; The Problems of Philosophy, p. 58.

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even general propositions (as mind- and language-independent, obtaining or non-obtaining, states of affairs). The logical particle ‘‘’’ stands for a relation of ‘implication’ between propositions. But propositions in this sense were abandoned in Principia (1910), never to reappear in Russell’s philosophy. Quite obviously the notion of a ‘‘proposition’’ changes in Russell’s formulations of the principle of acquaintance after Principia. Moreover, by the time of the logical atomism lectures, acquaintance with a universal is no longer construed simply as a relation with a universal as one of its relata. ‘‘Understanding a predicate,’’ Russell writes, ‘‘is quite a different thing from understanding a name . . . To understand a name you must be acquainted with the particular of which it is a name, and you must know that it is the name of that particular. You do not, that is to say, have any suggestion of the form of a proposition, whereas in understanding a predicate you do.’’40 Russell had come to maintain that universals have only a predicable nature. Consequently, words for universals (properties and relations) must always occur in predicate positions and never in subject positions.41 Pears is mistaken when he says that Russell’s logical atomism adopted an epistemic principle of acquaintance that ‘‘fixes the nature’’ of logical atoms as sense-data. In fact, Russell abandoned sense-data in 1918 when he embraced neutral monism, and this occurred only a few months after completing the atomism lectures. In the atomism lectures themselves, Russell was on the verge of embracing neutral monism. Consider the following: I feel more and more inclined to think that it [neutral monism] may be true. I feel more and more that the difficulties that occur in regard to it are all of the sort that may be solved by ingenuity . . . One is the question of belief and the other sorts of facts involving two verbs. If there are such facts as this, that I think may make neutral monism rather difficult, but as I was pointing out, there is the theory that one calls behaviorism, which belongs logically with neutral monism, and that theory would altogether dispense with those facts containing two verbs, and would therefore dispose of that argument against neutral monism.42

In My Philosophical Development, Russell recalled that his adoption of neutral monism required the abandonment of a conscious subject or ‘self’ and this put an end to his theory that ‘sense-data’ are the immediate objects of mental acts of sensation. Russell came to hold that it is no longer possible to distinguish the cognitive act of sensation from the physical sense-datum. To take one of Russell’s examples, the sensation that we have when we see a patch of color simply is that patch of color, an actual 40 42

‘‘The Philosophy of Logical Atomism,’’ p. 205. Ibid., pp. 279–280.

41

Ibid., pp. 205, 225–226.

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constituent of the physical world, and part of what physics is concerned with.43 The results of this change were striking. Russell explained that ‘‘new problems, of which at first I was not fully conscious, arose as a consequence of the abandonment of ‘sense-data.’ Such words as ‘awareness,’ ‘acquaintance,’ and ‘experience,’ had to be re-defined, and this was by no means an easy task.’’44 Russell attempts to replace ‘acquaintance’ and offers an analysis of knowing in sympathy with behaviorism which he regards as the natural ally of neutral monism.45 Neutral monism is, in fact, a theory within logical atomism. Undue emphasis on acquaintance obscures this. Tully writes: ‘‘The most notable feature of Russell’s conversion to neutral monism from logical atomism (as he called his position in 1914) was the abandonment of acquaintance or awareness as the cornerstone of his metaphysics and epistemology.’’46 If a primitive relation of acquaintance with sense-data were definitive of Russell’s logical atomism, then Russell’s adoption of neutral monism would have to be a ‘‘conversion’’ in the sense of an abandonment of logical atomism in favor of neutral monism. But logical atomism was not abandoned in favor of neutral monism. Russell makes it explicit that neutral monism is a theory within logical atomism.47 Russell had been grappling with James’s 1903 paper ‘‘Does Consciousness Exist?’’ for many years. James rejected the relational character of sensation. He contended that those who still cling to consciousness as an entity ‘‘are clinging to a mere echo, the faint rumor left behind by the disappearing ‘soul’ upon the air of philosophy.’’ The phrase ‘‘neutral monism’’ suggests that both mind and matter are composed of some ‘‘neutral stuff’’ which, in itself, is neither. Spinoza held that there is only one substance and that laws of mind and laws of matter are two among infinitely many different systems of attributes under which it can be comprehended. But Russell’s neutral monism was more directly connected with his logical analysis of space and time. In The Principles of Mathematics, Russell offered a logical reconstruction of the notion of ‘‘matter’’ in rational dynamics, abandoning the notion of a substance persisting through time, and offering new accounts of motion and continuous change informed by Weierstrass’s work on limits.48 Russell concluded that Zeno would be right to conclude that change is not a transition in time from one state to a consecutive state. Russell explains that ‘‘Weierstrass, by strictly banishing all infinitesimals, has at last shown that we live in an 43 45 46 47 48

My Philosophical Development, p. 101. 44 Ibid. Bertrand Russell, Philosophy (New York: W. W. Norton & Co., 1927), pp. 67, 129, 172. Russell did not embrace behaviorism, however. Robert Tully, ‘‘Three Studies of Russell’s Neutral Monism,’’ Russell 13 (1993): 7. See, for instance, Philosophy, p. 248; History of Western Philosophy, pp. 812, 833; ‘‘Logical Atomism,’’ p. 342. Bertrand Russell, The Principles of Mathematics, 2nd ed. (Oxford: Blackwell, 1937), p. 471.

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unchanging world, and that the arrow, at every moment of its flight, is truly at rest. The only point where Zeno probably erred was in inferring (if he did infer) that, because there is no change, therefore the world must be in the same state at one time as at another.’’49 In Principles, Russell accepted a perdurance theory of time according to which events stand in temporal ordering relations. He rejected the theory that material objects endure, change their properties in a (consecutive) succession of moments. When Russell came to adopt neutral monism he welded it to his theory of time. Russell’s early views on the philosophy of space and time are important, for they explain how he came to weld his neutral monism with Einstein’s theory of relativity. Tully surmises that Russell’s neutral monism evolved significantly. He argues that it is only in late incarnations of the theory that Russell came to think of his ‘‘neutral stuff’’ in terms of physical events of space-time.50 To be sure, in Russell’s The Analysis of Mind (1921) the connections to his perdurance theory of time are not made, though Russell’s preface hints that his neutral monism is a form of physicalism.51 But neutral monism is as much a philosophy of matter (physical continuants changing through time) as it is a philosophy of mind. It is not surprising that Russell would offer neutral monism as a philosophical ally of general relativity. In his essay ‘‘Logical Atomism,’’ Russell summarized the position on matter and time as follows: Every event has to a certain number of others a relation of ‘compresence’; from the point of view of physics, a collection of compresent events all occupy one small region in space-time. One example of a set of compresent events is what would be called the contents of one man’s mind at one time – i.e., all his sensations, images, memories, thoughts, etc., which can coexist temporally . . . We will define a set of compresent events as a ‘minimal region.’ We find that minimal regions form a fourdimensional manifold, and that, by a little logical manipulation, we can construct from them the manifold of space-time that physics requires. We find also that, from a number of different minimal regions, we can often pick out a set of events, one from each, which are closely similar when they come from neighboring regions, and vary from one region to another according to discoverable laws. These are the laws of the propagation of light, sound, etc. We find also that certain regions of spacetime have quite peculiar properties; these are the regions which are said to be occupied by ‘matter.’ Such regions can be collected, by means of the laws of physics, into tracks or tubes, very much more extended in one dimension of space-time than in the other three. Such a tube constitutes the ‘history’ of a piece of matter; from the point of view of the piece of matter itself.52 49 50 51 52

Ibid., p. 347. R. E. Tully, ‘‘Russell’s Neutral Monism,’’ in Nicholas Griffin, ed., The Cambridge Companion to Bertrand Russell (Cambridge: Cambridge University Press, 2003), p. 359. Bertrand Russell, The Analysis of Mind (London: Allen & Unwin, 1921), pp. 5f. ‘‘Logical Atomism,’’ p. 342.

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The ‘‘neutral stuff’’ are events which compose space-time.53 It is worth quoting as well from Russell’s book Philosophy (1927): The physical world, both through the theory of relativity and through the most recent doctrines as to the structure of the atom, has become very different from the world of everyday life, and also from that of the scientific materialism of the eighteenth-century variety. No philosophy can ignore the revolutionary changes in our physical ideas that the men of science have found necessary; indeed it may be said that all traditional philosophies must be discarded . . . The main point for the philosopher in the modern theory is the disappearance of a ‘thing.’ It has been replaced by emanations from a locality.54 In a word, ‘matter’ has become no more than a convenient shorthand for stating certain causal laws concerning events . . . But if matter has fared badly, mind has faired little better. The adjective ‘mental’ is one which is not capable of any exact significance. There is, it is true, an important group of events, namely percepts, all of which may be called ‘mental.’ But it would be arbitrary to say that there are no ‘mental’ events except percepts, and yet it is difficult to find any principle by which we can decide what other events should be included. Perhaps the most essential characteristics of mind are introspection and memory. But memory in some of its forms is, as we have seen, a consequence of the law of conditioned reflexes, which is at least as much physiological as psychological, and characterizes living tissue rather than mind. Knowledge, as we have found, is not easy to distinguish from sensitivity, which is a property possessed by scientific instruments. Introspection is a form of knowledge, but turns out on examination to be little more than a cautious interpretation of ordinary ‘knowledge.’ Where the philosopher’s child at the Zoo says ‘‘There is a hippopotamus over there,’’ the philosopher should reply: ‘‘There is a coloured pattern of a certain shape, which may perhaps be connected with a certain system of external causes of the sort called ‘hippopotamus.’’’ (I do not live up to this precept myself.) In saying that there is a coloured pattern, the philosopher is practicing introspection in the only sense I can attach to the term, i.e., his knowledge-reaction is to an event situated in his own brain from the standpoint of physical space . . . Thus ‘mind’ and ‘matter’ are merely approximate concepts, giving convenient shorthand for certain approximate laws. In a completed science, the word ‘mind’ and the word ‘matter’ would both disappear, and would be replaced by causal laws concerning ‘events,’ the only events known to us otherwise than in their mathematical and causal properties being percepts, which are events situated in the same region as a brain and having effects of a particular sort called ‘knowledge-reactions.’55

According to Russell’s neutral monism, selections of events have orderings that constitute a physical continuant in space-time; other selections have 53 54

See Bertrand Russell, ‘‘Philosophy in the Twentieth-Century’’ (1924), The Collected Papers of Bertrand Russell, vol. 8, p. 465. Philosophy, p. 97; see also pp. 146, 213f., 276f. 55 Ibid., p. 280.

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orderings that constitute a conscious mind. Russell’s ‘‘neutral stuff’’ are events; space-time is itself a construct of such events, and in this sense they are ‘‘neutral’’ between matter (physical continuants) and mind (conscious entities). Among the orderings relevant to the series of events that constitute a conscious mind there are those which, as a series, involve ‘‘physiological inference’’ (reaction patterns grounded in associative learning). ‘‘Knowing,’’ on Russell’s account, is an activity – a way of reacting to the environment.56 Russell’s neural monism involved a naturalization project that anticipates some of the most contemporary naturalization studies in the philosophy of mind. In view of the many different versions of acquaintance in Russell’s philosophy, it is clear that an empiricist principle that a mind (in the traditional sense) is acquainted with a sense-datum (as a mental entity) cannot play the role Pears assigns to it in defining logical atomism. Indeed, no version of Russell’s principle of acquaintance can play the role of defining his logical atomism. Why then does ‘acquaintance’ seem to have such a central position in Russell’s logical atomism lectures? Russell discusses ‘acquaintance’ in the lectures because he was imagining what a logically perfect or ideal and complete theory – the endpoint of an analysis and reconstruction – would be like. The syntactic forms found in Principia were taken as a guide to the sort of syntactic forms that would be present in the language of the perfect theory. Of course, the final and complete theory might contain linguistic forms not in Principia’s formal language because it would adopt new non-logical axioms appropriate to the contingent theories of physics, mind, psychology, etc., couched within it. Unlike Principia, the language of the perfect theory might contain genuine (logically proper) names aside from variables, and adopt primitive predicate constants and function constants pertaining to these various applied fields.57 Russell imagines that, as a necessary condition for understanding the ideal theory, one would have to have acquaintance with the universals indicated by the primitive predicate constants and the objects that are the referents of the logically proper names used in the contingent empirical theories that compose the ideal theory. ‘‘All analysis,’’ he says, ‘‘is only possible in regard to what is complex, and it always depends, in the last analysis, upon direct acquaintance with the objects which are the meanings of certain simple symbols.’’58 Russell therefore speculates in his logical

56 57 58

Ibid., p. 17. ‘‘The Philosophy of Logical Atomism,’’ p. 201. In Principia the only singular terms are variables. Ibid., p. 194.

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atomism lectures that the logically proper names of an ideal theory of knowledge and theory of mind would be indexicals like ‘‘this’’ and ‘‘that,’’ which refer to ephemeral momentary experiences – sensible particulars such as a patch of color or a sound. These, together with acquaintance with relations between universals (which yield synthetic a priori knowledge), would be the building blocks for the construction of knowledge. Russell’s examples of the philosophy of logical atomism, however, do not corroborate the interpretation that atomism involves ontological reduction to known (epistemic) entities and a zoological inventory of logical forms based on the entities with which a mind may be acquainted. They show that logical atomism was a conception of philosophy as analysis, reconceptualization, and reconstruction. Russell’s atomism was not a theory about an empiricist analysis of factual language – any more than Einstein’s theory of space-time is an analysis of the nineteenth-century factual language of physics or Weierstrass’s reconstruction of limits is an analysis of the use of the word ‘‘limit’’ in the calculus. Logical atomism is a form of structural realism. The ontology of the old theory is abandoned (or obviated), but the reconstruction for it retains (at most) the structures given by the laws of the old ontological framework (just as Maxwell’s equations for electromagnetic waves in an aether are retained in Einstein’s no-aether theory of relativity). In The Analysis of Matter (1927), Russell says that knowledge of physics is confined to structural and mathematical properties, not intrinsic natures of physical substances.59 In his book Philosophy Russell put it starkly: ‘‘Our knowledge of the physical world is purely abstract: we know certain logical characteristics of its structure, but nothing of its intrinsic character.’’60 Russell’s inventory of logical forms takes place only after the eliminativistic analysis and reconstruction of the structures of the old theory. Theories of mind and knowledge are themselves subject to analysis and reconstruction. Russell’s early theory of acquaintance cannot play a role in determining what must be in the ontological inventory, for that very theory later became subject to reconstruction. Russell’s early theory that acquaintance is a dyadic relation between a mind and sense-data, universals, and complexes was not the foundation of his philosophy of logical atomism. It was a feature of an epistemic theory he couched within his logical atomist research program – a theory he abandoned in favor of neutral monism. It is important not to lose sight of this. The lectures on the philosophy of logical atomism are just Russell’s attempts to set out a philosophical

59 60

Bertrand Russell, The Analysis of Matter (London: Kegan Paul, 1927), pp. 264, 287. Philosophy, p. 296.

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method. Subtheories within it, especially epistemic theories concerning sense-data and acquaintance, come and go. We must not hold Russell’s logical atomism hostage to the theses he advanced when working out this or that epistemic theory. Logical atomism is a research program. We must not define logical atomism as if it is wedded to any of the theoretical constructions couched within it – be it Russell’s early theory of acquaintance, Principia’s theory of classes, the multiple-relation theory of judgment, the construction of physical continuants from sense-data, neutral monism, or whatever. Russell’s ideas on epistemology in the logical atomism lectures changed when his eliminativism widened. But in spite of the rather radical changes Russell made from The Problems of Philosophy to Human Knowledge: Its Scope and Limits, none mark any change whatsoever in what he meant by his philosophy of logical atomism. The logical independence of the facts that are truth-makers In order to understand Wittgenstein’s influence on the logical atomism lectures, we must understand some of the technical constructions of Principia’s recursive theory of truth and falsehood. Unfortunately, Principia’s theory of truth has not been well understood. To see why, it is important to recall a bit of the historical development of Russell’s notion of a proposition and its role in his philosophy of logic. Prior to Principia, Russell maintained that logic is the science of propositional structure. Russell’s propositions are states of affairs – considered as intensional objects. That is, propositions may logically imply one another and yet be distinct. Some states of affairs obtain (are true) and others do not obtain (are false). Obtaining (truth) and nonobtaining (falsehood) are accepted as primitive unanalyzable properties. States of affairs are mind- and language-independent entities that contain ordinary objects, rocks, trees, mountains, and the like, as constituents. In Russell’s early philosophy, logic is the general theory of propositions. Propositions are purely logical objects. Hence, Russell accepts the thesis that every well-formed formula of the language of logic can be nominalized to form a singular term for a proposition. Implication is adopted as a primitive logical relation marked by the horseshoe sign. The horseshoe sign is flanked by terms to form a formula. It must not be conflated with the modern horseshoe for ‘‘if . . . then . . .’’ which is flanked by formulas to form a formula. Russell originally used the horseshoe sign to stand for a relation of implication. For clarity of exposition, let us stretch and boldface the horseshoe when it is used as a relation sign. Where ‘‘x’’ and ‘‘y’’ are individual variables, Russell allows

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‘‘x ) y’’ to be a well-formed formula of his language of the logic of propositional structure. Adopting the use of brackets to mark a nominalizing transformation, Russell allows the transformation of the formula ‘‘x ) y’’ into a term ‘‘{x ) y}.’’ This term is read ‘‘x’s implying y’’ and should not be conflated with the formula ‘‘x implies y.’’ The term can then occur in the position of ‘‘y’’ in ‘‘x ) y’’ to form the formula ‘‘x ) {x ) y}.’’ This is read ‘‘x implies x’s implying y.’’ It is convenient, however, to allow subject position to itself mark the nominalizing transformation and use dots as punctuation. In this way, we can write the convenient ‘‘x . ). x ) y’’ instead of the cumbersome ‘‘x ) {x ) y}.’’ On the early version of the theory of propositions, Russell allows the nominalization of any formula, including those containing bound variables of quantification. For instance, the formula ‘‘(x)(x ) y)’’ which says ‘‘everything implies y’’ can be nominalized to make the singular term ‘‘{(x)(x ) y)}’’ which is read ‘‘Everything’s implying y’’. Putting this in the position of ‘‘y’’ in the formula ‘‘x ) y’’ yields the formula ‘‘x ) {(x)(x ) y)}.’’ More conveniently this can be written ‘‘x .). (x)(x ) y).’’ This is read ‘‘x implies everything’s implying y.’’ Inference rules in the logic of propositions must be crafted so as to respect that grammar of the language. One must not let an individual variable such as x occur on a line of a proof. For example, the inference rule modus ponens is not the incoherent rule: From x and x ) y infer y.

It is all too easy to slight this fact by misunderstanding Russell’s use of the letters ‘‘p’’ and ‘‘q’’. One might think it acceptable to formulate the rule of modus ponens as follows: From p and p ) q infer q.

But this is every bit as incoherent as before. Russell uses ‘‘p’’ and ‘‘q’’ as individual variables, not special ‘‘propositional variables.’’ The appearance of legitimacy comes from the modern formulation of modus ponens as From P and P  Q infer Q.

But in the modern use, the letters ‘‘P’’ and ‘‘Q’’ are schematic for formulas and the modern sign ‘‘’’ is flanked by formulas to form a formula. Matters are even more confusing since Principia uses the horseshoe sign ‘‘’’ in the modern sense. Principia abandons Russell’s early ontology of propositions and the signs ‘‘’’ and ‘‘v’’ are adopted as primitive statement connectives. The logical signs of Principia are statement connectives in the modern sense. In Russell’s early logic of propositions, one can offer definitions as follows:

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  ¼ df  ) f f ¼ df ðxÞðyÞðx ) yÞ  &  ¼ df  ð )  Þ:61

Here  and  are any singular terms. This is certainly impossible in a modern sentential calculus. When Russell embraced his early ontology of propositions, he dallied with the theory that belief is a relation between a mind and a proposition. But in 1907 he published a paper entitled ‘‘On the Nature of Truth’’ which explored, without endorsement, the strengths and weaknesses of a ‘‘multiplerelation’’ theory of belief. On the multiple-relation theory, there are no propositions and belief is analyzed in a new way. In Principia, the multiplerelation theory plays a central role in the philosophical explanation of the order component of the order\type indices on the predicate variables of the formal language of the system. Unfortunately, its role in Principia has been missed. For many years Principia has been interpreted as adopting ´ Vicious Circle Principle (VCP) as the justification for embracPoincare’s ing a ramified and type-stratified hierarchy of entities (propositional functions). This interpretation became orthodoxy. The principle is this: (VCP) Whatever involves an apparent [bound] variable must not be a value of that variable.

The orthodoxy holds, in Russell’s view, that the logical paradoxes (of classes and propositional functions) have a source in common with semantic paradoxes such as the Epimenides Liar. Both arise from violation of the VCP. The weakness of this interpretation is that it is incompatible with the historical evolution of ramified types and with central features of Principia’s introduction. Russell in fact separated logical from semantic paradoxes, offering quite different solutions for the kinds of paradoxes in 1906.62 Principia itself offers philosophical justifications for ramified type structure quite independently of the VCP. Principia’s formal language adopts order\type indices on its variables. The type component is justified in the introduction by a ‘‘Direct Inspection Argument.’’ The order component is justified by means of a recursively defined hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ whose foundation is the multiple-relation theory of judgment.

61 62

Gregory Landini, Russell’s Hidden Substitutional Theory (New York: Oxford University Press, 1998). I use a boldface & to distinguish it from the statement connective conjunction. See Gregory Landini, ‘‘Russell’s Separation of the Logical and Semantic Paradoxes,’’ in ´ Philippe de Rouilhan, ed., Russell en heritage, Revue internationale de philosophie 3 (2004): 257–294.

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The Direct Inspection Argument assumes a nominalistic semantics for Principia’s predicate variables. Whitehead and Russell write: a direct consideration of the kind of functions which have functions as arguments and the kinds of functions which have arguments other than functions will show, if we are not mistaken, that not only is it impossible for a function ’^ y to have itself or anything derived from it as argument, but that if y y^ is another function such that there are arguments a with which both ‘‘’a’’ and ‘‘ya’’ are significant, then y y^ and anything derived from it cannot significantly be arguments to ’^ y. This arises from the fact that a function is essentially an ambiguity, and that, if it is to occur in a definition proposition, it must occur in such a way that the ambiguity has disappeared, and a wholly unambiguous statement has resulted.63

The idea is that a predicate variable can occur in a subject position (argument position) of another predicate variable only if in the semantics this position represents a predicate position in a formula. According to the nominalistic semantics, the type component of the order\type indices on predicate variables indicates the nature of the formulas of the language that can stand in for it. Whitehead and Russell give the following example: Take, e.g., ‘‘x is a man,’’ and consider ‘‘is a man.’’ Here there is nothing definite which is said to be a man. A function, in fact, is not a definite object which could be or not be a man; it is a mere ambiguity awaiting determination, and in order that it may occur significantly it must receive the necessary determination.64

The notion that a function is an ‘‘ambiguity awaiting determination’’ seems obscure until we realize that this is simply an archaic phrase for the notion that a predicate variable is undetermined until given an assignment (‘‘determination’’) in the semantic interpretation of the formal language. Whitehead and Russell are speaking about predicate variables and are concerned with the semantics. If a nominalistic semantics were to assign the formula ‘‘. . . is a man’’ to the variable ’ and ‘‘. . . is mortal’’ to the variable y, then it would assign the ungrammatical ‘‘. . . is a mortal is a man’’ to ’(y). However, if it assigns ‘‘Every thing is such that . . . it . . .’’ to ’, then the result is the grammatical ‘‘Everything is such that it is mortal.’’ Type superscripts on ’((o)) and y (o) track the semantic requirement that the sort of formula that the semantics can assign to y (o) must occupy a predicate position in the formula assigned to ’((o)). By appeal to this informal semantics, Principia endeavors to philosophically justify and explain the type component of the order\type indices on its predicate variables.65

63 64

A. N. Whitehead and Bertrand Russell, Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925), vol. 1, p. 47. Ibid., p. 48. 65 A formal definition of order\type is given in Chapter 5.

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The introduction of Principia explicitly abandons propositions. It adopts a multiple-relation theory of judgment and a hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ Writing on behalf of the orthodoxy, Goldfarb says that the multiple-relation theory ‘‘plays no role in Russell’s explanation of his formal system.’’66 Church went so far as to say that the multiple-relation theory was a late addition not well integrated into the system of Principia. He holds that an ontology of propositions is ‘‘clearly demanded by the background and purpose of Russell’s logic . . . in spite of what seems to be an explicit denial by Whitehead and Russell in Principia.’’67 In truth, Principia’s explanation of types and orders is not founded upon the VCP. Russell regards the VCP as a regulative principle which guides the search for a solution. The solution offered in Principia is its thesis that only individual variables of the work are genuine. Predicate variables are given a nominalistic semantics. This generates a philosophical explanation of the type component of the order\type indices adorning the predicate variables. To philosophically justify the order component of the order\type indices, Russell offers a ‘‘no-propositions’’ recursive definition of ‘‘truth’’ and ‘‘falsehood.’’68 The abandonment of propositions plays a central role in Russell’s philosophical justification of the order component of the order\type indices on the predicate variables of Principia. To appreciate this, however, it is important to pause to consider Russell’s reasons for abandoning propositions. Some interpretations maintain Russell abandoned propositions because of his growingly robust sense of reality – his stand against Meinongian objects of intentional inexistence such as the round square. Others find troubles with Russell’s account in The Principles of Mathematics of the unity of a nonobtaining (false) proposition.69 The search for such reasons is, however, in vain. The propositions Russell embraced prior to Principia were not postulated to account for the intentionality of mental acts. What Meinong calls an ‘‘objektiv,’’ on the other hand, is postulated as part of an account of intentionality (the aboutness of the mental). To be sure, in Principles Russell embraced a theory according to which there are denoting concepts. When a denoting concept occurs predicatively (if you will) in a proposition,

66

67 68 69

Warren Goldfarb, ‘‘Russell’s Reasons for Ramification,’’ in Wade Savage and C. Anthony Anderson, eds., Rereading Russell, Minnesota Studies in the Philosophy of Science 12 (Minneapolis: University of Minnesota Press, 1989), pp. 24–40. Alonzo Church, ‘‘Comparison of Russell’s Resolution of the Semantic Paradoxes with that of Tarski,’’ Journal of Symbolic Logic 41 (1976): 747–760. Landini, Russell’s Hidden Substitutional Theory. See, for instance, Bernard Linsky, ‘‘Why Russell Abandoned Russellian Propositions,’’ in A. D. Irvine and G. A. Wedeking, eds., Russell and Analytic Philosophy (Toronto: University of Toronto Press, 1993), pp. 193–209.

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Russell says that the proposition is ‘‘about’’ the object(s) denoted by the concept.70 But Russell hastens to add that this sense of ‘‘aboutness’’ is a purely logical notion, not a psychological or mental notion.71 It is not, therefore, a notion appropriate for an account of the intentionality of the mental. The logical notion of aboutness is not what Meinong (and Brentano) took as the mark of the mental. In ‘‘On Denoting,’’ Russell abandoned denoting concepts and with them he abandoned his early theory that there is a logical notion of aboutness. He still retained his early theory that belief is a dyadic relation between a mind and a proposition. But this too would soon be jettisoned. By 1906, Russell was imagining doing without general propositions. Only quantifier-free formulas (of the formal language) can be nominalized to form terms for propositions. Hence, belief, in the case when something general is believed, cannot be construed as a dyadic relation to a proposition. There is nothing Meinongian whatever in the postulation of false (nonobtaining) propositions as part of logic as the science of structure. Nor is there any problem Russell saw with the notion of the unity of a proposition. According to Russell, universals (properties and relations) have a predicable nature. The unity (and so the existence) of a proposition, whether obtaining or nonobtaining, is due to there being a universal occurring predicationally in it. The existence of the proposition is not to be conflated with the truth (or obtaining) of the proposition. With an ontology of propositions (states of affairs) assumed, truth (obtaining) and falsehood (nonobtaining) must be taken as primitive unanalyzable properties.72 The abandonment of propositions, on the other hand, opens the way to defining notions of ‘‘truth’’ and ‘‘falsehood.’’ Russell’s technical conception of a fact, invented for his new multiple-relation theory, is itself part of the abandonment of propositions. The notion of a fact is part of the analysis of ‘‘truth’’ as correspondence. The truth of a belief, in the simplest case, is analyzed as the existence of a fact that corresponds to the belief. The falsehood of the belief is the absence of a corresponding fact. The occurrence of a relating relation in a fact does not make the fact true. The notion ‘‘true fact’’ is incoherent. The occurrence of the relating relation is what makes it the case that there is such a fact. Facts and propositions are like men and dinosaurs: they never coexisted. Why then did Russell come to abandon his early theory of propositions in Principia? In his paper ‘‘On the Nature of Truth,’’ Russell explained that 70 72

The Principles of Mathematics, pp. 53, 64. 71 Ibid., p. 53. Russell likens them to the whiteness and redness of roses. See Bertrand Russell, ‘‘Meinong’s Theory of Complexes and Assumptions,’’ Essays in Analysis, ed. D. Lackey (London: Allen & Unwin, 1973), p. 75.

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the chief difficulty in embracing belief as a relation between a mind and a proposition is the existence of paradoxes ‘‘analogous to that of the Liar.’’73 Russell’s comment is odd. The propositional Epimenides Liar is the contingent paradox of the existence of a person who asserts that all propositions he believes are mistaken, and whose other beliefs are indeed all mistaken.74 Why would the Epimenides provide reason for abandoning propositions? In fact, Russell had considered and put aside worries about a propositional Epimenides paradox as early as June 1905 in a manuscript entitled ‘‘On Fundamentals.’’ Russell’s theory of propositions is well insulated against any propositional liar paradox because ‘belief,’ ‘assertion,’ and the like are not part of the logic of propositions. In papers of 1905 and 1906 on the theory of propositions, Russell separated logical and semantic paradoxes and offered different solutions for them.75 No semantic notions are involved in the pure language of Russell’s logic of propositions. And certainly nothing in the logic of propositions demands that belief or assertion is a relation between a mind and a proposition. Unfortunately, this is rarely appreciated. For example, Goldfarb holds that paradoxes akin to the Epimenides Liar can be generated in Russell’s logic of propositions. Goldfarb attempts to formulate a paradox with (x)(Ax )  x), maintaining that we need only let A be a formula that is uniquely satisfied by the proposition {(x)(Ax )  x)}.76 But any such attempt along these lines is doomed to failure. The letter A is being used as a stand-in for some formula of the object-language of the theory. But there is no formula of the object-language of Russell’s quantificational logic of propositions that can play the role Goldfarb assigns to it. In fact, the Russellian logic of propositions (i.e., quantification theory, including quantification over propositions) is quite consistent.77 What, then, was the paradox of propositions ‘‘analogous to the Liar’’ that Russell had in mind? The answer was long buried in Russell’s unpublished manuscripts. Russell had worked out an extension of his quantificational logic of propositions to form a substitutional theory of propositional structure. 73 74 75

76 77

Bertrand Russell, ‘‘On the Nature of Truth,’’ Proceedings of the Aristotelian Society 7 (1907–1908): 28–49. This differs from the Statement Liar – the paradox that arises with the statement, ‘‘This sentence is false.’’ See Bertrand Russell, ‘‘On the Substitutional Theory of Classes and Relations,’’ manuscript received by the London Mathematical Society on 24 April 1905. Published in Essays in ´ Analysis, pp. 165–189. See also Russell’s ‘‘Les paradoxes de la logique,’’ Revue de metaphysique et de morale 14 (1906): 627–650. The English manuscript is titled ‘‘On ‘Insolubilia’ and Their Solution by Symbolic Logic’’ and is printed in Essays in Analysis, pp. 190–214. Goldfarb, ‘‘Russell’s Reasons for Ramification,’’ p. 29. The proof of consistency simply tracks the proof of the consistency of modern quantification theory.

Logical atomism

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The theory began in 1905 on the heels of the theory of definite descriptions, and it lasted in one form or another until Russell’s official abandonment of propositions in 1910. Russell’s substitutional theory is entirely type-free, and adopts only one style of variables – namely, individual variables. The theory is able to emulate the structures of a type-stratified language of predicate variables (with such variables allowed in subject and well as predicate positions). That is, it recovers the structure of a simple typetheory of attributes in intension without assuming that every open formula comprehends an attribute. In this way, Russell hoped that the substitutional theory would save the logicist program by providing a genuine solution of Russell’s paradoxes of classes and attributes. In 1906, however, Russell discovered a new paradox of propositions unique to substitution. I have called it Russell’s ‘‘po/ao paradox.’’ It was this paradox that caused him to abandon propositions.78 Russell’s po/ao paradox of substitution involves no semantic notions such as ‘‘truth’’ or ‘‘designation.’’ Nonetheless, Russell thought it is ‘‘analogous’’ to the semantic paradox of Epimenides. That is, he thought that by investigation of solutions of the Epimenides he might find a solution of the paradox of substitution. Ultimately, he concluded that the only way to salvage the substitutional theory from the po/ao paradox is to adopt a theory whose genuine variables come with indices appropriate to different ‘‘orders’’ of propositions. This led Russell to abandon the substitutional theory. The theory had succeeded in emulating a type-stratified theory of classes within a type-free calculus that has only individual/entity variables. But retrofitting the substitutional theory with order indices on its individual/entity variables proved unacceptable to Russell. Just as the substitutional theory built types into logical form of a no-logical-assumptionof-attributes/classes (‘‘no-classes’’) theory, Russell endeavored to find some deeper analysis that would build the structure of types of attributes and the structure of orders of propositions into the grammar of a ‘‘no-logical-assumption-of-attributes/classes’’ and ‘‘no-propositions’’ theory. The desire to avoid an ontology of orders of propositions and emulate the structure of orders by recursive definitions of ‘‘truth’’ and ‘‘falsehood’’ motivated Russell to abandon the substitutional theory. Principia offers a recursive definition of ‘‘truth’’ and ‘‘falsehood.’’ The recursive definition philosophically explains the order component of the order\type indices on the individual and predicate variables of the formal language. The notion of ‘‘truth’’ as applied to syntactically complex formulas is defined in terms of a notion of ‘‘truth’’ as applied to less syntactically complex formulas. By

78

Landini, Russell’s Hidden Substitutional Theory.

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appealing to the truth conditions involved in a nominalistic semantics for predicate variables, Principia endeavors to make both order and type part of the significance conditions of its ‘‘structured’’ variables. Principia’s multiple-relation theory of judgment plays a central role in the atomic (base) case of the recursive definition of ‘‘truth’’ and ‘‘falsehood.’’ The recursive definition of ‘‘truth’’ and ‘‘falsehood’’ is intended to provide a means of recovering the structural characteristics (though not the ontology) of a ramified hierarchy of orders of propositions in an order-free (and type-free) no-propositions theory. Central to the recursive definition is Russell’s abandonment of his former ontology of propositions and conversion to the view that the logical particles are statement connectives; they are not signs that stand for relations or properties (of propositions). In Principia, Whitehead and Russell write: Owing to the plurality of the objects of a single judgment, it follows that what we call a ‘proposition’ (in the sense in which this is distinguished from the phrase expressing it) is not a single entity at all. That is to say, the phrase that expresses a proposition is what we call an ‘incomplete’ symbol; it does not have meaning in itself, but requires some supplementation in order to acquire a complete meaning.79

‘‘A proposition,’’ Whitehead and Russell continue, ‘‘like such phrases as ‘the so and so,’ where grammatically it appears as subject, must be broken up into its constituents if we are to find the true subject or subjects.’’80 In ‘‘On the Nature of Truth,’’ Russell intimates that a statement flanked by ‘‘. . . is true’’ is a disguised definite description. He writes that his new theory of ‘‘truth’’ is ‘‘an extension of the principle applied in my article ‘On Denoting’ [Mind, October, 1905].’’81 This strongly suggests that the theory of definite descriptions is directly involved in Russell’s new definition of truth. Russell is not explicit as to how the analysis is ‘‘like’’ the analysis of definite descriptions, nor does he say what definite description is disguised. But his intent seems clear. The recursive characterization of ‘‘truth’’ and ‘‘falsehood’’ is a correspondence theory. The notion of ‘‘truth0’’ applicable to formulas that are atomic (involving no logical constants or quantifiers) is defined in terms of correspondence between a belief and a fact. ‘‘Falsehoodo’’ is defined as the absence of any corresponding fact. When an atomic formula ‘‘R(c1,. . ., cn)’’ is flanked by the expression ‘‘. . . is true0’’ the formula is to be construed as a disguised definite description. It should be noted that unlike modern formal semantic theories, this allows a formula itself and not a name of a formula to be

79 80

A. N. Whitehead and Bertrand Russell, Principia Mathematica to *56 (Cambridge: Cambridge University Press, 1964), p. 44. Ibid., p. 48. 81 Russell, ‘‘On the Nature of Truth,’’ p. 48n.

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flanked by ‘‘. . . is true.’’ To state the base of the recursion generally, we have the following: [R(c1, . . ., cn)] is true if and only if there is a unique fact consisting of the relation R relating c1, . . ., cn in proper order.82

In this way, we can see how the theory of definite descriptions was to play a central role at the base of the recursive definition of ‘‘truth’’ and ‘‘falsehood.’’ Russell does not take up the question of how to formulate a truthdefinition for cases where formulas contain free variables. He did not anticipate Tarski’s ingenious notion of a denumerable sequence of objects (in the domain of an interpretation) satisfying an open formula. Nevertheless, with the base of the recursion for atomic statements established by the multiple-relation theory, he goes on to give truth-conditions for quantified formulas. Whitehead and Russell write: That the words ‘‘true’’ and ‘‘false’’ have many different meanings, according to the kind of proposition to which they are applied, is not difficult to see. Let us call the sort of truth which is applicable to ’a ‘‘first truth.’’ . . . Consider now the proposition (x).’x. If this has truth of the sort appropriate to it, that will mean that every value ’x has ‘‘first truth.’’ Thus, if we call the sort of truth applicable to (x).’x ‘‘second truth,’’ we may define ‘‘[(x) .’x] as second truth’’ as meaning ‘‘every value for ’x has first truth,’’ i.e., ‘‘(x).(’x has first truth).’’83

The passage is informal and sketchy. Moreover, there is no clause for molecular formulas. But the idea seems to be this. Where p is an atomic formula containing exactly n-many subordinate elementary propositional clauses, [ p] truee.n if and only if it is not the case that [p] is truee.n1. Where p and q are atomic statements, [p  q] is truee.n if and only if either [p] is falsee.a or [q] is truee.b, where a and b are some numerals appropriate to the molecular complexity of p and q and such that a þ b ¼ n. Where Ax is a quantifier-free formula, [(x)Ax] is true1.1; e.n if and only if every entity x is such that [Ax] is true1.o; e.n. The truth-conditions given for general statements is important. According to the recursive characterization of the Principia, the truth of a general statement lies in its correspondence with several facts. Whitehead and Russell are explicit: And generally, in any judgement (x). ’x, the sense in which this judgment is or may be true is not the same as that in which ’x is or may be true. If ’x is an elementary

82

83

That is, the fact consisting of the relation R relating c1, . . ., cn in proper order exists. The vexing problem of order which has come to be called the ‘‘direction problem’’ will be addressed anon (pp. 55ff.). Principia Mathematica to *56, p. 42.

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Wittgenstein’s Apprenticeship with Russell

judgment, it is true when it points to a corresponding complex. But (x). ’x does not point to a single corresponding complex: the corresponding complexes are as numerous as the possible values of x.84

The recursive characterization of ‘‘truth’’ and ‘‘falsehood,’’ the abandonment of propositions, and the multiple-relations theory of judgment are central features of Russell’s philosophical explanation of order in the ramified type-theory of Principia. To better understand how the recursion was to be implemented when predicate variables are involved, let us examine the recursion in a bit more detail. The plan is that the order indices on the predicate variables track the hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ as applied to statements (formulas) of a fixed hierarchy of languages. For instance, language L1 contains formulas with bound and free individual variables and no formulas with higher-order variables. Language L2 contains L1 and allows formulas with bound and free predicate variables of order 1, but no higher. To illustrate how order indices on predicate variables are explained in the semantics, let us introduce fine-grained order indices. For instance, instead of Principia’s predicate variable, ’ðoÞ

whose type is (o) and order is 1, consider the following variable: 1:m; e:a ðon oÞ

’

:

This is a predicate variable of type (o) which in the nominalistic semantics is such that only formulas that contain exactly m-many bound individual variables and a-many quantifier-free propositional subordinate clauses can be substituted for its occurrences. Arguments to this predicate variable are individual variables of the form oxo. Following Russell’s example, the recursively defined hierarchy of truth and falsehood contains clauses such as the following. Where A is a well-formed formula containing the variable 1..n; e.b (o\o) ’ free, and where d is the number of occurrences of the bound predicate variable in predicate positions in the formula A, we have [(1.n; e.b ’(o\o))A] is true z.w; . . .; 1..m; e.a iff Every formula A* is true z.w; . . .; 1..m þ (nd); e.a þ (bd) where A* is obtained by replacing each occurrence of the variable 1..n; e.b’(o\o) in A by a formula B containing exactly n-many bound individual variables, and b-many elementary (molecular) subformulas.

84

Ibid., p. 46.

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51

Consider the following example: ½ð1:1; e:1 ’ðonoÞ Þðo xo Þð1:1; e:1 ’ðonoÞ ðo xo Þ  1:1; e:1 ’ðonoÞ ðo xo ÞÞ is true

2:1; 1:1; e:0

Here the formula A is ðo xo Þð1:1; e:1 ’ðonoÞ ðo xo Þ  1:1; e:1 ’ðonoÞ ðo xo ÞÞ

Let B be a formula which contains a free individual variable ovo and which contains exactly one bound variable of order\type o and one elementary (quantifier-free) subformula. For instance, let B be the formula  ðo zo Þð1:0; e:1 Gðono; onoÞ ðo zo ;o  o ÞÞ

The result of the replacement is this:   ðo xo Þ  ðo zo Þð1:0; e:1 Gðono; onoÞ ðo zo ; o xo ÞÞ   ðo zo Þð1:0; e:1 Gðono; onoÞ ðo zo ; o xo ÞÞ

This is formula A* and it is true 2.1–1; 1.1 þ (1  2); e.o þ (1  2). That is, it is true 2.0; 1.3; e.2, for it has two elementary subformulas, three first-order quantifiers, and zero second-order quantifiers. Russell’s recursive definition of ‘‘truth’’ and ‘‘falsehood’’ next applies to the formula A*. But to apply the recursion, this formula must be defined in terms of an equivalent all of whose quantifiers prefix a quantifier-free formula. The definitions that accomplish this in Principia are given in its section *9 which defines subordinate occurrences of quantifiers in terms of formulas with all quantifiers initially placed. The prenex formula for which the truthconditions of A* are to be given is this: ðo xo Þðo zo Þð9o yo Þ 



1:0; e:1

Gðono; onoÞ ðo zo ; o xo Þ  1:0; e:1 Gðono; onoÞ ðo yo ; o xo Þ



Let us call this ‘‘prenex(A*).’’ The recursion then continues, but now objectually instead of nominalistically. The notion that the prenex(A*) is true 2.0; 1.3; e.2 is recursively defined to mean (to use the modern Tarski-style locution) that every entity satisfies (with respect to the variable oxo) the formula ðo zo Þð9o yo Þ 



1:0; e:1

Gðono; onoÞ ðo zo ; o xo Þ  1:0; e:1 Gðono; onoÞ ðo yo ; o xo Þ



(Or, as Russell would put it, this expression is true 2.0; 1.2; e.2 for every entity oxo.) In turn, this is defined to mean that every entity satisfies (with respect to the variable ozo) the formula ð9o yo Þ 



1:0; e:1

Gðono; onoÞ ðo zo ; o xo Þ  1:0; e:1 Gðono; onoÞ ðo yo ; o xo Þ



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where oxo is fixed. (Or, as Russell would put it, this expression, with oxo fixed, is true 2.0; 1.1; e.2 for every entity ozo.) Next, we have that some entity satisfies (with respect to oyo) the expression 



1:0; e:1

Gðono; onoÞ ðo zo ; o xo Þ  1:0; e:1 Gðono; onoÞ ðo yo ; o xo Þ



where oxo and ozo have been fixed as above. It is, in Russell’s locution, true 2.0; 1.0; e.2. To satisfy this formula, of course, is to fail to satisfy the formula 1:0; e:1

Gðono; onoÞ ðo zo ; o xo Þ 1:0; e:1 Gðono; onoÞ ðo yo ; o xo Þ

Or, in Russell’s locution, the above formula is true 2.0; 1.0; e.2 just when this formula is false e.2. And the material conditional is false e.2 just when (assuming oxo, ozo, and oyo are fixed as above 1.0; e.1G(o\o, o\o)) (ozo, oxo) is truee.1 and 1.0; e.1G(o\o, o\o) (oyo, oxo) is falsee.1. At last, the base of the recursion is given by means of Russell’s multiple-relation theory of judgment (discussed anon). Russell had nothing as sophisticated as the formal semantics found in Tarski’s pioneering work in the 1930s. But his intent is clear. The hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ does the work of explaining the order component of the order\type indices on Principia’s predicate variables. Its predicate variables are ‘‘structured’’ by these conditions of significance, and in this respect they are not genuine variables. The only genuine variables of Principia are its individual variables. The discovery that Principia’s hierarchy of senses of ‘‘truth’’ and ‘‘falsehood,’’ not the VCP, is the philosophical explanation of ramification has striking consequences for understanding Russell’s logical atomism and its relationship to Wittgenstein’s Tractatus. Central to the recursive account of truth in Principia is the abandonment of propositions, the abandonment of the thesis that the logical particles stand for relations or properties (as they had in Russell’s logic of propositions), and the adoption of a recursive correspondence theory according to which only ‘‘atomic’’ facts (as it were) are truth-makers. Negated statements are not made true by negative facts, and general statements are not made true by general facts. Indeed, all the facts that are explicitly embraced by Principia are ‘‘atomic.’’ That is, as far as the truth-conditions for the formulas of the formal language of Principia go, the universals that inhere predicatively in the facts that are truthmakers are logically independent. This thesis will at first appear shocking. The predicate variables of Principia have commonly been interpreted to range over a hierarchy of ‘‘propositional functions’’ in intension regimented by type and order, and

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53

these were frequently identified with Russell’s universals.85 On such a view, universals may be logically related. But we now see that Russell did not intend a realist interpretation of Principia’s predicate variables; he intended a nominalistic interpretation of predicate variables (and a realist interpretation of individual variables). In Russell’s intended interpretation, Principia is not a theory postulating the existence of a hierarchy of order- and typestratified entities. The structure of the hierarchy of order\types is built into the nominalistic semantics. To be sure, at the time of Principia, Russell allows that at least some universals are necessary existing logical individuals and he found himself acquainted with several. But the lesson he drew from his paradox of predication was that logic is powerless to offer any general comprehension principles that determine what universals exist. One of the tasks of Principia was to demonstrate how logic can get along without assuming such principles. Universals are in the range of Principia’s individual variables and are not stratified by orders or types. There are no logical relations between the universals that inhere in the facts that provide the truth-conditions for formulas of Principia. If there were, Principia’s recursive definition of ‘‘truth’’ and ‘‘falsehood’’ would collapse. Of course, this does not assure that Russell held that no universals stand in logical relations. But if Russell accepts such universals, they cannot not play any role whatsoever in the facts that provide the truthconditions for formulas of the language of Principia.86 We now see that Wittgenstein’s much celebrated Tractarian thesis of the logical independence of atomic facts was an idea he arrived at by study of Russell’s work in Principia. It is of utmost importance, therefore, to not lose sight of this when evaluating the extent to which Wittgenstein’s ideas are in alliance with Russell’s or in opposition. Acquaintance with logical objects Facts are structured complexes. The structure is very important. Distinct facts can have exactly the same constituents. In Principia, Whitehead and Russell use hyphens to compose names of facts. For instance, the expression ‘‘a-in-the-relation-R-to-b’’, or more tersely ‘‘a-R-b,’’ is used to name a fact. For example, there is ‘‘Desdemona-in-the-relation-loves-to-Cassio’’. A better expression which emphasizes the point that such an expression

85 86

See W. V. O. Quine, ‘‘Russell’s Ontological Development,’’ Journal of Philosophy 63 (1941): 657–667. Observe that Principia’s *13.01, which defines the ‘‘identity’’ sign to mean indiscernibility with respect to predicative propositional functions, does not stand for the universal (relation) identity.

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is a term and not a formula is ‘‘Desdemona’s loving Cassio.’’87 As we see, the quite different fact b-R-a (Cassio’s loving Desdemona) has exactly the same constituents but is structured differently. On Russell’s multiple-relation theory, belief (judgment, understanding) is not a dyadic relation between a mind and a proposition (state of affairs), but a multiple relation whose relata (in the atomic case) include the constituents of a would-be corresponding fact. Russell holds that when a person m has a belief (or makes a discursive judgment), say that a bears R to b, there is a belief (understanding or judgment) complex, a m-believing

R b

consisting of the mind m related by ‘belief’ to the entities ‘loves’ and a and b. This complex (or fact) is structured. We can form a name for this complex as follows: ‘‘m-believing-with-respect-to-(a, R, b).’’

On the other hand, when a person m has a belief that b bears R to a there is a complex (or fact) with the same constituents but structured differently. It has the picture, b m-believing

R a

We can form a name for this complex with ‘‘m-believing-with-respect-to-(b, R, a).’’

Now it is likely that Russell originally thought the definite description that the formula ‘‘aRb’’ disguises when it is flanked by ‘‘. . . is true’’ is

87

Unfortunately, confusion on this point is prevalent. It is found, for example, in Herbert Hochberg, ‘‘Propositions, Truth and Belief: The Wittgenstein–Russell Debate,’’ Theoria 66 (2000): 3–40. See also Thomas Ricketts, ‘‘Wittgenstein Against Frege and Russell,’’ in Erich Reck, ed., From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy (Oxford: Oxford University Press, 2002), pp. 217–251. Russell’s multiple-relation theory is characterized as holding that there is an attribute J of judgment such that ðxÞðyÞðRÞðJðx; y; R; zÞ :: x judges that y bears R to zÞ: Russell is then characterized as abandoning his early view that there is a attribute of judgment J that is a dyadic relation to a proposition, so that ðxÞðpÞðJðx; pÞ :: x judges that pÞ: Russell’s expression ‘‘J(m, a, R, b)’’ thereby gets misconstrued as a formula. Russell intends it to be a term (name) for a judgment-complex.

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‘‘the fact corresponding to m-believing-with-respect-to-(a, R, b).’’

Accordingly, truth consists in there being a complex a-in-the-relation-R-to-b

which corresponds to the belief complex m-believing-with-respect-to-(a, R, b).

That is, there is a fact consisting of a, loves, and b structured in the right way.88 Falsehood consists in there being no corresponding fact. Russell’s multiple-relation theory faces a difficulty that has come to be called the ‘‘narrow direction problem.’’89 G. F. Stout objected that the theory could not distinguish the belief that a loves b from the quite different belief that b loves a.90 It must be understood, however, that the problem is not to explain what distinguishes belief complexes. There is no special difficulty with belief complexes in this regard. On Russell’s view, facts (complexes) are structured entities. The complex a-loving-b differs from the complex b-loving-a because the entities a and b occur in different positions in them. For a similar reason, the belief complexes m-believing-with-respect-to-(a, loves, b) m-believing-with-respect-to-(b, loves, a)

differ in structure. The narrow direction problem does not challenge the assumption that complexes are structured entities. The problem is to explain how it is that the first belief complex (above) points only to a-loving-b as its would-be corresponding fact, while the second points only to b-loving-a as its would-be corresponding fact. Russell was quick to acknowledge the problem. In ‘‘On the Nature of Truth and Falsehood’’ he says that in the belief complex m-believing-withrespect-to-(a, loves, b), the subordinate relation loves ‘‘must not be 88

89

90

Hochberg claims that my account assumes that Russell took ‘correspondence’ as a primitive relation. See Hochberg, ‘‘Propositions, Truth and Belief. See also Herbert Hochberg, ‘‘The Role of Subsistent Propositions and Logical Forms in Russell’s 1913 Philosophical Logic and in the Russell–Wittgenstein Dispute,’’ in Ignacio Angelelli and Maria Cerezo, eds., Studies on the History of Logic: Proceedings of the III Symposium on the History of Logic (Berlin: Walter de Gruyter, 1996), pp. 317–341. I do not take correspondence to be a primitive relation. The multiple-relation theory is the basis of a recursive definition of ‘‘truth’’ and ‘‘falsehood.’’ In the base case of the recursion, the ‘correspondence’ (truth-) conditions of a belief-complex are defined as the existence of a fact consisting of certain constituents of that complex. The expression seems due to Nicholas Griffin. See his ‘‘Russell on the Nature of Logic,’’ Synthese 45 (1980): 117–188. See also Nicholas Griffin, ‘‘Russell’s Multiple-Relation Theory of Judgment,’’ Philosophical Studies 47 (1985): 213–247. G. F. Stout, ‘‘The Object of Thought and Real Being,’’ Proceedings of the Aristotelian Society 11 (1911): 187–208.

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abstractly before the mind but must be before it as proceeding from a to b rather than from b to a.’’ On this view, the subordinate relation as it enters into the belief state has a ‘‘sense’’ and in the corresponding complex it must have the same sense.91 Russell soon came to admit, however, that this proposal does not succeed. As he later put it in his manuscript Theory of Knowledge (1913), relations do not come with a ‘hook’ at one end and an ‘eye’ at another.92 When a relation occurs as a substantive (as is the case when it occurs as a constituent in a belief complex) it is not relating, and the so-called ‘‘sense’’ of the subordinate relation is lost. In The Problems of Philosophy, Russell thought he had the narrow direction problem solved by employing the idea of a partial isomorphism of structure. Though a relation does not come with a hook and an eye, the complex in which it occurs as a relating relation is structured. Russell writes: If Othello believes truly that Desdemona loves Cassio, then there is a complex unity, ‘Desdemona’s love for Cassio,’ which is composed exclusively of the objects of the belief, in the same order as they had in the belief, with the relation which was one of the objects occurring now as the cement that binds together the objects of the belief. On the other hand, if Othello believes falsely, there is no such complex unity composed of the objects of the belief.93

A belief complex such as m-believing-with-respect-to-(a, loves, b) has a structure – an order of occurrence of its constituents generated by the relating relation ‘belief’. It is in virtue of this that a-loving-b is its appropriately corresponding fact, for this fact alone has the constituents a, b, and ‘loves,’ and has a structural ordering of these constituents that is partly isomorphic to their structural order in the belief complex. A picture may help: a

a m-believing

R b

R b

It is the partial isomorphism of structure that assures that the belief complex points to exactly one would-be corresponding complex. In Theory of Knowledge Russell rejected this plan. The notion of the order of the constituents of a complex relies upon a linear or spatial analogy. In

91 92

93

Bertrand Russell, ‘‘On the Nature of Truth and Falsehood,’’ Philosophical Essays (London: Longmans, 1910), ch. 7, p. 158. The Collected Papers of Bertrand Russell, vol. 7, Theory of Knowledge: The 1913 Manuscript, ed. Elizabeth Ramsden Eames in collaboration with Kenneth Blackwell (London: George Allen & Unwin, 1984), p. 86. The Problems of Philosophy, p. 128.

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the complex, a-loving-b, it is not proper to say that the constituent a occurs ‘‘first,’’ and b occurs ‘‘second.’’ Complexes do indeed have structures, but the structure is not always spatial and certainly it does not track the linear order of the expressions of a statement. In Theory of Knowledge, Russell gave a final attempt to solve the narrow direction problem. He introduced what he calls ‘‘position relations.’’ An entity a has a determinate position in a complex even though one may not legitimately order the positions by ‘‘first,’’ ‘‘second,’’ and so on. For example, Russell tells us that in the complex a-similarity-b, there is only one position that both a and b occupy. He calls this complex ‘‘symmetrical.’’ The complex a-loving-b, on the other hand, is such that a and b occupy two distinct positions. The complex is ‘‘unsymmetrical.’’ It is structurally possible for distinct a and b to occupy one another’s positions in the complex. Accordingly, Russell calls it ‘‘homogeneous’’ with respect to these positions. In a complex such as Socrates’ being human, the positions occupied by Socrates and Humanity are quite different. The complex is unsymmetrical. But in this case it is not structurally possible for Socrates to occupy the predicational position of Humanity in the complex. Socrates is not a universal (and so lacks a predicational nature). The complex is both unsymmetrical and heterogeneous (non-homogeneous). Russell calls such complexes ‘‘nonpermutative.’’ A complex is ‘‘permutative’’ with respect to the two of its constituents when those constituents occupy distinct positions and when those positions are homogeneous.94 Russell’s account relies upon type* distinctions.95 Concrete complexes cannot occur as constituents of entities that are not complex, and universals are of a different type* from concrete complexes (facts) and entities that are not complex. Russell’s characterization of unsymmertrical and non-permutative complexes depends upon distinctions of type*. But it is important to note that types* are not the types or orders of Principia. Russell had done away with order\types of entities by building the structure of such a theory into Principia’s informal nominalistic semantics for its predicate variables. The notion of types*, unlike the notion of order\ types, was perfectly legitimate to Russell. Indeed, in Principles Russell held that the difference between universal and particular is an unanalyzable and primitive logical notion. Universals have both a predicable and an individual nature, and this is essential to the viability of the multiple-relation theory quite independently of the direction problem. The difference in 94 95

Collected Papers, vol. 7, pp. 122ff. Pierdaniele Giaretta claims that my account of the multiple-relation theory assumes that Russell excludes type* differences. This is not correct. See Pierdaniele Giaretta, ‘‘Analysis and Logical Form in Russell: The 1913 Paradigm,’’ Dialectica 51 (1997): 273–293.

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type* is not coded into logical grammar. Universals are values of the individual variables (type o) of Principia. In addition to the new notion of an entity’s ‘‘position in a complex,’’ Russell amends the multiple-relation theory in Theory of Knowledge by maintaining that in mental complexes of belief, judgment, and understanding, there occurs a cognitive act of logical intuition (acquaintance) with logical forms. As we saw, Russell tells us that the complex a-similarity-b is symmetrical and therefore has only one position; it is identical with the complex b-similarity-a.96 The complex a-before-b, on the other hand, is distinct from the complex b-before-a, but all these concepts have the same logical form – namely, the form of a dual complex (of two-place predication). Russell maintains that position relations are not determined by the logical form of a complex but by the relating relation of the complex. An object has a position in a complex. The logical form of a complex (fact) is not a template which shows where the terms are to be fitted together to form the purportedly corresponding complex. Logical forms play a different role. They are introduced to provide the content of our understanding of the classification of complexes into those that are ‘permutative’ and those that are ‘nonpermutative.’ Belief or judgment requires an understanding of the form of the purported corresponding complex. In Russell’s view, it is acquaintance with logical form that is essential for, and logically prior to, understanding words such as ‘‘predicate,’’ ‘‘relation,’’ ‘‘dual complex,’’ and logical words such as ‘‘or,’’ ‘‘not,’’ ‘‘all’’, and so on.97 For example, understanding the sentence ‘‘a is red’’ requires more than being acquainted with a, and the universal ‘redness.’ It presupposes an understanding of predication – the form of a monadic complex (fact). According to the multiple-relation theory, when the statement ‘‘a is red’’ is flanked by ‘‘. . . is true’’ it is a disguised definite description. Extrapolating from the modifications of the theory in Theory of Knowledge, the definite description now is this: ‘‘the fact corresponding to m-believing-with-respect-to-(a, redness, P),’’

where P is the logical form of monadic predication. The statement ‘‘a is red’’ (and so also the belief complex) is true if and only if there is a complex consisting of a and the universal Redness – namely, a’s-redness. In this simple case, the direction problem does not arise. The corresponding complexes are non-permutative.

96

Collected Papers, vol. 7, p. 122.

97

Ibid., p. 101.

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In contrast, consider the truth-conditions for ‘‘a loves b.’’ The would-be corresponding complex is permutative, and consequently Russell’s characterization of the notion of ‘‘correspondence’’ is indirect. The truthconditions of this statement cannot be given simply by stating that it is true just when there is a fact consisting of a and b and the relating relation loves. The order is central. To capture the order, Russell makes an appeal to his new view that the relation ‘loves’ fixes two position relations C1 and C2. Accordingly, the statement ‘‘a loves b,’’ when it occurs flanked by the predicate ‘‘is true,’’ disguises the following definite description: ‘‘the  such that a C1  and b C2 .’’98

In short, ½a loves b is true if and only if ð9Þða C1  :&: b C2  : :  ¼ Þ:

At first blush, this may seem to simply push the problem back. The truthconditions for ‘‘a C1 ’’ and ‘‘b C2 ’’ must be given. But according to Russell, these do not involve the direction problem. That is, [a has C1 in ] is true if and only if there is a fact consisting of a and C1 and . The complex asserted to exist is non-permutative. Thus the truth-conditions which involve indirect correspondence with permutative complexes such as ‘a-loving-b’ will be defined recursively from the truth-conditions that involve direct correspondence with non-permutative complexes such as ‘a-C1-’ and ‘b-C2-.’ Russell’s account assumes that there is a type* distinction between a fact (complex)  and an entity a that is not a complex. The relata of a position relation C1 must differ in type*. A fact cannot have a position in an entity that is not complex. With type* distinctions in place and logical forms taken to be constituents of belief complexes, the narrow direction problem cannot arise. Principia’s recursive correspondence theory of truth did not address the problem of the truth-conditions for ascriptions of propositional attitudes. Neither did Russell’s unfinished manuscript Theory of Knowledge. The multiple-relation theory was only needed to set out the truth-conditions of the atomic statements at the base of the recursion. Thus, questions as to the nature of logical forms as objects severally before the mind do not immediately show themselves. Russell does owe an account of belief ascriptions, however. To properly define the correspondence (truth-) conditions of ‘‘S believes that a loves b,’’ Russell must form a definite description which captures the structural order of the belief complex: m-believing-with-respect-to-(a, loves, b, R2). 98

Ibid., p. 147.

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(I use R2 as the logical form of dyadic predication.) Of course, the narrow direction problem arises once again.99 But one can apply Russell’s solution to the narrow direction problem in such cases. We are to find position relations E1, E2, E3, and E4 fixed by the relating relation ‘believes.’ The statement ‘‘S believes that a loves b,’’ when it is flanked by the truth-predicate, would, on this view, be interpreted as the following disguised definite description: ‘‘the fact  such that a has E1 in  and b has E2 in  and ‘loves’ has E3 in  and R2 has E4 in .’’

In this way, Russell hoped at last to have solved the narrow direction problem. The question of the ontological nature of logical forms, however, vexed Russell. This is particularly the case for those grounding our understanding of logical words. In Theory of Knowledge, Russell was tentative when it came to the question of what sort of entities logical forms are to be. In a chapter entitled ‘‘Logical Data,’’ he notes that ‘‘logical objects cannot be regarded as entities, and therefore what we shall call ‘acquaintance’ with them cannot really be a dual relation.’’100 There must be kinds of acquaintance. Universals, concrete particulars, and complexes differ logically, and ‘‘relations to objects differing in logical character must themselves differ in logical character.’’101 Whatever the ultimate analysis, Russell decides that something like logical experience or logical intuition is required for the understanding of logical words like ‘‘particular,’’ ‘‘universal,’’ ‘‘relation,’’ ‘‘dual complex,’’ and ‘‘predicate.’’ Russell regards logical intuition as a kind of immediate knowledge obtained through a process of abstraction and generalization carried to its utmost limit. Every logical notion is a summum genus – the limit of this process of abstraction. Finally, Russell comes to a decision in Theory of Knowledge. ‘‘If possible,’’ he writes, ‘‘it would be convenient to take as the form something which is not a mere incomplete symbol.’’ Logical forms are to be identified as abstract general facts ‘‘with no constituents.’’ Russell explains: the form of all subject-predicate complexes will be the fact ‘something has some predicate’; the form of all dual complexes will be ‘something has some relation to something.’ . . .[T]he logical nature of this fact is very peculiar. For ‘something has some relation R to something’ contains no constituent except R, and ‘something has some relation to something’ contains no constituent at all. It is therefore suitable to serve as the ‘form’ of dual complexes. In a sense, it is simple, since it cannot be analyzed. At first sight, it seems to have structure, and therefore not to be simple; but it is more correct to say that it is a structure.102

99 100

For a dissenting opinion, see Hochberg, ‘‘Propositions, Truth and Belief.’’ Collected Papers, vol. 7, p. 97. 101 Ibid., p. 100. 102 Ibid., p. 114.

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In Russell’s view, abstraction and generalization directs the mind in such a way that it becomes acquainted with a logical form. Construed as abstract facts with no constituents, logical forms are puzzling entities. But in any event, it is clear that logical forms do not contain ontological counterparts of logical particles, quantifiers, or variables: they contain no constituents at all. Russell is explicit. He remarks that ‘‘a molecular form is not even the form of any actual particular; no particular, however complex, has the form ‘this or that,’ or the form ‘not-this’.’’103 Consider the belief ascription made by a person S that m believes that every human is mortal. Extrapolating from the Theory of Knowledge, Russell’s approach would take the ascription to be true in virtue of the existence of the following corresponding belief complex: m-believing-with-respect-to-(humanity, mortality, GEN).

A constituent of this belief state is the logical object GEN, acquaintance with which grounds our a priori knowledge of the logical notion of the subordination of one universal under another. The logical form GEN is to be identified as the abstract general fact some-universal’s-being-subordinate-to-some-universal.

The person believes truly, however, when every human is mortal. The truth-conditions of the statement ‘‘Every human is mortal’’ are rendered by Principia’s recursive definition, not by correspondence with this logical form (construed as an abstract general fact). In this way, Russell’s account of the content of our understanding of logical notions is compatible with Principia’s recursively defined hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ A similar account applies where the understanding of logical particles is involved. Consider an ascription made by a person S that m believes a is red or b is white. The ascription is true in virtue of the existence of a complex belief-fact, namely m-believing-with-respect-to-(a, redness, b, whiteness, OR).

The logical notion OR which is a constituent of this belief state is not a relation that the logical particle ‘‘v’’ stands for. The logical particles of Principia do not stand for relations. They stood for relations when Russell embraced an ontology of propositions. But Principia is a ‘‘no-propositions’’ theory. The logical form OR is rather a logical object, acquaintance with which is to account for our a priori knowledge of the nature of disjunction. Russell was uncertain as to what sort of object it might be, but his

103

Ibid., p. 132.

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comments in Theory of Knowledge suggest that he identifies it with an abstract general fact such as the following: some-universals-being-such-that-one-or-another-of-them-is-exemplified.

It is important to realize that it is not in virtue of the existence of a molecular fact containing an ontological analog of the logical particle ‘‘v’’ that a person m believes truly that a is red or b is white. Rather, the person believes truly because a is red or b is white. That is, the truthconditions of ‘‘a is red or b is white’’ are given in accordance with Principia’s recursive definition. There are no molecular facts containing ontological counterparts of logical particles. Principia’s recursive theory of truth maintains that the existence of many facts is what provides truth-makers for general statements. No general facts are truth-makers. The recursive theory of truth in Principia requires that all facts (that are truth-makers) be atomic and logically independent. The exemplification of one universal (which occurs in such a fact that is a truth-maker) never excludes exemplification of another. The purpose of the multiple-relation theory in Principia was to provide the basis for a hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ The hierarchy depends upon the view that general facts (if there are such) are not truth-makers. Without these features of the recursion, Principia’s philosophical explanation of order indices on its predicate variables would be lost. Russell’s Theory of Knowledge manuscript inclines toward embracing abstract general facts. But in his 1918 logical atomism lectures Russell goes much further. He argues on behalf of both partly general facts and negative facts.104 These positions are quite compatible with Principia’s recursive definitions of ‘‘truth’’ and ‘‘falsehood.’’ Unfortunately, their compatibility has gone largely unnoticed. Interpreters frequently read Russell’s later inclinations toward general facts and negative facts into Principia. The mistake lends itself to the view that in Principia Russell held that the logical particles are relation signs and that general facts are the truth-makers for general statements. The interpretation obliterates the role that the recursive definition of ‘‘truth’’ and ‘‘falsehood’’ plays in Principia’s philosophical explanation of order indices on predicate variables. It ignores the recursive nature of its definition of ‘‘truth’’ and ‘‘falsehood’’ and, accordingly, it must construe the Vicious Circle Principle as providing the philosophical justification for ramified types.105 This undermines a proper understanding of the criticisms Wittgenstein had of Russell’s theory. 104 105

Russell, ‘‘The Philosophy of Logical Atomism,’’ p. 211. Richard McDonough, The Argument of the Tractatus (Albany: State University of New York Press, 1986), p. 19.

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Russell’s version of the multiple-relation theory in his Theory of Knowledge manuscript was surely intended to be consistent with Principia’s recursive theory of truth. In the manuscript, Russell reluctantly embraced a theory that explains knowledge of logical notions as a form of acquaintance between a mind and certain logical forms. He assumes tentatively that there is a faculty of pure logical intuition which acquaints the mind with the logical forms grounding our understanding of notions such as ‘‘all,’’ ‘‘some,’’ ‘‘not,’’ ‘‘or,’’ and ‘‘if . . . then.’’ He reified logical forms, identifying them (albeit tentatively) with abstract general facts. This was his plan for an account of our a priori knowledge of logical notions and for our grasp of abstract logical structure. But Russell’s identification of logical forms with abstract general facts could not reasonably have been entertained by Russell if it undermined the recursive nature of the correspondence theory of truth set forth in Principia. It is perfectly coherent for Russell to hold a recursive definition of truth which requires that no facts contain ontological counterparts of the logical connectives, and yet maintain that the mind can be acquainted with purely logical objects such as universal, particular, not, or, dyadic relation, all and some, and so on. In the logical atomism lectures, Russell embraces more than the fully abstract general facts ‘‘with no constituents’’ that Theory of Knowledge identifies as logical forms. He embraces general facts such as Every man’s being mortal, which has the universals Humanity and Mortality among its constituents. Indeed, Russell remarks that in his 1914 lectures at Harvard, he was inclined to accept negative facts in his ontology – in spite of its ‘‘nearly producing a riot.’’ Russell gives in to this inclination in his logical atomism lectures. In his 1919 paper ‘‘On Propositions: What They Are and How They Mean’’ he embraces neutral monism and discusses his inclination for negative facts as follows: There might be an attempt to substitute for a negative fact the mere absence of a fact. If A loves B, it may be said, that is a good substantial fact; while if A does not love B, that merely expresses the absence of a fact composed of A and loving and B, and by no means involves the actual existence of a negative fact. But the absence of a fact is itself a negative fact; it is the fact that there is not such a fact as A loving B. Thus, we cannot escape from negative facts in this way.106

This is a striking change of mind. Russell himself had tried to escape from negative facts in just this way. His multiple-relation theory maintained that the sentence ‘‘A loves B’’ is false when there is no corresponding fact! There is no appeal whatsoever to negative facts as truth-makers in Principia, and

106

Bertrand Russell, ‘‘On Propositions: What They Are and How They Mean,’’ Logic and Knowledge: Essays 1901–1950, ed. R. C. Marsh (London: Allen & Unwin, 1977), p. 288.

