Repairing Bertrand Russell’s 1913 Theory of Knowledge (History of Analytic Philosophy) 3030663558, 9783030663551

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HISTORY OF ANALYTIC PHILOSOPHY

Repairing Bertrand Russell’s 1913 Theory of Knowledge Gregory Landini

History of Analytic Philosophy Series Editor Michael Beaney Humboldt University Berlin King’s College London Berlin, Germany

Series Editor: Michael Beaney, Professor für Geschichte der analytischen Philosophie, Institut für Philosophie, Humboldt-Universität zu Berlin, Germany, and Regius Professor of Logic, School of Divinity, History and Philosophy, University of Aberdeen, Scotland. Editorial Board Claudio de Almeida, Pontifical Catholic University at Porto Alegre, Brazil Maria Baghramian, University College Dublin, Ireland Thomas Baldwin, University of York, England Stewart Candlish, University of Western Australia Chen Bo, Peking University, China Jonathan Dancy, University of Reading, England José Ferreirós, University of Seville, Spain Michael Friedman, Stanford University, USA Gottfried Gabriel, University of Jena, Germany Juliet Floyd, Boston University, USA Hanjo Glock, University of Zurich, Switzerland Nicholas Griffin, McMaster University, Canada Leila Haaparanta, University of Tampere, Finland Peter Hylton, University of Illinois, USA Jiang Yi, Beijing Normal University, China Javier Legris, National Academy of Sciences of Buenos Aires, Argentina Cheryl Misak, University of Toronto, Canada Nenad Miscevic, University of Maribor, Slovenia, and Central European University, Budapest Volker Peckhaus, University of Paderborn, Germany Eva Picardi, University of Bologna, Italy Erich Reck, University of California at Riverside, USA Peter Simons, Trinity College, Dublin Thomas Uebel, University of Manchester, England. More information about this series at http://www.palgrave.com/gp/series/14867

Gregory Landini

Repairing Bertrand Russell’s 1913 Theory of Knowledge

Gregory Landini Department of Philosophy University of Iowa Iowa City, IA, USA

ISSN 2634-5994     ISSN 2634-6001 (electronic) History of Analytic Philosophy ISBN 978-3-030-66355-1    ISBN 978-3-030-66356-8 (eBook) https://doi.org/10.1007/978-3-030-66356-8 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover image © Bertrand Russell by Hugh Cecil, vintage bromide print, mid-late 1910s This Palgrave Macmillan imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

The book is dedicated to Francesco Orilia for his enduring friendship and his important research on the nature of indexicals, time, relations, and facts.

Series Editor’s Foreword

During the first half of the twentieth century, analytic philosophy gradually established itself as the dominant tradition in the English-speaking world, and over the last few decades, it has taken firm root in many other parts of the world. There has been increasing debate over just what ’analytic philosophy’ means, as the movement has ramified into the complex tradition that we know today, but the influence of the concerns, ideas and methods of early analytic philosophy on contemporary thought is indisputable. All this has led to greater self-consciousness among analytic philosophers about the nature and origins of their tradition, and scholarly interest in its historical development and philosophical foundations has blossomed in recent years, with the result that history of analytic philosophy is now recognized as a major field of philosophy in its own right. The main aim of the series in which the present book appears, the first series of its kind, is to create a venue for work on the history of analytic philosophy, consolidating the area as a major field of philosophy and promoting further research and debate. The ’history of analytic philosophy’ is understood broadly as covering the period from the last three decades of the nineteenth century to the start of the twenty-first century, beginning with the work of Frege, Russell, Moore and Wittgenstein, who are generally regarded as its main founders, and the influences upon them, and going right up to the most recent developments. In allowing the ’history’ to extend to the present, the aim is to encourage engagement with contemporary debates in philosophy, for example, in showing how the concerns of early analytic philosophy relate to current concerns. In focusing on analytic philosophy, the aim is not to exclude comparisons with vii

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other-earlier or contemporary-traditions or consideration of figures or themes that some might regard as marginal to the analytic tradition but which also throw light on analytic philosophy. Indeed, a further aim of the series is to deepen our understanding of the broader context in which analytic philosophy developed, by looking, for example, at the roots of analytic philosophy in neo-Kantianism or British idealism, or the connections between analytic philosophy and phenomenology, or discussing the work of philosophers who were important in the development of analytic philosophy but who are now often forgotten. Bertrand Russell (1872-1970) is undoubtedly one of the central figures in analytic philosophy-arguably, the central figure in its early developmentas evidenced by the eight volumes on his work that have already appeared in this series. One of the main origins of analytic philosophy lies in Russell’s and Moore’s rebellion against British idealism at the turn of the twentieth century. A second major source is Frege’s invention and application of quantificational logic in the last two decades of the nineteenth century, upon which Russell built, beginning with The Principles of Mathematics (1903) and culminating in Principia Mathematica (1910-13), co-authored with A.  N. Whitehead (1861-1947). Russell’s work then increasingly addressed epistemological issues and in the last phase of his life focused more and more on social and political affairs. As far as analytic philosophy is concerned, his most productive and influential period was between 1900 and 1930, but even in this period there were many changes in his thinking, and one of the most difficult and controversial tasks in the history of analytic philosophy has been to make sense of these changes. In the present book, Gregory Landini identifies three main phases in Russell’s thinking after his analytic turn, which he calls the Principles era (1900-08), the Principia era (1910-17) and the Neutral Monist era (1918 onwards). He focuses here on the Principia era, which begins with the publication of the first volume of Principia and includes The Problems of Philosophy (1912), his ’Theory of Knowledge’ manuscript (1913), and Our Knowledge of the External World as a Field for Scientific Method in Philosophy (1914). The full title of this last book is significant for Landini’s project. For what characterizes the Principia era, he argues, is a certain philosophical agenda: to repudiate abstract particulars and the kinds of (metaphysical) necessity that they are thought to require. Repudiating these is seen by both Russell and Landini as liberating us, negatively, from the prisons of dogmatisms about necessity and, positively, by revealing hitherto unknown

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structural possibilities. Central to Russell’s 1914 book is the conception of logic as the essence of philosophy, where ’logic’ is here understood as the synthetic a priori study of relational structures (which Landini calls ’cpLogic’-’comprehension principle logic’). Given his ’acquaintance epistemology’ (common to both his Principles and Principia eras), the key philosophical task then becomes to explain our knowledge of logic. This task is what Russell undertook in those sections of his ’Theory of Knowledge’ manuscript that develop the multiple relation theory of judgement by adding structureless logical forms as objects of acquaintance. That manuscript was abandoned, it has often been thought, because of objections that Wittgenstein raised. Landini revisits this theory and suggests that it can be repaired once we appreciate what his philosophical agenda really was at the time. Landini has made significant contributions over the last three decades to the history of analytic philosophy. He published books Russell’s Hidden Substitutional Theory in 1998 and Wittgenstein’s Apprenticeship with Russell in 2007 and his monograph on Frege’s Notations appeared in this series in 2012. In 2018 a collection of papers marking the centenary of Russell’s lectures on the philosophy of logical atomism, co-edited with Landon Elkind, was also published in this series. Landini’s work has challenged many of the standard accounts of the history of analytic philosophy, and he has deepened his revisionary reading in the present book. He argues, for example, that Wittgenstein’s objection to Russell’s multiple relation theory of judgement, according to which it cannot assure that logic rules out judging nonsense, was not the reason for Russell’s abandoning the acquaintance epistemology of his 1913 manuscript. What was problematic, which Wittgenstein brought out to him, was the tension between Russell’s appeal to logical forms and his philosophical programme of repudiating abstract particulars. Landini’s diagnosis of this is insightful, and in exploring repairs to the multiple relation theory in detail, he also offers a critique of views held by other scholars-in his denying, for example, that problems of representation are relevant to the multiple relation theory. This book will certainly reset the agenda for the next phase in the history of analytic philosophy. November 2020 

Michael Beaney

Preface

Russell’s original scientific method in philosophy had the agenda of undermining the arguments of the metaphysicians for the indispensability of abstract particulars and kinds of necessity in the sciences. Philosophy so conceived offers the critical tool that exposes the dogmas parading as necessities that create prisons that fetter the mind. The research program grew out of the methods of Principia Mathematica, whose uniquely non-­ Fregean logicism aimed to free all the branches of mathematics from the metaphysician’s abstract particulars (numbers, sets/classes, spatial figures) and specialized notions necessity (arithmetic, geometric) governing them. In his book Scientific Method in Philosophy (aka: Our Knowledge of the External World as a Field for Scientific Method in Philosophy), Russell says that logic is the essence of philosophy. I hold that he had in mind the cpLogic (impredicative comprehension principle logic) of Principia. Impredicative comprehension assures the existence of properties and relations and thus makes cpLogic a unique synthetic a priori science. It is this that affords philosophy its distinctive value in enabling one to expose all non-logical necessities as frauds. Principia’s logic together with new empirical results in the sciences enables the genuinely scientific method in philosophy that holds promise for solving all philosophical problems. My book endeavors to repair and revive the research program for scientific philosophy of what I call Russell’s Principia Era (1910–1917). It includes the books: Principia Mathematica (vol 1, 1910; vol. 2, 1912, vol. 3, 1913; vol. 4, 1914 (unfinished)), The Problems of Philosophy (1912), Theory of Knowledge (1913), and Scientific Method in Philosophy (1914). xi

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There is a hybrid acquaintance epistemology in Problems. We find a coherentist justification for empirical (and non-phenomenal statements), and a foundationalist justification for synthetic a priori knowledge. Acquaintance with universals and particulars, and a multiple-relation theory of belief (judgment), is the foundation upon which it rests. The acquaintance epistemology reached an impasse in the summer of 1913 and was put into limbo. The later parts of Theory of Knowledge deconstructing notions of “necessity” in the empirical sciences were put into the book Scientific Method in Philosophy, which became the sequel to Problems. The Principia era ended when Russell adopted the physicalism of his unique brand of neutral monism which abandons acquaintance epistemology and the multiple-relation theory of judgment (mrtj). The research program of scientific philosophy continued, but in neutral monism physical laws of material continuants persisting in time are rewritten in terms of events involving contingent transient physical particulars some of which are among those series of events that constitute the perceptual states of one or another perceiver. Perceivers themselves are series of events composed of physical transient particulars realizing the new empirical behaviorist psychological laws. Unfortunately, Russell’s neutral monism never found any viable epistemology for the mathematical logic essential to his original scientific philosophy. Reviving Russell’s Principia era requires more than repairing its acquaintance epistemology. One must become reacquainted with Principia’s logicist thesis which embraced the revolution within mathematics that presents it as a science of relational structures studied by investigating relations independently of the contingencies of their exemplification. One must understand that Principia rejects the post-­ modern view that a “logic” consists in rules of inference designed to capture the wffs that come out invariant with respect to a chosen definition of “validity.” One must explain how it is that Principia can accept so-called rival logics (such as modal, intuitionistic, and relevant entailment), while offering a genuinely universal logic. My earlier work toward understanding the connection between Principia and the mrtj was published in 1991 as “A New Interpretation of Russell’s Multiple-Relation Theory of Judgment.” My focus was then on truth-makers and how the mrtj of Theory of Knowledge fits with the recursive definition of “truth” offered in Principia’s informal introduction. The logical particles are not relation signs and thus have no analogs occurring in any truth-maker. No conjunctive, disjunctive, negative, or general facts

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are truth-makers. The fundamental thesis of the Principia era is that when it comes to facts as truth-makers, and when the facts in question do not contain other facts as constituents, the existence of a fact does not entail anything about the existence (or non-existence) of any other. Every interpretation of Theory of Knowledge must be couched within a background interpretation of Principia. Admittedly, disputes between interpretations of Principia may forever remain unsettled. There is controversy at every turn. Ignoring Whitehead’s Realist semantics in the introduction to vol. 2 and informed only by the nominalistic semantic efforts of Russell’s introduction to Principia’s vol. 1, Church endeavored to “improve” the work, coding ramification into syntax. Church’s interpretation remains influential, and yet he rejected the mrtj as disconsonant with the logic and philosophy of Principia. In contrast, I hold that the syntax of Principia is that of simple-type theory. This continues the efforts at revising the history set out in my books Russell’s Hidden Substitutional Theory (1998) and Wittgenstein’s Apprenticeship with Russell (2007). My colleagues, Richard Fumerton and Ali Hasan, were very encouraging and ever ready to critically discuss many aspects of acquaintance epistemology. They convinced me that an internalist epistemology based on acquaintance with universals is still viable and important in spite of the sea of epistemological externalists and evolutionary reconstructivists that populate the field of epistemology today. Russell’s work doesn’t much engage with debates in epistemology between internalist foundationalism as opposed to externalist reliabilism. I will not endeavor to solve the Sellarsian dilemma without appeal, as internalists often do, to acquaintance with abstract particular meaning propositions. My rehabilitation of Russell’s acquaintance epistemology focuses on avoiding abstract particulars (e.g., logical forms, meaning propositions) in the mrtj. In writing this book, I owe much gratitude to Nino Cocchiarella. He established that there is one and only one way that Church’s Realist interpretation of Principia as a theory of ramified types of universals can meld with the mrtj. One must allow acquaintance with r-types of universals and accept Theory of Knowledge as a failed endeavor. Bernie Linsky’s devotion to Church’s r-types was a catalyzing foil. I benefited also from Kenneth Blackwell concerning the ties between Theory of Knowledge and Scientific Method in Philosophy. Nicholas Griffin and Russell Wahl also called for Principia to be a fixed point for all interpretations, revealing why Theory of Knowledge cannot facilely appeal to universals to solve its problems of

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direction and compositionality. Griffin, who was first to canonically articulate the problems themselves, noticed that Wittgenstein’s objection to the mrtj—that nonsense be excluded without further premises—perplexingly transforms a contingent psychological issue into an issue for logic to address. This led me to construe his objection as an advocacy of showing. The book would have been markedly different if not for Katarina Perović, who in posing questions about the vexing maps of Theory of Knowledge, rediscovered Russell’s separation of a permutative relation’s adicity from the number of position relations it determines. Thanks to Gülberk Koç Maclean for criticisms of my pivotal chapter on Acquaintance. Milan Soutor, a Fulbright Fellow studying at the University of Iowa in 2014, was a catalyzing agent studying unity, judgment, and truth primitivism. Jolen Galaugher encouraged the project while on a 2013 SSHRC postdoc at the University of Iowa to study Russell’s substitutional theory. Frann, Austin, and Ansel are without which not. Iowa City, IA

Gregory Landini

Contents

1 Introduction and Overview  1 2 What Is Logic? 77 3 Facts About Principia117 4 Acquaintance177 5 Direction239 6 Compositionality279 7 Scientific Philosophy’s Necessity325 Bibliography377 Index393

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CHAPTER 1

Introduction and Overview

Bertrand Russell’s Scientific Method in Philosophy advanced a research program whose agenda is to free every science from a metaphysics of abstract particulars and kinds of necessity. It offers an unparalleled revolution in philosophy, which remakes the field as a critical study of all the kinds of necessity (possibility) that there are. Russell put it as follows (SMP, p. 111): Philosophy is the science of the possible.

First heralded in the 1911 paper “Analytic Realism,” Russell hoped to expound the program in great detail in a 1913 book he tentatively called Theory of Knowledge. Scientific philosophy is intimately connected to the unique Whitehead-Russell logicism whose agenda was to free the branches of mathematics from abstract particulars (numbers, spatial figures) and kinds of necessity (arithmetic, geometric). The connection is implicit in the following (AR, p. 139): The true method, in philosophy as in science … has inspired analytic realism, and it is the only method, if I am not mistaken, with which philosophy will succeed in obtaining results as solid as those obtained in science.

The research program was committed to the acquaintance epistemology central to the account of synthetic a priori knowledge that Russell gave in © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8_1

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his 1911 book The Problems of Philosophy.1 The program was clearly incipient in Problems, but the multiple-relation theory of belief central to its acquaintance epistemology stalled and its epistemological parts were put into limbo. The debut of the research program, without its epistemology, finally arrived in the 1914 book Scientific Method in Philosophy.2 In the chapter “Logic as the Essence of Philosophy,” we find (OKEW, p. 33): Every philosophical problem, when it is subjected to the necessary analysis and purification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical.

Only logic enables the distinctive kind of philosophical criticism and analysis that has the great value of freeing the mind from the prisons imposed by misguided dogmatic conceptions of necessity. Russell’s inspiration for Analytic Realism originates from the logicism first set out in his 1903 book The Principles of Mathematics. In collaboration with Alfred N. Whitehead the book was originally to have a second volume. Instead, their work led to Principia Mathematica. The essence of Russell’s philosophy can only have been the logic of Principia. This presents logic as the synthetic a priori science.3 It is synthetic because it assures the necessary existence of properties and relations. (For this reason I shall call it cpLogic—i.e., comprehension principle logic.) It is a priori because it studies relational structures by studying the way relations order their fields independently of the contingencies of their exemplification. Principia’s section *12 offered impredicative comprehension axiom schemas:

12.1  f   f ! x  x x 

12.11  f   f ! xy  x , y xy  ,



where f! is not free in the wff φ. These are impredicative in virtue of there being no restrictions concerning bound predicate variables in φ.4 It is precisely impredicative comprehension that transforms logic into an informative science well out of reach of combinatorial algebra of fixed operations. The Whitehead-Russell logicism combined two independent revolutions: one within mathematics; and another within logic. The revolution

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in mathematics takes no stand on cpLogic. It originated with Cantor, Weierstrass, von Staudt, Pieri, and the non-Euclidean geometers who made it viable to hold that relations of order are the subject of all branches of mathematics. The revolution in logic originated with Gottlob Frege’s 1879 Begriffsschrift, where impredicative comprehension (of functions) made its debut. Frege did not employ schemas, and he worked in a language of levels of mathematical function terms, not a language of simple types of bindable predicate variables. He achieved the comprehension of functions by accepting a rule for the uniform substitution of complex function terms in axioms and theorems. f



x    fx  gx.



     fxy  gxy.

f

x

y

Principia embraces Frege’s revolutionary discovery of impredicative comprehension in logic, but rejects functions for an ontology of many-one relations. In either case, it is precisely by means of impredicative comprehension that the theorem of induction governing the ‘ancestral’ relation is captured within cpLogic. Induction requires no uniquely arithmetic intuitions governing numbers as abstract particulars. Logicism is an ontological thesis, not an epistemic thesis. It was committed neither to the axiomatization of logical truth nor of mathematical (or arithmetic) truth. It is unfortunate that logicism is so often misconstrued as an epistemic program that requires a consistent recursive axiomatization. Both Frege and Russell knew they hadn’t found such an axiomatization. It is also unfortunate that Frege’s logicism is so often taken as if it were the paradigm of logicism. Frege rejected the revolution in mathematics, presenting numbers in his Grundgesetze (1893) as logical abstract particulars (objects in his technical sense). The Whitehead-­Russell logicism rejected Frege’s logicism aiming to be the flagship of the revolution. Whitehead and Russell became wedded to the revolution within mathematics against abstract particulars after attending a 1900 Congress in Paris where Peano and the Italian school of mathematicians were presenting work. In his 1901 “Mathematics and the Metaphysicians,”5 Russell reports on the revelation he had. He explains (MM, p. 75).

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One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, … All pure mathematics—Arithmetic, Analysis, and Geometry—is built by combinations of the primitive ideas of logic.

The triumph is the discovery that mathematics is not a study of abstract particulars. The revolutionary mathematicians themselves have revealed the true nature of their field. Their subject matter is relational structures and thus their field studies relations. Whitehead-Russell logicism aims to reflect the discovery by mathematicians that all along they themselves were studying relational structures. Principia’s logicist thesis is simply that revolutionary mathematicians are doing cpLogic (i.e., studying relational structures) when they do mathematics. It imposes no change to genuine (revolutionary) mathematical scientific practice. It reflects that practice. Russell did not think of his logicism as inventing the revolution within mathematics, but rather as embracing it. In Russell’s view, Cantor was a pioneer. He revealed that relational order has replaced the notion of quantity in mathematics. The following passage is illuminating (MM, p. 92). The solution of the problems of infinity has enabled Cantor to solve also the problems of continuity. …The notion of continuity depends on that of order, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. … All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. …This improvement also is due to Cantor, and it is one which has revolutionized mathematics.