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no such appeal occurs in Problems of Philosophy or The Theory of Knowledge. What then is the proper explanation of Russell’s new inclination toward an ontology of general facts and negative facts, and how does it fit with Principia’s requirement that the logical particles do not stand for relations that occur in facts? The answer is that by 1914 Russell had lost confidence in the multiple-relation theory. With the multiple-relation theory in trouble, Russell has to revise the base clause of Principia’s recursive definition of the hierarchy of sense of ‘‘truth’’ and ‘‘falsehood.’’ It is in considering such revisions that Russell becomes inclined toward general facts and negative facts. Russell is famous for saying in his logical atomism lectures that the truth-conditions of a general formula (x)Ax require more than the truth of each among Ax1 through Axn. It requires the general fact that x1, . . ., xn are all the entities.107 But it certainly does not follow that this general fact is itself the truth-maker for the general formula. Russell seems ready to abandon the multiple-relation theory at the base of the recursion, but he is surely not prepared to abandon Principia’s recursive theory of truth itself. As we saw, that theory provided the philosophical justification for Principia’s hierarchy of orders. It requires the thesis that the logical particles do not stand for relations (or properties) and demands that general facts are not truth-makers for general formulas. Indeed, Russell explicitly states in his logical atomism lectures that the logical particles do not stand for relations that occur as constituents of facts.108 ‘‘In a logically perfect language,’’ he writes, ‘‘the words in a proposition should correspond one by one with the components of the corresponding fact, with the exception of such words as ‘or,’ ‘not,’ ‘if,’ ‘then,’ which have a different function.’’109 We can now recognize the unfortunate impact that Russell’s tentative suggestions concerning general facts and negative facts in the logical atomism lectures have had. Undue emphasis on general facts and negative facts has inappropriately distanced Wittgenstein’s Tractarian ideas from Russell. The difference between Russell and Wittgenstein is wrongly characterized by saying that Russell, unlike Wittgenstein, considered philosophy to be a special ‘‘zoological’’ taxonomy of logical entities – general facts, universals, particulars, negative facts, etc. This characterization leaves out the very essence of Russell’s logical atomism – namely, its conception of philosophy as an eliminativistic reconstruction of structure. Logical analysis enables a reconceptualization of our concepts, and offers a reconstruction that avoids philosophical and ontological conundrums by building structure into logical grammar. Wittgenstein adopted the view implicit in 107 109

Russell, ‘‘The Philosophy of Logical Atomism,’’ pp. 183, 234. Ibid., p. 196; further evidence is found at pp. 197, 211.

108

Ibid., p. 211.

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Principia’s recursive definition of truth and falsehood that all facts are atomic and logically independent. What explains Russell’s departure from his earlier view in favor of a theory of acquaintance with logical objects was his underdeveloped philosophy of mind. He was therefore vulnerable, as we shall see, to Wittgenstein’s demand for more thoroughly eliminativistic analysis in this quarter. Russell’s paralysis What then had Russell learnt from Wittgenstein for the logical atomism lectures? In 1924, Ramsey wrote to Wittgenstein with the following assessment of the influence he had on Russell. ‘‘Of all of your work,’’ says Ramsey, ‘‘he seems now to accept only this: that it is nonsense to put an adjective where a substantive ought to be which helps in the theory of types.’’110 Ramsey was not far off. It must be understood that the seemingly small point is quite important. Its importance is lost because of a tradition of interpretation which took Principia’s type theory to apply to universals. Principia’s propositional function signs cannot occupy positions open to individuals, so it seemed that within type-theory universals cannot occupy positions open to individuals. But the tradition of interpretation is mistaken. Principia’s theory of order\types of propositional functions is not a theory of order\types of entities. Universals are entities for Russell; they are among the values of Principia’s individual variables, not its predicate variables. Though Russell felt strongly disposed toward the view that universals are logical entities whose existence is logically necessary, the paradox of predication had forced him to conclude that formal logic must get along without any general axioms comprehending universals. We are, in Russell’s view, acquainted with universals (especially sensible qualities), but this does not render logical truths concerning what universals there are. Russell’s multiple-relation theory was fairly well worked out in Theory of Knowledge. Nonetheless, Russell had abandoned the multiple-relation theory by the time of his logical atomism lectures. Reflecting on the matter some forty years later in My Philosophical Development, Russell explained that ‘‘I had, later, to abandon this theory because it depended upon the view that sensation is an essentially relational occurrence – a view which . . . I abandoned under the influence of William James . . . I abandoned this theory, both because I ceased to believe in the ‘subject,’ and because I no longer thought that a relation can occur significantly as a term, except 110

Ludwig Wittgenstein, Letters to C. K. Ogden, with an Appendix of Letters by Frank Plumpton Ramsey, ed. G. H. von Wright (Oxford: Blackwell, 1973), p. 84.

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when paraphrase is possible in which it does not so occur.’’111 In short, it was Russell’s adoption of neutral monism, together with the thesis that universals have only a predicable nature, that led him to abandon the multiple-relation theory. It seems odd that Russell doesn’t mention Wittgenstein’s influence. Russell’s letters of 1912–1913 suggest a different picture. It is quite certain from the letters that Wittgenstein’s criticism played a significant role in Russell’s abandonment of the multiple-relation theory. On 27 May 1913 he wrote to Ottoline Morrell of the progress on his new Theory of Knowledge book: we were both cross from the heat – I showed him a crucial part of what I have been writing. He said it was all wrong, not realizing the difficulties – that he had tried my view and knew it wouldn’t work. I couldn’t understand his objection – in fact he was very inarticulate – but I feel in my bones that he must be right, and that he has seen something I have missed. If I could see it too I shouldn’t mind, but as it is, it is worrying, and has rather destroyed the pleasure in my writing.112

What did Russell feel in his bones was right? Was it Wittgenstein that caused Russell to abandon the multiple-relation theory or was it, as Russell reports, his conversion to neutral monism? The correct answer is that it was both. In the lectures on logical atomism, Russell wrote of ‘‘the impossibility of putting the subordinate verb on a level with its terms as an object term in the belief,’’ and concludes that the multiple-relation theory was ‘‘a little unduly simple’’ because it does treat the object verb as if one could put it as just an object like the terms, ‘‘as if one could put ‘loves’ an a level with Desdemona and Cassio as a term for the relation ‘believe.’’’113 Russell confesses that it was Wittgenstein who convinced him that in statements of judgment or belief there are two verbs and that both must occur in predicate positions. That universals have both a predicable and an individual nature had long been central to Russell’s philosophy. So one might well expect Russell to object to Wittgenstein’s idea that universals have only a predicable nature. But the thesis offers Russell exciting possibilities that he felt must be worked through. Russell held that an ever more thoroughly eliminativistic analysis (revealing that all proper contexts are extensional) might well avoid the axiom of reducibility and dissolve many of the outstanding problems plaguing Principia’s logicism.

111 112 113

Russell, My Philosophical Development, p. 181. The Selected Letters of Bertrand Russell, ed. Nicholas Griffin (Boston: Houghton Mifflin Co., 1992), p. 459. ‘‘The Philosophy of Logical Atomism,’’ p. 226.

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The thesis that universals have only a predicable nature requires a new nonrelational reconstruction of the notion of a mind standing in a relation of acquaintance with a universal. Accepting neutral monism was, therefore, difficult for Russell. It finally came late in 1918, a few months after Russell completed his lectures on logical atomism.114 Russell had been coming to grips with neutral monism since 1907, and we now know that it was discussed in early chapters of Russell’s Theory of Knowledge manuscript which Wittgenstein read. Russell and Wittgenstein surely would have had many conversations about the doctrine. Meanwhile, Wittgenstein was becoming convinced that a thoroughly eliminativistic analysis of all logical and semantic notions was needed. This would quite naturally attract him to neutral monism as well. But what, precisely, were Wittgenstein’s criticisms of the multiplerelation theory? In a passage that has become infamous, Wittgenstein wrote the following in a letter to Russell in June of 1913: I believe it is obvious that, from the prop[osition] ‘A judges that (say) a is in the Rel[ation] R to b’, if correctly analyzed, the prop[osition] ‘aRb v bRa’ must follow directly without the use of any other premiss. This condition is not fulfilled by your theory.

In the Tractatus, Wittgenstein put the criticism as follows (TLP 5.5422): The correct explanation of the form of the proposition ‘A makes the judgement p,’ must show that it is impossible for a judgment to be a piece of nonsense. (Russell’s theory does not satisfy this requirement.)

What are the ‘‘other premises’’ upon which Russell’s multiple-relation theory allegedly relies? In what sense does Russell’s theory not rule out judgments of nonsense? Consider the belief ascription, ‘‘S believes that Socrates is mortal.’’ On Russell’s theory, the following is to be the fact (belief-state) that makes the ascription true: m-believing-with-respect-to-(Socrates, Mortality, P)

where P is the logical form of monadic predication. But according to Wittgenstein, nothing in Russell’s theory shows that Mortality is a universal with a predicable nature and that Socrates is a particular. Russell will need added premises to assure this. When Wittgenstein first raised the objection Russell thought himself fortified because he had already improved the theory set out in the Problems. In working on a solution to the order problem, Russell had included P (the logical form of predication) as an 114

Russell, My Philosophical Development, p. 100.

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object to which a mind may be acquainted. Moreover, Russell held that acquaintance with the universal Mortality itself yields an acquaintance with P (an awareness of its predicable nature). Wittgenstein holds that logical notions are pseudo-concepts. The difference between a universal and a particular is a logical difference which has to be shown in formal grammar itself. There is no faculty which yields this information about universals, and it is no help to offer an explanation of our knowledge of logical notions by reifying them as logical forms with which we are acquainted. On Wittgenstein’s view, it is not at all incidental that the expression ‘‘. . . is mortal’’ occurs in a predicate position in ‘‘S believes that Socrates is mortal.’’ For it is precisely in virtue of its occurrence in predicate position that shows in the grammar of natural language that Mortality is a universal with a predicable nature. Nothing short of this will suffice. All theories of types (including Russell’s types*) must be done away with! It is important to realize that this is a powerful objection to the multiplerelation theory only if one adopts Wittgenstein’s extreme demands of analysis. In attempting to give truth-conditions for a statement through an analysis of ‘‘correspondence’’ between belief and fact, the multiplerelation theory relies upon a faculty of ‘logical experience’ with logical forms. This grounds understanding of the notions ‘predicate,’ ‘dual relation,’ ‘complex,’ and the like. For Russell, understanding logical form is presupposed in belief, and thus logical forms (construed if only tentatively, as abstract fully general facts) were included as constituents of beliefstates. But in Wittgenstein’s view, any such theory must use pseudoconcepts such as ‘‘universal,’’ ‘‘fact,’’ ‘‘logical form.’’ Allowing logical forms as constituents of a belief-complex accomplishes nothing in his view. Our understanding of the logical impossibility of a concrete particular occurring as a relating relation in a dual complex cannot be explained by appeal to our acquaintance with an extra entity – a logical form of predication construed as an abstract general fact before the mind. Wittgenstein writes (TLP 5.522): The ‘experience’ that we need in order to understand logic is not that something or other is the state of things, but that something is; that, however, is not an experience.

As Wittgenstein’s sees it, Russell’s multiple-relation theory relies on the employment of pseudo-concepts in its effort to exclude nonsense judgments. Russell reported to Wittgenstein that he was paralyzed by this criticism.115 As we saw, Russell understood him to be arguing that in a 115

Ludwig Wittgenstein, Letters to Russell, Keynes and Moore (Oxford: Blackwell, 1974), p. 24, letter R13, dated 22 July 1913.

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statement such as ‘‘S judges that a loves b’’ the subordinate verb ‘‘loves’’ must retain an assertoric force – it must, as Russell put it, appear in a complex as a relating relation. And yet it cannot actually be relating a and b, for nothing in the mere fact that S has this belief assures that it is the case that a loves b. In his logical atomism lectures Russell wrote that, in judgment, both verbs have got to occur as verbs, because if a thing is a verb it cannot occur otherwise than as a verb. The subordinate verb is functioning as a verb, and seems to be relating two terms, but as a matter of fact does not when a judgment happens to be false. That is what constitutes the puzzle about the nature of belief. The discovery of this fact is due to Mr. Wittgenstein.116

In Russell’s estimation, Wittgenstein has revealed ‘‘a new beast for our zoo.’’117 This is a beast indeed, for surely no atomic sentence can have two verbs occurring assertorically.118 It should be emphasized that Wittgenstein is offering a general objection to the very idea of a ‘‘theory’’ of truth and not just objecting to Russell’s theory. Wittgenstein’s demands have force against every theory that attempts to do semantics. To state the nature of the correspondence relations that characterize truth requires the employment of notions that Wittgenstein regards as pseudo-concepts. The nature of correspondence with fact that is involved in truth must be built into the proper syntactic expression of an assertion. Facts (complexes) cannot be named or described. A name of a fact, such as Russell’s ‘‘the-redness-of-this’’ or ‘‘a-in-the-relation-R-to-b,’’ fails to show the predicable nature of the relating property (or relation). Forming a definite description fares no better. In attempting to avoid the narrow direction problem and characterize the proper correspondence relation grounding truth, Russell had offered a description of the appropriate fact that is to be the truth-maker. But Russell’s description ‘‘the  such that a has C1 in  and b has C2 in ,’’

where the position relations C1 and C2 are fixed by the relation ‘loves,’ does not avoid the problem Wittgenstein sees. It relies upon the truth-conditions for ‘‘a has C1 in ,’’ and in giving these truth-conditions we saw that the predicate for C1 will not occur in a predicate position. Russell’s account has to rely upon an acquaintance with the logical form of dual predication to recover the predicable nature of the relation C1. Wittgenstein’s

116 118

Russell, ‘‘The Philosophy of Logical Atomism,’’ p. 225. 117 Ibid. Davidson’s theory comes close. See Donald Davidson, ‘‘On Saying That,’’ Synthese 19 (1968–1969): 130–146.

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criticism is that any characterization of the correspondence relation must ultimately say that there is a universal that relates the constituents of the purported fact that is to correspond to the belief state. But to speak of the relating relation – to put a sign for it in a subject position in a descriptive clause – is to fail to represent its universal nature. The notion of being a ‘‘universal’’ is a logical notion; it must be given in the syntax by keeping signs for universals in predicate positions. In the language of Principia, a metaphysical theory of the nature of universals has to be expressed through special metaphysical axioms concerning individuals. Such a theory of universals would involve the assumption of nonlogical predicate constants such U(x) for the property of being a universal and P(x, y) for exemplification. Using xo and yo as individual variables of order\type o, the theory would then adopt metaphysical axioms. For example, ð8xo Þð8yo ÞðPðxo ; yo Þ  Uðyo ÞÞ

would be an axiom assuring that if individuals xo and yo stand in the relation of predication, then yo is a universal. Wittgenstein found this to be an unsatisfactory result that is due to the insufficient depth of Russell’s logical analysis. He maintained that notions such as universal, particular, predication, and the like have logical content and are therefore pseudoconcepts that should be expressed by logical grammar, not by special predicate constants. Looking back on matters in 1916, Russell recalled how Wittgenstein severely objected to the Theory of Knowledge book project. He wrote the following to Ottoline Morrell: Do you remember that at the time when you were seeing Vittoz [Ottoline’s doctor] I wrote a lot of stuff about Theory of Knowledge, which Wittgenstein criticized with the greatest severity? . . . I saw that he was right, and I saw that I could not hope ever again to do fundamental work in philosophy. My impulse was shattered, like a wave dashed to pieces against breakwater. I became filled with utter despair [* I soon got over this mood], and I tried to turn to you for consolidation . . . I had to produce lectures for America, but I took a metaphysical subject although I was and am convinced that all fundamental work in philosophy is logical. My reason was that Wittgenstein persuaded me that what wanted doing in logic was too difficult for me. So there was no really vital satisfaction of my philosophical impulse in that work, and philosophy lost its hold on me. That was due to Wittgenstein more than to the war. What the war has done is to give me a new and less difficult ambition, which seems to me quite as good as the old one. My lectures have persuaded me that there is a possible life and activity in the new ambition. So I work quietly, and I feel more at peace as regards work than I have ever done since Wittgenstein’s onslaught.119 119

Russell, Autobiography, vol. 2, p. 66.

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Wittgenstein accuses Russell’s work in Theory of Knowledge as betraying the research program of logical analysis. It appeals to logical forms and universals as components of states of understanding and belief. Russell had forged ahead with the work, working at a remarkable speed of ten pages a day. His enthusiasm for the work is clear. But the pace found him with promissory notes (of the availability of a more exacting analysis) when dubious assumptions arose (such as the identification of logical forms with abstract general facts). The intellectual burdens of too many such promises ultimately proved too much, and Russell’s enthusiasm flagged. At times Russell found Wittgenstein’s philosophical tirades against the account barely intelligible, but it was clear enough to him that Wittgenstein was right that a much deeper logical analysis is needed. Russell described this as ‘‘an event of firstrate importance in my life,’’ which ‘‘affected everything I have done since.’’ Wittgenstein held that Russell’s constructions had not gone far enough and Russell was more than willing to believe this. Valiant as Principia was, it failed to salvage logicism. It endeavors to develop all of nonapplied mathematics from its replacement for a theory of classes and relations-in-extension which is based on an axiom of Reducibility. But Russell came to see that his philosophy of logic does not support Principia’s Reducibility axiom. Russell had conceded in Principia that the solution of the problems plaguing logicism was incomplete. He observes that some form of type-structure is correct, but hopes that a deeper analysis might show how to avoid Principia’s axiom of Reducibility. Wittgenstein offered Russell new ideas. What is required, according to Wittgenstein, is a more thorough analysis which extends Russell’s own technique of building structure into variables. On Wittgenstein’s view, universals have only a predicational capacity, and they are logically independent – exemplification of one never logically precludes exemplification of another. The appearance of a universal standing in logical relation is a sure sign that one’s analysis is incomplete. A complete analysis reveals the genuine universals involved (i.e., ‘‘material’’ properties and relations) and these are not logical entities – they are posits of the empirical science of physics. With Wittgenstein clamoring for an analysis of the logical difference between universals and particulars, Russell imagined a possible escape from the axiom of Reducibility. The task is daunting. It challenges one of Russell’s most sacred doctrines. Russell had long held that universals are capable of a twofold occurrence – a predicational occurrence and also an individual occurrence.120 If universals are not capable of occurring otherwise than predicationally, an entirely new way of thinking of metaphysics, logic, knowledge 120

The doctrine that universals are capable of a twofold occurrence was unchanged from Russell’s 1903 Principles of Mathematics through 1914.

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is required. Acquaintance and logical intuition would require a complete reconstruction. From this perspective it is not at all surprising to find that Russell turned to neutral monism and behavorism for help. Russell needed help. A wholly new set of analyses, more extensive than ever, would have to be uncovered. Retreat from Pythagoras We have seen that Russell’s philosophy of logical atomism is not founded upon an empiricist doctrine of acquaintance. Principia does not offer a program of ontological reduction to objects of acquaintance. Classes and numbers are not identified with ramified type-stratified attributes (propositional functions in intension). They are not identified (reduced) to any entities of Principia’s ontology. The same holds of propositions. Principia’s ‘‘no-propositions’’ theory is not a reductive identity thesis, with propositions identified with order\types of judgments arranged in a hierarchy. There are no entities countenanced in the Principia with which propositions are to be identified (logically constructed). In fact, there is no ‘‘theory’’ of ramified types of entities in Principia at all; ramified types are, to borrow a phrase from Wittgenstein’s Tractatus, ‘‘logical scaffolding.’’ As his endeavors to solve the paradoxes plaguing logicism became ever more extensive, Russell found himself retreating further and further from the mystical splendor generated by the Pythagorean (or better, Platonistic) conception that logic and pure mathematics are founded upon the residents of (Plato’s) heaven. In My Philosophical Development (1959), Russell wrote: The solution of the contradiction . . . seemed to be only possible by adopting theories which might be true but were not beautiful. I felt about the contradictions much as an earnest Catholic must feel about wicked Popes. And the splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze.121

Russell continues: Wittgenstein maintains that logic consists wholly of tautologies. I think he is right in this, although I did not think so until I read what he had to say on the subject. Mathematics has ceased to seem to me non-human in its subject matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal . . . I cannot any longer find any mystical satisfaction in the contemplation of mathematical truth.122

121

Russell, My Philosophical Development, p. 212.

122

Ibid., pp. 119, 212.

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Russell goes on to describe himself as reluctantly submitting to truths which were to him repugnant – in particular, the view that logic and mathematics consist of tautologies. One can find Russell advocating the view that logical truths are tautologies in The Analysis of Matter (1927) and in his introduction to the second edition of The Principles of Mathematics (1937), though in the latter work his enthusiasm for the view seems to have waned. Carnap’s conventionalist approach in The Logical Syntax of Language is rejected outright in Russell’s introduction. Carnap imagines two logical languages, one of which admits the multiplicative axiom and the axiom of infinity as ‘‘analytic,’’ while the other does not. In Russell’s view, Carnap makes analyticity ‘‘arbitrary’’ (pragmatic) and to be settled by choice of a formal linguistic calculus. Russell explains: I cannot myself regard such a matter as one to be decided by our arbitrary choice. It seems to me that these axioms either do, or do not, have the characteristic of formal truth which characterizes logic, and that in the former event every logic must include them.

Russell rejected Carnap’s account of ‘‘analyticity’’ and the thesis that logical truth (logical necessity) is ‘‘linguistic’’ in the sense of its being conventional – the adoption of grammatical rules governing the nondescriptive particles of a given language form. He would have nothing of Carnap’s conventionalist account of ‘‘analyticity’’ and logic, and rejects Carnap’s watering-down of the notion of logical form to linguistic form. Russell nevertheless describes the view that logic consists of tautologies as a doctrine that logic and mathematics are ‘‘linguistic.’’ He writes: Logical constants must be treated as part of the language, not as part of what the language speaks about. In this way, logic becomes much more linguistic than I believed it to be at the time when I wrote the Principles.123

This suggests that, under the influence of Wittgenstein, Russell came to believe that logic is linguistic. Monk, for example, writes: ‘‘Since his conversion by Wittgenstein to a linguistic view of logic, Russell had regarded mathematics not as a body of knowledge but as a set of tautologies.’’124 This is a serious misunderstanding. Imagine how misleading it would be to say that because Principia’s theory of classes makes class symbols fac¸on de parler, the theory of classes is ‘‘linguistic.’’ To be sure, in Principia Russell rejected his early view that logical constants stand for relations between entities. But modern logic 123 124

Russell, The Principles of Mathematics, 2nd ed., p. xi. Ray Monk, Bertrand Russell: The Ghost of Madness (New York: The Free Press, 2001), pp. 56, 202.

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agrees with that and yet has little patience for a conception of logic as linguistic. Russell did not come to think that logic is linguistic; he has simply rejected his early account of the ontology and metaphysics of logic. Is no less misleading to characterize Wittgenstein’s Tractatus as construing logic as ‘‘linguistic.’’ The mistake is prevalent. The thesis that Wittgenstein held a ‘‘linguistic’’ conception of logic is congenial to those who wish to set the stage for an account of Wittgenstein’s later philosophy and marginalize Russell. For instance, Hacker writes: When the metaphysics of logical atomism fell apart, it became clear that the very idea of logical form amounted to no more that the grammar of expressions, the rule for their use – in particular, their combinatorial possibilities and the circumstances that license their employment. ‘Logical form’ does not reveal the objective logical structure of things, since they have no logical structure. They can be said to have a nature or essence, but that is determined by grammar, by the rules for the use of the expressions in question, which lay down what it makes sense to say, and is not answerable to reality for truth or correctness.125

We shall find that Wittgenstein’s Tractarian account of logical necessity as tautologyhood makes logic part of metaphysical scaffolding. It does not make logic ‘‘linguistic’’ in the sense of being a part of linguistic or cognitive practices or ‘‘rules for the use of expressions.’’ Russell’s retreat from Pythagoras is not a retreat from a metaphysical conception of logic. The influence of Wittgenstein caused a retreat from the postulation of logical entities in favor of a new metaphysics. In Principia, the logical connectives are statement connectives flanked by statements to form statements. The logical particles do not stand for relations between propositions, and no facts (that are truth-makers) contain ontological counterparts of tilde, the conditional, the generality sign, etc. But Russell still viewed logic as a synthetic a priori science of metaphysical structure. In The Theory of Knowledge structures are reified as ‘logical forms’ and identified as abstract general facts that have no constituents. Logical notions such as predication, universal, fact, particular, negation, generality, and the like stand for logical forms with which we are immediately acquainted by a faculty of ‘logical intuition.’ Wittgenstein persuaded Russell that the problems Principia faced in relying upon its axioms of infinity and reducibility might be met by adopting a deeper analysis of notions such as universal, particular, fact, and the like. It was for this reason that in the 1920s Russell investigated Wittgenstein’s idea that the proper way to characterize the notion of logical truth (logical necessity) as ‘‘truth in virtue of form’’ is by means of the notion of tautology. 125

P. M. S.Hacker, Wittgenstein’s Place in Twentieth-Century Philosophy (Blackwell, 1996), p. 80.

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Faced with the paradoxes plaguing Logicism, Russell had come to hold that the formulas of mathematical languages are to be reconceptualized and recovered in formulas of the proper formal language for logic – formulas which, by means of Russell’s logical constructions, eschew Platonic (Pythagorean) ontological commitments to mathematical objects such as numbers, classes, relations-in-extension, and functions. In Principia, Russell’s desire to build the structure of orders of propositions into a ‘‘no-propositions’’ recursive theory of truth led him to reject his former conviction that the logical particles are signs for logical relations obtaining between mind- and language-independent structures (propositions). Logical truth or ‘‘truth in virtue of logical form’’ thus turns on the structures that are rendered by the logical particles. Now the logical particles, Russell tells us, are ‘‘concerned with syntax.’’ In this respect, he speaks of his having written an ‘‘epitaph on Pythagoras.’’ As he puts it: ‘‘the propositions of logic and mathematics are linguistic,’’ and they ‘‘are concerned with syntax,’’ and ‘‘all the propositions of mathematics and logic are assertions as to the correct use of a certain small number of words.’’126 But the correct use of the logical particles of the language for logic is not, in Russell’s estimation, determined by our having adopted certain linguistic rules in using the language. Russell rejected Carnap’s conventionalist position that there are different formal systems for logic with different logical particles. As we saw, in My Philosophical Development Russell described the evolution of his thinking on mathematical logic as a ‘‘retreat from Pythagoras,’’ for he came to believe ‘‘logic and mathematics are linguistic’’ and ‘‘tautologous.’’ But we now see that this is quite different from the homophonic forms of the thesis advocated by Carnap and by interpreters of Wittgenstein’s later philosophical ideas. There is another wrinkle to the story of Russell’s retreat from Pythagoras. Russell’s reconstruction of consciousness within neutral monism had striking consequences for his theory of knowledge and for knowledge of logic and mathematics in particular. Russell came to express these consequences as revealing a ‘‘linguistic’’ component to logic and mathematics. But, more accurately, it reflects what we now call the naturalization project in the philosophy of mind. Russell made several efforts toward reconstructions of ‘belief,’ ‘truth,’ ‘knowledge,’ ‘perception,’ ‘matter,’ and ‘mind.’ His neutral monism eventually led him to a naturalized epistemology, and this in turn led him to a new appreciation for Hume’s empiricism. He was pushed to a psychological conception of ‘‘knowing’’ as an activity. Inference is a ‘‘habit’’ built out of stimulus-response patterns governed by 126

Bertrand Russell, ‘‘Is Mathematics Purely Linguistic?’’ in Essays in Analysis, ed. D. Lackey (London: Allen & Unwin, 1973), p. 306.

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the Law of Effect.127 Dewey was also advocating a naturalized epistemology and theory of ‘‘logic’’ which developed from his pragmatism. On Dewey’s view, the traditional notion of an organism’s ‘representation’ of the world as being ‘true’ or ‘false’ is to be abandoned in favor of notions of organism–environment states of equilibrium or disharmony (‘‘inquiry’’) with accompanying degrees of ‘‘success.’’128 Recovering the traditional notion of ‘truth’ within naturalism proved to be quite important to Russell, and it was the focus of some stormy exchanges with Dewey. Russell’s program of logical atomism began in moderation, with logic and knowledge of logic exempted from analysis. By the 1930s Russell had concluded that knowledge of logic must succumb, and this drives his thought toward naturalism in psychology and Humeanism in epistemology. Understanding this evolution in Russell’s thought has important consequences for interpreting Russell’s logical atomism and its relationship with Wittgenstein’s Tractatus. Wittgenstein was Russell’s student, his apprentice. The Tractatus consists of ideas on how to find constructions which would realize the full import of Russell’s new logical atomist conception of philosophy. Russell’s logical atoms (simples) were never fixed as sense-data, and both sense-data and the primitive relation of ‘acquaintance’ were abandoned in 1918. Principia’s recursive theory of truth anticipates both the Tractarian thesis that the logical particles do not represent relations between propositions and the logical independence of facts. Once logical atomism is freed from the cloud of interpretations that model it as a form of reductive empiricism, we can begin to see that Wittgenstein belongs at Russell’s side.

127 128

Russell, Philosophy, p. 27. See Tom Burke, Dewey’s New Logic: A Reply to Russell (Chicago: University of Chicago Press, 1994).

3

My fundamental idea

No interpretation of the Tractatus can be adequate unless it reconciles two central themes of the work. The first is the Grundgedanke (fundamental idea) of the Tractatus. Wittgenstein states it as follows (TLP 4.03): My fundamental idea is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts.

The fundamental idea, by its name alone, has rights to being the focal point of the Tractatus. The second central theme is the Doctrine of Showing. In a letter to Russell of 1919, Wittgenstein responded to some questions Russell had raised about the Tractatus. He writes: Now I’m afraid you haven’t really got hold of my main contention, to which the whole business of logical propositions is only corollary. The main point is the theory of what can be expressed (gesagt) by propositions – i.e., by language (and, what comes to the same, what can be thought) and what cannot be expressed by propositions, but only shown (gezeigt); which, I believe, is the cardinal problem of philosophy.1

Wittgenstein’s letter says that distinction between showing and saying is the ‘‘main contention’’ of the work. These two themes must somehow be fit together. On Anscombe’s interpretation, the ‘‘picture theory of meaning’’ is the central principle of the Tractatus. Both Showing and the fundamental idea are subsumed under it. Anscombe argues that the picture theory entails the thesis of truth-functionality – namely, the doctrine that all meaningful nonatomic statements are built up truth-functionally from atomic statements.2 The consequence of this, she says, is that Wittgenstein has a large program to carry out. He has to establish that contexts such as identity, quantification, logical necessity, ascriptions of propositional attitudes, etc., 1 2

Ludwig Wittgenstein, Letter to Russell, Keynes and Moore, ed. G. H. von Wright with the assistance of B. McGuinness (Oxford: Blackwell, 1974), p. 71 (19 August 1919). G. E. M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 2nd ed. (New York: Harper & Row, 1959), p. 79.

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that apparently do not fit into the truth-functional mold, and which are not simply ungrammatical, can be recovered. Favrholdt has a similar project of deriving the main Tractarian themes from a central principle.3 He maintains that the focal point of the Tractatus is extensionality – namely, that all sentences are truth-functions of elementary propositions. The picture theory, he goes on, is entailed by the thesis of extensionality, and indeed so also is Wittgenstein’s treatment of contexts of identity, quantification, logical necessity and possibility, and even ascriptions of propositional attitudes. In McDonough’s more recent contribution, it is argued that all the main ideas of the Tractatus are entailed by the Grundgedanke itself, which he interprets as the doctrine that the logical connectives do not refer (together with a characterization of logic as tautologous).4 The difficulty with all such interpretations is that it seems quite impossible to derive the Tractarian themes from the principles deemed to be fundamental. Discussing McDonough’s project, Carruthers puts the problem nicely: One telling point against him [McDonough], is that the argument he attributes to Wittgenstein simply helps itself to the thesis that all propositions are tautologies; whereas this clearly needs arguing for, and indeed presupposes a whole program of analysis which must receive its justification from elsewhere. Another point is that even granting McDonough’s account of the nature of tautologies (which is, in fact, only partly accurate . . .), his attempt to demonstrate all other features of TLP is extremely weak. For example, his argument to show that all genuine propositions are contingent . . . can only succeed by making assumptions about what it is to say something which would need to be independently argued for; and once such an argument is provided, we can in fact derive the thesis that all genuine propositions are contingent without having to appeal to the nature of tautologies.5

In McDonough’s view, the ‘logical constants’ discussed in the fundamental idea include only the usual logical particles of the predicate calculus. The weight of Wittgenstein’s project then hinges their treatment as truthfunctions. As Carruthers points out, this notion cannot bear the weight. Anscombe and Favrholdt face similar difficulties. The entailments cannot be forged. More recent ‘‘therapeutic’’ interpretations searching for a central principle have focused on Wittgenstein’s notion of an ‘‘elucidation’’ and his metaphor of climbing a ladder and then throwing it away. But what then is the role of the Grundgedanke? The therapeutic approach cannot simply 3 4 5

David Favrholdt, An Interpretation and Critique of Wittgenstein’s Tractatus (Copenhagen: Munksgaard, 1965). Richard McDonough, The Argument of the Tractatus (Albany, N.Y.: State University of New York Press, 1986). Peter Carruthers, Tractarian Semantics (Oxford: Basil Blackwell, 1989), p. 185.

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turn its back on what Wittgenstein explicitly took to be the fundamental idea of the work. Showing and the Grundgedanke must be reconciled. But how? The simple answer is that the two are identical. As we shall see, the identification yields a powerful new approach for interpreting the Tractatus. Showing as radical Russellianism In his preface to the Tractatus, Wittgenstein credits Frege’s work, writing that ‘‘I am indebted to the great works of Frege, and to the writings of my friend Mr. Bertrand Russell for much of the stimulation of my thoughts’’ (TLP, p. 3). The emphasis Wittgenstein placed on Frege has been interpreted as slighting Russell. Such a reading is surely overreaching.6 There is, however, an important connection between Showing and Frege’s distinction of concept and object. Frege held that concepts are not objects, and this embroiled him in an exchange with Benno Kerry that centered on the question of whether the concept horse is a concept. In Frege’s view, concept expressions are predicational and as such are incomplete. They must be combined with singular terms to form an assertion. Singular terms, on the other hand, do not have this feature. Frege holds that the linguistic distinction has an ontological analog. Concepts are incomplete or unsaturated; objects are saturated. Frege admits that the ontology he intends cannot be articulated without the appearance of self-refutation. Frege hoped that his readers would grant him a ‘‘pinch of salt’’ when he used ordinary language to distinguish concept and object.7 The situation bears a similarity to Wittgenstein’s Tractarian idea that there are aspects of reality that cannot be expressed in language but can nonetheless be revealed. Geach maintains that Wittgenstein’s doctrine of showing grew out of a sympathy for Frege’s point of view.8 Geach suggests that it is to Frege that Wittgenstein’s doctrine of showing owes its greatest debt, not to Russell. In Frege’s dispute with Kerry, Russell famously sided with Kerry. The very statement of the thesis that a concept is not an object refutes it. Russell’s doctrine of the unrestricted variable demands that any proper formal system for logic adopt one style of genuine variable – the ‘‘individual’’ variable or ‘‘logical subject’’ variable. To deny that an entity is a logical subject (‘‘object’’), Russell wrote, must always be false or 6 7 8

Frege was largely unappreciated until Russell brought to light the importance of his work. Wittgenstein is simply adding to Russell’s efforts in this quarter. Gottob Frege, ‘‘On Concept and Object,’’ Translations from the Philosophical Writings of Gottlob Frege, ed. Peter Geach and Max Black (Oxford: Basil Blackwell, 1977), p. 54. P. T. Geach, ‘‘Saying and Showing in Frege and Wittgenstein,’’ in Essays in Honor of G. H. von Wight, J. Hintikka, ed., Acta Philosophical Fennica 28 (1976), p. 55.

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meaningless.9 At first, this seems to make Frege’s concept/object distinction attractive as a model of Showing. But once we identify the Doctrine of Showing with the Tractarian fundamental idea, we see that the model is too weak. The identity of Showing with the fundamental idea yields a new way of understanding Showing. According to the Tractatus, all (and only) notions with logical and semantic content are pseudo-concepts that are shown – i.e., built into structured variables. This makes it impossible to construe Showing in a Fregean way. Nor is it possible to capture the connection by the Fregean inspirations that generate Conant’s view that Showing is a distinction between constative uses of language, in which a formula states what is the case, and elucidatory uses of language, in which an apparently constative is employed to dissolve a philosophical puzzle.10 Showing is not the vague doctrine that there are certain aspects of reality that cannot be expressed in language but must be conveyed through certain sorts of elucidatory employments of language. Our identification of Showing with the fundamental idea fixes what ‘‘aspects of reality’’ Wittgenstein targeted. He targeted the logical and semantic aspects. And Wittgenstein’s plan is not simply that they be conveyed through ‘‘elucidatory employments of language’’ but that they be built into structured variables (if only in the language of an ideal completed scientific theory of the world). Interpretations of the Tractatus have not failed to notice that there is some connection between Wittgenstein’s Showing, Principia’s ramified type-theory, and the Frege/Russell conception of logic. Jaakko and Merrill Hintikka have suggested that the central Tractarian themes derive from the doctrine that language is a ‘‘universal medium’’ which must contain its own metatheory. One cannot, while using a given language, render the semantics of that language, for to do so would rely on a given network of meaning-relations obtaining between the language and the world, and so presuppose them in stating what they are. On the Hintikkas’ view, the origin of Wittgenstein’s distinction between showing and saying lies in his conviction that semantics is ineffable. The Hintikkas trace their notion of language as a ‘‘universal medium’’ to van Heijenoort’s famous distinction between ‘‘logic as language’’ and ‘‘logic as calculus.’’11 On the ‘‘logic as language’’ view attributed to Frege 9 10

11

Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), pp. 46, 449, 507. The expression ‘‘constative use of language’’ is from James Conant, ‘‘Elucidation and Nonsense in Frege and Early Wittgenstein,’’ in Alice Carey and Rupert Read, eds., The New Wittgenstein (London: Routledge, 2000), p. 179. Merrill B. Hintikka and Jaakko Hintikka, Investigating Wittgenstein (Oxford: Blackwell, 1986), p. 4.

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and Russell, logic is a universal and all-encompassing language with content in its own right. Van Heijenoort correctly observes that Frege advocated a ‘‘syntactic’’ approach to logical truth and validity which construes these notions in terms of deductive closure under the axioms of a formal system. He contrasts this with the ‘‘semantic’’ approach to logic of Tarski, where logical truth and validity are construed in terms of invariant truth throughout all admissible interpretations.12 Van Heijenoort aligns the semantic approach with the algebraic tradition of Boole, Peirce, and Schro¨der. This makes Frege appear to reject the semantic tradition when, in distinguishing his work from that of Boole and his followers, he spoke of them as offering only a ‘‘calculus ratiocinator’’ and not also a ‘‘lingua characteristica.’’ On the semantic approach, one sets out a formal system that is subject to various semantic interpretations that are given from a metatheoretic perspective. For instance, in the Boolean calculus, ‘‘x þ y’’ can be interpreted as the union of two sets, or as the logical sum of two formulas. The formal system expresses a ‘‘content’’ only when a particular interpretation over a domain is given. Followers of van Heijenoort think that according to Frege and Russell, logic is a language which contains its own semantic metatheory. Their conception of logic as a language rejects the modern distinction between the formal calculus of logic and the metasystem. The rejection makes metasystemic research into such issues as decidability and semantic completeness impossible. The van Heijenoort interpretation, however, makes questionable history. It assumes that the new logic which the semantic tradition and the Frege/Russell tradition were working with was quantification theory. But the new logic set out by Frege in his Begriffsschrift (1879) is not just a quantification theory; it is a theory of functions. Indeed, his Grundgesetze der Arithmetik (1893) it a theory of functions and their extensions. Logic in Russell’s Principles of Mathematics (1903) is theory of propositions and their structures. The proper contrast between the semantic tradition and the Frege/Russell conception of logic does not lie in van Heijenoort’s distinction between ‘‘logic as uninterpreted system’’ (or calculus) and ‘‘logic as language.’’ The distinctness of the formal systems of Frege and Russell is that they embody comprehension principles postulating the existence of purely logical entities. The ‘‘content’’ Frege and Russell saw in their respective formal systems for logic lies in the comprehension principles, not in advancing (per impossibilia) a language that somehow contains its own metatheory. This undermines hope of tracing Wittgenstein’s showing versus saying distinction to a conception 12

Jean van Heijenoort, ‘‘Logic as Calculus and Logic as Language,’’ Synthese 17 (1967): 324–330.

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of language as a universal medium that is to be found in the Frege/Russell conception of logic. Notions such as ‘‘identity,’’ ‘‘universal,’’ ‘‘particular,’’ ‘‘object,’’ ‘‘value,’’ which Wittgenstein maintains must be shown and not said, cannot be explained as consequences of a conception of language as a universal medium. Wittgenstein’s showing versus saying distinction does owe its origins to Russell’s conception of logic, however. The conceptual linchpin is the notion of the variable in logic. Russell held that any proper calculus for logic must adopt only one style of genuine variables – individual/entity variables. By constructing other ‘‘variables with structure’’ (e.g., predicate variables, class variables, etc.), Russell hoped to solve the paradoxes plaguing Logicism by building distinctions (of type and later order\type) into formal grammar. The structured variable shows what otherwise would have to be said by predications involving pseudo-predicates such as ‘‘x is an object,’’ ‘‘x is a class,’’ ‘‘x is a number,’’ and so on. Russell originally imagined reconstructing the structure of a type-theory of classes by means of a substitutional theory which builds structure into variables in a syntactic way. To understand his method, perhaps a brief note on an analogous method in elementary mathematics is helpful. The equation mþ3¼2

has no solution in the natural numbers. But we can replace this equation with an equation of two variables, a and b, as follows: ða  bÞ þ ð3  0Þ ¼ ð2  0Þ:

This equation has solutions in the natural numbers. For example, it is solved when a ¼ 0 and b ¼ 1. This approach employs new definitions for ‘‘addition,’’ ‘‘multiplication,’’ and ‘‘identity.’’13 Once the new definitions are introduced, we can go on to pretend that the original equation has a solution when m is the ‘‘negative integer’’ 1. But properly speaking there are no such entities as ‘‘negative integers.’’ Similarly, an ordinary unproblematic equation such as m þ 2 ¼ 3;

which does have a solution in the natural numbers, can be replaced by the new equation: ða  bÞ þ ð2  0Þ ¼ ð3  0Þ: 13

ða  bÞ ¼ ðc  dÞ ¼ df ða þ dÞ ¼ ðb þ cÞ ða  bÞ þ ðc  dÞ ¼ df ða þ cÞ  ðb þ dÞ ða  bÞ  ðc  dÞ ¼ df ðac þ bdÞ  ðad þ bcÞ

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This has a solution when a ¼ 1 and b ¼ 0. We then engage in the pretense that the original unproblematic equation has the ‘‘positive integer’’ þ1 as its solution. But again there is no such entity as a positive or a negative integer. The structure, not an ontology of positive and negative integers, is recovered by using relations on pairs of natural numbers. Russell’s substitutional technique for solving the paradoxes of attributes and classes is kindred in spirit to the approach to positive and negative integers. The language of substitution has only one style of genuine variables, namely individual variables. To emulate special structured variables for quantification over attributes in intension, multiple individual variables are used. Consider the following example. The type-theoretic expression ð8’ðoÞ Þ’ðoÞ ðxo Þ

says that all attributes ’(o) are exemplified by xo. Instead of a special bindable predicate variable ’(o) with a type index (o), Russell’s substitutional theory employs two individual variables a and b, writing ðaÞðbÞð9yÞða=b; x!y :&: yÞ:

This type-free substitutional statement says that for all entities a and b, there is an obtaining entity (proposition) y which is exactly like a except that x occurs in it wherever b occurs in a. Russell’s logical atomism was born in his work on substitution and became part of his philosophical methodology thereafter. Principia abandons the substitutional technique, but not the idea of structured variables that motivated it. There are no orders or types of entities (propositional functions) countenanced in the Principia. Quite the contrary, it offers a construction more extensively eliminativistic than the substitutional theory. In Principia, the ontology of propositions is abandoned and the logical particles no longer stand for relations among propositions. Although predicate variables (adorned with suppressed order\type indices) are adopted as part of the language, only individual variables are regarded as genuine variables. The predicate variables are ‘‘variables with structure’’ whose significance conditions give the variables ‘‘internal limitations.’’ That is, the admissible range of a variable is internally limited by its significance conditions and not said by statements of the language of the formal theory. Principia’s Introduction is not part of its formal theory. It gives a nominalist semantic interpretation of the predicate variables of the formal theory. It is this nominalistic interpretation that does the work of building orders and types into the significance conditions of the variables. Russell’s philosophical method of building ‘‘variables with structure’’ is adopted by the Tractatus and extended widely. It is worth examining some

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quotes from Russell and Wittgenstein side by side. During the era of Russell’s substitutional theory, we find the following: the range of significance must somehow be given with the variable; this can only be done by employing variables having some internal structure for such as are to be of some definite logical type other than individuals . . . But then we have to assume that a single letter, such as x, can only stand for an individual; and that can only be the case if individuals are really all entities, and classes, etc., are merely a fac¸on de parler.14

In his 1908 paper, ‘‘Mathematical Logic as Based on the Theory of Logical Types,’’ Russell gave the doctrine a semantic twist which set the stage for the nominalistic interpretation of predicate variables in Principia. We find: The difficulty which besets attempts to restrict the variable is that restrictions naturally express themselves as hypotheses that the variable is of such or such a kind, and that, when so expressed, the resulting hypothesis is free from the intended restriction . . . Thus a variable can never be restricted within a certain range if the propositional function in which the variable occurs remains significant when the variable is outside that range. But if the function ceases to be significant when the variable goes outside a certain range, then the variable is ipso facto confined to that range, without the need of any explicit statement. This principle is to be borne in mind in the development of logical types.’’15

The variable is ‘‘internally limited by its range of significance.’’16 Russell’s method is to solve philosophical problems by eliminating certain questionable ontological commitments and employing structured variables. The limits are given by the structure of the variables and not made by statements of the formal language of the theory. They are ‘‘internal’’ to the signs of the language itself. In the Tractatus we find the following: We can now talk about formal concepts, in the same sense that we speak of formal properties. (I introduce this expression in order to exhibit the source of the confusion between formal concepts and concepts proper, which pervades the whole of traditional logic.) When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign for this object. (A name shows that it signifies an object, a sign for a number that it signifies a number, etc.) . . . So the expression for a formal concept is a propositional variable in which this distinctive feature alone is constant. (TLP 4.126)

14 15

16

Bertrand Russell, ‘‘On ‘Insolubilia’ and Their Solution by Symbolic Logic,’’ Essays in Analysis, ed. D. Lackey (London: Allen & Unwin, 1973), p. 205. Bertrand Russell, ‘‘Mathematical Logic as Based on the Theory of Types,’’ in Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. Robert C. Marsh (London: Allen & Unwin, 1977), p. 73. Ibid., p. 72.

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Thus, the variable name ‘‘x’’ is the proper sign of the pseudo-concept object. Whenever the word ‘‘object’’ (‘‘thing,’’ ‘‘entity,’’ etc.) is rightly used, it is expressed in logical symbolism by the variable name . . . The same holds for the words ‘‘complex,’’ ‘‘fact,’’ ‘‘function,’’ ‘‘number,’’ etc. They all signify formal concepts and are presented in logical symbolism by variables. (TLP 4.1271)

Wittgenstein accepts Russell’s idea of building structure into variables. It is part of his notion that ‘‘formal concepts’’ are properly expressed by variables. In interpreting the Grundgedanke of the Tractatus, we must resist the tradition of interpretation that characterizes the logical constants narrowly. Wittgenstein’s conception of the ‘logical constants’ includes much more than the truth-functions (statement connectives), quantifiers, and identity sign of predicate logic.17 He held that words like ‘complex,’ ‘fact,’ ‘function,’ ‘number,’ etc., all signify formal concepts, and are represented in conceptual notation by variables. The notion of a ‘‘formal concept’’ here means to cover any expression that involves logical (or semantic) content. Wittgenstein writes (TLP 4.126): I introduce the expression in order to exhibit the source of the confusion between formal concepts and concepts proper, which pervades the whole of traditional logic. When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign of this object . . . Formal concepts cannot, in fact, be represented by means of a function, as concepts proper can.

Wittgenstein uses ‘‘logical constant’’ (in his statement of the Tractarian ‘‘fundamental idea’’) synonymously with his use of the expressions ‘‘formal concept’’ and ‘‘internal property (relation).’’ That is, the logical constants include all and only notions with logico-semantic content. Among such notions, Wittgenstein included ‘identity,’ ‘universal,’ ‘particular,’ ‘name,’ ‘fact,’ ‘necessity,’ ‘possibility,’ and all semantic notions such as ‘truth,’ ‘reference,’ ‘belief,’ representation,’ ‘content,’ and the like. We can now understand how it is that Wittgenstein took Showing to be the ‘‘main point’’ of the Tractatus while at the same time expressing his ‘‘fundamental idea’’ in terms of the logical constants. The Tractatus advocates an uncompromising extension of Russell’s method of building structure into variables. Russell’s logical atomism was a research program that grew out of this method. It is an eliminativistic program which makes philosophy the analysis of logical form. The Tractatus endeavors to perfect and complete this program by extending the eliminativism widely: all (and 17

McGuinness notices this. He argues that Wittgenstein’s original notion of a logical constant was ‘‘anything that had been supposed to be a logical object.’’ See Brian McGuinness, ‘‘The Grundgedanke of the Tractatus,’’ in G. Vesey, ed., Understanding Wittgenstein (London: Macmillan, 1974).

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only) logico-semantic distinctions are to be built into the logical syntax (the ‘‘variables’’) of an ideal language for science. In reading the Tractatus it is essential to keep in mind that it assumes the viability of many of Russell’s analyses – if only in broad outline. Consider, for instance, Wittgenstein’s discussion of Kant’s problem of incongruous counterparts (i.e., that it is a synthetic and yet necessary truth that a righthand glove cannot be made to coincide spatially with a left-hand glove). Wittgenstein writes (TLP 6.36111): Kant’s problem about the right and left hand, which cannot be made to coincide, exists even in two dimensions. Indeed it exists in one-dimensional space

---o—x--x—o---a b in which two congruent figures, a and b, cannot be made to coincide unless they are moved out of this space. The right hand and the left hand are in fact completely congruent. It is quite irrelevant that they cannot be made to coincide. A right-hand glove could be put on the left hand, if it could be turned around in four-dimensional space.

Concerning this argument Fogelin writes: This is one of the few arguments in the Tractatus that strikes me as just awful. It is surely obvious that Kant’s central point is that a right-hand glove and a left-hand glove cannot be made to coincide in a three-dimensional space. For this reason he calls them incongruent. Here it will not help to offer – as Wittgenstein does – an alternative definition of congruency.18

Unfortunately, Fogelin misses the fact that Wittgeinstein’s remark is part of an eliminativistic program of conceptual analysis whose intent is precisely to offer a reconstruction of the notion of congruence. Wittgenstein’s discussion of Kant on incongruous counterparts speaks loudly as an endorsement of Russell’s eliminativism. Indeed, Wittgenstein’s position in the Tractatus is simply borrowed from Russell. In Russell’s view, Kant’s discussion of incongruent counterparts relies upon an erroneous importation of ordinary language notions of superposition and motion into the properly metrical notion of congruence – as if congruence requires that there must be a continuous series of equal figures moved through physical space and leading from A to B. Russell wrote: No motion will transform abcd into a tetrahedron metrically equal in all respects, but with the opposite sense. In this fact, however, there seems to my mind, to be

18

Robert Fogelin, Wittgenstein, 2nd ed. (London: Routledge & Kegan Paul, 1987), p. 90.

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nothing mysterious, but merely a result of confining ourselves to three dimensions. In one dimension, the same would hold of distances with oppose senses; in two dimensions, of areas. It is only to those who regard motion as essential to the notion of metrical equality that right and left-handedness form a difficulty.19

Fogelin is correct in holding that motion and superposition are certainly part of the ordinary conception of congruence used by Euclid, Kant, and others, but this is irrelevant to its reconceptualization (its new ‘‘definition’’) within the new research program of logical analysis. A similar, though less compelling, case can be made with respect to Wittgenstein’s discussion of color incompatibility. In the Tractatus, Wittgenstein’s comments on color include the following (TLP 6.3751): For example, the simultaneous presence of two colors at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of color. Let us think how this contradiction appears in physics; more or less as follows – a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are in different places at the same time cannot be identical. (It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The statement that a point in the visual field has two different colors at the same time is a contradiction.)

It is certainly a formidable task to find an eliminativistic analysis of the ordinary notion of ‘‘color’’ which separates out the logical components from the material, but the Tractatus is committed to it – if only at the limit of scientific inquiry. The issue of color (assuming it is the physical property of surface reflectance) is complicated because it concerns the application of arithmetic in the theory of the measurement of a (perhaps) continuous physical process (surface reflectance). In the Tractatus, Wittgenstein attempts to tackle the problem of color incompatibility as a problem of the incompatibility of surface reflectance potentials, i.e., the impossibility of a particle (photon) having different velocities at the same time. This has seemed inadequate, since it simply would avoid the problem of color incompatibility only to confront the equally difficult problem of explaining why it is a logical impossibility for a material particle have two velocities (be at two positions in space) at the same time.20 An examination of Russell’s Principles again reveals Wittgenstein’s debt. Russell wrote:

19 20

Russell, Principles of Mathematics, pp. 417f. Ramsey objected that Wittgenstein explains the allegedly necessary connections within the color system by appeal to the allegedly necessary connections within space and time, but no argument is given that these spatio-temporal relations are logically necessary. See Frank Ramsey, Critical Notice of the Tractatus, Mind 32 (1923): 465–478.

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The most fundamental characteristic of matter lies in the nature of its connection to space and time. Two pieces of matter cannot occupy the same place at the same moment, and the same pieces cannot occupy two places at the same moment, though it may occupy two moments at the same place. That is, whatever, at a given moment, has extension, is not an indivisible piece of matter: division of space always implies division of any matter occupying the space, but division of time has no corresponding implication. (These properties are commonly attributed to matter: I do not wish to assert that they do actually belong to it.) By these properties, matter is distinguished from whatever else is in space. Consider colors, for example: these possess impenetrability, so that no two colors can be in the same place at the same time, but they do not possess the other property of matter, since the same color may be in many places at once.21

Wittgenstein’s aphorism on color is certainly not a paraphrase of Russell’s Principles as was his entry on incongruous counterparts. Wittgenstein wrote: ‘‘Space, time, and color (being colored) are forms of objects’’ (TLP 2.0251); ‘‘Objects are what is unalterable and subsistent; their configuration is what is changing and unstable’’ (TLP 2.0272). These are among the most cryptic passages in the Tractatus. But Russell’s work offers our best hope of deciphering them. In Principles, Russell prefaces his discussion of matter by noting that he is not concerned with the nature of the matter that actually exists, he is rather interested in the philosophical problem of matter – the problem of ‘‘the analysis of rational Dynamics considered as a branch of pure mathematics, which introduces its subjectmatter by definition, not by observation of the actual world.’’22 Russell is not speaking of an enduring material object, but defining the notion of a material point as extensionless, occupying exactly one place at a time. Russell holds that the impenetrability of colors, as with the impenetrability of matter, is a logical feature of the analytic reconceptualization of space, time, and matter. That these passages of the Tractatus are closely tied to Russell’s work encourages the interpretation that the key to unraveling Wittgenstein’s ideas lies with a proper understanding of Russell. Interpretations of the Tractatus emphasizing its picture theory often call attention to Hertz’s Principien der Mechanik (1894). In the preface to his book, Hertz suggests that physics constructs mathematical models (Bilder) of reality, representing the essential features of the physical world by the relations that hold in the model. Wittgenstein says that a picture must have the same logical (mathematical) multiplicity as the fact it pictures, and Hertz says that a system that is the model of another must satisfy the condition ‘‘that the number of co-ordinates of the first system is equal to the number of the

21

Russell, The Principles of Mathematics, p. 467.

22

Ibid., p. 465.

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second.’’23 The isomorphism between linguistic picture and fact is akin, Wittgenstein says, to the isomorphic relations between the gramophone record, the musical idea, the score, and the sound-waves (TLP 4.014, 4.0141). Similarly, Hertz says that the relation of a dynamical model to the system of which it is regarded as the model is precisely the same as the relation of the images which our mind forms of things to the things themselves. Wittgenstein himself invites us to compare his comments with those of Hertz on dynamical models (TLP 4.04). The many similarities with Hertz are important.24 But it equally important to understand that in Principles, well before Wittgenstein got his feet wet, Russell had already called attention to Hertz. Russell regarded Hertz’s work as an example of conceptual analysis. Working within a kinematic theory that embraces Maxwell’s equations for the propagation of electromagnetic waves in the aether, Hertz offers a systematic reconceptualization of Newtonian dynamics that avoids forces acting at a distance. Hertz’s ‘‘picture theory’’ (his notion of a dynamical model) was employed as an aid in achieving this reconceptualization of ‘‘force.’’ Hertz’s work offers an example of an approach that is eliminativistic.25 Hertz’s work is not historically relevant to Wittgenstein’s Tractatus simply (or even primarily) because it embraces a form of the picture theory. It is relevant because it is an example of a reconstruction which employs picturing as an eliminativistic tool. On the present identification of the Tractarian fundamental idea with Showing, the picture theory is demoted.26 The notion of picturing plays a role in the Tractatus not unlike the one it plays in Hertz’s work; it is but one of several tools for building logical semantic notions into structured variables. What then was original with Wittgenstein? Russell’s science of logical form is an eliminativistic analysis that builds structure into variables. It involves the supplanting of one language and ontology by another within which the successes of the former have been retained but completely reconceptualized. The supplanting framework dissolving all philosophical problems, of course, does not presently exist. It is an ideal, a limit of

23 24 25

26

Heinrich Hertz, The Principles of Mechanics (New York: Dover, 1956), p. 175. See James Griffin, Wittgenstein’s Logical Atomism (Seattle: University of Washington Press, 1964), pp. 100ff. In a surprising move, Timm Lampert has argued that Wittgenstein’s comments on color incompatibility assume Hertz’s conception of physical theory, replete with its assumption of the aether! See Timm Lampert, Wittgenstein’s Physikalismus: Die Sinnesdatenanalyse des Tractatus Logico-Philosophicus in ihren historischen Kontext (Paderborn: Mentis, 2000). McGuinness has seen this. See ‘‘The Grundgedanke of the Tractatus.’’

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scientific and conceptual analysis, obtainable in principle. Russell’s method of solving philosophical problems by building structure into variables became, in the hands of the youthful Wittgenstein, the Doctrine of Showing. Russell’s eliminativism was moderate; it exempted certain logical notions and knowledge of logic. Wittgenstein hoped to find analyses of these notions. In the language of the logically ideal scientific theory, a language in which the analysis of logical form is complete, all (and only) logico-semantic notions are built into logical syntax of structured variables. Ideal versus ordinary language Our identification of Wittgenstein’s doctrine of showing with the Tractarian fundamental idea has very important consequences for understanding the role that the notion of an ‘‘ideal’’ or ‘‘logically correct’’ language plays in the Tractatus. In his introduction to the Tractatus, Russell put matters as follows: In order to understand Mr. Wittgenstein’s work, it is necessary to realize what is the problem with which he is concerned. In the part of this theory which deals with symbolism he is concerned with the conditions which would have to be fulfilled by a logically perfect language.

Russell took the picture theory to pertain only to the language of an ideal and complete scientific theory. In the language of the ideal (‘‘logically perfect’’) theory, logical notions are built into the syntax itself. In some passages Wittgenstein corroborates this. He writes (TLP 4.002): Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes.

Ordinary language disguises thought, and therein, in Wittgenstein’s view, is the source of philosophical conundrums and confusions of which the whole of philosophy is full (TLP 3.324). Wittgenstein writes that ‘‘in order to avoid these errors we must make use of a symbolism that excludes them . . . that is to say, a sign-language that is governed by logical grammar – by logical syntax’’ (TLP 3.325). An interpretation that attributes the Russellian quest for an ideal language to Wittgenstein is not currently popular. The prevailing wisdom in interpreting the Tractatus takes the work as endeavoring to delimit the very conditions for the possibility of representation – i.e., to set limits so as to reveal that philosophical problems are generated by a violation of the conditions for meaningfulness that underlie all language and thought.

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Wittgenstein supposedly held in the Tractatus that picturing is a necessary condition for the very possibility of thought and representation. Stern puts the view as follows: ‘‘‘Logical form’ is reserved for those aspects of representation that must appear in any representation whatsoever. Wittgenstein introduces logical form as a generalization of the notion of pictorial form.’’27 ‘‘Wittgenstein held that philosophical analysis will show us the rules we tacitly accept,’’ Stern continues, ‘‘the underlying structure of our language, and so enable us to avoid the misunderstandings that had generated traditional philosophical controversies. He thought that one could do so by specifying rules for translating from our ordinary language into a new symbolism that would clearly display the rules governing the underlying structures of ordinary language.’’28 On this interpretation, Wittgenstein is concerned with the ‘‘logical form’’ of every language. Hacker puts the point starkly: One deep misapprehension runs through Russell’s introduction [to the Tractatus]: namely, that Wittgenstein was concerned with elaborating the conditions for a logically perfect language. This was no trivial misunderstanding, since the point of the book was to elaborate the logico-metaphysical conditions for any possible language. It was a treatise on the essential nature of any form of representation whatever – a metaphysics of symbolism.29

Hacker continues, ‘‘Frege and Russell had held that natural languages are defective . . . and are only partial guides to the objective logical structures of reality . . . whereas the Tractatus had argued that every possible language must, as a condition of sense, mirror (on analysis) the logical forms of what is represented.’’30 If this is Wittgenstein’s conception of logical form, it is certainly out of step with Russell’s. But an interpretation that severs Wittgenstein’s conception of logical form from that of Russell is not viable. Wittgenstein’s ideas on language and representation must be set in the context of his view that ‘‘formal concepts’’ are ‘‘represented in conceptual notation by variables’’ (TLP 4.1272). This forges a bond with Russell, for it was Russell who inaugurated the method of using variables with structure to solve philosophical problems. Moreover, the most important new tool of Russell’s method was his 1905 theory of definite descriptions. In his article ‘‘On Denoting’’ Russell drew the lesson that ordinary grammatical form

27 28 29 30

David Stern, Wittgenstein on Mind and Language (New York: Oxford University Press, 1995), p. 38. Ibid., p. 49. P. M. S. Hacker, Wittgenstein’s Place in Twentieth-Century Analytic Philosophy (Oxford: Blackwell, 1996), p. 69. Ibid., p. 80.

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may be misleading as to logical form. In a letter to Russell, Wittgenstein endorses the lesson. He writes that ‘‘your theory of descriptions is quite undoubtedly right, even if the individual primitive signs in it are quite different from what you believe.’’31 Similarly, in the Tractatus Wittgenstein wrote (TLP 4.0031): All philosophy is a ‘critique of language’ (though not in Mauthner’s sense). It was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one.

Wittgenstein’s statement that philosophy is a ‘‘critique of language’’ is conspicuously placed side by side with an endorsement of Russell’s discovery about logical form. Wittgenstein was certainly not unaware of the significance Russell placed upon his theory of incomplete symbols. Ramsey later called it ‘‘a paradigm of philosophy.’’32 The paradigm requires that we ally Wittgenstein’s thesis that formal concepts are shown by variables with Russell’s ideas on building structure into variables. Logical form is not the underlying condition for meaningfulness in thought and language; it is rather a reconceptualization of the ‘‘formal concepts’’ that are rightly (if naively) employed in ordinary languages but wrongly understood by speculative philosophy. In Russell’s view, philosophical analysis involves supplanting one linguistic framework by another in such as way as to retain (wherever possible) the successes of the old framework, and through a reconceptualization and reconstruction reveal the sources and solutions of the philosophical conundrums the former framework generated. The Tractatus offers a program of eliminativistic reconstruction, not a metaphysics of symbolism. Wittgenstein calls attention to ordinary language when he writes that ‘‘we need to understand the logic of our language,’’ and that ‘‘all the propositions of our everyday language, just as they stand, are in perfect logical order’’ (TLP 5.5563). Appearances to the contrary, passages such as this do not support the thesis that Wittgenstein was concerned with the formal conditions for representation in thought and language. These entries are not statements of disagreement with Russell – as if Russell thought the nonphilosophical use of ordinary language is somehow flawed. It is the philosophical use of ordinary language that is flawed. Obviously the work of Cantor, Weierstrass, Dedekind, and Frege and 31

32

Letter of November 1913, in Notebooks 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe, 2nd ed. (Oxford: Blackwell, 1979), p. 129. Wittgenstein objected to Russell’s use of the identity sign in his theory of definite descriptions. Frank Ramsey, ‘‘Philosophy,’’ The Foundations of Mathematics and Other Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (New York: Harcourt, Brace & Co., 1931), p. 263n.

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Hertz on logical form does not employ the ordinary language use of words like ‘‘infinite,’’ ‘‘limit’’ and ‘‘continuity,’’ ‘‘number,’’ or ‘‘force.’’ Their works are revisionist. They offer reconstructions based on the logical analyses of infinity, of limits and continuity, of numbers, force, and space and time. It is the philosopher’s use of the concepts of ordinary language that gives rise to metaphysical muddles. The muddles are dissolved once the proper analyses of these concepts is set out. It is from the perspective of an eliminativistic reconceptualization of all logico-semantic notions that we come to understand Wittgenstein’s discussion of ordinary language. It is from the perspective of an eliminativistic reconstruction that the preface of the Tractatus must be understood. In the preface we find the following: The book deals with the problems of philosophy, and shows, I believe, that the reason why these problems are posed is that the logic of our language is misunderstood. The whole sense of the book might be summed up in the following words: what can be said at all can be said clearly, and what we cannot talk about we must pass over in silence. Thus the aim of the book is to set limits to thought, or rather – not to thought, but to the expression of thoughts: for in order to be able to set limits to thought, we should have to find both sides of the limit thinkable (i.e., we should have to be able to think what cannot be thought). It will therefore only be in language that the limits can be set, and what lies on the other side of the limit will simply be nonsense.

Wittgenstein writes that ‘‘It [philosophy] must set limits to what can be thought; and in doing so, to what cannot be thought. It must set limits to what cannot be thought by working outward through what can be thought. It will signify what cannot be said, by presenting clearly what can be said’’ (TLP 4.114–4.115). The emphasis on language has inspired an interpretation that Wittgenstein offers a program that explores the limits of language and thought – a program which aims to bring about an insight into the nature of our language, an insight that would make it clear that traditional philosophical problems and the solutions that have been proposed to them are literally nonsense. But the notion of ‘‘setting limits from within’’ is again adopted from Russell’s work. Once again the idea originates with Russell’s work to build the structure of type-theory into syntax (and thereby evade an ontology of types of entities). In his early substitutional theory, Russell endeavored to build logical distinctions of type into the formal syntax by using ‘‘variables with structure.’’ There are no type restrictions on the variables. The logical reconstruction of statements about classes renders the limits that block the paradoxes of classes (and attributes) from within formal grammar. The language of Principia employs predicate variables adorned with order\type indices, but Russell

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hastens to add that the variables are not restricted but limited from within by their significance conditions. The limitations on the variables that block the paradoxes are part of the logical reconceptualization of classes, propositions, attributes in intension, numbers, and the like. In this way, Russell labored to explain that the distinctions of type and order – the limitations they impose – are not properly limits outside of which there is another realm. This provides the proper background for understanding Wittgenstein’s thesis that philosophy must set limits to language. The limits are from within, not restrictions or boundaries imposed from outside. Wittgenstein’s phrase ‘‘setting the limits of language’’ reveals his debt to Russell. He is extending the compass of Russell’s program of building structure into variables. Wittgenstein advocates building structure into variables so that all distinctions between would-be logical entities become part of the logical syntax of a ‘‘no-logical-entities’’ analysis. Understood in this way, the picture theory of the Tractatus applies only to Wittgenstein’s notion of a logically perfect language. It was not intended to place strictures of picturing on the admissible devices of representation, reference, and communication in natural languages. In Wittgenstein’s view, all logical (and semantic) notions are to be reconceptualized so that (where proper) they will be shown by the syntactical forms of a logically perfect language – the language of the ideal and complete scientific theory at the limit of inquiry. Russell’s introduction to the Tractatus characterized the point spot on.

Sub specie aeternitatis The identification of Showing with the Tractarian fundamental idea aids greatly in the clarification of both. What makes Showing seem to be a perplexing doctrine is that its connection with the Grundgedanke has been underappreciated. Objectors will be quick to call attention to the fact that the Tractatus places ‘‘the mystical,’’ ethics, God, and value as matters under the purview of Showing – they ‘‘show themselves’’ (sich zeigen). Wittgenstein wrote: ‘‘There are, indeed, things that cannot be put into words. They show themselves. They are what is mystical’’ (TLP 6.522). In his early attempts to publish the Tractatus, Wittgenstein sent the manuscript to Ludwig von Ficker and subsequently wrote the following in a letter to him: the book’s point is an ethical one. I once meant to include in the preface a sentence which is not in fact there now but which I will write out for you here, because it will perhaps be a key to the work for you. What I meant to write, then, was this: My

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work consists of two parts: the one presented here plus all that I have not written. And it is precisely this second part that is the important one. My book draws limits to the sphere of the ethical from the inside as it were, and I am convinced that this is the ONLY rigorous way of drawing those limits. In short, I believe that where many others today are just gassing, I have managed in my book to put everything firmly into place by being silent about it. And for that reason, unless I am very much mistaken, the book will say a great deal that you yourself want to say. Only perhaps you won’t see that it is said in the book. For now, I would recommend that you read the preface and the conclusion, because they contain the most direct expression of the point of the book.33

This letter presents another theme which must be reconciled with Showing and the Tractarian fundamental idea: Wittgenstein says that the point of the book is ethical. Wittgenstein’s inclusion of notions of God, ethics, and value among the notions that show themselves lends itself to an interpretation that makes Showing appear quite distinct from the Tractarian fundamental idea. Indeed, it lends itself to the construal of Showing as a mystical mode of access to the deep and important questions of human existence. This encourages interpretations of Showing that make it a mysticism which reveals that what is really important lies beyond the limits of the sayable. Engelmann put this as follows: He [Wittgenstein] has something of enormous importance in common with the positivists; he draws the line between what we can speak about and what we must be silent about just as they do . . . Positivism holds – and this is its essence – that what we can speak about is all that matters in life. Whereas Wittgenstein passionately believes that all that really matters in life is precisely what, in his view, we must be silent about.34

Presented with passages like these, one might be inclined to agree with Pears that the Doctrine of Showing is ‘‘a baffling doctrine bafflingly presented.’’35 He writes: If we try to deduce the meaning of the doctrine and its justification from the things that he puts on the list of what can be shown and not said,’’ writes Pears, ‘‘we do not get much help. For the list includes such disparate items as the existence of a named object (TLP 4.1211, 5.535), the identity of the world with my world (TLP 5.62), and the value of anything that has value (TLP 6.4–6.421).36

The fact that Wittgenstein viewed the Tractatus as having something to reveal about ethics does not, however, undermine our identification of 33 34 35

Quoted from G. H. von Wright, Wittgenstein (Oxford: Blackwell, 1982), p. 83. Paul Engelmann, Letters from Ludwig Wittgenstein, with a Memoir (Oxford: Blackwell, 1967), p. 97. David Pears, The False Prison, vol. 1 (Oxford: Clarendon Press, 1987), p. 143. 36 Ibid.