Russell has in mind cpLogic understood as the study of all the kinds of relational structures there are—a study conducted independently of the contingencies of whether the relations in question are exemplified; and one that embraces the impredicative comprehension of ever new relations (and properties). Later in The Principles of Mathematics, he would write (PoM, p. 429): …when Logic is extended, as it should be, so as to include the general theory of relations, there are, I believe, no primitive ideas in mathematics except such as belong to the domain of Logic.

And also we find (PoM, p. 419):

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Quantity, in fact, though philosophers appear still to regard it as very essential to mathematics, does not occur in pure mathematics, and does occur in many cases not at present amenable to mathematical treatment. The notion which does occupy the place traditionally assigned to quantity is order; and this notion, we saw, is present in both kinds of non-quantitative Geometry [projective and descriptive]. But the purity of the notion of order has been much obscured by the belief that all order depends on distance—a belief which … we have seen to be false.

Whitehead-Russell Logicism rejects the metaphysician’s thesis that intuitions of abstract particulars are required in mathematics. Even Kant, whose Transcendental Aesthetic hoped to end metaphysical speculation, fell into the error. Geometry had already abandoned that view when it accepted non-Euclidean non-metrical projective and descriptive spaces. Cantor undermined the view in the field of number. Principia boldly rejects the thesis that the Dedekind-Peano Postulates, which were conjured by intuitions of natural numbers as abstract particulars, have absolute authority. A similar orientation to geometry challenged Euclid’s parallel postulate. Principia’s volumes appeared in 1910, 1912, and 1913. Whitehead hoped to complete its fourth volume on geometry in 1914. The First World War interrupted his work, and so also did Einstein’s theories of Relativity which, in tying acceleration, mass, and time to the geometry of space, seemed to challenge applications in physics of the mathematical theory of the measurement of magnitudes. Whitehead came up with his own physics of universal relatedness, but he ever completed volume 4. His work notes were dutifully burned at his direction. The study of Whitehead’s physics and its relation to mathematical geometry remains a program for further research.6 Russell never completed Theory of Knowledge. His work notes on the multiple-relation theory and acquaintance were published in 1984 as Theory of Knowledge: The 1913 manuscript. His work has been thought to be a lost cause. We shall see that this is not so. Russell’s original scientific method in philosophy, replete with its logicism and its acquaintance epistemology, is viable. It remains important today, for it exposes the pseudo-problems involved in metaphysical necessities governing abstract particulars. At the same time, it facilitates the proper study of so-called rival logics and invariance notions of de re necessity just as readily as it facilitated the study of non-Euclidean geometry. It is my thesis that Principia, Problems, Theory of Knowledge (ms), and Scientific Method in

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Philosophy all fit together in a unique Principia era.7 The chapters ahead open the way to a revival of its original scientific method in philosophy.

The Principia Era I divide Russell’s work into three main eras: the Principles era (1900–1908), the Principia era (1910–1917), and the Neutral Monist era (1918–). The transition between the eras is a topic of ongoing research in the field of Russell studies. I hold that the transition from the Principles era of propositions to the no-propositions Principia era centers on one and only one problem—the unique family of diagonal (non-semantic) paradoxes of propositions that I call Russell’s “po/ao paradoxes.” Likewise, I hold that the transition from the Principia era to the neutral monist era centers as well on one and only one problem—the elimination of abstract particulars from Russell’s acquaintance epistemology. Russell’s agenda against abstract particulars in science (exempting the science of logic) predates Principia. The role of logic does as well. In a letter to Lucy Donnelly of 8 March 1907, Russell wrote: I think a philosophy course without logic is an absurdity—you might as well have a medical course without physiology. All philosophy is based on logic consciously or unconsciously; and it seems to me one of the chief purposes of a philosophical education is to make people conscious of their logic and of how it affects their general views.8

Impredicative comprehension is essential to all of Russell’s eras of philosophy, though he hoped it may be emulated. Russell’s logic is not a study of a deductive consequence relation between wffs (under an invariance conception of validity). In the Principles era, the Principia era, and I dare say, even in the neutral monist era, logic is viewed in the very same way.9 It is the synthetic a priori study of relational structures, studied by investigating relations independently of the contingencies of their exemplification. The misguided assumption still widespread in metaphysics today is that there must be essentialisms of abstract particulars and that both “logic,” in any conception, and mathematics are up to their neck in them.10 With the modern development of many new so-called rival logics (modal, relevant, paraconsistent, etc.), one might worry that no logic can enjoy a privileged ontological status above the civil wars of the

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metaphysicians.11 Williamson expresses the concern that logic fares no better than rival schools of metaphysics and therefore has no pride of place (Williamson 2013, p. 147): …readers may prefer to use the word “logic” differently. … but whatever advantages may accrue to their way of using “logic,” they will not include isolating some claims that are in principle metaphysically uncontroversial. There are none.

Nota Bene: By “logic,” the post-modern philosophers assume that there is a deductive consequence relation of ‘implication’ between abstract particulars (if only statements). This popular post-modern notion makes “logic” a set of rules (or even processes) of transition governed by shapes (or syntax) that preserve valid inference, where “valid” is semantically defined as the invariance of a selected value in all admissible interpretations over any (non-empty) domain. The post-modern conception of “logic,” buttressed by Tarski’s success in formal semantics, construes logical necessity as logical truth and presents both as invariant truths in virtue of their structure. This ignores the existence of cpLogic altogether. Comprehension introduces truths about relations that introduce structure. Its instances are not true in virtue of their own structure. Russell’s conception of logic does not make it a metaphysics of abstract particulars and thus cpLogic does not warrant Williamson’s criticism. It is entirely above the metaphysical fray over ontologies of abstract particulars. Characteristic of the Principia era is the thesis that abstract particulars are not the subject matter of any science, be it empirical science or the science of mathematical logic. It is precisely cpLogic, together with new empirical results in science, that undermines indispensability arguments for abstract particulars and kinds of metaphysician’s necessity. It is for this reason that Russell says, both in 1914 and in 1924, that logic is what is fundamental in philosophy. In 1924 he wrote (LA, p. 323): I hold that logic is what is fundamental in philosophy, and that schools should be characterized rather by their logic than by their metaphysics.

Indispensability arguments for abstract particulars and kinds of necessity are due to an impoverished logic. I draw the following lesson: Complexity of logical form is inversely proportional to ontological baroqueness.

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There is an inverse relationship between complexity of logical form and speculative metaphysical theories of abstract particulars with their distinctive kinds of necessity. If one’s logical forms are impoverished then one will readily fall prey to indispensability arguments in favor of kinds of necessity governing ontologically baroque theories of abstract particulars (e.g., numbers, triangles, lines, points, sets, mereological sums, possible worlds, propositions, meaning propositions, golden mountains, and even gods). Without the kinds of structures that become known to us a priori through cpLogic, we are susceptible to the confusions that lead metaphysicians to embrace abstract particulars. The Whitehead-Russell logicism and its cpLogic does have pride of place in philosophy programs. In his neutral monist era, Russell wrote books which nicely match up as replacements for the books that belong to his Principia era. The Problems of Philosophy is replaced by An Outline of Philosophy (1927). The unfinished parts of Theory of Knowledge on acquaintance and the multiple-relation theory were replaced by Russell’s behaviorism-sympathetic book The Analysis of Mind (1921). The relation of ‘acquaintance’ had been the foundation of the epistemology of both the Principia era and also the earlier Principles era. In these eras, but not in the neutral monist era, Brentano’s Principle of Intentionality is accepted and captured as the thesis that facts constituted by a relation of ‘acquaintance’ between a subject and an object are generated by Minds engaged in acts of selective attention to a this (be it universal or particular). The relation of ‘acquaintance’ (and its converse, ‘presentation’) embodies selective attention. The same selective attention occurs in sensation, when a fact constituted by a relation of ‘sensation’ is generated between a subject and a particular sense-datum. The relation of ‘acquaintance,’ distinctive of both the Principles era and the Principia era, is abandoned in the neutral monism era which tries to replace ‘acquaintance’ and ‘knowing-that’ with behavorist-inspired events of noticing (reacting appropriately to a stimulus) and knowing-how. In the 1929 edition of Scientific Method in Philosophy (aka: Our Knowledge of the External World as a Field for Scientific Method in Philosophy), Russell simplified the title to Our Knowledge of the External World. With the relation of ‘sensation’ abandoned, he tried to retrofit the construction into his neutral monism. It is best to consider The Analysis of Matter (1927) as the proper neutral monist replacement for the work. The title on the outer board of the 1914 first edition was simply “Scientific Method in Philosophy.” That is the title I’ll use. It reminds us that the book belongs to the Principia era and focuses its efforts not narrowly on epistemology, but on

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illustrating the methods of analytic realism. The 1918 Lectures on Logical Atomism are transitional and belong to no stable era at all. They witness Russell in flux—in between his Principia era and his neutral monist era. Russell was engaging with Wittgenstein’s 1913 Notes on Logic. (Ludwig Wittgenstein had been Russell’s student at Cambridge since 1911 and came to be thought of as a protégé.) The 1924 “Logical Atomism” is its neutral monist replacement. In the Principia era, by stipulative definition, “universal” (concept, property, relation) is used to refer to those distinctive entities postulated to exist that have the unique and indefinable capacity to unify a fact. (Every fact is unified by exactly one universal.) Likewise by stipulative definition a “particular” (among which are facts) is an entity that lacks the capacity to unify anything. Whosoever does not countenance such entities may offer their own theory, but they aren’t speaking about universals, particulars, and facts in Russell’s sense at all.12 Mind you, Russell’s notion of the capacity to unify a fact is not the notion of grounding the unity of a complex (fact), where the language of “grounding” means to suggest that the metaphysical default state is one in which there are no complex unified entities (such as facts). The assumption of such a metaphysical default leads advocates to feel that the story of so-called grounding is what explains the existence of the unified fact, since the metaphysical default is for there to be no such fact. (See MacBride 2011). Russellians regard that default assumption, and the need for such a “grounding,” as committing the fallacy of complex question. The complex fact that 0 < 1 involves the relation ‘less-than’ whose default metaphysical state is one of exemplification. It requires no metaphysical grounding at all. One should not be beguiled by Bradley’s regress arguments which, perhaps, commit the same fallacy by assuming that the metaphysical default state is that there are no complex entities.13 (See Perović 2016a). There are, of course, cases where one might say that a complex entity does come about contingently and thereby demands explanation—say in the case of the fact of Socrates’s loving Xanthippe. But that fact comes about because of the intentionality of Socrates which brings the relation ‘loves’ to hold between them. What brings about such a fact is not what unifies it. Let us call those universals robustly Russellian that exist independently of their exemplification and that have a two-fold nature (occurring in a way that unifies or occurring without unifying). Principia’s impredicative comprehension principle is synthetic and known a priori. It secures the logically necessary existence of robustly Russellian universals.

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Comprehension of a universal, as we shall see, does not assure that we are acquainted with it. But obviously with ‘acquaintance’ accepted as a primitive dyadic relation, acquaintance with a universal can occur only with a robustly Russellian universal. Perhaps there are universals that can only inhere in a fact in a way that grounds its unity and which nevertheless exist independently of being exemplified. That would be very strange, for then such a universal cannot occur in a fact of self-identity. Russell does not embrace such universals in his Principia era, but entertains such universals in his neutral monist era. I’ll call them non-robustly Russellian. Principia’s comprehension principles do not assure the existence of universals postulated by theories in physics such as the magnitudes ‘space-­ time,’ ‘mass,’ ‘acceleration,’ ‘electric,’ ‘magnetic,’ and so forth. They are known to exist only a posteriori. This is perfectly compatible with Principia’s assurance a priori of the existence of all sorts of geometric properties of mathematical space. Mathematical logic, including its study of various non-Euclidean geometries, requires no physical universals at all for its study. In special relativity and general relativity, Einstein’s hypothesizes physical connections between relations of ‘mass,’ ‘acceleration,’ and ‘electro-magnetic’ magnitudes. So conceived, they are all outside of the geometry of space as studied in mathematical logic. The general theory of the measurement of magnitudes by the rationals and reals remains in the purview of mathematical logic. In Chap. 7, I argue that time is not studied in any branch of mathematical logic. Armstrongian universals are distinctly non-Russellian, for they exist only if exemplified.14 Armstrong directs his work to the study of such universals postulated a posteriori as empirical hypotheses of a physical theory. No universal can have both a Russellian and Armstrongian nature. All the same, nothing prevents Russell’s scientific method in philosophy from appealing also to Armstrongian universals in its efforts to dissolve a given indispensability argument for this or that kind of necessity or kind of abstract particular. If scientific method in philosophy were to embrace the view that Armstrongian universals are among those with which we are not acquainted, one may wonder how to determine which are Russellian and which are Armstrongian. The way to decide might be by appeal to Armstrong’s conception of physical laws. He holds that physical law involves an unanalyzable relation between universals—that is, a primitively causal relation supporting de re counterfactuals between universals which assure, for example, that everything is such that if it were F it would (causally) be G. Armstrong imagines that the physical

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law itself only exists if the universals involved are exemplified at some interval of the causal history of the universe (sub specie aeternitatis). Hence, Armstrongian universals assure that physical laws are contingent. Russell and Armstrong accepted Eternalism, but Armstrong’s conception of a universal requires it. Russell fares better since Eternalism is not required of scientific philosophy. If there are uniquely biological universals, they include just those of a modern molecular biology. Respecting Darwinian ideas, one expects there to be no autonomous laws governing natural selection. The whole of speciation consists in brute contingencies of the natural history of organisms.15 More pressing is the status of the universals involved with intentionality. Contrary to such figures as Dennett, Dawkins, Millikan, and Armstrong himself, the research program for a bio-naturalization of intentionality remains a failure. The Dretske misrepresentation and Fodor disjunction-­ problems remain unresolved and (if one stops equivocating on the meaning of the “belief” and “know”) there is no longer grounds for hoping that universals involved with intentionality emerge from complex organizations of contingently exemplified physical + biological universals. There is no good reason to reject the robustly Russellian universals of the Principia era that are involved in intentionality. The multiple-relation theory requires us to be acquainted with robustly Russellian relation(s) of ‘belief.’ It holds that in ascribing beliefs to others a given ‘belief’ relation exists unexemplified and can occur in a given fact without unifying that fact. Universals such as ‘belief’ relations, like physical relations, exist unexemplified and are robustly Russellian. As we shall see, they differ from physical (and biological) universals in just the way the universals of Ethics differ from them. Autonomous laws governing the universals of intentionality, if they exist, are in the same camp as the laws of Ethics. I argue below that they are logically necessary and knowable a priori but their structural features are hidden because our acquaintance with them requires that they be exemplified in us. Scientific method in philosophy is defined by its agenda against abstract particulars and kinds of non-logical necessity. In light of this, it is important to keep in mind that, although there are variants of Russell’s Analytic Realism in the different eras of his work, none embrace abstract particulars in the sciences—excepting the science of logic. The title “logical atomism” appeared in the paper “Analytic Realism” and is the popular name of Russell’s philosophy. In My Mental Development, the expression “logical atomism” is used as if “analytic realism” were synonymous with it:

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…in the years 1899–1900, I adopted the philosophy of logical atomism and the techniques of Peano in mathematical logic.

In truth, Russell’s uses of the word “atom” change meaning. In his 1914 rewriting of laws governing material continuants in terms of transient particulars, Russell speaks as if contingently existing sense-data (and unsensed transient physical particulars) are the atoms of his logical reconstruction. Sense-data are concrete particulars. Clearly, Russell’s enthusiasm in his 1918 Lectures on Logical Atomism for some of Wittgenstein’s ideas should not be construed as if he were embracing a science of atoms as abstract particular sense-data! That would miss the point of his research program entirely. Russell speaks as if his 1918 research program is concerned to uncover the logical forms of facts (including atomic, negative, and general facts). None of these new ideas belong to the Principia era. They are manifestations of Russell’s misunderstanding of Wittgenstein’s Doctrine of Showing—the doctrine that in any correct language for empirical science an atomic statement’s formal expression shares (and thereby shows/pictures) the logical form of the fact that would exist were it to be true. Wittgenstein’s Tractatus Logico-Philosophicus proclaimed that there are genuinely logical atoms. Russell’s logical atomism was moving in the exactly opposite direction, eliminating abstract particulars from the science of logic. In the neutral monist era of the 1920s (and in the 1945 A History of Western Philosophy) the expression “logical atomism” is also used, but now in a way incompatible with the 1918 ideas.16 It is best to avoid the phrase. In my Frege’s Notations: What They Are and How They Mean (2012) I argued that Frege had anticipated the scandals of the metaphysicians of abstract particulars (such as “variable numbers,” points at infinity and so on) well before Russell’s paradoxes of abstract particulars were discovered. He sharply distinguished functions from objects and explained the variables used in mathematics in terms of his theory of the unsaturatedness of functions. Frege’s first-level functions fξ and gξ can compose to form ­ ­functions f(gξ) and g(fξ). For example, ξ+1 composes with ξ2 to form the functions ξ2 + 1 and (ξ + 1)2. But a function sign may never occur in the argument position of another function sign. Functions are extensional. There is no need for Frege to introduce extensions (classes, aggregates) as entities that capture the extensional contexts of mathematics. The naïve view was that appeals to extensions simply shadow appeals to functions themselves. Extensions are supposed to be innocuous, so that nothing

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new (it was naïvely thought) could result in working with extensional entities. Frege had skirted that entire issue by his commitment to the extensional nature of functions. Nevertheless, he did not embrace the revolution within mathematics against abstract particulars.17 He never doubted that numbers are logical objects and he never doubted that geometry studies non-logical spatial objects (circles, triangle, etc.). His approach to securing numbers as logical objects, though outlined in his Grundlagen (1884), awaited the theory of value-ranges of his Grundgesetze der Arithmetik (1893). Numbers are just those logical objects that are correlated uniquely with second-level numerical concepts. (Concepts are functions whose values are always truth-values). Frege held, though with trepidation, that it is a logical truth that functions are correlated uniquely with what he called value-ranges and introduced them by his axiom Basic Law Vb: fx  gx , z f z  zgz. x ,

This axiom, by itself, introduces a non-homogeneous second-level function z´ Φz that goes from first-level functions fξ to objects z´ fz that are value-ranges. Together with extensionality, this yields: ,

(Correlation)  ⊢x⁀ z fz  = fx.



In this way, the fundamentally second-level function fx ≈ x , y gy,





of one-to-one correspondence between functions f, g, gets correlated with the first-level function: ,

,

x  z fz  x , y y  z gz,

,

,



between the objects z fz and z gz. Thereby, we have the first-level function u ≈ v, where:

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u  v  df x  u  x , y y  v.



By double correlation, Frege puts: 0  df z  z    . ,





This identifies the object that is 0 as the value-range uniquely tied to the numerical second-level concept x

    x.



Similarly, Frege identifies the object that is 1 by double correlation: 1  df z  z    . ,





He arrives at this by correlation with the second-level existential numeric quantifier concept:      y   x  y  . y

x



Impredicative comprehension enables him to introduce the strong ancestral function Px x being mortal is quite different. The proposition named by a nominalization of the wff is construed as this: ‘Every term is true which results from substituting ‘any term’ for a in ‘a’s being human implies a’s being mortal’.