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Showing with the Grundgedanke. Without a doubt, the notions of ethics and value present a challenge to the present interpretation. Shortly after he met him in The Hague in December of 1919 to discuss the Tractatus, Russell wrote the following to Ottoline Morrell: I felt in his book a flavor of mysticism, but was astonished when I found that he has become a complete mystic. He reads people like Kierkegaard and Angelus Silesius and he seriously contemplates becoming a Monk.37

Wittgenstein’s interest in Schopenhauer, James, and Kierkegaard has proved to be a bountiful resource for interpretations of the Tractatus. But the broad success of our identification of Showing with the Tractarian fundamental idea forms a template for interpreting the Tractatus as a work in alliance with Russell. If Wittgenstein’s views on ethics can be understood as echoing what Russell was saying on ethics at the time, this surely trumps interpretive experiments which call upon Schopehauer, James, Kierkegaard, and the like. Ethical truths seem to be synthetic necessities which do not contain logico-semantic content. But in his Notebooks 1914–1916, Wittgenstein writes that ‘‘Ethics does not treat of the world. Ethics must be a condition of the world, like logic.’’38 What sense can be made of this? The answer lies in Russell’s Spinozism. In 1912 Russell published a paper entitled ‘‘The Essence of Religion’’ – a piece excerpted from a failed book project called Prisons which Russell wrote in 1911 while his affair with Ottoline Morrell was new. Russell wrote: Religion consists in union with the universe. Formerly, union was achieved by assimilating the universe to our own conception of the Good . . . we must find a mode of union which asks nothing of the world, and depends solely upon ourselves . . . The moralist divides the world into good and bad . . . But besides this dualistic attitude, there is another, wholly compatible with it, but monistic: an attitude which ignores the differences between the good and the bad, and loves all alike. This is the essence of religion; but because it has not been clearly distinguished from the moralist’s attitude, it has been supposed, wrongly, to require the belief that the world is good . . . Every such demand [that the world shall conform to our standards] is an endeavour to impose Self upon the world . . . The essence of religion is the union with the universe achieved by subordination of the demands of Self . . . This subordination is not complete if it depends upon a belief that the universe satisfies some at least of the demands of Self.39 37 38 39

Quoted from Brian McGuinness, Wittgenstein: A Life (Berkeley: University of California Press, 1988), p. 279. Notebooks 1914–1916, 2nd ed., p. 77. Bertrand Russell, ‘‘Prisons I’’ (Morrell Papers, University of Texas at Austin). Quoted from Kenneth Blackwell, The Spinozistic Ethics of Bertrand Russell (London: Allen & Unwin, 1985), p. 111.

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The same ideas emerge in Russell’s discussion of ‘‘true philosophic contemplation’’ in The Problems of Philosophy (1912). Russell wrote: ‘‘By thus making a barrier between subject and object, such personal and private things become a prison to the intellect. The free intellect will see as God might see, without a here and now, without the hopes and fears, without the trammels of customary beliefs and traditional prejudices, calmly, dispassionately, in the sole and exclusive desire of knowledge – knowledge as impersonal, as purely contemplative, as it is possible for man to obtain.’’40 Self-interest and subjectivity are a ‘‘prison’’ in Russell’s view, because they shut out the possibility of impersonal contemplation of the world – a Spinozistic ‘‘intellectual love of God’’ interpreted as an attitude produced by viewing the world sub specie aeternitatis.41 According to the paper ‘‘The Essence of Religion,’’ the ‘‘infinite self’’ is universal and impartial and peace comes to this self through harmony or agreement with the whole, by means of its experience of ‘‘wisdom.’’ Russell approvingly calls this ‘‘mysticism,’’ though he warns that it is misguided to interpret it as a ‘‘perception of new objects.’’ It is, instead, ‘‘a different way of regarding the same objects, a contemplation more impersonal, more vast, more filled with love than the fragmentary, disquiet consideration we give to things when we view them as a means to help or hinder our own purposes.’’42 Russell maintains that constructing a metaphysics tailored to the values of humans is a fundamental source of error in both religious thinking and in philosophy. This position is set out nicely in Russell’s ‘‘On Scientific Method in Philosophy’’: The ethical element which has been prominent in many of the most famous systems of philosophy is, in my opinion, one of the most serious obstacles to the victory of scientific method in the investigation of philosophical questions. Human ethical notions, as Chuang Yzu perceived, are essentially anthropocentric, and involve, when used in metaphysics, an attempt, however veiled, to legislate for the universe on the basis of the present desires of men. In this way, they interfere with that receptivity to fact which is the essence of the scientific attitude towards the world.43

Russell contrasts this self-interested approach with his new philosophy of logical atomism – a philosophy which takes a global and non-self-interested perspective. In Russell’s view, logical analysis has cleared away centuries of 40 41 42

43

Bertrand Russell, The Problems of Philosophy (London: Oxford University Press, 1912), p. 160. Blackwell, The Spinozistic Ethics of Bertrand Russell, pp. 160f. Bertrand Russell, ‘‘The Essence of Religion,’’ in The Collected Papers of Bertrand Russell, vol. 12, ed. Richard Rempel, Andres Brink, and Margaret Moran (London: Allen & Unwin, 1985), pp. 112–122. Bertrand Russell, ‘‘On Scientific Method in Philosophy,’’ in Mysticism and Logic and Other Essays (Totowa, N.J.: Barnes & Noble, 1976), p. 82.

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metaphysical muddles that were introduced because of an emotional attachment to a particular ethical view. It is emotional detachment that Russell finds praiseworthy in the mysticism of Spinoza. In ‘‘Mysticism and Logic’’ Russell wrote: Good and bad, and even the higher good that mysticism finds everywhere, are the reflections of our own emotions on other things, not part of the substance of things as they are in themselves. And therefore an impartial contemplation, freed from all preoccupation with Self, will not judge things good or bad, although it is very easily combined with that feeling of universal love which leads the mystic to say that the whole world is good.44

Russell demanded a scientific (analytic) approach to philosophical problems – an approach which embraces atheism and an abandonment of spiritual, emotional, and religious perspectives. Nonetheless, he grappled to find some analog of the mystical that might accrue to the scientific and intellectual contemplation of the world. What this analog might be is explored early on in Russell’s ‘‘A Free Man’s Worship’’ (1903) and it was a recurring focus of discussion during Russell’s relationship with Ottoline Morrell. Eventually Russell settled on a new meaning for ‘‘the mystical’’ and illustrated it by Spinoza’s intellectual contemplation of the universe sub specie aeternitatis. Russell confessed to Ottoline that Wittgenstein ‘‘detested’’ his paper ‘‘The Essence of Religion.’’ In Wittgenstein’s view he ‘‘had been a traitor to the gospel of exactness and wantonly used words vaguely; also that such things are too intimate to print.’’ Russell would surely agree that Leopardi had put the ideas better in his poems ‘‘L’infinito’’ and ‘‘La ginestra o fiori del deserto.’’ Russell and Ottoline were fond of reading Leopardi together while working on Prisons.45 But Wittgenstein’s remarks do not show that he disagreed with Russell’s ideas.46 They reflect his objection to Russell’s having expressed them. In fact, it is the very same Spinozistic ideas that appear in Wittgenstein’s Notebooks 1914–1916. Wittgenstein wrote that ‘‘the good life is the world seen sub specie aeternitatis . . . The thing seen sub specie aeternitatis is the thing

44 45

46

Bertrand Russell, ‘‘Mysticism and Logic,’’ in Mysticism and Logic, p. 27. See Ottoline Morrell, Ottoline at Garsington: Memoirs of Lady Ottoline Morrell, 1915–1918, ed. Robert Gathorne-Hardy (London: Faber & Faber, 1974), p. 226. Russell quotes from ‘‘La ginestra’’ in Power: A New Social Analysis (New York: W. W. Norton & Co., 1938), p. 33. He quotes ‘‘L’infinito’’ in full in his book The Impact of Science on Society (London: Allen & Unwin, 1952), p. 98, prefacing it by writing: ‘‘This point of view is well expressed in a little poem by Leopardi and expresses, more nearly than any other known to me, my own feeling about the universe and human passions.’’ Ray Monk, Bertrand Russell: The Spirit of Solitude (New York: The Free Press, 1996), p. 280.

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seen together with the whole of logical space.’’47 Wittgenstein goes on to say that ‘‘In order to live happily I must be in agreement with the world. And that is what ‘being happy’ means.’’48 Wittgenstein’s comments on ethics and God in the Notebooks and the Tractatus echo Russell’s Spinozistic theme of a contemplation of the facts of the world sub specie aeternitatis. The proper ethical attitude is one of harmony or peace of mind with the world. Wittgenstein wrote: ‘‘If there is any value that does have value, it must lie outside the whole sphere of what happens and is the case . . . And so it is impossible for there to be propositions of ethics. Propositions can express nothing of what is higher’’ (TLP 6.41, 6.42). He goes on to say, ‘‘it is clear that ethics cannot be put into words. Ethics is transcendental. (Ethics and aesthetics are one and the same.)’’ (TLP 6.421). Wittgenstein’s Tractatus aims to illustrate the attitude of harmony. It is shown by the conspicuous absence of any statements in the work which would attempt to metaphysically ground truths governing what is good or evil, permissible or obligatory. Such statements (and Russell’s own statements in ‘‘The Essence of Religion’’) only reflect a ‘self’ in disharmony with the world’s facts. The allegiance to Russell’s Spinozism is striking in the following passage (TLP 6.43): If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts – not what can be expressed by means of language. In short, the effect must be that it becomes an altogether different world. It must, so to speak, wax and wane as a whole. The world of the happy man is a different one from that of the unhappy man.

There are no ethical facts. But harmony is shown in one’s attitudes toward the world as a whole. Wittgenstein continues as follows: ‘‘We feel that even when all possible scientific questions have been answered, the problems of life remain completely untouched. Of course then there are no questions left, and this is itself the answer’’ (TLP 6.52). On this view, there can be no ethical theory. To espouse any theory of ethics is to be in disharmony. The conspicuous absence in the Tractatus of an ethical philosophy is itself what shows its ethical position. The proper ethical perspective is that of silence, because silence is a manifestation of one’s harmony. Compare the following passage from Russell’s ‘‘Mysticism and Logic’’: Although, as we saw, mysticism can be interpreted so as to agree with the view that good and evil are not intellectually fundamental, it must be admitted that here we are no longer in verbal agreement with most of the great philosophers and religious teachers of the past. I believe, however, that the elimination of ethical 47 48

Notebooks 1914–1916, 2nd ed., p. 77. See also TLP 6.45. Notebooks 1914–1916, 2nd ed., p. 75.

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considerations from philosophy is both scientifically necessary and – though this may seem a paradox – an ethical advance.49

We need to look no further than Russell to understand Wittgenstein on ethics. Wittgenstein’s comment to von Ficker that the Tractatus concerns ethics may well appear to be a sort of mysticism. In truth, it quite naturally reveals his agreement with Russell’s Spinozism.50

Russell’s rejection of Showing Our thesis is that the Tractatus contains Wittgenstein’s ideas on how to extend and perfect logical atomism – Russell’s eliminativistic program for a scientific philosophy based on the analysis of logical form.51 Wittgenstein’s doctrine of Showing is but a radical form of Russell’s eliminativism. But why then would Russell reject Showing? There is a problem shared by Wittgenstein’s attempt to make a distinction between showing and saying and Frege’s attempt to make a distinction between concept and object. Frege’s distinction leads to a conundrum that is similar to that which the Doctrine of Showing leads – namely, that the fundamental idea, and all attempts to justify it, will (when expressed literally) violate it. Frege’s conundrum is local. Wittgenstein’s is global. Every logical distinction is such that its expression via anything but a structured variable is self-undermining. How then can one philosophically justify the introduction of such structured variables? Russell couldn’t accept Frege’s local conundrum. And he certainly bridled when Wittgenstein embraced it globally. Russell rejected Wittgenstein’s extreme eliminativism. The extreme eliminativism of the Tractatus produces a mysticism about logical form that Russell found unacceptable and tangential to the theses of atomicity and extensionality which he regarded as the central tenets of the work. Russell characterizes the Tractarian picture theory as maintaining that a proposition (at least in the ideal language for science) represents by being a picture or morphic model (Bild) of the facts which it represents. Wittgenstein wrote: ‘‘What any picture, of whatever form, must have in common with reality, in order to be able to depict it – correctly or 49 50

51

‘‘Mysticism and Logic,’’ p. 27. Correspondence suggests that the Spinozistic title of the Tractatus was proposed by Moore: von Wright, Wittgenstein, p. 97. The original title was the German Logische-Philosophische Abhandlung. McGuinness glimpses this. He writes: ‘‘This will give us a true ‘logical atomism’ with no two particulars differing in type, no difference of type, for example, between mental and physical particulars: neutral monism.’’ See Brian McGuinness, ‘‘Pictures and Form,’’ in Approaches to Wittgenstein: Collected Papers of Brian McGuinness (London: Routledge, 2002), pp. 61–81.

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incorrectly – in any way at all, is logical form, i.e., the form of reality’’ (TLP 2.18). As Russell put it, ‘‘a map clearly conveys information, correct or incorrect; and when the information is correct, this is because there is a similarity of structure between the map and the region concerned.’’52 In My Philosophical Development, Russell wrote that for a time he was prepared to accept Wittgenstein’s picture theory, but not his mysticism according to which ‘‘propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to represent it – the logical form.’’53 Russell criticizes Showing as a form of mysticism concerning the relationship between language and the world. In his efforts to distance himself from what he regarded as an untenable mysticism, Russell caused readers of the Tractatus to miss important connections between the work and his own research program. In his introduction to the Tractatus, Russell wrote (TLP, xxi): Everything, therefore, which is involved in the very idea of the expressiveness of language must remain incapable of being expressed in language . . . This inexpressible contains, according to Mr. Wittgenstein, the whole of logic and philosophy. The right method of teaching philosophy, he says, would be to confine oneself to propositions of the sciences, stated with all possible clearness and exactness, leaving philosophical assertions to the learner, and proving to them, whenever he made them, that they are meaningless . . . What causes hesitation is the fact that, after all, Mr. Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit. The whole subject of ethics, for example, is placed by Mr. Wittgenstein in the mystical, inexpressible region. Nevertheless, he is capable of conveying his ethical opinions. His defence would be that what he calls the mystical can be shown, although it cannot be said. It may be that this defence is adequate, but for my part, I confess that it leaves me with a certain sense of intellectual discomfort.

Russell goes on in his introduction to say that he cannot agree with Wittgenstein’s remark that ‘‘propositions cannot represent logical form: it is mirrored in them . . . What expresses itself in language we cannot express by means of language. Propositions show the logical form of reality’’ (TLP 4.121). Russell hoped for a loophole through which one might escape from Tractarian mysticism. In My Philosophical Development,

52 53

My Philosophical Development, p. 113. Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 114.

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Russell recalled his fundamental disagreement with Wittgenstein on this point: ‘‘To be able to represent the logic form, we should have to be able to put ourselves with the propositions outside of logic, that is, outside the world’’ (Tractatus 4.12). This raises the only point on which, at the time when I most nearly agreed with Wittgenstein, I still remained unconvinced. In my introduction to the Tractatus, I suggested that, although in any given language there are things that language cannot express, it is yet always possible to construct a language of higher order in which these things are said. There will, in the new language, still be things which it cannot say, but which can be said in the next language, and so on ad infinitum. This suggestion, which was then new, has now become an accepted commonplace of logic.54

It is Wittgenstein’s extreme eliminativism that Russell is criticizing because it generates an untenable mysticism. Russell’s eliminativism, his logical atomism, was moderate. It exempted certain foundational logical notions and knowledge of logic itself. Showing is extreme because, unlike Russell’s atomism, it is resolute and makes no such exemption. Wittgenstein agrees with Russell that ‘‘logic is the essence of philosophy,’’ but he holds that Russell was mistaken in thinking that philosophy is itself a science founded upon knowledge of logical form. This is an in-house debate among allies. The Tractatus makes several suggestions designed to reveal the possibility, in principle, of a radically eliminativistic analysis and reconstruction – an eliminativistic analysis that envelops knowledge of logic itself. These suggestions, however, will themselves be in violation of the eliminativistic ideal because they will employ what the Tractatus regards as logical and semantic pseudo-concepts as if they were genuine concepts. This situation causes a serious tension in Wittgenstein’s work. On the one hand, the eliminativism inaugurates a research program requiring that we realize the ideal by finding the constructions that build logico-semantic notions into formal grammar in such a way that their content is shown, not said by predicate expressions. On the other hand, the extreme form of eliminativism Wittgenstein seeks – where all logico-semantic concepts are pseudo-concepts – leaves one employing the structures of the ideal framework, with complete silence observed when it comes to their explanation and justification. Any attempt at justification of the new framework would be self-refuting. Wittgenstein closes the Tractatus with: Wovon man nicht sprechen kann, daru¨ber muß man schweigen.55

54

Ibid.

55

‘‘Whereof one cannot speak, thereof one must be silent.’’

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Pure adherence to the eliminativistic ideal would be to content oneself with speaking in the new framework and to otherwise be silent, letting the syntax do the work of showing or wearing its logico-semantic relations on its sleeve. Showing is the thesis that all (and only) logico-semantic notions are pseudo-concepts that would be shown in the syntax of an ideal language for empirical science. This is an eliminativistic claim that the old ontological framework, in which notions with logico-semantic content are taken as primitive, can (in principle) be supplanted by an (ideal) framework within which these notions are reconceptualized and constructed anew. But what is the status of the eliminativistic claim itself, and how can it be supported? If all logico-semantic notions are pseudo-concepts which, in an ideal language, would be shown because built into logical syntax, then the very basis from which such an ideal language could be sought and justified seems wholly undermined. Russell knew the dark consequence of Wittgenstein’s radical position – it threatens to reduce his eliminativistic conception of philosophy as the science of logical form to philosophy as mysticism based on the pronunciations of an oracle.

Kicking away the ladder Wittgenstein’s doctrine of Showing originates from Russell’s eliminativism. Russell was quick to point out, however, that a thoroughly eliminativistic approach to knowledge of logic is self-undermining. Wittgenstein, working still as Russell’s apprentice, was undaunted. He simply embraced an ouroboric conception of philosophy.56 In Wittgenstein’s view, philosophy reaches its completion and perfection when it vanishes at the ideal end point of its own analyses. The Tractatus itself makes a distinction between what is sinnlos and what is unsinnig. The distinction helps to explaining the doctrine of showing. It softens the self-refutation that seems to arise if we take it at its word that its own entries are ‘‘meaningless.’’ In the Prototractatus, Wittgenstein wrote: Tautologies und Contradiction sind (sinnlos) nicht unsinnig. Sie geho¨ren zum Symbolismus und zwar a¨hnlich wie der 0 in die Arithmetik.57 56 57

The ouroboros symbol depicts a serpent swallowing its tail and forming a circle. It symbolizes ideas of self-reference, completion, and unity. ‘‘Tautologies and contradictions are sinnlos, not unsinnig. They are part of the symbolism, just as 0 is in Arithmetic.’’ See Ludwig Wittgenstein, Prototractatus, ed. B. F. McGuinness, T. Nyberg, and G. H. von Wright, with a translation by B. F. McGuinness and D. F. Pears (Ithaca: Cornell University Press, 1971), p. 38.

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(Similar entries occur in the Tractatus at 4.461 and 4.4611.) In understanding the doctrine of showing one should distinguish between the admirably ‘‘meaningless’’ (sinnlos) expressions of logic, mathematics, the picture theory, etc., which show the scaffolding of the world, and despicably ‘‘meaningless’’ (unsinnig) statements which, though they may be perfectly grammatical in natural language, are violations of logical form. Heidegger’s ‘‘The nothing nothings’’ is unsinnig. Russell’s ‘‘Classes are logical fictions’’ and Wittgenstein’s ‘‘Numbers are exponents of operations’’ are sinnlos. The attempt to exonerate the Tractatus from self-incrimination by appeal to its distinction between what is sinnlos and what is unsinning has come under heavy fire recently. The distinction between sinnlos and unsinning is often conjoined with Ramsey’s view that the Tractatus aims to set out the formal preconditions for meaningful representation in thought and in any language. The conjunction is unstable. The sinnlos sentences that show proper logical grammar would seem to violate the conditions for meaningful representation no less than do the unsinnig propositions. Resolutely speaking, they are both gibberish. Therapeutic interpretations demand a resolute reading. They focus on the comments in the preface and the entries at the end of the Tractatus at 6.53 and 6.54 where Wittgenstein proclaims that his own statements are themselves nonsense and introduces the metaphor of throwing away a ladder once climbed. Wittgenstein wrote (TLP 6.54): My propositions serve as elucidations [Erlau¨tern] in the following way: anyone who understands me eventually recognizes them as nonsensical [unsinnig], when he has used them – as steps – to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up.) He must transcend these propositions, and then he will see the world alright.

These passages define what the therapeutic interpretation calls the ‘‘frame’’ of the work. The purpose of the Tractatus, on the therapeutic reading, is to liberate its readers from philosophy. The therapeutic interpretation demands that readers of the Tractatus be resolute in taking its expressions to be meaningless and yet elucidatory. Courageous readers are supposed to accept that the work elucidates what is important about logic, philosophy, God, ethics, and value by being quite literally meaningless. The complicated historical evolution of early drafts of the Tractatus may never be fully sorted out. But whatever the history is, it will not be accommodated by a boldly therapeutic reading.58 There is, after all, a Grundgedanke in the Tractatus. In ignorance of the revolution that has

58

Hacker has given a scathing, and telling, criticism of the therapeutic interpretation. See P. M. S. Hacker, ‘‘Was He Trying to Whistle It?’’ in Alice Cary and Rupert Read, eds., The New Wittgenstein (London: Routledge, 2000), pp. 353–388.

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occurred in understanding Russell’s philosophy, one is free to imagine that Showing might involve elucidations by use of meaninglessness. But this freedom is no longer viable. Wittgenstein’s inherited the view that there can be no theory of types of entities. It was Russell’s view that the limits of type-theory must be built into logical grammar by means of variables with structure, and not said by predicates which assert restrictions on the variables. The Tractarian fundamental idea is inextricably tied to Russell’s conception of the variables of logic. It says that all and only ‘‘formal concepts’’ (logical and semantic concepts) are properly shown by means of structured variables. This is incompatible with a therapeutic interpretation of Showing. A therapeutic interpretation cannot hope to explain why the Tractatus endorses so much of Russell’s philosophy. Reading the Tractatus as ouroboric, on the other hand, does justice to the resoluteness demanded by the therapeutic interpretation and yet also preserves the identity between Showing and the fundamental idea. It is this that explains the comments in the Tractatus in the preface and at the end (the so-called ‘‘frame’’ of the work). Wittgenstein agrees with Russell that logic is the essence of philosophy, but he disagrees with Russell that some fundamental logical notions and knowledge of logic must be exempted from analysis. Since the statements in the Tractatus employ concepts that have logical and semantic content, they are themselves pseudo-statements. Anticipating Russell’s criticism that ‘‘Mr. Wittgenstein says a good deal of what cannot be said,’’ the Tractatus contains the metaphor of climbing a ladder and then throwing it away when one reaches the top. The ‘‘Tractarian ladder’’ and the idea of statements as ‘‘elucidations’’ is Wittgenstein’s official defense against the self-refuting nature of Showing. Wittgenstein borrowed the ladder simile used in Mauthner’s discussion of absolute skepticism. Mauthner was speaking of the consistency of the position of a skeptic who exploits knowledge of logic to show that such knowledge is itself unreliable. We find the following: Just as it is not impossible for the man who has ascended to a high place by a ladder to overturn the ladder with his foot after the ascent, so also it is not unlikely that the skeptic after he has arrived at the demonstration of his thesis by means of an argument proving the non-existence of proof, as it were by a step-ladder, should then abolish this very argument.59

59

F. Mauthner, Beitra¨ge zu einer Kritik der Sprache (Stuttgart, 1901), p. 2, as quoted in Max Black, A Companion to Wittgenstein’s Tractatus (Cambridge: Cambridge University Press, 1964), p. 377.

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The radical skeptic challenges knowledge of logic itself. Wittgenstein’s radical eliminativism seems to do the same, for in proclaiming that logic is not a genuine science with a subject matter, it undermines knowledge of logic. In The Problems of Philosophy Russell made the following comments concerning radical skepticism: When, however, we speak of philosophy as a criticism of knowledge, it is necessary to impose a certain limitation. If we adopt the attitude of a complete skeptic, placing ourselves wholly outside all knowledge, and asking, from this outside position, to be compelled to return within the circle of knowledge, we are demanding what is impossible, and our skepticism can never be refuted. For all refutations must begin with some piece of knowledge which the disputants share; from blank doubt no argument can begin. Hence the criticism of knowledge which philosophy employs must not be of this destructive kind, if any is to be achieved. Against this absolute skepticism, no logical argument can be advanced. But it is not difficult to see that skepticism of this kind is unreasonable.60

Russell responds to the radical skeptic who challenges logic by conceding that there is a limit to philosophical criticism. Wittgenstein responds differently. He writes (TLP 6.51): Skepticism is not irrefutable, but obviously nonsensical, when it tries to raise doubts where no questions can be asked. For doubt can exist only where a question exists, a question only where an answer exists, and an answer only where something can be said.

Wittgenstein refuses to concede to a limit. Russell maintained that the radical skeptic’s question as to why she should be compelled to accept knowledge of logic is unanswerable. Her skepticism is irrefutable, though unreasonable. Russell hoped to ground knowledge of logic in an epistemic theory of self-evidence and logical intuition. Wittgenstein strongly objected. He writes that ‘‘Self-evidence, which Russell talked about so much, can become dispensable in logic, only because language itself prevents every logical mistake – What makes logic a priori is the impossibility of illogical thought’’ (TLP 5.4731). As Wittgenstein sees it, the radical skeptic who challenges logic has failed to ask any question. ‘‘If a question can be asked it must be answerable. The riddle does not exist’’ (TLP 6.5). Wittgenstein holds that we may proceed with logical analysis. This is the ladder. Ultimately, the analysis will envelop logic. But on Wittgenstein’s ouroboric conception of philosophy, this does not undermine the process. It completes it. There is no limit to logic because logic is the scaffolding of both thought and the world.

60

The Problems of Philosophy, p. 150.

4

Logic as if tautologous

Norway provided Wittgenstein with the solitude he thought necessary for work on logic. Together with his close friend David Pinsent he had found a place to vacation at O¨ystese, a tiny village in a little bay on the Hardangerfjord with majestic hills rising behind. When he returned to Cambridge on 2 October, he told Russell of his firm intention to leave Cambridge and live in Norway. Russell wrote to Lucy Donnelly on October of 1913: Then my Austrian, Wittgenstein, burst in like a whirlwind, just back from Norway, and determined to return there at once to live in compete solitude until he has solved all the problems of logic. I said it would be dark, and he said he hated daylight. I said it would be lonely, and he said he prostituted his mind talking to intelligent people. I said he was mad, and he said God preserve him from sanity. (God certainly will.)1

The ideas germinated in the year Wittgenstein spent in Norway would become a centerpiece of Wittgenstein’s philosophy of logic – the doctrine that logic consists of tautologies. August 1913 dates the following entry in Pinsent’s diary: ‘‘It is probable that the first volume of Principia will have to be re-written, and Wittgenstein may write himself the first eleven chapters. That is a splendid triumph for him!’’2 The catalyst for Wittgenstein seems to have been that Russell came to know of Sheffer’s result that all the quantifier-free formulas of Principia’s sentential calculus can be expressed via one logical connective – the Sheffer stroke.3 Russell received a copy of a paper from Sheffer on 15 April 1913. Russell read a paper to the Cambridge Moral Sciences Club in April 1912 entitled ‘‘On Matter.’’ On the back page of a later draft of Russell’s paper entitled ‘‘On Matter: The Problem Stated,’’ there are jottings in Russell’s and Wittgenstein’s hands which reveal that they were discussing truth-tables and 1 2 3

Letter to Lucy Donnelly, quoted from Brian McGuinness, Wittgenstein: A Life (Berkeley: University of California Press, 1988), p. 184. Quoted from McGuinness, Wittgenstein, p. 180. H. Sheffer, ‘‘A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants,’’ Transactions of the American Mathematical Society 14 (1913): 481–448. Peirce’s unpublished papers show that he had discovered this result earlier.

107

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Sheffer’s stroke.4 (See Fig. 4.1, below.) In his ‘‘Notes on Logic’’ composed in Norway, and in the Tractatus, Wittgenstein employs the Sheffer dagger instead of the stroke. He suggests that it has a special significance, and speaks as if it reveals inner logical connections between the logical particles. The dagger is the dual of the stroke.5 The relationship is displayed in the table below. Principia

dagger

stroke

p pq pvq p&q (p v q) (p & q)

p#p (p # q) (p # q) p # q p#q (p #q)

pjp p j q p j q (p j q) (p j q) pjq

We find the following passage in the Tractatus (TLP 5.1311): When we infer q from p v q and p, the relation between the propositional forms of ‘p v q’ and ‘p’ is masked, in this case, by our mode of signifying. But if instead of ‘p v q’ we write, for example ‘p # q .#. p # q’ and instead of ‘p’, ‘p # p’ (p # q ¼ neither p nor q), then the inner connection becomes obvious. (The possibility of inference from (x).fx to fa shows that the symbol (x).fx itself has generality in it.)6

It is unclear why Wittgenstein thought the that the dagger reveals inner or internal logical connections – as if the internal nature of the logical connectives in the inference (disjunctive syllogism) p v q; p =\ q

is unmasked when expressed by ðp # qÞ # ðp # qÞ; p # p =\q:

Nicod’s reduction of the propositional calculus to one rule and one axiom may provide a clue. In a letter to Russell from Norway dated 30 October 1913 Wittgenstein proclaims to have found such a reduction: ‘‘One of the consequences of my new ideas will – I think – be that the whole of Logic 4 5

6

See McGuinness, Wittgenstein, p. 160. The dagger symbol does not occur in Sheffer’s paper, and Wittgenstein used the stroke sign p/q to mean what we now mean by the dagger. I have rewritten quotes accordingly. Principia used a dot for conjunction. I use &. This entry also occurs in Wittgenstein’s Prototractatus at 5.04112 and in the Vienna and Engelmann typescripts of the Tractatus.

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109

Fig. 4.1. Verso, turned upside down, of folio 1, ‘‘On Matter: The Problem Stated,’’ Bertrand Russell Research Center, McMaster University, Hamilton, Ontario, Canada. The Sheffer stroke appears as p  q and is defined as p v q. Close scrutiny suggests Russell’s handwriting p on the left, with Wittgenstein’s handwriting W (Wahrheit) for true and F (Falsch) for false.

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follows from one Pp only!’’7 Wittgenstein worked with the Sheffer dagger. Nicod later worked with the Sheffer stroke. Nicod’s one inference rule is this: From p j (q j r) and p, infer r.

Modus ponens is an immediate instance of this rule, putting q for r. Since Wittgenstein was also looking for a reduction to one rule, it is natural that he might imagine disjunctive syllogism to be that rule. If we consider what would be a straightforward analog for the dagger of Nicod’s rule for the stroke we get: From p# (q # r) and q # q, infer r.

Disjunctive syllogism is an instance of this rule, putting (p # q) for p and p for q and q for r. Neither this, nor disjunctive syllogism, works to provide one inference rule for the dagger, however. A successful single rule is this: From (p # q) # r . # . s and p, infer r.8

Wittgenstein failed to find the reduction that Nicod discovered in 1916.9 But his investigations convinced him that he had discovered that logic must be decidable. The ramified type-structure of Principia left Russell a puzzle as to the nature of logical truth (logical necessity). Before Principia, Russell conceived of logic as the science of propositional structure, and he maintained that any true and fully closed formula in the language of the logic of propositions is logically true (logically necessary). The ramification in Principia’s no-propositions theory undermines this account of logical truth. The logical truths of the intended model of Principia’s ramified type-structures do not include all instances of the axiom schema of Reducibility, the infinity axiom, or the multiplicative axiom (axiom of choice). Yet all of these are fully general and (presumably) true. So how can one get at the notion of logical necessity as truth in virtue of form? Russell had reached an impasse and Wittgenstein was eager to offer an escape: logical truth (logical necessity) is tautologyhood.

7

8 9

Ludwig Wittgenstein, ‘‘Extracts from Letters to Russell 1912–1920,’’ in Notebooks 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe, 2nd ed. (Oxford: Blackwell, 1979), pp. 123, 126. See Thomas Scharle, ‘‘Axiomatization of Propositional Calculus with Sheffer Functors,’’ Notre Dame Journal of Formal Logic 6 (1965): 209–217. Jean Nicod, ‘‘A Reduction in the Number of the Primitive Propositions of Logic,’’ Proceedings of the Cambridge Philosophical Society 19 (1917): 32–41. The paper was read before the society on 30 October 1916.

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What is logic? How can one get at the notion of logical truth as truth in virtue of form? In his book Introduction to Mathematical Philosophy (1919), Russell confesses ignorance. He explicitly notes that it would not do to fix upon some particular formal system for logic and proclaim that the logical truths are just those deducible within the system. In one system of deduction, a given logical truth may be adopted as an axiom, and yet in another it is a theorem from a different set of axioms. Nor, he notes, would it suffice to make derivability of a contradiction from its denial the criterion for logical truth. Such derivability will be dependent upon the particularities of the deductive system in question.10 Russell then introduces Wittgenstein, who was working on the matter: The importance of ‘‘tautology’’ for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I don’t know whether he has solved it, or even whether he is alive or dead.11

Wittgenstein had been working on the problem since 1913, but with the onset of the war, his communication with Russell ceased. Russell does not endorse Wittgenstein’s idea that the notion of ‘‘truth in virtue of logical form’’ that characterizes the propositions of logic (and therefore, in Russell’s view, of nonapplied mathematics generally) can be explicated by means of Wittgenstein’s notion of ‘‘tautology.’’ But he notes enthusiastically that it may bear fruit. In The Analysis of Matter (1927), however, Russell accepted Wittgenstein’s view that the new criterion of logical truth is tautologyhood. He writes: We must ask ourselves, therefore: What is the common quality of the propositions which can be deduced from the premises of logic? The answer to this question given by Wittgenstein in his Tractatus Logico-Philosophicus seems to me the right one. Propositions which form part of logic, or can be proved by logic, are all tautologies – i.e., they show that certain different sets of symbols are different ways of saying the same thing or that one set of symbols says part of what the other says . . . Such propositions, therefore, are really concerned with symbols. We can know their truth or falsehood without studying the outside world, because they are only concerned with symbolic manipulations. I should add – though here Wittgenstein might dissent – that all pure mathematics consists of tautologies in the above sense.12

Interestingly, Russell does not endorse Wittgenstein’s notion of tautologyhood in the introduction to the second edition of The Principles of Mathematics (1937). He remarks: ‘‘The fundamental characteristic of logic, obviously, is that which is indicated when we say that logical 10 11 12

Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1919), p. 203. Ibid., p. 204. Bertrand Russell, The Analysis of Matter (London: Kegan Paul, 1927), p. 171.

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propositions are true in virtue of their form . . . I confess, however, that I am unable to give any clear account of what is meant by saying that a proposition is ‘true in virtue of its form’.’’13 Now more measured, Russell seems to back away from his earlier enthusiasm for Wittgenstein’s idea. But what was Wittgenstein’s idea? In the Tractatus we find Wittgenstein’s account of logical necessity summarized as follows. ‘‘The certainty, possibility or impossibility of a situation is not expressed by a proposition, but by an expression’s being a tautology, a proposition with sense, or a contradiction’’ (TLP 5.525). Russell came to describe himself as reluctantly agreeing with Wittgenstein that logical truths are tautologies. But, as we shall see, the agreement is more appearance than substance. Logic as if decidable What Wittgenstein thought he had discovered in Norway is that logical notions are pseudo-notions. Logical truths do not have a logical ‘‘content’’ which the mind comes somehow to know. This entails that there can be no formal system of logic – no set of axioms (chosen because they embody selfevident logical content) from which logical truths are deducible. There cannot be an a priori science of logic, for logic has no subject matter. There are no objects for it to study. Consequently, there can be no theory of deduction. If logical notions are pseudo-concepts, the science of deduction must be supplanted. Wittgenstein hoped to do this by finding a form of representation in which all and only logical equivalents have exactly one and the same expression. In a November 1913 letter to Russell, he wrote: The great question is now: How should a notion be constructed which will make every tautology recognizable as a tautology in one and the same way? This is the fundamental problem of logic.14

The tautologous nature of an expression would thereby be shown immediately in the symbol. Consider the following passages from the Tractatus: It is always possible to construe logic in such a way that every proposition is its own proof. (TLP 6.1265) All the propositions of logic are of equal status: it is not the case that some of them are essentially primitive propositions and others essentially derived propositions. Every tautology itself shows that it is a tautology. (TLP 6.127)

Of course, if there is a form of expression in which all and only logical equivalents have exactly one and the same expression, then the translation 13 14

Bertrand Russell, The Principles of Mathematics, 2nd ed. (New York: W.W. Norton & Co., 1937), p. xii. ‘‘Extracts from Letters to Russell 1912–1920,’’ Notedbooks 1914–1916, 2nd ed., p. 129.

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of a given formula into such a notation would form a decision procedure for logic. Wittgenstein’s did not balk at this consequence. His philosophical commitment to the decidability of logic is clear. In the Tractatus one finds: It is the peculiar mark of logical propositions that one can recognize that they are true from the symbol alone, and this fact contains in itself the whole philosophy of logic. And so too it is a very important fact that the truth or falsity of non-logical propositions cannot be recognized from the proposition alone. (TLP 6.113) One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol . . . The proof of logical propositions consists in the following process: we produce them out of other logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course, this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies. (TLP 6.126)

In Wittgenstein’s view, Russell was mistaken in thinking that deductive inference relies on the application of logical truths discovered by a science of logic. The inference rules of a formal system for logic would have to lie outside the formal system. But any would-be justification of an inference rule for the formal system would employ pseudo-notions such as ‘‘implication,’’ ‘‘truth,’’ ‘‘negation,’’ and the like. Wittgenstein writes (TLP 5.452): (Thus in Russell and Whitehead’s Principia Mathematica there occur definitions and primitive propositions given in words. Why this sudden appearance of words? It would require a justification, but none is given, or could be given, since the procedure is illicit.)

If all logical (and semantic) notions are pseudo-concepts, a system of formal deduction must be eliminated in favor of the view that a formula’s status as a logical truth, a contingent truth, or a contradiction should be shown by the syntax of its very expression in the ideal language for empirical science. An early attempt at finding such a notation is embedded in the following story told by Russell of the odd event of Wittgenstein’s first meeting the Whiteheads: Whitehead described to me the first time that Wittgenstein came to see him. He was shown into the drawing room during afternoon tea. He appeared scarcely aware of the presence of Mrs. Whitehead, but marched up and down the room for some time in silence, and at last said explosively: ‘‘A proposition has two poles. It is apb.’’ Whitehead, in telling me, said: ‘‘I naturally asked what are a and b, but I found I had said the wrong thing. ‘a and b are indefinable,’ Wittgenstein answered in voice of thunder.’’15 15

Bertrand Russell, The Autobiography of Bertrand Russell, vol. 2, 1914–1944 (Boston: Little, Brown & Co., 1968), p. 139.

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aqb

a pb

b

Fig. 4.2

a

a pb

a

aaq bb

apb

b

b

Fig. 4.3

In Wittgenstein’s ab-Notation, the truth-conditions for the conditional p  q are pictured in Fig. 4.2. The b-pole depicts falsehood and the a-pole truth. One may use any two mutually exclusive poles in place of a and b. In the Tractatus, W (Wahreit) and F (Falsch) are used. An illustration of the technique is exemplified in Fig. 4.3 for the formula p .. q  p. Following the b-pole, one sees that it leads to the a-pole of p and the b-pole of q  p, which yields the b-pole of p. Thus, the b-pole leads to contradictory assignments to p. The formula is a tautology. Something quite similar to the ab-Notation was independently discovered much later by Martin Gardner, who calls it a system of ‘‘shuttles.’’ Gardner does not discuss the Tractatus, but his notation significantly improves the ab-Notation because shuttles offer a uniform way to present the many connecting lines associated with the a and b poles. Gardner does not label his shuttles as a and b poles, and he splits up complex formulas into several shuttle diagrams. But these are not essential parts of the technique. Moreover, though Gardner does not employ it, we can use the device of twisting the shuttle lines to represent negation. A double twist represents

Logic as if tautologous

∼q ⊃ p

a pb

115 aq b

a b

a b

p .⊃. ∼q ⊃ p

Fig. 4.4

double negation. To illustrate the technique, consider the shuttle diagram of p .. q  p (Fig. 4.4). The poles of q are twisted for negation. Wittgenstein never wrote out the general rules for checking complex formulas for validity by the ab-Notation. Indeed, he refused to write out general rules for Russell, writing: ‘‘I’m upset that you did not understand that rule for the signs in my last letter, since it bores me unspeakably to explain it! . . . Please think the matter over yourself, I find it awful to repeat a written explanation, which I gave the first time with the greatest reluctance. So another time!’’ In truth, Wittgenstein had given no general explanation of rules needed to use the ab-Notation to check compound formulas. Gardner made an effort to state some of the important rules for applying the technique and remarks that they would indeed be tedious to state fully.16 Wittgenstein’s ab-Notation offers a decision procedure for propositional tautologies. But Wittgenstein was after much more, as we see in the following letter to Russell from Norway: I can’t myself say quite clearly yet what tautologies are, but I’ll try to give a rough account. It is the peculiar (and most important) characteristic of non-logical propositions, that their truth cannot be seen in the propositional sign itself . . . But the propositions of logic – and they alone – have the property of expressing their truth or falsehood in the very sign itself. I haven’t yet succeeded in getting a notation for identity which satisfies this condition; but I don’t doubt that such a notation must be discoverable. For compounded propositions (elementary propositions) the ab-Notation is adequate.17 16 17

Martin Gardner, Logic Machines and Diagrams, 2nd ed. (Chicago: University of Chicago Press, 1982), pp. 60–79. See ‘‘Extracts from Letters to Russell 1912–1920,’’ Notebooks 1914–1916, 2nd ed., p. 128.

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Wittgenstein is after for a decision procedure that would be extendable to quantification theory with identity. The idea is to find a notation for logic which would be such that once a formula is expressed in the notation one could discern from the notation itself whether or not it has the status of a logical truth. In his ‘‘Notes on Logic,’’ Wittgenstein speaks as if he has found a way to extend the ab-Notation to quantification theory. He writes: The application of the ab-Notation to apparent variable propositions becomes clear if we consider that, for instance, the proposition ‘‘for all x, ’x’’ is to be true when ’x is true for all x’s and false when ’x is false for some x’s. We see that some and all occur simultaneously in the proper apparent variable notation. The notation is: for ðxÞ’x : a-ðxÞ-a’xb-ð9xÞ-b and for ð9xÞ’x : a-ð9xÞ-a’xb-ðxÞ-b Old definitions now become tautologous.18

One can only guess how this proposal was supposed to proceed. But it is clear that Wittgenstein is attempting to convince Russell that the ab-Notation’s decision procedure can, in principle, be extended to quantification theory. This is corroborated in the following letter: Of course the rule I have given applies first of all only for what you call elementary propositions. But it is easy to see that it must also apply to all others. For consider two Pps in the theory of apparent variables *9.1 and *9.11. Put then instead of ’x, (9y).’y. y ¼ x and it becomes obvious that the special cases of these two Pps like those of all the previous ones become tautologous if you apply the ab-Notation. The ab-Notation for Identity is not yet clear enough to show this clearly but it is obvious that such a Notation can be made up. I can sum up by saying that a logical proposition is one of the special cases of which are either tautologous – and then the proposition is true – or self-contradictory (as I shall call it) and then it is false. And the ab-Notation simply shows directly which of these two it is (if any).19

The reference to Principia’s *9 is important. Wittgenstein claims that if one examines *9.1 and *9.11 (the two axioms for quantification theory in *9 of Principia) we can see that it is ‘‘obvious’’ that the logical truths of quantification theory are tautologous. To see how he might have come to this conclusion, we have to understand *9. Principia renders quantification theory in two ways: first at section *9 and then again in *10. The system of *10 parallels that of modern quantification theory. But Principia regards it as a convenience justified by the system of *9 which is required by the philosophical foundation for

18 19

‘‘Notes on Logic 1913,’’ Appendix I of Notebooks 1914–1916, 2nd ed., p. 96. ‘‘Extracts from Letters to Russell 1912–1920,’’ Notebooks 1914–1916, 2nd ed., p. 126.

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ramified type-theory. The quantification theory of section *9 defines subordinate occurrences of quantifiers in terms of occurrences of quantifiers prefixing the whole formula. Where p is quantifier-free and y does not occur free in the formula ’ and x does not occur free in the formula y, Principia’s definitions include the following: *9.01 *9.02 *9.03 *9.04 *9.05 *9.06 *9.07 *9.08

(x)’x ¼df (9x) ’x (9x)’x ¼df (x) ’x (x) ’x v p ¼df (x)(’x v p) p v (x) ’x ¼df (x)(p v ’x) (9x) ’x v p ¼df (9x)(’x v p) p v (9x) ’x ¼df (9x)(p v ’x) (x)’x v (9y)yy ¼df (x)(9y)(’x v yy) (9y)’y v (x)yx ¼df (x)(9y)(’y v yx)

Principia’s *9 endeavors to demonstrate that each line of any proof in the system of *10 can by recovered in the system of *9 which works by employing universal and existential generalizations on tautologies.20 Given a formula, say (9y)(Ay  (x)Ax), where A is quantifier-free, the definitions of *9 pull subordinate quantifiers to initial placements. We arrive at (9y)(x)(Ay  Ax). Next remove the quantifiers and we have Ay  Ax:

This is not a tautology. But in this example, it easy to see that we can start our derivation from the tautology Ay  Ay and work our way back to the original formula by generalization and the definitions provided by *9. By existential generalization, we get (9y)(Ay  Ax). By universal generalization we have (x)(9y)(Ay  Ax); by a rule implicit in *9 which tells us when we may switch quantifiers, we arrive at (9y)(x) (Ay  Ax). Then by employing the definitions of *9 we arrive at our formula (9y)(Ay  (x)Ax). By proceeding in this way, the system of *9 can recover all the principles and inference rules of the quantification of Principia’s *10. Captivated by the results of *9, Wittgenstein came to believe that logical truths of quantification theory are in a sense generalized tautologies. Russell could hardly disagree with this. But the adequacy of *9 as a quantification theory certainly does not entail that there is a decision procedure for the logical truths of quantification theory. The deductive

20

This is not to say that logic itself consists of generalized tautologies, not at least if standard second-order calculi count as capturing logic. The comprehension principle of standard second-order logic is not a generalized tautology. Ramsey seems to recognize this in exempting Principia’s Axiom of Reducibility.

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techniques of *9 provide no general recipe for deciding whether or not there is a tautology from which to begin a derivation. In 1936 Church proved that quantification theory is undecidable.21 Principia’s *9 is semantically complete with respect to the logical truths of quantification theory.22 It is therefore undecidable. For a time, Russell confessed that he agreed with Wittgenstein’s conception of logic as tautologous.23 But his conception of a ‘‘tautology,’’ unlike that of Wittgenstein, did not connote decidability. It was divorced from the idea of finding a notation in which all and only logical equivalents have exactly the same representation. The undecidability of quantification theory is a significant blow to Wittgenstein’s conception of logic. As we shall see, it undermines Wittgenstein’s hope of finding a notation in which all and only logical equivalents have one and the same representation.24 Scaffolding It is common today to explain the notion of tautologyhood by reference to a truth-table and even to credit Wittgenstein with the invention of truthtables. But truth-tables were not Wittgenstein’s innovation.25 It is difficult to say precisely when they were discovered. Certainly they are related to proofs of the independence of the axioms of postulate systems. Independence proofs designate a desired value and employ the idea that a complex formula’s value is a function of the values of its subcomponents. If the desired value is possessed by all but one of postulates of the system, and is preserved 21 22 23 24

25

Alonzo Church, ‘‘A Note on the Entscheidungsproblem,’’ Journal of Symbolic Logic 1 (1936): 40–41. See Gregory Landini, ‘‘Quantification Theory in *9 of Principia Mathematica,’’ History and Philosophy of Logic 21 (2000): 57–78. Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 119. Oddly, Hacker writes, ‘‘There can be no doubt that Wittgenstein’s explanation of the tautologousness of the propositions of logic has had a profound effect upon the general understanding of logic.’’ See P. M. S. Hacker, ‘‘Was He Trying to Whistle It?’’ in Alice Carey and Rupert Read, eds., The New Wittgenstein (London: Routledge, 2000), p. 356. This piece of misinformation is repeated again and again and has become part of the folklore of Wittgenstein. It appears in works intended for a wide readership such as A. C. Grayling, Wittgenstein: A Very Short Introduction (Oxford: Oxford University Press, 2001), p. 32; and it is found in popular books such as David Edmonds and John Eidinson, Wittgenstein’s Poker (New York: HarperCollins, 2001), p. 232. It also appears in more serious works aimed at history such as Anthony Kenny, Wittgenstein (Cambridge, Mass.: Harvard University Press, 1973). See also Anthony Kenny, ed., Oxford Illustrated History of Western Philosophy (Oxford: Oxford University Press, 1997), p. 260. Indeed, it even occurs in scholarly academic works such as Matthieu Marion’s Wittgenstein, Finitism and the Foundations of Mathematics (Oxford: Clarendon Press, 1998), p. 3. It also creeps into Alan Janik’s ‘‘Weininger and the Two Wittgensteins,’’ in D. Stern and B. Szabados, eds., Wittgenstein Reads Weininger (Cambridge: Cambridge University Press, 2004), p. 67.

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by the system’s inference rules, then all theorems will have the designated value. In this way the independence of a given postulate of a system can be proved. Edward Huntington read a paper to the American Mathematical Society in September 1903 which employs independence proofs for axioms of propositional logic.26 Russell cites Huntington’s article in the Principia, mentioning the independence proofs approvingly.27 Of course, Frege invented the idea of a truth-function – a function from truth-values to truth-values. In his Begriffsschrift (1879) a truth-functional calculus is set out. Frege could not have missed the fact that if f(z,y) and g(x) are functions, then the composition f(g(x),y) of these functions is itself a function, for this is a commonplace mathematical result. If f(z,y) is the truthfunction for the conditional, and g(x) for negation, then Frege’s ‘ f(g(p),q) is akin to the modern P ! Q. It would be quite obvious to Frege both that the truth-conditions of any such composition function can be displayed in a table and that there is an algebraic decision procedure to determine which such compounds yield the value t, for all truth-values as argument. The likely reason Frege did not make this a point of emphasis in his work is that the achievement he awarded his Begriffsschrift, in contrast with Boole’s algebra of logic, is that it goes well beyond such simple algebraic results of a set of simple functions (operations). The Begriffsschrift set out a quantification theory which permits the comprehension of new functions and quantification over them. Frege maintained that, unlike Boole’s algebra, his incorporation of comprehension principles makes logic genuinely informative. The comprehension of functions reveals new logical structures. Boole’s Calculus for Logic appeared in 1848. Extensions of it to form an algebra of relatives were advanced by Schro¨der and (independently) Peirce. In his manuscript ‘‘Qualitative Logic’’ of 1886, Peirce set out a system of logic and employed a propositional reading of Venn diagrams to justify his rules and principles.28 Peirce used diagrammatical and truthconditional techniques to determine whether a given proposition belonged to logic. In Mu¨ller’s 1909 Abriss of Schro¨der’s Vorlesungen u¨ber die Algebra der Logik, we find a truth-table: x72 Two-Valued Systems. If , , , . . . are general symbols for the domain of these systems, i.e., such that they can only mean 0 or 1, then each expression put together

26 27 28

Edward Huntington, ‘‘Sets of Independent Postulates for the Algebra of Logic,’’ Transactions of the American Mathematical Society 5 (1904): 288–307. A. N. Whitehead and Bertrand Russell, Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1927), vol. 1, p. 206. Charles Sanders Peirce, ‘‘Qualitative Logic’’ (1886), in The Writings of Charles Sanders Peirce, vol. 5, 1884–1886, ed. Christian Kloesel (Bloomington: Indiana University Press, 1993), p. 369.

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[composed] from these, e.g.,  þ  also gets one of the two values 0 or 1, as long as the values of the individual symbols are determined. Thus, one can easily check [verify] in two-valued calculi the correctness of each and every theorem or axiom of the domain, e.g., of the Axiom VIIx, ð þ zÞð þ zÞ ¼ z by successively [in succession] assigning to the individual symbols appearing therein [in the theorem] the value 0 and 1 for all possible combinations – as shown, for instance, in the following diagram [scheme], where the values belonging together are placed in columns under one another.29

This is an explicit tabular method used as a decision procedure. Since ð v Þ & ð v zÞ :  : z

always gets the value 1, it is a truth of logic. Perhaps one day the historical record will be set straight and Wittgenstein will no longer be given credit for inventing truth-tables.30 But one shouldn’t correct one historical error only to offer another – attributing to Wittgenstein the innovation of using truth-conditions to represent a 29

Eugen Mu¨ller, Abriss der Algebra der Logik; repr. in Ernst Schro´´der, Algebra der Logik, vol. 3 (New York: Chelsea Publishing Co., 1966), p. 708. The translation was aided by Tuomas Manninen. The original reads: ‘‘x 72. Zweiwertige Symbole. Sind , , , . . . allgemeine Symbole fu¨r die Gebiete dieses Systems, d. h. solche, deren jedes nur entweder 0 oder 1 bedeuten kann, so wird auch jedem daraus zusammengesetzten Ausdruck, z. B.  þ , einer der beiden Werte 0 und 1 zukommen, sobald die Werte der Einzelsymbole bestimmt sind. ‘‘Man kann somit auch leicht die Richtigkeit eines jeden beliebigen Gebietstheorems oder axioms – z. B. des Axioms VIIx  þ Þ ¼  ð þ Þð im zweiwertigen Kalkul nachpru¨fen, indem man den darin vorkommenden Symbolen einzeln die Werte 0 und 1 nacheinander in allen mo¨glichen Zusammenstellungen beilegt, – etwa wie das folgende Schema andeutet, worin die zusammengeho¨rigen Werte kolonnenweise untereinander stehen. ¼ 0011 ¼ 0101 ¼ 1010 þ ¼ 0 1 1 1 þ ¼ 1 0 1 1 ð þ Þð þ Þ ¼ 0 0 1 1 ¼ :’’

30

Some steps at correcting the historical record are made in McGuinness, Wittgenstein, p. 162. See also John Shosky, ‘‘Russell’s Use of Truth-Tables,’’ Russell 17 (1997): 11–26, and also Irving Anellis, ‘‘The Genesis of the Truth-Table Device,’’ Russell 24 (2004): 55–70.

Logic as if tautologous p

q

r

1

t

t

t

2

f

t

t

3

t

f

t

4

f

f

t

5

t

t

f

6

f

t

f

7

t

f

f

8

f

f

f

121

p

q 5

7

6

1 2

3 8 4

r

Fig. 4.5

propositional formula.31 John Venn did this in the 1880s. The general idea of a diagrammatic procedure for checking syllogistic and class inferences is quite old and dates at least to Euler (1761) if not also to the Stoics. John Venn supplanted Euler’s limited techniques.32 His propositional diagrams are very instructive, for they picture truth-conditions by means of a spatial representation. This is particularly illuminating when it comes to Wittgenstein’s philosophical views. Truth-table and Venn diagrammatic representations have a relationship that is given, for three propositional letters, in Fig. 4.5. Wittgenstein’s idea that a proposition has two poles is nicely represented spatially in Venn’s diagram by the fact that the formation circles have an inside region and an outside region. Venn’s propositional diagrams capture truthtables. Fixing on the ordering of the values as in the above truth-table, Wittgenstein’s Tractatus represents p  q as: 1; 2; 3; 4 ðt; t; f; tÞðp; qÞ:

Venn’s spatial method represents it as: p

31

32

q

This occurs in McGuinness, Wittgenstein, p. 162. It is also found in Gordon Baker, Wittgenstein, Frege, and the Vienna Circle (New York: Blackwell, 1988), p. 86. More recently, it is suggested by P. M. S. Hacker, Wittgenstein’s Place in Twentieth-Century Analytic Philosophy (Oxford: Blackwell, 1996), p. 280. John Venn, ‘‘On the Diagrammatic and Mechanical Representation of Propositions and Reasonings,’’ Philosophical Magazine (July, 1880).

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On the Venn diagrammatic method, one shades just those regions whose value in the tabular method get an f. Thus, our example shades area 3 alone. Venn showed how propositional deductive inferences can be checked by overlaying pictorial diagrams of the premises and checking the resulting diagram for the truth-conditions of the conclusion. For instance, let us apply the technique to check the following argument for validity. We have: p

q

p ⊃ (q & r ) ∼r ∼p r

The analogous technique with a truth-tabular representation is illustrated below: 1, 2, 3, 4, 5, 6, 7, 8 (t, t, f, t, f, t, f, t) (p, q, r) (f, f, f, f, t, t, t, t) (r, r, r) (f, f, f, f, f, t, f, t) (p, p, p)

Pairing the two premises together requires that we look only at those columns that both get the value t, namely, 6 and 8. The argument is valid, since in every such column the conclusion also gets the value t. Of course, for arguments with many primitive propositional signs, Venn’s diagrams become increasingly impractical. Rectangular graphs extending Venn’s diagrams for n-many terms were presented by Allan Marquand and published in 1881.33 Modern truth-tables are little more than a notational variant. The overlapping circles of Venn’s propositional diagrams are particularly illuminating for Wittgenstein’s conception of logic. (Indeed, the Tractatus and Notebooks 1914–1916 use a spatial analogy to represent negation in just the way found in the Venn diagrams.)34 Venn’s spatial diagrams require that the primitive propositional symbols are assigned to the fundamental regions of the diagram. (The circles must not be tagged by compound formulas.) Otherwise, the diagrams would no longer show the proper structural (logical) relationships. Consider the following: 33 34

See Gardner, Logic Machines and Diagrams, p. 43. Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 30.

Logic as if tautologous p

123 q

∼p p⊃q p⊃q

The diagram fails to represent the proper inferential relations because p  q is not logically independent of p and q. This feature corresponds nicely with the Tractarian thesis of the logical independence of all (atomic) states of affairs (TLP 2.061). Determinateness of sense (truth-conditions) requires the logical independence of atomic statements. At Tractatus 6.12 Wittgenstein writes that ‘‘tautologies show the logical form of language and the world.’’ That an expression is a tautology or a contradiction is a formal notion built into the syntax of the ideal language. Tautologies and contradictions are to be shown syntactically in the logically perfect language; they are ‘‘limiting cases – indeed the disintegration – of the combination of signs’’ (TLP 4.466). Wittgenstein writes that ‘‘tautologies and contradictions show that they say nothing’’; they ‘‘lack sense’’ but are ‘‘part of the symbolism’’ (TLP 4.461). Venn’s propositional diagrams nicely illustrate Wittgenstein’s ideas. In a logically perfect language the status of an expression as tautologous, contradictory, or contingent is built into (shown by) the syntactic conditions for the representation of genuine assertions. A propositional tautology shades nothing, and contingencies shade some but not all areas. In Venn diagrams, tautologies and contradictions are not genuine statements. A genuine statement is made by shading some, but not all, areas of the diagram. All tautologies have nothing shaded; they are just the overlapping circles – the scaffolding. Indeed, in Venn’s propositional diagrams, formulas that are logically equivalent have exactly one and the same representation. The same Venn diagram represents (p & q), p v q, (p  q), etc. At Tractatus 5.513 Wittgenstein wrote that ‘‘it is manifest that ‘q .&. p v p’ says the same thing as ‘q,’ and ‘p v p’ says nothing.’’ This is realized in the Venn’s propositional diagrams (Fig. 4.6). The Venn diagram q .&. p v p is the same as the diagram for q. The tautology p v p does nothing to the shading. Venn’s propositional diagrams also show the relationship Wittgenstein saw between propositional logical necessity and tautologyhood. In every contingent proposition some regions of the diagram are shaded and some are not. The contingency of a genuine atomic statement (one that is neither tautologous or contradictory) is displayed immediately by the fact that its spatial diagram depicts it with an inside region and an outside region,

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p

q

Fig. 4.6

exactly one of which must be shaded. The contingency of a statement is given as an immediate structural feature of its representation in the Venn diagrams. The representation of a statement is the inverse image of the representation of its negation; one simply inverts the shading. Indeed, we saw that all and only logical equivalents are represented in exactly one and the same way. In the ideal system of representation (which we have glimpsed with the Venn diagrams) logical necessity, contingency, and logical falsehood are part of the symbolism itself. Tautologies are not statements of the ideal language because ‘‘the propositions of logic describe the scaffolding of the world, or rather they represent it’’ (TLP 6.124). Wittgenstein wrote: ‘‘A tautology leaves open to reality the whole – the infinite whole – of logical space; a contradiction fills the whole of logical space leaving no point of it for reality. Thus neither of them can determine reality in any way’’ (TLP 4.463). Wittgenstein did not invent truth-tables. Nor did he invent the idea of representing formulas by their truth-conditions. We found that in Venn’s diagrams. Nor, indeed, did he teach Russell the thesis that the logical particles do not stand for relations between entities. The logical particles in Principia are statement connectives, not predicate expressions as in Russell’s earlier logic of propositions. Wittgenstein hoped to make a quite different contribution. He hoped to demonstrate that a deductive calculus for logic can be supplanted by a representational system in which all and only logical equivalents have exactly one and the same expression. The representation of quantifier-free sentences in terms of their truth-conditions (or, alternatively, Venn’s representation) offers just such a notation. As Wittgenstein sees matters, systems that employ different logical particles ‘‘&,’’ ‘‘v,’’ ‘‘,’’ ‘‘,’’ etc., hide their formal (‘‘internal’’) nature. Wittgenstein attempted to exploit the truth-table representation of propositions as evidence for his view that a proper representation would reveal that tautologies and contradictions are scaffolding.

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Wittgenstein’s N-operator Logical representations such as Venn’s propositional diagrams which depict quantifier-free sentences in terms of their truth-conditions provide a notation in which all logical equivalents have exactly one and the same representation. This notation builds tautologies and contradictions into scaffolding. Wittgenstein hoped to extend this result to quantification theory with identity. Logically equivalent quantified propositions are to have exactly one and the same representation. For example, both (x)fx and (9x)fx are to have the same expression; similarly, (x)(fx & gx) is to have the same expression as (x)fx & (x)gx. Wittgenstein’s plan for achieving this was to invent a notation that would reveal the internal connections between universal quantification and conjunction, and between existential quantification and disjunction. Wittgenstein maintains that the concept all is not properly expressed by a universal quantifier. He writes (TLP 5.521): I dissociate the concept all from truth-functions. Frege and Russell introduced generality in association with logical product or logical sum. This made it difficult to understand the propositions ‘(9x)fx’ and ‘(8x)fx’ in which both ideas are embedded.

In this entry, Wittgenstein criticizes Russell and Frege for having failed to reveal the proper internal connections between the quantifiers and the propositional connectives. He writes (TLP 5.451): once negation has been introduced, we must understand it both in the propositions of the form ‘‘p’’ and in propositions like ‘‘(p v q),’’ ‘‘(9x)  fx,’’ etc. We must not introduce it first for the one class of cases and then for the other, since it would then be left in doubt whether its meaning were the same in both cases, and no reason would have been given for combining the signs in the same way in both cases.

The definitions of Principia’s section *9, definitions such as *9.01, *9.02, and *9.07, define tilde and disjunction signs for quantifiers in terms of different tilde and disjunctions signs as applied to quantifier-free formulas. The definitions were an unavoidable result of Principia’s philosophical explanation of the order component of the order\type indices on its predicate variables. The order component is explained by reference to the hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ as applied to formulas of the intended nominalistic semantics for the predicate variables of the calculus.35 The philosophical foundation of order lies in the hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ The hierarchy is generated from the

35

See chapter 2.

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nominalistic semantics which requires that the truth-conditions for quantified formulas depend upon the logically prior and independent truth-conditions for quantifier-free formulas. For this to be possible, the quantification theory of *9 must define subordinate occurrences of quantified formulas in terms of an equivalent formula in prenex formal form. Wittgenstein objected to Principia’s recursive definition of ‘‘truth.’’ His replacement is symbolism which reveals the internal connections between the use of quantifiers and the propositional logical particles. Wittgenstein adopts the view that the concept all is given by the use of a free variable.36 If Wittgenstein’s plan is to eliminate universal quantification in favor of conjunction and eliminate existential quantification in favor of disjunction, one might naturally ask why he did not use a truth-table approach to quantification theory. The general form of a truth-function is: ð1 ; . . .; 2 n Þðp1 ; . . .; pn Þ:

Each  stands for a truth-value and n is schematic, so that the form represents an arbitrary large number of argument positions. Now, in order to represent ‘‘(9x)fx,’’ one might think to put: ðt1 ; . . .; t2n 1 ; f2 n Þðfx1 ; . . .; fxn Þ:

The free variables do the work of generality. Similarly, for (x)fx one would have: ðt1 ; f2 ; . . .; f2 n Þðfx1 ; . . .; fxn Þ:

In this representation, (9x)fx has the same expression as (x)fx. Indeed, for (x)(fx & gx) we have ðt1 ; f2 ; . . .; f2 n Þðfx1 ; . . .; fxn ; gx1 ; . . .; gxn Þ:

This also represents the logically equivalent (x)fx & (x)gx. An important benefit of this approach is that it inherits all the merits of truth-tabular (or Venn-diagrammatic) representations of propositional truth-conditions. Tautologies are shown as scaffolding and this would extend to (9x)(fx v fx). There are insuperable difficulties in this approach, however.37 In order to express quantification in the form ( 1, . . .,  2n)(p1, . . ., pn), we must 36

37

See F. P. Ramsey, ‘‘Facts and Propositions,’’ in The Foundations of Mathematics and Other Logical Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (London: Harcourt, Brace & Co., 1931), p. 153. Fogelin argues that Wittgenstein would have objected to this approach because it is incompatible with TLP 4.128, 5.453, 5.553, which reject ‘‘privileged numbers’’ in logic. See Robert Fogelin, Wittgenstein, 2nd ed. (London: Routledge & Kegan Paul, 1987), p. 64. But given that n is used schematically for any arbitrary large (though finite) number, I don’t see how the approach is committed to privileged numbers.

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know where to position the t’s and f’s among the ’s. The positions, for arbitrarily large n, can always be known in cases where we have only monadic predicates. But when polyadic predicates are allowed this is not the case. For the quantified formulas involving polyadic predicates, there is no computable relationship in general between an increase in the number of objects in the range of the quantifiers and the positions of new t’s among the ’s in the truth-table representations. Once relation signs are allowed, the truth-table representation for n þ 1 cannot be determined from the truth-table for n. Surely Wittgenstein would have known these problems with truth-table representations of quantified formulas. Indeed, assuming that he knew them, we are led to the thesis that Wittgenstein introduced his N-operator notation to avoid the weaknesses of truth-tabular representations. The N-operator is offered as a notation in which all and only logical equivalents of quantification theory with identity have exactly one and the same representation. Before we explain how Wittgenstein thought that the N-operator notation accomplishes this striking result, we do well to ask how it recovers the result for quantifier-free expressions. How does the N-operator notation provide a symbolism in which all propositional (quantifier-free) logical equivalents have exactly one and the same representation? Tautologies and contradictions should be scaffolding, as in the Venn-diagrammatic and truth-table notations. Wittgenstein says little about the nature of operations. He does, however, distinguish operations from what he calls ‘‘functions’’ in the following passages: The occurrence of an operation does not characterize the sense of a proposition. Indeed, no statement is made by an operation, but only by its result, and this depends on the bases of the operation. (Operations and functions must not be confused with each other.) (TLP 5.25) A function cannot be its own argument, whereas an operation can take one of its own results as its base. (TLP 5.251)

These passages are far from clear. Surely a function f, like an operation, can take one of its own results, e.g., the value fx, as its base (as its own argument). So TLP 5.251 does not seem to distinguish functions from operations. Evidently, Wittgenstein has in mind the notion of a propositional function, not the mathematical notion of a function. Where ’ is a predicate variable (propositional function sign), Principia’s grammar does rule out ’(’yˆ) and ’(’) as ungrammatical. In Principia, an ontology of mathematical functions is not primitive. The primitive signs of the language are propositional function signs (i.e., predicate variables), not function signs. Descriptive mathematical function signs are defined by means of

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definite descriptions which use predicate variables. Likely, Wittgenstein’s point is simply that, unlike Principia, he adopts the mathematical notion of a function as a primitive. That is, his notion of an operation is just the mathematical notion of a function. Wittgenstein introduces his N-operator notation in the following rather cryptic passage which ties the notation to the truth-tabular presentation of a proposition. He writes:  NðÞ  is the negation of all the values of So instead of ‘(——T)(, . . .)’ I write NðÞ. the variable . (TLP 5.502)

Wittgenstein’s comments suggest that the N-operator notation is akin to Sheffer’s dagger notation. The two-placed Sheffer dagger p # q could be written as #(p, q), and this parallels Wittgenstein’s N(p, q). Indeed, for any propositional formula of a language with the logical signs, v, &, , , we can find an N-operator analog for the formula by first translating the formula into dagger notation and then replacing each occurrence of the dagger with N. (More conveniently, we can translate to dagger and tilde, and replace each with N.) But the differences between the N-operator notation and the Sheffer dagger are quite significant. If the sign N(p1, p2, . . ., pn) is a function sign in the mathematical sense, then it is a term, not a formula. This is important. The expression #(p, q) is a formula. Wittgenstein’s N(p, q) is a term which names (or represents) the truthconditions for such a formula. It is akin to the truth-tabular representation of a formula which pictures truth-conditions. Unfortunately, this is but the first of several puzzles that arise in attempting to understand the N-operator notation. The following passages are of concern: What values a propositional variable may take is something that is stipulated. The stipulation of values is the variable. (TLP 3.16) To stipulate values for a propositional variable is to give the propositions whose common characteristic the variable is. The stipulation is a description of those propositions. The stipulation will therefore be concerned only with symbols, not with their meaning. And the only thing essential to the stipulation is that it is merely a description of symbols and states nothing about what is signified. How the description of the propositions is produced is not essential. (TPL 3.317) When a bracket expression has propositions as its terms – and the order of the terms  ‘’ is a inside the brackets is indifferent – then I indicate it by a sign of the form ‘ðÞ.’ variable whose values are terms of the bracketed expression and the bar over the variable indicates that it is the representative of all its values in the brackets. E.g., if  has the three values P, Q, R, then

Logic as if tautologous

129  ¼ ðP, Q, RÞ: ðÞ

What the values of the variables are is something that is stipulated. The stipulation is a description of the propositions that have the variable as their representative. How the description of the terms of the bracketed expression is produced is not essential. We can distinguish three kinds of description: 1. Direct enumeration, in which case we can simply substitute for the variable the constants that are its values; 2. giving a function fx whose values for all values of x are the propositions to be described; 3. Giving a formal law that governs the construction of the propositions, in which case the bracketed expression has as its members all the terms of a series of forms. (TLP 5.501)  ¼ p (not p); if it has two values, then NðÞ  ¼ If  has only one value, then NðÞ p  q (neither p nor q). (TLP 5.51)

Wittgenstein’s intent in using the bar over  is not clear. Is the Tractatus introducing several syntactically distinct N-operator symbols at 5.501, or is it that we have a one-placed symbol? Does Wittgenstein mean that in the one-placed expression N() the variable can take several values at once? Or does he mean there can be several different N-operator expressions N(), N(1 2), N(1 2 3), and so on? The best answer is that the syntactic expression N() of the N-operator is one-placed. Wittgenstein allows, however, that we can place in the position of  a list, or a recipe, or a schema, which determines what are to be the base(s) of the operation. Thus, for example, when  is assigned to the list ‘‘p, q, r’’ we may write N(p, q, r) as if we had a three-placed expression N(1 2 3), with p, q, and r in the respective places. Wittgenstein’s intent breaches syntax as we know it, and it is uncomfortable to have to get along with a wink and a nod. In any event, on Wittgenstein’s view, the positions of p and q in the value of N(p, q) are clearly supposed to be unordered so that the sign N(p, q) is somehow the same as the sign N(q, p). Indeed, we shall see that N(NN(p, q), r) is to be regarded, in some sense, as the same as N(p, q, r).38 The N-operator notation is a picture of the truth-conditions of a formula – something akin to a propositional Venn diagram. Realizing this, we can see what Wittgenstein had in mind in thinking that N-operator symbolism recovers the characteristic features of truth-table representations – namely, that all and only propositional logical equivalents have the same representation and tautologies are scaffolding. We have the following rules of operation: (1) N(1, . . ., n) ¼ N(i, . . ., j), 1  i  n, and 1  j  n. (2) N(. . ., . . ., . . .) ¼ N(. . .. . .). 38

´ ˆ me Sackur, Formes et faits (Paris: J. Vrin, 2005), p. 138. This is also noted in Jero

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(3) N(. . .NN(1, . . ., n). . .) ¼ N(. . .1, . . ., n, . . .). (4) N(. . .N(. . ., . . ., N, . . .). . .) ¼ N(. . . .). (5) NN(, N(1, . . ., n)) ¼ N(N(, N1), . . ., N(, Nn)).39 The addition of an N to both sides of any among (2)–(5) is allowed as well.40 These rules assert the sameness of certain practices of operation. They are not, therefore, identity statements. Most of the rules are straightforward. Clause (1) tells us that arguments to N may be permuted. Clause (2) permits the deletion of repeated arguments. Clause (3) says that if NN(1, . . ., n) is an argument to the N-operator, then we can delete this argument and add 1, . . ., n to the arguments. Clause (4) tells us that we may delete or insert any argument of the form N(. . ., . . ., N, . . .). Finally, (5) applies to distributed forms. For example, NN(p, N(q, r)) ¼ N(N(p, Nq), N(p, Nr)). This is the analog of the rule of distribution for expressions p .v. q & r and p v q .&. p v r. What explains distribution is the fact that both sides are picked out by N(N(, )). Hence, when N(, ) is regarded as picking out the list N(, 1), . . ., N(, n), the form yields NðNð; N1 Þ; . . .; Nð; Nn ÞÞ:

The same form also yields NNð; Nð1 ; . . .; n ÞÞ

when  picks out N(1, . . ., n). The difference in the two lies only in the designation of values. By application of (1)–(5) we can see how Wittgenstein thought that the N-operator recovers the features of truth-table representations and Venn’s propositional diagrams. The N-operator notation gives a picture of the truth-conditions. Where A is a quantifier-free formula of Principia, let t[A] be its expression in dagger notation.41 The following examples illustrate the transformations that show that logical equivalents have exactly one and the same representation in N-notation.

39

40 41

Wittgenstein’s N-operator anticipates, and has much in common with, the ‘‘primary algebra’’ of George Spencer Brown. Brown points out the important difference between his symbolism and the dyadic Sheffer stroke, and he intimates that all and only logical equivalents have the same notation. He admits to having been influenced by the Tractatus, but he does not discuss the N-operator. See George Spencer Brown, Laws of Form (New York: Julian Press, 1967), pp. 109, 115. For convenience let us use ‘‘NN(1, . . ., n)’’ instead of ‘‘N(N(1, . . ., n)).’’ The dagger expression of ‘‘p’’ is ‘‘p # p’’ but we keep the tilde for convenience.

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Example 1: t½p & q ¼  p # q NðNp; NqÞ ¼ NðNq; NpÞ by ð1Þ  q # p ¼ t½q & p Example 2: t½p v p ¼  ðp # pÞ NNðp; pÞ ¼ NNp by ð2Þ ¼ NðNp; NpÞ by ð2Þ  p # p ¼ t½ p & p Example 3: t½q :&: p v  p ¼  q # ðp # pÞ NðNq; NNNðp; NpÞÞ ¼ NðNq; Nðp; NpÞÞ by ð3Þ ¼ NNq by ð4Þ  ð q # qÞ ¼ t½ q

This example reveals what Wittgenstein had in mind at TLP 5.513 when he wrote that it is ‘‘manifest that ‘q: p v p’ says the same thing as ‘q’, and that p v p says nothing.’’ Example 4: t½p :  : q  r ¼  ð p # ð q # rÞÞ NNðNp; NNðNq; rÞÞ ¼ NNðNp; Nq; rÞ by ð3Þ ¼ NNðNq; Np; rÞ by ð1Þ ¼ NNðNq; NNðNp; rÞÞ by ð3Þ  ð q # ð p # rÞÞ ¼ t½q:  : p  r Example 5: t½p :v: q & r ¼  ðp : # :  q # rÞ NNðp; NðNq; NrÞÞ ¼ NðNðp; qÞ; Nðp; rÞÞ by ð5Þ; ð3Þ ¼ NðNNNðp; qÞ; NNNðp; rÞÞ by ð3Þ  ð ðp # qÞ : # :  ðp # rÞÞ ¼ t½p v q :&: p v r

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Example 6: t½p v q :&: r  r ¼  ðp # qÞ #  ð r # rÞ N½NNNðp; qÞ; NNNðNr; rÞ ¼ N½Nðp; qÞ; NðNr; rÞ by ð3Þ ¼ NNðp; qÞ by ð4Þ  ðp # qÞ ¼ t½p v q Example 7: t½p ¼ p NNp ¼ NfNp; NðNp; Nq; pÞg by ð4Þ ¼ NfNp; NðNp; NNðNq; pÞÞg by ð3Þ ¼ NNfNðNp; pÞ; NðNp; NðNq; pÞÞg by ð5Þ ¼ NðNp; NðNq; pÞÞ by ð4Þ and ð3Þ  q # p : # :  p ¼ t½q  p :&: p

We saw that tautologies drop out of Venn’s diagrams. All logical equivalents have the same Venn diagram, and so p v q .&. r  r has the same diagram as p v q. Example 6 reveals that the same result is embedded in the N-operation. The tautologous conjunct adds nothing; and every subordinate occurrence of N(. . .p. . . Np . . .) drops out. When expressed by N-notation, every atomic propositional sign contains within itself its logical (truth-conditional) relationships with all the propositional signs. Wittgenstein wrote (TLP 3.42): A proposition determines only one place in logical space; nevertheless the whole of logical space must already be given by it. (Otherwise negation, logical sum, logical product, etc., would introduce more and more new elements – in coordination.) (The logical scaffolding surrounding a picture determines logical space. The force of a proposition reaches through the whole of logical space.)

The N-operator sign for p implicitly contains every propositional sign q, r, s, . . . of the entire language. In N-operator notation, the sign for p is taken to be the same as the sign for its logical equivalent p :&: q  q .&. r  r .&., . . . This is what Wittgenstein had in mind in saying that the whole of logical space is ‘‘given by’’ every propositional sign. If we interpret the N-operator as picturing truth-conditions and follow TLP 5.51, then all and only tautologies have their truth-conditions pictured by NN(. . ., . . .N, . . .). This is easily demonstrated. If a formula has this form it clearly has the truth-conditions of a tautology. Suppose a formula A is a tautology. A formula in conjunctive normal

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form42 is tautologous if and only if every conjunct is a disjunction containing some formula and its negation. Thus the conjunctive normal form of A can be expressed as follows: p1 v  p1 v C1 : & : p2 v  p2 v C2 : & :; . . .; : & : pn v  pn v Cn :

When the truth-conditions of this are pictured in N-notation, the result is: NðNðp1 ; Np1 ; C1 Þ; Nðp2 ; Np2 ; C2 Þ :; . . . ; : Nðpn ; Npn ; Cn ÞÞ:

By rule (4) we can eliminate all of the components except one, and arrive at: NNðpi ; Npi ; Ci Þ;

where 1  i  n. This has the form NN(. . ., . . .N , . . .). In this way, calculation by means of the N operator forms a decision procedure for tautologyhood. In Wittgenstein’s view, propositional logic is a practice of calculating with equations using the N-operator. We can now see how he imagined this to be the case. Upon first receiving the Tractatus, Russell wrote to Wittgenstein asking several questions. In his remarks, Russell reveals that he thinks of the N-operator as a sort of generalized Sheffer dagger. At TLP 6 Wittgenstein wrote:  NðÞ .  The general form of a truth-function is ½ p; ; This is the general form of a proposition.

Russell asks the following question about this passage:  NðÞ .’  Yes, this is one way. But could one not do ‘General truth-function ½ p; ;  equally well by making NðÞ‘at least one value of  is false,’ just as one can do equally well with p v q and with p  q as fundamental? I feel as if the duality of generality and existence persisted covertly in your system.43

Wittgenstein’s reply was this:  may also be made to mean p v q v r v . . . But this You are right that NðÞ doesn’t matter! I suppose you don’t understand the notation . It does not mean ‘‘for all values of  . . .’’44

Rules (1)–(5) are equally valid whether we regard N as akin to a generalized Sheffer dagger or its dual as a generalized Sheffer stroke. The 42

43 44

A formula is in conjunctive normal form if and only if it is a conjunction (possibly degenerate), each conjunct of which is a disjunction (again possibly degenerate) of propositional letters or their negations. A single propositional letter or its negation or a disjunction of such is a degenerate conjunction. Bertrand Russell, ‘‘Letter of 13 August 1919,’’ Russell 10.2 (1990): 108. Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 131.

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truth-conditions of tautologies will be pictured differently, of course. If we interpret the N-operator as picturing truth-conditions akin to a generalized stroke, then all and only tautologies have their truth-conditions pictured by Nð. . . ; . . . N; . . .Þ:

Perhaps it was because (1)–(5) apply equally on either approach that Wittgenstein felt justified in his reply that Russell’s concern doesn’t matter. Quantification and the N-operator Wittgenstein’s goal, of course, was to find a symbolism for quantification theory with identity in which all logical equivalents have exactly one and the same representation. If one takes the sign ‘‘¼’’ for identity to be a primitive relation sign like ‘‘R,’’ then the logical (tautological) nature of statements such as ‘‘(x)(x ¼ x)’’ and ‘‘x ¼ x’’ is lost; the statements appear as though they have the same form as contingent statements such as ‘‘(x)Rxx’’ and ‘‘Rxx,’’ and special axioms have to be introduced to justify inferences governing the identity sign. In Wittgenstein’s view, identity is a logical notion and therefore must be shown in the formal grammar of an ideal language. But how can identity be built into logical grammar? In a letter to Russell dated December 1913 Wittgenstein wrote: ‘‘The question of the nature of identity cannot be answered until the nature of tautologies is explained. But that question is the fundamental question of all logic.’’45 It took some time for Wittgenstein to settle on a plan. In the language of Principia Mathematica, the identity sign is defined as follows:

13:01 x ¼ y ¼ df ð’Þð’x  ’yÞ:46

One might at first think that Wittgenstein would be very attracted to this definition. Employing the definition, the offensive ‘‘(x)(x ¼ x)’’ becomes ‘‘(x)(’)(’x  ’x)’’ which looks much more like a tautology on the grounds that it is but a universal generalization of a tautologous form – namely, ’x  ’x. Similarly, if we apply the definition of the identity symbol to ‘‘(y)(9x)(x ¼ y)’’ we arrive at ‘‘(y)(9x)(’)(’x  ’y).’’ This too is arrived at by generalization on a tautologous form ‘‘’x  ’x.’’ The advantages of this approach to the problem of showing that identity is part of logical grammar are clear. But Wittgenstein rejected it, writing (TLP 5.5302): 45 46

Ludwig Wittgenstein, ‘‘Notes Dictated to G. E. Moore in Norway,’’ Notebooks 1914–1916, 2nd ed., p. 129. This expression omits Principia’s notion of predicativity which is not germane to the present discussion.

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Russell’s definition of ‘¼’ is inadequate, because according to it we cannot say that two objects have all their properties in common. (Even if this proposition is never correct, it still has sense.)

This passage wants clarification. Though Whitehead and Russell were realists concerning universals, they intended that the attribute (propositional function) variables in the Principia be interpreted in terms of a nominalistic semantics. Accordingly, the propositional function variables in *13.01 have an intended interpretation quite different from what Wittgenstein had in mind when he spoke of ‘‘material properties.’’ Moreover, in Russell’s 1912 paper ‘‘On the Relations of Universals and Particulars,’’ Russell himself argued that two distinct particulars might have all their properties in common.47 Wittgenstein’s comment therefore seems to saddle Russell with a position he did not hold.48 A better interpretation is needed, and a simple one is at hand. When properly understood, TLP 5.5302 is not criticizing definition *13.01 as it functions in Principia. Wittgenstein was pointing out that in his efforts to build identity into grammar he could not employ a definition of the identity sign akin to that of Principia. That is, if the identity sign is defined to mean coexemplification of all the same material properties, the definition would not be adequate. Russell certainly would agree. Wittgenstein’s answer to the problem of building identity into scaffolding is his informal semantic interpretation of quantifiers and variables. Each free individual variable has an assignment which makes it such that its referent is distinct from every other free individual variable. Though Ramsey found translation procedures between the language of exclusive quantifiers and predicate logic with identity, there has been little progress toward the development of a deductive system for exclusive quantifiers. Carnap worried that such a system encounters serious problems with respect to keeping track of the exclusiveness of the variables when quantifiers are instantiated and generalized.49 In our Appendix A, a deductive system is set out which addresses these problems by introducing quantificational expressions such as (8xz1, . . ., zn)A, where the variables z1, . . ., zn are free 47

48

49

Bertrand Russell, ‘‘On the Relations of Universals and Particulars,’’ in Logic and Knowledge: Essays by Bertrand Russell 1901–1905, ed. R. C. Marsh (London: Allen & Unwin, 1977), p. 118. Russell corroborates this in a marginal comment to his copy of F. P. Ramsey, The Foundations of Mathematics and Other Logical Essays. Annotating Ramsey’s comment that ‘‘the definition makes it self-contradictory for two things to have all their elementary properties in common,’’ Russell wrote: ‘‘Don’t agree.’’ The Collected Papers of Bertrand Russell, vol. 10, A Fresh Look at Empiricism 1927–1942, ed. John G. Slater (London: Routledge, 1996), p. 106. Rudolf Carnap, The Logical Syntax of Language, 2nd ed. (London: Routledge & Kegan Paul, 1956), p. 50.

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and must include all variables free in A except x. In a universal instantiation of (8xz1, . . ., zn)A, x must be replaced by a variable distinct from each among z1, . . ., zn. For the present discussion, it will suffice to examine the role Wittgenstein’s exclusive reading of free individual variables plays in his thesis that the N-operator recovers quantification theory. The following passages in the Tractatus concerning the N-operator are important to our discussion. We shall find that they can be understood as consistent with the goal of representing all and only logical equivalents of quantification theory in exactly one and the same way. If  has as its values all the values of a function fx for all values of x, then  ¼ (9x)fx. (TLP 5.52) NðÞ  NðÞ .  This is the general form of a The general form of a truth-function is ½ p; ; proposition. (TLP 6) What this says is just that every proposition is a result of successive applications to  (TLP 6.001) elementary propositions of the operation NðÞ.

Reading TLP 5.52, one may take the syntax of the expression N(fx) to indicate that the N-operator operates on every closed formula of the form fx. Fogelin, for example, takes N(fx) to form a formula that is the possibly infinite conjunction of the negations of every proposition of the form fx.50 Thus, N(fx) signifies the formula N(fa, fb, fc, fd, . . .) where a, b, c, and so on are all the names of the language. In this way, Fogelin maintains, Wittgenstein intended to express (9x)fx, which is the same as (x) fx. This amounts to a substitutional interpretation of quantification, and it not clear that this was Wittgenstein’s intent. The existence of the expression (9x)fx for quantification must not assure the existence of names for the objects in the range of the quantifier. Wittgenstein is explicitly committed to the rejection of any view that would make logic depend upon the existence of an actual infinity of contingent particulars. For similar reason, it is doubtful that Wittgenstein intended a substitutional account of quantification. It would commit logic to the actual existence of an infinite conjunctive or disjunctive formula. Moreover, if there are nondenumerably many entities in the range of the quantifier, there cannot be a name for every object. (Wittgenstein may, however, have rejected the Cantorian proofs of the nondenumerability of entities. Real numbers, classes, and the like, after all, are not entities for Wittgenstein.) In any event, given Fogelin’s reading, we shall want to know how to capture the truth-conditions of (9x)fx. It will be necessary to be able to confine the scope of the generality of the variable. Otherwise, NN(fx) 50

Fogelin, Wittgenstein, pp. 57 and 62.

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would render the truth-conditions of (x)fx. Wittgenstein was not unaware of the issue of confining the scope. He wrote (TLP 4.0411): If, for example, we wanted to express what we now write as ‘‘(x).fx’’ by putting an affix in front of ‘‘fx’’ – for instance by writing ‘‘Gen.fx’’ – it would not be adequate: we should not know what was being generalized. If we wanted to signalize it with an affix ‘‘g’’ – for instance by writing ‘‘f(xg)’’ – that would not be adequate either: we should not know the scope of the generality sign. If we were to try to do it by introducing a mark into the argument places – for instance by writing ‘‘ðG; GÞFðG; GÞ’’ – it would not be adequate: we should not be able to establish the identity of the variables. And so on. All these modes of signifying are inadequate because they lack the necessary mathematical multiplicity.

Perhaps we might introduce braces to confine scope. Then NN(fx) which expresses (x)fx is distinguished from N[N(fx)] which expresses (x) fx and (9x)fx. Interestingly, Fogelin maintains that Wittgenstein’s N-operator notation must be inadequate to express the formulas of ordinary quantification theory because it cannot capture the right scope distinctions. Fogelin maintains that the generality of the variables must be confined within one and the same scope. On Fogelin’s reading, the N-operator notation can express homogenous multiply general formulas, such as (x)(y)fxy and (9x)(9y)fxy, but not mixed multiply general formulas such as (9x)(y)fxy. In order to arrive at (9x)(9y) fxy (where the quantifiers are interpreted  Thus, N(fxy) inclusively) Fogelin puts fxy for Wittgenstein’s  in NðÞ. expresses (9x)(9y) fxy. This is just equivalent to (x)(y)fxy, and, as we see, the scope of the generality has not been confined. Fogelin does think that it is possible to express the homogenous multiply general formula (9x)(9y) fxy by, as he puts it, bringing (9x)(9y)fxy under the operator N.51 He does not say how this is to be done. As we saw, this requires confinement of scope, otherwise N(N(fxy)) expresses (x)(y) fxy. Using braces to confine generality, N[N(fxy)] expresses (x)(y) fx and so (9x)(9y)fxy. But if we use braces, what forces us to confine all variables with one and the same scope? Why not allow innermost brackets to confine the right-most variable only? Then N[N[fxy]] would capture (9x)(y)fxy after all. The explanation lies in Fogelin’s reading of Wittgenstein’s comment that N() expresses (9x)fx, when  is assigned all propositions of the form fx. Fogelin concludes that in any embedded occurrence, the expression fxy always represents faa, fab, fba, fac, and so on. Thus, on 51

Ibid., p. 79.

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Fogelin’s reading, fxy plays the same role in N[N(fxy)], which we used to express (9x)(9y)fxy, and in NN(fxy), which expresses (x)(y) fxy. This is why Fogelin thinks that the scope of the variables will always be confined together.  Geach agrees with Fogelin that the original N-operator notation NðÞ is expressively inadequate for quantification theory. Geach’s remedy is to extend of the notional resources of the Tractatus. Geach takes the N-operator to be flanked by class terms, and he modifies the original Tractarian notation to accommodate this view. Geach’s enhancement of the Tractarian notation commits the Tractatus to an ontology of classes of formulas. Geach acknowledges that his interpretation runs against Wittgenstein’s rather unequivocal rejection of the view that a theory of classes is part of logic. ‘‘Wittgenstein was exaggerating,’’ says Geach, ‘‘when he said that the theory of classes is altogether superfluous in mathematics (6.031) for he cannot get on without classes of propositions. But his class theory is very rudimentary: it has no null class, nor has it classes of classes; and by the same token it can never generate any Russellian Paradoxes.’’52 On Geach’s construal, N(fx) is a bad notation and better replaced by N(x:fx) where x:fx represents a class of quantifier-free sentences each of which is of the form fx. For example, to express (9x)(y)fxy, Geach puts N(N(x:N(y:N(fxy)))). To see how this is arrived at, consider the propositional model of (9x)(y)fxy (interpreted inclusively) in an n-object domain (where a1, . . ., an are names): fða1 ; a1 Þ&; . . .; & fða1 ; an Þ : v: fða2 ; a1 Þ&; . . .; & fða2 ; an Þ : v; . . .; v: fðan ; a1 Þ&; . . .; & fðan ; an Þ:

Geach’s notation represents each disjunct f(ai, a1) &, . . ., & f(ai, an) by N(y:N(fai,y)). So he has: Nðy : Nfða1 ; yÞÞ : v; . . .; v : Nðy : Nfðan ; yÞÞ:

Next Geach represents this disjunction and arrives at his N(N(x:N(y:N (fx, y)))). Consider the more simple example of (x)fx which Geach expresses by N(x:N(fx)). Geach considers the possibly infinite class x:(N(fx)) of closed formulas (sentences) N(fa), N(fb), N(fc), etc., each obtained by replacing x in N(fx) by a name. Then he considers N(x: (N(fx))). This tells us to take as bases of N the sentences which are members of the class x:N(fx), and so we arrive at the possibly infinite conjunction N(fa) & N(fb) & N(fc) & . . . Then, discharging the many N’s in this formula, one gets the possibly infinite (fa) & (fb) & (fc) & . . . As Geach 52

Peter Geach, ‘‘Wittgenstein’s Operator N,’’ Analysis 41 (1981): 169.

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observes, in the last step no one instance of the operator N is applied infinitely many times. Fogelin argues that Geach’s enhancement of the Tractarian notion could not have been adopted by Wittgenstein. Fogelin reads Geach’s class x: N(fx) as ‘‘a set of propositions that is the result of possibly infinitely many (unordered) applications of the operator N to a possibly infinite set of propositions.’’53 That is, Fogelin sees x:N(fx) as the class of closed formulas each member of which is obtained by putting a name for x in fx and applying N to the result. If the class is infinite, then there have been infinitely many applications of N. Thus, even in Geach’s expression of (x)fx as N(x:N(fx)), Fogelin objects on the grounds that it is committed to possibly infinitely many applications of N. Fogelin therefore rejects Geach’s approach, pointing out that at TLP 5.32 we find: All truth-functions are results of successive applications to elementary propositions of a finite number of truth-operations.

In Fogelin’s view, Geach’s proposal permits infinitely many applications of the N-operation and also violates the demand of successiveness. ‘‘If the set of base operations is infinite,’’ writes Fogelin, ‘‘then nothing will count as the immediate predecessor of the final application of the operation N in the construction of a universally quantified proposition.’’54 Geach rightly pointed out that Fogelin misreads his class notation x: N(fx). It does not require infinitely many applications of the N-operation. But Fogelin is unrepentant. His criticism of Geach is based on an interpretation of the role fx plays in x:N(fx). He rejects Geach’s reading of x: N(fx) because he maintains that N(fx) in x:N(fx) cannot be used to pick out formulas N(fa), N(fb), and so on. In Fogelin’s view, innermost N in Geach’s x:N(fx) is not a constituent of the propositional function N(fx) but must operate on each of the propositions picked out by fx.55 He concludes that Geach has not found a way to enhance the Tractarian N-operator notation that would secure its expressive adequacy for quantification theory. Fogelin holds that only fx, fxy, and so on, where f is a primitive predicate letter of the language, can pick out formulas. Geach rejects this view. This is the crux of the dispute between them. What then of Fogelin’s objection that Geach violates Wittgenstein’s criterion of successiveness? At TLP 6, Wittgenstein gave the general form  NðÞ .  Wittgenstein then writes at TLP of a truth-function as follows: ½ p; ; 6.001 that this general form says that ‘‘every proposition is the result of  successive applications to elementary propositions of the operation NðÞ.’’ He follows up with this (TLP 6.01): 53

Fogelin, Wittgenstein, p. 80.

54

Ibid., p. 81.

55

Ibid.

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Therefore the general form of an operation O0 ð Þ is  NðÞ Þ   NðÞ  0 ð Þ ð¼ ½ ; ; ½; This is the most general form of transition from one proposition to another.

Now it is far from clear that Wittgenstein means to say that every oper NðÞ  0 ð ation O0 ð Þ has the general form ½; Þ for this would make every operation a transition between propositions. Perhaps there are other sorts of operations. We will take up this matter again in our discussion in chapter 5 of number. The safe interpretation is that Wittgenstein is offering  NðÞ  0 ð ½; Þ as the general form of an operation O0 ð Þ which enables the ‘‘transition from one proposition’’ to another. Anscombe interprets the general form to be the general term of a con NðÞ  is a particular secutive series of truth-functions. She says that ½ p; ; example of Wittgenstein’s statement of the ‘‘general term’’ of a formal (consecutive) series.56 Wittgenstein wrote: If an operation is applied repeatedly to its own results, I speak of successive applications of it. (‘‘O’O’O’a’’ is the result of three successive applications of the operation ‘‘O’  to ‘‘a’’.) In a similar sense, I speak of successive application of more than one operation to a number of propositions. (TLP 5.2521) Accordingly, I use the sign ‘‘[a, x, O’x]’’ for the general term of the series of forms a, O’a, O’O’a, . . . This bracketed expression is a variable: the first term of the bracketed expression is the beginning of the series of forms, the second is the form of a term x arbitrarily selected from the series, and the third is the form of the term that immediately follows x in the series. (TLP 5.2522)

In these passages, Wittgenstein is speaking of the general term of a series. Indeed, he writes at TLP 5.2523 that the concept of successive applications of an operation is equivalent to the concept ‘‘and so on.’’ Moreover, Wittgenstein observed that operations may be repeated to form a consecutive series of expressions. In the case of a one-placed operation O(x), Wittgenstein writes ½x; ; OðÞ

as an abbreviation for ½O0 ðxÞ; Ov ðxÞ; Ovþ= ðxÞ

where x is the first term of the series. Anscombe concludes that

56

See G. E. M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 2nd ed. (New York: Harper & Row, 1959), p. 132.

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 NðÞ  ½ p; ;

can be replaced by ½ p; Nn ðpÞ; Nnþ1 ðpÞ

where the ‘‘first’’ term of the series is some set of atomic propositions picked out by p. Thus, for example, if p picks out just p and q, then Anscombe hopes to show that a series of truth-functions of p and q can  be generated by ‘‘successive applications’’ of the operation NðÞ. A serial ordering is one that is transitive, asymmetric, and connected. A serial ordering is not always a consecutive ordering, however. Rationals are serially ordered but are not consecutive – there is no rational immediately after 1/2. Moreover, the real numbers are serially ordered even though they are nondenumerable. Now, a ‘‘general term of a series’’ (if there is one for the series) relies upon the existence of a consecutive ordering. For instance, the general term of the consecutive numeric series 0, 1, 4, 9, 16, 25, . . .

is n2. The general term of the series relies on the consecutive ordering of the natural numbers, for we are to consecutively replace n in n2 by 0, then by 1, then 2, and so on. By identifying Wittgenstein’s ‘‘general form of a truthfunction’’ as a ‘‘general term of a formal series,’’ Anscombe argues that Wittgenstein was committed to holding that the truth-functions are correlated one to one with the natural numbers. She goes on to point out, however, that since Wittgenstein allows that there can be infinitely many bases, this conflicts with Cantor’s power-class theorem.57 If there are @o many propositions there will be nondenumerably many truth-functions and hence they cannot be in one–one correspondence with the natural numbers. Anscombe concludes that Wittgenstein’s claim of successiveness is mistaken.58  NðÞ  is the general term of the series Wittgenstein does not say that ½ p; ; of truth-functions. He says it is the general form of a truth-function. Wittgenstein explicitly says that the truth-functions can be serially

57

58

Ramsey also interprets Wittgenstein as allowing that there can be infinitely many bases. See F. P. Ramsey, ‘‘The Foundations of Mathematics,’’ in The Foundations of Mathematics and Other Logical Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (London: Harcourt, Brace and Co., 1931), p. 7. Anscombe rightly notes that there is reason to believe that Wittgenstein did not accept Cantor’s power-class theorem. Wittgenstein embraced predicative quantification and Cantor’s proof requires impredicative quantifiers. In Principia (1910) impredicative quantification is avoided only by relying on an axiom of Reducibility. Wittgenstein rejected Reducibility.

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ordered, at TLP 4.45 and 5.1. But this by no means entails that they have a general term (and thus are consecutively ordered). Wittgenstein calls a series that is ordered by an internal relation a ‘‘series of forms,’’ and he says that the number series is ordered by an internal relation (TLP 4.1252).  NðÞ  to be Hence, if Anscombe is right that Wittgenstein intended ½ p; ; the general term of a consecutive series of truth-functions, then we should expect there to be some internal relation which determines what selections to make and in what consecutive order to make them.59 Indeed, if one attempts to offer a general term for a consecutive series of truth-functions, one obviously cannot rely upon the consecutive ordering of the natural numbers. How would such a general term determine which propositions are to be picked out first and offered to the N-operator and which come  NðÞ  next? No principle for selection derives from Wittgenstein’s ½ p; ; interpreted as if it were the general term of the series of truth-functions. We do well to recall, however, Wittgenstein’s use of the bar. Perhaps he  NðÞ  to allow that  stands in for a (yet to be intended the bar in ½ p; ; determined) recipe for selecting the bases of the operation. In some instances of the schema – say in the case of the sixteen dyadic truth-functions – the bar stands in for a recipe which provides a consecutive ordering. In other instances, the bar stands in for a selection method which does not order the selections consecutively.60 Anscombe works her way through the dyadic truth-functions (i.e., those based on two propositional bases p and q). She takes N(p, q) so that we get the truth-function (t,f,f,f)(p, q). The next selection she makes is N(p, q) so that applying N() we get the truth-function (t,t,t,f)(p, q). She goes on to apply N() to get N(N(p, q), N(q, p)) and this yields (f,f,f,f)(p, q), and by various selections she eventually arrives at all the dyadic truthfunctions. Oddly, she does not follow the ordering of dyadic truthfunctions Wittgenstein set out at TLP 5.101. For any finite n, there are several ways to consecutively order the n-adic truth-functions. Based on n-many propositions there will be p-many (namely, 2n) rows of the truthtable, and 2p many truth-functions. Wittgenstein himself computes this at TLP 4.42. A typical algorithm for writing out the rows is to alternate t and f in the first row 2n many times, then alternate in twos, then fours, then eighths, etc., eventuating in the last row where 1/2 (2n) of t’s occur and then the same number of f’s. Alternatively, one can start with these

59

60

A general term of a consecutive series of natural numbers relies on the natural ordering of the numbers. This ordering is derived from an internal relation, since number is a formal concept in Wittgenstein’s view. I came to this idea during an interesting conversation with Dick Schmitt concerning the N-operator.

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and work in the other direction. But there are 16! many ways to order them.61 Each consecutive ordering will determine a distinct selection of bases to the N-operator, but none can be viewed as being given by an internal relation. Anscombe is mistaken. Wittgenstein’s assertion that all truth-functions can be reached by ‘‘successive’’ applications of the N-operator does not require Anscombe’s identification of his notion of ‘‘the general form of a truth-function’’ with the notion of the general term of a consecutive series of truth-functions. When Wittgenstein said that every truth-function is the result of successive applications of the N-operator, he may have simply meant that the N-operator is expressively adequate. As long as we reject Anscombe’s account, nothing in Geach’s embellishment of the N-operator conflicts with Wittgenstein’s criterion of successiveness. Fogelin maintains that the expressive adequacy of the N-operator notation entails the decidability of polyadic predicate logic. This is clear in his criticism of Geach. Fogelin writes: At 6.126 Wittgenstein tells us that ‘‘one can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol’’ . . . Since one can decide whether a proposition belongs to logic by calculating the logical properties of the symbol, the truth of a proposition of logic will not depend upon the disposition of independent logical objects. In is in this way that Wittgenstein’s commitment to a decision procedure goes hand in hand with his ‘‘fundamental idea’’ that ‘the logical constants are not representatives.’ (4.0312)62

In Fogelin’s view, the purpose of the N-operator notation was to reveal that the logical constants are not relation signs. Thus, he holds that if the N-operator is expressively adequate then it will show that the logical constants are not relation signs. Fogelin thinks that this entails decidability. But how does it entail it? On the modern view of logic, the logical particles do not stand for relation signs and this is certainly compatible with undecidability. Geach’s enhancement makes the N-operator notation expressively complete, but Geach rejects the connection Fogelin tries to forge between the apparatus of expressing quantified sentences by the N-operator and the issue of decidability. Fogelin is correct, however. There is a connection between the N-operator notation and Wittgenstein’s conviction that logic is decidable. But Fogelin has not properly established what the connection is. The N-operator notation is adequate to express any formula of propositional logic. It inherits this from the expressive adequacy of the Peirce/Sheffer stroke (and dagger), 61 62

There are sixteen different dyadic truth-functions (including tautology and contradiction). The number of permutations is thus 16! Robert Fogelin, ‘‘Wittgenstein’s Operator N,’’ Analysis 42 (1982): 127.

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which was proved by Post in 1920.63 The question of the expressive adequacy of the N-operator notation for (formulas of) quantification theory, however, is quite another matter. On the present interpretation, Wittgenstein intended the N-operator notation to provide an apparatus that represents all and only logical equivalents in one and the same way. This forges a connection between expressive adequacy and decidablity. If the N-operator notation succeeds in expressing every logical equivalent in exactly one and the same way, then it has to fail to be expressively adequate, for polyadic predicate logic is not decidable. Unfortunately, neither Geach’s nor Fogelin’s interpretations show how Wittgenstein could have thought that the N-operator notation provides a means of expressing all and only logical equivalents of quantification theory in exactly one and the same way. A new interpretation is needed. How could Wittgenstein have thought that the N-operator notation is adequate for the expression of quantification? Recall that many-placed N-operator expressions are obviated by the fact that Wittgenstein allows himself to view the expression N(p, q, r) as composed of a one-placed operator sign N() with the variable  assigned to p and q and r. At the same time, he allows himself to view it as composed of a triadic operator sign N(1, 2, 3) with p and q and r at their respective positions. This is  can be illegitimate. But let us proceed anyway. The bar above  in NðÞ used to abbreviate an n-placed expression N(1, . . ., n), where each variable among 1, . . ., n themselves may pick out several propositions. Consider, then, the mixed and multiply general sentence (9x)(y)fxy with the quantifiers exclusive. In an n-many element domain (where n any is a finite cardinal) we have: fðx1 x2 Þ &; . . .; & fðx1 xn Þ : v; . . .; v : fðxn x1 Þ &; . . .; & fðxn xn1 Þ:

The free variables are to be read exclusively. Expressed with the N-operator notation we have: NNðNðNfðx1 x2 Þ; . . .; Nfðx1 xn ÞÞ; . . .; NðNfðxn x1 Þ; . . .; Nfðxn xn1 ÞÞÞ;

where the free variables are read exclusively. This has the same truthconditions (over an n-element domain) as: ðx1 Þ; ðx2 Þ; . . .; ðxn Þðfðx1 x2 Þ &; . . .; & fðx1 xn Þ : v; . . .; v : fðxn x1 Þ &; . . .; & fðxn xn1 ÞÞ;

63

Emil Post, ‘‘Introduction to a General Theory of Elementary propositions,’’ Ph.D. diss., Columbia University, 1920. See also his ‘‘Introduction to a General Theory of Elementary Propositions,’’ American Journal of Mathematics 43 (1921): 163–185.

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where the quantifiers are understood exclusively. This approach avoids the problem of having to determine the positions of t’s and f’s in a truth-table representation ð1; . . .; 2 v Þðp1; . . .; pv Þ;

where v ¼ 2n. This, then, was Wittgenstein’s way of expressing (9x)(8xy)fxy by means the N-operator notation. The original Tractarian notation is expressively adequate to express heterogeneous multiply general quantification over an arbitrarily large finite domain. Be this as it may, the N-operator falls well short of its purpose of reconstructing quantification (even over an arbitrarily large finite domain) in such a way that all logical equivalents have exactly one and the same representation. Wittgenstein’s hopes are realizable without the N-operator notation for the monadic predicate calculus (although Wittgenstein did not know this). Monadic predicate logic is decidable. Thus, for any formula of monadic predicate logic, we can find a pattern (for the positions of the t’s and f’s) that is preserved in the truth-table expression of the formula when the number of elements in the domain of the quantifiers increases. Of course, to find the pattern requires us to find a decision procedure. It is now well known that a formula A of monadic predicate logic with p-many distinct predicate letters is logically true (logically valid) if and only if its propositional model in a 2p-many element domain is a tautology. We can therefore, find the pattern of t’s and f’s for a given formula (with p-many distinct predicate letters) by looking at its propositional model in a domain of 2p. Perhaps it seemed to Wittgenstein that the expressive adequacy of quantification by the N-operator notation (over an arbitrarily large finite domain) is all he needs. In such an N-notation, all and only logical truths of predicate logic have the ‘‘tautologous’’ form: NNð. . . ; . . . N; . . .Þ:

Consider the simple case of ‘‘(y)(9x)(fx  fy)’’ with the quantifiers inclusive. We have: fx1  fx1 :v:; . . .; :v: fxn  fx1: & :; . . .; : & : fx1  fxn :v:; . . .; :v: fxn  fxn

where the free variables are to be read inclusively. Expressed with the N-operator notation and applying our rules, we have: NðNðNfx1 ; fx1 . . .; Nfxn ; fx1 Þ; . . .; NðNfx1 ; fxn . . .; Nfxn ; fxn ÞÞ:

All the subcomponents can be dropped by rule (4) except one, and we have: NNðNfx1 ; fx1 . . .Þ:

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This is a tautologous form. Likely this is what Wittgenstein was up to. But a careful reflection reveals that in polyadic predicate logic we shall not be able to apply our rules governing the N-operator effectively unless the finite number n is fixed. As we have presented it, the N-operator notation employs a schematic use of the letter ‘‘n.’’ But this undermines our ability to apply the rules governing the use of N-operator notation. In short, if we cannot apply the rules when ‘‘n’’ is employed schematically, then it is no longer the case that all and only logical equivalents have the same expression in N-operator notation. Expression in the N-operator notation will not be a decision procedure even for monadic predicate logic. Wittgenstein failed to reconstruct a formal (internal) connection between the universal quantifier and conjunction; and he failed to reconstruct such a connection between the existential quantifier and disjunction. He fails to provide a notation in which all and only logical equivalents have one and the same form. The Tractarian N-operator notation postulates the existence of a construction to fit a research paradigm, and presents itself as reaching that goal. But it is little more than bravado based on a belief that logic must be decidable. Wittgenstein seems never to have pursued the matter further. He was content to leave the considerable toil of investigating the Entscheidungsproblem to Ramsey.64

64

Frank Ramsey, ‘‘On a Problem of Formal Logic,’’ The Foundations of Mathematics and Other Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (Harcourt, Brace & Co., 1931), pp. 82–111.

5

Tractarian logicism

Logicism is the thesis that mathematics is reducible to logic, or so it is often said. Unfortunately, this characterization is misleading. Frege’s logicism concerned arithmetic, not mathematics in general; and, as we shall see, reduction, in the sense of a derivation from axioms, is not essential to logicism. Frege’s logicism is the thesis that all arithmetic truths are logical truths. Russell held a more encompassing form of logicism according to which all mathematical truths, including those of analysis and geometry, are logical truths. Russell’s more encompassing form is based upon his acceptance of the arithmetization of the branches of nonapplied mathematics. Arithmetization reconstructs the branches as theories of order and structure, not theories of magnitude. The theory of order is part of the logic of relations. Relational order is involved in modern dynamics insofar as it employs conceptions of continuity and change over time in its account of motion in space.1 For this reason, Russell imagined that logicism plays a central role in the solution of Zeno’s famous dynamical paradoxes and Kant’s antinomies of space. Logicism arose in connection with the quest for rigor in the deductive methods of mathematics and logic. The quest uncovered a new quantification theory quite distinct from the categorical systems dating back to Aristotle and medieval logicians. Boole, Peirce, and Schro¨der’s methods construed logic as an algebra. Frege started afresh, importing the notion of variables and functions into logic. Both schools founded modern quantification theory. But Frege’s Begriffsschrift discovered something more than a new quantification theory. Frege’s pioneering work showed that the comprehension of new functions is part of logic. Quantification theory does not make logic ampliative – the conclusion of a deduction cannot get beyond what is analytically contained in the premises. Comprehension, on the other hand, does just this. Frege’s ideas eventually sparked a firestorm

1

Bertrand Russell, Our Knowledge of the External World, 2nd ed. (London: Allen & Unwin, 1926), chs. 5, 6, 7.

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of debate between proponents of rival conceptions of the nature of logic, necessity, and analyticity. Frege and Russell held different conceptions of the nature of logic. Interpretations of their work continue to be rigorously debated. Followers of van Heijenoort argue that in Frege’s view ‘‘logic contains its own meta-theory’’ because it is itself an interpreted universal language (a characteristica lingua universalis), not a formal calculus (in the Boolean tradition) which admits of different interpretations. Hylton interprets Russell in a similar vein.2 Both are questionable.3 But we need not take up the debate here. What is important for the present is that both assumed that there is a consistent axiomatic system within which all and only logical truths are deducible. This is a ‘‘deducibility assumption’’ which, put in modern terms, is the thesis that there is a consistent axiomatic calculus for logic that is semantically complete (with respect to the formulas that are logically true in the Tarski semantics for the language of the calculus). Since Frege and Russell held that arithmetic truths are logical truths, their deducibility assumption entails that all arithmetic truths can be reached deductively from the axioms of a would-be semantically complete calculus for logic. It is important to understand that the deducibility assumption is not part of logicism proper. It is a component of both Frege’s and Russell’s plan for establishing that logicism is true. For instance, mathematical induction seems to provide an example of nonlogical and uniquely arithmetic necessary truth. To account for its necessity, Kant’s Transcendental Aesthetic postulated the existence of a pure aesthetic (sensory) intuition of time which forces all empirical experiences to be temporally structured into successive and consecutive episodes. Frege and Russell hoped that by deducing mathematical induction within an axiomatic system for logic – a system austere enough to preclude the intrusion of nonlogical intuitions from creeping into its inferences – they would reveal that no special Kantian intuitions are required to account for arithmetic truth (and arithmetic necessity). The only intuitions needed are logical intuitions. It is now well known that first-order predicate calculi are semantically complete. But the axiom system for logic that Frege proposed, and those Russell proposed, were not first-order predicate calculi. They offered what are sometimes called ‘‘higher-order’’ calculi. There are different forms of such calculi. The language of a first-order predicate calculus does not 2 3

Peter Hylton, Russell, Idealism and the Emergence of Analytic Philosophy (Oxford: Oxford University Press, 1990). See Gregory Landini, Russell’s Hidden Substitutional Theory (New York: Oxford University Press, 1998).

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include bindable predicate variables. A ‘‘standard second-order calculus’’ permits bindable predicate variables but requires that such variables always occur in predicate positions. Where ’ and y are predicate variables, ‘‘’(y)’’ is ungrammatical in such a calculus. Russell’s famous paradox of the attribute that an attribute G exemplifies if and only if G does not exemplify itself cannot be formulated within the language of a standard second-order calculus. Russell’s paradox arises in a different calculus – a naive second-order calculus with nominalized predicates. What is called ‘‘simple type-theory’’ is a second-order calculus with nominalized predicates that avoids Russell’s paradox by a type-regimentation of the language of the calculus. Now, standard second-order calculi are type-free and consistent. They are, however, semantically incomplete. That is, no consistent axiomatization of a standard second-order predicate calculus is semantically complete with respect to the formulas of its language that are logically true (in the Tarski semantics for the language of the calculus). The semantic incompleteness result extends to second-order logic with nominalized predicates. It is important to understand that these results do not jeopardize logicism. They certainly do destroy Frege’s and Russell’s original plans for demonstrating the truth of logicism. But logicism is not wedded to a deducibility thesis. Indeed, logicism is a metaphysical thesis about the nature of arithmetic truth (and in Russell’s form, about mathematical truth in general). Logicism cannot be shown to be false because of the failure of the deducibility assumption.4 Wittgenstein rejects all theories of deduction because a theory of deduction would present logic as if it were a genuine science. He therefore rejects the deducibility assumption made by both Frege and Russell. But it would be incorrect to conclude that Wittgenstein rejected logicism. As we have seen, logicism does not entail the deducibility assumption. Nor does it entail the existence of a semantically complete axiomatic theory of logic. To be sure, the Tractatus rejects the logical objects that epitomize Frege’s logicism. But again Wittgenstein’s rejection is not ipso facto a rejection of logicism. If it were, Russell could not be counted as a logicist. Logicism certainly does not entail that the logical particles stand for logical relations or that there are logical objects such as classes, 4

Go¨del showed that arithmetic truth is not consistently recursively axiomatizable. Every axiomatic theory S of arithmetic which is consistent and adequate to represent every recursive numeric function is such that there is a sentence G of the language of the theory such that neither G nor G is provable in S. But the result that concerns us here is the semantic incompleteness of standard second-order logic. For a discussion of the impact of Go¨del’s result for logicism, see Geoffrey Hellman, ‘‘How to Go¨del a Frege-Russell: Go¨del’s Incompleteness Theorems and Logicism,’’ Nouˆs 15 (1981): 451–468.

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propositions, or propositional functions in intension. In Principia, the logical particles do not stand for relations; and a calculus for logic is reconstructed without assuming that there are logical truths which assure the subsistence of classes or propositions. Principia endeavored to formulate an eliminativistic reconstruction of logic without the ontology of classes or propositions, or the assumption that every well-formed formula comprehends an attribute (or as Russell put it, a ‘‘propositional function’’ in intension with both an individual and predicable nature). Wittgenstein’s doctrine of Showing demands that logic is not a science of logical objects. But this does not separate him from the logicist camp any more than Principia’s no-classes and no-propositions theory separates Whitehead and Russell. Logicism is not wedded to the assumption of logical objects. It is, therefore, mistaken to think that Wittgenstein cannot have been a logicist because he rejected the notion that there are purely logical objects which are the subject matter of a science of logic. As we shall see, the Tractatus does not reject logicism. Quite the contrary, it advocates a new and more radically eliminativistic form of logicism than that found in Principia.

Ramified types as scaffolding in Russell and Wittgenstein The formal system of the first edition of Principia embodies a ramified type structure. The primitive signs are v, , (,), 0 (prime). Predicate variables and individual variables come with order\type symbols. The individual variables are xo, yo, zo, and their primes (so that we cannot run out), and the predicate variables are ’t, y t,  t, and their primes. An order\type symbol can be defined recursively as follows: (1) o is an order\type symbol. (2) If t1, . . ., tn are order\type symbols, then (t1, . . ., tn) is an order\type symbol. (3) There are no other type symbols. The notion of the order of an order\type symbol is defined as follows: (1) The order\type symbol o has order 0. (2) An order\type symbol (t1, . . ., tk) has order n þ 1 if the highest order of the order\type symbols t1, . . ., tk is n. The order of an order\type symbol can be determined by counting parentheses from left to right, adding þ1 for every left parenthesis and –1 for every right parenthesis. The order is the highest (necessarily non-negative) integer obtained in the counting process.5 A variable is predicative if its 5

William Hatcher, The Logical Foundations of Mathematics (Oxford: Pergamon Press, 1982), p. 106.

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order is the order of its order\type index. On the present characterization, all and only variables of the formal language of Principa are predicative.6 The atomic formulas of Principia are of the form ’ðtl; ...; tnÞ ðxtll ; . . .; xn tn Þ:

The formulas (wffs) are the smallest set K containing all atomic wffs such that if A, B, C are wffs in K and xt is an individual variable free in C, then so are (A), A v B, and (xt)C. The primitive language of the theory does not allow complex predicate terms formed by circumflexing variables in formulas.7 The axiom schemata for Principia’s quantification theory are given in two ways. The first is at *9 and is intended to be sensitive to the philosophical explanation of the order\type indices on predicate variables. For technical convenience, however, Principia sets out the simpler quantification theory of *10. The axiom schemata (with *10) are as follows. A v A . . A B . . A v B A v B . . B v A A v (B v C) . . B v (A v C)8 B  C . . A v B . . A v C (xt)A  A[yt j xt], where yt is free for free xt in A. (xt)(B v A) . . B v (xt)A *10.12 where xt does not occur free in B. *12.1n (Reducibility) ð9’ðtl; ...; tnÞ Þðxl tl ; . . .; xntn Þð’ðtl; ...; tnÞ ðxl tl ; . . .; xn tn Þ

*1.2 *1.3 *1.4 *1.5 *1.6 *10.1

 Aðxl tl ; . . .; xn tn ÞÞ; where ’(tl, . . ., tn) is not free in A. The inference rules are: Modus Ponens: From A and A  B, infer B Universal Generalization: From A, infer (8xt)A 6 7 8

See Landini, Russell’s Hidden Substitutional Theory. I count individual variables as predicative for convenience. Circumflex is only used in Principia as an aid in explaining the system. It is not part of the formal theory. See Landini, Russell’s Hidden Substitutional Theory, p. 265. This is not independent and thus should be dropped. See P. Bernays, ‘‘Axiomatische Untersuchung des Aussagenkalku¨ls der PM,’’ Mathematische Zeitschrift 25 (1926): 305–330.

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Definitions include the following: ð9xt ÞA ¼df  ðxt Þ  A A  B ¼df  A v B A & B ¼df  ð A v :BÞ A  B ¼df ðA  BÞ &ðB  AÞ

This completes the formal system. As we formulate it, the order\type indices on the variables of Principia’s ramified type-theory look like those of simple type-theory. In simple typetheory, what we have called ‘‘order\type symbols’’ are called ‘‘type symbols.’’ Simple type-theory ignores order altogether. If we want variables whose indices display the order component of an order\type symbol, we could replace individual variables xo with oxo, replace predicate variables ’(o) with 1’(o\o), replace predicate variables ’((o)) with 2’(1\ (o\o)), and so on. But order in Principia always tracks the order of the type symbol. This is because the formal language of Principia’s ramified-type theory allows only predicative variables – i.e., variables whose order is the order of the type symbol. All and only variables of Principia are predicative, and the only terms are its variables. For this reason, there is no need to display the order component of a predicate variable. It is given with the order of the type symbol.9 The difference between Principia’s ramified types and simple type-theory lies in what comprehension principles are legitimate. In simple type-theory, *12.1n is not a ‘‘reducibility’’ axiom, since orders are irrelevant. In Principia’s ramified type-theory, *12.1n is a reducibility axiom. We can see this by observing that *12.1n strays significantly from the predicative principles of comprehension that are validated by the semantics Russell intended for Principia. To see this, recall that Russell intended a nominalistic semantics for Principia’s predicate variables. The only genuine variables of Principia are its individual variables of order\type o. Only individual variables are to be interpreted objectually. All entities countenanced in the work, whether universal, particular, or complex (facts), are on a par as individuals. There are no types or orders of entities. Russell’s nominalistic semantics comes, however, at a high price. It generates a ramified-type structure. The informal semantics relies on a hierarchy of languages the formulas of which are to be the admissible substituends for predicate variables of a given order\ type. The hierarchy is recursively defined by means of the admissible senses of ‘‘truth’’ and ‘‘falsehood’’ as applied to the formulas. The nominalistic semantics interprets a formula such as 9

See Landini, Russell’s Hidden Substitutional Theory.

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ð1 ’ðonoÞ Þ1 ’ðonoÞ ðo xo Þ

of a language L2 as asserting that every formula A(z) of a fixed first-order language L1 is such that A(xjz) is satisfied (given z is free for free x in A). It is essential that L1 be a language whose formulas do not themselves contain bound predicate variables, else the truth-conditions of the semantics would be viciously circular. We can go on to build a new language L3, by introducing order-indexed predicate variables 2’(1\(o\o)) which are nominalistically interpreted over formulas of L2. The process continues to introduce variables 3’(2\(1\(o\o))), and 4’(3\(2\(1\(o\o)))), whose significance conditions are fixed by the nominalistic semantics. It is the truth-conditions of the nominalistic semantics of Principia’s first edition that philosophically justifies and explains the orders of the order\type indices on the predicate variables of the system. The nominalistic semantics of Principia validates comprehension principles for predicative type-theory. Consider the following predicative axiom schema for comprehension: ð9’p Þðx1 t1 ; . . .; xn tn Þð’p ðx1 t1 ; . . .; xn tn Þ  AÞ;

where ’(t1, . . ., tn) is not free in A, and the order of the order\type index p is that of the truth-conditions of the formula A (rendered by the recursively defined hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ of the nominalistic semantics). The semantics makes it difficult to justify a comprehension principle in which the order\type symbol of the predicate variable strays from the sense of ‘‘truth’’ for the formula A. Recall that impredicative quantification is excluded in Principia. To take Russell’s famous example, consider the following: x satisfies all the conditions of every great general.

According to predicative type structure, satisfying all the conditions of every great general cannot be among the conditions that x satisfies. The notion of all conditions is disallowed. In Principia we have ðy ðoÞ ÞðGðzo Þ zo y ðoÞ ðzo Þ : : y ðoÞ ðxo ÞÞ:

The formula contains a quantifier binding a predicative variable y (o) whose is order 1. Thus the formula has the truth-conditions of order 2. (It is capable only of truth2 or falsehood2.) With *12.1n (Reducibility), however, we can put this formula in the position of A and arrive at the following:

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ð9’ðoÞ Þðxo Þð’ðoÞ ðxo Þ  ðy ðoÞ ÞðGðoÞ ðzo Þ zo y ðoÞ ðzo Þ: :y ðoÞ ðxo ÞÞÞ:

We see that ’(o) has an order\type index whose order is 1 in spite of the fact that the formula that comprehends it has truth-conditions of order 2. In this way, Principia’s comprehension principle *12.1n strays markedly from comprehension principles of predicative type-theory. It is in this sense that *12.1n is a ‘‘reducibility’’ axiom. Russell explained in his lectures on logical atomism that ‘‘the theory of types is really a theory of symbols, not of things. In a proper logical language, it would be perfectly obvious.’’10 This attitude is sometimes characterized as a change of heart brought about by the influence of Wittgenstein. But the influence goes in the other direction. The demand of type-freedom was the consequence of Russell’s unwavering adherence to the doctrine that any proper calculus for logic must adopt only one style of genuine variables (individual variables). This has come to be known as Russell’s ‘‘doctrine of the unrestricted variable.’’ The doctrine dates back to Russell’s Principles of Mathematics (1903) and led to his construction of the structure of a type-theory of classes (and attributes in intension) in both the substitutional theory (1905–1908) and in Principia. The purpose of the constructions is to enable pure logic to proceed to the development of arithmetic (and pure mathematics) without relying on assumptions of the logical existence of classes or attributes – assumptions that (because of Russell’s paradoxes) cannot be part of logic. Wittgenstein agreed with Russell’s demand that a ramified type-theoretical structure without the ontology of ramified and typed entities must be constructed within a type-free language for logic. But he advocates a philosophical explanation of the structure of predicative types that is quite different from Principia. Unfortunately, the similarities and differences have not been properly understood. Part of the confusion stems from the longstanding interpretation that Principia embraced a ramified type-theory of entities (propositional functions). This obliterated Principia’s nominalistic semantics. The fog of this confusion has finally cleared. A predicative comprehension is acceptable to Wittgenstein as long as predicate variables are rendered by a nominalistic semantics. For this is compatible with his thesis that logical truths are tautologies. For example, the semantics would treat the formula ð9’ðoÞ Þðxo Þð’ðoÞ ðxo Þ  ðyo ÞRðo;oÞ ðxo ; yo ÞÞ

10

Bertrand Russell, ‘‘The Philosophy of Logical Atomism,’’ Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. R. C. Marsh (London: Allen & Unwin, 1977), p. 267.

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as saying that some formula A that is capable of truth1 (and falsehood1) is such that its substitution for the variable ’(o) yields a formula that is true1. Since (yo)R(o,o) (xo, yo) is among the possible substitution instances for ’(o), and ðxo Þððyo ÞRðo;oÞ ðxo ; yo Þ  ðyo ÞRðo;oÞ ðxo ; yo ÞÞ

is a generalized tautology, Wittgenstein finds it perfectly acceptable. But Reducibility is quite another matter. Wittgenstein rejected Reducibility. It cannot be construed as a tautology. Moreover, though he agreed with Russell that a predicative type-structure must be built into the significance conditions of the predicate variables, he strongly disagreed with Principia’s use of the multiple-relation theory to generate the recursive hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ Wittgenstein’s doctrine of showing makes every semantic theory illicit. Truth-conditions must be pictured by syntax of the structured variable alone, not set out in a semantic theory. Principia’s multiple-relation theory of judgment – which forms the base case for its recursive definition of ‘‘truth’’ and ‘‘falsehood,’’ must say what can only be shown. The theory puts words for universals in subject positions. It must embrace pseudo-predicates with logical content such as ‘‘. . . is a universal,’’ and ‘‘. . . is a fact.’’ Russell’s amendments to the theory in his ‘‘Theory of Knowledge’’ project of 1913 makes matters even worse. They embrace still more logical pseudo-objects such as abstract general facts and logical forms. Wittgenstein’s Tractatus endeavors to avoid the constructions of Principia’s semantics altogether by using structured variables. He writes (TLP 3.333): The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition ‘F(F(fx))’ in which the outer function F and the inner function F must have different meanings, since the inner one has the form ’(fx) and the outer one has the form y(’(fx)). Only letter ‘F’ is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of ‘F(F(fu))’ we write ‘ð9’Þ: Fð’ðuÞÞ : ’ðuÞ ¼ Fu’: That disposes of Russell’s paradox.

The plan is to build the structure of predicative order\types into structured variables so that the variables themselves picture the structure of their admissible values. The variable pictures the structure of its values without need of any semantic theory. The syntactic complexity of the

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structured variable is a ‘‘prototype’’ which pictures the structure that all of its admissible values are to have. Wittgenstein hopes to find a notation in which the structure, not the ontology, of types and orders is shown by the syntax of the variables.11 In the Tractatus, Wittgenstein does not even attempt to offer a notation which realizes the goal. Borrowing from Frege, however, we can imagine a notation of structured variables that does justice to type structures.12 Frege’s Begriffsschrift introduced different levels of functions. The notations for the levels have important similarities to simple type structures. On Frege’s view, a first-level function f yields a value for an object x taken as its argument. As Frege puts it, the argument ‘‘falls under’’ the first-level function. A first-level function, on the other hand, can ‘‘fall within’’ a second-level function when it ‘‘mutually saturates’’ it. Many second-level functions (and higher-level functions) are quantifier concepts for Frege. For example, Frege takes the quantifier concept (8x)fx to be a second-level function under which first-level functions may fall. (To distinguish the Fregean language, we use the symbol 8 for the universal quantifier.)13 For example, the first-level function of self-identity falls within this secondlevel function, because every object x is self-identical. To introduce a structured variable that ranges over second-level quantifier concepts, put Mf. The lower-case Greek  appears as a subscript in Mf to remind us of the bound individual variable  in expressions of second-level quantifier concepts. Then we can quantify over second-level quantifier concepts with the notation (8M)Mf. For example, we have: ð8MÞM f  ð8xÞfx:

Since function signs can only appear in predicate (function) positions, and quantifier concepts are extensional, we have: ’x x yx : ’y : M ’ M M y  :

For instance, we have: ’x x yx: ’y : ð8xÞ’x  ð8xÞyx ’x x yx: ’y : ð8xÞ  ’x  ð8xÞ  yx

11

12 13

Sackur notices the conection between Wittgenstein’s idea of structured variables (as prototypes) and Russell’s use, during the era of his substitutional theory of propositions, ´ ˆ me Sackur, Formes et faits (Paris: J. Vrin, 2005), of the ‘‘prototype of a matrix.’’ See Jero pp. 89, 121. I do not mean to suggest that Frege himself embraced such variables in his Grundgesetze. Accordingly, ’x  x yx now abbreviates (8x)(’x  yx).

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and so on. Next we have third-level quantifier concepts. For instance, we can represent ð8xÞ’x  ð9xÞyx

as a context resulting from the mutual saturation of the quantifier concept (8x)fx with the concept O’ ’ (9x)yx. Accordingly, we represent a notation perspicuous of the quantificational structures of third-level concepts as follows: Y’[O’]. The occurrence of the subscripted Greek ’ indicates that a first-level function variable ’ is bound in the quantifier concept. The pattern continues as we ascend the levels. A brief chart of one-place expressions might help: Russellian types o

x ’(o) ’((o)) ’(((o)))

Fregean levels x ’ Mf Y’[O’]

Fregean levels are captured by using structured variables which display the higher-level concepts as quantifier concepts. Such structured variables are not exploited in the notation of Principia’s ramified type-structure. Typestructures are shown in the Fregean notation. By keeping function symbols in function position, the structured variables provide prototypes of the structure of their values. Is a similar technique of prototypes (structured variables) possible for orders and ramification? We saw that orders are philosophically explained in Principia by Russell’s informal nominalistic semantics which offers a recursively defined hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ founded upon the multiple-relation theory of judgment. In this way, Russell endeavored to make Principia’s predicate variables unrestricted; they are ‘‘limited from within by their conditions of significance.’’ Wittgenstein cannot accept this semantic approach. Principia’s semantic theory endeavors to say what should be shown (pictured) by the syntax of structured variables. The Doctrine of Showing rejects all semantic theories. There can be no ‘‘theory’’ of truth. Wittgenstein hopes to achieve the structure of predicative type-theory by employing syntactically structured variables (prototypes) which picture the structure of their admissible values. He offers nothing whatever to achieve this. We had good fortune with types by employing Fregean structured variables, but in the case of order\types it is far from clear that what Wittgenstein wants can be accomplished. In any case, we see that both Principia and the Tractatus agree in rejecting orders and types of entities. They disagree only on how this is to be done.

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Equations versus tautologies Wittgenstein’s views on how to philosophically ground orders and types are obviously allied with Russell’s endeavors to solve the problems plaguing logicism by ‘‘building structure into the variables.’’ Unfortunately, the alliance had not been appreciated and Wittgenstein has been interpreted as abandoning logicism. Perhaps the central argument for the position that Wittgenstein was not a logicist comes from his rejection of the thesis that arithmetic consists of tautologies.14 As Ramsey put it: W[ittgenstein] and I think it is wrong to suppose with Russell that mathematics is more complicated formal logic (tautologies); and I am trying to make definite the vague ideas we have of what it does consist of.15

Wittgenstein’s Tractarian approach construes arithmetic as ‘‘equations,’’ not tautologies. Curiously, Russell had a tendency to ignore this feature of Wittgenstein’s work. It is not mentioned in his Introduction to Mathematical Philosophy (1919). In the first printing of the second edition of The Principles of Mathematics (1937) Russell continued to ignore it. He observed that Principia’s multiplicative axiom and axiom of infinity are fully general and explained that this shows that truth and full generality (in the calculus for pure logic) do not form a sufficient condition for logical truth. What, then, he asked, is sufficient for a statement to belong to logic (mathematics)? Russell answers as follows: ‘‘In order that a proposition may belong to mathematics, it must have a further property: according to Wittgenstein it must be ‘tautological,’ and according to Carnap it must be ‘analytic.’’’16 Wittgenstein had by now become vociferous in his objection to Russell’s repeated mischaracterization. When G. H. Hardy read a paper at the Moral Sciences Club in Cambridge in 1940, he attributed to Wittgenstein the view that mathematics consists of tautologies. Reportedly, Wittgenstein attended and denied having ever held the view, pointing to himself and saying in an incredulous tone of voice: ‘‘Who I?’’17 At last, in a letter to Stanley Unwin of 17 July 1947, Russell instructed that

14 15

16 17

See Friedrich Waismann, Wittgenstein and the Vienna Circle, ed. Brian McGuinness, trans. Joachim Schulte and Brian McGuinness (Oxford: Blackwell, 1979), p. 35. Frank Ramsey, letter to Moore of 6 February 1924, in Josef Rothhaupt, ed., Farbthemen in Wittgensteins Gesamtnachlass: philologisch-philosophische Untersuchungen im La¨ngschnitt und Querschnitten, Monografien Philosophie 273 (Weinheim: Beltz, 1966), p. 46. Bertrand Russell, Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), p. ix. See Wolfe Mays, ‘‘Recollections of Wittgenstein,’’ in K. T. Fann, ed., Ludwig Wittgenstein: The Man and His Philosophy (New York: Dell Publishing Co., 1967), p. 82.

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the word ‘‘some’’ replace ‘‘Wittgenstein’’ in the passage, altering subsequent printings of the second edition of Principles.18 Two fundamental convictions led Wittgenstein to conclude that arithmetic consists of equations rather than tautologies: (1) he was convinced that identity is a pseudo-predicate, and (2) he accepted ramified typestructure. The difficulties imposed by requiring predicativity will be discussed anon. For the present, let us focus on the problem of identity. We shall argue that Wittgenstein came to hold that the Frege/Russell construction of cardinals is undermined by his thesis that identity is a pseudo-predicate. In the Notebooks, we find Wittgenstein investigating how to recover Frege/ Russell definitions of cardinals without illicitly employing identity: 21.10.14. [Isn’t the Russellian definition of nought nonsensical? Can we speak of a class x^ðx 6¼ xÞ at all? – Can we speak of a class x^ðx ¼ xÞ either? For is x 6¼ x or x ¼ x a function of x? – Must not 0 be defined by means of the hypothesis ð9Þ:ðxÞ x? And something analogous would hold of all other numbers. Now this throws light on the whole question about the existence of numbers of things. ^ðuÞgDef: 0 ¼ a^fð9Þ:ðxÞ x:a ¼ u 1 ¼ a^fð9Þ::ð9xÞ:x :y:z y;x y ¼ z:a ¼ u^ðuÞgDef: [The sign of equality in the curly brackets could be avoided if we were to write: ðuÞ fðxÞ xg: 0 ¼ u^d The proposition must contain (and in this way shew) the possibility of its truth. But not more than the possibility. [Cf. 2.203 and 3.02 and 3.13.] By my definition of classes ðxÞ:  x^ðxÞ is the assertion that xðxÞ is null and the definition of 0 is in that case 0 ¼ a^½ðxÞ:  a Def.19

At first blush, it will appear that uses of identity that Wittgenstein would find illicit are everywhere present in Principia. For example, Marion cites *54.02, which defines the cardinal number 2 as follows: 2 ¼df y^ð9xÞð9zÞðx 6¼ z & y ¼ e^ðe ¼ x v e ¼ zÞÞ:20

But Principia’s definition of 2 as a class of couples does not involve an illicit use of identity. It can be recovered with exclusive quantifiers thus:

18

19 20

Kenneth Blackwell, ‘‘The Early Wittgenstein and the Middle Russell,’’ in Irving Block, ed., Perspectives on the Philosophy of Wittgenstein (Cambridge, Mass.: MIT Press, 1981), p. 27n3. Ludwig Wittgenstein, Notebooks 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe, 2nd ed. (Oxford: Blackwell, 1979), p. 16. Mathieu Marion, Wittgenstein, Finitism, and the Foundations of Mathematics (Oxford: Clarendon Press, 1998), p. 53.

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2 ¼df y^ð9xÞð9zx Þðx 2 y :&: z 2 y & ð8wxy Þðw 2 = yÞÞ:

If Wittgenstein is justified in maintaining there are illicit uses of identity in the Frege/Russell account of number, he cannot have had these constructions in mind.21 To discover whether illicit uses of identity are required by the Frege/ Russell account of cardinal numbers, one must first understand that cardinal numbers can be given a logical analysis within an early Fregean theory of levels of functions (where functions are construed extensionally). We must then investigate whether this theory can be reconstructed with exclusive quantifiers that eliminate the identity sign. The ‘‘no-classes’’ theory of Principia is but a technically complicated means of emulating the language of classes while tracking the early Fregean analysis of number in terms of second-level functions. In his Begriffsschrift (1879) Frege showed how the notion of the ancestral can be given within logic itself. This is the foundation for a theory of cardinal number. But in his Die Grundlagen der Arithmetik (1884) Frege shifted gears, announcing a new theory of cardinals as objects in terms of extensions of functions. The theory of extension is developed in Grundgesetze der Arithemtic, volume 1 (1893), as a general theory founded on the now infamous Basic Law V: zðfzÞ ¼ zðgzÞ  ð8xÞðfx  gxÞ:

The course-of-values of f equals the course of values of g if and only if the functions f and g are coexemplifying. Cardinal numbers are then identified as certain courses of values. That is, where z´ (fz) is the course of values of the function f, the object which is its cardinal number is given as A˚z´ (fz). In the Grundgesetze, Frege explains the evolution of this thought thus: One reason why the execution appears so long after the announcement is to be found in the internal changes in my Begriffsschrift, which forced me to discard an almost completed manuscript. These improvements may be mentioned here briefly . . . The introduction of courses of values of functions is a vital advance, thanks to which we gain greater flexibility.22

The passage suggests that the shift to courses of values required extensive changes to notations developed the Begriffsschrift. The early ‘‘almost

21 22

I use superscripts to mark exclusivity. Thus (9zx) is read ‘‘some z other than x.’’ See Appendix A for a deductive system of exclusive quantifiers. Gottlob Frege, Grundgesetze der Arithmetik, vol. 1 (Darmstadt: Georg Olms Verlag, 1962). (First Published Jena, 1893.) The translation is from Gottlob Frege, The Basic Laws of Arithmetic: Exposition of the System, ed. Montgomery Furth (Berkeley: University of California Press, 1964), p. 7.

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completed’’ manuscript must have been a theory of cardinal numbers as second-level concepts.23 Statements of cardinality are fundamentally quantificational statements. The notion of the cardinality of a sortal concept f is the second-level concept of being in one–one correlation with f. This is essential to counting. Let us define as follows: ’x xy yy ¼dfð9kÞð11Relxy ½kðx; yÞ :&:ð8xÞð’x  ð9yÞðyy :&: kðx; yÞÞÞ :&: ð8yÞðyy  ð9xÞð’x & kðx; yÞÞÞÞ:

Attributes ’ and y are in one–one correspondence if and only if there is a one–one function k, that correlates every entity that has the attribute ’ with a unique entity that has y, and vice versa. This definition employs the notion of a one–one relation which involves the identity sign. The notion of a dyadic relation being one–one is defined in Frege’s Begriffsschrift as follows: a

b

c

(a = e) f (b, a) f (b, e)

δ = I f (δ, ε) ε

The definiens is on the left, unlike the modern notations for definitions that places it on the right. Observe that the variables  and e in the definiendum above and below the ‘‘I’’ bind the occurrences in f(, e). They remind us that we have a second-level quantifier concept, and this keeps the function sign f in predicate position. In a more modern notion we have: 11Relxy ½fðx; yÞ ¼df ð8x; y; zÞ ðfðx; yÞ & fðx; zÞ : : y ¼ zÞ & ð8x; y; zÞðfðx; yÞ & fðz; yÞ :: x ¼ zÞ:

Now the definiens uses the identity sign. But this is not an illicit use since it can be easily eliminated by means of exclusive quantifiers. We put: 11Relxy ½fðx; yÞ ¼dfð8xÞð8yx Þð8zxy Þðfðx; yÞ   fðx; zÞ :&: fðx; xÞ   fðx; yÞÞ &ð8xÞð8yx Þð8zxy Þðfðy; xÞ   fðz; xÞ :&: f ðx; xÞ   fðy; xÞÞ:

From here it is straightforward to develop the Frege/Russell cardinals as second-level functions. We simply track Frege’s Grundgesetze notations, reworking them in terms of what we surmise were the early Begriffsschrift constructions. Let us write Begriffsschrift* for clarity. 23

Unfortunately, there is no known copy of Frege’s earlier manuscript. Likely it was destroyed in the wars.