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Its constituents include the denoting concept ‘every term,’ the concepts ‘human,’ ‘mortal,’ and the relation of substitution. According to Principles, denoting concepts have a twofold occurrence in propositions, as do concepts generally. They may occur as term (of the proposition—i.e., as a constituent in it not providing unity) and they may occur as concept (providing unity). The denoting concept ‘Every man’ occurs as concept in the proposition ‘Every man is mortal.’ It violates logical form to imagine the proposition ‘Socrates is mortal’ as the outcome of substituting Socrates for ‘Every man’ in ‘Every man is mortal.’ One must substitute entities occurring as term, not as concept. In “On Denoting,” Russell’s “Gray’s Elegy Argument” against denoting concepts has this problem as its source. Russell could not find an intelligible logical form for the way denoting concepts occur as concept. The difference between occurring as concept (which grounds the unity of a proposition) and occurring as term remains quite sound. In the Principia era, it defines what a universal (concept, relation) is. Concerns over Bradley’s regress should be dismissed. Bradley commits the fallacy of complex question in asking how it is that there are complex entities. There is no reason to hold that the absence of complex entities is a natural metaphysical default state. When Russell speaks of the universal as having the unique capacity to ground unity, he does not mean to be agreeing with Bradley’s concern that the default state is the absence of complex entities. A universal grounds the unity of a proposition by occurring in it as concept. There is no default state of a universal not so occurring. Note that when a relation occurs as concept in a proposition it by no means follows that the proposition is true. Taking the proposition ‘Desdemona loves Cassio,’ one may say that she is related (tied, unified) to Cassio by the relation ‘love’ occurring in it as concept. It is a sheer equivocation to take it to mean she is related by love to Cassio.1 If Russell used the word “fact” to mean a true proposition, he cannot mean “fact” in the 1910, no-­propositions, sense. Once propositions are abandoned so also is abandoned the notion of “occurring as concept.” Only with 1910-facts does being related by the relation ‘loves’ become the same thing as being related by love. In either case, fact or proposition, there are no problems of infesting unity. By 22 December 1905, Russell thought his substitutional theory was the complete philosophical solution of the paradoxes relevant to his logicism. No semantic paradoxes of propositions (e.g., paradoxes of assertion or belief or truth-predicates, or of the Liar) infest the substitutional theory of propositions. Russell made it clear in 1906 that he thought that all

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semantic paradoxes (e.g., involving equivocal notions of denoting, naming, defining) are irrelevant to the formal logic of propositions. Here are some of Russell’s ebullient comments: [Note added 5th February 1906] … I now feel hardly any doubt that the no-­ classes theory affords the complete solution of all the difficulties…(OT, 7 March 1906) … It affords what at least seems to be a complete solution of all the hoary difficulties about the one and the many; for, while allowing that there are many entities, it adheres with drastic pedantry to the old maxim that, “whatever is, is one.” (STCR, 14 April 1906) There seems reason to hope that the method proposed in this article avoids all the contradictions, and at the same time preserves Cantor’s results… (InS, Sept 1906)

Whitehead was enthusiastic about Russell’s substitutional theory. As late as 16 June 1907 we find Whitehead saying “I agree that the substitutional theory is the proper explanatory starting point.” Russell was ready to go forward with the second volume of The Principles of Mathematics. The edifice would be based on substitution. All that was left to do, Russell reports in his Autobiography, was to “write the book out” (A, vol. 1, p. 152). Yet substitution was abandoned by 1908. What happened? The question of what caused the transition from the Principles era to the Principia era has been subject of intensive debate among historians. Unfortunately, there remains to this day no consensus as to the answer. Nevertheless, answering this question has a direct impact on interpretation of Russell’s Theory of Knowledge research program. One has to take a stand. I hold that there is one reason and one reason alone that Russell ended his Principles era.2 In April/May 1906, Russell discovered that his substitutional emulation of simple types central to the Principles era logic of propositions faced a unique family of diagonal paradoxes of propositions formulable only by the methods of the substitutional theory itself. I have called this family of diagonal paradoxes of propositions, Russell’s “po/ao paradoxes.”3 I found them in 1985 in Russell’s many manuscripts of 1906. The diagonal paradoxes that arise in the substitutional theory are unique to the techniques of substitution. They emerge because the ontology of propositions as structured entities enables the emulation of functions that conflict with Cantor’s power result that there can be no function

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from objects onto (simple-type-regimented) properties of those objects. The po/ao paradoxes require only Cantor’s syntactic diagonal techniques. With propositions abandoned in 1908 due to the po/ao paradoxes, Russell’s translation program was abandoned. The syntax of simple impredicative types stands alone. Russell’s approach to emulation took a semantic turn. In his introduction to Principia he offered a nominalistic (modern substitutional) account of simple-type regimented bindable predicate variables. It was always impredicative simple types that was to be emulated, first by the techniques of substitution and later, in the introduction to Principia, by the semantic techniques of a nominalistic semantics for bindable predicate variables. Unfortunately, Russell’s nominalistic semantics forced him to be concerned about vicious circles in the recursive truth-definitions. The syntax of Principia has a simple-type scaffold, with impredicative comprehension, but Russell’s nominalistic semantics must put: (i φ )χ( i φ!) is true iff every wff B of Li is such that χ[B/i φ!] is true. The wffs of Li must not include the bound variable i φ! else the truth-­ conditions would be viciously circular. This is the source of his considerations of “order” and “ramification.” They are, one and all, unwanted by-products of Russell’s informal nominalistic semantics which relies upon the truth-definitions of wffs of the various languages that serve as the nominalistic substituents. They are features of the semantics and are not coded into Principia’s formal grammar which remains that of impredicative simple-type theory. Unfortunately, Whitehead and Russell never formally set out the syntax of simple-type regimentation. The proper approach would have been to first give the syntax of simple types and then suppress the indices under conventions. In Principia, they tried to indicate the syntax via a practice of typical ambiguity and never restored simple-type indices anywhere in the work. As a result, they left the door open for aberrant interpretations which “fix” Principia’s grammar, inventing a syntax of ramified types (e.g., Church’s r-types) with order and type regimented variables that aligns with Russell’s attempts at a semantics in his informal introduction. Not recognizing that Russell intended a nominalistic semantics, Church’s “fixing” of Principia grammar interferes with Principia’s impredicative comprehension axiom schemas *12.1.11. In the hands of Church, who was misled by the informal semantic explanations of Principia’s

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introduction, impredicative comprehension appears as if it were an untoward add on—an ad hoc Reducibility axiom (not a schema) wholly unmotivated by the semantics which he gave precedence. In truth, it was Russell’s nominalist semantics that was the untoward add on. By 1919 Ramsey had convinced Russell that his nominalistic semantics cannot validate Principia’s impredicative comprehension axiom schemas and, with intellectual honesty, he thereby abandoned the semantics. He did not abandon Principia’s formal theory. Ramsey tried his own hand at a nominalistic semantics which, he thought, would validate impredicative comprehension. Leaving the finitary syntax of Principia entirely intact, he imagined a substitutional (modern nominalistic) semantics embracing infinite conjunctions and disjunctions. Russell demurred. Whitehead’s attitudes are rarely discussed. In fact, he was never required to endorse, nor even asked to endorse, the nominalistic semantics of the introduction of the first edition. Whitehead railed at Russell’s 1925 introduction to the second edition of Principia, which explored experiments changing the syntax and evaluating Wittgenstein’s ideas of ramification and a radical extensionality to obviate impredicative comprehension. Both introductions consisted largely in experimental interpretations composed by Russell alone. Whitehead vehemently disavowed Russell’s introduction to the second edition in a 1926 letter to Mind. More importantly still, a careful look reveals that Whitehead never agreed with the introduction to the first edition! In fact, in the first edition’s vol. 2, we find Whitehead with a Realist (objectual) semantics for the various simple-type regimented individual variables and the comprehension principles of cpLogic. Universals are included among the ranges of the lowest simple type of individuals in both Russell’s nominalistic semantics (which applied only to bindable individual variables of higher than lowest type) and Whitehead’s fully Realist semantics. Whitehead goes further (PM, vol. 2, p. xii): We often speak as though the type represented by small Latin letters were not composed of functions. It is, however, compatible with all we have to say that it should be composed of functions. It is to be observed, further that, given the number of individuals, there is nothing in our axioms to show how many predicative functions of individuals there are, i.e., their number is not a function of the number of individuals: we only know that their number is ≥ 2 Nc

‘ Indiv

, where “Indiv” stands for the class of individuals.

Nothing in the formal system prevents the interpretation that the number of entities (individuals) over which individual variables of given simple′ type range can be much greater than 2 Nc Indiv , where Nc‘Indiv is the

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number of entities (individuals) over which the simple-type regimented individual variables of the next lower type range (PM, vol. 2, vii). The Whitehead-­Russell disagreement over the semantic interpretation doesn’t at all impact the formal syntax and axioms of Principia which remain that of simple types with fully impredicative comprehension. The difference, however, is that Russell’s semantic ideas fail, while Whitehead’s succeed. The textual evidence that the formal syntax of Principia is that of simple (impredicative) types abounds, once one looks. For example, section *12 offers impredicative comprehension axiom schemas *12.1.11. There is no other means of comprehension in the work. Circumflex is explicitly said not to be a predicate term forming operation at Principia’s p.  19. (Church’s interpretation ignores this.) Principia explicitly said (p.  162) that the letters φ, ψ, f, and g are not bindable. It says that only φ! ψ!, f! and g!, and so on, with the exclamation, are bindable. There are no bindable non-predicative predicate variables whose order is higher than the order of its simple type. (Church’s interpretation ignores this.) Church’s syntax undermines the secondary scopes needed at *20.07 and *20.071 for the no-classes theory (as we shall see). The evolution of Principia reveals that its syntax is that of simple (impredicative) types too. Russell’s substitutional theory clearly aimed to emulate simple types. The substitutional theory was abandoned in 1908 precisely because Russell came (mistakenly as we shall see) to hold that it couldn’t succeed without imposing an untenable ad hoc ontology of orders of propositions. The logical particle “Ͻ” of the Principles era is a relation sign for implication. It is flanked by terms to form a wff. In stark contrast, the sign “⊃” of Principia is not a relation sign. It is flanked by wffs to form a wff. In Principia one cannot write “x ⊃ y” which is ungrammatical. In contrast, the expression “ x Ͻ y,” which says that x implies y, is perfectly well-formed. Of course, it is ungrammatical in Principia to write “~x ∨ y” and “x ∨ y.” But in Principles era one can put x∨y =df x Ͻ y .Ͻ. y This says that x’s implying y implies y. Every lower-case letter of English is regarded as an individual variable in the Principles era. There are no special “propositional” variables of the pure logic of propositions. Any wff φ of the formal language of logic can be nominalized—that is, transformed into a term “ { φ}” by using nominalizing braces. But subject positions are sufficient to mark the nominalizing transformation. Thus, for example, a wff such as “x .Ͻ. y Ͻ x” abbreviates the tedious expression “x Ͻ  {y Ͻ x}.”

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Similarly, the wff “(x)(ψx) .Ͻ. ψy” abbreviates “ {(x)(ψx)} Ͻ  {ψy}.” It says that everything’s being a ψ implies y’s being a ψ. In Principles, Russell often spoke of a proposition as logically unasserted if it occurs as term in a proposition. But eventually he came to realize that since he “finds it impossible to divorce logical assertion from truth,” he must take assertion as a psychological matter irrelevant to logic.4 After all, the logically true proposition  {(x)(x=x)} occurs unasserted in the proposition  {(x)(x=x) .Ͻ. y=y}. Nonetheless, translation of Principia’s language of simple-type-regimented individual variables into the Principles era language of substitution is very straightforward. Consider, this (∃ φ)(x)(φ!x ≡ x = x). Disambiguation yields wffs such as the following and their translations into substitutional language: (∃ φ (o))(xo)(φ (o)(xo) ≡ x o = x o) (∃ p, a)(x)(p

x ≡ x = x) a

(∃ φ ((o)))(x (o))(φ ((o))(x (o)) ≡ x (o) = x (o)) (∃ s, t, w)(p, a)(s

p, a .≡. p = t • a = w) t, w

(∃ φ (o, o)))(xo, yo) (φ (o, o)(x o, y o) ≡ x o = y o) (∃s, t, w)(x, y)(s

x, y ≡ x=y). t, w

Any wff in the primitive object language of Principia’s simple-type theory can be translated into the simple-type free language of substitution. The 1905 theory of definite descriptions is the centerpiece. All the usual conventions governing the scope of a definite description apply. The case of multiple substitutions is defined in terms of a carefully crafted series of single substitutions. We find: p

x x  df (q )( p ! q ) a a

s

p, a  p, a   df ( q )  s !q . t, w  t, w 

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The paradoxes relevant to logicism are solved by the simple-type scaffolding emulated by the substitutional techniques of using multiple variables. An expression “F(F)” cannot be translated since it would require P

p, a !q a

which is ill-formed. Cantor’s paradox of the greatest Cardinal, the Russell paradoxes (of classes and universals), the Burali-Forti paradox of the greatest ordinal, are all ill-formed in the substitutional syntax. The substitutional theory provides a proof of the infinity of individuals. Infinity is readily proved because propositions are themselves individuals and there are no simple types. One can prove: (x)(  {x Ͻ x} ≠  { x Ͻ  { x Ͻ x }} ).5 In a letter as late as 5 January 1908, Whitehead remarks: By the bye won’t we have to prove in some section that the numbers 1, 2, … ℵ0: got in any one type by starting from one entity of that type, and proceeding in the approved manner, are the same numbers as are got by doing the same thing in another type and then transforming into the original type. Probably you have done this—but it has slipped my memory—just give me the reference if it is all right.

In fact this result cannot be proved in Principia. The letter suggests that Whitehead was still under the impression that Russell’s substitutional theory was alive and well. And more startling still, we find the following in Principia’s “Note to Section D” (PM, vol. 1, p. 301): …the relation +1, i.e., (in virtue of the notion of *38) the relation of x +1 to x, where x is supposed to be a finite cardinal integer … leads to the descriptive function (+1) ‘x, i.e., x+1. … It correlates with any class α of numbers the class obtained by adding 1 to each member of α, i.e., (+1) “α. This correlation may be used to prove the that the number of finite integers is infinite (in one of the two sense of the word “infinite”); for if we take as our class α all the natural numbers including 0, so that the natural number can be correlated with a proper part of themselves.

This passage is completely incongruous with the work. The sense of a class being “infinite” in the Note is that of Dedekind infinity (a one-to-one mapping of the class of inductive cardinals 0, 1, 2… onto one of its proper

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subclasses, say 1, 2, …). No such mapping is one-to-one if the inductive cardinals are finite; and it is clear in vol. 1 in comments after (*22.351, *24.1), well before the appearance of section *38 and the Note to Section D, that Principia’s axioms don’t provide for infinity. I surmise that the Note is a vestige of earlier work and was supposed to have been removed! The quantification theory of propositions of the substitutional theory is perfectly consistent in spite of what we might regard as its embodying an ontological self-reference. It has the axiom schema: (x)(ψx) Ͻ ψα, for any term α. For example, an instance is this: (x)(x=x) .Ͻ.  {(x)(x=x)} =  {(x)(x=x)}. No paradox of propositions arises from this. In fact, in embracing his substitutional theory, Russell agreed with Peano that semantic paradoxes involving “defines,” “names,” and “denotes” are not genuine paradoxes at all. They were regarded by Russell as simply equivocations which can be summarily dismissed. He was quite explicit about this.6 He explains that there can be no univocal notion of “defines” or “names,” since one must fix the apparatus of the language in question and speak of “defines-in-L,” “names-in-L,” and “denotes-in-L.” This dispatches the Berry, König-­ Dixon, and the Richard as pseudo-paradoxes. (The Berry is the most famous: “The least number not nameable in less than nineteen syllables” seems to name just such a number if and only if it does not.) The ever popular 1908 Grelling (which Russell never discussed) would be dismissed also as an equivocation. It involves a predicate “is heterological” that denotes a property H that a predicate expression exemplifies if and only if it denotes a property that it does not exemplify.7 Quine (1966, p. 9) attempted to evade the equivocation with his predicate …yields a falsehood when appended to its own quotation.

But once we remove all the ellipsis, Quine is offering the expression …is the predicate of the false wff obtained by appending this predicate to its own quotation.

Far from a neo-Grelling, this offers only just a variant of the indexical Liar: “This statement is false.”

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Indexicals occur in no science and thus such Liars are irrelevant. In any case, wffs made with indexicals have no truth-conditions. An indexical requires a context of utterance to get a reference. There is no way to simultaneously utter something while referring to the context of that very utterance.8 The ontology of propositions requires ‘truth’ and ‘falsehood’ as primitive properties (though neither has a predicate in formal language of the substitutional theory). Accordingly, paradoxes of ‘truth’ and ‘falsehood’ are treated quite differently from the paradoxes Russell dismissed as equivocations. Russell cannot classify “truth” as a semantic notion so long as he has an ontology of propositions. Russell’s is correct that “defines,” “names,” and “denotes” make no sense apart from expression of a language fixed in advance. The notion of “truth” has no analogous natural feature. Tarski’s formal recursive definition of “true-satisfaction” does. It requires “true-in-L.” He has: (Convention T) ┌ p ┐ is true-in-L iff p. Tarski’s formal semantics for true-satisfaction requires a fixed language L be given in advance. It requires that ┌ p ┐ is a structural description in L of the wff p. Tarski’s formal semantics of denumerable sequences of objects satisfying open wffs of a language L can only apply in the context of formal delineation of the expressions of the language in question. But this has no bearing on the truth of propositions. Happily, there is no reason to fear that contingent Liars might arise in the formal language of the substitutional theory. The substitutional theory is not plagued by contingent non-­ indexical Liar-like paradoxes. Propositional liars have the form: (q)( θq . ≡ . q =  {(p)(θp Ͻ ~p)} ). Such propositional liars are not diagonal.9 They do not infest the pure logic of propositions because there is no wff θq of the pure language of the substitutional theory satisfying the above. They cannot be formed in the formal logic of propositions since they rely on θq involving cases such as “s believes q,” or “s asserts q,” and so on, which involve contingent psychological notions of belief or assertion. Thus, the contingent Liar paradoxes of propositions (if any) and the semantic paradoxes equivocating on notions of “denotes,” “names,” and “defines” are one and all irrelevant to Russell’s substitutional theory. With all these successes, the substitutional theory is very attractive. What happened?

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The central papers on substitution are these: “On the Relation of Mathematics to Logic,” (Nov. 1905) “On Substitution,” 22 December (manuscript) 1905 “On Some Difficulties in the theory of Transfinite Cardinals and Order Types” (OT) Dec 1905 “On the Substitutional Theory of Classes and Relations” (STCR), May 1906 “On ‘Insolubilia and their Solution by Symbolic Logic” (InS) June 1906 “Mathematical Logic as Based on the Theory of Types,” (ML) July 1907.