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aða  zðBzÞÞ24

A zðBzÞ ¼dfGrundgesetze

Cardy yy ½’  ¼dfBegriffsschrift

’x xy yy



0 ¼dfGrundgesetze A zðz 6¼ zÞ Cardy y6¼y ½’ 

0 ½’  ¼dfBegriffsschrift

This involves the identity sign, but it is readily eliminated: 0 ’ ¼df ð9yÞðð8xÞ  yx :&: Cardy yy ½’ Þ:

We next parallel Frege’s definition of the relation P of preceding immediately in the cardinal number series: Preceding Immediately 



aPb ¼dfGrundgesetze ð9yÞð9yÞðyy :&: b ¼ A zyz :&: a ¼ A z½yz & z 6¼ yÞ ax y x Py bx y x ¼dfBegriffsschrift ð9yÞð9yÞðyy :&: bx ’x ’ Cardz yz ½’  :&: ax ’x ’ Cardz yz & z6¼y ½’ Þ:

This involves the identity sign. But it is readily eliminated with exclusive quantifiers: ax y x Py bx y x ¼df ð9yÞð9zÞðyz :&: bx ’x ’ Cardy yy ½’  : & : ð9Þð8xz Þðx  yx :&: ax ’x ’ Cardy y ½’ ÞÞ:

Next we have the ancestral relations and the definition of natural number: Strong Ancestral a5b ¼dfGrundgesetze

ð8FÞðaPz z Fz :&: ðFz & zPy : z;y: FyÞ :: FbÞ

ax y x 5y bx x ¼dfBegriffsschrift ð8FÞðax y x Py zx y x z F’ ½zx ’x  :&: ðF’ ½zx ’x  :&: zx y x Py yx y x :z;y : F’ ½yx ’x Þ :: F’ ½bx ’x Þ: Weak Ancestral a b ¼dfGrundgesetze

a5b :v: a ¼ b

ax y x y bx y x ¼dfBegriffsschrift 24

ax y x 5y bx y x :v: ax ’x ’ bx ’x :

This uses the sign  without subscripts for equinumerosity defined over all entities. We have omitted its Grundgesetze definition for convenience of exposition.

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Natural Number (finite cardinal) 0 z

Nz ¼dfGrundgesetze

N’ ½nx ’x  ¼dfBegriffsschrift

0x y x y nx y x :

The individual natural numbers are defined by Frege by counting cardinals themselves. To recover this for second-level functions, we need heterogeneous relations of equinumerosity.25 But these pose no serious difficulties: Counting Cardinals 1 ¼dfGrundgesetze



A zðz ¼ 0Þ

1 ½’  ¼dfBegriffsschrift CardM Mxyx y 0xyx ½’  ‘ 1 ½’  ’ ð9xÞðCardy y¼x ½’ Þ: 2 ¼dfGrundgesetze



A zðz 1Þ

2 ½’  ¼dfBegriffsschrift CardM Mxyx y 1xyx ½’  ‘ 2 ½’  ’ ð9xÞð9zÞðx 6¼ z :&: Cardy y¼x :v :y¼z ½’ Þ: 3 ½’  ¼dfBegriffsschrift CardM Mxyx y 2xyx ½’  $ ¼dfGrundgesetze



A zð0 zÞ

ð@0 Þ ½’  ¼dfBegriffsschrift

CardM 0xyx y Mxyx ½’:

Frege uses the sign $ for the cardinal number of natural numbers; we use the more modern @0 (aleph null). The following theorems are expected: ‘ 0 ’ ’ ð8yÞ  ’y ‘ 1 ’ ’ ð9xÞð’x & ð8yx Þ  ’yÞ ‘ 2 ’ ’ ð9xÞð9yx Þð’x & ’y :&: ð8z 25

xy

Þ  ’zÞ:

For counting cardinals we need heterogeneous relations of equinumerosity such as CardMMxyx y 1xyx [’]. The general form of Mxy x y 1xy x is Fy [My ]. Hence we need the following: ’x x M Fy ½M y   ¼ dfBegriffsschrift ð9RÞð11Relx M ½R’ ðx; M ’ Þ : & : ð8xÞðfx  ð9MÞðFy ½M y   :&: Ry ðx; M y  ÞÞÞ :&: ð8MÞðFy ½M y    ð9xÞðfx :&: Ry ðx; M y  ÞÞÞÞ: 11Relx M ½Ry ðx; M y  Þ ¼dfBegriffsschrift ð8xÞð8MÞð8OÞðRy ðx; M y  Þ :&: Ry ðx; O y  Þ :: M y  y O y  Þ :&: ð8xÞð8yÞð8MÞðRy ðx; M y  Þ :&: R’ ðy; M y  Þ :: x ¼ yÞ CardM

Fy½My

½’  ¼dfBegriffsschrift

’x x M Fy ½M y  :

As before, the occurrences of the identity sign in the heterogeneous relations of equinumerosity are easily eliminated by Wittgenstein’s exclusive quantifiers. They pose no problems.

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Our mating of Wittgenstein’s exclusive quantifiers with a Frege/Russell theory of cardinals as second-level functions yields a theorem of mathematical induction: Mathematical Induction ‘ ð8FÞðF’ ½0x’x & ð8mÞð8nÞðF’ ½mx ’x  :&: N’ ½mx ’x  :&: mx y x Py nx y x  :: F’ ½nx ’x Þ ::ð8zÞðN’ ½zx ’x   F’ ½zx ’x ÞÞ:

So far we have found no insurmountable hurdles in avoiding illicit uses of identity. Definitions of cardinal addition and cardinal multiplication are straightforwardly introduced in this theory. We have: Addition m þ n ¼dfGrundgesetze 





A zð9FÞ ð9GÞðm ¼ A zFz :&: n ¼ A zGz :&: Fz z  Gz :&: z  aFa [ aGaÞ ½mx Fx þFG nx Gx  ½’  ¼dfBegriffsschrift ð9FÞð9GÞðm x ’x ’ Cardy Fz z  Gz :&: Cardy

Fy

½’  :&: nx ’x ’ Cardy

Fy v Gy

Gy

½’  : & :

½’ Þ:

Multiplication m · n ¼dfGrundgesetze 





A zð9FÞ ð9GÞðm ¼ A zFz :&: n ¼ A zGz : & : z  aFa · aGaÞ ½mx Fx XFG nx Gx  ’ ¼dfBegriffsschrift ð9FÞð9GÞðmx ’x ’ Cardy

Fy

½’  :&: nx ’x ’ Cardy

ð9RÞððRðx; zÞ: xz: Fx & GzÞ : : & : : Carduy

Gy

½’  : & :

Rðu;yÞ

½’ ÞÞ:

The definition for multiplication relies on the notion of one–one correspondence between entities exemplifying a monadic attribute and those exemplifying a dyadic relation. We have: Cardyz

Ryz

½’  ¼dfBegriffsschrift

’x xyz Rðy; zÞ

’x xyz Rðy; zÞ ¼dfBegriffsschrift ð9kÞð11Relxyz ½kðx; y; zÞ :&: ð8xÞð ’x  ð9y; zÞðRðy; zÞ :&: kðx; y; zÞÞ & ð8y; zÞðRðy; zÞ  ð9xÞð’x :&: kðx; y; zÞÞÞ 11Relxyz ½kðx; y; zÞ ¼dfBegriffsschrift ð8xÞð8y; zÞð8w; uÞðkðx; y; zÞ & kðx; w; uÞ :: y ¼ w & z ¼ uÞ & ð8x; uÞð8y; zÞðkðx; y; zÞ & kðu; y; zÞ :: x ¼ uÞ:

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But as before, these pose no difficulties for exclusive quantifiers. We can embrace arithmetic theorems such as 1 þ 1 ¼ 2 and 2 · 3 ¼ 6 in logical notations. They become (respectively): ½1x y x þy 1x x  ’ ’ 2 ½’  ½2x y x · y 3x x  ’ ’ 6 ½’ 

Arithmetic truths, on this account, are logical truths. The central difficulty of the Frege/Russell cardinals (and the natural numbers as inductive cardinals) lies in the problem of assuring the infinity of the natural number system.26 If there are, say, only two objects, then 3 ’ ’ 4 ’ :

Three ‘‘equals’’ four! Unless there are infinitely many entities, this analysis of finite cardinals will have arithmetic collapse for some large natural numbers. Frege’s theory of classes (courses of values) overcomes this difficulty by correlating each function with a unique class of objects and each function of functions with a class of classes of objects. By means of the correlation, Zero (a second-level function of first-level functions) is correlated with an object 0 (the class of all Zero-membered classes), One (a second-level function of first-level functions) is correlated with an object 1 (the class of One-membered classes), and so forth. The infinity of functions is transformed into an infinity of objects (classes), and the infinity of objects (numbers themselves) is thereby assured. Of course, Frege’s correlation fell to Russell’s paradox of the class of all classes not members of themselves. Russell avoids the paradoxes of classes (and attributes) by means of Principia’s type-theoretical scaffolding. But Principia’s type-theory requires a special axiom of infinity of individuals (objects). Expressed in terms of second-level concepts, the axiom of infinity is this: ð8MÞðN’ ½Mx ’x   ð8yÞðMx y x  ð9yÞ  yyÞÞ

This says that no concept y whose cardinal number is finite (inductive) is universally exemplified.27 26

27

Potter and Noonan and others have objected that this approach cannot accommodate counting finite cardinals. For instance, the number of primes less than 10 is four. But they are mistaken. Frege need only introduce further heterogeneous relations of correspondences. See Michael Potter, Reason’s Nearest Kin: Philosophies of Mathematics from Kant to Carnap (Oxford: Oxford University Press, 2000), p. 115; Harold Noonan, Frege (Oxford: Polity, 2001), p. 109. Marion is mistaken in thinking that the formulation of this axiom requires an illicit use of identity. See Mathieu Marion, Wittgenstein, Finitism, and the Foundations of Mathematics (Oxford: Clarendon Press, 1998), p. 61.

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We have seen that, although the definitions for the constructions of the Frege/Russell cardinals involve the identity sign, these occurrences can be eliminated in favor of exclusive quantifiers. One may well wonder, therefore, why Wittgenstein rejected the Frege/Russell conception of arithmetic as consisting of logical truths (or ‘‘tautologies’’ as Wittgenstein sees them). Wittgenstein’s comments on Principia’s troubles with infinity suggest that at one time he was not only very attracted to the view that a proper analysis of arithmetic reveals its truths to be tautologies, but also that he thought that his elimination of identity might be the very result that facilitates it by showing how to avoid Principia’s infinity axiom. In an early entry from October 1914 in his Notebooks, Wittgenstein wrote: The question about the possibility of existence propositions does not come in the middle but at the very beginning of logic. All the problems that go with the Axiom of Infinity have already to be solved in the proposition ‘‘(9x)(x ¼ x).’’28

Wittgenstein may well have originally thought that his treatment of identity ‘‘solves’’ the problem facing cardinals as second-level concepts by assuring the infinity of logical simples. To be sure, both Wittgenstein and Russell (at the time of Principia) considered it to be contingent whether there are finitely or infinitely (@0) many individuals.29 And indeed, Wittgenstein holds such statements are pseudo-propositions since ‘‘identity’’ and ‘‘individual’’ are pseudopredicates. Russell’s anecdote on the matter is illuminating: Wittgenstein will not permit any statement about all the things in the world. In Principia Mathematica, the totality of things is defined as the class of all those x’s which are such that x ¼ x, and we can assign a number to this class just as to any other class, although of course we do not know what is the right number to assign. Wittgenstein will not admit this. He says that such a proposition as ‘‘there are more than three things in the world’’ is meaningless. When I was discussing the Tractatus with him at The Hague in 1919, I had before me a sheet of white paper and I made on it three blobs of ink. I besought him to admit that, since there were these three blobs, there must be at least three things in the world; but he refused, resolutely. He would admit that there were three blobs on the page, because that was a finite assertion, but he would not admit that anything at all could be said about the world as a whole. This was connected with his mysticism, but was justified by his refusal to admit identity.30

28 29

30

Notebooks 1914–1916, 2nd ed., p. 10. Indeed, by the time of the logical atomism lectures, Russell had changed his mind and had come to hold that universals are not logical entities but causal structures posited as a part of empirical science. Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 116.

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Nonetheless, like the Notebooks, the Tractatus proclaims that the ‘‘solution’’ of the problem of Principia’s axiom of infinity is found in the proper theory of identity. We find: And now we see that in a correct conceptual notation pseudo-propositions like ‘a ¼ a,’ ‘a ¼ b  b ¼ c .. a ¼ c,’ ‘(x)(x ¼ x),’ ‘(9x)(x ¼ a),’ etc. cannot even be written down. (TLP 5.534) This also disposes of all the problems that were connected with such pseudopropositions. The solution of all the problems that Russell’s ‘‘axiom of infinity’’ brings with it can be given at this point. What the axiom of infinity is intended to say would express itself in language through the existence of infinitely many names with different meanings. (TLP 5.535)

Calling attention to the subjunctive mood used in TLP 5.535, Fogelin hopes to explain how Wittgenstein could at once be proposing a solution to Russell’s problems with the axiom of infinity while holding that statements of number are pseudo-statements. He writes: ‘‘Here using the subjunctive, Wittgenstein leaves open the question whether any correct conceptual notation must satisfy this demand.’’31 Fogelin’s attention to the subjunctive is far from convincing. Wittgenstein’s comments strongly suggest that he thought that infinity is shown by the conceptual notation of exclusive quantifiers. This is certainly compatible with Wittgenstein’s thesis that statements of the number of objects cannot be said in a proper conceptual notation. To be sure, Wittgenstein maintained that the number of ‘‘individuals’’ of Principia’s ramified type-theory is not a matter for logic to decide. Principia’s individual variables are genuine and wholly unrestricted; they range over every entity. ‘‘Every’’ means every here, for there is no type-theory of entities in Principia. Even though he included universals among individuals, Russell had come reluctantly to hold that logic cannot assure that there are infinitely many individuals. The variables of Wittgenstein’s N-operator, however, are not akin to Principia’s individual variables. This makes it possible for exclusive quantifiers to show the infinity of logical simples. Wittgenstein’s system of exclusive quantifiers shows what an axiom of infinity attempts to express. Recall that our infinity axiom for the conception of natural numbers as second-level concepts is this: ð8MÞðN’ ½Mx ’x   ð8yÞðMx y x  ð9yÞ  yyÞÞ:

Wittgenstein may well have construed this statement of infinity as a generalized tautology. Every instance of this axiom is a theorem of the logic of exclusive quantifiers. Consider the following: 31

Robert Fogelin, Wittgenstein, 2nd ed. (London: Routledge & Kegan Paul, 1987), p. 74.

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ð8yÞð0 y   ð9yÞ  yyÞ:

Replace 0y  with (8y) yy and we arrive at ð8yÞðð8yÞ  yy  ð9yÞ  yyÞ:

This is a theorem of classic logic and of the logic of exclusive quantifiers. Next consider: ð8yÞð1 y   ð9yÞ  yyÞ:

Replace 1y  by (9x)(yx & (8zx) yz). We arrive at: ð8yÞðð9xÞðyx & ð8zx Þ  yzÞ  ð9yÞ  yyÞ:

This is also a theorem of exclusive quantifiers. The very expression of y’s being exemplified by exactly one logical simple presupposes that there is more than exactly one simple.32 No general proof of infinity is forthcoming. But exclusive quantifiers nonetheless validate an infinity axiom. It seems then that a Frege/Russell approach to cardinals should be attractive to Wittgenstein. Why then does he maintain that arithmetic consists of equations, not tautologies? What is it about his thesis that identity is a pseudo-relation that undermines the Frege/Russell view? In Ambrose’s lecture notes, we get a clue to Wittgenstein’s answer: The criterion for sameness of number, namely that the classes concerned are 1–1 correlated, is, however, peculiar. For no correlation seems to be made. Russell had a way of getting around this difficulty. No correlation need actually be made, since two things are always correlated with two others by identity. For there are two functions, the one satisfied by only a, b and the other only by c, d, namely x ¼ a .v. x ¼ b and y ¼ c .v. y ¼ d . . . We can then construct a function satisfied only by the pairs ac and bd, that is, a function correlating one term of one group with one term of the other, namely, x ¼ a. y ¼ c .v. x ¼ b . y ¼ d, or the function x ¼ a .v. y ¼ d: x ¼ b .v. y ¼ c. These correlate a with c and b with d by mere identity when there is no correlation by strings or other material correlation. But if ‘‘¼’’ makes no sense, then it is no correlation.33

Wittgenstein objects to the use of identity in the notion of one–one correspondence which is the foundation of the Frege/Russell conception of the cardinal number. But we have seen that the usual uses of identity with

32 33

See Appendix A. Wittgenstein’s Lectures 1932–1935, ed. Alice Ambrose (Amherst, Mass.: Prometheus Books, 2001), p. 150. This matter concerned Wittgenstein early on as well. Hints can be found in his 1914 notes dictated to Moore from Norway. See Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 117.

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cardinals can be recovered with exclusive quantifiers. So what is the objection? The problem Wittgenstein sees must lie with his conception of material properties and relations. If there is a one–one correspondence between the entities exemplifying a material property F and the entities exemplifying a material property G, then there must be a relation which grounds the correspondence. Suppose that just a and b exemplify F and just c and d exemplify G. Suppose one accepts a comprehension axiom for relations as part of logic such as the following: ð9’Þð’ðx; yÞ :x y Aðx; yÞÞ;

where ’ is not free in the formula A and A may contain the identity sign. One is assured the existence of a relation R that establishes a one–one correlation function between F and G. The relation R is such that Rðx; yÞ :x y : x ¼ a & y ¼ c :v: x ¼ b & y ¼ d:

Without the identity sign as part of the language, the above is not an instance of any logical comprehension axiom for relations. Without identity, logic alone cannot be assured that there is such a relation R. But if Wittgenstein is arguing that Russell’s account requires that logic comprehends the existence of a correlating relation, he is mistaken. On the Frege/ Russell account, logic itself does not have to assure the existence of such a correlation relation whenever material properties F and G are in one–one correspondence. Where a, b, c, and d are physical objects, there might well be a non-logical spatial relation (or some other material relation) that assures the one–one correspondence. There may well be a relation R such that Rða; cÞ & Rðb; dÞ & ð8z

ab

Þð8v

abz

Þ  Rðz; vÞ:

The existence of material correspondence relations correlating F’s and G’s is not part of arithmetic (or logic); it is part of the application of arithmetic to the material world. The applications are contingent. Wittgenstein’s point, however, may well have been about the application of arithmetic. This would give him reason to reject the marriage of his exclusive quantifiers with the Frege/Russell natural numbers. Material properties and relations are logically independent. The elimination of identity has the consequence that any particulars may well share all material properties and relations and yet be distinct. In cases where this happens, arithmetic (if construed with Frege/Russell) would have no application. The relevant one–one correspondences would not exist. If this is correct, we can see why Wittgenstein would have come to conclude that

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the Doctrine of Showing forces the rejection of the Frege/Russell analysis of number. The application of arithmetic cannot be founded upon the notion of one–one correspondence. Wittgenstein is caught between a rock and a hard place. If he abandons his stand against identity, then his doctrine of showing is false, for there is at least one genuine logical relation, namely identity. On the other hand, if he rejects identity and accepts the logical independence of material properties and relations, then the Russell/Frege approach (which he thinks would make arithmetic consist of tautologies) cannot provide an account of the application of arithmetic. The doctrine of showing again would seem to be false. The concept of number is a ‘‘formal’’ concept that must, like the tautologies of logic, be shown. This is a serious impasse. Wittgenstein felt its sting. Writing to Russell from Norway in 1913, he says that ‘‘identity is the very Devil and immensely important; very much more so that I thought . . . I have all sorts of ideas for a solution of the problem but could not yet arrive at anything definite. However, I don’t lose courage and go on thinking.’’34 Wittgenstein felt that he had to reject an analysis of arithmetic that makes it consist of tautologies. But we must not conclude from this that he rejected logicism. Quite the contrary, Wittgenstein came to hold a more radically eliminativisitic form of logicism than Principia – an analysis of arithmetic that relies neither on impredicative quantification nor on illicit uses of identity. Without it, his Doctrine of Showing is lost. Numbers as exponents of operations The historical evolution of Wittgenstein’s thesis that arithmetic consists of equations rather than tautologies reveals that he belongs squarely in the logicist camp.35 The Tractatus tells us that number is a formal concept. Formal concepts contain logico-semantic content. Thus, predicate expressions for formal concepts are pseudo-expressions. Showing demands that all such pseudo-concepts be shown by the use of structured variables. In the Tractatus, Wittgenstein emphasizes the importance of the use of structured variables when he announces that ‘‘number’’ is a ‘‘formal’’ concept akin to concepts such as ‘object’ and ‘identity.’ He writes (TLP 4.1272): So the variable name ‘‘x’’ is the proper sign of the pseudo-concept object. Whenever the word ‘‘object’’ (‘‘thing,’’ ‘‘entity,’’ etc.) is rightly used, it is expressed in logical

34 35

Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 123. Pasquale Frascolla sees this and takes Wittgenstein to hold a ‘‘no-classes logicism.’’ See his paper, ‘‘The Tractarian System of Arithmetic,’’ Synthese 112 (1997): 353–378.

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symbolism by the variable name. For example, in the proposition ‘‘there are 2 objects which . . .,’’ it is expressed by ‘‘(9x,y). . .’’ Wherever it is used otherwise, i.e., as a proper concept word, there arise nonsensical pseudo-propositions. So one cannot, e.g., say, ‘‘There are objects’’ as one says ‘‘There are books.’’ Nor ‘‘There are 100 objects’’ or ‘‘There are @o objects.’’ And it is nonsense to speak of the total number of all objects. The same holds for the words ‘‘complex,’’ ‘‘fact,’’ ‘‘function,’’ ‘‘number,’’ etc. They all signify formal concepts and are presented in logical symbolism by variables, not functions or classes (as Frege and Russell thought).

The passage criticizes Russell, saying that ‘number’ is ‘‘represented in conceptual notation by variables, not by functions or classes.’’ This is rather odd. Fair enough, the logicism of Frege’s Grundgesetze embraced ranges-of-values (or loosely speaking ‘‘classes’’) as logical objects. But Russell’s logicism hoped to recover arithmetic (and mathematics in general) without ontological commitments to classes, propositions, or the assumption that every open formula determines an attribute (propositional function in intension). Number is treated as a formal (logical) concept in the Principia. So what is Wittgenstein getting at here? Wittgenstein agreed with the lesson Russell had exacted from the paradoxes facing logicism. The lesson was that neither classes nor propositional functions (understood as intensional entities with both a predicable and an individual nature) can be assumed as purely logical objects. In the Principia, Russell had hoped to generate logicism without the assumption of classes. Wittgenstein shares this orientation. He writes (TLP 6.031): The theory of classes is completely superfluous in mathematics. This is connected with the fact that the generality required in mathematics is not accidental generality.

Russell’s paradox shows that there is no purely logical ground for the postulation of classes. Classes are not logically necessary entities, and truths about them (if it so happens that there are classes) are at best accidental (contingent). Therefore, classes are of no use for the logicist foundations of arithmetic. Principia espoused an eliminativistic reconstruction of a logic of classes without the ontology. But Principia failed. Its reconstruction of the structure of a theory of classes relies on nonlogical axioms such as Reducibility. In Wittgenstein’s view, Russell’s mistake in Principia was in attempting to find a structural replacement for a theory of classes. Wittgenstein’s Tractarian account of arithmetic is a new attempt at logicism – a logicism which does not endeavor to recover the structure of a theory of classes. Wittgenstein’s idea is that operation and calculation underlie both logic and arithmetic. He writes (TLP 6.021): A number is the exponent of an operation.

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We have encountered the Tractarian notion of an ‘‘operation’’ in our discussion of the N-operator.36 The Tractatus (6.01) gives ½; NðÞ’ð Þ

as the general form of an operation O’ðÞ which enables the transition from one proposition (formula) to another. Wittgenstein notes that this is just the same as ½ ; ; NðÞ:

All logical truths are supposedly expressible in one and the same way by means of the N-operator notation. Logic consists of calculations made by means of the N-operator. Arithmetic, in turn, consists of calculations made by exponents which track the repetition of a given operation. Some operations may be repeated to form a consecutive series of expressions. This is where the Tractarian idea of ‘‘numbers as exponents of operators’’ comes in. In the case of a one-placed operation O(x), Wittgenstein writes ½x; ; OðÞ

as an abbreviation for ½O0 ðxÞ; Ov ðxÞ; Ovþ= ðxÞ

where x is the first term of the series. The Tractatus gives a recursive characterization which I will rewrite below for clarity.37 The symbols 0 and þ/ (successor) are primitive. O0’ x ¼ x O

vþ=’

Def:; v’

x ¼ O’O x

Def:;

So in accordance with these rules, which deal with signs, we write the series x; O0’ x; O0þ=’ x; O0þ=þ=’ x; . . .

Therefore, instead of ‘[x, , O’]’, I write ‘½O0’ x; Ov’ x; Ovþ=’ x’

And I give the following definitions 1 ¼ 0þ= 2 ¼ ð0þ=Þþ= 3 ¼ ðð0þ=Þþ=Þþ=

Def:; Def:; Def:;

(and so on). (TLP 6.02) 36 37

See chapter 4. The Tractatus uses þ1 instead of our þ/, and it puts the definiendum on the right.

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The schematic letter v stands in for numerals 0, 1, 2, 3, and so on. Consider, for example, Wittgenstein’s expression for the general term of the series of ‘‘integers’’ (TLP 6.03): ½0; ; þ=:

The operation here is this: ðþ=Þ’x ¼ xþ=:

The first expression the operator sign is to have as its argument is not a proposition sign, but the numeral 0. Applying the characterization of an operation we have the following: ðþ=Þ0’ 0 ¼ 0 ðþ=Þvþ=’ 0 ¼ ðþ=Þ’ðþ=Þv’ 0

The series of integers is then ðþ=Þ0’ 0; ðþ=Þv’ 0; ðþ=Þvþ=’ 0; . . .

And by applying the operation (þ /)’x this is just 0; 0þ=; ð0þ=Þþ=; . . .

Here we have an operation on numerals which starts from the numeral 0 and generates a consecutive series of numerals. In contrast, N was an operation on propositions (formulas) and does not generate a consecutive series. It should be observed that Wittgenstein’s use of the identity sign in equations does not violate his thesis that identity is a logical pseudopredicate to be built into exclusive quantifiers. Wittgenstein glosses his use of the identity sign in such cases by remarking that it gives ‘‘rules which deal with signs.’’ An expression such as O2þ=’ ðxÞ ¼ O3’ ðxÞ

is not intended to be a statement, though it is called an ‘‘equation.’’ The expression informs us that the practice of carrying out the calculation ’ ’ O2 þ/ (x) is the same as the practice of carrying out the calculation O3 (x). The equation does not express an identity between entities. We discovered

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a similar circumstance with the use of the identity sign in connection with Wittgenstein’s N-operator.38 According to the Tractatus, numbers are exponents of operations. Wittgenstein illustrates this by taking the example of 2 · 2 = 4. He recursively defines multiplication and offers the following proof (TLP 6.241): The proof of the proposition 2 · 2 ¼ 4 runs as follows: ðOv · u Þ’x ¼ ðOv Þu’ x Def: ðO2 · 2 Þ’x ¼ ðO2 Þ2’ x ¼ ðO2 Þ1þ=’ x ¼ ðO2 Þ2’ x ¼ O1þ=’ ðO1þ= Þ’x ¼ ðO’OÞ’ ðO’OÞ’x ¼ O’O’ O’O’x ¼ O1þ=þ=þ=’ x ¼ O4’ x

Observe that the expression (O’O)’ (O’O)’x appears in the proof. This is designed to remind us of the grouping of brackets in (1 + 1) · (1 + 1) = 4. It appears then, that the proof is designed to reveal that the principle of the association of multiplication vanishes. Frascolla maintains, however, that the proof is flawed because the expression (O’O)’ (O’O)’x is undefined. To solve this problem, Frascolla offers an embellishment of the Tractarian recursive definitions.39 There is, however, a simple solution of the problem raised by Frascolla. It is corroborated explicitly in the text of the Tractatus. At TLP 6.241 Wittgenstein puts the operation sign O2’  in the position of O in O1 þ /’ x to produce (O2)1 þ /’x. The underlying principle behind this is this: ðO’Þ’x ¼ O’x:

If we accept this simple principle the rest of the proof goes though without difficulty. Consider the expression (O’O)’x. The argument place of the operation sign (O’O) has been omitted. The operation expression (O’O) can be written as O’(O’) so as to reveal its argument place. The notation (O’O) is intended to make us notice that we can take O’(O’) as a new compound operator, and apply 6.02 to it (since O in 6.02 is intended to be schematic for any operator). Once we see that (O’O)’y is more perspicuously written as (O’O’)’y, we can apply 6.02 and observe that O’O’ is the same as O1 þ /’. Thus, we have (O1 þ /’)’y. In short, 6.02 assures that:

38 39

See chapter 4. See Pasquale Frascolla, Wittgenstein’s Philosophy of Mathematics (London: Routledge, 1994).

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ðO’OÞ’y ¼ ðO1þ=’ Þ’y:

Now, putting (O’O)’x in the position of y, we get: ðO’OÞ’ðO’OÞ’x ¼ ðO1þ=’ Þ’ ððO’OÞ’xÞ:

But we have seen that ðO’OÞ’x ¼ ðO’O’Þ’x; ¼ O’O’x:

By 6.02 we found that O’O’x ¼ O1 þ /’x. Hence we have: ðO’OÞ’ðO’OÞ’x ¼ ðO1þ=’ Þ’ðO1þ=’ xÞ ¼ O1þ=’ ðO1þ=’ xÞ:

There is no flaw in Wittgenstein’s proof at 6.241. Wittgenstein’s proof doesn’t specify any particular operation. It is illuminating to apply the schematic O to the numerical operation (þ /)’x, whose first argument is the numeral 0. We arrive at the following: ðþ=Þ2 · 2’ 0 ¼ ðþ=Þ4’ 0:

Here we have a operation on numerals. This best explains Wittgenstein’s remark to Waismann that ‘‘logical operations are performed with propositions, arithmetical ones with numbers.’’ To be sure, the Tractatus says that all variables can ultimately be constructed as propositional variables. We find (TLP 3.314): An expression has meaning only in a proposition. All variables can be construed as propositional variables. (Even variable names.)

This suggests that in Wittgenstein’s view, all operators can ultimately be constructed from the N-operator. Tractatus 6.01 might be read as corroborating this. The passage is not explicit, however. It may well be that at 6.01 Wittgenstein meant to be speaking only of the general form of a propositional operation, which of course is built up from the N-operator. The expression ½ ; ; NðÞ

is schematic for such a propositional operation. The schema does not designate a specific propositional operation because the recipe for choosing the bases to be operated upon (indicated by the bar) must be given. If the recipe offers infinitely many propositions as bases, the operation does not generate a consecutive series. Hence, 6.02 will not apply to it, for that

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applies only to operations that generate consecutive series. The schema O’x in Wittgenstein’s definition of multiplication clearly involves those operations that do generate consecutive series. It has been argued, however, that the Tractarian conception of numbers as exponents of operations is intended to reconstruct an adjectival approach to cardinal number.40 In explaining Wittgenstein’s definition of multiplication and calculation of 2 · 2 = 4, Potter offers a propositional operation. We have the following for ‘‘there are exactly n ’’s’’: ð0 9xÞAx ¼df  ð9xÞAx ð1 9xÞAx ¼df ð9xÞðAx & ð0 9yÞðAy :&: x 6¼ yÞÞ ð2 9xÞAx ¼df ð9xÞðAx & ð1 9yÞðAy & x 6¼ yÞÞ ðnþ= 9xÞAx ¼df ð9xÞðAx & ðn 9yÞðAy & x 6¼ yÞÞ:41

These recursive definitions can be captured with Wittgenstein’s exclusive quantifiers. We have: ð0 9xz1; ...; zn ÞAx ¼df  ð9xz1; ...; zn ÞAx ð1 9xz1; ...; zn ÞAx ¼df ð9xz1; ...; zn ÞðAx & ð0 9yx z1; ...; zn ÞAyÞ ð2 9xz1; ...; zn ÞAx ¼df ð9xz1; ...; zn ÞðAx & ð1 9yx z1; ...; zn ÞAyÞ ðnþ= 9xz1; ...; zn ÞAx ¼df ð9xz1; ...; zn ÞðAx & ðn 9yx z1; ...; zn ÞAyÞ:

Now letting p be the statement that there are no ’’s, Potter argues that ’ Wittgenstein’s ‘‘On p’’ will have the same sense as the proposition that there are n ’’s. On Potter’s view, the adjectival nature becomes clear if the operation O in question is construed as taking any statement that there is a certain number of ’’s to the statement that there is one more ’ than that. Potter doesn’t characterize the propositional operation, but using exclusive quantifiers we can put: ð Þ O’ðm 9yÞ’y ¼ ðmþ= 9yÞ’y: ’

Thus, on Potter’s reading, On (0Ey)’y says that there are n many ’’s. It will be worth while to perform a calculation of 2 · 2 = 4 to illustrate the adjectival reading. If we transform using (**) we get:

40 41

Potter, Reason’s Nearest Kin. See David Bostock, Logic and Arithmetic: Natural Numbers (Oxford: Oxford University Press, 1974), p. 10.

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O4; ð0 9yÞ’y ¼ O’½O’½O’½O’ð0 9yÞ’y ¼ O’½O’½O’ð1 9yÞ’y ¼ O’½O’ð2 9yÞ’y ¼ O’ð3 9yÞ’y ¼ ð4 9yÞ’y ¼ ð9vÞð’v & ð9uv Þð’u & ð9zvu Þð’z & ð9xvuz Þð’x &  ð9yvuzx Þ’yÞÞÞÞ:

This asserts that there are exactly four objects that are ’. Numeric superscripts are exponents of a variable binding operator, and it is in this adjectival sense that Potter explains Wittgenstein’s comment that a number is the exponent of an operation. Whether as an operation on propositions (formulas) or as an operation on numerals, the characteristic nature of Wittgenstein’s treatment of cardinal numbers as exponents of an operation comes to the fore. Wittgenstein hopes that by adopting the notion of operation (recursive function) as a primitive formal notion, we can avoid what otherwise would be a realm of substantive arithmetic truths governing an ontology of numbers. The laws are supposedly shown in the syntax of the recursively characterized operation. Laws of association, commutation, and distribution, for instance, are supposed to be shown in simple calculations which use the recursive characterization of addition. The laws, which otherwise would have to be proved by use of mathematical induction, are supposedly shown when number is understood as an exponent of an operation. Consider addition. No definition of addition or proof of 2 + 2 = 4 occurs in the Tractatus.42 But we can imagine introducing addition with ðOuþv Þ’x ¼ ðOu Þ’ðOv’ xÞ Def:

There is an interesting typescript found by von Wright in Vienna in 1965 which is a forerunner of the Tractatus.43 This typescript is not a carbon copy of the Gmunden or Engelmann typescripts. The Gmunden and Englemann typescripts contain the recursive definition of multiplication found in the Tractatus. This is absent from the Vienna typescript. Instead we find the following presented as a proof of 2 þ 2 ¼ 4 (Vienna 6.241):

42 43

Potter mistakenly surmises that a proof of 2 þ 2 ¼ 4 occurs in one of the typescripts of the Tractatus. See Potter, Reason’s Nearest Kin, p. 185. G. H. von Wright, Wittgenstein (Oxford: Blackwell, 1982), p. 75.

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So lautet der Beweis der Satzes 2 þ 2 ¼ 4: 1 þ 1 þ 1 þ 1 ¼ 4 Def: 1 þ 1 ¼ 2 Def: 2 þ 2 ¼ 4:

Undoubtedly, this was originally intended to be taken with the following (Vienna 6.231): It is a property of affirmation that it can be construed as double negation. It is a property of ‘1 þ 1 þ 1 þ 1’ that it can be construed as ‘(1 þ 1) þ (1 þ 1).’

Curiously, Vienna 6.231 is found in the Tractatus without the entry offering a proof of 2 þ 2 ¼ 4. In any event, Vienna 6.241 does not successfully demonstrate 2 þ 2 ¼ 4. The expression ‘‘1 þ 1 þ 1 þ 1’’ is an abbreviation for ‘‘1 þ (1 þ (1 þ 1)),’’ so we only get ‘‘1 þ (1 þ 2)’’ and not ‘‘2 þ 2.’’ The principle of the association of addition is needed. In a modern development of arithmetic, this requires a proof by mathematical induction. It is interesting that the Vienna typescript does not contain the entry in Tractatus proclaiming that numbers are exponents of operations, though it does contain the recursive entry 6.02. It may be that at the time Wittgenstein was still thinking of matters in terms of a reconstruction of arithmetic identity in terms of logical equivalence. In Vienna 6.231 Wittgenstein likens the association of addition to double negation. We have seen that in accordance with the N-operator, all logical equivalents were to be given one and the same representation. Thus, double negation, the association of conjunction, and the like would vanish as substantive principles. If one construes arithmetic identities in terms of logical truths, one could explain the sort of thing Wittgenstein might have had in mind. The notion of the relative product of two dyadic relations R and S is this: m(R/S)n ¼df (9u)(mRu & uSn).

We can then put: m(þ1)2n ¼df m(þ1)/(þ1)n.

On this construal 2 þ 2 ¼ 4 becomes: (9u)(m(þ1)2 u .&. u(þ1)2 n) m,n m(þ1)4n.

This equivalence is grounded in the association of conjunction, and, as we noted, Wittgenstein takes this to have vanished with the N-operator.

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This explanation, however, relies on the following theorem of the logic of relations that establishes that the operation notation is equivalent to that of relative product: mR/Sn .. n ¼ S’(R’m).44

Wittgenstein cannot rely on this logical equivalence to explain Vienna 6.231 and Vienna 6.241. In Principia, operations (functions) are contextually constructed from the theory of relations by means of the theory of definite descriptions. That is, where R is a dyadic relation, Principia’s *30.01 and *30.1 take the functional expression R’x to be contextually defined with: R’x ¼df (y)(xRy).

Thus Principia can readily establish the connection between operation and relative product. Wittgenstein rejects Principia’s approach to functions as relations. So it seems clear that he must be taking operations (functions) and their powers as primitive. In the Tractatus, operations are certainly primitive. For example, þ1’x is an operation, not a ‘‘function’’ (in the sense of Principia’s defined descriptive functions). So we must seek a different explanation of Wittgenstein’s view that principles such as the association of addition are shown by operations and do not require proof by mathematical induction. A Fregean conception of finite cardinals as second-level concepts enabled one to arrive at the thesis that the principle of mathematical induction is a part of logic. Formulated in our notation of structured variables, the idea is to put: N’ ½nx ’x  ¼df ð8FÞðF’ ½0x’x & F’ ½ax ’x  :&: ½ax y x Py bx y x  :a; b : F’ ½bx ’x  : :: : F’ ½nx ’x Þ:

This enables a proof of mathematical induction: ð8FÞðF’ ½0x’x & ðF’ ½ax ’x  : & : ½ax y x Py bx y x  :&: N’ ½ax ’x : :a; b : : F’ ½bx ’x Þ :: ð8nÞðN’ ½ax ’x   F’ ½nx ’x ÞÞ:

But the viability of the proof turns on either accepting impredicative quantification, as Frege did, or accepting Principia’s axiom of Reducibility. Wittgenstein rejected impredicative quantification as well as Reducibility. So he has to reject the Fregean approach. How then does Wittgenstein hope to proceed? 44

In Principia the proof is through the theory of definite descriptions. The function sign S’(R’m) abbreviates the definite description: (z)((y)(mRy)Sz). Wittgenstein’s Tractatus, on the other hand, takes functions (i.e., operators) as primitive.

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It is likely that Wittgenstein’s account of numbers as exponents of an operation was inspired by Russell’s work on the ‘‘powers of a relation.’’ Recursive axioms for addition and multiplication, together with axioms for identity, successor, and mathematical induction, suffice (within a predicate calculus) for modern axiomatic theories of elementary arithmetic. Given Russell’s tutorship, Wittgenstein could not have failed to know this. In Russell’s 1901 paper ‘‘On the Logic of Relations,’’ a recursive definition of the notion of the powers of a relation is accepted as if it were a primitive logical notion. Russell’s recursive definition amounts to this: P0u ðxÞ ¼ x PseqðvÞ ðxÞ ¼ PðPv ðxÞÞ:45

The definition is set in the context of a characterization of a class of progressions whose generating relation is R. If u is progression in this class, 0u is just defined to be the first element of the progression u, and where v is a member of u, Russell defines seq(v), i.e., v þ /, as the entity y such that vRy, when the domain of R is restricted to the class of entities u. Russell remarked that the notion of the ‘‘powers of a relation’’ is indispensable to the general theory of arithmetic as it applies to any progression. Russell argued that the general theory of progressions is independent of any particular series of entities which is a progression. The general form of all the arithmetic of finite numbers can then be derived without appeal to any particular progression. One can prove the isomorphism of progressions, define multiplication and addition for members of any of the progressions whose generating relation is R, and prove all the usual formal arithmetic laws such as commutation and association for addition and multiplication. Wittgenstein seems to be emulating Russell’s notion of the ‘‘powers of a relation’’ when he introduces 6.02 and his notion of ‘‘the powers of an operation.’’ The definitions at TLP 6.02, the definition of multiplication at TLP 6.241, and the definition of addition that one would wish that Wittgenstein had rendered, echo Principia theorems *301.21 and *301.23 and *301.5 which characterize þ 1 (successor), addition, and multiplication (respectively) by means of the notion of powers of relation. In their summary of *301, Whitehead and Russell write: In this number, we have to exhibit the powers of a relation R, i.e., the various members of Potid’R, as of the form R , where is an inductive cardinal. We have already had R2 ¼ RjR and R3 ¼ R2jR. What we need is a definition which shall give Rv þ c1 ¼ RvjR. 45

Bertrand Russell, ‘‘On the Logic of Relations,’’ Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. R. C. Marsh (London: Allen & Unwin, 1977), p. 15.

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Numerical superscripts on a relation sign track the number times we have the relation taken as a relative product with itself. The expression RjR is a class theoretical form of the relative product of a relation R with itself. That is, for any a and b, we have that a(RjR)b is equivalent to (9x)(aRx & xRb). Principia defines the notion of the powers of a relation without any appeal to a system of numbers. This is achieved because the constructions are couched in a system capable of emulating an adequate theory of classes and relations-in-extension. The theory relies ultimately upon its axiom of reducibility. Wittgenstein accepts ramification, but rejects Reducibility. Moreover, he does not accept an ontology of numbers. So again we are back to the question as to what Wittgenstein had in mind. We are on the right trail. Principia comments at *95 that if we took the notion of the powers of a relation and numbers as given, then the notion of ‘‘the general term of a series’’ (and in particular the notion of the series of finite cardinals) could be defined. This seems to be precisely the idea behind Wittgenstein’s approach to arithmetic in the Tractatus. The appeal to recursion is a first step in understanding Wittgenstein’s view of arithmetic in the Tractatus. Of course, the superscripts on relation signs in Principia’s account of the powers of a relation are bindable variables. Wittgenstein uses superscripts in his recursive definitions at TLP 6.02 and 6.241 as schemas for numerals, and as such they cannot be bound. All the same, the parallel with Principia is striking. Wittgenstein is taking the notion of the power of an operation (a function) and recursive definition of functions as primitive notions.46 TLP 6.02 is instance of a general recursion rule. With recursion a primitive, Wittgenstein generates the notion of a consecutive ordered series (of expressions) paralleling the ancestral construction which generates a consecutively ordered a series of objects. Wittgenstein maintains that he can thereby express the general term of the series R, R2, R3, . . . of relative products. He writes (TLP 4.127): The order of the number-series is not governed by an external relation but by an internal relation. The same is true of the series ‘aRb’ ‘ð9xÞ : aRx : xRb’ ‘ð9x; yÞ : aRx : xRy : yRb; ’ and so forth. (If b stands in one of these relations to a, I call b a successor of a.) 46

In modern terms, this assumes all initial recursive functions together with the rules of recursion and substitution and the restricted -operator. These rules enable the formation of complex recursive functions from the initial ones.

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One instance of the relation R (above) is þ/. Now, in Wittgenstein’s view, the general term of a consecutive series is a purely formal notion which is properly expressed in logical notion by a variable. He writes: In order to express the general term of a series of forms, we must use a variable, because the concept ‘term of that series of forms’ is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the ones above is incorrect; it contains a vicious circle.)

The ‘‘vicious circle’’ Wittgenstein alludes to here is just the impredicative quantifiers allowed in Frege and Russell’s early works. In the ramified theory of Prinicpia, impredicative quantifers are excluded.47 Impredicative quantifiers violate the Vicious Circle Principle. Principia recovers mathematical induction and a definition of the ‘‘powers of a relation’’ by appeal to its Axiom of Reducibility. Russell was never satisfied with Principia’s reliance on Reducibility and hoped to find a means of avoiding it. In the Tractatus, Wittgenstein’s solution is to take the notion of the ‘‘general term of a series’’ (recursion) a formal notion. It is to be an ‘‘internal relation’’ expressed with a structured variable, for instance, by introducing a new variable ½0; ; þ=:

The general term of the series of integers is to be built into the variable instead of being defined by means of a theorem of mathematical induction. In this way, Wittgenstein hopes to show that a principle of mathematical induction is obviated by his account of numbers as exponents of operators. Indeed, he hopes to reveal that proofs by mathematical induction are wholly unnecessary once number (in the sense of recursion or the powers of an operation) is adopted as primitive. On Wittgenstein’s view, the ‘‘truths’’ of arithmetic wrought by proofs by mathematical induction of, e.g., the association, commutation, and distribution of addition and multiplication are properly part of the ‘‘scaffolding’’ which is shown in every arithmetic calculation. The Tractarian account of arithmetic has been criticized for its inability to express even the most pedestrian arithmetic statements. In Wittgenstein’s recursive definitions the exponents are schematic letters for numerals, and as such they are not open to quantifiers. The expression ð9pÞðOvþm’  ¼ Op’ Þ

is a pseudo-statement. Simple arithmetic seems to require that arithmetic identity statements and quantification over numbers are well formed. For 47

Anscombe seems confused on this point, writing that Principia violates the Vicious Circle Principle in its definition of the ancestral. See G. E. M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 2nd ed. (New York: Harper & Row, 1959), p. 128.

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example, Wittgenstein’s approach makes it impossible to express inequality, which normally would be defined by x5y ¼df ð9zÞðNumberðzÞ : & : 0 6¼ z :&: x þ z ¼ yÞ:

Similarly, it has seemed impossible to recover division, nonidentity statements, and the like.48 These objections may not be telling, however. With recursive functions (operations) as primitive, many arithmetic calculations can be recovered through the theory of recursion itself. There may be hope for a partial vindication of some of Wittgenstein’s ideas on number. Anscombe goes so far as to say that Wittgenstein’s characterization of a formal series of terms is itself enough to generate arithmetic. Deferring the details to Quine’s paper ‘‘On Derivability,’’ she writes: Quine shows that a certain notion, practically the same as that of a formal series of expressions, enables us to define ancestral relations without any such quantification as ‘every property that . . .’ or ‘some one of the relations . . .’ at all. This definition, which may be called a fulfilment of Wittgenstein’s intentions, accordingly avoids any risk of a vicious circle, such as might arise if some of the properties or relations covered by the quantifications employed had themselves to be specified in terms of the ancestral relation.49

Unfortunately, it not clear how Quine’s work could come to the aid of Wittgenstein. Quine’s constructions require that identity be adopted as a genuine relation.50 Wittgenstein has closed off this approach from the onset. Frascolla’s recent work shows more promise. He has formulated a system W for the Tractarian theory of operations. Frascolla shows that for any numerals m and n, of the language of the equational fragment of Peano Arithmetic (PE), the following holds: ‘W Om’ x ¼ On’ x if and only if ‘PE m ¼ n:51

Frascolla’s system W is a purely equational theory. Signs for logical connectives, quantifiers, and axioms governing them are not included in its language.52 Frascolla’s new work offers an exciting new approach toward a recovery of some form of arithmetic in the Tractatus. But more is needed to comprehend Wittgenstein’s thesis that logic and arithmetic are scaffolding and not genuine sciences whose body of truths are applied. We have argued that Wittgenstein is offering a radically eliminativistic approach to logicism which adopts the notion of finite (ordinal) 48 49 50 51

See Potter, Reason’s Nearest Kin, p. 183. Anscombe, An Introduction to Wittgenstein’s Tractatus, p. 130n. W. V. O. Quine, ‘‘On Derivability,’’ Journal of Symbolic Logic 2.3 (1937): 113–119. Frascolla, ‘‘The Tractarian System of Arithmetic,’’ p. 365. 52 Ibid., p. 364.

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number – more exactly, the notion of recursion (powers of an operation) as a primitive formal notion. This view is corroborated by the exchanges between Wittgenstein and Ramsey. In 1923, while he was a schoolteacher in the Austrian village of Puchberg, Wittgenstein annotated a copy of the Tractatus during discussions of the work with Ramsey. In one marginal comment Wittgenstein wrote that ‘‘the fundamental idea of math[ematics] is the idea of calculus represented here by the idea of operation,’’ and he goes on to say that ‘‘the beginning of logic presupposed calculation and so number.’’53 This is often taken as evidence that Wittgenstein abandoned logicism.54 But the contrary is the case. The marginal note reveals that Wittgenstein’s Tractatus attempts a new eliminativistic reconstruction of arithmetic and logic – one that unifies them both in the notions of operator and calculus. Recall that Wittgenstein thinks his adoption of recursion (numeric powers of operations) does not require an ontology of numbers. On Wittgenstein’s view, the numerals used to indicate the powers of the operation do not name numbers. In fact, the numeral ‘‘0’’ when it occurs in Wittgenstein’s statement that [0, , þ/] is the general term of the series of ‘‘integers,’’ not a name of an integer at all. In Wittgenstein’s view, the series 0, 0 þ/, (0 þ /) þ/, . . .

is a series of numerals and not a series of numbers. Just as the transitions of the N-operator in the calculus of logic manipulate formulas, the operations that form the transitions in the calculus of arithmetic manipulate numerals. In neither case is there any ontological commitment. Appeal to recursive operations in the modern sense cannot provide a straightforward answer to the puzzle of Wittgenstein’s account of arithmetic in the Tractatus. It must be accompanied by an assertion that recursion is itself the practice of calculating. To see this, we must return to the notion of an operation. When Wittgenstein takes the notion of ‘‘number’’ (that is, the notion of an exponent/power of an operation) as a primitive formal notion, he takes himself to have provided for the notion general term of a consecutive series – a characterization of a practice of transition. Wittgenstein imagines the N-operator, which enables our transition from one proposition sign to another, as generating the tautologies of logic; he imagines the idea

53 54

Casimir Lewy, ‘‘A Note on the Text of the Tractatus,’’ Mind 76 (1967), p. 421. A recent example may be found in Marion, Wittgenstein, Finitism, and the Foundations of Mathematics, p. 44. See also Michael Wrigley, ‘‘A Note on Arithmetic and Logic in the Tractatus,’’ Acta Analytica 21 (1998): 129–131. Pasquale Frascolla, ‘‘The Early Wittgenstein’s Logicism: Rejoinder to M. Wrigley,’’Acta Analytica 21 (1998): 133–137.

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of transition from one numeral to another as generating the ‘‘equations’’ of arithmetic. Wittgenstein wrote (TLP 6.24): The method by which mathematics arrives at its equations is the method of substitution. For equations express the substitutability of two expressions and, starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations.

An ‘‘equation’’ embodies the process of transition from one numeral to another; and, by substitution of expressions, it enables the process to continue. In Wittgenstein’s view, number is understood as the power of a recursive operation involved in a calculation. It is here that the unity of logic and arithmetic resides. Wittgenstein wrote in the Tractatus (TLP 6.22): The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.

‘‘Mathematics,’’ he continued, ‘‘is a method of logic’’ (TLP 6.234). The logical method is that of the calculation of the internal properties of operations that are shown in their signs. Such calculations occur with the N-operator of logic and with the numeral operators of arithmetic. Waismann reports that Wittgenstein gave the following explanation of what it is that mathematics and logic have in common: What is right about Russell’s idea is that in mathematics as well as in logic, we are dealing with systems. Both systems are due to operations. What is wrong about it is the attempt at construing mathematics as a part of logic. The true analogy between mathematics and logic is completely different. In mathematics, too, there is an operation that corresponds to the operation which generates a new sense from the sense of a given proposition, namely the operation which generates a new number from given numbers. That is, a number corresponds to a truth-function. Logical operations are preformed with propositions, arithmetical ones with numbers. The result of a logical operation is a proposition, the result of an arithmetical one is a number.55

Wittgenstein’s logicism does not ‘‘reduce’’ arithmetic to logic. But on Wittgenstein’s view, both logic and arithmetic form practices of calculation involving operations (with numeric powers) that generate transitions among signs (not logical or arithmetic objects). The essence of Wittgenstein’s approach is that of calculation (or, better, transition). We can see this by imagining Wittgenstein’s reply to one of Ramsey’s objections to the Tractarian account of arithmetic. One of 55

Waismann, Wittgenstein and the Vienna Circle, p. 218.

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Ramsey’s concerns was the problem of recovering statements about the world which contain mathematical components. Ramsey offers the following example: The square of the number of ’’s is greater by two than the cube of the number of y’s.

Ramsey thinks that we cannot help but analyze this by writing ð9m; nÞð^ y’y e m : ^ : y^yy e n :^: m2 ¼ n3 þ 2Þ:

The difficulty Ramsey sees is that this is an empirical proposition and yet it seems to contain the proposition m2 ¼ n3 þ 2, which is a pseudo-proposition in Wittgenstein’s view. Ramsey is concerned that the application of arithmetic to the world is undermined by Wittgenstein’s approach. But Wittgenstein might reply to Ramsey by maintaining that the subordinate clause m2 ¼ n3 þ 2 has an adjectival role. The statement would then be analyzed as: ðm 9xÞ’x & ðn 9xÞ’x :&: ðm2 9xÞyx y ðn3þ2 9xÞyxÞ:

The arithmetical part consists then in calculating based on operators. Accordingly, Ramsey’s example does not undermine Wittgenstein’s thesis that arithmetic, like inference in logic, does not involve the application of a body of special truths governing logical entities. Wittgenstein writes (TLP 6.1262): Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases.

The legitimacy of the practice of logical inference lies in the tautological nature of the transition – a legitimacy that can be shown by calculation in accordance with the N-operator. Similarly, arithmetic does not involve applying a body of arithmetic truths governing an ontology of numbers. We find (TLP 6.211): Indeed, in real life a mathematical proposition is never what we want. Rather we make use of mathematical propositions only in inferences from propositions which do not belong to mathematics to others that likewise do not belong to mathematics.

Wittgenstein viewed recursion as a practice of calculating. Wittgenstein characterizes this practice by speaking of operations (and implicitly recursion), but all descriptions of operations as existing, and all statements identifying operations, are pseudo-statements that are shown in the practice of calculation itself. Interestingly, writing on number in his introduction to the Tractatus, Russell wrote that ‘‘I cannot see any point on which it is wrong.’’ Russell’s

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introduction only remarks that the theory of number is in need of ‘‘greater technical development,’’ insofar as ‘‘it is only capable of applying to finite numbers’’ (TLP, p. xx). This seems surprising. How could the many problems with Wittgenstein’s account of elementary arithmetic have passed Russell’s vetting? Wittgenstein had sent a version of the Tractatus while in an Italian prison in Cassino and it was in Russell’s hands sometime between the end of June and early August of 1919. It is not certain what version of the Tractatus Russell had before him. Russell’s introduction may have been based on something closer to what von Wright has called the Prototractatus, which he dates circa 1919.56 The Prototractatus does not contain the definition of multiplication at TLP 6.241 (and the proof of 2 · 2 ¼ 4). That was added in 1922 to the Ostwald printing. Moreover, in a letter dated 13 August 1919, Russell asked Wittgenstein to answer a list of questions concerning the Tractatus. The following was included: ‘‘6.03 general form of an integer: [0, ,  þ 1].’’ You only get finite ordinals. You deny classes, so cardinals collapse. What happens to @0? If you said classes were superfluous in logic I would imagine that I understood you, by supposing a distinction between logic and mathematics; but when you say they are unnecessary in mathematics I am puzzled. E.g., something true is expressed by Nc‘Cl‘ ¼ 2Nc‘. How do you re-state this prop?

Wittgenstein answered many of the questions in a letter of 19 August, but there was no reply to this question.57 Russell’s introduction to the Tractatus avoids heavy criticism of its account of number, but it is far away from giving it a pass. In sum, Wittgenstein’s conviction that identity is a pseudo-predicate and his objection to impredicative quantification led him to abandon the Frege/Russell approach to number. He appeals to his N-operator to argue that an axiomatic science of logic can be supplanted by the practice of calculating whether a formula is a tautology. Likewise he appeals to the notion of operators and their numeric exponents to argue that a science of arithmetic can be supplanted by the practice of calculating the outcomes of equations. Both fail. Ramsey was unaware that logic is not decidable and so did not recognize the failure of the N-operator. But soon after his initial enthusiasm, he came to see that Wittgenstein’s account of arithmetic in terms of equations is inadequate. ‘‘I have spent a lot of time developing such a theory,’’ he wrote, ‘‘and found that it was faced with

56

57

Russell’s reference to 6.03 as stating the general term of an integer reveals that he was not looking at the Prototractatus, for the general term of an integer is given at 6.02 in that typescript. See Wittgenstein, Notebooks 1914–1916, 2nd ed., p. 131.

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what seemed to me to be insuperable difficulties.’’58 Ramsey rightly concluded that Wittgenstein’s strictures against using the identity sign were too austere. As we shall see in the next chapter, Ramsey came to reject Wittgenstein’s most prized thesis concerning identity and its corollary that arithmetic (and pure mathematics) consists of equations, and worked instead toward a demonstration that it consists of tautologies.

58

Frank Ramsey, ‘‘The Foundations of Mathematics,’’ in The Foundations of Mathematics and Other Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (New York: Harcourt, Brace and Co., 1931), p. 17.

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Oracular philosophy contented neither Russell nor Ramsey. ‘‘The philosopher must drag beliefs into the light of day,’’ Russell once wrote, ‘‘and see whether they still survive. Often it will be found that they die on exposure.’’1 The Tractatus promises solutions for the problems of philosophy. It is a disappointment that it offers only programmatic gestures toward solutions. It was left to Russell in the 1925 second edition of Principia and Ramsey in ‘‘The Foundations of Mathematics’’ (1925) and ‘‘Mathematical Logic’’ (1926) to develop and assess the viability of some of Wittgenstein’s ideas for rectifying Principia. To evaluate Wittgenstein’s idea that Reducibility would not be needed in a system in which there has been a complete analysis and reconceptualization of all nonextensional contexts, Principia’s second edition offers a new introduction and three new appendices A, B, and C. The new introduction and appendixes of the second edition were written by Russell alone. Whitehead resoundingly disowned the new introduction, publishing a note in Mind that Russell alone was responsible for it.2 Interpretations typically portray Russell’s new introduction as abandoning the system of the first edition and endorsing Wittgenstein’s Tractarian ideas. For example, Monk writes: Russell’s new Introduction suggests major revisions to the original theory of Principia: it dropped the axiom of reducibility and adopted Wittgenstein’s system of logic in the Tractatus, according to which propositions are analysable into truthfunctional compounds of an initial stock of ‘‘elementary propositions.’’ The consequences of these changes were drastic: large parts of mathematics, in particular the theory of real numbers (‘‘Dedekind section’’), were no longer provable from

1 2

Bertrand Russell, Philosophy (New York: W.W. Norton & Co., 1927), p. 5. Whitehead may have seen some of it, but had little patience for Wittgenstein’s aphorisms. See Victor Lowe, Alfred North Whitehead: The Man and His Work, vol. 2 (Baltimore: Johns Hopkins University Press, 1990), pp. 273–278. Whitehead eventually wrote an assessment of Principia in ‘‘Indication, Class, Number, Validation,’’ Mind 43 (1934): 281–297; ‘‘Corrigenda,’’ Mind 43 (1934): 543.

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within the system. This meant, in effect, abandoning Volumes II and III of the original work.3

Monk concludes: In fact, from the point of view of Russell’s original hopes for mathematical logic, the second edition of Principia Mathematica represents a major step backwards. For the parts of mathematics that cannot be proved using the new system are precisely those that had first inspired his Herculean endeavors to reduce mathematics to logic . . . One senses that Russell’s heart was not fully in this attempted return to technical work.4

This is far from historically accurate. Compare what Russell actually says in the introduction to the second edition: Dr. Leon Chwistek took the heroic course of dispensing with the axiom of reducibility without adopting any substitute; from his work, it is clear that this course compels us to sacrifice a great deal of ordinary mathematics. There is another course recommended by Wittgenstein for philosophical reasons. This is to assume that functions of propositions are always truth-functions, and that a function can only occur in a proposition through its values. There are difficulties in the way of this view, but perhaps they are not insurmountable. It involves the consequence that functions of functions are extensional. It requires us to maintain that ‘‘A believes p’’ is not a function of p . . . We are not prepared to assert that this theory is definitely right, but it has seemed worth while to work out its consequences in the following pages. It appears that everything in Vol. 1 remains true (though often new proofs are required); the theory of inductive cardinals and ordinals survives; but it seems that the theory if infinite Dedekind and well-ordered series largely collapses, so that irrationals and real numbers can no longer be adequately dealt with. Also Cantor’s proof that 2n > n breaks down unless n is finite.5

Russell expressed himself unequivocally in writing that ‘‘we are not prepared to assert that this theory is definitely right, but it has seemed worth while to work out its consequences in the following pages.’’ Russell did not abandon the first edition in favor of the Wittgensteinean constructions he discussed in the second edition. Quite to the contrary, his purpose was to evaluate them. His conclusion was measured. When technically worked out, the Tractarian ideas destroy much that is important. Russell’s approach to avoiding Reducibility in second edition of Principia is to reconstruct Wittgenstein’s idea that a complete analysis would yield a thoroughly extensional empirical science in which a ‘‘function can only

3 4 5

Ray Monk, Bertrand Russell: The Ghost of Madness 1921–1970 (New York: The Free Press, 2001), p. 46. Ibid., pp. 47f. A. N. Whitehead and Bertrand Russell, Principia Mathematica, 2nd ed. (Cambridge: Cambridge University Press, 1925, 1957), vol. 1, p. xiv.

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occur in a proposition through its values.’’ Wittgenstein never seems to have seen Russell’s notes for the second edition. Ramsey, however, read Russell’s notes in 1923 and discussed them with him.6 In Ramsey’s 1926 paper ‘‘Mathematical Logic,’’ Russell’s technique in the second edition for recovering a proof of mathematical induction without the Reducibility axiom is described as ‘‘ingenious.’’ Nonetheless, Monk characterizes Ramsey’s review of the second edition of Principia as ‘‘terse’’ and ‘‘unenthusiastic’’ and says that ‘‘Ramsey had little respect for the plans for a new edition of Principia.’’7 Monk’s interpretation is generated without attention to the technical constructions of the first and second editions of Principia. Attention to those constructions suggests that Wittgenstein would have been chagrined by the results of Russell’s second edition concerning his work. Russell concluded that the Tractarian aphorisms on Reducibility yield only marginal success. Ramsey’s review of Russell’s second edition is measured in tone because he hopes to defend Wittgenstein’s ideas against Russell’s conclusions. Ramsey wrote: The principle of mathematical methods which appear to require the Axiom of Reducibility are mathematical induction and Dedekind section, the essential foundations of arithmetic and analysis respectively. Mr. Russell has succeeded in dispensing with the axiom in the first case, but holds out no hope for a similar success in the second. Dedekindian section is thus left as an essentially unsound method, as has often been emphasized by Weyl, and ordinary analysis crumbles to dust. That these are its consequences is the second defect in the theory of Principia Mathematica, and, to my mind, an absolutely conclusive proof that there is something wrong. For as I can neither accept the Axiom of Reducibility nor reject analysis, I cannot believe in a theory which permits me with no third possibility.8

In Ramsey’s view, Russell’s second edition works out Wittgenstein’s ideas in an unsatisfactory way. Mathematical induction is saved, but analysis is lost. Ramsey hopes that some other interpretation of Wittgenstein’s views

6

7 8

Russell’s Autobiography, vol. 2 (Boston: Little, Brown, & Co., 1968), pp. 245, 249, makes it clear that Nicod was also aware of the plans for a second edition. In a letter, he asked Russell to include the result that Principia’s *1.4 p v q .. q v p can be proved from *1.2, *1.3, *1.5, *1.6 if *1.3 is altered to q . . q v p. This result was in his paper ‘‘A Reduction in the Number of Primitive Propositions of Logic,’’ Proceedings of the Cambridge Philosophical Society 19 (1917): 32–41. Russell didn’t mention this, but instead included Nicod’s result reducing the propositional calculus to one axiom (schema) and one inference rule. Nicod died of tuberculosis in February of 1924. Monk, Bertrand Russell: The Ghost of Madness 1921–1970, p. 45. Frank Ramsey, ‘‘The Foundations of Mathematics,’’ in The Foundations of Mathematics and Other Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (Harcourt, Brace and Co., 1931), p. 29.

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might be more successful. In this chapter, we shall evaluate Ramsey’s interpretation of the oracle as compared with Russell’s. Extensionality and neutral monism As Russell saw it, Wittgenstein’s Tractatus advocates ‘‘the principle of extensionality’’ and the principle of truth-functionality (which Russell calls ‘‘the principle of atomicity’’). Russell is right in this, for both are immediate consequences of Wittgenstein’s Doctrine of Showing (once Showing is identified with the Tractarian fundamental idea). Showing demands that all (and only) logico-semantic concepts are shown in the grammar of the logically ideal language. On this view, the breach of truthfunctionality that is commonplace in natural languages must not be taken as reflecting anything ultimate about the ontological structure of the world. Contexts that are not truth-functional (such as the intensional contexts presented by ‘necessity’ and ‘possibility,’ and the intentional contexts presented by ‘believes,’ ‘asserts,’ and ‘says’) embody logico-semantic notions and therefore, by the lights of Wittgenstein’s ‘‘fundamental idea,’’ present formal notions that should be rendered by variables. In the ideal language at the limit of scientific inquiry, semantic relationships are built into syntax. Concepts that are not truth-functional would therefore disappear leaving only extensional contexts and truth-functional complex formulas. A sentence w in which a sentence D occurs is a truth-functional context of D if and only if any sentence E which is materially equivalent to D may be substituted for any occurrence of D in w without alteration of the truthvalue of w. For instance, let the context w be D and suppose that D  E, then D  E. A context w in which a predicate expression D(v1, . . ., vn) occurs with terms (variables or constants) 1, . . ., n occurring in the positions of v1, . . ., vn (respectively) is an extensional context of D(v1, . . ., vn) if and only if for any n-placed predicate expression E such that ðv1 ; . . .; vn ÞðDðv1 ; . . .; vn Þ  Eðv1 ; . . .; vn ÞÞ;

any occurrence of D(v1, . . ., vn) in w may be replaced by E(v1, . . ., vn) with the terms (variables or constants) 1, . . ., n occurring in the positions of v1, . . ., vn (respectively) without altering the truth-value of w. For example, let the context w be (x)’x. We see that (x)(’x  yx) is sufficient to assure (x)yx. On the other hand, suppose that ðxÞðð9yÞðHy & CxyÞ  ð9yÞðKy & CxyÞÞ

says that every entity is a circulatory system that employs a heart if and only it is a circulatory system that uses a kidney. Suppose there are

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properties H* (of being a circulatory system that uses a heart) and K* (of being a circulatory system that uses a kidney). That is, suppose: ðxÞðH x  ð9yÞðHy & CxyÞÞ ðxÞðK x  ð9yÞðKy & CxyÞÞ:

Clearly, if one allows a primitive context H ¼ H

of identity, the context is not be an extensional context of H*. Replacement fails, for the property H* is not identical with the property K*. This cannot be a counterexample to Wittgenstein’s thesis of extensionality, however. He not only rejects the view that identity is a genuine relation, he also rejects the view that there are material properties H* and K*. There must be no logical particles or quantifier expressions in formulas that comprehend material properties and relations. What of the context ‘‘S believes that an ant’s circulatory system uses a heart’’? In symbols this is: S believes ðzÞðAz  H xÞ:

It certainly does not follow that S believes (z)(Az  H*x), i.e., that an ant’s circulatory system uses a kidney. The context is not extensional. Replacement of coexemplifying predicate expressions fails. But again Wittgenstein is ready with the reply that there are no material properties H* and K*. In evaluating whether a given counterexample is telling against Wittgenstein’s thesis of extensionality, one must respect the fact that his thesis concerns material properties and relations. Moreover, one must respect his thesis that material properties and relations have predicable nature only. No fact has the form ‘G(H)’ or ‘R(F, G)’ wherein material properties or relations occur nonpredicationally. In the ideal language, predicates must occur in predicate positions. Even with these qualifications, however, Wittgenstein has a difficult road to travel. Assume that ‘‘F’’ is a material predicate. In the context ‘‘S believes Fx’’ this material predicate occurs in a predicate position; and yet, (x)(Fx  Gx) does not suffice for replacement salva veritate. What analysis did Wittgenstien have in mind which would reveal, appearances to the contrary, that contexts such as belief are extensional? Wittgenstein’s demand of extensionality likely drew him to advocate an elimination (and reconstruction) of the soul (mind and ‘self’). The thesis that propositional attitude contexts are nonextensional surely derives from thinking of the soul or mind as a simple entity capable of standing in a relation to a meaning (an entity which is the thought content shared by users of a given language). The abandonment of ‘soul’ and meanings as entities is an important ally if one is searching for an extensional,

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truth-functional account of propositional attitudes. In the Tractatus, Wittgenstein offers the following entries: At first sight it looks as if it were also possible for one proposition to occur in another in a different way. Particularly with certain forms of propositions in psychology, such as ‘‘A believes that p is the case’’ and ‘‘A has the thought p,’’ etc. For if they are considered superficially, it looks as if the proposition p stood in some kind of relation to object A. (And in modern theory of knowledge (Russell, Moore, etc.) these propositions have actually been constructed in this way.) (TLP 5.541) It is clear, however, that ‘‘A believes that p,’’ ‘‘A has the thought that p,’’ and ‘‘A says that p,’’ are of the form ‘‘‘p’ says p’’: and this does not involve a correlation of a fact with an object, but rather the correlation of facts by means of the correlation of their objects. (TLP 5.542) This shows too that there is no such thing as the soul – the subject, etc. – as it is conceived in the superficial psychology of the present day. Indeed, a composite soul would no longer be a soul. (TLP 5.5421)

Wittgenstein says ‘‘this shows that there is no such thing as the soul.’’ The expression ‘‘shows’’ occurring in 5.5421 makes it appear as if he regards the abandonment of the soul as the consequence of some argument previously given (say, at 5.542). But Wittgenstein’s intent is better expressed in the words: ‘‘this approach reveals that on my view there is no such thing as the soul.’’ The abandonment of the mind or soul as a single entity is the beginning point of Wittgenstein’s gestures toward an eliminativistic analysis of propositional attitude contexts, not the result of some argument against the existence of a simple ‘soul’ that is hinted at in the Tractatus. Russell had retained a unitary ‘self’ (or ‘soul’) in The Problems of Philosophy (1912). Because of this it may seem that Wittgenstein’s ideas, though not themselves explicitly committed to neutral monism, may have played a role in Russell’s conversion to neutral monism. Neutral monism also abandons the soul or unitary ‘self’.9 Entries similar to those of the Tractatus occur in the notes on logic that Wittgenstein dictated to Moore in 1915. Russell saw the notes in 1915 before he adopted neutral monism. Be this as it may, there are strong reasons to think that the influence goes the other way. As early as The Principles of Mathematics (1903) Russell was attracted to a conception of time (and change) that is called the ‘‘static,’’ ‘‘perdurance,’’ or ‘‘four-dimensionalist’’ theory. Russell hoped to address McTaggart’s puzzles about time by conceiving of a physical

9

Nicholas Griffin, ‘‘Russell and Wittgenstein on the Logical Form of Belief Statements,’’ in R. Harre´ and J. Shosky, eds., Russell–Wittgenstein: Logical Form and the Project of Philosophy (Prague: Filosofia, 2006), pp. 69–103.

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continuant as a series of its temporal phases. In Our Knowledge of the External World (1914) Russell endeavored to reconstruct the laws of physical continuants from sense-data and sensibilia. This is a form of four-dimensionalism. Physical objects which continue through time (continuants) are orderings of their appearances (sense-data). What of the conscious ‘self ’ and sensation? Russell does not address the matter. But he was well aware of William James’s 1904 paper entitled ‘‘Does Consciousness Exist?’’ James maintained that consciousness is best modeled as a stream or continuous flow. The parts of the stream (momentary states of consciousness) live in acts of abstraction from the stream. Russell published papers on James’s neutral monism in 1907 and 1914.10 It is now known that the first chapters of Russell’s unfinished book Theory of Knowledge (written between 1912 and 1913) were discussions of neutral monism. In those papers, Russell expressed his attraction to the thesis, but he found it to have serious difficulties. By 1918, however, Russell adopted neutral monism and abandoned his earlier notion of sensation as a relation between a ‘self’ and a sense-datum. The relation of acquaintance as a primitive dyadic relation is also abandoned, and Russell now embarked on a project of constructing both the physics of continuants and the laws of psychology (consciousness) in terms of different orderings of phases. Russell endeavored to develop entries 5.541, 5.542, and 5.5421 of the Tractatus in Appendix C of his second edition of Principia. Russell’s analysis assumes that there is no unitary soul or ‘self.’ He writes that a person is a relation A, which serially orders facts in its field. That is, a person is series of events (facts).11 This is, at minimum, a four-dimensionalist analysis of a person. Though Russell does not mention neutral monism in Appendix C, it is reasonable to assume that it lies in the background.12 Accordingly, a soul (mind, consciousness) is itself taken as a series of events. As always, it is difficult to be sure what interpretation best suits Wittgenstein’s oracular remarks in the Tractatus. Russell would certainly have discussed both four-dimensionalism and neutral monism with him. Moreover, Russell is surely the best authority we have on what Wittgenstein had in mind. The Doctrine of Showing of the Tractatus

10 11 12

Bertrand Russell, ‘‘On the Nature of Acquaintance,’’ Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. R. C. Marsh (London: Allen & Unwin, 1977). A. N. Whitehead and Bertrand Russell, Principia Mathematica to *56 (Cambridge: Cambridge University Press, 1964), p. 403. Favrholdt cites Russell’s comment that ‘‘this series of events is part of the series of events that constitute the person,’’ and mistakenly concludes that ‘‘Russell’s interpretation . . . has no consequences concerning the concept of a subject [or soul].’’ See David Favrholdt, An Interpretation and Critique of Wittgenstein’s Tractatus (Copenhagen: Munksgaard, 1965), p. 109.

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requires a reconstruction of propositional attitude reports that supports extensionality. Since Russell’s interpretation is explicitly couched within four-dimensionalism (and so likely is a form of neutral monism), we should not balk at the possibility that Wittgenstein had imagined some such version when he boldly conjectured that an extensional account can, in principle, be formulated. In examining the viability of the Tractarian view that, when properly analyzed, ‘‘A believes p’’ does not undermine truth-functionality, Russell is not advocating the view. He endeavors only to establish the minimal result that the principle of extensionality (and of truth-functionality) is not refuted, when strictly interpreted, by the analysis of such sentences as ‘‘A believes p.’’ Not surprisingly, Russell’s account had difficulties. The difficulties indicate to some interpreters of Wittgenstein that Russell was working out his own ideas and misinterpreted the oracle. But Russell was not working out his own ideas. He was assessing and developing Wittgenstein’s, so far as they were intelligible to him. Unfettered by the usual requirement of stating a clear position, Wittgenstein’s pronunciations stand the test of time by having the dubious merit of being ineffable. As Russell later put it, ‘‘he himself, as usual, is oracular and emits his opinion as if it were a Czar’s ukase, but humbler folk can hardly content themselves with this procedure. I have examined the problem at length in An Inquiry into Meaning and Truth (pages 267ff), but the conclusion at which I arrived is somewhat hesitant.’’13 Wittgenstein’s abandonment of the ‘soul’ was not offered on the basis of having found a new reconstruction or reconceptualization of propositional attitude contexts. He simply pronounces that if one abandons the ‘soul’ some reconceptualization of propositional attitudes will be uncovered. Wittgenstein’s pronouncements were grounded by his research program – his intuition that the Doctrine of Showing is correct. Russell starts his analysis with the simple example ‘‘A asserts that Socrates is Greek.’’ He hopes to work his way up to the more complicated cases of belief and aboutness. An assertion (or utterance token), he says, is a series of noises in temporal succession. A word such as ‘‘Socrates’’ is a class of series (ordered tokens) containing the sound type so˘k0 ra tez0 . That is, a word is a class of similar noises. Accordingly, in the present case, we first have a token noise which is a member of  (i.e., a so˘k0 ra tez0 noise type), then a token noise which is a member of  (an ˘ız noise type), and lastly a

13

Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 88. Not surprisingly, the passages from the 1940 Inquiry echo Russell’s ideas in Appendix C of the second edition of Principia.

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token noise which is a member of  (a grek noise type). Where x#y is just the ordered pair of x and y, Russell offers the following: A asserts that Socrates is Greek¼df ð9x; y; zÞðx 2  & y 2 : & :z 2  & x # y [ x # z [ y # z  AÞ:

This just says that there are noise tokens x, y, z which are members of the words ‘‘Socrates,’’ and ‘‘is,’’ and ‘‘Greek’’ (respectively) and that ðaÞðbÞðaðx # y [ x # z [ y # zÞb:  :aAbÞ:

That is, the so˘k0 ra tez0 noise (itself an event or fact) comes immediately before the ˘ız noise, and the ˘ız noise comes immediately before the grek noise in the series of events A which constitutes the person. On this construal, we see clearly that ‘‘A asserts that Socrates is Greek’’ is not a sentential context of ‘‘Socrates is Greek,’’ and it cannot breach truthfunctionality, since it does not present a sentential context for which another sentence may be substituted. Nor, indeed, is it a context which could breach extensionality, since it contains no predication ‘‘. . . is Greek’’ for which a coextensive predicate could be placed. Turning next to Russell’s attempt to analyze ‘‘A believes p’’ on behalf of Wittgenstein, it is useful to examine Russell’s work notes for what would become Appendix C of the 1925 second edition of Principia. In his 1923 paper ‘‘What is Meant by ‘A Believes p’?’’ Russell begins by taking a belief as an ‘‘occurrence’’ (event or fact) in the series of events that is a person.14 He says it is called a ‘‘belief’’ because of certain correspondence relations it has to fact. The nature of a belief as an event, he continues, does not matter. Russell then takes T as the correspondence relation that obtains when a belief corresponds to the fact f that makes the belief true; similarly, he takes F as a noncorrespondence relation that obtains when a belief fails to correspond to the fact f. In order to define the notion of a proposition as a nonlinguistic entity, Russell notes that various different occurrences (events/facts) are said to be beliefs in the same proposition. Two beliefs are beliefs in the same proposition when they are related by correspondence or noncorrespondence to the same fact, namely, the fact that makes them both true or both false. A proposition is then defined as the class of all the beliefs having one of the two relations, correspondence or noncorrespondence, to a given fact: Prop ¼df a^ð9f Þða ¼ y^ðyTf Þ :v: a ¼ y^ðyFf ÞÞ: 14

Bertrand Russell, ‘‘What is Meant by ‘A believes p’?’’ in The Collected Papers of Bertrand Russell, vol. 9, Essays on Language, Mind and Matter 1919–1926, ed. John G. Slater (London: Unwin Hyman, 1988), p. 159.

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For example, the proposition ‘Socrates is Greek’ is the class of all beliefs that correspond to the fact Socrates’ being Greek. Finally, Russell has: A believes y^ðyTf Þ ¼df 9!ð^ yðyTf Þ \ C’AÞ A believes y^ðyFf Þ ¼df 9!ð^ yðyFf Þ \ C’AÞ:

Returning to our example, ‘‘A believes that Socrates is Greek’’ is analyzed as the statement that there is a belief (event) common to both the class of all beliefs that correspond to the fact Socrates’ being Greek and to C’A (the field of the relation A).15 A person A believes that Socrates is Greek if and only if the belief (event) which corresponds to the fact Socrates’ being Greek is among the series of events that constitutes the person A. Russell regards this as a first start. He knew full well that this analysis of ‘‘A believes p’’ is troublesome in that it allows that a fact can be named. He wrote: The reference to facts and truth or falsehood can however be eliminated. Obviously beliefs can be defined by their intrinsic properties, and perhaps two beliefs in the same proposition can be defined by their intrinsic similarities. We still, however, have a statement which has a fact for a subject, the fact of being A’s belief. This reference can, however, be eliminated. Take ‘‘A believes B killed C.’’ This says that in A there are occurrences called ‘‘ideas’’ of B and killing and C, related in a certain way. Here the proposition does not appear, and ‘‘A believes p’’ is not a function of p.16

In the example above, we have: A believes yˆ(yTSocrates’s being Greek) ¼df 9!ð^ yðyTSocrates’s being GreekÞ \ C’AÞ:

This uses the expression ‘‘Socrates’ being Greek’’ as a name of a fact. On Russell’s view, a genuine name must be a simple sign – an individual variable of the formal language. A definite description would not be possible either. The multiple-relation theory of judgment was abandoned because Russell came to believe that universals have only a predicable nature (and cannot also be subjects). A definite description of the purportedly corresponding fact is impossible. Indeed, it was precisely for this reason that Russell sought a new analysis of ‘truth’ as correspondence and dallied with negative facts and noncorrespondence as a relation in 1917. As we saw in chapter 2, a definite description would have to place a universal word in a subject position. Russell felt that Wittgenstein was 15 16

That is, the domain together with the range of the relation (series of events) that constitute the person A. ‘‘What is Meant by ‘A believes p’?’’ p. 159.

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right in maintaining that the difference between a universal and a particular is itself a logical difference. The nominalistic semantics of predicate variables offered in the first edition of Principia relies on a recursive analysis of ‘‘truth’’ as correspondence whose base clause is grounded by the multiple-relation theory. The multiple-relation theory allows universals to be logical objects which can stand in a relation of acquaintance. By the time of the logical atomism lectures, universals have only a predicable nature. Universal words cannot occupy subject positions. A new correspondence theory of belief is therefore needed. The presentation in Principia’s second edition aims to rectify these difficulties by avoiding names of facts as subjects. In a given belief predicating a property of Socrates, an idea (or thought) of Socrates and an idea of ‘being Greek’ stand in a relation. Russell calls this relation that of ‘‘predication.’’ He writes that ‘‘this is the relation which holds between our thought of the subject and the thought of the predicate when we believe that the subject has the predicate. It is wholly different from the relation that holds between the subject and the predicate when the belief is true.’’17 Thus, where  is the set of ideas that have the same meaning as a Socrates idea, and  is the class of ideas that have the same meaning as an idea of being Greek, and P stands for predication, Russell has: A believes that Socrates is Greek ¼df ð9x; yÞðx 2  & y 2  :&: xPy & x 2 C’A : & : y 2 C’AÞ:

Ideas with the same meaning are such as to be substitutable in ascriptions of belief without alteration of the truth of the ascription. On Russell’s analysis, the logical analysis of ‘‘A believes p’’ reveals that that it is not a sentential context of p. Thus it contains no sentential context which, by replacement of a materially equivalent sentence, could breach truthfunctionality. Moreover, there is no occurrence of a predicate ‘‘. . . is Greek’’ for which to substitute a coextensive predicate in an attempt to breach extensionality. Another example of an apparently non-truth-functional context is ‘‘Socrates is a constituent of ‘Socrates is Greek.’’’ We know that Socrates is Greek if and only if Plato is Greek,

and yet the replacement of ‘‘Plato is Greek’’ in ‘‘Socrates is a constituent of ‘Socrates is Greek’’’ transforms a truth into a falsehood. Russell observes that in this case there are three different matters that must first be

17

Whitehead and Russell, Principia Mathematica to *56, p. 404.

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separated. The question may concern the fact (state of affairs), or it may concern the belief, or finally it may concern a token of an utterance type (verbal proposition). The case of the token utterance (or sequence of sounds) is uninteresting and Russell quickly dispatches it as follows. Russell takes the word ‘‘Socrates’’ to be a class l of series (ordered tokens) each of which contains an instance of the sound type so˘k0 ra tez0 . The utterance type ‘Socrates is Greek’ is a class u of series (ordered tokens) each of which contains a so˘k0 ra tez0 noise then an ˘ız noise then a grek noise. Russell’s analysis of the fact that the word ‘‘Socrates’’ is a constituent of the utterance type ‘‘Socrates is Greek’’ is as follows. Each token of the utterance type ‘‘Socrates is Greek’’ contains a part which is a token of the utterance type ‘‘Socrates.’’ That is, if P is a series so˘k0 ra tez0 ˘ız grek which is an instance of the utterance type ‘‘Socrates is Greek,’’ then there is a series Q which is a token of the utterance type ‘‘Socrates’’ and so a member of l which is part of the P series. The case of the Socrates being a constituent of the fact ‘Socrates is Greek’ is analyzed quite differently. Socrates is a continuant which is analyzed as a series of events. Socrates doesn’t occur in any single event. Rather, a momentary particular ‘Socratesphase’ occurs. Russell takes a fact to be an event as well. The properties or relations occurring in such events are phases and are particular to them. Thus the notion of a continuant Socrates occurring in a fact is entirely reconceptualized within this ontology. Russell tells us that an event may resemble another in respect of particulars (that occur in them) or in respect of the n-adic relations (occurring in them) or both. Russell calls the first sort of resemblance ‘‘particular resemblance,’’ and the others ‘‘n-adic relation resemblance.’’ In certain cases, any two events that have particular resemblance to a third event have particular resemblance to each other. Russell uses this to reconstruct the notion of an event c being about exactly one particular (who persists though time). Suppose events c, c*, c** have particular resemblance and are related as above. Then the particular (for example, Socrates) that such events are about is constructed as the class of events that have particular resemblance to c. Thus the notion of the continuant Socrates being a constituent of an event ‘Socrates is Greek’ is reconstructed as the notion that the event is a member of the class of events which constitute Socrates. The analysis of ‘‘Socrates is a constituent of the belief ‘Socrates is Greek’’’ is analogous. A belief-event is a momentary event containing thought phases. To say that Socrates (the continuant) is a constituent of a belief is reconceptualized in Russell’s framework of ordered events. A beliefevent is about the continuant Socrates if it is a member of the class of belief-events constituting a certain idea (a class of psychical events each of which refers to a Socratesphase).

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Russell intends only a very rough sketch whose aim is to reveal that truth-functionality is not breached by the reconstruction. He summarizes his discussion by distinguishing ‘‘propositions considered factually’’ and ‘‘propositions as vehicles of truth and falsehood.’’ That is, he distinguishes statements in which sentences occur as subordinate clauses that are to be analyzed by means of reference to facts (events) and those in which the subordinate sentence acts as a vehicle of truth and falsehood (such as when flanked by a logical particle). Russell concludes that it is a proposition considered factually that is involved in the analysis of such statements as ‘‘A believes p’’ and ‘‘p is about A’’ (i.e., ‘‘A is a constituent of p’’). Thus these statements do not properly form contexts that are not truth-functional. Russell’s analysis fails to fulfill Wittgenstein’s austere demand that all notions with logico-semantic content are to be shown, for it employs semantic notions such as ‘‘thought,’’ ‘‘sameness of meaning,’’ and ‘‘predication’’ which would have to be built into formal grammar. But Russell explained: ‘‘It is not necessary to lay any stress upon the above analysis of belief, which may be completely mistaken. All that is intended is to show that ‘A believes p’ may very well not be a function of p, in the sense in which p occurs as a truth-function.’’18 Russell intended only a preliminary investigation whose purpose was simply to reveal that Wittgenstein’s theses of extensionality and truth-functionality are not obviously false. When he returned briefly to the issue in his 1940 book, An Inquiry into Meaning and Truth, Russell rejected Carnap’s (quasi-behaviorist) proposal that belief is a relation between a person and an expression. But he reaffirmed that his analysis in the second edition of Principia had shown that contexts of propositional attitudes cannot be used to form a telling argument against either extensionality or truth-functionality.19 The oracle on Reducibility In the second edition of Principia, Russell endeavored to work through some of Wittgenstein’s ideas for avoiding the Axiom of Reducibility. The axiom schema of Reducibility plays a central role in the proof of mathematical induction. In Principia’s ramified type-theory, the natural numbers are each of type ((o)), and are characterized by the following:

18 19

Ibid. Bertrand Russell, An Inquiry into Meaning and Truth (London: Allen & Unwin, 1940, 1966), p. 268.

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ððoÞÞ is a member of every class ðððoÞÞÞ containing 0ððoÞÞ and the successor of every class ððoÞÞ which is a memberof  ðððoÞÞÞ :20

Since this expression contains a quantifier over classes of type (((o))) it has truth-conditions of order 4. To prove the principle of mathematical induction, however, this very expression has to be taken as comprehending a third-order function of ((o)). Only in this way can the class which is its extension be in the range of the quantifier ‘‘every class (((o))).’’ In an analogous way, constructions needed for analysis and the real numbers rely upon Reducibility. In fact, Reducibility also plays a role in assuring the adequacy of the definition of the identity sign in Principia. Russell has: 

13:01 xt ¼ yt ¼df ð’ðtÞ Þð’ðtÞ ðxt Þ  ’ðtÞ ðyt ÞÞ:

From this we get: xt ¼ yt :  : Aðxt Þ  Aðyt Þ:

Reducibility assures that xt ¼ yt will support full replacement of the signs in all contexts A. Without Reducibility some contexts A would be excluded from the range of the predicate variable ’(t). Quite independently of any considerations of Wittgenstein or Ramsey, Russell had misgivings about Principia’s reliance on the axiom of Reducibility. In Principia he observes that it might be removed from his logicist foundation for mathematics: That the axiom of reducibility is self-evident is a proposition which can hardly be maintained . . . In the case of the axiom of reducibility, the inductive evidence in its favour is very strong, since the reasoning which it permits and the results to which it leads are all such as appear valid. But although it seems improbable that the axiom should turn out to be false, it is by no means improbable that it should be found to be deducible from some other more fundamental and more evident axiom.21

Whether one regards a given principle is a ‘‘truth of logic’’ depends on what one takes logic to be. Having discovered the paradox of the attribute R an attribute F exemplifies if and only if F does not exemplify itself, Russell resigned himself to the belief that the science of logic must reconstruct the structure of a ramified type-theory of attributes without any principles comprehending attributes in intension (‘‘propositional functions’’). The

20

21

Principia uses special lower-case Greek letters as class variables. But, of course, these variables are not genuine since there are no classes countenanced in the work. See chapter 5 for the definition of an order\types symbol. Principia Mathematica to *56, p. 60.

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construction in Principia embraces a nominalistic semantics for predicate variables. In Principia, the order indices on predicate variables are philosophically explained in terms of recursively defined hierarchy of orders of truth and falsehood couched in a nominalistic semantics for predicate variables. Principia offers a pragmatic justification for Reducibility: it facilitates the theory of classes (and arithmetic). For this reason Russell called the axiom ‘‘the axiom of classes.’’ Indeed, Russell argued that if an ontology of classes were assumed, then Reducibility would be validated. The argument is as follows. The comprehension of classes (of individuals) would be rendered by an axiom schema thus: ð9cðoÞ Þðzo Þðzo 2 cðoÞ :  : Bzo Þ;

where c(o) is not free in B, and 2 is adopted as a primitive relation sign. There are no restrictions on what bound variables occur in the formula B. Now consider the following predicative comprehension principle: ð9’ðoÞ Þðzo Þð’ðoÞ ðzo Þ  Azo Þ;

where ’(o) is not free in A and no variables in A have higher order\type than (o). The expression zo 2 c(o) meets this restriction on A. Thus we arrive at ð9’ðoÞ Þðzo Þð’ðoÞ ðzo Þ  Bzo Þ;

which is Reducibility (in the monadic case). Similar argument can be made for n-adic cases. Unfortunately, Russell’s pragmatic justification for the Axiom of Reducibility is insufficient to warrant assuming it as a part of logic – given that is, a nominalistic interpretation of predicate variables. If it could be known that there must be some logical conception of a class, Russell’s pragmatic reasoning would be persuasive. But this is the very question his logicism was supposed to address. Russell’s paradox places the burden of proof on the logicist who maintains that there is a logical notion of a class. The paradox suggests that there can be no logical conception of a class. Principia was supposed to demonstrate that a logical conception of a class is recoverable in spite of the paradox. After Principia, Russell came to see that a nominalistic semantics philosophically grounds comprehension principles far weaker than Reducibility. The recursive truth-definition grounds only the comprehension schema for the theory of predicative order\types.22 The nominalistic semantics of Principia validates only principles of predicative type-theory. Thus, from the perspective of Principia’s nominalistic semantics for predicate variables Reducibility is an ad hoc principle, added on in 22

See chapter 5.

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order to emulate classes and relations-in-extension in the system. In this regard, the principle seems to fare no better than an outright postulation of the existence of classes. In his 1919 book, Introduction to Mathematical Philosophy, Russell continues to express the hope that Reducibility might be avoided.23 And now he is convinced that it is not a purely logical principle. He admits that given its dependence upon Reducibility, Principia’s theory of classes is ‘‘not finally satisfactory’’ and is ‘‘less complete’’ than the theory of descriptions.24 Given Russell’s early concerns with Reducibility, it seems likely that his comments encouraged Wittgenstein to search for a replacement. It is clear enough that Reducibility is a significant comprehension principle and not a tautology in Wittgenstein’s sense. In the Tractatus, we find (TLP 6.1232): Propositions like Russell’s ‘axiom of reducibility’ are not logical principles, and this explains our feeling that, even if they were true, there truth could only be the result of a fortunate accident.

We saw that the quantification theory of section *9 of Principia brought Wittgenstein to conjecture that the logical truths of quantification theory are arrived at by means of (generalizations from) tautologies. Principles for the comprehension of attributes (and functions) are certainly not logical truths of simple quantification theory and are not generalized tautologies. Indeed, on the conceptions of logic offered by both Frege and Russell (prior to Principia) the presence of such comprehension principles in logic is precisely what makes logic informative. In his efforts to avoid types (and orders) of entities, Principia’s nominalistic semantics tried to emulate comprehension principles for attributes without making the assumption that every open well-formed formula of the language of logic comprehends an attribute in intension (with both a predicable and an individual nature). Given Russell’s paradox, such assumptions cannot be part of logic. Wittgenstein viewed the matter another way. For Wittgenstein, tautologyhood is the mark of a proposition of logic. Since comprehension principles are not tautologies, comprehension principles for attributes cannot be part of logic.25 It wasn’t long until he thought he had a counter-model of Reducibility. In a letter to Russell of 1913 Wittgenstein wrote: 23 24 25

Bertrand Russell, Introduction to Mathematical Philosophy (London: Alan & Unwin, 1919, 1953), p. 193. Ibid. Note that one cannot arrive at Reducibility theorems by starting from a generalized tautology such as Ax  x Ax and applying existential generalization to arrive at (9’)(’x  x Ax). This inference is illicit. It violates the distinction between a schematic letter A and a predicate variable ’.

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Imagine we lived in a world in which nothing existed except @0 things and, over and above them ONLY a single relation holding between infinitely many of the things and in such a way that it did not hold between each thing and every other thing and further never held between a finite number of things. It is clear that that axiom of reducibility would certainly not hold good in such a world.26

It is difficult to understand what Wittgenstein had in mind in this cryptic passage. In the Tractatus, Wittgenstein is content to say the following (TLP 6.1233): It is possible to image a world in which the axiom of reducibility is not valid. It is clear, however, that logic has nothing to do with the question whether our world really is like that or not.

In any event, it is important to emphasize that whether or not Reducibility is counted as part of logic depends essentially on what one takes to be the nature of logic. The worlds imagined by Wittgenstein in which Reducibility is false are obviously worlds which satisfy his conception of logic. The same point applies to Ramsey, who also attempted a countermodel of Reducibility. Ramsey writes: All the primitive propositions of Principia Mathematica are tautologies, except the Axiom of Reducibility, and the rules of deduction are such that from tautologies only tautologies can be deduced, so that were it not for one blemish the whole structure would consist of tautologies.27

For Ramsey, the comprehension principle of Reducibility is not a generalized tautology (or arrived at by means of generalization from tautologies). Hence, he echoes Wittgenstein in proclaiming that if it is true it is little more than a ‘‘happy accident.’’ In 1925, Ramsey wrote: It is clearly possible that there should be an infinity of atomic functions, and an individual a such that whichever atomic function we take, there is another individual agreeing with a in respect of all the other functions, but not in respect of the function taken. Then (’)(’!x  ’!a) could not be equivalent to any elementary function of x.28

To explain Ramsey’s idea, assume that G, y, y, w . . . are all the atomic functions that a exemplifies; and assume that a, b, c, d, e, and so on are all distinct. Then consider this array: 26 27

28

Ludwig Wittgenstein, Letters to Russell, Keynes and Moore, ed. G. H. von Wright and B. McGuinness (Oxford: Blackwell, 1974), p. 42. Frank Ramsey, ‘‘Mathematical Logic,’’ in The Foundations of Mathematics and Other Logical Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (London: Harcourt, Brace & Co., 1931), p. 76. See also ‘‘The Foundations of Mathematics’’ in the same volume, p. 11. Ramsey, ‘‘The Foundations of Mathematics,’’ p. 57.

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Ramsey next shows that x ¼ a, that is (’)(’!x « ’!a) cannot be coextensive with any atomic function G, y, y, w . . . For suppose it were coextensive with G, then since Gc, we have a ¼ c which contradicts the assumption of Ramsey’s counter-model that a, b, c, etc., are distinct. If it is coextensive with y then we have a ¼ b which contradicts the assumption, and so on. Ramsey then observes that if x ¼ a is not equivalent to any atomic (quantifier-free) function, then it is not equivalent to any first-order function either. Ramsey’s counter-model of Reducibility does not apply to a realist interpretation of Principia’s predicate variables.29 Indeed, there is no compelling reason to find Reducibility objectionable when couched in a logical realist conception of attributes. To see this, we need only observe that, unlike logical realism, Ramsey’s counter-model assumes the logical independence of atomic attributes. Independence requires that if F and G are atomic, then it is possible for an entity to exemplify both. If F is an atomic attribute of a, there is an entity b in Ramsey’s world such that Fb. Suppose, however, that independence did not obtain and that there is an atomic attribute G such that (z)(Gz  Fz). By hypothesis, b has every atomic attribute of a except F. But now Ramsey’s world would demand that Gb, and we have a contradiction. So Ramsey’s model assumes independence.30 Ramsey’s argument cannot apply against Principia under a realist semantics of its predicate variables which takes there to be attributes

29

30

Crabbe´ shows that Reducibility is (deductively) independent of the other axioms of ´ ´ Logique et analyse 21 (1978): Principia. Marcel Crabbe,´ ‘‘Ramification et predicativit e,’’ 399–419. But our concern is whether in an objectual semantics, every model of the axioms of Principia (minus Reducibility) is a model of Reducibility. Potter attempts to give a counter-model to the so-called ‘‘strong’’ form of Reducibility. Potter follows Church in imagining that Principia has two different accounts of predicativity: a weak notion in the introduction and a strong form at *12. The strong form reduces all predicative n-place functions to elementary n-place functions. See Michael Potter, Reason’s Nearest Kin: Philosophies of Mathematics from Kant to Carnap (Oxford: Oxford University Press, 2001), p. 161. This interpretation is mistaken. There is only one form of predicativity in Principia. See Gregory Landini, Russell’s Hidden Substitutional Theory (New York: Oxford University Press, 1998). Potter’s counter-model assumes a form of the independence of attributes inconsistent with ramification.

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(propositional functions) in intension. The lowest order\type on a predicate variable is (o), and quite clearly comprehension in Principia does not obey logical independence. For instance, by comprehension we have: ð9’ðoÞ Þðxo Þð’ðoÞ ðxo Þ  y ðoÞ ðxo ÞÞ:

Indeed, there is simply no such thing as Ramsey’s ‘‘atomic propositional function’’ in the formal language of Principia. Thus there is no such entity in the intended realist semantics for Principia. Ramsey’s argument would have to suppose that Principia had definitions such as: ð’ðoÞ ÞA’ðoÞ ¼df ð8F a ÞðA½ð8x1 o ; . . .; 8xn o ÞðF a ðx1 o ; . . .; 8xn o Þj’ðoÞ Þ & A½ð9x1 o ; . . .; 9xn o ÞðF a ðx1 o ; . . .; xn o Þj’ðoÞ Þ;

where F a is an atomic function variable, and ’(o) occurs in a predicate position in A with respect to variables v1o, . . ., vno and the variables x1o, . . ., xno replace v1o, . . ., vno respectively. But quite clearly this is not in the system of Principia. On the other hand, in a nominalistic semantic interpretation of Principia’s predicate variables, Ramsey’s argument seems telling. Principia’s introduction offers just such a nominalistic semantic interpretation of predicate variables. The nominalistic semantics employs a recursively defined hierarchy of senses of ‘‘truth’’ and ‘‘falsehood’’ whose base case relies on Russell’s multiple-relation theory of judgment. On this view, a judgment (or belief) is a complex (fact) containing the constituents of the would-be corresponding fact that makes it true. Among the constituents is a universal (a relation or property) which inheres predicatively in the would-be corresponding fact. Thus, universals form the foundation for the hierarchy of senses of ‘‘truth’’ and ‘‘falsehood.’’ This requires the logical independence of the universals that inhere in facts (complexes) that form the truth-makers for the formulas of Principia. Ramsey’s ‘‘atomic functions’’ meet this criterion, and thus he has shown that Russell’s nominalistic semantics cannot validate Principia’s Axiom of Reducibility. Are some relational universals logically irreflexive for Russell? Consider a universal P such that (x)(Px  T(x, b)) where T is a logically irreflexive relation.31 Suppose then that P is an atomic property of a in Ramsey’s world. Then if F is an atomic function of a in Ramsey’s world, and P is not F, Ramsey’s world has it that Pb. But this is impossible because of the logically irreflexive nature of R. Following Wittgenstein, 31

It might be thought that ‘x left of y with respect to z’ is logically irreflexive where x and y are concerned. But this presumes a particular nonlogical theory of space-time, and so the irreflexivity is not a matter of logic but of physics.

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Ramsey maintains that there cannot be any atomic functions (universals) that are logically irreflexive. If a purported universal is logically irreflexive, or asymmetric, etc., then it is so only in virtue of a logical structure yet to be uncovered by analysis. Wittgenstein’s Doctrine of Showing demands that all logical features are formal and shown by the logical syntax of an ideal formalism. The thesis of the independence of atomic facts (and of atomic functions) is the consequence of Showing. Russell did not embrace Showing, but he did intend a nominalistic semantics for Principia’s predicate variables and a recursive definition of truth and falsehood. The foundation of this edifice was the universals inhering in facts, and thus Russell had arrived at a thesis very near to the logical independence of universals and (atomic) facts. Because of this, Russell would have to consider Ramsey’s counter-model of Reducibility a serious problem for his nominalistic semantics. The informal nominalistic semantics offered to explain the order component of Principia’s predicate variables legitimates only a predicative comprehension principle. The Axiom (schema) of Reducibility is thus unjustified by the semantics. This is why Russell came to think it of the utmost importance to find a way to avoid Reducibility. Slipshod notations? In Principia’s second edition, Russell evaluated some of Wittgenstein’s ideas for a neo-Principia without Reducibility. Russell reconstructed Wittgenstein’s ideas by modifying the formal system of the first edition. Let us use the name ‘‘PrincipiaW’’ for the system Russell built in order to evaluate Wittgenstein’s ideas. The system has a formal grammar quite different from that of Principia, and it adds extensionality principles. The new system occurs in the new introduction and in Appendix B of Principia’s second edition. Unfortunately, the revisions to the formal grammar that Russell envisioned for the reconstruction of Wittgenstein’s ideas have been largely missed. The new formal system is difficult to uncover from the introduction because Russell brashly intersperses discussions of his intended nominalistic semantics with statements concerning what are to be the new axioms and grammar of the system. In an important discussion of Appendix B of the second edition – where Russell attempted to recover a proof of mathematical induction without Reducibility – John Myhill admitted that he did not understand the new grammar. He wrote: Appendix [B] is a thorny text indeed, full of slipshod notations which it taxed the patience of this reader to correct, and I am not at all sure I got to the heart of his error since there are so many superficial ones which can be corrected in various

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ways (and one does not know which is the right correction, i.e., which one represents just what Russell had in mind).32

Fortunately, a clearer picture of Russell’s intent has recently come to the fore. The ‘‘slipshod notations’’ Myhill found and his negative assessment of Russell’s findings were the result of his misunderstanding of the new grammar. In PrincipiaW, predicate variables include Greek letters ’, y, y (and we add their primes so that we cannot run out). For clarity of exposition, let us put type indices superscripted on the right-hand side of variables and predicate terms. Order indices are numerical superscripts on the lefthand side of the variables and predicate terms. (For convenience, order superscripts can be omitted when the order is the order of the type.) The order\types of the first edition are dropped. In the first edition, all and only predicate variables are predicative (the order is the order of the type). In the new grammar introduced for the purposes of assessing Wittgenstein’s ideas, order is separate from the order of the type. An order superscript is ‘‘o’’ if and only if the type superscript is ‘‘o.’’ But aside from that, Russell does not demand that the order is the order of the type. Orders can now be higher than the order of the type. (Indeed, it is even open whether the order could be lower than its type.)33 The system PrincipiaW introduces a schema of extensionality. Where a (t) P and bQ(t) are used for any terms of the system, and mvt is used for an individual variable or predicate variable of the system, we can formulate extensionality as follows: (EXT) ðm vt Þða PðtÞ ðm vt Þ  b QðtÞ ðm vt ÞÞ :  : A½a PðtÞ   A½b QðtÞ ja PðtÞ ; where bQ(t) replaces some (not necessarily all) free occurrences of aP(t) in A, mvt is an individual variable or a predicate variable, and bQ(t) and aP(t) are predicate terms (variables or lambda abstracts).

Russell is clear in his endorsement of extensionality. He writes: According to our present theory, all functions of functions are extensional, i.e., yÞ  fðy y^Þ: ’x x yx :  : fð’^

32

33

John Myhill, ‘‘The Undefinability of the Set of Natural Numbers in the Ramified Principia,’’ in George Nakhnikian, ed., Bertrand Russell’s Philosophy (New York: Barnes & Noble, 1974), p. 25n. In this respect, the system seems akin to that set out by Copi. See Irving Copi, The Theory of Logical Types (London: Routledge & Kegan Paul, 1971).

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. . . Consequently, there is no longer any reason to distinguish between functions and classes, for we have, in virtue of the above, y ¼ y y^: ’x x yx :  : ’^ We shall continue to use the notation yˆ(’y) which is more convenient than ’yˆ; but there will no longer be any difference between the meaning of the two symbols.34

Given the extensionality of the system, Russell regards class and relationin-extension abstracts as themselves complex predicate terms. It is more convenient, however, to use Church’s lambda expressions to form complex predicate terms. Thus, we define the membership sign and the use of class and relation-in-extension expressions as follows: 5dl Pl tl ; . . .;dn Pn tn 4 2

a

fbl vl tl ; . . .;bn vn tn : Agðtl; ...; tnÞ

¼df a ½l bl vl tl ; . . .;bn vn tn Aðtl; ...; tnÞ ðdl Pl tl ; . . .;dn Pn tn Þ:

Then we add an axiom schema of lambda-conversion: a

½lvl tl ; . . .; vn tn Aðtl; ...; tnÞ ðdl Pl tl ; . . .;dn Pn tn Þ:  : A½dl Pl tl jvl tl ; . . .;dn Pntn jvn tn :

The well-formed formulas (wffs) and terms are defined in the usual way.35 In the new grammar, an argument to a predicate variable of type (t) must be of type t, but its order can even be greater than the order of the predicate variable to which it is argument! This is a striking new feature. Another striking feature of the grammar of PrincipiaW is that predicate variables (and predicate terms) of different order can meaningfully flank the identity sign. This will certainly appear strange. For a realist semantics for the system it suggests that attributes (‘‘propositional functions’’ as intensional entities) of different orders may nonetheless be ‘‘identical.’’ On a realist semantics, attributes with different orders must surely be distinct. The explanation for both these striking features is that Russell intended a new nominalistic semantics for the predicate terms and variables of PrincipiaW. (For convenience, one can imagine ‘‘identity’’ in the

34 35

Principia Mathematica, 2nd ed., p. xxxix. A recursive definition of the terms and formulas follows: (i) A variable or constant of given type and order is a term of that type and order. (ii) Where aP(t1, . . ., tn) is a term, and b1Q1t1, . . ., bnQntn are terms, then aP (t1, . . ., tn)(b1Q1t1, . . ., bnQntn) is a wff. (iii) Where A is any wff, and bxt an variable, then (bxt)A is a wff. (iv) If A and B are wffs, then (A), (A v B), are wffs. (v) Where A is a wff in which the variables x1t1, . . ., xntn occur free, and a is an order symbol for a number greater than or equal to the order of the type (t1, . . ., tn), then a [lx1t1, . . ., xntnA] (t1, . . ., tn) is a term. (vi) These are the only terms and wffs.

Principia’s second edition

211

system interpreted as indiscernibility by the lights of the ideography of the formal system.) The identity sign in PrincipiaW is defined just as in the 1910 Principia.36 Thus we put: a

PðtÞ ¼ b QðtÞ ¼df ðc ’ððtÞÞ Þðc ’ððtÞÞ ða PðtÞ Þ:  : c ’ððtÞÞ ðb QðtÞ ÞÞ;

where c ¼ max(a, b) þ 1.

This definition of identity will allow for full substitutivity. This is because the predicate of being coextensive with bQ(t), that is, b

½lvt ðxt ÞðvðtÞ ðxt Þ 

b

QðtÞ ðxt ÞÞðtÞ ;

will be in the range of the quantifier c’(t), since c ¼ max(a, b) þ 1. By the extensionality axiom, coextensivity is sufficient for full substitutivity. Thus, identity flanked by predicate terms and variables will be sufficient for their full substitutivity. This result will not, however, apply to individual variables. Presumably Russell also intends the following axiom, though he did not state it: x ¼ o yo :  : Aðo xo Þ  A½o yo jo xo ;

o o

where oyo replaces some (not necessarily all) occurrences of oxo in A. In Appendix B, Russell attempted a proof of mathematical induction without assuming a Reducibility principle. Go¨del found a flaw in one of the lemmas (*89.16) and wondered if it could be avoided.37 Myhill showed that mathematical induction cannot be recovered if we simply drop Reducibility and append extensionality and predicative comprehension principles to the system of Principia’s first edition.38 I have shown that in the new syntax of PrincipiaW, the flaw in the proof of *89.16 can be patched and mathematical induction recovered without assuming Reducibility.39 This was the positive result on behalf of Wittgenstein’s ideas that Russell wanted to herald in the new edition of Principia. But, as we saw, his overall assessment of Wittgenstein’s idea that a more thoroughly extensional system would avoid the need for Reducibility was much more negative. Cantor’s power-class theorem and real analysis is not recoverable. Reducibility would be needed after all.

36 37 38

39

Principia Mathematica, 2nd ed., p. xxxvii. Kurt Go¨del, ‘‘Russell’s Mathematical Logic,’’ in Paul A. Schilpp, ed., The Philosophy of Bertrand Russell, vol. 1 (Evanston, Ill.: Harper Torchbooks, 1963), p. 145. More exactly, the ramified type system of Principia as reconstructed in K. Schu¨tte’s Beweistheorie (Berlin: Springer verlag, 1960). See Myhill, ‘‘The Undefinability of the Set of Natural Numbers,’’ p. 27. Gregory Landini, ‘‘The Definability of the Set of Natural Numbers in the 1925 Principia Mathematica,’’ Journal of Philosophical Logic 25 (1996): 597–615.

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The derivability of mathematical induction in the second edition relies heavily upon the extensionality principle (EXT) of the new system. In particular, it follows from this principle that if predicates are coextensive over order m entities, then they are coextensive over entities of every order. Hazen and Davoren finds this consequence unacceptable, and ‘‘incompatible with the philosophical motivation’’ for the system of the second edition. Russell’s philosophical motivation, of course, was to avoid the assumption of an Axiom of Reducibility. Hazen and Davoren write: this [that is, (EXT)] would only follow if for every higher level function a coextensive function already existed at a lower level. And this is simply Russell’s Axiom of Reducibility, which he was not assuming in the introduction and Appendix B, and which is not valid on the intended semantics.40

This misunderstands Russell’s intentions. To be sure, Russell was not assuming the Reducibility axiom (schema) of the first edition of Principia as an axiom (schema) in the second edition. His plan was to work without any such axiom, revise the grammar, and adopt extensionality principles (which would be justified by a nominalistic semantics for the predicate variables and predicate terms). But surely he did not mean to rule out the possibility that an analog of reducibility (set in the new grammar) is derivable in the new system. Indeed, this is just the sort of result Russell was hoping for. In fact, on several occasions Russell expressed a hope that some logical truth might be found from which the results obtained via Reducibility might be deduced; and in the second edition he expressed hope that some logical truth perhaps more restricted in scope than Reducibility might enable the recovery of real analysis.41 Hazen and Davoren are mistaken about Russell’s philosophical motivations. Nevertheless, (EXT) does yield an analog of Reducibility in the system I sketched above. The argument is simple. Leaving out unimportant type and order indices for convenience, the following is a theorem in the system: ðn ’Þð½l ð9 n yÞðn yx x xÞðn ’Þ  ½l x x xðn ’ÞÞ:

It follows straightforwardly from (EXT) that these two lambda abstracts are intersubstitutable in all contexts. Now the following is also a theorem:

40

41

A. P. Hazen and J. M. Davoren, ‘‘Russell’s 1925 Logic,’’ Australasian Journal of Philosophy 78 (2000): 549. See also A. P. Hazen, ‘‘A Constructive Proper Extension of Ramified Type Theory (The Logic of Principia Mathematica, Second Edition, Appendix B),’’ in Godehard Link, ed., One Hundred Years of Russell’s Paradox (Berlin: De Gruyter, 2004), pp. 449–480. Principia Mathematica, 2nd ed., vol. 1, p. xlv.

Principia’s second edition

213

ðnþm ’Þð½l v v vðnþm ’ÞÞ:

By (EXT) we substitute and arrive at: ðnþm ’Þ½ln  ð9 n yÞðn yv v n vÞðnþm ’Þ:

Then by lambda-conversion, we have a form of Reducibility for PrincipiaW: ðnþm ’Þð9 n yÞðn yv v

nþm

’vÞ:

This certainly raises questions about (EXT). The derivability of Reducibility in PrincipiaW from (EXT) obviates Russell’s work in Appendix B toward the rectification of mathematical induction. Contrary to Russell’s assessment, PrincipiaW would be able to recover Cantor’s power-class theorem and real analysis. More dramatically still, using the technique of the above argument, we see that in the new grammar the hierarchy of orders drops out. For instance, we know that the following is a theorem: ðm yððtÞÞ Þð0ðtÞ 2

m

yððtÞÞ :&: ðvðtÞ ÞðvðtÞ 2 myððtÞÞ  S‘vðtÞ 2 myððtÞÞ Þ:  :

mþ1

NððtÞÞ  myððtÞÞ Þ:

Mathematical induction is the stronger principle: ðm yððtÞÞ Þð0ðtÞ 2 myððtÞÞ :&: ðvðtÞ ÞðvðtÞ 2 myððtÞÞ & vt 2 ::

mþ1

ððtÞÞ

N

m ððtÞÞ

 y

mþ1

NððtÞÞ :  : S‘vðtÞ 2 myððtÞÞ Þ

Þ:

The usual proof of mathematical induction is blocked because we cannot instantiate the variable my((t)) above to the class m þ 1{vt: vt 2 0(t) .&. vt 2 m þ 1 ((t)) N }. But by following the lesson just learned, we know that 0ðtÞ 2

m ððtÞÞ

y

:&: ðvðtÞ ÞðvðtÞ 2

m ððtÞÞ

y

 S‘vðtÞ 2

m ððtÞÞ

y

Þ ::

mþ1

NððtÞÞ 

m ððtÞÞ

y

is coextensive over order my((t)) with [ly yx x yx]. Thus, by (EXT) it is coextensive with it over every order. We have: ðmþ1 yððtÞÞ Þð0ðtÞ 2 ::

mþ1

ððtÞÞ

N

mþ1 ððtÞÞ

y



:&: ðvðtÞ ÞðvðtÞ 2

mþ1 ððtÞÞ

y

mþ1 ððtÞÞ

y

 S‘vðtÞ 2

mþ1 ððtÞÞ

y

Þ

Þ:

The proof of mathematical induction now goes through. Indeed, by the above lesson, any theorem which is a universal statement with its quantifier ‘‘restricted’’ to an order m can be shown to hold where the quantifier is not so restricted because it will be coextensive over order m with a logical function [ly yx x yx].

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These considerations reveal that (EXT) is too strong to have been what Russell intended for PrincipiaW. All the same, Russell uses extensionality in his rectification of mathematical induction in Appendix B. So an interpretation of the second edition cannot simply drop it from the system or restrict it to propositional functions of lowest type. What seems wrong is to apply extensionality in cases where the predicate expressions in question express purely logical notions. Perhaps the following better represents what Russell had in mind: (EXT)* ð9vt Þa PðtÞ ðvt Þ & ð9vt Þ aPðtÞ ðvt Þ & ð9vt Þb QðtÞ ðvt Þ ^ ð9vt Þ a QðtÞ ðvt Þ :: ðm vt Þða PðtÞ ðm vt Þ :  :b QðtÞ ðm vt ÞÞ :  : A½a PðtÞ  ! A½b QðtÞ ; where bQ(t) replaces some (not necessarily all) free occurrences of aP(t) in A, mvt is an individual variable or a predicate variable, and bQ(t) and aP(t) are predicate terms (variables or lambda abstracts).

This modification blocks the above derivation in PrincipiaW of a form of Reducibility. At the same time, it lets through my patch of Russell’s 89.16 based on (EXT)* and makes sense of Russell’s work in the second edition to rectify mathematical induction.42

Ramsey’s extensional functions Ramsey separated the paradoxes into two groups, maintaining that only one is relevant to the logicist concern to capture mathematics within logic. Paradoxes such as Richard’s paradox, the Ko¨nig/Dixon, the Grelling, the Liar, and the Berry paradoxes essentially involve semantic notions concerning ‘‘reference,’’ ‘‘naming,’’ or ‘‘truth.’’ These paradoxes are not relevant to logicism. Ramsey wrote: the second set of contradictions are none of them purely logical or mathematical, but all involve some psychological term, such as meaning, defining, naming or asserting. They occur not in mathematics, but in thinking about mathematics; so

42

Hazen and Davoren are not so sanguine about extensionality principles in the 1925 Principia. Indeed, they argue that they have found a model of the system in which instances of (EXT) come out false. See Hazen, ‘‘A Constructive Proper Extension of Ramified Type Theory,’’ p. 476. Their argument is based on the result that in a standard second-order calculus one can define the truths of first-order arithmetic. Unfortunately, the result does not hold in ramified second-order logic. See George Boolos and Richard Jeffrey, Computability and Logic, 2nd ed. (Cambridge: Cambridge University Press, 1980), pp. 206–211.

Principia’s second edition

215

that it is possible that they arise not from faulty logic or mathematics but from ambiguity in the psychological or epistemological notions of meaning and asserting.43

In Ramsey’s view, only simple type-theory is needed, for this is sufficient to block Russell’s paradoxes of classes and predication (and kindred paradoxes). Ramsey’s rejection of ramification has been interpreted as a form of realism with respect to attributes and the advocacy of simple-type theory. The formal system for simple type-theory drops considerations of order from the notion of an order\type symbol set out in Principia’s first edition. By dropping orders one can justify adopting the impredicative comprehension principle ð9’ðtl; ...; tnÞ Þðxl tl ; . . .; xn tn Þð’ðtl; ...; tnÞ ðxl tl ; . . .; xn tn Þ  AÞ;

where ’(t1, . . ., tn) is not free in A. In appearance this looks like Principia’s *12.1n, but in the context of the ontology of a type-theory of attributes, it is not a Reducibility principle. On a realist view, attributes are intensional entities that are stratified into types. Orders have no place in such a realism. As Quine frequently pointed out, attributes are independent of mind and language. They don’t contain ontological counterparts of quantifier expressions. Order is a feature of the linguistic representation of an attribute, not a feature of the attribute itself. A realist semantics for predicate variables avoids ramification. This interpretation of Ramsey neglects the historical evolution of Principia. It leaves Ramsey in ignorance of the parameters under which Russell worked to find a solution of the paradoxes of classes and attributes. Ramsey’s 1926 paper ‘‘Mathematical Logic’’ suggests, to the contrary, that he understood well enough that the source of ramification in Principia’s first edition was its nominalistic semantics. In fact, Ramsey is not advocating simple type-theory and impredicativity. Like Russell, he endeavors to recover the structure of a type-theory of classes/attributes without the ontology. Like Russell, he intends a nominalistic semantics for Principia. But Ramsey thinks he has found a nominalistic semantics that does not generate ramification. Though Ramsey’s untimely death cut short his offering a detailed discussion of the semantic interpretation he had in mind, there are ample hints that he intended a semantics which embraces infinite conjunctions and disjunctions. Ramsey’s plan is to interpret predicate variables by appeal to functions which can take objects to infinitary formulas. 43

Ramsey, ‘‘The Foundations of Mathematics,’’ p. 77.

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Wittgenstein’s Apprenticeship with Russell

In his introduction to the second edition, Russell examined the idea of infinite conjunctions and disjunctions as a means of avoiding Reducibility. Omitting type indices on his variables for convenience, Russell investigates the issue of infinite conjunctions and disjunctions in the context of a nominalistic semantics. First, there are variables ’ with type indices and no order indices, and these are to range over quantifier-free formulas of a first-order language L1. Next, where A[] is any quantifier-free formula of L1 in which  occurs in a predicate position, Russell offers the following definition: ð’1 ÞAð’1 Þ ¼df ð’ÞðA½ðxÞ’x^ yÞ & ð’ÞðA½ð9xÞ’x^ yÞ:

This defines the use of a variable ’1 ranging over all first-order formulas of L1 containing exactly one quantifier. Similarly, one can define the use of a variable ’2 ranging over all first-order formulas of L1 containing exactly two quantifiers, and so on. Then, instead of introducing a new variable ’1 (note the subscript) ranging over all first-order formulas (expressed by 1 (o) ’ above), one could try to get along with an infinite conjunctions and disjunctions as follows: ð’1 ÞAð’1 Þ :¼: ð’1 ÞA½’1  & ð’2 ÞA½’2  & . . . & ð’n ÞA½’n  & ð’nþ1 ÞA½’nþ1 ; etc:

But after his investigation, Russell rejected this way of developing Wittgenstein’s ideas.44 He writes: It is obvious that, in practice, an infinite conjunction or disjunction such as the above cannot be manipulated without assumptions ad hoc. We can work out the results for any segment of the infinite conjunction or disjunction and we can ‘‘see’’ that these results hold throughout. But we cannot prove this, because mathematical induction is not applicable. We therefore adopt primitive propositions, which assert only that what we can prove in each case holds generally. By means of these it becomes possible to manipulate variables such as ’1.

Russell considers introducing variables such as ’1, y 1, y1, etc. as primitive. He does not define them in terms of infinite conjunction and disjunction. He then extends the language L1 to the language L2 which includes these new variables. The variable ranges over all first-order formulas of L1. Russell observes that the truth-conditions for any formula using bound or free occurrences of the variables ’1, y 1, y1 are the same as those of some formula that does not use them. The reason is that ’1x expresses (in the semantics) one from among of the formulas, (’1)A[’1], (’2)A[’2], 44

In spite of this, Michael Potter maintains that Russell’s second edition embraced infinite conjunctions and disjunctions. See Potter, Reason’s Nearest Kin, p. 204.

Principia’s second edition

217

(’3)A[’3], and so on, where A[] is any quantifier-free formula of L1 in which  occurs in a predicate position and contains x free. So by exploring any given case, we can see that ð’ÞAð’Þ  Að’1 Þ

holds once we are given a particular formula A(’1). It would be possible, Russell notes, to add this as an axiom schema. From this, one could prove 

A0

ð’ÞAð’Þ  ð’1 ÞAð’1 Þ:

Reducibility is not needed here. But similar considerations will not ground 

A

ð’ÞAð’Þ  Að’2 Þ:

The variable ’2 is introduced into the language L3 and ranges over formulas of L2. Now ’2x will be interpreted in the semantics as ranging over formulas such as (’1)A(’1) and (9’1)A(’1), where A contains x free. In virtue of A0 Russell notes that we can allow that the formulas ’2x ranges over will have the same truth-conditions as (’)A(’) and (9’)A(’). Russell then demonstrates that A˚ will not always hold. This, he decides, was the source of Principia’s need for an Axiom of Reducibility. Russell rejects the idea of appealing to infinite formulas. Instead, he offers PrincipiaW as the system which best represents Wittgenstein’s idea that a function can occur only through its values. The result of Russell’s investigation was that one can rectify mathematical induction but cannot recover real analysis. Ramsey rejected Russell’s assessment, hoping that a different development of Wittgenstein’s ideas might bear more fruit. Ramsey’s ideas developed out of his reading of Russell’s work notes for the second edition. Though Russell did not pursue the idea of employing infinite conjunctions and disjunctions, Ramsey does. In ‘‘The Foundations of Mathematics,’’ Ramsey writes: In this lies the great advantage of my method over that of Principia Mathematica. In Principia the range of ’ is that of functions which can be elementarily expressed, and since ð’Þ:fð’!^ y; xÞ cannot be so expressed it cannot be a value of ’!; but I define the values of ’ not by how they can be expressed, but by what sort of senses their values have, or rather, by how the facts their values assert are related to their arguments. I thus include functions which could not even be expressed by us at all, let alone elementarily, but only by a being with an infinite syllogistic system. And any function formed by generalization being actually predicative, there is no longer any need for an Axiom of Reducibility.45

In his paper ‘‘Mathematical Logic,’’ Ramsey ties the idea to Wittgenstein’s conception of quantification. He writes: 45

‘‘The Foundations of Mathematics,’’ p. 42.

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Wittgenstein’s Apprenticeship with Russell

On Wittgenstein’s theory a general proposition is equivalent to a conjunction of its instances, so that the kind of fact asserted by a general proposition is not essentially different from that asserted by a conjunction of atomic propositions. But the symbol for a general proposition means its meaning in a way different from that in which a symbol for an elementary proposition means it . . . Hence the orders of propositions will be characteristics not of what is meant, which is alone relevant in mathematics, but of the symbols used to mean it . . . Applying this mutatis mutandis to propositional functions, we find that the typical [order] distinctions between functions with the same arguments apply not to what is meant, but to the relation of meaning between symbol and object signified. Hence they can be neglected in mathematics.46

Ramsey goes on in the paper to say that it is by accepting Wittgenstein’s theory of general and existential propositions that we could ‘‘get rid of the Axiom of Reducibility.’’47 The plan Ramsey adopts from Wittgenstein is that a general proposition binding a predicate variable can be semantically interpreted in terms of assignments that involve an infinite conjunction. To take an example, consider the formula ð’ðoÞ ÞðB  ’ðoÞ ðyo ÞÞ;

where B is some closed first-order formula. In Russell’s nominalistic semantics, the nominalistic truth-conditions for the formula will be circular unless the predicate variable ’(o) ranges over formulas of a fixed language L, whose expressions do not contain such predicate variables. The order of the order\type symbol is 1 and so the formulas that are substitution instances for it must be formulas of language L1 which can contain at most bound individual variables. Ramsey also interprets predicate variables such as ’(o) nominalistically. But, unlike Russell, he allows countable infinite conjunctions and disjunctions of the formulas (which contain at most bound individual variables) to be substituends for the variable ’(o). Call this language $L1. On Ramsey’s semantics, there is an interpretation function R (based on an interpretation of $L1) that assigns entities in the domain to the individual variables and extensional functions to the predicate variables of the language of simple type-theory. For example, consider again the formula ð’ðoÞ ÞðB  ’ðoÞ ðyo ÞÞ:

Let us assume that the object a is assigned to the variable yo. Ramsey’s semantics interprets the formula as saying that for every assignment

46

‘‘Mathematical Logic,’’ p. 77.

47

Ibid., p. 79.

Principia’s second edition

219

function fe there is some formula A of $L1, such that fe(a) ¼ A and dB  Ae is true. Since for some of the functions fe the formula A will be an infinite conjunction, the limitations of Russell’s nominalistic semantics do not apply to Ramsey’s nominalistic semantics. If the formula ð’ðoÞ ÞðB  ’ðoÞ ðyo ÞÞ

is true in the Ramsey semantics, then so is B  ð’ðoÞ ÞðB  ’ðoÞ ðyo ÞÞ:

This is because there is an extensional function fe that assigns a to an infinite conjunction of formulas of $L1 which is semantically equivalent to (’(o))(B  ’(o)(yo)). In this way, Ramsey hoped to generate a nominalistic semantics which does not generate ramification and thereby validates *12.1n. Ramsey believes that in his infinitary nominalistic semantics, *12.1n is not a Reducibility axiom, and in fact is consistent with Wittgenstein’s demand that logic consist of tautologies. Once we understand that Ramsey intended an infinitary semantics, we come to an understanding of his conception of a ‘‘propositional function in extension.’’ This plays a central role in his treatment of identity. In a letter of 1923 to his mother, Ramsey wrote that he had reported his knowledge of Russell’s work notes for the second edition of Principia to Wittgenstein. He put the response vividly: ‘‘He [Wittgenstein] is, I can see, a little annoyed that Russell is doing a new edit[ion] of Principia because he thought he had shown Russell that it was so wrong that a new edition would be futile. It must be done altogether afresh.’’48 It is worth pointing out that the reason Wittgenstein thought Principia needed to be ‘‘done afresh’’ escapes Monk’s biography of Russell. But the reason is clear. Wittgenstein took his elimination of identity to be a central achievement of the Tractatus, and it was precisely because of the elimination of identity that he maintained that logicism must reconceptualize arithmetic truths in terms of equations rather than tautologies. The rejection of identity would thoroughly alter the constructions of the first edition of Principia. Ramsey corroborates this interpretation in his letter to Wittgenstein: I went to see Russell a few weeks ago, and am reading the manuscript of the new stuff he is putting in to the Principia. You are right that it is of no importance; all that it really amounts to is a clever proof of mathematical induction without using the

48

Frank Ramsey, letter to Wittgenstein of 20 September 1923, in Ludwig Wittgenstein: Letters to C. K. Ogden, with an Appendix of Letters by Frank Plumpton Ramsey, ed. G. H. von Wright (Oxford: Blackwell, 1973), p. 78.

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axiom of reducibility. There are no fundamental changes; identity is just as it used to be.49

The same criticism of Russell appears in Ramsey’s review of the second edition for Nature.50 Russell admitted that for a time he accepted Wittgenstein’s views on identity but soon came to the conclusion that it made mathematical logic impossible and, in fact, that Wittgenstein’s criticism is invalid.51 Though Ramsey’s initial letters are in sympathy with Wittgenstein on identity, it wasn’t long until he too had a change of heart. Ramsey’s diary entry of February 1924 reports that in his meeting with Russell to discuss the second edition, Russell was ‘‘rather good against W’s [Wittgenstein’s] identity.’’52 In his 1925 work ‘‘The Foundations of Mathematics,’’ he wrote that he had spent a lot of time developing Wittgenstein’s construal of identity and the theory of mathematics as equations but found it to be ‘‘faced with insuperable difficulties.’’53 Identity must be recovered in some way. Ramsey hoped to find some compromise between Wittgenstein and Russell on identity. Ramsey’s paper ‘‘The Foundations of Mathematics’’ attempts to recover the uses of Principia’s identity sign by introducing a more radically extensional conception of a propositional function. Principia’s first edition had the following definition: x ¼ y ¼df ð’!Þð’!x  ’!yÞ:

Ramsey accepts Wittgenstein’s claim that atomic states of affairs are logically independent. On this view, predicate variables of lowest type range over the universals that are the constituents of logically independent (atomic) states of affairs. With lowest type predicate variables interpreted ranging over material properties, Principia’s definition of identity would not be viable. Entities occurring in atomic states of affairs may be distinct and yet share all their material properties. Unlike Wittgenstein, however, Ramsey thought that a foundation for arithmetic will require a proxy for the theory of classes. ‘‘By using these variables [functions in extension],’’ he

49 50 51

52 53

Frank Ramsey, letter to Russell of 20 February 1924, in Wittgenstein, Letters to C. K. Ogden, p. 84. Frank Ramsey, review of the second edition of Principia Mathematica, Nature 116 (1925): 127–128. See Russell, My Philosophical Development, p. 115. Russell endorses Wittgenstein against identity in Our Knowledge of the External World, 2nd ed. (London: Allen & Unwin, 1926), p. 212. A footnote to ‘‘unpublished work of Wittgenstein’’ in the first edition (1914) was altered in subsequent editions to reference the Tractatus. Cited in Monk, The Ghost of Madness, p. 509. Ramsey, ‘‘The Foundations of Mathematics,’’ p. 17.

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wrote, ‘‘we obtain the system of Principia Mathematica, simplified by the omission of the Axiom of Reducibility, and a few corresponding alterations. Formally it is almost unaltered; but its meaning has been considerably changed.’’54 Ramsey rejects Wittgenstein’s demand that arithmetic consist of equations, not tautologies. The notations of Principia remain unaltered, though the semantics has changed their meanings. Principia wouldn’t have to be ‘‘done afresh’’ after all. Ramsey introduces the notion of a propositional function in extension ’ex, which is a function from individuals to propositions. ‘‘Such a function of one individual,’’ Ramsey tells us, ‘‘results from any one-many relation [function] in extension between propositions and individuals.’’55 He then introduces the expression ‘‘(’e)(’ex  ’ey),’’ as a new way of capturing identity. One will be immediately struck by two problems. First, Ramsey’s symbol ‘‘’ex’’ is a term, not a formula, and as such it cannot flank the biconditional sign.56 The biconditional sign is a statement connective, not a relation sign. Thus, the expression ‘‘(’e)(’ex  ’ey)’’ is not well formed as an object-language expression.57 Second, Ramsey is unclear as to what he means by a ‘‘proposition.’’ Clearly he does not mean to introduce an ontology of Russellian propositions understood as intensional entities. What then can be made of Ramsey’s propositional functions in extension? The answer is that Ramsey was not introducing a new objectlanguage sign, but introducing a new nominalistic semantic interpretation of the first-order predicate variables ’(o) of Principia’s first edition. To remind ourselves that Ramsey intends a function sign, let us use the sign fex instead of the sign ’ex. Thus, Ramsey’s semantics for Principia’s predicate variables ’(o) exploits the function fe from individuals to propositions (formulas of the language L1 that contain at most bound individual variables). It is assumed in Ramsey’s semantics that fe is a function – i.e., for any individual x and any formulas p and q of the language L1, if fe(x) ¼ p and fe(y) ¼ q, then q is equivalent to q. Ramsey is not suggesting that we alter the syntax of Principia, replacing x ¼ y ¼df ð’!Þð’!x  ’!yÞ

by the definition x ¼ y ¼df ð’e Þð’e x  ’e yÞ:

54 56 57

Ibid., p. 56. 55 Ibid., p. 52. As we have seen, Principia’s logical particles are statement connectives. See Peter Sullivan, ‘‘Wittgenstein on ‘The Foundations of Mathematics,’ June 1927,’’ Theoria 61 (1995): 124.

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Ramsey intends that the syntax of Principia remain unaltered. It is the semantic interpretation that is to be changed in accord with the conception of a propositional function in extension. Restoring lowest order\type indices to Principia’s (’!)(’!x  ’!y), we get ð’ðoÞ Þð’ðoÞ ðxo Þ  ’ðoÞ ðyo ÞÞ:

On Ramsey’s view, this is to be semantically interpreted so that it comes out true if and only if for every extensional function fe and all individuals x and y (in the domain), there are formulas A and B of L1 such that fe(x) ¼ A and fe(y) ¼ B and dA  Be is true. Under Ramsey’s semantics, if the formula p assigned to the variable x is the same as the formula assigned to y, then we get the tautology dA  Ae. On the other hand, if the individual x is not y, then there is certainly a function fe and some formula B of L1 such that fe(x) ¼ B and fe(y) ¼ B. Hence we get the contradiction dB  Be. Accordingly, the semantics makes (’(o))(’(o)(xo)  ’(t)(yo)) come out as either a tautology or a contradiction. Ramsey concludes that his semantics offers a compromise between Russell and Wittgenstein on identity. Ramsey observes that his interpretation of identity agrees with Wittgenstein that the identity of indiscernibles is not a logical truth. Principia’s axiom of infinity asserts that there are an infinity of discernible individuals, but as Ramsey puts it, ‘‘on my system, which admits functions in extension, the Axiom of Infinity asserts merely that there are an infinite number of individuals.’’58 Ramsey’s semantics offers a new meaning for the axiom of infinity. His semantics of extensional functions suggests a way to show that although infinity must be assumed as a axiom, it is either tautologous or contradictory. Ramsey’s semantics interprets (9x)(x ¼ x), i.e., ð9xo Þðð8’ðoÞ Þð’ðoÞ ðxo Þ  ’ðoÞ ðxo ÞÞÞ;

as saying that there is some individual x in the domain of the interpretation such that for every function fe there is some formula B of L1 such that fe(x) ¼ B and dB  Be is true. Because the last clause is a tautology, Ramsey concludes that (9x)(x ¼ x) is either a tautology or the domain of interpretation is empty. If the domain is empty, then (9x)(x ¼ x) has no semantic interpretation at all and so is ‘‘absolute non-sense.’’59 Next, Ramsey explains that his semantics makes ‘‘There is at least 1 individual,’’ ‘‘There are at least 2 individuals,’’ 58 59

Ramsey, ‘‘The Foundations of Mathematics,’’ p. 210. Ramsey, ‘‘The Foundations of Mathematics,’’ p. 60.

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‘‘There are at least n individuals,’’ ‘‘There are at least @0 individuals,’’ ‘‘There are at least @1 individuals,’’

and so on either tautologies or contradictions.60 The series starts out tautologous and somewhere it becomes contradictory. Consider ‘‘There are at least 2 individuals.’’ We have: (9x)(9y)(x 6¼ y), which is (9x)(9y)((8’(o))(’(o)(xo)  ’(o)(yo)). This holds in Ramsey’s semantics if and only if there are entities x and y and some function fe and some formulas A and B of L1 such that fe(x) ¼ A and fe(y) ¼ B and dA  Be is false. Now if there are two entities there is a function fe that is such that fe(x) ¼ A and fe(y) ¼ A, and dA  Ae is false. It is false because (A  A) is a tautology. So our formula is made true by a tautology. If, on the other hand, there is only one entity, then Ramsey’s semantics will make it false because of a contradiction. The semantics says there is a function fe which assigns a formula A to this one entity and that dA  Ae is false. Ramsey concludes that Wittgenstein was right that Principia’s axiom of infinity is a pseudo-proposition, but he hastens to point out that his semantic interpretation of the axiom of infinity makes it either a tautology or a contradiction. In this way, Ramsey hopes to rehabilitate identity, making it part of Wittgenstein’s notion of logical scaffolding. Meanwhile, Russell seems to have been moving in the opposite direction with respect to the question of infinity. At the time of Principia’s first edition, the presence of the axiom of infinity was not something that was of utmost concern to Russell. The axiom of infinity first appears at *120.03 of Principia. When it is needed, it is added without much ado as an antecedent to conditionals of Principia’s arithmetic of inductive cardinals. Nonetheless, Whitehead and Russell write that ‘‘it seems plain that there is nothing in logic to necessitate its truth or falsehood, and that it can only be legitimately believed or disbelieved on empirical grounds.’’61 This may seem surprising. If Principia relies on a contingent principle to reach arithmetic from logic, then its logicism fails. How could Whitehead and Russell have missed this? The answer is that Russell included universals among individuals of Principia’s lowest type and considered them to be purely logical entities whose existence is, as it were, logically necessary. All the same, Russell knew that, though we are contingently acquainted with many universals, nothing in logic (as

60

61

Marion erroneously characterizes Ramsey as holding that these are all tautologies. See Mathieu Marion, Wittgenstein, Finitism and the Foundations of Mathematics, (Oxford: Clarendon Press, 1998), p. 69. Principia, 2nd ed., vol. 2, p. 183.

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Principia’s ramified types construes it) can assure a logical theorem that there are infinitely many universals.62 In the logical atomism lectures (1918), Russell’s conception of universals as logical individuals was in flux. He was moving toward Wittgenstein’s position that universals are material properties and relations which have only a predicable nature. They are not logical individuals but contingent entities posited as a part of an empirical scientific theory.63 On this view, the existence of even one individual, to say nothing of infinitely many, can no longer be glossed by appeal to universals. The axiom of infinity is now an empirical principle. In Introduction to Mathematical Philosophy (1919) Russell felt it a ‘‘defect of logical purity’’ that his calculus for pure logic has theorems that assure the existence of even one individual.64 The axiom schemata of quantification theory in Principia (a version of which is now commonplace in modern classical quantification theory) permits the proof of a theorem that is satisfied only if there is at least one individual. Principia has:  10:1 ðxÞAx  Ay=x; where y is free for x in A The following theorem is immediate: ‘ ðxÞAx  ð9xÞAx: If we take Ax to be a tautologous form such as Bx  Bx or the form x ¼ x, the antecedent will be a theorem and so by modus ponens we have an existential theorem: ‘ ð9xÞðBx  BxÞ: Pure logic would be committed to the existence of at least one object. In Principia, Whitehead and Russell trace the source of the existential commitment to the presence of free variables. They write: The assumption that there is something . . . is implicit in the proposition *10.1 that what is true always is true in any instance . . . The assumption that there is something is involved in the use of the real variable, which would otherwise be meaningless.65 62

63

64 65

On the present interpretation, Principia’s eliminativistic construction of a hierarchy of types does not embrace an ontological hierarchy of attributes (universals) in intension. See Landini, Russell’s Hidden Substitutional Theory. See Nino Cocchiarella, ‘‘Russell’s Theory of Logical Types and the Atomistic Hierarchy of Sentences,’’ in C. Wade Savage and C. A. Anderson, eds., Rereading Russell (Minneapolis: University of Minnesota Press, 1989), pp. 41–62. Reprinted in Nino Cocchiarella, Logical Studies in Early Analytic Philosophy (Columbus: Ohio State University Press, 1987), pp. 193–221. Introduction to Mathematical Philosophy, p. 203. Whitehead and Russell, Principia Mathematica to *56, p. 226.