The paper ML waited for a long time at The American Journal of Mathematics, only appearing in 1908. None of these papers reveal the hidden story of what happened to the substitutional theory. The hidden story10 is that in April/May 1906, Russell discovered a family of diagonal po/ao paradoxes that violate Cantor’s power result. They are unique to the techniques of substitution because they rely on diagonal functions that arise from the theory’s ability to prove that identical propositions have identically ordered constituents. They do not have the form of Liars or commit semantic equivocations. Cocchiarella (1980) surmised that there would be such a paradox,11 and visiting the Russell Archives at McMaster University, I first uncovered an early version of the po/ao in an April–May 1906 manuscript titled “On Substitution” (OS). It has significantly different formulations depending on which functions of propositions are invoked to violate Cantor’s diagonal power result. All three of the versions below appear in Russell’s work notes: Principles Appendix B Version x !q .•. (∃p)(∃a)(x)(∃q)( p a s x q ≡  {(∃r)(∃c)(x =  {(s)(r Ͻ s)} .•. ~(r ))}). c c OS Version (∃p)(∃a)(x)(∃q)( p

x x !q .•. q ≡  {(∃r)(∃c)( x =  {r Ͻ c} .•. ~(r ) )}). a c

Hawtrey Version12 (∃p)(∃a)(x)(∃q)( p

x b x !q .•. q ≡  {(∃r)(∃c)(x =  {r ! s} .•. ~(r ) )}). a c c

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The three versions are not at all alike. Indeed, in InS Russell thought that he had solved all of them by means of his elimination of general propositions. And indeed he had! This was to have been his grand solution to the po/ao paradox(es) and the vindication of his substitutional technique. He boldly maintained that his substitutional theory “avoids all known paradoxes while at the same time preserving nearly the whole of Cantor’s work on the transfinite” (InS, p, 213). Obviously, InS does not eliminate quantified wffs. The elimination of general propositions advocated in InS anticipated Principia’s section *9 which defines subordinate quantifiers in terms of quantifiers initially placed. (The work notes are in Russell’s CP, vol. 5.) In the earlier 1905 substitutional theory, one may transform any wff φ into a term  { φ} for a proposition. The 1906 paper InS changes this, maintaining that only a quantifier-free wff φ can be nominalized to form a term  { φ} for a proposition. Thus, for example, where φ is quantifier-free, InS has: (x) φx Ͻ φy =df (∃x)(  { φx} Ͻ {φy}). This allows the occurrence of quantified wffs to occur in subordinate contexts of the logical particles without having to nominalize the wffs. Looking at the above formulation of the po/ao paradox(es), this is quite safe. Now in the 1905 substitutional theory, we find the following: Principle of Substitution x (∃y)(a out  { φy}) Ͻ (x)(  { φa} ! { φx}).13 a This principle, together with the axiom x (p, a)(x)E!(ιq)( p !q), a yields the theorem: x (∃p, a)(x)(p ≡  { φx}), a where p and a are not free in the wff φ. The 1906 InS version makes the theory too weak because it restricts the above theorem so that φ is quantifier-­free. With the techniques of *9, however, Russell felt it would be safe to go further and introduce what he called “mitigating axioms” into InS. He has: x (∃p, a)(x)(p ≡ φx), a

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where any quantifiers in φ are defined so that they are pulled to initial positions by the definitions of *9. This cannot revive the Appendix B version of the po/ao because its clause s x =  {(s)(r Ͻ s)} c is ill-formed in InS. But the mitigating axiom does revive the OS and Hawtrey versions. In July of 1907, Russell was chagrined. InS was in print and proclaimed that it had the complete solution, but he came to realize that his mitigating axioms were too strong. They revive the po/ao paradox. He quickly composed the paper ML which endorsed a new substitutional theory retrofitted with a tentative stop gap—orders of propositions and an axiom of order-Reducibility for them. In the 1907 manuscript “On Types” the desirability of having an Appendix on substitution is mentioned. By the time it appeared in 1908 he had changed his mind—again. He knew that orders of propositions were untenable—wholly impossible to philosophically justify and entirely arbitrary since several different ad hoc methods for orders of propositions would do. The reduction of orders of propositions requires axioms such as: x x (rn, bo)(∃p1, ao)(xo)(p1 o ≡ rn o ). ao bo This reduction axiom adopts the grammatical rule that the following is a wff: pm

xz ! qw, bn

where z = n and m = w. It all was philosophically unacceptable to Russell and he abandoned substitution. The appendix on substitution, long planned as central to solving the paradoxes, never materialized. Curiously, Russell gave up the project of InS just as he was about to succeed. It just needs different mitigating axiom schemas. When the substitutional theory had seemed viable, Russell was thinking that Principia’s simple-type regimented language should develop out of the simple-type free (and certainly order-free) substitutional language itself. He hadn’t imagined independently articulating the Principia language of simpletype regimentation and using it to help him find the safe mitigating axioms for InS. If he had, he might well have realized that he need to only accept a mitigating axiom schema such as this: x (∃p, a)(x)(p ≡ φx), a

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where the wff φ is a translation from Principia’s simple-type regimented language into the language of substitution (and where any quantifiers in φ are defined so that they are pulled to initial positions by the definitions of *9). No version of the diagonal po/ao paradox can be revived by such a mitigating axiom scheme. The wffs φ that generate the diagonal paradox have no analogs in the language of Principia’s simple-type regimentation. x They require the substitutional wff p !q. At the same time, the mitigating y axioms give the substitutional theory all the power of Principia. The Appendix on the InS should be restored to Principia.

Principia’s Simple Types The formal grammar of Principia was never set out in the work independently of its convention of simple-type ambiguity. As a result, the many informal semantic comments Russell offered in its Introduction are all too easily misread as if germane to the account of its formal grammar. It is my view that all issues pertaining to ramified types (r-types) are unwelcome products of such a misreading. They don’t belong to the formal theory. The primitive signs of the language of Principia are ∨, ~, (, ), and ∃. The symbol ∀ for universal quantification is due to Gentzen’s work in the 1930s and is not used in Principia. A type symbol of the simple-type theory is recursively defined (in modern times) as follows: (i) o is a type symbol. (ii) If t1,…, tn are type symbols, then (t1,…, tn) is a type symbol. (iii) There are no other type symbols. The notion of order appropriate to a simple-type symbol is useful in comparing rival interpretations. It can be recursively defined as follows: (i) The type symbol o has order 0. (ii) A type symbol (t1,…, tn) has order m+1 if the highest order of any of the type symbols t1,…, tn is m.14 The individual variables of Principia’s formal simple-type theory are x1t , …, xnt , and among these are the individual variables of lowest simple type, namely, x1o , …, xno (informally xo, yo. zo). Predicate variables are those individual variables whose simple-type symbol is not o. The only terms in

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Principia are the individual variables of whatever simple type. Atomic well-formed formulas are of the form: x 1

t ,tn 

( x1t1 ,…, xntn ).

Where t ≠ o, instead of the predicate variables x t, y t, z t, it is convenient to use letters (informally φ t, ψ t, χ t, f t, g t) as predicate variables. Accordingly, using a predicate variable  1

t ,tn 

 t1 ,tn 



t1 1

, the above is this:

tn n

( x ,…, x ).

The practice of typical ambiguity is to suppress type indices on the variables with conventions of uniform restoration understood. With typical ambiguity in place, however, it is vitally important to distinguish bindable object-language predicate variables from schematic letters. In Principia, type ambiguity uses φ!, ψ!, χ!, f!, g! for its bindable object-language predicate variables and they always come with the exclamation (shriek!). When letters occur without the shriek, as with φ, ψ, χ, f, g, they are schematic for wffs (well-formed formulas). Thus, with typical ambiguity, an atomic wff of some or other simple type is expressed as follows:  φ!(x1, …, xn). This, as we noted, must not be conflated with  φ(x1, …, xn). This is schematic for a wff in which free object-language variables x1, …, xn occur. A schematic letter φ for a wff obviously cannot be bound in a quantification. The formulas (wffs) are the smallest set K containing all atomic wffs such that if φ and ψ are wffs in K, and if x t is an individual variable free in formula θ, then ~(φ), (φ ∨ ψ), and (x t)(θx t) are wffs. Where p, q, and r are schematic for quantifier-free formulas and where φ, ψ, are schematic for all wffs, quantifier-free or otherwise, the axiom schema are as follows: *1.2 p ∨ p .⊃. p *1.3 q . ⊃. p ∨ q *1.4 p ∨ q . ⊃. q ∨ p *1.5 p ∨ (q ∨ r) . ⊃. q ∨ (p ∨ r)15 *1.6 q ⊃ r . ⊃. p ∨ q .⊃. p ∨ r *9.1 φx ⊃ (∃x) φx *9.12 φx ∨ φy .⊃. (∃z) φz

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*12.1n (∃f )(x1,…, xn)( f!(x1,…, xn) ≡ φ(x1,…, xn)), where f! is not free in the wff φ.16 Written with type symbols restored under the convention of typical ambiguity, the above is: *12.1n (∃ f  1

t ,tn 

)( x1t1 ,…, xntn )( f  1

t ,tn 

( x1t1 ,…, xntn ) ≡ φ ( x1t1 ,…, xntn )),

where f  1 n  is not free in the wff φ ( x1t1 ,…, xntn ). Removing the unnect ,t t ,t essary use of f  1 n  as a predicate in favor of y  1 n  , this is t ,t

*12.1n (∃ y  1

t ,tn 

)( x1t1 ,…, xntn )( y  1

t ,tn 

( x1t1 ,…, xntn ) ≡ φ ( x1t1 ,…, xntn )),

where y  1 n  is not free in the wff φ ( x1t1 ,…, xntn ). Note that obviously the schematic letter φ is for a wff and thus receives no simple-type restoration. There is no doubt that the comprehension axiom schemas are impredicative since they impose no restrictions whatsoever on which wffs are instances of the schematic φx and φxy occurring in them. One may wonder why it is that we do not find *12.n in Principia. We find only the following in Principia’s vol. 1. t ,t

*12.1 (∃f )(f!x ≡x φx)

*12.11 (∃f )(f!xy ≡x, y φxy).

Interestingly, there is the following comment (PM, vol. 1, p. 167): In dealing with relations of more than two terms, similar assumptions would be needed for three, four, … variables. But these assumptions are not indispensable for our purposes, and are therefore not made in this work.

One must not be misled by the comment “this work.” Not all the volumes of the work were completed. There was to have been Whitehead’s vol. 4 on geometry. In his 1914 Our Knowledge of the External World as a Field for Scientific Method in Philosophy, Russell wrote (OKEW, p. 61): 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 this was only necessary to admit relations of two terms.

It is quite clear that had we seen vol. 4, we would have seen provisions, at least for triadic relations and for notations of triadic relations-e. Whitehead clearly knew that triadic (and higher) relations in intension are needed in geometry and that one must make provisions for notations for the

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no-relations in extension (no-relations-e) theory for triadic and higher relations-e. Whitehead communicated just this point to Russell in a 27 April letter of 1905. (See letter below.) In his 1924 “Logical Atomism” Russell reveals that he understood this well enough (LA, p. 332): How far it is necessary to go up the series of three-term, four-term, five-­ term … relations I don’t know. But it is certainly necessary to go beyond two-termed relations. In projective geometry, for example, the order of points on a line or of planes through a line requires a four-term relation.

So there is no question that Principia intended *12.n. The reason we only find *12.1.11 is that it was sufficient for the analysis of serial order and for all the topics of vol. 1–3. The opening of vol. 4 would have offered *12.n. And now it is the epistemic question that comes to the fore. Is there particular natural number n, such that knowledge of relations of adicity n is sufficient for all knowledge? Even if there are contingently no facts of more than, say, a googolplex of constituents, the answer seems unknowable. Derived inference rules are allowed in Principia and frequently legitimated by the theorems it proves. But the fundamental inference rule in Principia are the following, where φ and ψ are any wffs, quantifier-free, or otherwise: *1.1 Modus Ponens: From φ and φ ⊃ ψ, infer ψ *10.1 Universal Generalization: From φx t, infer (x t) φx t Switch: From ( x t1 )(∃ y t2 ) φ( x t1 , y t2 ) infer (∃ y t2 )( x t1 ) φ( x t1 , y t2 ), where there is a logical particle in the wff φ on one side of which all free occurrences of x t1 occur and on the other side of which all free occurrences of y t2 occur.17 Definitions in Principia are always stipulative. They introduce notational conveniences which may be everywhere replaced by appeal simply to definition. They include the following: *1.01 p ⊃ q =df ~p ∨ q *2.01 p • q =df ~(~p ∨ ~q) *4.01 p ≡ q =df (p ⊃ q) • (q ⊃ p) where p and q are any wffs quantifier-free or otherwise. I use dots symmetrically for convenience, with the greater number of dots indicating the main connective.18 In addition, the quantification theory employs the following definitions of section *9. For the present, I will continue to write

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out type indices for clarity. Where p is quantifier-free and not containing x t, Principia’s definitions include the following: *9.01 ~(x t) φx t =df (∃x t) ~ φx t.

*9.02 ~(∃x t) φx t =df (x t) ~ φx t.

Where x t does not occur free in the quantifier-free formula p, Principia has: *9.03 (x t) φx t ∨ p =df (x t)(φx t ∨ p). *9.04 p ∨ (x  t) φx t =df (x t)(p ∨ φx t). *9.05 (∃x t) φx t ∨ p =df (∃x t)(φx t ∨ p). *9.06 p ∨ (∃x t) φx t =df (∃x t)(p ∨ φx t). Where the wff φ x t1 does not contain the variable y t2 free and the wff ψ y t2 does not contain x t1 free, Principia has the following definitions for pulling quantifiers to an initial position in pairs: *9.07 ( x t1 ) φ x t1 ∨ (∃ y t2 )ψ y t2 =df ( x t1 ) (∃ y t2 )(φ x t1 ∨ ψ y t2 ). *9.08 (∃ x t1 ) φ x t1 ∨ ( y t2 )ψ y t2 =df (∃ x t1 )( y t2 )(φ x t1 ∨ ψ y t2 ). *9.0x ( x t1 ) φ x t1 ∨ ( y t2 )ψ y t2 =df ( x t1 )( y t2 )(φ x t1 ∨ ψ y t2 ). *9.0y (∃ x t1 ) φ x t1 ∨ (∃ y t2 )ψ y t2 =df (∃ x t1 )(∃ y t2 )(φ x t1 ∨ ψ y t2 ). The omission of *9.0x and *9.0y seems to be an oversight. This completes the system. Whitehead and Russell knew that the axiom schema of the convenient quantification theory of section *10 would be derived from *9 by metalinguistic (strong) induction on the length of a wff. Only the base cases, where φ and ψ are quantifier-free, for such an inductive proof are proved in Principia. They are the following two cases: *10.1 (x t) φx t⊃ φy t, where y t is free for x t in the wff φ. *10.12 (x t)(p ∨ φx t) .⊃. p ∨ (x t) φx t, where x t is not free in p.19 Indeed the theorems of section *9 only prove the base cases where φ and ψ are schematic for quantifier-free wffs. But we can see that there is no difficulty for such a proof.20 Principia has explicit existence theorems as instances of the impredicative comprehension schema *12.n and it points out explicitly (in comments at *24) that *10.1 (alternatively *9.1.11) for quantification theory assure the existence of individuals of every simple type, including the lowest. One may wonder how this fits with Russell’s comment in The Problems of Philosophy that all existence is known a posteriori (PoP, p.  75). Since

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Problems is in the Principia era, he can only have meant to be speaking of the existence of particulars—his example, after all, concerned the Emperor of China. What Principia calls “individuals” (of a given type) are not particulars. Principia makes no commitment to any particulars. Universals with which we are acquainted are among the individual variables (of lowest type). Individual variables of higher simple type (on a Realist semantics for Principia) ranges over universals too. Particulars are never the subject matter of logic. While Principia’s formal theory assures that it is a purely logical matter that there are individuals (e.g., universals) in every simple type, including the lowest, it by no means assures that it is a logical matter that there are infinitely many individuals in any simple type. Whitehead, as we shall see, had a strong Realist intuition that there are infinitely many individuals (universals) in every simple type. All the same, it remains far from epistemically certain that logic assures an infinity of logical individuals in the lowest simple type (or in any simple type). So far as Principia is concerned, the matter is unsettled.21 Now in his book 1919 Introduction to Mathematica Philosophy, written after the Principia era, Russell has departed from the stand he took with Whitehead in Principia on this matter. He wrote: “There does not even seem any logical necessity why there should even be one individual—why, in fact there should be any world at all” (IMP, p. 203) In a footnote he continues: “The primitive propositions in Principia Mathematica are such as to allow the inference that at least one individual exists. But I now view this as a defect in logical purity” (Ibid.). What explains these comments is that Russell was then exploring his neutral monist idea that universals can only occur in such a way as to unify a fact. Acquaintance has been thereby abandoned. This requires a quite different semantic interpretation of Principia’s formal quantification theory—one which takes the variables of lowest type to range over particulars alone. Since quantification cannot be committed to particulars, Russell accepts that it must be revamped so that it can be conducted without free variables and without allowing any existential theorems. Curiously, Russell’s 1925 second edition to Principia does not itself attempt to realize the so-called logical purity. It offers but one official amendment to its formal cpLogic—the replacement of its section *9 with a new section *8 offered as a system of deduction without free variables.22 Perhaps it was intended as a step toward the goal of obtaining the “logical purity.” But perhaps not! The goal is nowhere even mentioned. The

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method of deduction without free variables is welcome. Russell’s conception of “logical purity” is not. Deduction without free variables takes as an axiom the following: Closure*9.1 (ψn, … ψ1) (ym,…, y1)(x)(φx ⊃ (∃z) φz), where ψ!1, … ψ!n and y1,…, ym are all the variables besides x and z occurring free in the wff φ. The new system of *8 was perhaps the first quantification theory without free variables, but it does not avoid existential theorems—not even those framed with individual variables of lowest type. The technical problem of conducting deduction to avoid existential theorems involving individual variables of lowest type is made complicated by the fact Russell does not accept vacuous quantification and that in both of his systems of *9 and *8, subordinate occurrences of quantifiers are defined in terms of quantifiers initially placed. Even when p is quantifierfree, the system cannot accept, p .⊃. (x) φx ⊃ p, since it is defined as the existential: (∃x)(p .⊃. φx ⊃ p). In any case, a concern against free variables is one thing, a blanket rejection of existence theorems known a priori is quite another. There is no good reason to reject the Whiteheadian Realist semantics which allows universals among the values of its individual variables of lowest type.

No Classes and No Relations-e Principia means what it says in denying that there are classes and relations­e (i.e., relations in extension). That is the very foundation of its distinctive logicist agenda of embracing the revolution within mathematics. One cannot hope to “repair” the work by introducing classes and relation-e into it, That would be to destroy its very raison d’être. Sadly, so many (perhaps beginning with Gödel and Quine) who are oblivious to its agenda sought to do just that. They reinterpret the work so that it is a theory of simple types of classes and construed relations in extension as if they were classes of ordered n-tuples. That is not a repair. It is a destruction. Even on a Realist interpretation of Principia’s simple-type-regimented monadic individual variables, they don’t range over classes. There are no classes. To understand the theory, one must recognize the centrality of quantificational scope and the conventions governing its many incomplete

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symbols. When we do, some quite wonderful surprises emerge. Consider the following: *13.01 x = y = df (φ)(φ!x ⊃ φ!y) *13.02 x ≠ y = df ~(x = y) *14.01 [ιx φx][ψ(ιx φx)] = df (∃x)(φy ≡y y = x .•. ψx). Though Principia does not adopt it, the following is also quite useful **14.01 [ιx φx][ψx] = df (∃x)(φy ≡y y = x .•. ψx). (I use ** to indicate an addition or correction to Principia.) The benefit of stipulative definition *14.01 is that it enables Russell to drop the scope marker [ιx φx][…] in favor of ψ(ιx φx) with the convention that the scope intended is the smallest possible in the context ψ in question. It should be noted as well that one must use *14.01 (or **14.01) before using *13.01 and *13.02. A definition framed with the genuine terms that are individual variables cannot apply to expressions of definite descriptions which are not genuine terms at all. Thus, one cannot apply *13.02 to a clause such as “ιx φx ≠ y.” The definition *14.01 has to be applied first because definition *13.02 is formed with individual variables. Thus one must not think that “ιx φx ≠ y” and “~(ιx φx = y)” are equivalent, unless of course E!(ιx φx).23 A convention is adopted on dropping scope markers for smallest scope which assures that “~(ιx φx = y)” means “~[ιx φx][ ιx φx = y].” The difference between a schema and an object-language bindable predicate variable is important to the contextual definition *14.01 as well. When, in dropping a scope marker and writing ψ(ιx φx) what is intended is some wff ψ in which the scope marker is to be restored in the smallest scope possible. For example, an instance of the schema ψ(ιx φx) is f !(ιx φx) ⊃ p. Since f ! is a genuine (object-language bindable) predicate variable, this has to be interpreted as [ιx φx][f!(ιx φx)] ⊃ p. That renders the smallest scope. Compare f (ιx φx) ⊃ p. The scope cannot be determined because f is schematic for a wff. All the same there is a special temporary convention that was introduced into *14 to help the reader understand scope. It does not apply in the work in

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general and should never have been introduced at all. Its effect actually obfuscates a proper understanding. It says that in writing ψ(ιx φx) a primary scope is intended and in writing f {ψ(ιx φx)} a secondary scope is intended. Consider, for example: *14.21 ⊢ ψ(ιx φx) ⊃ E!(ιx φx). I strike it out because it is not a properly general theorem schema and uses the temporary convention.24 If we neglect the temporary convention, we find that it has instances that are obviously false. Consider this: ~(∃y)( y = ιx φx) ⊃ E!(ιx φx). The following are proper theorem schemas that remove the temporary convention: **14.21 ⊢ [ιx φx][ψ(ιx φx)] ⊃ E!(ιx φx) !**14.21 ⊢ ψ!(ιx φx) ⊃ E!(ιx φx). The latter works because ψ! is an object-language bindable predicate variable, and thus the smallest scope is the primary scope. One can also prove: ιx φx = ιx φx ⊃ E!(ιx φx) i.e. [ιx φx][ιx φx = ιx φx] ⊃ E!(ιx φx) ιx φx ≠ ιx φx ⊃ E!(ιx φx) i.e. [ιx φx][ιx φx ≠ ιx φx] ⊃ E!(ιx φx) These follow because the definition of the identity sign is made with individual variables and thus cannot apply to definite description expressions.25 The case of *14.21 has to be corrected. But in contrast the important case of *14.18 is quite proper as it stands. We find: *14.18 ⊢ E!(ιx φx) .⊃. (x)ψx ⊃ ψ(ιx φx).26 Whitehead and Russell explain (PM, vol. 1, p. 180): The above proposition shows that, provided ιx φx exists, it has (speaking formally) all the logical properties of symbols which directly represent objects. Hence when ιx φx exists, the fact that it is an incomplete symbol becomes irrelevant to the truth-values of logical propositions in which it occurs.