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Neither the new introduction to the second edition of Principia nor its new Appendices explicitly discussed the question of forming a quantification theory that includes the empty domain. But the sentential calculus and quantification theory of section *8 set out in Appendix A of Principia’s second edition set forth quantification theory without free variables. This indicates that Russell was working toward remedying the ‘‘defect of logical purity.’’ Russell’s system antedates Quine’s quantification theory without free variables by some fifteen years.66 The elimination of free variables, however, does itself generate a quantification theory that includes the empty domain. Nonetheless, just as Quine showed that modification of his system of quantification without free variables accommodates the empty domain, it is possible to modify *8 in a way to include the empty domain of individuals.67 Russell’s second edition and Ramsey’s reconstructions of the Tractarian ideas both fail to vindicate the oracle. Not surprisingly, Russell’s assessment of Tractarian ideas in the Principia’s second edition did not please Wittgenstein. He regarded the elimination of identity to be a great achievement in the Tractatus. It is ignored in the second edition of Principia. This explains Wittgenstein’s cold reaction to Russell’s work on the second edition. In light of their early support, how disappointing it must have been for Wittgenstein to find both Russell and Ramsey abandoning him.68 Over the years, Russell’s wit has left us with many a poignant quip. In his Introduction to Mathematical Philosophy, written while in prison for his pacifistic stance against the First World War, he offers the following. Dedekind’s theory of irrational and real numbers, Russell tells us, brought into prominence the idea of dividing all terms of a series into two classes of which the one wholly precedes the other. Dedekind divided the ratios into two classes, according as their squares are less than 2 or not. All the terms of the one class are less than all in the other. There is no maximum (i.e., the greatest term of the ordering) to the class of ratios whose square is less that 2, and there is no minimum to those whose square is greater than 2. Between these two classes is the Dedekind ‘‘gap’’ where 2 ought to be, but there is no term. Russell wrote: From the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving there was no rational limit to the ratios whose square is less than 2,

66 67 68

W. V. O. Quine, Mathematical Logic (Cambridge, Mass.: Harvard University Press, 1940). Gregory Landini, ‘‘Quantification Theory in *8 of Principia Mathematica and the Empty Domain,’’ History and Philosophy of Logic 25 (2005): 47–59. See Frank Plumpton Ramsey, Notes on Philosophy, Probability and Mathematics, ed. Maria Carla Galavotti (Naples: Bibliopolis, 1991), pp. 336–346.

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some allowed themselves to ‘‘postulate’’ an irrational limit which was to fill the Dedekind gap. Dedekind . . . set up the axiom that the gap must always be filled, i.e., that every section must have a boundary.

Russell then quips: ‘‘The method of postulating what we want has many advantages; they are the same as the advantages of theft over honest toil.’’69 Russell improved Dedekind’s idea by constructing the irrational number 2 as the lower section of a Dedekind cut. Analysis and logical reconstruction is, in Russell’s view, intellectually honest in a way that postulation to satisfy the constraints of a research program is not. It is illuminating in this respect to observe Russell’s portrait of the characters of Ramsey and Wittgenstein. Russell wrote: The way in which he [Ramsey] approaches problems is extraordinarily different. Wittgenstein announces aphorisms and leaves the reader to estimate their profundity as best he may. Some of his aphorisms, taken literally, are scarcely compatible with the existence of symbolic logic. Ramsey, on the contrary, is careful, even when he follows Wittgenstein most closely, to show how whatever doctrine is concerned can be fitted into the corpus of mathematical logic.70

Wittgenstein’s Tractatus set out a research program that is generated by his Doctrine of Showing. But the Tractatus itself offers little more than oracular postulations of the existence of constructions that achieve its ends – exclusive quantifiers, the N-operator, numbers as exponents of operators, extensionality, elimination of the soul, and the like. In the end, Wittgenstein offers very little of the constructive toil Russell so lauded. The promised advantages that the Tractatian postulations have over Principia’s constructions vanish when they are brought to the light of day. The joys the Tractatus elicits are stolen.

69

70

Russell, Introduction to Mathematical Philosophy, p. 71. To find a logical definition of irrationals, Russell says, ‘‘we must disabuse our minds of the notion that an irrational must be the limit of a set of ratios.’’ Russell, My Philosophical Development, p. 126.

7

Logic as the essence of philosophy

Structure is central to logical form and logical analysis. Wittgenstein held that ‘‘It was Russell who performed the service of showing that the apparent logical form of a proposition need not be its real one’’ (TLP 4.0031). This overreaches. The notion is also found in Frege’s analysis of cardinal number and in the work of many others. Russell’s 1905 theory of definite descriptions simply offered a new tool for research. It was certainly an important tool. Ramsey aptly described the theory of definite descriptions as ‘‘a paradigm of philosophy.’’ But what is logical form? What precisely was Russell’s paradigm? According to Russell’s conception of philosophy, metaphysical conundrums arise because ordinary (and quasi-scientific) notions such as ‘‘space,’’ ‘‘time,’’ ‘‘matter,’’ ‘‘motion,’’ ‘‘limit,’’ ‘‘continuity,’’ ‘‘change,’’ and the like are hybrid notions whose logico-semantic components have not been separated from their empirical/physical components. Analytic philosophy aims at a separation of these components, accomplished by means of a logical analysis running side by side with advancements and empirical discoveries in physical science. In the process, a new more exacting account of the world emerges. Abandoning the ontological speculations of philosophers working in darkness, the new theory offers a reconceptualization of the issues involved and a solution of philosophical problems. Consider the following statement: (P) The temperature of O is 98 degrees and rising.

We all know what this means – in some sense of ‘‘knowing meaning.’’ But we are equally unclear what ontology its truth commits us to. Shall we conclude that if (P) is true, the number 98 is rising? In the hands of a naive philosopher, the English surface grammar might inspire a metaphysics of entities that are temperatures and even numbers which rise. The philosophy of logical analysis admonishes philosophers of this sort. It enjoins philosophers to sort out the interplay of physical, logical, and arithmetic notions involved in the ordinary language statement. The analysis certainly does not neglect facts about how (P) is used in ordinary language to 227

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communicate. But it involves much more. Logical analysis must ask ‘‘What is temperature in the physical sense?’’ not simply ‘‘How is the word ‘temperature’ used by a given linguistic community?’’ Hand-waving more than a little bit, physics tells us that the temperature of an object is the kinetic energy of the molecules composing the object relative both to fixed parameters of pressure and volume and a reference frame. When confined to a tube that fixes a standard of pressure and volume, the molecules of mercury are known to be predictably sensitive to the kinetic motions of the molecules in the objects they are in contact with. Increase in the molecular motions of the mercury causes its volume to increase in the tube. By scribing marks on such a tube containing mercury and fixing a number of marks distanced from one another (as in the Celsius scale, when the tube is immersed in ice and in boiling water), we can assign a numeric scale from 0 to 100. Now if cardinal numbers are second-level quantifiers, and we let ot fx abbreviate the expression x is a mark that is reached at time t after the cessation of the expansion of mercury in a standard tube in contact with O,

then we can say (98x) ot fx. There are ninety-eight marks that are reached after the cessation at time t of the expanding of mercury in a standard tube in contact with O. That is, we say that the temperature of O at t is 98 degrees. To say the temperature of O is rising at t brings in the notion of rate of change relative to the change in time. That is, we need to bring in the notion of the second derivative of a function. Accommodating this will further complicate the technical expression of the simple ordinary language statement. But in any event, once the proper ‘‘logical form’’ is given – i.e., once we write the technically accurate sentence that makes perspicuous the interplay of physical and mathematical properties – the technical language will obviously dismantle the inclination our naive philosopher may have toward the adoption of an ontology of temperatures or rising numbers. Russell and Wittgenstein presented one and the same conception of philosophy as logical analysis. But differing conceptions of the nature of logic and how it is ‘‘known’’ generate different conceptions of what is to be analyzed and what is to be the primitive foundation from which analysis is to begin. What is analyzed? Which concepts are those whose structure is to be analyzed? Which analyses should be reductive identifications as opposed to eliminativistic reconstructions? Reductive identifications afford a preservation of truth of ordinary statements. Eliminations seem to make ordinary statements false (or meaningless). But this does not do justice to the differences. An eliminativistic analysis does not have to proclaim that ordinary language statements are false. And neither reductivists nor eliminativists need be construed as analyzing or preserving what

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ordinary people mean (or intend to mean) when they speak. In fact, both reductive and eliminativistic methodologies may embrace the thesis that ordinary language is perfectly right in itself since it is blithely free of any commitment to a philosophical ontology. How, then, does one assess the success of a given eliminativistic or reductive analysis? If analysis is not endeavoring to recover the (or any of the) connotations of the ordinary language sentence, what is it recovering? Russell and Wittgenstein shared a straightforward answer. Analysis is warranted wherever there appear to be necessities that are not logical. Misunderstanding of the hybrid nature of notions such as ‘‘exists,’’ ‘‘number,’’ ‘‘class,’’ ‘‘matter,’’ ‘‘continuity,’’ ‘‘space,’’ ‘‘time,’’ ‘‘motion,’’ and the like gives rise to the philosophical postulation of metayphysical forms of necessity and essentialism that are not of a logical nature. Analysis and reconstruction aim at separating the logico-semantic components of these notions from the physical components. Once separated, nonlogical necessities and essences vanish. The only necessity is logical necessity; and logical necessity is fundamentally a matter of form or ontological structure, not something grounded in the inner metaphysical essence of a special realm of objects. Russell and Wittgenstein share the thesis that all necessity is logical necessity. In the Tractatus we find (TLP 6.375): Just as the only necessity that exists is logical necessity, so too the only impossibility that exists is logical impossibility.

Any purportedly necessary connection that is nonlogical requires a logical analysis that separates the logico-semantic components of the concepts involved from the empirical (material) components. Wittgenstein criticized Russell for offering some constructions that fall short of the goal. Belief contexts, relations between universals, and semantic contexts generally call for much deeper reconstructions than Russell had hitherto found. Wittgenstein held that no genuinely atomic statement has a logical contrary. At the limit of analysis and reconstruction all facts are atomic (‘‘elementary’’), and all contexts are extensional. Wittgenstein suggests that even ordinary predicate expressions would disappear at the limit of philosophical reconstruction. At TLP 2.0272 we find him saying that an elementary state of affairs is a configuration of objects.1 It is unclear to what extent Wittgenstein’s passage alludes to Russell’s four-dimensional theory of time according to which matter (continuants persisting in time) are reconstructed out of configurations of events. On such a view, 1

James Griffin, Wittgenstein’s Logical Atomism (Seattle: University of Washington Press, 1964), p. 78.

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exemplification of a physical property is itself a series of events in spacetime. Whatever was Wittgenstein’s conception of universals as properties of matter, it is clear that he regards the universals (material properties and relations) that inhere in atomic facts to be logically independent of one another. If there are universal words for material properties and relations, they must occur in predicate positions only. All material contexts are extensional. The exemplification of one universal never excludes the exemplification of any other. There are no nonlogical necessary relations among material properties and relations. The appearance of such relationships raise the red flag; they beg for further analysis – the only necessity being logical necessity. Wittgenstein inherited a program of logical analysis and reconstruction from Russell and extended it broadly. Wittgenstein’s Tractatus maintained that all and only those concepts that have (in all or in part) logical or semantic content must be analyzed. All and only logical and semantic components (‘‘formal concepts’’) are to be built into structured variables. The material concepts that remain are part of empirical science. We have argued that this is the Doctrine of Showing and the Grundgedanke of the Tractatus. Understood in this way, Showing is an extreme form of Russellian eliminativism. It proclaims that all (and only) logical and semantic notions (‘‘formal notions’’) are pseudo-concepts that in an ideal language for empirical science are shown with structured variables. At the eliminativistic ideal there are no ontological predicates; ontology is scaffolding. In Wittgenstein’s view, logic (and knowledge of logic) is the shared scaffolding of the world and of thought. Russell did not accept this extreme eliminativism. Indeed, it has won little support even from strong advocates of the analytic research program in philosophy. It is illuminating to see what became of that research program in the hands of two of its most famous disciples, Carnap and Quine. The wonderful exchange between Carnap and Quine on ontology reveals how these prominent disciples of the Russell–Wittgenstein conception of philosophy as an analysis of logical form lost their way. Ontology as meaningless The Tractatus is ouroboric. It suggests that completed analysis would yield an end to philosophical ontology, an end to metaphysics, and the disappearance of all philosophical questions. But this seems circular and self-undermining. The Tractatus has a problem. Its own eliminativistic constructions threaten to destroy themselves. The Tractarian simile of a ladder to be kicked away once climbed offers little consolation. Ramsey’s quip nicely sums up the situation: ‘‘But what we can’t say we can’t say, and

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we can’t whistle it either.’’2 Wittgenstein’s extreme Doctrine of Showing is either self-refuting or reduces philosophy to oracular pronunciation. Russell had expressed a similar concern in his introduction to the Tractatus. Perhaps, Russell mused, there is some escape from Wittgenstein’s conclusion that ultimately one must be silent – that the proper method of philosophy is to adamantly refuse to use logico-semantic (metaphysical) pseudo-predicates. The escape Russell had in mind was to pursue a ‘‘loophole through a hierarchy of languages.’’ Carnap was also discontent with Wittgenstein’s conception of philosophy as oracle. He wrote: ‘‘he [Wittgenstein] seems to me to be inconsistent in what he does. He tells us that one cannot make philosophical statements, and that whereof one cannot speak, thereof one must be silent; and then instead of keeping silent, he writes a whole philosophical book.’’3 In the end, Carnap would embrace the loophole Russell suggested. The Tractarian eliminativistic program was to demonstrate that by using structured variables one can (in principle) build ontological categories (the logico-semantic content of pseudo-predicates) into the syntax of an ideal scientific language. Carnap reinvented this program as one whose goal was simply the elimination of metaphysics – the meaninglessness of ontology. Carnap’s endeavors to realize the elimination of metaphysics evolved significantly. In his early work he turned to empiricism and the verificationist foundation for physics he had sketched in his Logical Structure of the World (Der logische Aufbau der Welt) (1928). The operational definitions rendering empirical significance conditions for theoretical terms were supposed to exclude the statements of speculative metaphysics. Carnap’s Aufbau and his 1932 paper ‘‘The Elimination of Metaphysics through Logical Analysis of Language’’4 employed the newly uncovered logical tools of Principia to attempt an empiricist reduction with an exactness and logical precision of which Berkeley could only dream. Carnap attempted to demarcate metaphysics from science by employing a formal reductive analysis of the meaning of theoretical terms. Theoretical terms get their meaning in virtue of their inferential (logical) connections to terms referring only to observations (or in its extreme form, to sense experience). The ‘‘verification theory of meaning,’’ as it came to be called, held that the meaning of a theoretical assertion is given by the evidentiary conditions that would establish (in principle) its empirical confirmation or disconfirmation in 2

3 4

Frank Ramsey, ‘‘General Propositions and Causality,’’ in The Foundations of Mathematics and Other Logical Essays by Frank Plumpton Ramsey, ed. R. B. Braithwaite (London: Harcourt, Brace and Co., 1931), p. 238. Rudolf Carnap, Philosophy and Logical Syntax (London: Kegan Paul, 1935). Rudolf Carnap, ‘‘The Elimination of Metaphysics through Logical Analysis of Language,’’ in A. J. Ayer, ed., Logical Positivism (London: Macmillian, 1959), pp. 60–81.

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observation. In Carnap’s early view, speculative metaphysical theories are unsinnig in a verificationistic sense – they are meaningless because they fail to have proper evidentiary relationships to observation. Observation is the foundation for what is (empirically) meaningful. Carnap’s Aufbau revealed the formidable difficulties inherent in radical empiricism which assumes that there is a conceptual or inferential reduction of each theoretical statement to a group of observation (or sensory) statements which confirm or disconfirm it. Duhem’s work, later adopted by Quine, sealed its fate.5 Examination of the scientific practice of the verification (confirmation or disconfirmation) of a hypothesis revealed that only in the context of a large body of theory does a given hypothesis confront the tribunal of observational evidence. Hence the empirical meaning of a given theoretical term of a hypothesis cannot be adequately characterized in terms of potential observational evidence alone, nor can it be specified for the hypothesis taken in isolation. The cognitive meaning of a statement in a language is reflected in the totality of its deductive relationships to all other statements in the language. In The Logical Syntax of Language (Der logische Syntax der Sprache) (1934) Carnap employed a quite different approach to the elimination of metaphysics. He distinguishes between formal and material modes of speech. Object-language (material mode) statements about the world must be distinguished from the metalanguage (formal mode) statements about a given linguistic system – statements that are apt to be misconstrued by philosophers given to metaphysical speculation. For instance, ‘‘Red is necessarily darker than yellow’’ is not an ontological statement about the essence of the property Redness. It is a metalinguistic statement about the grammar of a linguistic system of color predicates. In his ‘‘Empiricism, Semantics and Ontology,’’ Carnap advocated the adoption of many-sorted formal languages whose different styles of variables represent the differences in ontological category. Predicates with ontological or semantic content are supplanted by variables of a given linguistic system. The choice of linguistic system, in turn, is a matter to be decided pragmatically. Carnap’s views on ontology to fall into two groups. The first form emphasizes a verificationist analysis of meaning whose foundation is positivism and empiricism. This form is nicely illustrated in Carnap’s essay ‘‘The Elimination of Metaphysics through the Logical Analysis of 5

A major difficulty, pointed out by Duhem, is that a theoretical statement can be submitted to an empirical (observational) test only when it is taken together with a host of auxiliary theoretical statements pertaining to the testing conditions. Another difficulty is that a rigid distinction between ‘‘observable’’ and ‘‘theoretical’’ is difficult to sustain – even when observations are characterized as immediate sense experiences. Even first-person sensory reports can be corrupted by theoretical assumptions.

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Language,’’ and in ‘‘Pseudo-Problems of Philosophy.’’ The second form is found in The Logical Syntax of Language and its distinction between the ‘‘material’’ and the ‘‘formal’’ modes of speech. It is found in ‘‘On the Character of Philosophical Problems’’ and in ‘‘Empiricism, Semantics and Ontology.’’ In an important discussion of Carnap’s view on ontology, Norton calls the second form a ‘‘formalizability approach’’ to metaphysics and offers compelling arguments that it does not rest upon any form of verificationism. In Norton’s view, the formalizability approach is founded upon Carnap’s ‘‘hierarchical conception of philosophy’’ and a principle of tolerance.6 Norton observes that what he calls ‘‘the hierarchical conception of philosophy’’ is nicely illustrated by John Wisdom’s dictum that ‘‘Philosophy is concerned with acquiring new knowledge of facts, not knowledge of new facts.’’ It is the view that philosophy, in some sense, concerns analysis. Norton goes on to say that this does not imply that as an investigation of structure (logical form) philosophy concerns only language or concepts. He includes the logical atomists among the philosophers who held the hierarchical view. Most importantly, he correctly explains that the hierarchical view is not committed to (or motivated by) empiricism.7 Carnap’s verificationist attempts to realize the meaninglessness of ontology show little in common with Russell and Wittgenstein. It is eliminativism that is driving logical atomism, not empiricism. On the other hand, Carnap’s ‘‘hierarchical conception of philosophy’’ was surely influenced by the eliminativistic current in the program of logical atomism of Russell and Wittgenstein. It is especially important to keep this in mind in investigating Carnap’s debate with Quine on ontology. Russell’s eliminativism did not have as its goal the elimination of ontology (metaphysics) itself. Russell’s logical atomism proclaimed that philosophy provides a conceptual clarification of the concepts of the mathematical and empirical sciences. It provides an account of the inferential relationships between a theory and its evidentiary basis. In Russell’s view, a logically proper scientific language (and theory) will eventually supplant the old ontological theories that improperly infuse science with doctrines of speculative metaphysics. Russell’s plan was to solve problems in a given theory by abandoning the ontology of that theory and offering a logical reconstruction of its laws within a new ontological framework. But the Tractatus dares to dream of a final eliminativistic framework in which all ontological categories are built into structured variables. This dream captivated Carnap. What Carnap saw 6 7

Bryan Norton, Linguistic Frameworks and Ontology: A Reexamination of Carnap’s Metaphilosophy (The Hague: Mouton, 1977), p. 13. Ibid., p. 32.

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in the Tractatus was a program for eliminating metaphysics, and all other features of logical atomism were made subservient to this goal. It is from this perspective that we must view Carnap’s revamping of the Tractatarian ‘‘fundamental idea’’ in terms of his distinction between the object-language (‘‘material mode’’ of speech) for a given science and the metalanguage (‘‘formal mode of speech’’) in which the structure of the object-language is articulated. In Carnap’s book The Logical Syntax of Language, wide latitude is allowed in the building of an object-language for a given scientific investigation, provided it meets certain formal constraints concerning its terms and formulas, proper axioms, and inference rules. This is Carnap’s ‘‘Principle of Tolerance.’’ In ‘‘Empiricism, Semantics and Ontology’’ the principle is introduced as follows: The acceptance or rejection of abstract linguistic forms, just as the acceptance or rejection of any other linguistic forms in any branch of science, will finally be decided by their efficiency as instruments, the ratio of the results achieved to the amount and complexity of the efforts required. To decree dogmatic prohibitions of certain linguistic forms instead of testing them by their success or failure in practical use, is worse than futile; it is positively harmful because it may obstruct scientific progress. The history of science shows examples of such prohibitions based on prejudices deriving from religious, mythological, metaphysical, or other irrational sources, which slowed up the developments for shorter or longer periods of time. Let us learn the lessons of history. Let us grant to those who work in any special field of investigation the freedom to use any form of expression which seems useful to them; the work in the field will sooner or later lead to the elimination of those forms which have no useful function. Let us be cautious in making assertions and critical in examining them, but tolerant in permitting linguistic forms.8

Ontological predicates or universal words (Allwo¨rter) which exhaust an ontological kind are either rejected by Carnap as meaningless pseudopredicates, or reformulated as metalanguage predicates about the objectlanguage linguistic forms. Carnap couldn’t accept the Tractarian ladder – the notion that logic is scaffolding. Instead he offered a conventionalism about logic and mathematics and a pragmatism that manifests itself as a principle of tolerance for the development of different formal languages for science. In this way, Carnap hoped to rehabilitate what he takes to be the Tractarian distinction between statements that are sinnlos (admirably meaningless because they have logical or mathematical content that is to be shown in the syntax of an ideal language) and those that are unsinnig (despicably meaningless because they irredeemably construe logical or semantic predicates as designating genuine properties or relations with ‘‘material content’’).

8

Carnap, ‘‘Empiricism, Semantics and Ontology,’’ in Meaning and Necessity: A Study in Semantics and Modal Logic, enlarged ed. (Chicago: University of Chicago press, 1956), p. 221.

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In fact, many of the ‘‘universal words’’ (words that exhaust an ontological category) of Carnap’s Logical Syntax of Language were words that the Russell–Wittgenstein research program had attempted to built into logical grammar. The idea of building them into structured variables originated with Russell. ‘‘Existence’’ is a pseudo-predicate. It is embedded in the individual variables of quantification, not expressed by a predicate. It follows that ‘‘object(x),’’ ‘‘thing(x),’’ and ‘‘individual(x)’’ are pseudo-predicates, for they each would be tantamount to a predicate ‘‘exists(x).’’ Similarly, Principia had offered eliminative analyses of ‘‘number’’ and ‘‘class’’ by building structure into special variables that reflect the ontological elimination of classes and numbers. Wittgenstein’s radical extension of Russell’s eliminativistic program hoped to find eliminativistic reconstructions for pseudo-predicates like ‘‘universal,’’ ‘‘truth,’’ ‘‘fact,’’ ‘‘identity,’’ and all words introducing nonextensional contexts. The Tractatus follows Russell’s eliminativistic program of employing ‘‘variables with structure.’’ Carnap’s doctrine of universal words as ‘‘quasi-syntactical’’ predications without empirical content is of a piece with this research program. In ‘‘Empiricism, Semantics and Ontology,’’ Carnap maintains that predicates such as ‘‘number,’’ ‘‘class,’’ and ‘‘attribute’’ are pseudo-predicates that are properly given by special variables of quantification of a formal system (linguistic framework). Ontological categories are to be eliminated by adopting formal languages with different styles of variables for the would-be ontological predicates. This is a striking testament to Carnap’s indebtedness to Russell’s research program. Unfortunately, Carnap saw Russell’s eliminativism through a glass darkly. The paradigm Carnap found was Principia’s regimented language of many-sorted variables. Instead of following Russell and Wittgenstein in adopting an eliminativistic reconstruction from within one canonical formal language that adopts individual variables as its only genuine variables, Carnap hoped to realize the meaningless of ontology by adopting conventionalism and setting out different formal systems or ‘‘linguistic frameworks’’ with many sorts of variables – distinct styles of variables for the ontological categories. Carnap’s theory of logical necessity (analyticity or ‘‘truth in virtue of logical form’’) makes analyticity relative to a system of linguistic conventions and meaning postulates which govern a specified set of primitive signs adopted in a linguistic framework. Carnap then considers the choice of a framework to be a pragmatic matter concerning the fruitfulness of a given system for scientific inquiry. Carnap distinguished between ‘‘internal questions’’ framed within a given linguistic framework and ‘‘external questions’’ about the linguistic framework itself. Internal questions of existence are not ontological questions because their formulation employs predicates which, by the construction

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of the given linguistic framework, are genuine. For example, while ‘‘cat(x)’’ is a genuine predicate of a formal framework for an applied physics, the expression ‘‘physical object(x)’’ is a pseudo-predicate not expressible in the system because the would-be ontological category of physical objects is given with the use of the variables, x and y, etc. It is only after the formal language and axioms of a given (applied) theory have been established that one can ask and answer existence questions. One can ask ‘‘Are there cats?’’ in the physical object framework, and the statement ‘‘There are cats’’ is contingent. But ‘‘Are there physical objects?’’ is a pseudo-question relative to that framework. Carnap treats this pseudo-question just as Russell treats ‘‘Does something exist?’’ That is, the notion ‘‘physical object(x)’’ is treated as a pseudo-predicate of the physical object-language in just the way ‘‘exists(x)’’ is a pseudo-predicate by the lights of Principia. Adopting a linguistic framework which has a special style of numeric variables n, m, etc., the statement ‘‘Are there prime numbers over 100?’’ is a genuine existence question for Carnap, and its answer is analytic relative to the given linguistic framework. It is a genuine existential statement by the lights of the framework because ‘‘Prime(m)’’ is a genuine predicate of the linguistic system. On the other hand, ‘‘Are there numbers?’’ is a pseudo-statement because ‘‘Number(m)’’ is not a genuine predicate of the framework. Number is built into the special style of variable m. To take yet another example, consider a formal system with special styles of bindable variables ’, y, y which can occur in predicate position only. The quantification theory of such a formal system will have the axiom schema ð8’ÞA’  Aðy=’Þ;

where y is free for ’ in the formula A. From this and classical derivations we get the theorem ð9’Þð’x x ’xÞ:

Countenancing attribute quantifiers and a semantics for them is now an analytic consequence of adopting the formal system itself. But in such a framework ‘‘attribute(x)’’ is a pseudo-predicate. One can ask ‘‘Are there ’’s?’’ because ‘‘(9x) ’x’’ is formulable in the language. But the question ‘‘Are there attributes?’’ is not formulable at all. Carnap does allow ‘‘external questions’’ about the viability and fruitfulness of a given linguistic framework for inquiry. External questions concern the success of a given theory (or research program based on the theory) that is couched within the linguistic framework. In Carnap’s view, these questions must be answered pragmatically. The question of which is ‘‘correct’’ or ‘‘true of the world’’ is a pseudo-question. The

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question of which works best to achieve a set of practical ends is a legitimate and substantive question. For example, ‘‘Are there numbers?’’ is meaningful if it is understood to ask an external question about the scientific usefulness of a linguistic framework with a special style of numeric variables. The framework has been very useful in engineering, for example. But this external question is not an ontological question. Observe that a linguistic framework with general variables x and y instead of special variables m and n may well count ‘‘number(x)’’ as a genuine predicate. If so, the existence question ‘‘Are there numbers?’’ is legitimate. But when couched in such a linguistic framework it is regarded by Carnap as an existential question, not an ontological question. Recall Quine’s famous dictum: To be is to be the value of the variables of quantification.

In considering the question as to what are to count as existential commitments made in a language (or theory expressed within it), one must discover what particles in the language do the work of the apparatus of the bound variables of quantification. The very notions of ‘‘name,’’ ‘‘singular term,’’ ‘‘predication,’’ and the like are inseparably tied to the apparatus of quantification. Thus, without appeal to a quantificational apparatus, the notion of a statement making an existential commitment is empty. Carnap wholly agrees. Carnap does not agree, however, that the quantificational apparatus of the language sets the parameters of the ontological commitments of the language (or the theory couched within it). He takes it to set the parameters of what counts as an existence statement in the language. For Carnap, ‘‘ontological’’ commitments are not existential commitments. Ontology traditionally concerns the postulation of metaphysical entities with essential natures – entities that give rise to metaphysical necessities – not the mundane existential statements of what there is. For Carnap, it is the linguistic framework (the special styles of variables of the quantification theory couched in it) that determines what is to be understood as a mundane existential statement. Ontology is to be meaningless. Ontology as structured variables The differences between Carnap’s views on ontology and Russell’s views on logical atomism are illuminating. For each would-be ontological category Carnap employs a different style variable whose range is sealed off from every other. Carnap has something like the type-theoretical language of Principia in mind. But Carnap was unaware of the eliminativism working in Russell’s philosophy of logical atomism. It was Russell’s reconstructions, framed from within a formal system with only one style of genuine

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individual variables, that gave rise to the introduction of differently structured variables. The unfortunate result of Carnap’s conventionalism is that the constructive component of logical atomism is lost in Carnap’s work. Once conventionalism is adopted, the considerable toil of providing the ‘‘logical fictions’’ which enable one to abandon the old theory’s ontology and recover its structure is love’s labor lost. Consider the numeric predicate ‘‘even.’’ If our logical forms are constrained by the language of first-order predicate logic (a calculus in which only individual variables are bindable), this notion is expressed as follows: EvenðxÞ ¼df ð9yÞðnumberðyÞ :: 2  y ¼ xÞ:

Carnap sets out a language that adopts a special style of variables m, n, etc. for the ontological category of numbers and linguistic rules for the formation of singular numeric terms. Thus, ‘‘number(y)’’ is a pseudo-predicate of this language. We have: EvenðmÞ ¼df ð9nÞð2  n ¼ mÞ;

In the hands of Carnap’s conventionalism, this is all that is left of the Russell/ Wittgenstein program of building ontological structure into variables. To illustrate the point, let us use structured variables to represent a Fregean hierarchy of levels of concepts. On this view, the ontological structure of cardinal numbers cannot be properly represented within the confines of the language of first-order predicate logic. Higher levels of concepts may ‘‘mutually saturate’’ one another, and this is properly represented by means of special variables. Analyzing cardinal numbers as second-level functions, the mutual saturation involved in a cardinal number’s being arithmetically even is represented by the following: E’ ½m ’ :

This is defined to mean: ð9nÞðN’ ½nx’x :: ½nx y x Xy 2x x  ’ ’ m ’ Þ:9

Thus the grammar of ‘‘Even(x)’’ does not properly reflect the ontological structures involved. The expression ‘‘4 is even’’ is perfectly correct ordinary English, but the philosophers who conclude that ‘‘Even(4)’’ is perspicuous of the ontology have been misled by the surface grammar. On the Fregean view, the ontology is perspicuously expressed by E’[4’]. This represents a mutual saturation of the second-level quantifier concept 4’ and the third-level quantifier concept E’[O’]. Moreover, in Frege’s view this logical form is 9

Recall the definition of multiplication from chapter 5.

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not properly represented by ‘‘G(F)’’, with ‘‘F’’ and ‘‘G’’ property expressions. The number 4 is not a special sort of metaphysical abstract object, with the property evenness essential to it. Nor is it a property with a special essence. Cardinal numbers are quantificational structures. Ordinary language (and in particular a first-order canonical language) simply fails to make the ontology of quantificational structures perspicuous. Frege’s conception of levels of functions paved the way for an entirely new attitude toward ontological speculations based on the surface grammar of natural language. Natural language expressions such as ‘‘2 is solid’’ and ‘‘One times the moon is two’’ are not formulable in a language whose syntactic forms are sensitive to the ontological nature of cardinal numbers as quantifier functions. A logical analysis of cardinal numbers as second-level quantifier functions endeavors to show that the logical form (quantificational structure) of arithmetic statements is hidden in ordinary language. Frege introduces a new notation to make perspicuous his ontology of secondlevel quantifier functions which ‘‘mutually saturate’’ with first-level functions. The ontological structure of ‘‘falling within’’ or ‘‘mutual saturation’’ must be respected. First-level functions fall within second-level functions. Once the quantificational structures of the cardinal numbers (as second-level functions) is revealed, the necessity of arithmetic is revealed to be logical necessity. To be sure, Frege eschewed quantification over higher-level functions in his Grundgesetze (1893). Indeed, he may well have come to believe that such quantification is illicit. He proposed a theory that each function, no matter its level, is correlated one-to-one with a certain object called an ‘‘extension,’’ or ‘‘graph,’’ or ‘‘course-of-values’’ (Wertverlauf). In the Grundgesetze, a cardinal number can be correlated with a unique extension (object). But this by no means undermines the centrality of the notion of mutual saturation in Frege’s philosophy. The language of levels of functions properly represents the ontological structures of cardinal numbers as quantifier concepts. Firstlevel functions fall within second-level functions and this ‘‘mutual saturation’’ is essential to the nature of a concept having a given cardinal number. A firstorder language cannot represent the structures involved.10 Second-level constructions of cardinals were not fully appreciated by Carnap, though he attended lectures by Frege between 1910 and 10

Unfortunately, some interpreters of Frege take mutual saturation to drop out of his philosophical analysis of cardinals as objects. Frege’s letters suggest otherwise. Frege remarked that Russell’s definition, according to which the number of a class u is the class of all classes similar to u, ‘‘agrees completely with my definition.’’ But he went on to make it clear that ‘‘we must not then regard classes as systems; for the bearer of a number, as I have shown in my Grundlagen der Arithmetic, is not a system, an aggregate, a whole consisting of parts, but a concept, for which we can substitute the extension of a concept.’’ See Gottlob Frege, Philosophical and Mathematical Correspondence, ed. Gottfried Gabriel et al. (Chicago: University of Chicago Press, 1980), p. 139.

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1914.11 The constructions of Frege’s Grundgesetze subsumed them, and these were in terms of objects, not quantifier concepts. Russell’s paradox of classes had undermined the system of the Grundgesetze. Thus it was the recovery of Fregean constructions in the type-theory of Whitehead and Russell’s Principia Mathematica that influenced Carnap most. But the eliminativism of Principia is largely hidden by a tradition of interpretation that construes it as adopting an ontology of types and orders of entities (propositional functions). The ontological elimination of classes and propositional functions in Russell’s substitutional theory of 1906 was unknown to Carnap. It goes much further than Principia in illustrating Russell’s plan for building would-be ontological assumptions into structured variables of a linguistic framework. Consider again the notion of a natural number being even. Russell’s substitutional construction offers the following: Even½x; y; z ¼df ð9h; d; eÞðNðh; d; eÞ:&: ðs=twÞ½s=tw; rc :rc : ðh=de; xy Xxy 2xyÞðrcÞ ¼ ðs=twÞ½s=tw  x=yzÞ

Technical details of the substitutional theory aside, the numerical property of evenness is now construed as a three-placed relation of entities x, y, and z.12 According to the substitutional constructions, arithmetic evenness is

11 12

See Rudolf Carnap, Lecture Notes: 1910–1914, ed. Erich H. Reck and Steve Awodey (Chicago: Open Court, 2004). It should be noted that ‘‘&’’ is not sentential conjunction here. Some of the supporting definitions are: p=a  r=c ¼df p=a; x x r=c; x s=tw; rc rc Arc ¼df ð9p; aÞðp=a  r=c:&:s=tw; paÞ rc ð9p; aÞðp=a  r=c:&:ApaÞ 2rc ¼df ð9x; yÞðx 6¼ y:&: ðp=aÞ½p=a  r=c Simðo;oÞ ðp=aÞ½p=a; z :z : z ¼ x :v: x ¼ yÞ Nðq; p; aÞ ¼df ðl; m; n; oÞððs=twÞ½s=tw; rc rc 0rc 2 ðj=kuvÞ½j=kuv  l=mno & ðh; d; eÞððs=twÞ½s=tw  h=de 2 ðj=kuvÞ½j=kuv  l=mno  ðs=twÞ½s=tw; xy xy Sh=de ðx; yÞ 2 ðj=kuvÞ½j=kuv  l=mnoÞ:  :ðs=twÞ½s=tw  q=pa 2 ðj=kuvÞ½j=kuv  l=mnoÞ ðh=de; xy Xxy 2xyÞðrcÞ ¼ df ð9l; mÞð9j; bÞððs=twÞ½s=tw  h=de ¼ ðs=twÞ½s=tw; xy :xy : ðp=aÞ½p=a  x=y Simðo;oÞ ðp=aÞ½p=a  l=m & ðs=twÞ½s=tw xy 2xy ¼ ðs=twÞ½s=tw; xy :xy : ðp=aÞ½p=a  x=y Simðo;oÞ ðp=aÞ½p=a  j=b & ð9f; u; vÞðf=uv; xy :xy : x 2 ðp=aÞ½p=a  l=m:&:y 2 ðp=aÞ½p=a  j=b : : & : : ðp=aÞ½p=a  r=c Simðo;ðoÞÞ ðs=twÞ½s=tw  f=uvÞÞ:

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not a one-placed property! In the substitutional theory, there are no type restrictions on what kinds of entities can stand in the three-placed relation of evenness, though only some triads of entities will exemplify the relation. Russell’s eliminativism avoids a type hierarchy of entities, and it avoids a metaphysical ontological essentialism of numbers. There are no numbers. Sentences such as ‘‘Austin is even’’ are not formulable in the language of the reconstruction. Carnap is to be forgiven for not being aware of the constructions of Russell’s little-known substitutional theory. The paradigm he had of Russell’s technique of construction was Principia and Russell’s thesis that the surface grammatical forms of natural language may not be indicative of logical form.13 Carnap was under the spell of Wittgenstein’s oracular pronunciations that at the limit of analysis and reconstruction philosophical ontology vanishes. Russell’s more moderate views rely on certain ontological commitments as the basis for the constructions that eliminate others. Thus Carnap could not embrace Russell’s substitutional theory had he known of it. The theory depends on the ontological assumption of propositions as mind- and language-independent states of affairs. What Carnap distilled from Russell’s program of analysis was the distinctive styles of variables in Principia’s type-theory. Obviously, he cannot fully endorse Russell’s eliminativistic program of solving problems by supplanting the old ontology by a new ontology whose laws recover (where possible) the old laws by reconstruction. The elimination of all ontology was Carnap’s raison d’eˆtre. Carnap versus Quine on ontology Wang has shown that many-sorted languages are translatable into onesorted languages without loss of expressive power or provability (though the reverse does not hold).14 We can always express statements formed with multiple sorts of variables in a language with only one sort of variables (individual variables) by adopting appropriate predicates. For instance, by adopting ‘‘T(x)’’ for ‘‘x is a thing,’’ and ‘‘Cls(x)’’ for ‘‘x is a class,’’ and ‘‘N(x)’’ for ‘‘x is a number,’’ we can get along with one sort of variable. Indeed, Quine observes that even the language of type-theory succumbs. Quine writes: we can even abandon Russell’s notion of an hierarchical universe of entities disposed into logical types; nothing remains of type theory except an ultimate

13 14

See TLP 4.0031. Hao Wang, A Survey of Mathematical Logic (Peking: Science Press, 1962), pp. 323, 326.

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grammatical restriction on the sorts of repetition patterns which variables are allowed to exhibit in formulas. Yet formally our logic, refurbished as described, is indistinguishable from Russell’s theory of types plus Russell’s convention of typical ambiguity. Now the point of this logical digression is that even under the theory of types the use of distinctive styles of variables, explicitly or even implicitly, is the most casual editorial detail . . . It is a distinction which is not invariant under logically irrelevant changes of typography.15

Quine proposes that instead of using indexed letters as variables specifically for entities of type t one could adopt a predicate ‘‘T t(x)’’ to mean that ‘‘x is an entity of type t.’’ Thus instead of (8’t) A’t one has (8x)(T t(x)  Ax). Applying similar techniques to the type-theory of classes, Quine writes: This does away with Russell’s grammatical restriction which declared ‘‘xm 2 yn’’ meaningless where m þ 1 6¼ n. Sense is now made of ‘‘xm 2 yn’’ for all m and n. If m þ 1 6¼ n, then ‘‘xm 2 yn’’ merely becomes false. That we can suddenly be so cavalier with Russell’s grammatical restriction makes one wonder whether he needed to make it. He did not.16

Quine regards the use of many-sorted languages as a ‘‘trivial consideration’’ and a ‘‘casual and eliminable shorthand,’’ and ‘‘not invariant under logically irrelevant changes in typography.’’17 Quine dismisses many-sorted languages because he thinks one can always adopt a single style of variable together with predicates special to the ‘‘theory’’ in question. This begs the question against the positions of Frege, Russell, and Wittgenstein. On their view, Quine’s introduction of such predicates in a first-order language introduces pseudo-predicates. Wang’s translation result concerning many-sorted languages by no means justifies Quine’s conclusions. Wang’s result does not undermine Frege’s account of cardinals as second-level functions, and neither does it challenge Russell’s eliminativism. The difference between a one-sorted language with special predicate expressions (and proper axioms governing them) and a language which has structured variables is quite significant ontologically. Quine’s point does have purchase against Carnap, however. Frege and Russell made ontological assumptions. Frege was committed to an ontology of functions. Russell’s early substitutional reconstructions were founded upon a theory of propositions (as states of affairs) as logical entities, and this ontological commitment manifests itself both in the thesis that formulas of the language can be nominalized to form terms and in the thesis that the 15 16 17

W. V. O. Quine, ‘‘Carnap’s Views on Ontology,’’ in The Ways of Paradox and Other Essays (Cambridge, Mass.: Harvard University Press, 1976), p. 210. W. V. O. Quine, Set Theory and Its Logic (Cambridge, Mass.: Harvard University Press, 1980), p. 268. ‘‘Carnap’s Views on Ontology,’’ p. 208.

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logical particles express relations. Principia is committed ontologically as well – if only to logical forms. Wittgenstein’s Tractatus called for a radical eliminativism extending Russell’s program. Reconstructions were to be found for all ontological pseudo-predicates – all notions with logical and semantic content. Carnap heard the call, but his conventionalism obviates the extensive constructions Russell and Wittgenstein (at the time of the Tractatus) thought essential. Obviously, Carnap cannot resort to differences in ontology to justify the adoption of many-sorted languages. His program endeavors to realize the thesis that ontological questions are meaningless. Carnap’s approach attempts to realize the end of ontology (the end of metaphysics) by adopting conventionalism and a principle of tolerance in forming many-sorted (type-theoretical) languages. In the hands of Carnap, Russell’s method of building structure into the variables becomes little more than an adoption of a many-sorted language with different styles of variable whose ranges are sealed off from one another. Carnap’s conventionalism leaves him without an adequate reply to Quine’s contention that the use of different styles of variables is little more than a casual editorial detail. To reply to Quine, Carnap must explain why he singles out certain predicates to be pseudo-predicates and he must do this without conceding, after all, that some ontological concerns are not meaningless. The question Carnap faces is whether there is a principled distinction between genuine predicates and pseudo-predicates. Which predicates are in need of philosophical analysis? Quine has no patience for the view that universality is the sign of a pseudo-predicate – the view that a predicate is meaningful only by contrast to what it excludes, so that being true of everything would make a predicate meaningless. He writes: Surely self-identity, for instance, is not to be rejected as meaningless. For that matter, any statement of fact at all, however brutally meaningful, can be put artificially into a form in which it pronounces on everything. To say merely of Jones that he sings, for instance, is to say of everything that it is other than Jones or sings. We had better beware of repudiating universal predication, lest we be tricked into repudiating everything there is to say.18

The conceptual distance between Quine and the Russell/Wittgenstein program is striking in this passage. Predicates of identity and self-identity, on which Quine’s point relies, are paradigmatic pseudo-concepts, according to the Tractatus. Wittgenstein thought he had a clear principle for determining what concepts are pseudo-concepts. According to the Doctrine of Showing, any

18

W. V. O. Quine, ‘‘Ontological Relativity,’’ in Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), p. 52.

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concept that contains logical or semantic content is a pseudo-concept. But Wittgenstein’s criterion of a pseudo-concept presupposes that the notion of logical or semantic content is unproblematic. Wittgenstein’s doctrine that logic is tautologous does little to clarify what logic is and how we have epistemic access to it. In Russell’s moderate eliminativistic program, logic and knowledge of logic is exempted. Russell relies on a special faculty of logical acquaintance. Logic is given to us by acquaintance. For Carnap, fundamental questions emerge: What are the logical particles of a given language? Which notions are semantic for a given language? Carnap has no recourse but to appeal to the appearance of certain words in the statements that are analytic for the language. The pseudo-concepts are not simply those that are universally true of their objects, but rather those that are analytically true of them. If a predicate is analytically true of every entity of a category, then it is a pseudo-predicate which, if a many-sorted language were adopted, is given by a distinctive style of variable for entities of the category. If a predicate is not analytically true of its objects, then it is a genuine predicate. Many-sorted languages now play no essential role in Carnap’s thesis about ontology. Quine justly summarizes Carnap’s situation as follows: No more than the distinction between analytic and synthetic is needed in support of Carnap’s doctrine that the statements commonly thought of as ontological, namely, statements such as ‘‘There are physical objects,’’ ‘‘There are classes,’’ ‘‘There are numbers,’’ are analytic or contradictory given the language . . . The contrast which he wants between those ontological statements and empirical existence statements such as ‘‘There are black swans’’ is clinched by the distinction of analytic and synthetic. True, there is in these terms no contrast between analytic statements of an ontological kind and other analytic statements of existence such as ‘‘There are prime numbers above one hundred’’; but I don’t see why he should care about this.19

Quine rightly concludes that Carnap’s views on ontology – his distinction between external questions and internal questions – turns ultimately on his distinction between analytic and synthetic truth. Carnap relies on the analytic-synthetic distinction to single out the pseudo-predicates that he subsumes into the variables of a linguistic framework. Carnap’s thesis of the meaninglessness of ontology has thus boiled down to his conventionalist views on the nature of analyticity. Carnap’s notion of an analytic truth makes it relative to a linguistic framework – or better put, his notion defines analyticity by the verbal and physical practices of a given linguistic community at a time. This, however, is the undoing of the distinction. A notion of analytic truth that is relative to the practices of a

19

Quine, ‘‘Carnap’s Views on Ontology,’’ p. 210.

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linguistic community cannot bear the weight Carnap assigns to it. On such an account, analytic-in-L is a property of the oral and physical dispositions to behavior; and these are not different in kind from behavioral dispositions concerning ordinary synthetic a posteriori questions of existence – including ‘‘Are there black swans?’’ Quine correctly observes that if meaning is a property of behavior, the analytic-synthetic distinction collapses. But without the analytic-synthetic distinction, logical and semantic content is not different in kind from material content. The collapse destroys Wittgenstein’s thesis that only concepts with logical or semantic content are pseudoconcepts. The doctrine that ordinary language contains concepts in need of clarification – concepts that are a hybrid of logico-semantic content and material content – becomes empty. Philosophical analysis cannot, on Quine’s view, separate out these contents in its efforts to build the logical and semantic components into the syntax. There is no difference to separate. Conventionalism relativizes the distinction between logico-semantic content and material (empirical) content to the linguistic practices of a given community of language users. Quine rightly demonstrates that it undermines Carnap’s argument for the meaninglessness of ontology. Laudably, Quine rejected conventionalism. His philosophy, like Carnap’s, takes Russell’s analytic philosophy as its point of departure. Indeed, his naturalized epistemology was likely to have been inspired by Russell’s strident eliminativism in the 1920s and 1940s. But Quine’s arguments against the analytic-synthetic distinction hold logic hostage to an empirical theory which makes meaning a property of behavior – a theory according to which ‘‘meaning’’ is intelligible only relative to an empirically well-tested manual delineating a holistic translation of the observed verbal and physical speech dispositions of a given linguistic community into another. In Quine’s naturalization of logic and our knowledge of logic, there can be no difference in kind between logical and empirical truth. Return to Pythagoras In interesting paper, Alberto Coffa emphasizes the role of conventionalism in Wittgenstein’s middle philosophy. Coffa maintains that Wittgenstein and Carnap in the late 1920s and 1930s should be viewed as offering a thesis about ‘‘Strange Sentences.’’ He writes: in that period, and as far as I can tell, for the first time in the history of positivism, two of the leaders of positivism, Carnap and Wittgenstein, came to acknowledge the existence and significance of Strange Sentences, and then proceeded to develop a theory of them, i.e., an account of what they are and how they function. Instead of limiting themselves to nonsense, or to viewing them as truths or falsehoods of logic or of some generalized physics, they proceeded to recognize the idiosyncratic

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character of these claims and the inability of a philosophy to get off the ground unless it begins by recognizing the peculiar and essential role played by Strange Sentences in all forms of knowledge. Both Carnap and Wittgenstein use the name ‘‘syntax’’ to refer to their theories of Strange Sentences.20

The following are included in Coffa’s long list of Strange Sentences: Time is one-dimensional. Time is continuous. Space is three-dimensional. The metric structure of space is Euclidean. The mathematical continuum is composed of atomic elements. The only primitive data are relations between experiences. Sense qualities belong to the primitive data. A thing is a complex of sense-data. A thing is a complex of atoms. The three-dimensionality of the system of colors is an internal property. Every color is at a place. Every sound has a pitch. Every process is univocally determined by its causes (determinism). Coffa sees Wittgenstein’s theories of logical syntax and philosophical grammar as appropriating some of the conventionalist’s ideas concerning the axioms of geometry. ‘‘When one looks at Wittgenstein’s writings in the late 1920s and early 1930s from this prespective,’’ Coffa writes, ‘‘it is hard not to detect in his remark ‘The axioms of, say, Euclidean geometry are ´ ‘The axioms of geometry rules of syntax in disguise’ an echo of Poincare’s are definitions in disguise.’’’21 Coffa is quite right to emphasize the popularity of conventionalism in the late 1920s and 1930s. But one must take care not to read the Tractatus itself as if it embraced conventionalistic doctrines. Russell’s moderate eliminativism, not conventionalist approaches to non-Euclidean geometry, gave rise to Wittgenstein’s idea that the philosophical conundrums embodied in Strange Sentences are problems of ‘‘syntax.’’ The Tractarian thesis that logic (and knowledge of logic) is ‘‘scaffolding’’ is certainly not a form of conventionalism or pragmatism. Conventionalism enters when one has to confront the self-undermining nature of an extreme eliminativism such as is advocated in the Tractatus. The Tractarian thesis that logic and knowledge of logic is scaffolding won no followers. Both conventionalism and pragmatism were enlisted in an effort to fill the void. They are called upon to patch the wounds of an unbridled eliminativism which 20 21

Alberto Coffa, ‘‘Carnap’s Sprachanschauung Circa 1932,’’ Philosophy of Science Association 2 (1976): 210. Ibid., p. 211.

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deconstructs logic itself. In the 1930s, with the logicism of Frege’s Grundgesetze dead, logic came to be viewed as a branch of the mathematical study of formal systems. Since there are differing deductive systems formulable within mathematics, there are differing ‘‘logics.’’ Indeed, the ‘‘classical’’ system for logic set out in Principia now had rivals – for instance, there was Brouwer’s ‘‘intuitionistic logic’’ and the modal systems of logic advanced by C. I. Lewis. By 1936 Church demonstrated that there is no decision procedure for predicate logic. The Doctrine of Showing could no longer assume that logical ‘‘truths’’ are shown in the syntax of an ideal theory. The assumption entails decidability. In the context of these developments, the Tractarian thesis of a monolithic truth-functional logic as the ineffable scaffolding of thought and the world strained credulity. Carnap’s conventionalist revamping of the Tractatus came to the rescue. As Coffa puts it, the situation in logic in the 1930s was held to be no different from that of nonEuclidean geometry in the nineteenth century. Carnap’s plan was to embrace conventionalism and pragmatism just as had Poincare´ concerning non-Euclidean geometry. The extent to which Wittgenstein came to fully embrace the conventionalist ideas concerning logic and mathematics that were current in the 1930s is far from certain. Conventionalism is certainly out of sorts with the Doctrine of Showing of the Tractatus. Nonetheless, it seems quite clear that Brouwer’s intuitionism and the rise of alternative systems of logic and mathematics influenced Wittgenstein’s thinking in this period. Upon his return to Cambridge in 1929, he began to entertain modifications of the Tractarian account. His paper ‘‘Some Remarks on Logical Form’’ questions whether the Tractarian analyses are adequate to concepts such as length, color, brightness, and pitch – concepts that admit of gradations, continuous transitions, and combinations in various proportions. Unfortunately, Wittgenstein’s concerns are not easy to discern from his paper. The issue seems to be that difficulties arise with the application of logical and arithmetic concepts to the physical world in the practice of measurement. If physical processes are genuinely continuous, then how can they be measured? How can the apparatus of measurement of continuous physical processes (in short, the applied calculus) be built up from the meager tools of the Tractarian account of mathematics – tools which demand that complex concepts be built up by means of truth-functional compositions of atomic statements (each of which refer to discrete material properties that are logically independent from one another)? The Tractarian program would have it that the measurement of continuous processes is recovered by the practice of using operations (with numeric exponents). But it is hard to imagine that Weierstrass’s account of limits is recoverable via Tractarian operations. Similar problems arise in

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recovering applications of the calculus to motion – an application which Russell heralded as resolving Zeno’s dynamical paradoxes.22 Wittgenstein’s ideas in ‘‘Some Remarks on Logical Form’’ have been exploited by interpreters eager to argue for the bankruptcy of the program of logical atomism. Hacker characterizes Wittgenstein’s jottings as a striking conversion in his thinking: Here he [Wittgenstein] grappled with the problem of determinate exclusion, realizing, as he had not done in the Tractatus, that there are logical relations (of exclusion and implication) which are determined not by truth-functional composition, but by the inner structure of atomic propositions. At this stage, he suggested that this be budgeted for by abandoning the topic neutrality of the logical connections and drawing up truth-tables specific to the ‘‘propositional system’’ (Satzstem) to which the atomic propositions belong (viz., the ‘system’ of determinates of a given determinable). Thus, in the case of colour exclusion, the conjunction of ‘A is red’ (A being a given spatio-temporal point) with ‘A is blue’ (i.e., the truth-tabular assignment ‘TT’) is nonsense, giving the proposition greater logical multiplicity than the phenomena admit of, and must be excluded by rules of syntax. This concession, as he realized shortly thereafter, spelled the death-knell of the philosophy of logical atomism, and struck at the heart of the metaphysics and the conception of logic which he had advocated in the Tractatus.23

Wittgenstein is raising a problem that has force against the extreme positions of the Tractatus. His problem is by no means telling against Russell’s more moderate logical atomism. Indeed, objections to the Tractarian approach to the measurement of continuous processes does not reveal that Wittgenstein intended to abandon logical atomism – at least not if we construe logical atomism as an eliminativistic research program. Quite the contrary, an eliminativist methodology seems to be what is driving Wittgenstein’s study of the problem of measurement – and in particular his discussion of color incompatibilities. Wittgenstein’s new approach suggests that certain propositions are to be put together to form a logical system of concepts that are defined by its system of relationships. The concepts referred to by the propositions of such a system are not analyzable in isolation from the system. This does not imply, as Hacker thinks, that in spite of their being ‘‘atomic,’’ facts have an ‘‘internal structure.’’ The point is that it is the system as a whole that is applied in measurement of continuous physical processes. On the new account, it is this kind of system of concepts that explains color incompatibility, for the incompatibility 22

23

There is some reason to be concerned, however, that even with the modern definitions of limit and continuity, there remain difficulties of application. See Wesley C. Salmon, ed., Zeno’s Paradoxes (New York: Bobbs-Merrill Co., 1977). P. M. S. Hacker, Wittgenstein’s Place in Twentieth-Century Analytic Philosophy (Oxford: Blackwell, 1996), p. 77.

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comes from the system, not from the material properties and relations. In Wittgenstein’s view, it is the application of concepts as a system that is required for the measurement of continuous physical processes. This is certainly a departure from the Tractarian notion that an atomic proposition (in which material predicate expressions of the ideal physical theory occur) hooks onto the world by picturing (individually and independently) a fact. But it certainly does not mark a collapse of logical atomism – understood as a program of eliminativistic philosophical analysis. We have seen that Russell’s logical atomism should not be saddled with any of the particular constructions within it – be it Russell’s early doctrine of ‘acquaintance’ as a primitive relation, his theory of classes, the existence of sense-data, or whatever. A similar point applies to the more austere eliminativistic program inaugurated by the Tractatus. Though subsequent years will find Wittgenstein departing further and further from the Tractarian idea of one correct eliminativistic reconstruction of all logical and semantic concepts, the eliminativism persists. Austin’s ‘‘ordinary language philosophy’’ can even be included within the research program. Wittgenstein’s philosophical investigations on notions of ‘‘family resemblance,’’ ‘‘language games,’’ ‘‘form of life,’’ ‘‘rule following,’’ and the like are certainly not out of sorts with eliminativism – especially if one develops a version of eliminativism along Kuhnian lines. If a unification of Wittgenstein’s early, middle, and later philosophical work is possible, this is where it is to be found. There is, however, a dark side. Any form of radical eliminativism, be it the Tractarian form or otherwise, runs the grave risk of loosing its mooring. Appeals to ordinary language dissolutions of philosophical problems, ‘‘language games,’’ ‘‘forms of life,’’ and the like make it unclear to what extent we have an analyses of anything. It was with this concern in mind that Russell found Wittgenstein’s Philosophical Investigations unsatisfactory. He wrote: ‘‘Its positive doctrines seem to me trivial and its negative doctrines unfounded. I have not found in Wittgenstein’s Investigations anything that seemed to me interesting and I do not understand why a whole school finds important wisdom in its pages.’’24 Russell went on to say that Wittgenstein seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary. I do not for one moment believe that the doctrine which has these lazy consequences is true. I realize, however, that I have an

24

Bertrand Russell, My Philosophical Development (New York: Simon & Schuster, 1959), p. 216.

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overpoweringly strong bias against it, for, if it is true, philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea table amusement.25

For Russell, logical analysis must involve more than a discussion of speech acts and social practices of a given community. He quipped: ‘‘Adherents of WII [the later Wittgenstein] are fond of pointing out, as if it were a discovery, that sentences may be interrogative, imperative or optative as well as indicative.’’26 Russell objected to the emphasis on practice because he thought that it trivializes analysis by abandoning a quest for an understanding of world. In response to the descendents of Wittgenstein trumpeting the view that philosophy cannot be a science espousing theories, Russell writes: The only reason I can imagine for the restriction of philosophy to such triviality is the desire to separate it sharply from empirical science. I do not think such a separation can be usefully made. A philosophy which is to have any value should be built upon a wide and firm foundation of knowledge that is not specifically philosophical. Such knowledge is the soil from which the tree of philosophy derives its vigour. Philosophy which does not draw nourishment from this soil will soon wither and cease to grow.27

A focus solely on practice undermines the tradition of philosophy – a tradition of inquiry that Russell traces to Thales, which searches for an understanding of the world. Russell wrote: ‘‘I cannot feel that the new philosophy is carrying on this tradition. It seems to concern itself, not with the world and our relation to it, but only with the different ways in which silly people say silly things.’’28 Russell admitted that his own investigations (toward eliminativistic constructions) led him to an ever-widening retreat from Pythagoras. But he couldn’t retreat into ‘‘forms of life’’ or conventions of ordinary linguistic and social practice. The naturalization of logic leads to a story of sociobiology, cultural practices, goals, and endeavors, not to a philosophical understanding of the world. Nonetheless, it is hard to see how to stop Russell’s own retreat from Pythagoras from running into the abyss. Either it ends with Wittgenstein’s Tractarian mysticism according to which logic and semantics are ineffable ‘‘scaffolding,’’ or it advances a form of psychologism. Husserl, Frege, and Russell had won an early battle against psychologism – the thesis that logic and knowledge of logic is grounded in human practices and psychology. Radical eliminativism drops the torch. But so also does Russell’s own development of logical atomism from the 1920s onward. Russell’s neutral monist reconstruction of the notions of ‘‘consciousness’’ and ‘‘knowledge’’ soon engulfed logic and knowledge of logic. Consider the following passage from Philosophy (1927): 25

Ibid.

26

Ibid., p. 217.

27

Ibid., p. 230.

28

Ibid.

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To ‘‘understand’’ even the simplest formula in algebra, say (x þ y)2 ¼ x2 þ 2xy þ y2, is to be able to react to two sets of symbols in virtue of the form which they express, and to perceive that form is the same in both cases. This is a very elaborate business, and it is no wonder that boys and girls find algebra a bugbear. But there is no novelty in principle after the first elementary perceptions of form. And perception of form consists merely in reacting alike to two stimuli which are alike in form but very different in other respects.29

Russell goes on as follows: mathematical [deductive] inference consists in attaching the same reactions to two different groups of signs, whose meanings are fixed by convention in relation to their constituent parts, whereas induction consists, first, in taking something as a sign of something else, and later, when we have learned to take A as a sign of B, in taking A as also a sign of C . . . both kinds of inferences are concerned with the relation of a sign to what it signifies, and therefore come within the scope of the law of association.30

These passages are disappointing. They attempt to naturalize both inductive and deductive inferences by appeal to associative learning. We find Russell struggling, and failing, to prevent logic from collapsing into psychological processes such as the Law of Effect. Russell’s acerbic wit carried the day against Dewey’s naturalistic reconstructions of ‘‘logic,’’ ‘‘representation,’’ and ‘‘truth’’ in terms of notions such as ‘‘inquiry,’’ ‘‘equilibrium,’’ and ‘‘stability of organism-environment systems.’’31 But Russell’s own later naturalism does little to save logic from collapsing into psychology. The moral of our story is that Russell’s original program of logical atomism is worth resurrection. All necessity is logical necessity. The task of philosophy is to provide the reconstructions that reveal this. But the resurrection of logical atomism requires that a concession be made to Pythagoras and Plato. Logic and knowledge of logic cannot be submitted to a naturalistic or eliminativistic analysis of any kind. Logical atomism depends on the view that logic and mathematics have nothing whatever to do with human conventions, practices, or psychology. Russell’s moderate eliminativism deserves a new beginning which returns it to its roots as a science of logical analysis. Logic is the essence of philosophy. If we attempt to answer the question as to the nature of logic and our knowledge of logic, we will travel in a loop. Knowledge of logic is presupposed in any such account. Russell’s original logical atomism avoids the loop by exempting logic and knowledge of logic from eliminativistic reconstruction. Wittgenstein unabashedly embraced the loop. The ouroboric Tractatus 29 31

Bertrand Russell, Philosophy (New York: W. W. Norton & Co., 1927), p. 86. 30 Ibid. See Tom Burke, Dewey’s New Logic: A Reply to Russell (Chicago: University of Chicago Press, 1994).

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proclaimed that logic has no content and that there cannot be a genuine kind of knowledge (or science) of logic. Wittgenstein hoped that logic, and knowledge of logic, could be understood as scaffolding. Far from perfecting Russell’s philosophy, however, logical atomism collapses in the hands of Wittgenstein into mysticism. The wrong oracle yields the wrong philosophy and the constructions of Wittgensten’s Tractatus were almost uniformly wrong. Logical atomism cannot survive the will-o’-the-wisp pronounciations of Russell’s well-intentioned apprentice.

Appendix A: Exclusive quantifiers

Wittgenstein’s Tractatus is famous for its thesis that identity is not a relation. He seems first to have noticed, in his 1914 ‘‘Notes Dictated to Moore,’’ that identity theory can be adjoined to predicate calculus by adding the following as an axiom schemata: Ax  ð9yÞðx ¼ y :&: AyÞ:1

But instead of using this to generate a theory of identity, Wittgenstein used it to convince himself that identity is not a genuine relation. If it were a relation, it would be a logical relation since (x)(x ¼ x) would then be a logical truth. But such a truth would not be a generalized tautology and thus it would not have the characteristic feature he thought all logical truths must have. One might think that defining the identity sign in terms of indiscernibility would bring it into the fold. But we saw in chapter 4 that the only properties Wittgenstein countenances are material properties, and indiscernibility with respect to material properties is insufficient for identity. Material properties are logically independent; the exemplification of one never logically excludes the exemplification of another. Distinct objects might be indiscernible with respect to material properties. In the Tractatus, Wittgenstein finally settles on a plan. The following passages are central: Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all. (TLP 5.5303) Thus I do not write ‘‘f(a, b)  a ¼ b’’ but ‘‘f(a, a)’’ (or ‘‘f(b, b)’’); and not ‘‘f(a, b)  a ¼ b,’’ but ‘‘f(a, b).’’ (TLP 5.5351) And analogously I do not write ‘‘(9x)(9y)  f(x, y)  x ¼ y,’’ but ‘‘(9x)f(x, x)’’; and not ‘‘(9x,y)  f(x, y)  x 6¼ y,’’ but ‘‘(9x, y)f(x, y).’’ (So Russell’s ‘‘(9x, y)f(x, y)’’ becomes ‘‘(9x, y)f(x, y) .v. (9x)f(x, x).’’ (TLP 5.532) 1

Ludwig Wittgenstein, ‘‘Notes Dictated to Moore in Norway,’’ in Notebooks 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe, 2nd ed. (Oxford: Blackwell, 1979), p. 117. Quine reports that he first learned that this schema suffices for identity theory from Hao Wang. See W. V. O. Quine, Set Theory and its Logic (Cambridge, Mass.: Harvard University Press, 1980), p. 13.

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Thus, for example, instead of ‘(x) : fx  x ¼ a’ we write ð9xÞ  fx :  : fa :  ð9x; yÞ  fx  fy : ðTLP 5:5321Þ The identity sign, therefore, is not an essential constituent of conceptual notation. (TLP 5.532)

The challenge of building identity into scaffolding is formidable, however. Wittgenstein acknowledged the difficulties. Writing to Russell from Norway in 1913, he says that ‘‘identity is the very Devil and immensely important; very much more so that I thought . . . I have all sorts of ideas for a solution of the problem but could not yet arrive at anything definite. However, I don’t lose courage and go on thinking.’’2 Wittgenstein held that a proper symbolism – a symbolism in which logical notions are embedded in the syntactic forms – would build identity and difference into the employment of quantifiers and variables themselves. He thought he had accomplished this in the Tractatus. And he took this as one of his greatest achievements. The devil is in the details, however. Wittgenstein never set out a deductive system for exclusive quantifiers. Interpreters usually follow Wittgenstein in contenting themselves with offering a translation schema between a language of exclusive quantifiers and a language with identity. This is inadequate. Of course, Wittgenstein rejects systems of deduction. So we can understand why the Tractatus does not work out a deductive system of exclusive quantifiers. It is, nonetheless, interesting and important to examine whether a deductive system of exclusive quantifiers is viable. In what follows we find that it is. Wittgenstein’s passage at TLP 5.532 suggests that he takes a quantifier to exclude another, when and only when it is in the scope of the other. Consider, ð9xÞð9yÞ fðx; yÞ :v: ð9zÞ fðz; zÞ:

On Wittgenstein’s convention, the quantifiers (9x)(9y) in the first disjunct are exclusive. But since the quantifiers of the first disjunct are not in the scope of (9z) in the second disjunct, the quantifiers (9x)(9y) are not to be read as being exclusive of (9z). But using scope alone to track exclusivity raises an important difficulty for formulating a deductive system for exclusive quantifiers. In classical quantification theory, all instances of the following schema are axioms: ð1Þ ðxÞðB  AxÞ :  : B  ðxÞAx;

where x does not occur free in B. Consider the instance

2

Wittgenstein, Notebooks 1914–1916, p. 123.

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ð2Þ ðxÞðð9yÞFy  GxÞ :  : ð9yÞFy  ðxÞGx:

In the clause (x)((9y)Fy  Gx) the quantifier (9y) is within the scope of the quantifier (x) and is therefore, by Wittgenstein’s plan, to be read exclusively. But in the clause (9y)Fy  (x)Gx the two quantifiers are not read as exclusive. The material conditional is therefore not valid. To rectify this problem we abandon Wittgenstein’s idea that the scope of quantifiers determines exclusivity. Instead, we employ superscripted variables in quantificational expressions together with scope to determine exclusivity. By employing superscripts the exclusive nature of a quantifier when embedded in another’s scope is readily depicted. Let us use ‘‘8’’ for universal quantification. If we express (2) with a superscripted free variable, we get: ð2aÞ ð8xÞðð9yx ÞFy  GxÞ :  : ð9yx ÞFy  ð8xÞGx:

The superscripted free variable x reveals the exclusivity of the quantifiers in question. We now see that (2a) is not a proper instance of (1). The formula B is (9yx)Fy, and the variable x does occur free in it. On the other hand, in ð2bÞ ð8xÞðð8xÞð9yx ÞFy  GxÞ :  : ð9yx ÞFy  ð8xÞGx

the variable x does not occur free in (8x)(9yx)Fy. Hence the formula is valid. Similarly, ð2cÞ ð8xÞðð9yÞFy  GxÞ :  :ð9yÞFy  ð8xÞGx

is valid because we do not use scope of a quantifier alone as determinative of exclusivity. In (2c), the quantifier (9y) does not exclude among its values those of (8x). To illustrate the use of superscripted variables, the following are some translations: ðxÞðyÞRxy ðxÞð9yÞRxy

ð8xÞð8yx ÞðRxx :&: RxyÞ ð8xÞð9yx ÞðRxx :v: RxyÞ

ð9xÞðyÞRxy ð9xÞð9yÞRxy

ð9xÞð8yx ÞðRxx :&: RxyÞ ð9xÞð9yx ÞðRxx :v: RxyÞ

ðxÞðyÞðzÞRxyz ðxÞð9yÞð9zÞRxyz

ð8xÞð8yx Þð8zxy ÞðRxxx :&: Rxxy :&: Rxyx :&: Rxyy :&: RxyzÞ ð8xÞð9yx Þð9zxy ÞðRxxx :v: Rxxy :v: Rxyx :v: Rxyy :v: RxyzÞ

ð9xÞð9yÞðzÞRxyz ð9xÞð9yx Þð8zxy ÞðRxxx :&: Rxxy :&: Rxyx :&: Rxyy :&: RxyzÞ ð9xÞð9yÞð9zÞRxyz ð9xÞð9yx Þð9zxy ÞðRxxx :v: Rxxy :v: Rxyx :v: Rxyy :v: RxyzÞ

On the left are the usual inclusive expressions and on the right are their translations into exclusive quantifiers. The pattern should be clear.

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Appendix A

Wittgenstein’s Tractatus is content to hint toward a translation procedure from inclusive to exclusive quantifiers. The translation is obviously intended to exclude illegitimate occurrences of the identity sign and yet preserve the innocuous uses of identity (such as those in Russell’s theory of definite descriptions). Ramsey, on the other hand, was concerned to find a rigorous translation procedure and his manuscripts show him working on the problem. ‘‘Have you noticed,’’ he wrote to Wittgenstein in a letter of 12 November 1923, ‘‘the difficulty in expressing without ¼ what Russell expressed by (9x): fx  x ¼ 6 a? ’’3 In a subsequent letter of 20 December, Ramsey acknowledges having received a reply from Wittgenstein. He wrote: I didn’t think there was a real difficulty about (9x): fx  x 6¼ a i.e., that it was an objection to your theory of identity, but I didn’t see how to express it, because I was under the silly delusion that if an x and an a occurred in the same proposition the x could not take the value a. I had also a reason for wanting it not to be possible to express it. But I will try to explain it all in a fortnight from now.

In a footnote, Ramsey adds: Thanks for giving me the expression fa  (9x, y)  fx  fy: fa  (9x)  fx.