Since Principia’s general convention on omitting scope markers allows instances of ψ(ιx φx) in the above to have secondary scopes, a proof of

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*14.18 would require a strong induction in the meta-language on the length of a wff. Principia offers only what would be the base case of such an inductive proof. The general statement of the equivalence of primary and secondary scope is given as follows: *14.3 ⊢ ( p ≡ q . ⊃p, q. fp ≡ fq) .•. E!(ιx φx) :⊃: f {[ιx φx][ χ(ιx φx)]} ≡ [ιx φx][ f {χ(ιx φx)}]. The illicit nature of *14.3 is heralded by Whitehead and Russell themselves. They write: In this proposition, however, the use of propositions as apparent variables involves an apparatus not required elsewhere, and we have therefore not used this proposition in subsequent proofs.

They go on to apologize: …*14.3 introduces propositions (p, q, namely) as apparent variables, which we have not done elsewhere, and cannot do legitimately without the explicit introduction of the hierarchy of propositions which a reducibility-axiom such as *12.1.

This comment is clearly a hang-over from the days of the translation program and its agenda of philosophically explaining away the formal language of simple-type scaffolding by translation of Principia’s simple-type regimented notations into the simple-type free language of the substitutional theory of propositional structure. The comment requires the translation program because the substitutional grammar requires “x Ͻ y” as a wff and “Ͻ” as a relation sign. The grammar of Principia makes “x ⊃ y” ill-­formed. What is proper is this: **14.3 E!(ιx φx) .⊃. f {[ιx φx][ χ(ιx φx)]} ≡ [ιx φx][ f {χ(ιx φx)}], where f is a truth-functional context. The notion of a context f being truth-functional has to be expressed in the meta-language. It is this: p ≡ q .⊃. fp ≡ fq, for all wffs p and q. How does Principia accomplish the emulation of classes and relationse? Whitehead understood it well. The no-classes and no-relation-e ­

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theories reveals how to construct extensional contexts by appeal to scope distinctions. In a letter to Russell of 16 June 1907, Whitehead wrote: (1) Your transition from inten[s]ion to extension by means of f  { xˆ ( ψx)} = df (∃ φ!)( φ!x ≡x ψx .•. f(φ! zˆ )) Df is beyond all praise. It must be right. That peculiar difficulty which was worried about from the beginning is now settled forever. For expressions of classes of individuals of some or other simple type, Principia has: *20.01 [ xˆ ( ψx)][f  { xˆ ( ψx)}] =df (∃ φ!)(φ! x ≡x ψx .•. f (φ!))27 *20.02 x ∈ φ! =df φ!x. I have restored the scope marker [ xˆ ( ψx)][ … xˆ ( ψx)…] to*20.01 as it is warranted by the comments of Principia’s introduction (p. 80). It must be understood that occurrences xˆ ( ψx) without the determining scope marker are to be taken in smallest scope. Moreover, class expressions never have simple-type indices since their expressions are façons de parler. Free lower-case Greek α, β, μ, and so on are such that they stand in for the expressions zˆ( ψz) or for αˆ ψα. Hence, consider the following: *20.06 x ∉ α =df ~( x ∈ α). cls*20.06 β ∉ α =df ~( β ∈ α). Definitions for classes of classes of some or other relative type are the following: *20.08 [ αˆ ψα][f ( αˆ ψα)] =df (∃ φ)(φ!α ≡α ψα .•. f(φ!)) *20.081 α ∈ ψ! =df ψ!α. Bound lower-case Greek is defined in Principia by the following: *20.07 (α)fα = df (φ) f { xˆ ( φ!x)} cls*20.07 (α)fα =df (φ) f { µˆ ( φ!μ)} *20.071 (∃α)fα = df (∃ φ) f { xˆ ( φ!x)} cls*20.01 (∃α)fα = df (∃ φ) f { µˆ ( φ!μ)}. Principia allows definite descriptions of classes (of some or other relative type) as well: *20.072 [ια φα][ψα] = df (∃α)(φβ ≡β β = α .•. ψα). The usual conventions on scope markers for definite descriptions hold for definite descriptions of classes.

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The general principle governing the logical equivalence of primary and secondary scopes of a definite description for a class is the same as it is at **14.3. Thus we have: clsdesc *14.3 E!(ια φα) .⊃. f {[ια φα][ χ(ια φα)]} ≡ [ιx φx][ f {χ(ια φα)}], where f is a truth-functional context. However, it is quite important to note that when it comes to class expressions, the following is not a proper analog: f {[ xˆ φx][ χ( xˆ φx)]} ≡ [ιx φx][ f {χ( xˆ φx)}], where f is a truth-functional context. There is no antecedent clause here. In Principia there is no expression E!( xˆ φx) and *12.1 assures, if you will, the existence conditions needed to emulate a theory of classes. Nonetheless, the above is mistaken. The proper analog is more austere. It is this: cls*14.3 f {[ xˆ φx][ χ( xˆ φx)]} ≡ [ xˆ φx][ f {χ( xˆ φx)}], where f is a truth-functional context and χ is an extensional context.28 The convention that omission of the scope marker for most secondary scope is essential. Church’s influential reading missed this when he assumed that the letters φ, ψ, f and g, and so on, without the exclamation, are non-predicative object-language predicate variables. Church’s reading actually undermines the no-classes theory! If one takes f in *20.07 to be a non-predicative bindable object-language predicate variable, the class expression would be forced to have a primary scope. A primary scope in *20.07 is this: (α)fα =df (Γ)[ xˆ Γ!x][f ( xˆ Γ!x)]. That would render  φ!α ≡α ψα =df (Γ)[ xˆ Γ!x][ φ!(Γ!) ≡ ψ(Γ!)]. Such a definition won’t work. Russell gives the following secondary scopes:  φ!α ≡α ψα =df (Γ)( [ xˆ Γ!x][ φ! (Γ!)] ≡ [ xˆ Γ!x][ψ(Γ!)]). Church’s interpretation undermines the definitions for the elimination of class expressions.29 The fundamental theorem of section *20 is the following which is the simple-type replacement for what has come to be called “naïve abstraction.” *20.3 ⊢ x ∈ zˆ ψz ≡ ψx ˆ ≡ ψβ. cls*20.3 ⊢ β ∈ αψα

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The first is for classes of individuals (of any simple type). The second is the analogous theorem for classes of classes of any relative type. Only the first is proven. But the needed instance of *12.n for the second (when classes of classes of individuals are concerned) is given in Principia. We find: *20.112 ⊢ (∃g)( g!(ˆz φ!z) ≡ φ ψ(ˆz φ!z)). What is wanted, however, is the more general case for classes of classes of any relative type: cls*20.112 ⊢ (∃g)( g!α ≡α ψα). The proof of this theorem schema requires a metatheoretic strong induction on the length of a wff. The same point holds for many theorems in section *20 such as: *20.61 ⊢ (α)fα ⊃ fβ. i.e. ⊢ (α)fα ⊃ f ( xˆ Γx) cls*20.61 ⊢ (α)fα ⊃ f (β). i.e. ⊢ (α)f α ⊃ f ( σˆ Γσ) What is found are primary scope theorems that form the base case of the needed metalinguistic proof by strong induction on the length of a wff. Secondary scope pose no difficulty. Just as Principia is a no-classes theory, it is also a no-relation-e theory. Expressions for relations-e (relations in extension) are façons de parler, just as assuredly as are class expressions. Relations-e expressions are not emulating classes of ordered n-tuples. Moreover, relations-e notations are defined in terms of relation (in intension) notations which are primitive expressions such as φ!xy, φ!xyz, and so on. (Alternatively, φ!(x, y), φ!(x, y, z), etc.) A relation may well be non-homogeneous, holding between different simple types of individuals. So also a relation-e such as found in αRβ may be nonhomogeneous, holding between a class α of one relative type and a class β of a different relative type. A relation-e may be such as αTR, between a class α and a relation-e such as R or it may be between individuals (of some or other simple type) and classes or relations-e. Indeed, there are classes of relations-­e as well. Accordingly, many different definitions are needed. Principia only gives a few, assuming readers will see the patterns. Notations of (dyadic and homogeneous) relations-e of individuals (of whatever simple type) parallel contextual definitions of classes and are given as follows: ˆˆ ˆˆ *21.01 [ xy(ψxy)][f  { xy(ψxy)}] = df (∃ φ)(φ!xy ≡x, y ψxy .•. f(φ!)) *21.02 x { φ!}y = df φ!xy

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ˆˆ (φ!xy)} *21.07 (R)fR = df (φ)f { xy ˆˆ *21.071 (∃R)fR = df (∃ φ)f { xy(φ!xy)} *2.072 [ιR φR][ψ(ιR φR)] = df (∃R)( φS ≡S S = R .•. ψR) ˆ ˆ ψ(R,S)][f { RS ˆ ˆ ψ(R,S)}] *21.08 [ RS =df (∃ φ)(ψ(R,S) ≡R, S φ!(R,S) .•. f(φ!)) *21.081 P {ψ!}Q = df ψ!(P,Q). Further definitions are needed for the treatment relation-e of classes and relations-e for relations-e. But the patterns for dyadic relations-e are clear. Of course, Principia has φ!(x1, …, xn) for relations (in intension) of any adicity whatsoever and it surely intends the impredicative axiom schema *12.n. But how to write notions for relations-e that are triadic and higher? Whitehead raised this question in a letter to Russell of 27 April 1905. Speaking of Veblen’s work in geometry he goes on to say: He proves that Descriptive Geometry is the study of the properties of a single three-term relation, and the points are the field of this relation. Of course he does not quite know that this is his point of view, but it is the gist of it, and it throws a flood of light on the whole subject. Now this advance makes it urgent that we produce a notation suitable for three-term relations. In fact since four-term relations occur (harmonic relations etc.) we want a notation suitable for relations with any number of terms. …I should propose to keep xRy as a simplification in this instance of the general form, but otherwise use the new symbolism?

Whitehead is explaining the limitations of writing the variables on either ˆˆ φ!(x,y)}y. How then to express, side of a relation-e sign as in xRy and x { xy for example, a triadic relation-e? The solution is clear. Put: ∈ φ! =df φ!xy ∈ ψ! =df ψ!(P,Q) etc. The definitions at *21.02 and *21.081 can then be regarded as abbreviating (respectively) the above. The notation allows the expression of triadic relation-e as follows: ˆˆˆ φ!(x,y,z). ∈ R       ∈ xyz The plan extends to expressions of relations-e of any adicity. Definitions for union, intersection, and complement are too obvious to mention here. But it is important to take note that all definitions made

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with lower-case Greek for classes must be applied before applying the definitions for class and relations-e. A definition that is very useful is this: *24.03 ∃!α =df (∃x)(x ∈ α) cls*24.03 ∃!α =df (∃γ)(γ ∈ α). Definition *24.01 cannot be applied to cases such as ∃!ια φα. One must first apply definition *20.072 eliminating the definite description. This is because definite description expressions are not genuine terms. There is also the very important notation: *30. 01 [R ‘y][ψ(R ‘y)] =df [(ιx)(xRy)][ψ {(ιx)(xRy)}]. cls*30.01 [R ‘α][ψ(R ‘α)] =df [(ισ)(σRα)][ψ {(ισ)(σRα)}] One of the most important applications of *30.01 is the following Nc ‘α =df (ιγ)(γ Nc α). i.e. Nc‘α =df (ισ)(σ = γˆ (γ sm α)). This is the definite description “the cardinal number of α,” that is, the class of all classes γ similar to α. The expression “γ sm α” says that members of γ stand in a one-to-one onto correspondence to members of α. (For convenience, one can write this as γ ≈ α.) This notion is extremely ambiguous in relative type because the ‘sm’ relation-e of similarity may be descending, ascending, or homogeneous. A similarity relation-e involved in γ sm α is called descending if γ is any relative type lower than the relative type of α. It is homogeneous if they are of the same relative type, and ascending if γ is of any relative type higher than α. Whitehead uses the expression Noc for the homogeneous similarity relation. He uses N1c for a descending (by one step) similarity relation. Whitehead uses the following (PM, vol. 2, p. 6): Noc(β) ‘α This is “the class σ equal to the class of all classes γ each of which is similar to α and each of which is the same relative type as β.” In symbols: (ισ)(σ = γˆ (γ sm α .•. γ =β ∨ γ ≠ β)). These are expressions for relative types of classes. They have no bearing on simple types. If we want to talk about classes of individuals, Principia does afford notation (at *63–*65).30 One can put: Nc(βx) ‘αx

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This indicates that both α and β are classes of individuals of the same simple type. One can put: Nc(β xt ) ‘  xt  This indicates a difference in simple type. This is a one-step descending cardinal number of  xt  . There are no numerals “0” and “1” and “2” in Principia. As usual, contextual definitions are introduced that are relative type ambiguous. One has to repeat definitions. We find: *52.01 1 = df 1 = αˆ (∃x)(α = ι ‘x) *73.45 ⊢ 1 = βˆ (β sm ιΛ). The above theorem, however, indicates a different “1” of a higher relative type. *54.01 0 = df ιΛ *54.02 2 = df αˆ (∃x,y)(x ≠ y • α = ι‘x ∪ ι‘y) cls*54.02 2 = df αˆ (∃μ,ν)(μ ≠ ν • α = ι‘μ ∪ ι‘ν). One may wonder what happened to the Fregean approach to the definition of the inductive cardinals. Fregean approach would have required putting into vol. 1 the definition of μ + ν which occurs in Principia’s vol. c 2, and there we do find: *101.1 ├ 0 = Nc ‘Λ *101.2 ├ 1 = Nc ‘ι‘x *101.31 ├ 2 = Nc‘(ι‘ι‘x ∪ ι‘Λ). These do capture something of Frege’s view, but hardly the spirit. Frege’s approach would be: 0=df Noc ‘Λ f 1 =df N1c ‘ιΛ ˆ f 2 =df N2c ‘ β (f 1  f 1 c f

 



β)),

and so on. Note that to say that β comes before or equal to α in the consecutive series of inductive cardinals (natural numbers), Principia has: α

  1 c



β.31 This uses the ancestral ( 1) of the relation-e that is + 1 . Besides c

c

0 whose Fregean definition above involves the homogeneous relation-e that is Noc, the definition of 1 uses the descending relation N1c and other definitions above use the descending relation-e that is N2c. That is because one

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must be able to count individuals. Principia’s definition of cardinal addition is this: *110.02 μ + ν =df ξˆ (∃α, β)(μ = Noc ‘α • ν = Noc ‘β .•. ξ sm α + β). c

The definition yields the following theorem: *110.643 ⊢ 1 + 1 = 2 c ξˆ (∃α, β) (1 = Noc‘α • 1 = Noc‘β • ξ sm α + β) = ξˆ (∃μ)(2 = Noc‘μ • ξ sm μ). The Fregean approach yields an analogous theorem, though the scopes are primary: Frege*110.643 ⊢ f 1 + f 1 =f 2 c ⊢ [N1c ‘ιΛ][ N1c ‘ βˆ (f 1  f 1

  c



β)]

[ ξˆ (∃α, β) (N1c ‘ιΛ = Noc ‘α • N1c ‘ιΛ = Noc ‘β .•. ξ sm α + β) = ξˆ (∃α) (N1c ‘ βˆ (f 1  f 1 β) = Noc ‘α • ξ sm υ)].

  c



Why is this not in vol. 1? The likely answer is connected to the fact that in November 1910, Whitehead came to realize substantive new introductory remarks are needed to introduce conventions to stabilize the ambiguity of the expression “Nc ‘α.” This also required changes to the definition of cardinal addition and other operations.32 The original untenable definition was this: μ + ν =df ξˆ (∃α, β)(μ = Nc‘α • ν = Nc‘β .•. ξ sm α + β ). c

I suspect that there was an emergency rearrangement moving into vol. 1 material on selections and Zermelo’s axiom of choice which were originally and naturally slated for vol. 2, and moving the material *100–*106 of on cardinals originally and naturally slated for vol. 1 into vol. 2. (The definitions of *63–*65 on relative type, after all, set the stage for the discussions of cardinals of relative types and are used only in that discussion.) As it is, vol. 2 repeats discussion of 0, 1, and 2. It wasn’t until November 1910 that Whitehead discovered that the unruly ambiguity of “Nc ‘α” has broad implications for the expressions for theorems on cardinals. Even in distinct occurrences of it in the same wff its meaning can change. Mind you, this is not an issue of simple type ambiguity of Principia’s individual variables which is quite uniform. Nor does it concern the use of lower-case μ, α, β, and so on, for classes, nor the use of R, S, and so on for relations-e, which are also quite uniform. The unruly nature of “Nc ‘α” caused Whitehead a great distress. The production of

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vol. 2 was delayed a year. Russell had no role.33 Details of the impact on vol. 1 remain uncertain.

A New Axiom **105 Whitehead and Russell held that nothing within the revolutionary conception of mathematics assures that there are infinitely many inductive cardinals (natural numbers). This thesis is not a consequence of their logicism. The intuition of infinity comes solely from the metaphysicians that imagine primitively arithmetic notions of addition, order, and sequence of a distinctly arithmetic necessity governing numbers as abstract particulars. If it were epistemically plausible that there are infinitely many logical particulars, Whitehead and Russell certainly would have added an axiom to that effect to Principia. But even if they had, it still wouldn’t make it an arithmetic matter that there are infinitely many inductive cardinals. This point came as a shock to great many metaphysicians who, taking Frege’s logicism as a paradigm, demanded that Principia supply a proof of the infinity of natural numbers. Such a demand, which is fully appropriate for Frege’s logicism, is entirely inappropriate for the Whitehead-Russell logicism. Frege rejected the revolution within mathematics that Whitehead and Russell so lauded. There is nothing in the revolutionary Cantorian notion of number that should make the infinity of natural numbers a matter of arithmetic (mathematics). By the lights of the revolution, it is not an issue proper for mathematics to settle. Accordingly, given the revolutionary Cantorian notion of natural number as a finite (inductive) cardinal (understood in terms of one-to-one correspondence relations), not all the Peano/Dedekind postulates for natural numbers—postulates guided by intuitions of abstract particulars—can properly be regarded as guided by genuinely mathematical intuitions. Principia has: *120.101 ├ α ∈ NC induct ≡ (ξ∈μ ⊃ξ   1 .•. 0∈μ :⊃μ: α∈μ) c *120.12 ├ 0 ∈ NC induct *120.121 ├ α ∈ NξC induct ⊃   1 ∈ NξC *120.124 ├   1 ≠ 0

  c



c

*120.31 ├ ∃!(Nc‘   1 ) :⊃: Nc‘   1 = Nc‘   1 .⊃. Nc‘α = Nc ‘β c

c

c

The last is substantially weaker than Peano’s Axiom 4. But the following is provable:

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Infin ax :⊃: (α)(β)(α ∈ NC induct • α ∈ NC induct :⊃:   1 =   1 .⊃. α = β). c

c

*120.03 Infin ax =df (α)(α ∈ NC induct ⊃ ∃! α). In spite of its name, the wff that is infin ax is not an axiom of Principia. The work is no more guided by Dedekind/Peano metaphysical intuitions concerning numbers as abstract particulars than is modern geometry is guided by Euclidean intuitions about abstract particulars that are spatial figures. The revolutionary mathematics offers the light post, not intuitions of abstract particulars. Misunderstanding this, and taking Frege’s logicism as if it were the paradigm, critics often maintain that Principia abandoned logicism in favor of “If-Thenism,” i.e., the thesis that from proper axioms governing the entities that are particulars of a given field, only quantification theory is needed to derive the theorems of that field. For example, from the axioms of Euclidean geometry governing abstract particulars (lines, points, triangles, etc.), nothing more than quantification theory is needed to derive the theorems concerning these geometric particulars. From the Peano/Dedekind axioms for natural numbers as abstract particulars, nothing more than quantification theory is needed to derive elementary arithmetic governing these particulars. If-thenism applies to axiomatic astronomy as much as to mathematics. It is certainly not logicism. The criticism that Principia is an If-Thenism was pioneered by Putnam (1967) and Boolos (1994) among others.34 It is misguided because it entirely misses (or dismisses) the revolution within mathematics against abstract particulars. If-Thenism obviously cannot embrace the revolution. It is limited (even if it allows simple-type regimentation) to a quantification theory with identity. It does not embrace *12.n comprehension. Impredicative comprehension is needed for the constructions involved in the revolutionary view that mathematics is a science of relational structures. The If-Thenisit’s axioms for any field always govern particulars. Whitehead and Russell are not guided by metaphysical intuitions governing particulars. The ever-popular criticism that Principia’s logicism fails over the issue of the infinity is misguided. It is a question-begging perspective derived from the anti-revolutionary metaphysicians themselves. Metaphysicians imposing an ontology of abstract particulars upon the fields of mathematics do not have the authority to establish the agenda for what must be reconstructed if Principia’s logicism is to be successful. The

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question of what is properly a mathematical (arithmetic or geometric) necessity, quite obviously, is not to be determined by the very metaphysicians of abstract particulars which (as Russell put it in his A History of Western Philosophy, p, 829) are guilty of “muddles.” Principia’s isn’t shy about this matter. It notes that, in spite of there being no arithmetic assurance that there are infinity of many natural numbers, the study of every kind of structure, including the study of ‘Progressions,’ can be conducted. The properties that the domain of a relation has when it is an exemplified Progression can be fully defined without appeal to inductive cardinals and be fully studied independently of whether any relation is a progression (See PM, Vol. 2, CP. 245). It is not a theorem of Principia that the class of natural numbers are the domain of a progression. Thus, Principia does not follow Cantor in defining the notion of a progression by appeal to an assumption of infinitely many inductive cardinals (natural numbers). For a similar reason they do not follow Cantor’s definition of ℵ0. They write (PM, vol. 2, p. 260): Cantor defines ℵ0 as the cardinal number of any class which can be put in one-one relation with the inductive cardinals. This definition assumes that ν ≠ ν +c 1, when ν is an inductive cardinal; in other words, it assumes the axiom of infinity; for without this, the inductive cardinals would form a finite series, with a last term, namely Λ. For this reason among others, we do not make similarity with the inductive cardinals our definition. We define ℵ0 as the class of those classes which can be arranged in progressions…

There is no theorem in Principia that is ∃!ℵ0 . The fact is that the infinity of natural numbers is not, according to Principia, an issue within mathematics. This is a jolt to any metaphysicians guided by intuitions of numbers as abstract particulars. But it is hardly a surprise for the revolutionaries. The infinity of the natural numbers as inductive cardinals may (epistemically) be logically necessary; and it may (epistemically) not be logically necessary. But either way, this is not a mathematical issue. Likely, Whitehead held that infin ax is logically necessary (PM, Vol. 2, p. 183): It is important to observe that, although the axiom of infinity cannot (so far as appears) be proved a priori, we can prove that any given inductive cardinal exist in a sufficiently high type.

The import of the phrase “so far as appears” has often been missed. Principia quite frequently reports that monism is (epistemically) possible (vol. 1: notes to *22.351, *24.1, *50.33; vol. 2: notes to *101, *120.03, pp. 8, 183). The question is epistemic. Principia’s formal system should not render a

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logical theorem of the infinity of logical particulars (of lowest type) because it is not self-evident. But it certainly does not follow that Whitehead and Russell thought it is self-evident that it is not a logical necessity. The point is that we have no special epistemic access to what is, or is not, a logical necessity. We can be wrong, thinking something is logically necessary when it is not (which happened with Frege Basic Law Vb). And we can be wrong in thinking something is not logically necessary when it is (which happened with both Descartes and Hume concerning 1+1 = 2). Whitehead and Russell knew well enough that Principia’s formal axiomatic system was incomplete with respect to logic. They knew that there may (epistemically) be a logical necessity not captured among its theorems. If there are infinitely many universals in lowest simple type, then it is a logical necessity that there are infinitely many natural numbers (in every relative type). The infinity of the natural numbers, however, is not part of mathematics. In vol. 2 of Principia, there are striking new discoveries. We find the following presented as Cantor’s power-theorem: *117.66 Nc ‘Cl ‘α > Nc ‘α h*117.66 Noc ‘Cl ‘α > Noc ‘α.35 I’ve modified it with “h” for homogeneous. It should be noted that according to Principia, the power-class Cl ‘α is of higher relative type than that of α. So the above is a comparison of cardinals of different relative type. We are familiar with Cantor’s diagonal theorem: 

*116.72 ⊢ Nc ‘Cl ‘α = 2 Nc   h*116.72 ⊢ Noc ‘Cl ‘α = 2 No c Again, it is homogeneous cardinals that are needed for the theorem. There are also important unstated results that derive from Cantor’s diagonal techniques as applied to classes of individuals (rather than classes of classes). The following is provable if we restore simple-type indices to the individual variables: Noc ‘ Vxt  > Noc‘ Vxt . The homogeneous cardinal number of the universal class Vxt  of individuals of simple type (t) is larger than the homogeneous cardinal number of the universal class Vxt of individuals of simple type t. This is not about classes of classes. It is about classes of individuals of different simple type. You won’t find this as a theorem in Principia because Whitehead and Russell were loath to restore simple-type indices. But Whitehead knew. (See PM, vol. 2, vii.) In the case of individuals, we have:

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⊢ Noc ‘ Vxt  ≥ 2

N o c ′V

xt

.

This says that the cardinal number of the universal class Vxt  of individuals of simple type (t) is at least as large as 2 to the power of the cardinal number of the universal class Vxt of individuals of the next lower type t. It might well be greater. Whitehead’s view that it is epistemically possible that the universal class Vxt of individuals of a lower simple type t may be finite while the universal class Vxt  of individuals (not classes) of next higher simple type is infinite. To say this, and to prove it, we need to restore simple-type indices. Sadly, it has gone largely unnoticed. Whitehead does prove that where descending non-homogeneous relations of ‘similarity’ are involved, Cantor’s power-class theorem yields the result that some descending cardinals are empty: *105.26 ⊢ N1c t ‘α = Λ. Of course, nothing here says that an inductive cardinal is empty. Note that t ‘α is the universal class

µˆ (μ = α ∨ μ ≠ α). The theorem proves that N1c t ‘α, which is a cardinal number, equals the empty class Λ. Indeed, in consideration of simple types of individuals, if instead of t ‘α we have the universal class Vxt  of individuals, we transform *105.26 into the following: **105.26 ⊢ Nc( β xt ) ‘ Vxt  = Λ. This is important, but not explicitly mentioned in Principia since Whitehead and Russell never restored simple-type symbols. It follows from Cantor’s diagonal method paired with non-homogeneous relations between individuals of different simple type. That is, one gets the following CantorIndiv ⊢ ~(∃β)( β xt sm Vxt  ). This says that no-class β xt of individuals (each member of which is equal to or not equal to the individual x t) is similar to the universal class Vxt  of individuals (each member of which is equal to or not equal to the individual x(t)). You won’t find the expression of CantorIndiv anywhere in Principia. Obviously, its proof requires that simple-type indices be restored to the individual variables.

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Where homogeneous cardinals are concerned, all is well with Principia emulating a homogenous version of Hume’s Principle. Principia has: *103.14 ⊢ α sm β ≡ Noc ‘α = Noc‘β. All the same, Whitehead reveals that Cantor’s power-class theorem, with descending relations of similarity, entails that not all instances of Hume’s Principle are true: (Hume’s Principle) α sm β ≡ Nc ‘α = Nc ‘β. Whitehead writes (PM, vol. 2, p. 15): *100.321 ├ α sm β ⊃ Nc ‘α = Nc ‘β Note that Nc ‘α = Nc ‘β ⊃ α sm β is not always true. …For if the Nc concerned is descending, and α and β are sufficiently great, Nc ‘α and Nc ‘β may both be Λ.

Hume’s Principle has false instances in descending cardinals. The result carries over to Frege’s Grundgesetze theory of levels of functions since it fully embraces non-homogeneous functions and thus non-homogenous equinumerosity. Neo-Fregeans wedded to the idea that Hume’s Principle is somehow essential to the concept of cardinal number (as referring to an abstract particular) must therefore excise non-homogeneous ‘equinumerosity’ from Frege’s hierarchy of levels.36 Whitehead speaks of a descending cardinal Nc ‘α where α is “sufficiently great” so that it may be empty. He leaves it entirely unsettled when it is sufficiently great. It is open, therefore, to add a new axiom that answers the question definitively. The axiom I propose is this: **105 Λ = Nc( β xt ) ‘  xt  ⊃  xt  sm Vxt  . The axiom **105 maintains that for a descending cardinal number (as a class of classes that are each equal to or not equal to β xt ) the only case where Λ = Nc( β xt ) ‘  xt  is when  xt  sm Vxt  . In transposition, the axiom is this: ~(  xt  sm Vxt  ) ⊃ (∃β)( β xt sm  xt  ). This says that if α not similar to Vxt  then there is some class β xt of individuals of the next lower type that is similar to it. The new axiom **105 naturally belongs in section *105 on descending cardinals. It is an extension of ideas in the section applying them to classes of individuals of different simple type.

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I believe that Whitehead would be sympathetic to new axiom **105. It secures the infinity of natural numbers if we go up sufficiently high in relative type. The new axiom yields: ⊢ **105 Inf V t  .37 x

Thus, it yields: ⊢ **105 Inf V o . x

With axiom **105, we can see that if we go high enough in simple type, we can avoid having the wff that is Infin ax as an antecedent clause. We have: ⊢ **105 Infin ax (x((t)) ). That is a nice result, perhaps too nice for Whitehead to resist.38 Interestingly, one can use **105 to arrive at: ⊢ **105 (∃xo, yo)( xo ≠ yo) ⊃ Inf Vxo . Accordingly, we get: ⊢ **105 (∃xo, yo)( xo ≠ yo) ⊃ Infin ax (x(o) ). If there are (contingently) at least two individuals of lowest simple type, then by **105 we are assured contingently of the infinity of Vxo , that is, the universal class of individuals of simple type (o). The empirical census of individuals of lowest type that would be needed to assure that Inf Vxo is quite a bit easier to conduct than Whitehead thought!

Truth as Correspondence Recursively Defined To recover from the abandonment of the substitutional theory, Russell offered an introduction to Principia that tries to validate impredicative comprehension and simple-type regimentation by a nominalistic semantics based on a recursively defined hierarchy of levels of “truth” and “falsehood.” The nominalistic semantics failed. But once again Russell managed to transform events into something good: his multiple-relation theory and

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philosophical agenda against abstract particulars in epistemology were preserved. The role (and even the existence) of the recursive definition of “truth” of Principia had long been lost because Church’s influential interpretation altered the grammar of Principia to code an r-type (ramified type) syntax. Church thought Principia should embrace a metaphysics of propositions. Since Principia’s recursion is not explicit when it comes to the truth-­ conditions of quantifier-free molecular wffs, Church insists on the misinterpretation that Principia’s logical particles stand for relations between propositions. Principia’s recursion is actually quite explicit for quantified wffs (i.e., “general judgments”). We find the following (PM, p. 46): We shall use the symbol “(x). φx” to express the general judgment which asserts all judgments of the form “ φx.” Then the judgment “all men are moral” is equivalent to “(x). ‘x is a man’ implies ‘x is mortal,’” i.e. (in virtue of the definition of implication) to “(x). x is not a man or x is mortal.” As we have just seen, the meaning of truth which is applicable to this proposition is not the same as the meaning of truth which is applicable to “x is a man” or to “x is mortal.” And generally, in any judgment (x). φx, the sense in which this judgment is or may be true is not the same as that in which φx is or may be true. If φx is an elementary judgment, it is true when it points to a corresponding complex. But (x). φx does not point to a single corresponding complex: the corresponding complexes are as numerous as the possible values of x.

Observe that the passage speaks as though a quantified wff “(x) φx” were a judgment. Russell is not speaking of a judgment-fact generated by the general judgment that (x) φx. He means simply to be speaking of the wff “(x) φx” itself. The point is that in giving this wff’s truth-conditions we point to many facts, each contributing to its truth-conditions. That clearly reflects the intent to offer a recursive definition of “truth.” The recursion is given quite explicitly in the following (PM, p. 42): That the words “true” and “false” have many different meanings, according to the kinds of proposition to which they are applied, is not difficult to see. Let us take any function φ xˆ , and let φa be one of its values. Let us call the sort of truth which is applicable to φa “first truth.” (This is not to assume that this would be first truth in another context: it is merely to indicate that

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it is the first sort of truth in our context.) Consider now the proposition (x). φx. If this has truth of the sort appropriate to it, that will mean that every value of φx has “first truth.” Thus if we call the sort of truth that is appropriate to (x). φx “second truth,” we may define “ {(x). φx} has second truth” as meaning “every value for φ xˆ has first truth,” i.e. “(x). (φx has first truth).”

To recursively define “falsehood” as applied to quantified wffs, Principia writes (PM, p. 42): Similar remarks apply to falsehood. Thus “ {(x). φx} has second falsehood” will mean “some value for φ xˆ has first falsehood,” i.e. “(x).(φx has first falsehood),” while “{(∃x). φx} has second falsehood” will mean “all values for φ xˆ have first falsehood, i.e. “(x).(φx has first falsehood).” Thus the sort of falsehood that can belong to a general proposition is different from the sort that can belong to a particular proposition.

As we can see, the passages allow “… has first truth” and “… has second truth” to flank quantified wffs. Modern logicians following Tarski’s recursive definition of true-satisfaction take “… is true” to flank a quoted atomic wff of a given formal language L. That is, they demand: “ Rn (x1,…, xn)” is trueL. This is not Russell’s view. There are no quote marks and so it does not intend: “…” is true. Where the wff φ is quantifier-free, the definition has: [(x) φx] is true1 =df (x)( [ φx] is truee). If it were to require quote marks, the definiendum would incoherently quantify into quotation. The recursive definition of “truth” and “falsehood” for quantified wffs works together with the definitions of Principia’s section *9 on quantification theory. In that section, all subordinate quantificational wffs are defined by means of wffs in which all the quantifiers are initially placed. Thus, for example, where φ and ψ are quantifier-free, [(x) φx ∨ (∃y)ψy] is true1.2; e. 2 =df [(x)(∃y)(φx ∨ ψy)] is true1.2; e. 2

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This introduces fine grained distinctions.39 As Russell puts it: “…as was explained in *9, two propositions which do not contain the same number of apparent variables cannot be of the same type” (PM, p. 162). What then is the intended recursive definition for quantifier-free wffs and for atomic wffs? Principia’s informal semantic introduction doesn’t say. We find (PM, p. 44): Owing to the plurality of the objects of a single judgment, it follows that what we call a “proposition” (in the sense in which it is distinguished from the phrase expressing it) is not a single entity at all. That is to say, the phrase which 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.

The asterisk is to a footnote that cites Principia’s Chapter III on the theory of definite descriptions. In the simplest case, truth consists in a relation of correspondence between a fact and a judgment-fact. This suggests that if one is acquainted with the relation R, then when the wff: Rn(x1,…, xn) is flanked by “… is true” it acts as a disguised definite description for the purportedly corresponding fact that its truth-maker. Russell’s intent seems quite clearly to define “truth” as applied to an atomic wff in terms of the existence of a definite description of the would-be fact satisfying the definite description. “Falsehood” is then defined as the absence of any such fact. Recursion handles the rest. In Landini (1991), I imagined this:  Rn    [Rn(x1,…, xn)] is truem =df E!(ιf )( m- Bn + 2-  x1  corresponds to f ).    x   n I noted, however, that this has difficulties since it introduces m as a subject of the mind engaging in belief. To see the trouble, consider this: [p ∨ q] is truem =df p truem ∨ q is truem.

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It is implausible in the extreme to assume that such belief-facts are always available. No reference to any mind or subject should occur. Perhaps Russell intended just to give the following definition: M believes truly that Rn (x1,…, xn) = df  Rn    E!(ιf )(m- Bn + 2-  x1  corresponds to f ).    x   n This comports with what he says in the following passage (PM, p. 44): … the phase that expresses a proposition is what we call an “incomplete symbol”; it does not have meaning in itself, but require some supplementation in order to acquire a complete meaning. This fact is somewhat concealed by the circumstance that judgment itself supplies a sufficient supplement, and that judgment itself supplies no verbal addition to the proposition.

If so, all is well. He wasn’t intending that a mind be explicitly mentioned in the base case of his recursive definition of “truth.” The atomic (base) case of the recursion looks like this: [Rn(x1,…, xn)] is true =df E!(ιf )(f consist x1, …, xn and Rn with Rn providing its proper order). This facilitates the straightforward general recursion: [~p] is truee. 1 =df ~([p] is truee. 1 ) [p ∨ q] is truee. 2 =df [p] is truee. 1 ∨ [q] is truee. 1. [~(p ∨ q)] is truee. 2 =df ~([p ∨ q] is truee. 2). Quantifier-free molecular facts are anathema. The logical particles cannot be relation signs unless one accepts an ontology of propositions. And now with the base of the recursion made clear, we see that thrust to the fore are all the wonderful issues in Theory of Knowledge concerning how to properly form the definite description in the atomic case(s). Russell came to know very well the challenging difficulties in describing the “proper order.”

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There is no justification for resisting this simple approach to the recursion and holding that in Principia Russell held that quantifier-free molecular facts somehow contain ontological analogs of the logical particles—something he explicitly denies in Theory of Knowledge.40 Of course, Principia assures that there is a relation Σ! such that (φ, x, ψ, y)( Σ!(φ!, x, ψ!, y) ≡ φ!x ∨ ψ!y). After existential instantiation, to assert Σ!(φ!, x, ψ!, y) is equivalent to asserting φ!x ∨ ψ!y. But the existence of this four-term relation Σ! by no means makes “∨” into relation sign.41 Indeed, the very expression of comprehension of the relation Σ! uses the sign “∨” in such a way that it is not a relation sign. Moreover, it must not be thought that Principia took upon itself the task of emulating the Principles logic of propositions and thus that, in spite of its no-propositions theory, the logical particles of Principia somehow are relation signs as they were in the Principles era. Indeed, some held that the thesis that logical particles are not relation signs was a thesis originating with Wittgenstein.42 Russell was interpreted as agreeing with Wittgenstein in his logical atomism lectures when he wrote that “…the correspondence of a molecular proposition with facts is of a different sort from the correspondence of an atomic proposition with a fact” (LLA, p.  211). But Russell writes that analogs of the logical particles “and,” “or,” “not” are not constituents of facts (Russell, LLA, p. 196): …the components of the fact which makes a proposition true or false, as the case may be, are the meanings of the symbols which we must understand in order to understand the proposition. That is not absolutely correct, but it will enable you to understand my meaning. One reason it fails of correctness is that it does not apply to words which, like “or” and “not,” are parts of propositions without corresponding to any part of the corresponding facts.

It is a sheer misinterpretation to think that these are statements indicating a change in mind from Principia under the influence of Wittgenstein! Principia clearly rejects the Principles era ontology of propositions. It is not trying to emulate a logic of propositions by means of is multiple-­ relation theory. It instituted its recursive definition of “truth” long before Wittgenstein first appeared in Cambridge. Wittgenstein learned the position that logical particles are not relation signs from his apprenticeship with Russell.