In subsequent work notes, Ramsey distills the following rule of scope: Two different constants must not have the same meaning [reference]. An apparent [bound] variable cannot [have] the value of any letter occurring in its scope unless the letter is a variable apparent in that scope.4

Wittgenstein’s use of scope to show exclusivity applies to free variables as well. But as we saw, this encounters serious difficulties for forming a deductive system. In our system, the expression (9x)(fx .&. x ¼ 6 a) is rendered as (9xa)fx. The technique of using superscripts to mark exclusivity requires that we not allow expressions such as (8x)Rxy in the language. Instead, we have (8xv )Rxy. In this way quantification respects the exclusivity of the free variables. Axiomatization Predicate logic (hereafter PL) can be axiomatized as follows. The alphabet of the language of PL consists of the primitive signs (,), v, 0 , , f,  and the potentially infinite stock of individual variables x1, . . ., xn. The terms of the system are the individual variables. (Informally we use x, y, z.) The n-placed predicate letters are F n, where n is a natural numeral, and any 3 4

Ludwig Wittgenstein, Letters to C. K. Ogden, with an Appendix of Letter by Frank Plumpton Ramcey, ed. G. H. Von Wright (Oxford: Blackwell, 1973), pp. 82, 83. Frank Plumpton Ramsey, Notes on Philosophy, Probability and Mathematics, ed. Maria Carla Galavotti (Naples: Bibliopolis, 1991), p. 159.

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such letter followed by any finite number of primes. The atomic formulas of the language are of the form: ðPn ð1 ; . . .; n ÞÞ;

where Pn is an n-placed predicate letter and 1, . . ., n are individual variables. I use A and B and C as schemata for formulas of the objectlanguage. The formulas are defined recursively as the smallest class K containing the atomic formulas and such that (A), (A  B), and ((x)C) are in K whenever A, B, and C are in K. Outer brackets on formulas will be dropped for convenience. Bondage and freedom of occurrences of individual variables in formulas are understood in the usual way. The axiom schemata of the theory are as follows: Ax1 Ax2 Ax3 Ax4 Ax5 Rule 1 Rule 2

df(9) df(v) df(&) df()

A . . B  A A .. B  C: : A  B .. A  C B  A . . A  B (x)(B  Ax) .. B  (8x)Ax, where x is not free in B. (x)Ax  A[y/x], where y is a variable free for free x in A. (MP) modus ponens From A and A  B, infer B (UG) universal generalization From Ax, infer (x)Ax, where x is an individual variable. (9x)Ax ¼df (8x)Ax A v B ¼df A  B A & B ¼df (A  B) A  B ¼df (A  B) & (B  A)

This completes the system. Predicate logic with identity (hereafter PL¼) may be formulated by extension of PL. We choose a dyadic relation sign of the language of PL (hereafter we’ll use ¼) and add the following axiom schema: Ax6

Ax  ð9zÞðx ¼ z :&: AzÞ;

where z is free for x in A. All the usual laws of identity are then provable in this system. We now define a formal system Qe for exclusive quantifiers. The alphabet of the language of Qe is the same as that of PL¼ except for the addition of the primitive sign 8. The terms of the language of Qe are just those of PL¼. The atomic formulas of the language are of the form:

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Appendix A

ðPn ð1 ; . . .; n ÞÞ;

where Pn is an n-placed predicate letter and 1, . . ., n are individual variables. The formulas are defined recursively as the smallest class K containing the atomic formulas and such that (A), (A  B), ((8xz1, . . ., zn)C) are in K whenever A, B, and C are wffs in K and all the distinct variables occurring free in C except x are among z1, . . ., zn. Outer brackets on formulas will be dropped for convenience. Bondage and freedom of occurrences of individual variables in formulas are understood in the usual way. The axiom schemata of the theory are as follows: A .. B  A A .. B  C : : A  B .. A  C B  A .. A  B (8xz1, . . ., zn)(B  Ax) .. B  (8xz1, . . ., zn)Ax, where x is not free in B. A5 (8xz1, . . ., zn)Ax  A[y/x] where y is a variable free for free x in A, and distinct from z1, . . ., zn A6 (9x)Ax :: A(y1|x) v A(y2|x) v, . . ., v A(yn|x) v (9xy1, . . ., yn)Ax A7 (8xy, z1, . . ., zn)A  (8xz1, . . ., zn)A, where y is not free in A. Rule 1 (MP) modus ponens From A and A  B, infer B Rule 2 (UG) universal generalization From Ax, infer (8xz1, . . ., zn)Ax, where all the variables occurring free in A are among x, z1, . . ., zn.

A1 A2 A3 A4

dfð9Þ

ð9xzl; ...; zn ÞAx ¼ df  ð8 xzl; ...; zn Þ  Ax

The definitions, df(v), df(&), and df(), are as in PL¼. In the system Qe, quantifier notation (superscripts on bound variables) keep track of the exclusiveness of the variables. To illustrate the way quantifiers in the system Qe keep track of the exclusive variables, and in particular the use of axiom A6, we prove the following theorems. Their statement in classical inclusive quantification theory is paired with a proof in the system Qe. For convenience of exposition, let us use R as a dyadic relation letter and F and G as monadic predicate letters in the proofs below. We first write the theorem with inclusive quantifiers, and then its expression in our language of exclusive quantifiers. ‘ ðxÞðyÞRxy  ðyÞðxÞRxy ‘ ð8xÞð8yx ÞðRxy :&: RxxÞ  ð8yÞð8xy ÞðRxy :&: RyyÞ

Exclusive quantifiers

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

259

ð8xÞð8yx ÞðRxy :&: RxxÞ :  : ð8yx ÞðRxy :&: RxxÞ A5 ð8yx ÞðRxy :&: RxxÞ :  : Rxy & Rxx A5 Rxy :&: Rxx :  : Rxx logic Rxy :&: Rxx :  : Rxy logic ð8xÞð8yx ÞðRxy :&: RxxÞ :  : Rxy 1; 2; 4; syll: ð8xÞð8yx ÞðRxy :&: RxxÞ :  : Rxx 1; 2; 3; syll: ð8xÞðð8xÞð8yx ÞðRxy :&: RxxÞ :  : RxxÞ 6; UG ð8xÞð8yx ÞðRxy :&: RxxÞ :  : Ryy 7; A5; MP ð8xÞð8yx ÞðRxy :&: RxxÞ :  : Rxy & Ryy 5; 8; logic ð8yx Þ9 UG ð8xÞð8yx ÞðRxy :&: RxxÞ :  : ð8xy ÞðRxy & RyyÞ 10; A4; logic ð8yÞ11 UG ð8xÞð8yx ÞðRxy :&: RxxÞ  ð8yÞð8xy ÞðRxy :&: RyyÞ 12; A4; logic

‘ ð9xÞðFy y y ¼ xÞ :&: Gx x Fx :&: ð9yÞGy :: ð9xÞðGy y y ¼ xÞ ‘ ð9xÞðFx & ð8yx Þ  FyÞ :&: Gx x Fx :&: ð9yÞGy :: ð9xÞðFx & ð8yx Þ  GyÞ

1. Fx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx :: ð9xÞGx 2. ð9xÞGx :: Gx vð9yx ÞGy A6 3. Fx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx :: Gx v ð9yx ÞGy 1; 2; logic x 4. Fx :&: ð8y Þ Fy :&: Gx x Fx :&: ð9xÞGx :&: Gx : v : Fx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx :&: ð9yx ÞGy 3; logic x 5. Gy :: ðFx :&: ð8y Þ  Fy :&: Gx x Fx :&: ð9xÞGxÞ  ðGy :&: ð8yx Þ Fy :&: Gx x FxÞ logic 6. Gy :&: ð8yx Þ Fy :&: Gx x Fx :: Gy :&:  Gy logic 7. Gy :&:  Gy :: p & p logic 8. Gy :: ðFx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGxÞ  ðp &  pÞ 5; 6; 7; logic x 9. ð8y Þ8 UG 10. ð9yx ÞGy :  : ðFx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGxÞ  ðp &  pÞ 9; logic x 11. Fx :&: ð8y Þ Fy :&: Gx x Fx :&: ð9xÞGx :&: ð9yx ÞGy :: ðp &  pÞ 10; logic x 12. Fx :&: ð8y Þ Fy :&: Gx x Fx :&: ð9xÞGx :&: Gx 4; 11; logic 13. Fx :&: ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx ::  Gy x

x

8; logic

14. ð8y ÞðFx & ð8y Þ Fy :&: Gx x Fx :&: ð9xÞGx ::  GyÞ 13; UG

260

Appendix A

15. Fx & ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx :: ð8yx Þ  Gy 14; A4; logic 16. Fx & ð8yx Þ Fy :&: Gx x Fx :&: ð9xÞGx :: Gx & ð8yx Þð GyÞ 12; 15; 3; logic 17. ð8xÞðFx & ð8yx Þ Fy :&: Gx x Fx :: Gx & ð8yx Þð GyÞÞ 16; UG 18. ð9xÞðFx & ð8yx Þ Fy :&: Gx x FxÞ :: ð9xÞðGx & ð8yx Þð GyÞÞ 17; logic ‘ ð9xÞðFy y y ¼ xÞ :&: Fa :: Fy y y ¼ a ‘ ð9xÞðFx & ð8yx Þ  FyÞ & Fa :: Fa & ð8ya Þ  Fy

1. ð9xÞðFx & ð8yx Þ  FyÞ & Fa :: Fa & ð8ya Þ  Fy :v: ð9xa ÞðFx &ð8yx Þ  FyÞ A6 x x logic 2. Fx & ð8y Þ  Fy :  : ð8y Þ  Fy 3. ð8yx Þ  Fy   Fa A5 4. Fx & ð8yx Þ  Fy :  :  Fa 2; 3; syll 5. ð8xa ÞðFx & ð8yx Þ  Fy :  :  FaÞ 4; UG 6. ð9xa ÞðFx & ð8yx Þ  FyÞ :  :  Fa 5; logic 7. ð9xÞðFx & ð8yx Þ  FyÞ & Fa :: Fa & ð8ya Þ  Fy 1; 6; logic The last two proofs show the use of A6 in the treatment of multiple existential quantifiers. The next proof also illustrates an important feature of our system.

Ramsey/Hintikka translations Ramsey worked on exclusive quantifiers between 1923 and 1925.5 He arrived at a translation procedure from Wittgenstein’s language of exclusive quantifiers into PL¼, and a translation procedure in the converse direction. Some years later, Hintikka independently discovered a translation function from exclusive to inclusive quantifiers similar to Ramsey’s. Hintikka’s translation from PL¼ into exclusive quantifiers, however, differs from that of Ramsey in that it allows the presence of tautologies. Neither Ramsey nor Hintikka sets out a deductive calculus for exclusive quantifiers, and neither employs superscripted variables as we have above. Instead they follow Wittgenstein in taking the scope of the quantifiers to 5

Frank Plumpton Ramsey, ‘‘Identity,’’ in Notes on Philosophy, Probability and Mathematics, pp. 155–169.

Exclusive quantifiers

261

indicate exclusiveness. Accordingly, I shall rewrite their translation procedure into the language of Qe. The Ramsey/Hintikka translation procedure is as follows: ðT1Þ

ð9xzl; ...; zn ÞA may be replaced by ð9xÞðx 6¼ z1 & x 6¼ z2 &; . . .; & x 6¼ zn : & : AÞ

ðT2Þ

ð8xzl; ...; zn ÞA may be replaced by ðxÞðx 6¼ z1 & x 6¼ z2 &; . . .; & x 6¼ zn :: AÞ:6

The translation proceeds from innermost to the outermost quantifiers. Thus the translation rules apply even when (9x z1, . . ., znx)Ax or (8x z1, . . ., zn) Ax occurs as a subformula (consecutive part) of a larger formula. It is important to recall that these are not well-formed formulas of the language of Qe unless x, z1, . . ., zn are all the variables occurring free in A. Note as well that the Ramsey/Hintikka translation procedure must begin from formulas whose quantifiers have narrowest possible scope. Failing this, the translation procedure would yield wrong results. For example, consider applying it to the following: ð9xÞð8yx ÞðRxx & RxyÞ:

In order to yield a correct translation, the Ramsey/Hintikka translation procedure must begin from the formula ð9xÞðRxx & ð8yx ÞRxyÞ:

Clearly, ‘‘(9x)(y)(Rxx :&: x ¼ 6 y  Rxy),’’ which is equivalent to ‘‘(9x)(y)Rxy,’’ is the proper transcription. Misconstruing the Ramsey/Hintikka translation by taking A to be ‘‘Rxx & Rxy,’’ one arrives at the following: ð9xÞðyÞðx 6¼ y :  : Rxx & RxyÞ:

This is incorrect. The original implies (9x)Rxx. The above does not imply this. It is trivially true in a one-element domain. Ramsey also offered a procedure for expressing appropriate formulas for predicate logic PL¼ into the language of exclusive quantifiers. Ramsey’s procedure is to begin with inmost quantifiers and repeat applications of transformation rules such as the following:

6

Jaakko Hintikka, ‘‘Identity, Variables, and Impredicative Definitions,’’ Journal of Symbolic Logic 21 (1956): 225–245.

262

Appendix A

Rule(i) (where x and y are the only free variables in A) t½ðyÞAxy ¼ ð8yx ÞðAxx & AxyÞ t½ð9yÞAxy ¼ ð9yx ÞðAxx v AxyÞ:

Rule(ii) (where x, y, and z are the only free variables in A) t½ðzÞAxyz ¼ ð8zxy ÞðAxyx & Axyy & AxyzÞ t½ð9zÞAxyz ¼ ð9zxy ÞðAxyx v Axyy v AxyzÞ

and so on. In repeating applications of rules such as these we are to ignore free variables occurring as superscripts to quantifiers.7 For example: ðxÞðyÞð9zÞAxyz:

Transformation: 1. t½ð9zÞAxyz ¼ ð9zxy ÞðAxyx v Axyy v AxyzÞ xy x xy 2. t½ðyÞð9z ÞðAxyx v Axyy v AxyzÞ ¼ ð8y Þðð9z ÞðAxxx v Axxx v AxxzÞ & ð9zxy ÞðAxyx v Axyy v AxyzÞÞ ¼ ð8yx Þðð9zxy ÞðAxxx v AxxzÞ & ð9zxy ÞðAxyx v Axyy v AxyzÞÞ 3. t½ðxÞð8yx Þðð9zxy ÞðAxxx v AxxzÞ & ð9zxy ÞðAxyx v Axyy v AxyzÞÞ ¼ ð8xÞð8yx Þðð9zxy ÞðAxxx v AxxzÞ & ð9zxy ÞðAxyx v Axyy v AxyzÞÞ If the identity sign occurs in A, then we may eliminate it as follows. Replace every occurrence of v ¼ v by p  p, and each occurrence of u ¼ v, where u is a different variable from v, by (p  p). Then, whenever possible, eliminate the tautologies and contradictions in the contexts of their occurrence by replacement of logical equivalents. If elimination is possible, the result is a formula of Qe. If elimination is not possible, the original is a pseudoformula of PL¼ employing an illicit use of the identity sign. For example: ðxÞðx ¼ xÞ Transformation: ð8xÞðp  pÞ: This is a pseudo-proposition not expressible in the language of Qe. In contrast we have: ðxÞð9yÞðx 6¼ y :&: AxyÞ

7

The issue did not arise for Ramsey since he did not employ such superscripts.

Exclusive quantifiers

263

Transformation: 1. ð8xÞð9yx Þðx 6¼ x :&: Axx : v : ðp  pÞ :&: AxyÞ 2. ð8xÞð9yx Þððp  pÞ & Axx : v : ðp  pÞ :&: AxyÞ 3. ð8xÞð9yx Þððp  pÞ :&: AxyÞ 4. ð8xÞð9yx ÞAxy Consider one further example: ð9xÞðyÞðAy :  : x ¼ yÞ Transformation: 1. ð9xÞðð8yx ÞðAx :  : x ¼ x : & : Ay :  : x ¼ yÞ 2. ð9xÞðð8yx ÞðAx :  : p  p : & : Ay :  : ðp  pÞÞ 3. ð9xÞð8yx ÞðAx :&: AyÞ As we see from the last example, Russell’s theory of definite descriptions survives, even though, as Wittgenstein put it in a letter to Russell, ‘‘the individual primitive signs in it are quite different from what you believe.’’8 Usual proof techniques for the consistency and soundness of PL¼ may be employed to arrive at similar results on behalf of Qe. By means of the translation procedures, we can prove the following theorem by strong induction on the length of a proof: Bl ; . . . :; Bn ‘PL A ! t½Bl ; . . .; t½Bn  ‘Qe t½A: Any proof in PL (i.e., predicate logic without identity) can be transformed into a proof in the system Qe. Thus, Qe is semantically complete with respect to the logical truths of PL. Indeed, the formal system Qe legitimates a Copi-style natural deduction technique. It remains to consider the adequacy of Qe as a theory of identity. Proofs involving identity in PL¼ are not preserved by translation. We needed to use A6 of Qe to recover the use of laws of identity in PL¼. Moreover, there are clearly derivations in PL¼ that are not forthcoming in Qe. For instance, there is no proper analog of (x)(x ¼ x) or of (x)(9y)(x ¼ y) provable in Qe.

Infinity as scaffolding Models of the axioms of classical PL¼ require a nonempty domain, but not an infinite domain. Unlike the system PL¼, however, all models of the

8

Letter of November 1913 in Notebooks 1914–1916, 2nd ed., p. 129. Wittgenstein objects to Russell’s use of the identity sign in his theory of definite descriptions.

264

Appendix A

axioms of Qe have infinite domains. Let Av be Fv  Fv, where F is a predicate letter of Qe. The following theorem schema is valid: ‘Qe ð9z1 Þð9zz Þ; . . .; ð9xn z1; ...; zn1 ÞðAz1 & Az2 &; . . .; & Azn & ð8yz1; ...; zn Þ AyÞ

Proof: z1; ...; zn Þ Ay :: 1. Az1 & Az2 &; . . .; & Azn & ð8y zl;:...;zn ð8y Þ Ay logic 2. ð8yz1; ...; zn Þ Ay :  : Ay A5 3. Ay  ð9yÞ Ay A5; logic; dfð9Þ z1; ...; zn Þ Ay :: ð9yÞ Ay 4. Az1 & Az2 & . . .; & Azn & ð8y 1; 2; 3; syllogism z1; ...; zn1 ÞðAz1 & Az2 &; . . .; & Azn & 5. ð8z1 Þð8zz Þ; . . .; ð8n ð8yzl; ...; zn Þ  Ay :: ð9yÞ  AyÞ

4; UG:

z1; ...; zn1

ÞðAz1 & Az2 &; . . .; & Azn & 6. ð9z1 Þð9zz Þ; . . .; ð9n ð8yz1; ...; zn Þ AyÞ :: ð9yÞ Ay 5; logic 7.  ð9yÞ Ay since Av is a tautologous form z1; ...; zn1 ÞðAz1 & Az2 &; . . .; & Azn & 8.  ð9z1 Þð9zz Þ; . . .; ð9n z1; ...; zn ð8y Þ AyÞ 6; 7; logic:

In the system Qe, the following are theorems: ‘Qe ð8xÞ Ax  ð9xÞ Ax ‘Qe ð9xÞðAx & ð8yx Þ AyÞ  ð9xÞ Ax ‘Qe ð9xÞð9yx ÞðAx & Ay & ð8zxy Þ AzÞ  ð9xÞ Ax

and so on. The potential infinity of logical objects is shown by the potential infinity of distinct individual variables of the language of Qe. As discussed in chapter 5, when Qe is appended to a Fregean calculus that allows quantifiers over higher-level concepts and numbers are developed as second-level concepts, the above are equivalent (respectively) to the following: ‘Qe 0xAx  ð9xÞ Ax ‘Qe 1xAx  ð9xÞ Ax ‘Qe 2xAx  ð9xÞ Ax

and so on. Thus any model of Qe validates the following infinity statement: ðInfinityÞ

ð8MÞðN’ ½Mx ’x   ð8yÞðMx y x  ð9yÞ yyÞÞ:

Exclusive quantifiers

265

This is precisely the result Wittgenstein heralded at TLP 5.535 when he wrote: ‘‘What the axiom of infinity is intended to say would express itself in language through the existence of infinitely many names with different meanings.’’ Every distinct free variable (‘‘name’’ as Wittgenstein uses the expression in this passage) has a distinct value. The infinity of distinct variables in the system thereby shows infinity.9

9

There is an interesting recent system for exclusive quantifiers that does not have the infinity result. See Kai Wehmeier, ‘‘Wittgenstein’s Predicate Logic,’’ Notre Dame Journal of Formal Logic 45 (2004): 1–11.

Appendix B: Modality in the Tractatus

Russell held that logic is a science of logical form. But his views on the nature of logical form changed significantly as his ‘‘retreat from Pythagoras’’ widened. The changes are very important to understanding Russell’s analysis of logical necessity. They are no less important for understanding Wittgenstein’s conception of logical necessity in the Tractatus. Bradley writes that Russell’s [logic] is extensionalistic, having no room for modal notions especially the de re (essentialistic) ones, while Wittgenstein’s atomism is robustly modal and essentialistic.1 We shall find that this is far from the case. Russell’s analysis of logical necessity as universal truth Perhaps one of the most commonly criticized doctrines of Russell’s philosophical logic is his analysis of logical necessity in terms of full generality and truth. In Russell’s early view, every statement which is fully general and true is logically necessary. Russell’s thesis that logical necessity is properly analyzed in terms of full generality and truth must be evaluated with respect to his language for logic. Full generality, after all, is relative to a given formal language for logic. To prepare us to understand how Russell might think the analysis is plausible, consider a naive formal language for a theory of attributes with bindable predicate variables allowed in both subject and predicate positions. (This is called a ‘‘second-order logic with nominalized predicates.’’) The theory is naive because it assumes a comprehension principle for attributes that yields Russell’s paradox. In Russell’s view, logical necessity coincides with logical truth. In the language of modern predicate logic, where ‘‘F’’ is a predicate letter, it is common to speak of the formula ðxÞðFx  FxÞ

as logically true in Tarski’s sense that every denumerable sequence of objects in every nonempty domain satisfies the formula. No matter how ‘‘F’’ is interpreted in a given domain, the formula will be true in the domain. In 1

Raymond Bradley, The Nature of All Being: A Study of Wittgenstein’s Modal Atomism (Oxford: Oxford University Press, 1992), p. xviii.

266

Modality in the Tractatus

267

the naive theory of attributes, one might think to capture the reference to ‘‘every’’ interpretation of ‘‘F’’ mentioned in the Tarski semantics by turning ‘‘F’’ into a predicate variable and universally binding it. This yields the following: ð’ÞðxÞð’x  ’xÞ:

This is true and fully general (in the naive language of a second-order logic with nominalized predicates). One might be concerned, however, with ð9’Þð9xÞ ’x:

This too is fully closed and true. Is it logically true? If we assume (naively) that our theory of attributes is part of pure logic, with attributes as logical entities, then it is a truth of logic. To see this, consider the attribute I, such that ðxÞðIx  x ¼ xÞ:

This is the attribute of self-identity. Now it follows that I(I ). Hence, ð9’Þð9xÞ ’x is a truth of the theory of logical attributes. The naive theory of such attributes is inconsistent. It confronts Russell’s paradox of the attribute F that an attribute G exemplifies if and only if G does not exemplify itself. Because of the paradoxes associated with the assumption of logical entities such as attributes or classes, modern conceptions construe logic as having no ontological commitments to entities. Typetheory cannot come to the rescue of Russell’s analysis of logical necessity. The statement becomes ‘‘ð9’ðoÞ Þð9xo Þ ’ðoÞ ðxo Þ.’’ But if entities of type o are contingent particulars, the statement is again contingent. Modern logic draws the conclusion that logic should be free of ontological commitments to purely logical objects. The ‘‘proper’’ logical calculus, it is often thought, is the ‘‘first-order’’ predicate calculus; and indeed, it is a formulation of the predicate calculus which embraces the empty domain – so that no existential formula whatever is a thesis of the calculus. But this is a hasty generalization. In 1905, Russell thought he had found a genuine solution of the paradox in his substitutional theory of propositional structure. In virtue of this solution, he endeavored to preserve his analysis of logical necessity in terms of truth and full generality. In the context of a conception of logic as a science of propositional structure, Russell accepted the following thesis: Any proposition (state of affairs) that is fully general and obtaining, is a purely logical state of affairs.

Any fully general formula of the substitutional calculus for logic that is true (i.e., the proposition named by a nominalization of the formula obtains) is a logical truth.

268

Appendix B

Russell worked out a formal calculus for substitution by December of 1905 on the heels of the theory of definite descriptions. During the early part of the era of Russell’s substitutional theory, general as well as particular states of affairs were adopted as part of the ontology. Indeed, every well-formed formula of the formal calculus for logic can be nominalized to form a term that stands for a proposition. As we saw in chapter 2, the language of the substitutional theory adopts the horseshoe sign (which we’ll write as ‘‘)’’ for clarity of exposition) for a relation of implication. The sign is flanked by terms to form a formula. This is not the modern logical particle horseshoe ‘‘’’ which is used for ‘‘if . . . then . . .’’ The modern sign is flanked by formulas to form a formula. The difference is clear when nominalizing brackets are employed to transform a term into a formula of the language. Where A is a formula, {A} is a term. Thus, in the substitutional theory, the expression ‘‘{A} ) {B}’’ as well as ‘‘x ) y’’ is allowed. This reveals that the sign ‘‘)’’ is a relation sign and not a modern statement connective. It is convenient, however, to drop the nominalizing brackets. Using dots for punctuation, one may write ‘‘x . ). y ) x’’ instead of ‘‘x ) {y ) x}.’’ The substitutional language is wholly type-free, and employs only one style of variables – namely, entity variables. Nonetheless, the theory solved the Russell paradoxes (of classes and attributes) in exemplary eliminativistic fashion by finding a substitutional proxy for a type-regimented theory of attributes in intension. The type regimentation is built into the formal syntax of substitutional statements which reconstruct statements of the theory of attributes. The formula p=a; x!q

is a primitive of the language of the theory. It says that q is exactly (structurally) like p except at most that x occurs in q wherever a occurs in p. (More conveniently, Russell reads it as saying that q results from substituting x for a in p.) It is ungrammatical in the language to write p=a; x; y !q

for this would attempt to say that two entities x and y are both substituted for a in p. However, Russell defines the expression s=t; w; p; a !q

by appeal to a carefully crafted process of several single substitutions. This formula says that p is substituted for t and a is substituted for w (‘‘simultaneously’’) in s. In a similar way, Russell goes on to three, four, and any finite number of ‘‘simultaneous’’ substitutions.

Modality in the Tractatus

269

The substitutional theory succeeds in building the structure of a hierarchy of types of attributes into formal grammar. The number of substitutions replaces the notion of the type of an attribute. For instance, ’ðoÞ ðxo Þ

is supplanted by ð9qÞðp=a; x!q :&: qÞ:

The next type y ððoÞÞ ð’ðoÞ Þ

is supplanted by means of double substitutions: ð9qÞðs=t; w; p; a; !q :&: qÞ:

In this way, the substitutional theory emulates the structure of a typetheory of attributes in intension without its ontological assumptions. Now statements expressed in the language of simple type-theory are quite readily translated into the language of substitution. Consider the simple type-theoretical statement: ð’ðoÞ Þðxo Þð’ðoÞ ðxo Þ  ’ðoÞ ðxo ÞÞ:

The statement is emulated in the type-free substitutional language by ðp; aÞðxÞððqÞðp=a; x!qÞ ) ðqÞðp=a; x!qÞÞ:

Notice that (q)(p/a;x!q) is a definite description and obeys all the rules governing the scope of definite descriptions in Principia. By contextual definition, we arrive at: ðp; aÞðxÞð9qÞðp=a; x!r : r: r ¼ q : & : q ) qÞ:

Ascending type, consider the following expression of the language of simple type-theory: ð’ððoÞÞ Þðy ðoÞ Þð ’ððoÞÞ ðy ðoÞ Þ  ’ððoÞÞ ðy ðoÞ ÞÞ:

This is emulated in the substitutional theory by: ðs; t; wÞðp; aÞððqÞðs=t; w; p; a! qÞ ) ðqÞðs=t; w; p; a! qÞÞ:

Again observe that (q)(s/t,w; p,a!q) is a definite description. By contextual definition we have: ðs; t; wÞðp; aÞð9qÞðs=t; w; p; a! r : r: r ¼ q : & : q ) qÞ:

270

Appendix B

The type-theoretical statement ð9’ðoÞ Þð9xo Þ ’ðoÞ ðxo Þ

is emulated in substitution as follows: ð9pÞð9aÞð9xÞð9qÞðp=a; x!q :&: qÞ:

This says that there are some entities p, a, and x, and some entity q which obtains and is exactly like p except containing x as a constituent wherever p contains a. If p is the proposition {a ¼ a}, and x is the proposition {(x)(x ¼ x)}, then there is a unique proposition q, namely, fðxÞðx ¼ xÞg ¼ fðxÞðx ¼ xÞg;

and this proposition obtains. Given propositions, obtaining or not, are purely logical individuals, we can see that the existential commitment poses no difficulty for Russell’s thesis that logical necessity is properly analyzed in terms of full generality and truth (in the substitutional language of the logic of propositions). Indeed, it is a theorem of Russell’s substitutional theory that there are infinitely many necessarily existing entities (propositions). The substitutional theory met its end with the ‘‘no-propositions’’ theory of Principia. The cause was a little-known paradox of propositions which Russell discussed only in his unpublished manuscripts.2 Russell came to believe that the only way to avoid this paradox was to introduce into the substititutional theory a hierarchy of orders of propositions and an orderstratified grammar of restricted variables.3 He was dissatisfied with this. Not only would it require reducibility principles to recover arithmetic, it would have required Russell to abandon his longstanding commitment to the view that any proper calculus for pure logic must have one style of genuine variables – individual/entity variables. Russell’s substitutional theory had succeeded in emulating simple type-theory by building type structures into the formal grammar of substitution. But Russell needed a theory which built the structure of both types and orders into grammar. The successor of Russell’s substitutional theory was Principia. Russell thought that by abandoning his ontology of propositions and adopting a nominalistic semantics for predicate variables, he could provide a 2 3

Gregory Landini, Russell’s Hidden Substitutional Theory (New York: Oxford University Press, 1998). There is reason to believe that Russell was mistaken in this assessment and that a form of the substitutional theory can be salvaged. See Gregory Landini, ‘‘On ‘Insolubilia’ and Their Solution By Russell’s Substitutional Theory,’’ in Godehard Link, ed., One Hundred Years of Russell’s Paradox (New York: De Gruyter, 2004), pp. 373–399.

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philosophical justification for introducing a language with order\type indices on its predicate variables. This preserves Russell’s doctrine that any proper calculus for pure logic must employ only one style of genuine variables – individual variables. In Principia, the only genuine variables are the ‘‘individual’’ variables; there are no orders or types of entities. Principia does not offer a ‘‘theory’’ of types of entities. In eliminativistic fashion, the structure of order\types is built into the formal grammar (the conditions of significance) of its predicate variables. Universals, particulars, and facts are all on a par qua individuals in the Principia. The eliminativistic constructions of Principia destroy Russell’s early conception of logical truth – a conception that had been preserved by the eliminativistic techniques of the substiutional theory. Truth, together with full second-order closure in the ramified and type-regimented language of Principia, is not sufficient for logical truth. For example, the expression of the infinity of the universal class of individuals of lowest type is a fully closed second-order formula. But even if it is true, it is far from clear that it is a logical truth given the intended semantics of Principia. Individuals of lowest type include universals, and Russell suggests that universals are logical entities. Nonetheless, nothing in the ontology of Principia assures that there must be infinitely many universals. Principia’s eliminativistic constructions cannot recover a theorem of infinity. In contrast, the eliminativistic constructions of Russell’s earlier substitutional theory preserved a theorem of infinity and yet showed how to reconstruct arithmetic within logic without assuming any logical principle for comprehending attributes. During the era of the substitutional theory, Russell’s argument is strongest in favor of his analysis of the notion of logical necessity in terms of full closure and truth. After Principia, the analysis is seriously compromised. Logical necessity as logical truth In his paper ‘‘On the Notion of Cause’’ (1914) as well as in Introduction to Mathematical Philosophy (1919) and the lectures on logical atomism (1918), Russell persists in saying that (logical) necessity is to be analyzed as properly applying only to propositional functions.4 Yet in his Introduction to Mathematical Philosophy he regards the fact that Principia’s logic yields existential theorems a ‘‘defect in logical purity.’’5 Russell was quite well

4

5

The thesis also occurs in Russell’s debate with Frederick Copleston: ‘‘A Debate on the Existence of God,’’ in Bertrand Russell on God and Religion, ed. A. Seckel (New York: Prometheus Books, 1987), pp. 123–146. Bertrand Russell, Introduction to Mathematical Philosophy (London: Alan & Unwin, 1919, 1953), p. 203.

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aware that, given the no-attributes, no-classes, and no-propositions account of logic in the Principia, the statement ð9xo Þðxo ¼ xo Þ cannot count as logically necessary. In Principia itself Russell says that the statement of infinity, instances of the axiom (schemata) of reducibility, and the ‘‘multiplicative axiom’’ (or axiom of choice) are fully general and presumably true but cannot be regarded as logical truths. So it seems clear that Russell was well aware that the order\type regimentation imposed in Principia puts an end to his early conception that logical truths are fully general truths. Why then does he persist with the view in some of his writings? The answer is that in works written after Principia Russell’s analysis of logical necessity is restricted to first-order formulas. Russell argues that logical necessity has been misconceived. He has in mind Lewis’s ‘‘strict implication’’ in particular.6 In Russell’s view, a statement of predicate logic is logically necessary if and only if the universal secondorder closure of the statement, derived from it by changing all predicate constants into predicate variables, is true. Russell’s analysis is rarely understood and frequently criticized. Russell himself is partly at fault. In his Introduction to Mathematical Philosophy, he wrote: Another set of notions as to which philosophy has allowed itself to fall into hopeless confusions through not sufficiently separating propositions and propositional functions are the notions of ‘‘modality’’: necessary, possible and impossible. (Sometimes contingent or assertoric is used instead of possible.) . . . In the case of propositional functions, the three-fold division is obvious. If ‘‘’x’’ is an undetermined value of a certain propositional function, it will be necessary if the function is always true, possible if it is sometimes true, and impossible if it is never true.7

In ‘‘On the Notion of Cause,’’ Russell writes: A proposition is necessary with respect to a given constituent if it remains true when that constituent is altered in any way compatible with the proposition remaining significant.8

Statements such as these have obscured the fact that Russell’s position is that full generality (in the proper language for logic) is required for the notion of logical necessity (as applied to a first-order formula). His position is clearest in a paper ‘‘Necessity and Possibility’’ (circa 1905), but that

6 7 8

C. I. Lewis, A Survey of Symbolic Logic (Berkeley: University of California Press, 1918), pp. 291ff. Introduction to Mathematical Philosophy, p. 165. Bertrand Russell, ‘‘On the Notion of Cause,’’ in Mysticism and Logic (Totowa, N.J.: Barner & Noble, 1976), p. 134.

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paper is not widely known.9 There are hints in Russell’s work that a more careful thesis is what he advocates, but one must be on the lookout for them. For example, Russell considers the statement, ‘‘If all men are mortal and Socrates is a man, then Socrates is mortal.’’ He writes: we may substitute  for men,  for mortals, and x for Socrates, where  and  are any classes whatever, and x is any individual. We then arrive at the statement: ‘‘No matter what possible values x and  and  may have, if all ’s are ’s and x is an  then x is a ’’; in other words, the ‘‘propositional function ‘if all ’s are ’s and x is an  then x is a ’ is always true.’’ Here at last we have a proposition of logic.10

Russell’s discussion is in terms of classes  and , but the appeal to full generality is clear. Full generality is needed for logical necessity. Russell analyses quantification over classes by means of bound predicate variables. Moreover, Russell’s use of the convenient replacement of ‘‘Socrates’’ by an individual variable wants amendment. The original sentence is to give way to the first-order sentence ’x x yx :&: ½zSz½’z :: ½zSz½yz:

This is the logical form of the original. Next, the logical necessity of this form (i.e., its necessity with respect to all the variables, individual variables and predicate variables) consists in the truth of the fully general secondorder formula: ð’ÞðyÞðð’x x yxÞ :&: ½zSz½’z :: ½zSz½yzÞ:

This is Russell’s notion of logical necessity. (For convenience, order\type indices on the variables are suppressed.) It is important to keep in mind that full second-order generality is required for logical necessity. Russell was concerned to argue that it is mistaken to regard necessity and possibility as attributes of propositions. For pedagogical reasons, he explained his account of necessity (logical necessity) by introducing ‘‘degrees of necessity’’ depending on the degree of generality involved with closing the original first-order formula (‘‘propositional function’’) which is said to be necessary. ‘‘(9x)Fx’’ may therefore be read as saying that the formula ‘‘Fx’’ is ‘‘possible with respect to x.’’ Russell recognizes that this is a very weak notion of possibility, and not logical

9

10

Bertrand Russell, ‘‘Necessity and Possibility,’’ The Collected Papers of Bertrand Russell, vol. 4, Foundations of Logic 1903–1905, ed. Alasdair Urquard (London: Routledge, 1994), pp. 507–520. Introduction to Mathematical Philosophy, p. 197.

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possibility. We must bear this in mind in reading his logical atomism lecturers, when Russell writes: If you take ‘‘x ¼ x,’’ that is a propositional function which is true whatever ‘‘x’’ may be, i.e., a necessary propositional function. If you take ‘‘x is a man,’’ that is a possible one. If you take ‘‘x is a unicorn’’ that is an impossible one.11

How does Russell arrive at the result that ‘‘x is a unicorn’’ is impossible? Russell only means that it is impossible with respect to x, because (9x)(x is a unicorn), i.e., there are no unicorns. To ask whether it is logically possible we must first decide if the predicate ‘‘unicorn’’ is to be a primitive or a complex predicate. It certainly seems complex – perhaps it is the notion of a horse with a horn on its forehead. Replacing ‘‘x is a unicorn’’ with ‘‘x is a horse and x has a horn on its head,’’ Russell’s account requires that we next replace these predicates by variables ‘‘’’’ and ‘‘y’’ (respectively). So to ask whether ‘‘x is a unicorn’’ is logically possible we are to ask whether (9’)(9y) (9x)(’x & yx). We now find that it is logically possible after all. These considerations are of utmost importance in reading Wittgenstein’s criticisms of Russell on the notion of logical necessity (and logical truth). In the Tractatus we find: It is incorrect to render the proposition ‘(9x).fx’ in the words ‘fx is possible,’ as Russell does. (TLP 5.525) The mark of a logical proposition is not general validity. To be general means no more than to be accidentally true for all things. An ungeneralized proposition can be tautological just as well as a generalized one. (TLP 6.1231)

Anscombe writes that ‘‘Russell thought that ‘‘fx’’ is possible only if there is an actual case of an f .’’12 This is mistaken if it is intended as a criticism of Russell’s conception of logical possibility. For Russell, ‘‘fx’’ is logically possible if and only if (9’)(9x)’x. There is no commitment to any entity having f. There is rather a commitment to some entity satisfying some ’. As we have seen, his analysis of logical possibility of a first-order form requires its full second-order closure. It makes a travesty of Russell’s view to present it as if it analyzes logical necessity as truth plus full generality in the language of a first-order predicate calculus. Russell never held the view. Wittgenstein’s remarks warrant a more careful interpretation.

11 12

Bertrand Russell, ‘‘The Philosophy of Logical Atomism,’’ Logic and Knowledge: Essays by Bertrand Russell 1901–1950, ed. Robert C. Marsh (London: Allen & Unwin, 1977), p. 213. G. E. M. Anscombe, An Introduction to Wittgenstein’s Tractatus, 2nd ed. (New York: Harper & Row, 1959), p. 159.

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If A is a formula of predicate logic (first-order logic) containing no individual constants and containing only the distinct free variables x1, . . ., xm, and if A* results from the formula A by replacing (respectively) the distinct predicate letters F1, . . ., Fi, of A by distinct predicate variables ’1, . . ., ’i, then Russell’s reconstruction recovers &A as (’1), . . ., (’ i)(x1), . . ., (xm)A*. Russell’s intent is to make logical necessity, restricted to formulas of the (first-order) predicate calculus, coincide with the notion of logical truth as ‘‘truth in virtue of form.’’13 The notion of truth in virtue of the form of an expression of the language of (first-order) predicate logic is captured by means of full generality and truth of a second-order expression. Russell’s analysis of necessity parallels Tarski’s semantic definition of logical truth. Russell gets at Tarski’s idea of different interpretations of the predicates of a first-order formula (over domains of differing cardinality) by turning the predicate letters into predicate variables and universally closing the formula. The fact that Russell’s analysis parallels Tarski’s semantics for firstorder logical truth is rarely appreciated. All too often interpreters have followed Moore in charging Russell with ‘‘logical howlers’’ when it comes to modal notions.14 For example, Bradley argues that Russell cannot distinguish the following: ð1Þ &ðp :  : q  rÞ ð2Þ p :  : &ðq  rÞ:

As an example, he gives: (1a) All the books on the shelf are blue entails that if my copy of the Principles is a book on this shelf then my copy of Principles is blue. (2a) If all the books on the shelf are blue then my copy of the Principles is a book on this shelf entails that my copy of Principles is blue.

The first is true and the second is false.15 Let’s use the following: Pz: z a copy of Principles belonging to me Sz: z is a book on the shelf Bz: z is blue.

13

14

15

See Nino B. Cocchiarella, ‘‘Philosophical Perspectives on Quantification in Tense and Modal Logic,’’ in D. Gabbay and F. Guenthner, eds., Handbook of Philosophical Logic, vol. 2 (Dordrecht: D. Reidel, 1983), pp. 309–353. G. E. Moore, ‘‘External and Internal Relations,’’ Proceedings of the Aristotelian Society 21 (1919): 40–62. See also Raymond Bradley, The Nature of All Being: A Study of Wittgenstein’s Modal Atomism (Oxford: Oxford University Press, 1992), p. 22. Bradley, The Nature of All Being, p. 22.

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Thus we have (respectively): &fSz z Bz :  : ½xPx½Sx  ½xPx½Bxg Sz z Bz :  : &f½xPx½Sx  ½xPx½Bxg

Russell’s account can distinguish these. He has: ð1aÞR

ð’ÞðyÞðGÞð’z z yz :  : ½xGx½’x  ½xGx½yxg

ð2aÞR

Sz z Bz :  : ð’ÞðyÞðGÞf½xGx½’x  ½xGx½yxg:

There is no logical howler here. Consider the sentence ‘‘Ben Franklin is a cat.’’ Is this logically possible? We are inclined to say that it is. On Russell’s analysis (as well as on Tarski’s semantic characterization) of logical truth, we must begin by exacting the form. If we transcribe the statement into first-order logic, using ‘‘Fx’’ for ‘‘x invented bifocals’’ and ‘‘Cx’’ for ‘‘x is a cat,’’ we get: ½xFx½Cx:

Expanding the definite description, this yields, ð9xÞðFy y y ¼ x :&: CxÞ:

On Tarski’s account, this sentence is satisfiable. It is true in a certain interpretation of the predicates ‘‘F ’’ and ‘‘C’’ over a given domain. Just let ‘‘F’’ uniquely pick out the moon and ‘‘C’’ pick out objects orbiting the earth. Russell’s analysis parallels Tarski’s semantic characterization. Russell says that our sentence ‘‘(9x)(Fy  y y ¼ x .&. Cx)’’ is logically possible if and only if ð9’Þð9yÞð9xÞð’y y y ¼ x :&: yxÞ

is true. The latter is indeed true, since we can easily find a property ’ exemplified by exactly one entity (say, the moon), and a property y that it has (say, orbiting the earth). Both Tarski’s semantic approach and Russell’s approach require the existence of at least one entity (since classically understood the domains of a Tarski interpretation must be nonempty). Tarski’s account and Russell’s account are on a par. For Russell, the only necessity is logical necessity, and this coincides with logical truth. Logical truth is truth in virtue of logical form or structure. Russell’s account, as with Tarski’s semantic approach, makes necessity a fundamentally de dicto notion. Friends of essentialism’s de re modalities would bridle at the thought that the ‘‘possibility’’ of Franklin’s being a cat should turn on contingent facts about interpretations of the formula over a domain of actual entities. But this is because they have in mind a de re form

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of necessity and possibility whose natural ally is essentialism. Consider the following: The inventor of bifocals might not have invented bifocals.

Taken de re and using } as a statement operator for possibility, we have: ½xFx½ }f½xFx½x 6¼ zg:

Removing the scope markers, this is: ð9xÞðFy y y ¼ x :&:}ð9zÞðFy y y ¼ z :&: x 6¼ zÞÞ:

Can Russell’s account capture this de re notion of possibility? Quantified modal logic which accepts de re expressions as syntactically well formed is not committed to a form of essentialism that severs logical necessity from logical truth. But to preserve the connection, one must adopt what Cocchiarella calls the ‘‘primary semantics’’ for necessity. In Cocchiarella’s primary semantics, logical necessity coincides with logical truth (as understood in terms of the the Tarski-style semantics).16 Indeed, McKay has shown that when anti-essentialism is universally valid, de re and de dicto semantically collapse.17 The de re formula ð9xÞðFy y y ¼ x :&: } ð9zÞðFy y y ¼ z :&: x 6¼ zÞÞ

is semantically equivalent to the de dicto ð9xÞðFy y y ¼ x :&: } ð9vÞð9zÞðFy y y ¼ z :&: v 6¼ zÞÞ:

In the primary semantics, every de re formula is semantically equivalent a de dicto formula. Accordingly, Russell’s account of logical necessity can apply to de re modal statements. A Russellian can simply put ð9xÞðFy y y ¼ x :&: ð9’Þð9zÞð’y y y ¼ z :&: x 6¼ zÞÞ:

This is clearly true since there are two distinct entities only one of which invented bifocals. To understand Cocchiarella’s primary semantics, it is illuminating to begin with what Parsons calls the ‘‘Modal Thesis of Anti-essentialism.’’18 The modal thesis of anti-essentialism assures that if some entity has a 16 17 18

Nino Cocchiarella, ‘‘On the Primary and Secondary Semantics of Logical Necessity,’’ Journal of Philosophical Logic 4 (1975): 13–27. Thomas McKay, ‘‘Essentialism and Quantified Modal Logic,’’ Journal of Philosophical Logic 4 (1975): 423–438. Terence Parsons, ‘‘Essentialism and Quantified Modal Logic,’’ Philosophical Review 78 (1969): 35–52.

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property of logical necessity, then every entity has it of logical necessity. In short, the only attributes that an entity has of logical necessity are purely logical attributes such as Fyˆ  Fyˆ, yˆ ¼ yˆ, and the like. Parsons observes that even on the Kripke semantics for quantified modal logic, no essentialist formula is a thesis, for there are always ‘‘maximal model structures’’ in which no essentialist formula is true. In a maximal model structure, every possible Tarski-style interpretation of the predicate letters (over a given domain) is represented among the possible worlds in the model structure. For instance, where F is a primitive predicate letter of the language, ‘‘(9x) &Fx’’ may be invariantly true in all the worlds of a given nonmaximal model structure, but it cannot be invariantly true in the worlds of a maximal model structure because there is an interpretation over the domain which assigns the predicate letter ‘‘F ’’ to the empty set. Hence, there will be a possible world in the maximal model structure in which nothing is F. Parsons shows that quantified modal logic is not committed to any essentialist formula being a thesis (since none are universally valid).19 Of course, the presence of maximal models is not enough to assure the complete identification between logical necessity and logical truth. Universal validity (invariant truth in every possible world of every model structure) does not coincide with the Tarski-semantic conception of logical truth. The existence of maximal models assures that no essentialist formula is universally valid, but it does not assure that every anti-essentialist formula of quantified modal logic is universally valid. If a model structures C is not maximal, then a formula such as ‘‘(9x) &Fx’’ can be invariantly true in every world of the structure C. To achieve the result that logical necessity coincides with the semantic notion of logical truth for a firstorder language we need to turn to Cocchiarella primary semantics for logical necessity where every model structure is maximal. In the primary semantics for logical necessity, anti-essentialism is universally valid. No essentialist statement is true in any world. If we understand Russell’s approach properly – that is, in light of models in which anti-essentialism is universally valid – Cocchiarella reveals that Russell’s account of necessity is vindicated.20 Rescher writes that ‘‘where modal logic was concerned, Russell adopted Lord Nelson’s precedent and stolidly put his telescope to the blind eye.’’21 We now see that, where logical necessity of first-order logic is concerned, it was interpreters of Russell that had the blindness.

19 20 21

A thesis of a system is an axiom or theorem of that system. ‘‘On the Primary and Secondary Semantics of Logical Necessity,’’ p. 18. Nicholas Rescher, ‘‘Russell’s Modal Logic,’’ in George Roberts, ed., Bertrand Russell Memorial Volume (London: Allen & Unwin, 1979), p. 143.

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The primary semantics of logical necessity is central to a proper understanding of Russell’s analysis of the logical necessity of first-order formulas. It is no less central to a proper understanding of Wittgenstein’s Tractarian account of necessity. Like Russell, Wittgenstein held that all necessity is logical necessity. Moreover, he agrees with Russell that logical necessity is logical truth. He differs from Russell on how logical necessity is to be analyzed. In Wittgenstein’s view, logical necessity is tautologyhood. An ungeneralized proposition (formula) can be tautological just as well as a generalized one. Russell knew his account could not be applied to second-order formulas. Wittgenstein hoped that his thesis that all logical truths are tautologous would apply to all formulas, first-order or otherwise. In chapter 4, we offered an interpretation of Wittgenstein on logical necessity that explains his thesis that logical truths are tautologies. The thesis is clearest when applied to sentential logic (where no quantifiers or free variables occur). The representation of sentential forms in terms of Venn diagrams (or equivalently, representations in terms of truth-conditions) nicely captures the Tractarian idea that tautologies are ‘‘scaffolding’’ and have no genuine ‘‘truth-conditions.’’ But what would a sentential modal logic look like if it captured Wittgenstein’s Tractarian idea that logical necessity is tautologyhood? Clearly, an object-language modal operator & is antithetical to the Tractarian proscriptions on an ideal language for science. Accepting such an operator as part of the language is acceptable only if it is semantically interpreted to coincide with the notion of tautologyhood. Any modal system for Tractatus must, therefore, be understood in the context of Cocchiarella’s primary semantics. Von Wright once hoped to characterize the propositional modal theory of the Tractatus as the system he calls ‘‘M.’’ But in this system, necessity certainly cannot be interpreted as tautologyhood. Indeed the system M is deductively equivalent to the modern system T. The system T can be formulated as follows: 1: ‘ A; where A is a tautology: 2: ‘ &ðp  qÞ :  : &p  &q Rule 1 : From ‘ p and ‘ p  q; infer ‘ q Rule 2 : From ‘ p; infer ‘ &p

This system fails to make the proper connections between logical necessity and logical truth (tautologyhood). In T modalities do not collapse to firstdegree. Iterations such as &&p are not equivalent to &p. To capture the Tractatian idea of logical necessity one must be able to demonstrate that the logical modalities as well as the logical particles have a purely ‘‘formal content’’ (so that they are part of logical scaffolding) and are not signs for

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genuine (‘‘material’’) properties of the world. Cocchiarella22 has shown that Wittgenstein’s view of propositional logical necessity is realized in a modal propositional calculus that adds the following to the system T: 3:

‘  &p; where p is modal-free and not tautologous:

In Cocchiarella’s system, every sentence is provably equivalent to a modalfree sentence. In short, the statements &A (A is logically necessary) and }A (A is logically possible) syntactically represent that A is a tautology and that A is not a tautology (respectively). Of course, Wittgenstein had much more in mind than the analysis of logical truth (logical necessity) for sentential logic. He maintained that all (and only) logical truths are tautologous. Wittgenstein’s N-operator notation was intended to reveal the tautologous nature of logical truth in general. In ideal symbolism the contingency, logical possibility, logical necessity, or contradictoriness of A, even where A contains quantifiers and the identity sign, is shown by its syntactic expression. Logical truths of quantification theory are ‘‘generalized tautologies.’’ Wittgenstein’s thesis that all logical truths are generalized tautologies originated with Principia’s section *9. The semantic completeness of a first-order version of the quantificational theory of section *9 (without identity) corroborates the view – but only partly. There is a sense in which the logical truths of first-order logic are generalized tautologies. But second-order logic is not semantically complete. So we cannot say that the logical truths of secondorder quantification theory are generalized tautologies. Moreover, in saying that the logical truths of first-order quantification theory are tautologous, Wittgenstein meant that logic is decidable. We now know that this is false, even for first-order logic. Tractarian combinatorialism Some entries in the Tractatus suggest a combinatorial account of logical possibility. The truth-conditions of a contingent formula are said to depend upon the combination and separation of objects within logical space. Wittgenstein writes: Objects are what is unalterable and subsistent; their configuration is what is changing and unstable. (TLP 2.0271)

22

Nino B. Cocchiarella, ‘‘Logical Atomism and Modal Logic,’’ in Logical Studies in Early Analytic Philosophy (Columbus: Ohio State University Press, 1987), pp. 222–243. The axiom schemata above are taken from Nino B. Cocchiarella, ‘‘Logical Necessity Based on Carnap’s Criterion of Adequacy,’’ Korean Journal of Logic 5.2 (2002): 1–21.

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The configuration of objects produces a Sachverhalt. (TLP 2.0272) In a Sachverhalt objects fit into one another like the links in a chain. (TLP 2.03)

In interpreting such passages Wittgenstein’s thesis that logical truths are tautologous must be kept squarely in mind. The position that logical truths are tautologies is central to the Tractarian Grundgedanke and the Doctrine of Showing. Hence, an interpretation of Tractarian combinatorialism must be measured by its compatibility with the thesis that logical necessity is logical truth (tautologyhood). One must ask how the N-operator and the notion of tautology as scaffolding work together with combinatorialism. In Bradley’s book The Nature of All Being, passages of the Tractatus from 2.011 through 2.0141 are interpreted to advocate de re modal properties that objects have for combination with other objects. Wittgenstein writes: ‘‘If I am to know an object, though I need not know its external properties, I must know all its internal ones’’ (TLP 2.01231). He goes on to say that ‘‘if all objects are given, then at the same time, all possible states of affairs are also given’’ (TLP 2.0124). Bradley concludes from such passages that the ‘‘internal properties’’ of the Tractatus are essential properties had by objects de re. Indeed, Bradley goes so far as to maintain that an object’s capacity for combination does not track every logical possibility. He even allows that some possible worlds may contain more objects and some may contain fewer objects than those of the actual world. Bradley’s interpretation of Tractarian combinatorialism makes a reconciliation with its Grundgedanke impossible. The ‘‘internal properties’’ (and internal relations) are just logical or semantic notions which, at the limit of science, are shown in the ‘‘structured variables’’ of the grammar of the ideal eliminativistic theory. Structure is a fundamentally de dicto notion which pertains to facts and their logical relationships. Wittgenstein’s objects (logical atoms) have no logically essential properties at all. Indeed, the Tractarian Grundgedanke is that there are no logical properties and relations. How then is the combinatorialism embedded in the Tractarian picture theory to be reconciled with Wittgenstein’s account of (logical) necessity as tautologhood? A well-traveled road toward the unification of combinatorialism with the Tarski conception of logical truth is the construction of possible worlds as maximally consistent sets of (atomic) and logically independent states of affairs. All worlds have the same logical atoms, particulars, and material universals. States of affairs are configurations of the atoms. In his pioneering work, Meaning and Necessity, Carnap attempted to avoid ontological commitments to worlds and possible (nonobtaining) states of affairs by constructing worlds in terms of state-descriptions. Wittgenstein certainly

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would reject an ontology of logically possible states of affairs. This would threaten to make logical possibility into a genuine (material) property after all. Carnap’s state-description approach is in certain respects a welcome development toward reconciliation. Unlike modern combinatorial approaches, Carnapian or otherwise, Tractarian combinatorialism relies on results (assumed to be) established in later parts of the Tractatus. In particular, an infinity of logical simples (atoms) was to have been a part of the logical scaffolding that is built into exclusive quantifiers. Infinity is shown by the variables of every statement (of the ideal language for science). To see how Tractarian combinatorialism rests on this result, it is instructive to consider Ramsey’s concern that there is a tension between Tractarian combinatorialism and its analysis of logical necessity as tautologyhood. Ramsey calls in to question Wittgenstein’s comment at TLP 5.5261: ‘‘We can describe the world completely by means of fully generalized propositions, i.e., without first correlating any name with a particular object.’’ Ramsey examines the case where there are only two objects a and b and a single property f. What then shall we say of the fully general proposition that at least three distinct entities possess some property? A combinatorial account of possibility applied to a logical space of only two objects and one property is incapable of capturing the contingency of this fully general proposition. The totality of ‘‘possible worlds’’ (combinatorial possibilities) and the facts in them are these: w0 : w1 : fa w2 : fb w3 : fa; fb

The statement ‘‘fb’’ is true at w1, and ‘‘fa’’ is true at w2. But now consider the statement ð9fÞð9xÞð9yx Þð9zxy Þðfx & fy & fzÞ;

which says that three distinct objects have some property. This statement is false at worlds w0–w3. But it should be contingently false (and so true at some w). Thus its contingency cannot be captured combinatorially (since our logical space has two objects and one property). The answer to Ramsey’s objection lies in Wittgenstein’s treatment of quantification and identity via the N-operator. A ‘‘complete description’’ of, say, w1 would not be given by ð9fÞð9xÞð9yx Þðfx &  fyÞ:

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A complete description of w1 requires us to write: ð9fÞð9xÞð9yx Þðfx &  fy & ð8zxy Þ  fzÞ:23

So expressed, the very sign for the expression shows the possibility of there being a potential infinity of objects. Ramsey’s objection makes an assumption of a logical space with only two objects and one property. But the very statement of Ramsey’s assumption in exclusive quantifiers (and also N-notation) is self-refuting. Thus, Ramsey’s concern cannot be formulated. In a truth-table, the contingency of a compound quantifier-free formula A (and likewise its possibility or necessity) is built into the syntactic expression of every atomic propositional sign. Where quantifier-free formulas are concerned, we saw that this feature is recovered equally by Venn diagrams and by Wittgenstein truth-tabular representation of a formula. The Venn diagrams build }A into the symbolism of A itself because the areas generated by the overlapping regions of the diagram correspond to the rows of a truth-table for A. The Venn diagrams (and ab-Notation) show the contingency, logical possibility, logical necessity, or contradictoriness of a formula in the sign itself. We saw that the N-operator should be viewed as recovering his feature of the Venn diagrams. N-notation expression of ‘‘p’’ is supposed to be the ‘‘same’’ expression as any logical equivalent ‘‘p :&: q  q .&. r  r & . . .’’ Thus the potential infinity of other statement letters is built into the sign of every atomic statement (in N-notation). Now in Wittgenstein’s view, quantified statements, construed as potentially infinite truth-functions generated by the N-operator, also wear their contingency, logical necessity, or contradictoriness on their syntactic sleeves. Consider the simple case of (9x)fx, where f is a predicate letter. When expressed with the N-operator notation, we have NN(fx), which is the same as the more perspicuous NNðfx1 ; . . .; fxn Þ

(with n schematic for an arbitrarily large finite cardinal). The contingency is shown in the symbolism just as readily as the contingency of p v q is shown by a Venn diagram or by its N-operator expression as NN(p, q). What is also shown by the N-notation is the potential infinity of logical atoms. The N-notation exploits the existence of a potential infinity of distinct variables displayed with the N-operator representation of every quantified formula. Each distinct free variable, on Wittgenstein’s

23

Ultimately, the quantified statement is to be expressed in Wittgenstein’s N-notation. But we have avoided it for convenience of exposition.

284

Appendix B

exclusive-quantifier account of identity, is to have a distinct referent. The potential infinity of logical atoms is thereby built into the formal grammar. The same is to hold for more complicated formulas such as Ramsey’s ð9fÞð9xÞð9yx Þð9zxy Þðfx & fy & fzÞ:

This statement has a contingent logical form when expressed in N-notation. In Wittgenstein’s view, the expression of contingency via the N-notation coincides with a combinatorial account because every such contingent form in N-notation (expressing quantification) shows the (potential) infinity of logical simples. In short, the Tractarian combinatorial account of logical necessity is the combinatorialism of the truth-table – extended (as Wittgenstein hoped it could be) to a general higher-order quantification theory. We have seen in chapter 4 that each atomic formula has the ‘‘same’’ expression (via N-notation) as an expression which depicts a potential infinity of distinct variables. For example, (9x)fx, expressed with the N-operator notation, is NN(fx1, . . ., fxn). The different atomic statements fx1, . . ., fxn form a truthtable of 2n many rows. These are all the combinatorial possibilities of a domain of n-many entities having or lacking predicate f. But the logically correct language has a potential infinity of distinct predicate letters as well. In Wittgenstein’s view, identity for attributes is every bit as much a pseudopredicate as an identity predicate between individual variables. Thus, in Wittgenstein’s view, a given atomic statement has the ‘‘same’’ expression as that which expresses its expansion for all the distinct predicate letters. All logical equivalents are to have one and the same expression via the N-notation. The inclusion of a new predicate letter is just another expansion of what is supposed to be contained within each atomic statement sign. All the potentially infinite rows of the truth-table expansion for all the predicates and all the variables of the language are contained within each atomic propositional sign itself – in N-notation. This is what Wittgenstein meant by saying that every atomic statement reflects the whole of logical space.

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Index

acquaintance 28, 60, 63, 72, 76, 182, 249 analytic/synthetic distinction 244–245 Anscombe, G. E. M. 2, 29, 77, 140–143, 183, 274 attributes (propositional functions) 16–21, 150, 154, 171, 204, 215, 219, 266, 268, 269, 271, 284 in extension 219–221 Austin, J. L. 249 Awodey, Steve 240 Ayer, A. J. 1, 15, 16, 19

Davidson, Donald 69 Dedekind, Richard 27, 190, 225 definite descriptions 48–49, 54, 58, 59, 127, 198, 227, 268 denoting concepts 42, 44 Dewey, John 76, 251 Diamond, Cora 3 Doctrine of Showing 77, 79, 80, 82, 83, 90, 94, 95, 96, 100–106, 155, 170, 192, 196, 208, 226, 230, 231, 243, 247, 281 Duhem, Pierre 14, 232, 233

Baker, Gordon 121 Berkeley, George 30, 231 Bernays, Paul 151 Boole, George 25, 81, 119, 148 Bostock, David 176 Bradley, Raymond 266, 275, 281 Brentano, Franz 45 Brouwer, L. E. J. 247 Brown, George Spencer 130 Burke, Tom 76

Edmonds, David, and Eidinson, John 118 Einstein, Albert 28, 39 elimination 14–23, 40, 85, 89, 92, 93, 150, 170, 184, 193, 228, 233, 235, 239–240, 242, 244, 248, 271 extreme/radical 100–106, 230, 243, 245–247, 249–252 of metaphysics 231 elucidation 80 empiricism: logical 1 reductive 28–30, 76 Engelmann, Paul 108, 121 equations 158–159, 170, 185 ethics (value) 94–100 Euler, Leonhard 121 exclusive quantifiers: and finite cardinals 160–169, 226, 282, 283, Appendix A extensionality 78, 100, 192–193, 201, 208–214

calculation: practice of 185–186, 187 Cantor, Georg 27, 141, 190, 211, 213 Carnap, Rudolf 13, 73, 75, 158, 201, 230, 242–247 Carruthers, Peter 78 Church, Alonzo 44, 206, 210 Chwistek, Leon 190 Clarke, Ronald 8 class 16–21, 72, 138, 154, 165, 171, 204, 220 Cocchiarella, Nino 224, 277–280 Coffa, Alberto 245–247 color 87–88, 249 combinatorialism 280–284 Conant, James 1, 2, 3 conventionalism 244–245, 246, 247 Copi, Irving 263 Crabbe, Marcel 206

fact: complex 45, 48–49, 52, 53–65, 69 general 52, 60–64, 155 logical independence of 52, 123 negative 52, 62–64 Favrholdt, David 78, 195 Fogelin, Robert 86, 136–139, 143–144, 167

297

298

Index

formal concepts 84–85, 91, 92, 105, 170 Frascolla, Pasquale xi, 183, 194–196 Frege, Gottlob 1, 2, 3, 18, 79, 81, 100, 147, 149, 156, 160–170, 171, 179, 204, 227, 239–240, 242, 243 fundamental idea (Grundgedanke of the Tractatus) 77, 85, 234, 281 Gardner, Martin 114, 115, 116–118 Geach, P. T. 79, 138–139, 143–144 Giaretta, Pierdaniele 57 Go¨del, Kurt 149, 211 Goldfarb, Warren 44, 46 Grayling, A. C. 118 Griffin, James 229 Griffin, Nicholas xi, 6 Hacker, P. M. S. 2, 7, 91, 104, 118, 121, 248 Hardy, G. H. 158 Hatcher, William 150 Hazen, Allen P., and Davoren, Jen M. 212, 214 Hegel, Georg Wilhelm Frederich 25 Heidegger, Martin 104 Hellman, Geoffrey 149 Hertz, Heinrich 88–89 Hintikka, Jaako 260–261 Hintikka, Merrill B., and Hintikka, Jaakko 80 Hochberg, Herbert 54, 55 Hume, David 30 Huntington, Edward 119 Hylton, Peter 148

Liar Paradox 21 Linsky, Bernard 44 logic: a-b notation for 114–116 as essence of philosophy 27, 102 as language 80–81 connectives/particles x, 124, 143, 149 constants 77, 85 decidability of 112, 115, 117, 143–144, 146 grammar of 8 ideal/perfect langauge of 90–94 linguistic conception of 73–74, see also conventionalism modal 1, 277 necessity/truth 17, 110, 123, 229–230, 251, Appendix B scaffolding of 72 space 99 tautology 73, 74, 155, 158, 185, 187, 204, 219, 244, 253, 279–284, Ch. 4 logical atomism ix, xi, 1, 83, 85, 97, 100, 154, 166, 233, 237, 248, 251, 252, 271, Ch. 2 logical fictions 11–19 logical forms 58–62, 155 logical objects 150 logicism, 203, Ch. 5 deductive assumption of 148

Kant, Immanuel 18, 22, 86, 88–89, 107 congruent counterparts 86–87, 129, 148 Kenny, Anthony 118 Kerry, Benno 79 Kuhn, Thomas 20, 249

Manninen, Tuomas xi, 120 Marion, Mathieu 118, 223 Mauthner, Fritz 92, 105 Maxwell, James Clerk 39, 89 McDonough, Richard 78 McGuinness, Brian 9, 85, 100, 121 McKay, Thomas 277 McTaggart, J. M. E. 194 Meinong, Alexius 45 Monk, Ray 5–10, 73, 189–191 Moore, G. E. 9, 24, 275 Morrell, Ottoline 5, 6, 8, 9, 70, 110 Mueller, Eugen 119 multiple-relation theory of judgment 5, 6, 22, 40, 42, 45, 49, 54–69, 155, 199 Myhill, John 208 mysticism 1, 94, 95–100, 101–103

ladder (metaphor of the) 104, 105, 106, 230, 234 Lambert, J. H. 24 Lampert 89 Laudan, Larry 14 Leibniz, Gottfried Wilhelm 33 Leopardi, Giacomo 98 Lewis, C. I. 30, 106, 272

neutral monism 34–40, 66, 194–196, 250 Newton, Isaac 12–13 Nicod, Jean 10, 108 Noonan, Harold 165 N-operator 127–146, 167, 172, 173, 174–176, 178, 184, 186, 187, 226, 280, 281, 283–284 Norton, Bryan 233

identity 134–136, 159–170, 220–225, 253, 284 infinity 165, 166–168, 271, 282 James, William 35, 65, 195

Index operation 127–128 and powers of a relation 180, 184, 185 and recursion 179–182 numbers as exponents of 171–178, 226 practice 130 successive 139–143 oracle 103 Orilia, Francesco xi ouroboros 103, 230, 251 paradox: Epimenides Liar 42, 46, 47 of attributes 17 of classes 16–21 of propositions 21 po/ao 47 semantic vs. syntactic 214 Parsons, Terence 277–278 particulars 32 Pears, David 2, 29, 34, 95 Peirce, C. S. 25 picture theory 77, 88–89, 94, 100, 101, 249 Pinsent, David 9, 10, 107 Poincare,´ Henri 42, 247 Popper, Karl 1 Potter, Michael 165, 166, 176–177, 206 pragmatism 246–247 Principia Mathematica: axiom of infinity 158, 165–168, 222, 272 Reducibility, axiom of 10–11, 110, 117, 151, 152, 155, 182, 189–191, 201–214, 215–219, 272 rewriting 9–10, 107 section *9 51, 114, 116–118, 125, 126, 151, 204, 280 prisons 96–98 propositions 33–34, 40, 44–47 as states of affairs 34, 40, 267, 268 implication as a relation between 34, 40, 268 unity of 45 Quine, Willard van Orman 53, 76, 183, 215, 225, 230, 237, 241–245 radical skepticism 106 Ramsey, Frank 65, 87, 104, 117, 135, 141, 146, 184, 185–186, 188, 189–192, 214–226, 227, 230, 256, 260–263, 282–284 Reck, Erich 240 recursive definition of truth x, 40, 42, 44, 45, 48, 49–52, 59, 63, 76, 155, 157, 203 reduction 12–19, 228

299 Rescher, Nicholas 278 Ricketts, Tom 5 riddles ix, 106 ´ ˆ me 129, 156 Sackur, Jero Salmon, Wesley 248 Scharle, Thomas 110 Schmitt, Richard 142 Schopenhauer, Arthur 1, 2, 96 sense-data ix, 11, 15, 28–40, 76, 195 Sheffer, Henry 9, 10, 108, 128, 130, 133, 143 Shosky, John 120 Showing see Doctrine of Showing Shro¨der, Ernst 25 sinnlos/unsinnig 103–104, 234 Smullyan, Raymond ix Sommerville, S. 6 soul 193–196 Spinoza, Baruch 35, 96, 97, 98, 99 Stern, David 2, 91 Stout, G. F. 55 structural realism ix substitutional theory x, 1, 7, 12, 19, 46–47, 82–84, 93, 154, 240–241, 267–271 Tarski, Alfred 21, 48, 49, 52, 148, 149, 275, 276 therapeutic reading of Tractatus 78, 104–105 time (space-time) 28, 35, 36, 38, 229 truth-functionality 77, 192, 201 truth-table xi, 1, 118, 121, 124, 126–127, 128, 283, 284 Tully, R. E. 35, 36 types: order\types 42, 53, 57, 72, 82, 93, 150–153, 204, 273, 277 orders 42, 44, 50 predicative 153 ramification x, 7, 11, 72, 110, 150, 152–154 simple x, 6–8, 12, 42–43, 105, 149, 152, 215, 241–243, 267, 269–270 type* 57–58, 59, 68 universals 53, 60, 65–72, 155, 199, 223, 224, 230, 271 Urmson, J. O. 32 van Heijenoort, Jean 80, 81, 148 variable: doctrine of the unrestricted 19, 21, 154 genuine 52, 79, 82, 83, 152, 154, 271 internally limited 83, 84, 93, 157 many-sorted 235, 241–243, 244

300

Index

variable: (cont.) predicative 150 propositional 84 structured ix, 20, 22, 23, 80, 82, 83, 84, 85, 89, 90, 91, 92, 94, 100, 105, 230, 233, 235, 238, 281 Venn, John 121, 123, 125, 126, 129, 132, 279, 283 Vicious Circle Principle 42, 52, 62, 182 von Ficker, Ludwig 94, 100 von Wright, G. H. 65, 187, 279

Wahl, Russell xi Wang, Hao 241, 242 Wehmeier, Kai 265 Weierstrass, Karl 27, 35, 39, 247 Westfall, Richard 13 Whitehead, Alfred North 189 Williford, Kenneth xi Wisdom, John 212 Zeno 35, 36, 147, 248