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In all the chaos of 1918, we must keep squarely in mind that in the Principia era, no negative facts, no general facts, and no molecular facts (disjunctive facts, conjunctive facts) are truth-makers. In Theory of Knowledge, Russell had also been quite clear. He wrote (TK, p. 132): But a logical form, even an atomic form, is not a constituent of the particulars that have that form. Thus it is, in some sense, further removed from the particular than the universal is. Finally, 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’.

What Russell hadn’t achieved in Theory of Knowledge is avoiding cognitive analogs of the logical particles—analogs acquaintance with which ground our understanding of ‘all,’ ‘and,’ ‘or,’ and ‘not.’ That was a problem which he hoped Wittgenstein’s new ideas might lead to a solution. Admittedly, one needs an account of the judgment (belief) facts that are to be the truth-bearers for general belief. What, for example, is the truth-bearer when one believes that (x) φx? That concerns the challenging question of the problem of compositionality. But the pressing question for the present is why Russell should have come to think in his 1918 Logical Atomism Lectures, that negative facts and general facts are needed as truth-makers? The existence and significance of the recursive definition of truth in Principia has long been missed but it reveals that at that time Russell saw no need whatsoever to embrace general facts as truth-makers. Russell’s 1918 lectures mark a change of mind, but they do not give stand-­ alone arguments. We find the following: it would be a great mistake to suppose that you could describe the world completely by means of particular facts alone. Suppose that you had succeeded in chronicling every single particular fact throughout the universe, and that there did not exist a single particular fact of any sort anywhere that you have not chronicled, you still would not have got a complete description of the universe unless you also added: ‘These that I have chronicled are all the particular facts there are’ (LLA, p. 183) It is perfectly clear, I think, that when you have enumerated all the atomic facts in the world, it is a further fact about the world that those are all the atomic facts there are about the world, and that is just as much an objective fact about the world as any of them are. It is clear, I think, that you must admit general facts as distinct from and over and above particular facts. The same thing applies to ‘All men are mortal’. When you have taken all the

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particular men that there are, and found each one of them to be severally mortal, it is definitely a new fact that all men are mortal; how new a fact, appears from what I have said a moment ago, that it would not be inferred from the mortality of the several men that there are in the world. (LLA, p. 236)

Russell knew that to describe the facts of the world completely, one needs the concept ‘all.’ It is impossible to eliminate the concept ‘all  ’ in favor of a concept of ‘conjunction.’ When one articulates precisely which facts contribute to making an all-statement true, one will rely on indicating all and only those facts that are relevant to the truth-condition. That is correct, but it is entirely irrelevant to the metaphysical question of whether there are general facts. The question was not whether one can eliminate the concept ‘all.’ Neither is the question about what may justify one in inferring, merely from knowledge of several cases, that these are all the required cases. With definite descriptions of fact, one can state the truth-­conditions, say, for “All men are mortal,” that is, (x)(~Hx ∨ Mx). One has: (a1…an)((y)(~E!(ιf )(y CHf ) ∨ E!(ιf )(y CMf ) .⊃. y = a1 ∨…∨ y = an) : ⊃ : (x)(~Hx ∨ Mx ) ≡ ([~E!(ιf)(a1 CHf) ∨ E!(ιf)(a1 CMf)] • …• [~E!(ιf)(an CHf) ∨ E!(ιf)(an CMf)])).

Nothing here requires a general-fact as a truth-maker. Many facts equally contribute to its being true that all humans are mortal. It is difficult to understand how in 1918 Russell could have missed theses points—given that in the Principia era they were his own views! The only viable answer is that in 1918 Russell had abandoned his position that one can give a definite description of a fact. In the Lectures, Russell is explaining what ontological commitments are forced once one abandons the view that one can form definite descriptions of facts. Consider the following (OPw, p. 288): 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.

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It is amusing to realize that when Russell says, “there might be an attempt” he is likely speaking of his own attempt. In Problems, Russell wrote that “…a belief is true when there is a corresponding fact, and is false when there is no corresponding fact” (P, 129). Quite clearly, Russell’s multiple-­ relation theory maintained that the belief that A loves B is false when there is no fact of A’s loving B. There being no such fact is certainly not itself a new negative general fact. Interestingly, Russell’s tentative stand in the Lectures embracing general facts and negative facts didn’t last long. We do find a discussion of negative facts in the 1919 paper “On Propositions: What they are and how they mean,” and in The Analysis of Mind (1921). But we find nothing of negative facts or general facts in the paper Logical Atomism (1924). They are omitted entirely from An Outline of Philosophy (1927) and The Analysis of Matter (1927). Russell is decidedly unsympathetic in Inquiry Into Meaning and Truth (1940). He rejects not only negative facts but also general facts as truth-makers, writing: “…there is no one verifier for ‘all men are mortal’” (InQ, p. 256). In Human Knowledge: Its Scope and its Limits (1949), Russell says that “the world can be described without the word ‘not’ ” (HK, p. 500). Looking back in My Philosophical Development, he writes (MPD, p. 186): It is only the simpler sort of statement that has a single verifier; the statement “all men are mortal” has as many verifiers as there are men.

And Russell continues, explaining as follows: (MPD, p. 188): The complexity of the correspondence grows greater with the introduction of logical words such as “or” and “not” and “all” and “some.”

This passage reminds us of the recursive definition of truth as correspondence of the Principia era. The Principia era adopts a guiding principle inspired by the recursive definition of “truth.” Facts that are truth-makers are such that, if they do not themselves have facts as constituents, their existence is logically independent of one another. This fundamental thesis of the Principia era rules out negative facts, molecular facts, and general facts as truth-makers. The logical particles appearing in the wffs of comprehension have no analog constituents in the universals comprehended. For example, from comprehension, the following is a matter of pure logic:

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(∃θ)( θ!(ψ!) ≡ψ! (x, y, z)(ψ!xy • ψ!xz .⊃. y = z) .•. (∃ φ)(φ!x ≡x x = x .•. θ!(φ!))). Under a Realist interpretation, and after existential instantiation it follows that there is a fact that θ!(φ!) composed solely of universals. Indeed, it is a purely logical fact which cannot fail to exist. It is logically necessary that ‘identity’ has the property of being transitive. It would be misguided, however, to characterize the fact that θ!(φ!) as a “general fact.” Moreover, we are not acquainted with the property of being transitive and hence the fact that θ!(φ!) is not a truth-­maker. What facts are truth-makers depends on what belief-facts as truth-­bearer there are. There are, it must be understood, a great many facts that are not truth-makers. We are not acquainted with universals that we understand only by understanding the wffs involved in their comprehension. In this way, the fundamental thesis of the Principia era is protected. The same point enables the fundamental thesis of the Principia era to allow that there are logically interdependent facts. The metaphysics of the comprehension of universals offers no good arguments for molecular and/or general facts. The wffs expressing the exemplification conditions of a given universal do not have ontological counterparts that are constituents of the universal or of a fact in which it is exemplified. A “physical conjunction” of facts is unintelligible if “conjunction” is to have the same meaning as the notion of logical “conjunction.” Armstrong once gave arguments for so-called conjunctive facts in physics. The property of mass is an example of a universal whose contingent exemplification physically assures the contingent existence of other facts. Universals for elements on the Periodic Table offer further examples. Armstrong offers the example of a methane molecule. Perhaps there is a physical universal such that when it is instantiated through time, we get number of facts in time (events) involving electron orbitals of a methane molecule. Following Whitehead, a methane molecule is an event through time (not at a time), but even so it seems untoward to ask how many events are “conjoined” in a methane molecule. A methane molecule is a complicated event which involves protons and neutrons (and many elementary particles composing them) and electrons interacting with one another through time. Thus, it is best to hold that the exemplification of the physical relation involving electrons which hold the methane molecule together is many-placed relation. There need be nothing conjunctive about it. The exemplification of

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several many-place electron orbital multiple relations of bonding may be involved as well. But again this is not a conjunction of events, each of which is the exemplification of a dyadic relation of bonding. It is rather a complex multiple relation of causal entanglement that is exemplified:  en 1   e1      n + m —  .    — Causes e  e   m   n This is not a conjunctive fact (or event). Armstrong’s appeal to physical events and causal nexuses offer no good evidence for the intelligibility of “conjunctive” facts. An appeal to the contingent exemplification of many-­ place relations is always available for physics. What about negative facts? Is there any good evidence from physics? Perhaps as an example of a negative fact one might offer a physical vacuum.43 David Lewis tells an amusing story of how death occurs upon the event of the air in the lungs rapidly escaping when a person is thrown into a void. “This is how the void causes death,” Lewis writes in musing about its naïve description, “…it is deadly not because it exerts forces and supplies energy, but because it doesn’t.”44 Quite the contrary, we know that the air in one’s lungs escapes in a vacuum because the presence of repulsive forces of the molecules in the lungs are not sufficiently counterbalanced by the presence of such repulsive forces between molecules outside. The state requiring explanation is not air escaping, but air staying in one’s lungs in the first place due to the pressure of the air surrounding us. Of course, one must appeal to some or another disequilibrium of the pressures, but cases of disequilibrium are not cases of physical absence. For example, the notion of a physical vacuum in a box is not the notion of nothing being in the box.45 Any physical box must have walls whose delimitation requires mention of particles with a non-zero probability of being at positions in the box. Physical vacuums lend no credence to an ontology of negative facts.46 Negative facts, if they should exist, do not contain any ontological analog of the word “not.” Even in the Logical Atomism Lectures, Russell recognized that no facts contain ontological analogs of the logical particles “all,” some, “and,” “or,” and “not” (LLA, pp.  196, 207, 211). Indeed, he observed that negative facts are not properly characterized as truth-makers for negative statements involving the word “not” (or involving prefixes such as “un,” “ill,” “dis,” or suffixes such as “less,” and the like). Russell holds that a positive statement (i.e., one without any

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negation lurking anywhere) might well require a truth-maker that is a negative fact. And conversely, a negative statement might well have a nonnegative fact as its truth-maker. He notes (LLA, p. 182) that the very same truth-maker(s) for the true statement “Gravitation varies inversely as the square of the distance” make false the statement “Gravitation varies directly in proportion to the distance.” He says that the fact that is the truth-maker for “Socrates is dead” is what makes “Socrates is alive” false.47 What then are the constituents of a negative fact? Russell admits that he has no answer. “You could not give a general definition,” he admits, “if it is right that negativeness is ultimate” (LLA, p. 216). In the lectures, Russell wrestled with whether one can simultaneously avoid both general facts and negative facts. In avoiding one, he worries that the other crops up. Taking the case of “There is not a hippopotamus in this room,” he writes (LLA, p. 213): …it is quite clear there is some way of interpreting that statement according to which there is a corresponding fact, and that fact cannot be merely that every part of this room is filled up with something that is not a hippopotamus. You would come back to the necessity for some kind or other of fact of the sort that we have been trying to avoid.

It is difficult to understand the concern here. Perhaps Russell’s point was that a general negative statement would require, assuming that it doesn’t require a single general fact as truth-maker, the existence of negative (atomic) facts among its truth-makers. But the truth-condition for the statement “Everything in this room is not a hippopotamus” is surely not that for each, x there is a negative fact of x’s not being a hippopotamus in this room. The natural truth-condition for the general statement is simply given by saying that everything in this room is not a hippopotamus. Otherwise put, for each x that is a hippopotamus, there does not exist a fact consisting exactly of x and this room, with the relation of ‘containment’ exemplified.48 As long as one can form definite descriptions of facts, all is well. There, I think, lies his point. The 1918 Lectures find Russell exploring the consequence of his having abandoned the viability of forming definite descriptions of facts. The celebrated case Russell uses to argue for negative facts is the problem of assuring a falsehood-maker for the positive statement that “Socrates is alive.” He writes (LLA, p. 214):

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When, e.g., you have a false positive proposition, say “Socrates is alive,” it is false because of a fact in the real world. A thing cannot be false except because of a fact, so that you find it extremely difficult to say what exactly happens when you make a positive assertion that is false, unless you are going to admit negative facts.

Russell’s comment corroborates our interpretation that he was exploring the consequences of abandoning definite descriptions of facts. On the multiple-­ relation theory, it is quite expected that the belief-fact that Socrates is alive should not correspond to any fact. It by no means follows that no facts conspire to make “Socrates is alive” false. Interestingly, the case is importantly different from the hippopotamus case. One cannot very well say that what makes it false that Socrates is alive is that every fact containing Socrates fails to contain the property ‘being alive.’ Once Socrates is dead, there are no longer facts containing Socrates. It is more accurate, therefore, to say that every fact fails to contain an entity who is uniquely S with the property ‘being alive’ exemplified. Again forming definite descriptions is the key. Curiously, in an earlier lecture Russell says that “Socrates is alive” is made false by an occurrence long ago in Athens, which is the truth-maker for “Socrates is dead” (LLA, p.  182). What, we may well ask, was that occurrence in Athens? Naturally enough, the answer seems to be that it was the presence of coniine in Socrates’s blood due to his having consumed poison hemlock.49 That presence doesn’t seem to be a negative fact at all. But one might well realize that the presence of coniine is not the entire story. The presence of coniine still has to be relevantly connected to the absence of Socrates’s life. So the appeal to coniine simply pushes the problem back some steps. What this reveals, however, is our understandable ignorance about the nature of life. As Wittgenstein put it (TLP 6.4311): Death is not an event in life. There is a message there. Life is, in fact, what requires an explanation and not non-life. Happily, we can hand-wave to see the direction one needs to go. We first distinguish Socrates from his unique s-cells (where an s-cell requires his unique DNA). It is properly the s-cell that is alive, and Socrates is said to be alive only in a very derivative sense that certain important s-cells are alive. So take the case of the important s-cells in question. Let’s assume that cellular life is constituted by certain unique bio-chemical motion of molecules. Let’s call it AZ. Now obviously there are great many motions other than AZ that can be present in a cell, and a preponderance of any one (or

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many) of them constitutes the cell’s not being alive. No negative fact (event) is involved. But there are many motions other than AZ. So we are inclined to just say that some motions other than AZ occur predominantly. For a given cell, it is the fact (event) of the presence of some specific one or another such motion in that cell that is the truth-maker of it not being alive. In another cell it is NZi, in yet another cell it is NZo, and so on. Note that a low degree of the presence of, say, motion NZi is certainly not (logically or physically) incompatible with the presence of a preponderance of motions AZ constituting a cell being alive. Motions come in degrees and thus on the molecular biological view we are imagining, life motions concern the preponderance of an AZ motion. The point is simply that in a given dead cell we do not have a clue which among the great many motions other than AZ are present in it. It is philosophically misguided to insist that there is something common to all these other motions—an absence of life—at work. The proper thing to say is simply that no fact consists of this s-cell with a preponderance of motion AZ. Thus, what makes it false that Socrates is alive right after he drank the hemlock is that no fact consists of his important s-cells having the motion AZ. Statements of action and intent offer many more and equally vexing examples. Had Alcibiades admitted guilt in plotting to overthrow the Athenian democracy, perhaps Socrates’s death wouldn’t have happened in the way it did. But even if so, it certainly does not follow that Alcibiades’s omission (i.e., his failure to admit guilt) caused Socrates death. Alcibiades’s omission is not itself a negative fact with a causal power. It is best cashed out as a counterfactual of causation: if Alcibiades were to have confessed, there would have been no fact constituted by Socrates, hemlock, and the relation of ‘consuming.’ Counterfactuals require truth-conditions which we address in Chap. 7. Nothing demands that their truth-conditions invoke the existence of negative facts. Similarly, forming an intention to bring it about that one’s confession does not transpire is not an intention to bring about a negative fact. It requires only an intention to assure that every fact does not consist of one’s having the property of confessing. Once again the key to success is simply the viability of forming definite descriptions of facts. And again, the fundamental thesis of the Principia era is that where facts that are truth-makers are concerned and where a fact does not contain another as a constituent, the existence of one fact logically entails nothing about the existence or non-existence of any other.

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Notes 1. Falling prey to the equivocation, Lebens (2017, p. 95) calls it a “platitude about truth.” 2. One may wonder, as Stevens (2010) does, why I don’t include other issues as well. In some passages Russell suggests that other issues were also weighing on his mind—for example, an untoward ontology of false propositions, issues of unity, and contingent liar paradoxes of propositions formed from psychological states of assertion and belief. My answer is simply that his priority was the foundations of mathematics. Had the substitutional theory worked, there is no plausible way that such tangential philosophical concerns would have been significant enough to warrant abandoning his substitutional solution of the Russell paradox. 3. See Landini (1998a, 2007). The first to anticipate such a syntactic paradox was Cocchiarella (1980). Hylton (1980) thought he had anticipated that there would be diagonal semantic paradoxes of propositions involving ‘truth’ and ‘falsehood.’ In 1984, I found that Cocchiarella was correct, uncovering the po/ao paradoxes in Russell manuscripts. Semantic paradoxes, whether diagonal or akin to a propositional Liar, are irrelevant. No notion of “assertion,” ‘belief,’ or “truth” and “falsehood” is part of the object language of substitution. Unfortunately, Russell felt that he had to abandon propositions to evade the diagonal constructions. It turns out that he gave up just as he was about to succeed with his 1906 no-general propositions theory. See Landini (2022c). 4. See (PoM, p. 504). Committed to his claim that propositions are i­ ntensional entities of a metaphysics of linguistic meaning, Lebens (2017, p. 101) distorts Russell’s propositions into entities that are self-representations. 5. For details, see Landini (1998a). 6. See STCR, p. 185 and InS, p. 209. 7. It is plausible that for each language there is a unique Het property: (L)(∃H)(x)(Hx ≡ (∃ φ)(x denotes-L φ ⊃ ~ φx)). But nothing at all suggests that there is a unique Het property for every language. Thus, the equivocation in the Grelling cannot be repaired by holding: (∃H)(L) (x)(Hx ≡ (∃ φ)(x denotes-L φ ⊃ ~ φx)). 8. See Chap. 4 for a treatment of indexicals. 9. See Landini (2008) for a rejection of the Yablo paradox. 10. Linsky (1993) and Stevens (2004) have argued for multiple causes of the abandonment of propositions— especially emphasizing an alleged problem of propositional unity. But their alleged problem of propositional unity vanishes once one couches the matter of unity in terms of the distinction, clearly set out in Principles, between occurring as concept and occurring as

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term (of a proposition). It is by occurring as concept, that for example, the relation ‘loves’ provides the unity of the false proposition ‘Desdemona loves Cassio’ just as surely as it provides in the unity of the true proposition ‘Desdemona loves Othello.’ 11. Hylton (1980) imagined a diagonal semantic paradox and interpreted InS as addressing it with orders of propositions. Goldfarb (1989) imagined Liar paradoxes to be diagonal paradoxes of propositions arising in Russell’s quantification theory of propositions quite independently of the substitutional theory. 12. The letter is in the Russell Archives and Research Center. 13. The analog of this for facts reveals that all Slingshot arguments against facts are fallacious. See Landini (2014c). 14. There is a parenthesis counting algorithm for the order of a simple-type symbol. Count parentheses from left to right, adding one for each left and subtracting one for each right. The order of the simple-type symbol is the highest reached in the counting process (see Hatcher 1982, p. 106). 15. This was shown to be redundant. 16. Principia gives just two instances of *12.1n namely, *12.1 and *12.11. The latter assures the existence of dyadic relations in intension. There are no relations-e (relations in extension) according to Principia’s no-relations-in-extension theory. 17. This rule is implicit and we know of it by noting that it is explicit in the system of *8 of Principia’s 1925 second edition, a system which Russell offered as a replacement for *9. 18. Principia used dots asymmetrically when viable and it ordered the conjunction sign “p.q” as having wider force over all other of the above logical particles. This often enabled it to avoid using dots in connection with the conjunction sign • which was written as small as a punctuation dot. 19. The formal proof that every instance of these schemas are derivable from *9, were it to have been rigorously conducted, would have required metalinguistic induction on the length of a wff. 20. See Landini (2000). 21. Russell changed his mind in his 1919 book Introduction to Mathematical Philosophy and concluded, contrary to the epistemic uncertainty expressed in Principia, that he does know that it is not a logical truth that there are infinitely many logical individuals. 22. See Landini (2005). Russell wrote the new Introduction and appendices to the second edition without the collaboration of Whitehead. In these experiments, Russell explored some ideas he attributed to Wittgenstein, and found them only of limited success. There is good evidence that Whitehead did not want these experiments included in the second edition. See Lowe (1990).

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23. The difference plays a role at Principia *96.48. 24. See Landini (2013). 25. When the scope marker is omitted and a definite description expression repeats as in ψ!(ιx φx, ιx φx), the two are connected together. To indicate different scopes without using scope markers, one can simply put ψ!(ιx φx, ιy φy), with the convention that the left most description is to be primary. 26. For example: E!(ιx φx) .⊃. (x)~ψ!x ⊃ ~[ιx φx][ψ! (ιx φx)]). 27. There is no need to write f  (φ! zˆ ). Circumflex predicate constants are not a part of Principia’s formal language. 28. A context χ is an extensional =df (φ, ψ)(φ!x ≡x ψ!x .⊃. χ(φ!) ≡ χ(ψ!)). 29. Seeing this, Church might try modifying his rule of uniform substitution to evade the problem. But he knew full well that no rule of uniform substitution is ever stated in Principia. He takes this as an oversight, offering to fix it. I take Principia as offering axiom schemas which require no such rule of uniform substitution. 30. For instance, *63.01 t ‘x = df yˆ (y = x ∨ y ≠ x) cls*65.01 αβ = df α ∩ t ‘β, i.e., α∩ σˆ (σ = β ∨ σ ≠ β) And also there is the important notation R(β) where R is a relation-e whose domain consists of classes that are equal to β or not equal to β. 31. Frege doesn’t use + . By Fregean, I mean only that the inductive cardinals c

are defined in terms of Nc. 32. See Landini (2017a). 33. Quine (1985, p. 79) comments that Mrs. Whitehead spoke disapprovingly of Russell, complaining that “he would loll on the Riviera while Alty grubbed away at Principia.” Unbeknownst to Quine, Evelyn might have been remembering the 1911 episode. 34. Compare Coffa (1980) and Griffin (2000), Gandon (2012), Kraal (2014). 35. Whitehead’s expression of the theorem relies on his illicit definitions: *117.02 μ > Nc ‘α =df μ > Noc ‘α *117.03 Nc ‘α > μ =df Noc ‘α > μ . The definitions are illicit because the definiendum involved in each has already been defined. It is a small point, but one must be on the lookout for such issues. 36. See Landini (2017b). 37. For convenience of exposition, I omit the proof. See Landini (2022c), forthcoming. 38. This would completely obviate Whitehead’s appeal to “NC ind.” That notion relies on his untenable doctrine of “true-whenever significant” whereby we are instructed to find a relative type high enough to make true the wff he puts down as if it were a theorem!

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39. Consider the following example:  {(φ)(x)(~ φ!x ∨ φ!x)}] is true2.1; 1.1; e. b  {(1.1;e.a φ (o) )(x)(~ 1.1;e.a φ (o) (x) ∨ 1.1;e.a φ (o) (x))} is true2.1; 1.1; e. b iff For all wffs Aξ containing just two bound individual variables and just one elementary clause, { (x)(~Ax ∨ Ax)} is true2.1 − 1; 1.1 + (1 × d); e. b + (a × d) The index d is the number of occurrences that the predicate variable 1.1;e.1  φ (o) has in the wff, namely 2. In this recursion, let a = 1, and note that b is the number of distinct atomic wffs not containing 1.1;e.a φ (o) in : (x)(~ 1.1;e.1 φ (o) (x) ∨ 1.1;e.1 φ (o) (x)). In the present case, that number is 0. For example, let Aξ be the wff (∃y) Γe.1ξy then we have: (x)( ~(∃y)Γe.1xy ∨ (∃y)Γe.1xy)). In order to compute the sense of “truth” applicable to it, we have to apply the definitions of Principia’s system *9: ~(∃u) φu =df∗9.01 (u)~ φu (u) φu ∨ (∃v)ψv =df∗9.07 (u)(∃v)(φu ∨ ψv), where v is not free in the wff φ and u is not free in the wff ψ. Thus, we see that we have: (x)(u)(∃v)( ~Γe.2xu ∨ Γe.2xv), and this is true2.0; 1.3; e. 4. The truth-conditions walk downward recursively to a base case of atomic wffs. It is there, and only there, that Russell applied his multiple-relation theory. 40. See Klement (2015). 41. Contrast Klement (2015). 42. See, for example, Pears (1987) and Hylton (1980). 43. See Kukso (2006). 44. See David Lewis (2004). 45. For a discussion of what draws philosophers to embrace negative facts as truth-makers, see Molnar (2000). 46. It is not a mysterious negative fact of absence of water that can cause a plant to die. It is the presence of different pressures on a significant number of its cell walls. Cell walls can tolerate unequal pressures on their inside and outside, but they reach a limit and then breach. They breach because the inside of the cell wall has the presence of a magnitude of pressure and the outside has the presence of different magnitude not resisted by the wall’s strength. 47. The significance of this passage was pointed out to me in conversation with Katarina Perović.

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48. Note that in this case, unlike the case of “A loves B,” the definite description involved doesn’t land us in the problem of direction. No one could be justified in a concern over whether the room is contained in the hippopotamus, since not even a hippopotamus could swallow up a room. 49. The oddity that this passage does not mention negative facts was brought to my attention by Katarina Perović.

Bibliography Works

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Other Authors

Boolos, George. 1994. The Advantages of Theft over Honest Toil. In Mathematics and Mind, ed. Alexander George, 27–44. Oxford: Oxford University Press. Cocchiarella, Nino B. 1980. The Development of the Theory of Types and the Notion of a Logical Subject in Russell’s Early Philosophy. Synthese 45: 71–115. Coffa, Alberto. 1980. Russell and Kant. Synthese 45: 43–70. Goldfarb, Warren. 1989. Russell’s Reasons for Ramification. In Rereading Russell, ed. C. Wade Savage and C. Anthony Anderson, 24–40. Minneapolis: University of Minnesota Press. Griffin, Nicholas. 2000. Russell, Logicism and ‘If-Thenism’. Proceedings of the Canadian Society for History and Philosophy of Mathematics 13: 134–146. Hylton, Peter. 1980. Russell’s Substitutional Theory. Synthese 45: 1–31. Klement, Kevin. 2015. Acquaintance, Knowledge and Logic: New Essays on Bertrand Russell’s, Problems of Philosophy, pp. 189–229. Edited by D. Wishon and B. Linsky. Stanford: CSLI Publications. Kraal, Anders. 2014. The Aim of Russell’s Early Logicism. Synthese 7: 1493–1510. Kukso, B. 2006. The Reality of Absences. Australasian Journal of Philosophy 84: 21–37. Landini, Gregory. 1991. A New Interpretation of Russell’s Multiple-Relation Theory of Judgment. History and Philosophy of Logic 12: 37–69. ———. 1998a. Russell’s Hidden Substitutional Theory. Oxford: Oxford University Press. ———. 2000. Quantification Theory in *9 of Principia Mathematica. Vol. 21, 57–78. History and Philosophy of Logic. ———. 2005. Quantification Theory in *8 of Principia Mathematica and the Empty Domain. History and Philosophy of Logic 25: 47–59. ———. 2007. Wittgenstein’s Apprenticeship with Russell. Cambridge: Cambridge University Press. ———. 2008. Yablo’s Paradox and Russellian Propositions. Russell 28: 97–192. ———. 2013. Typo’s of Principia Mathematica. History and Philosophy of Logic 34: 306–334.

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———. 2017a. Whitehead’s (Badly) Emended Principia. History and Philosophy of Logic 37: 114–169. ———. 2017b. Frege’s Cardinals Do Not Always Obey Hume’s Principle. History and Philosophy of Logic 38: 127–153. ———. 2022c. Principia’s Logic as based on Russell’s Substitutional Theory of Simple Types. Lebens, Samuel. 2017. Bertrand Russell and the Nature of Propositions: A History and Defense of the Multiple-Relation Theory of Judgment. New York: Routledge. Lewis, David. 2004. Void and Object. In Causation and Counterfactuals, ed. J.  Collins, E.J.  Hall, and L.A.  Paul, 277–290. Cambridge: Cambridge University Press. Linsky, Bernard. 1993. Why Russell Abandoned Russellian Propositions. In Russell and Analytic Philosophy, ed. A.D.  Irvine and G.A.  Wedeking, 193–209. Toronto: University of Toronto Press. Lowe, Victor. 1990. Alfred North Whitehead: The Man and his Work, Vol II 1910–1947. Edited by J.B.  Schneewind (Baltimore: Johns Hopkins University Press). Molnar, George. 2000. Truthmakers for Negative Truths. Australasian Journal of Philosophy 87: 72–86. Pears, D.F. 1987. The False Prison: A Study of the Development of Wittgenstein’s Philosophy. Vol. I. Oxford: Clarendon Press. Putnam, Hilary. 1967. The Thesis that Mathematics Is Logic. In Bertrand Russell: Philosopher of the Century, ed. Ralph Schoenman, 273–303. London: Allen & Unwin. Quine, W. V. O. 1966. The Ways of Paradox and Other Essays by W. V. O. Quine (New York: Random House). First published as “Paradox,” Scientific American 206, 1962. ———. 1985. The Time of My Life: An Autobiography. Cambridge: MIT Press. Stevens, G. 2004. From Russell’s Paradox to the Theory of Judgement: Wittgenstein and Russell on the Unity of the Proposition. Theoria 70: 28–61. ———. 2010. The Russellian Origins of Analytical Philosophy: Bertrand Russell and the Unity of the Proposition. London: Routledge.

CHAPTER 4

Acquaintance

Whatever may be the right theory of the nature of intentionality, I agree with Quine that it must reject what he calls the “myth of the museum”— the myth that singular reference is the fundamental component of meaning, cognitively prior to and independent of quantification.1 When it comes to this matter, we needn’t take sides in the debate between Quine’s empiricist (without the dogmas) theory of the ontogenesis of reference2 and Chomsky’s rationalist theory of the innateness of a generative and transformational grammar. It is a lesson that one can draw from both Quine and Chomsky alike. It is a lesson that one can draw from Russell’s “On Denoting” and his debate with Meinongians over the apparatus that secures the determinate aboutness that is characteristic of intentionality. The lesson is that all predicational thought has a quantificational scaffold. I reject that old conundrum arising from the question: How do we think about what is not? It commits the fallacy of complex question. We don’t think about what is not. We think quantificationally. Quantificational thinking is not built upon quantification-free thinking. This will be of central importance to the solutions in Chaps. 5 and 6 of the problems of direction and compositionality. Quantificational thinking is not composed of singular and molecular antecedents brought together in some as yet mysterious predicational synthesis or by some equally mysterious abstraction from particular cases. Traditional empiricism notwithstanding, understanding quantification is not recovered by abstraction. In © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8_4

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fact, abstraction (recognizing patterns) is made possible only by the impredicative comprehension involved with the quantificational scaffold of intentionality. For it is here that we find the source of our understanding of the ancestral relation which is the heart of the notions of ‘any,’ ‘and so on,’ ‘general term of a sequence,’ and pattern recognition itself. I reject singular predicational thought altogether—that is, I reject the existence of predicational thoughts that do not have a quantificational scaffold. It is misguided to imagine that the use of reports de se and the use of indexicals in communication reveal that there is singular thought. The use of indexicals itself relies on a cognitive quantificational scaffold of intentionality together with a community-wide adoption of conventions. That is not to say that one must follow Quine in rejecting introspective awareness and the absolute determinacy of reference in thought.3 Quite to the contrary, impredicative quantificational thinking is precisely the scaffolding that secures the absolute determinacy of thinking, including introspective thinking. The quantificational scaffolding of intentionality is what secures its Poetic License. That is, there are no constraints whatsoever on what may be the so-called objects of thought—be they round-squares, the Russell class, the Russell property, or Desdemona’s love of Cassio. Any cognitively normal person has the power of determinate intentionality—she can control exactly what she is thinking about. No genuine contingent paradoxes can arrive from this.4 There is no good objection coming from the Pinocchio-­ style pseudo-paradox of Veronique who (on threat of contradiction) cannot think she herself to have a currently growing nose since her nose grows if and only if she asserts a falsehood.5 This can readily be dismissed. Such a person for which reality, cognition, and proboscis are lawfully linked is neither cognitively nor physically normal. Are there any such paradoxes? Prior (1961) thought so. We are working in a no-propositions framework, but it is useful to address his argument. Prior notes that for all wffs θp, the following is a theorem of a pure logic of propositions: θ(p)(θp Ͻ ~p). Ͻ. (∃p)(θp • p) • (∃p)(θp • ~p). In a contingent case where θq is “S thinks about q,” the theorem, together with θ(p)(θp Ͻ ~p), yields that (on pain of contradiction) S can’t control the object of her thought. There are two distinct propositions S thinks about. The lesson I draw from “On Denoting” unravels Prior’s puzzle. For a thought to be about x requires that x be the object of the

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quantificational apparatus of S’s intentionality. To think (descriptively) about the proposition q, one requires some wffs φ and ψ and the following: θq = df φx ≡x x = q .•. S thinks [ιxφx][ψx]. Now notice that θ(p)(θp Ͻ ~p) is simply not true. It says this: φx ≡x x = {(p)((φx ≡x x = p .•. S thinks [ιxφx][ψx]).Ͻ. ~p)} .•. S thinks [ιxφx][ψx]. So far as anyone knows there is no such wff φ. Prior certainly never gave us one.6 Alexius Meinong and his many followers held that the directedness of thinking requires a Principle of Independence (of Sein from Sosein).7 The Sosein of the object of intentionality is what directs it to that specific object and not another. This led to the famous dictum (Meinong 1904, p. 83): There are objects [of thought] of which it is true that there are no such objects.

Descriptive empirical phenomenology in the hands of Meinong characteristically proclaims that in respecting the act-object phenomenal description of the way experiences seem to us, one must accept that to think about something requires that there is something one thinks about. To think about the golden mountain requires there to be the golden mountain one thinks about. Because of this, “Meinongianism” has come to be defined as holding the principle that without further premise one can (at least sometimes) validly infer the de re “There is an object that is so and so about which S is thinking,” simply from the de dicto “S is thinking about an object that is so and so.” There is no other way, Meinongians hold, for intentionality to secure determinate aboutness. Unfortunately, for all its apparent liberalness, Meinongianism fails to capture poetic license and thus undermines its very raison d’être. To avoid contradictory ‘objects’ of thought, it requires limitations on what objects of thought there are. The proper way, and I think the only way, to capture the poetic license characteristic of intentionality is to maintain all predicational thinking has a quantificational scaffold. Meinongians often complain that a Russellian robust sense of reality is a prejudice in favor of the actual. The complaint, however, misses the point. To have a Russellian robust sense of reality is simply to reject Meinongian indispensability argument for

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intentionally inexistent object of thought. It commits a fallacy—a Meinongian fallacy. The task is to avoid Meinongianism while accepting Brentano’s Principle of Intentionality according to which aboutness is the characteristic mark of the mental. Mere rejection of Meinong’s indispensability argument, of course, is not enough. One must explain how thought is determinately directed. The task is daunting if one imagines that to succeed the very nature of intentionality itself must be explained. Happily, it is not the nature of intentionality that requires explanation in order to reply to Meinongians. Russell’s “On Denoting” and its quantificational theory of definite descriptions revealed the first steps. The use of descriptions presupposes intentionality. It does not endeavor to explain its nature. The next step is to explain how one is to avoid Meinongian fallacies and yet speak, de re, of a “shared” and “repeatable” content between several believers. One has to assure that the multiple-relation theory does not entail, that “…you can’t straightforwardly talk about that which is believed” (Soames 2014, p. 445). Quine (1956) was very sensitive to this concern. He agrees with Russell that we can get along perfectly well without invoking meanings (including meaning propositions) as abstract particulars shared in communication. A mind may be engaged in believing something, but we should not be led to conclude that there are semantic particulars that are metaphysical intensional objects of a dyadic relation of belief. That inference is just another case of committing the Meinongian fallacy. In believing, perhaps minds are engaged in the mental activity of parsing. But we should not be led to conclude that there are entities parsed. Quine knew (as Russell before him) that there must be a proper way to form de re quantification into propositional attitudes—a way that makes no use of the untoward ontology of interpretable semantic particulars that have seemed so very indispensable to a theory of intentionality. Some might go so far as to say that his work on “quantifying in” is an attempt at a “multiple-relation” theory. But it is certainly very far away from what Russell had in mind. Quine’s program against intensional entities tout court led him to a prohibition against bindable predicate variables. His impoverished inventory of logical forms pushes him to a behavioristically sympathetic thesis that construes de re quantification as an assertion that a particular satisfies an open wff. The Russellian answer is that the quantificational scaffolding of intentionality embraces impredicative comprehension. Binding predicate variables are central to Russell’s multiple-relation theory. Thus, what is shared, in

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addition to a quantificational apparatus, need only be an acquaintance with universals.

Acquaintance and Intentionality The fundamental epistemic notion of the Principia era is that there is a dyadic relation of ‘acquaintance,’ one of whose relata must be a subject (if not also a mind) as a distinctively mental entity. The relation is instrumental for all a posteriori knowledge and for Russell’s account of our synthetic a priori knowledge of cpLogic as the essence of scientific method in philosophy. In the opening chapter of Theory of Knowledge, we find (TK, p, 5): The purpose of what follows is to advocate a certain analysis of the simplest and most pervading aspect of experience, namely what I call ‘acquaintance.’ It will be maintained that acquaintance is a dual relation between a subject and an object which need not have any community of nature. The subject is ‘mental’; the object is not known to be mental except in introspection. The object may be in the present, in the past, or not in time at all; it may be a sensible particular, or a universal, or an abstract logical fact. All cognitive relations—attention, sensation, memory, imagination, believing, disbelieving, etc.—presuppose acquaintance.

Russell’s ‘acquaintance’ relation is importantly connected to Brentano’s famous Principle of Intentionality which makes psychological aboutness the “mark of the mental.” The connection between the two lies in that both emphasize the selective attention of a subject to a this presented. It is this feature that is characteristic of mind. Some of Russell’s early passages allude to Brentano’s principle and, it must be emphasized, it was adopted both in the Principles era of propositions and in the no-propositions Principia era. But to properly recognize them, we must look at his comments both during and after the Principia era. In The Problems of Philosophy, he writes (PoP, p. 42): The question of the distinction between act and object in our apprehending of things is vitally important, since our whole power of acquiring knowledge is bound up with it. The faculty of being acquainted with things other than itself is the main characteristic of a mind.

The connection of this passage to Brentano is not explicit, though the locution “main characteristic of a mind” is reminiscent of Brentano’s

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“mark of the mental” phrasing. The connection is corroborated when, in his neutral monist era, Russell says that he abandoned Brentano’s principle. In his 1919 “On Propositions: What they are and how they mean,” we find (OPw, p. 306): The first effect of the rejection of the subject is to render necessary a less relational theory of mental occurrences. Brentano’s view, for example, that mental phenomena are characterized by ‘objective reference,’ cannot be accepted in its obvious sense.

In his 1921 The Analysis of Mind, Russell admits that he had once held Brentano’s view: (AMi, p. 15): Until very lately, I believed, as he [Brentano] did, that mental phenomena have essential reference to objects, except possibly in the case of pleasure and pain. Now I no longer believe this, even in the case of knowledge.

The neutral monist era abandoned the subject and thereby abandoned relations of ‘acquaintance’ and ‘sensation.’ In My Philosophical Development, Russell admits this too (MPD, p. 185): My abandonment of the relational character of sensation led me to substitute “noticing” for “acquaintance.”

The behaviorist-inspired conception of noticing replaced the relation of acquaintance. Noticing this does not involve a relation to a subject, but is a complex event of stimulus together with the “appropriate” (adaptive?) behavioral response. “We may say generally that an object, whether animate or inanimate, is ‘sensitive’ to a certain feature of its environment if it behaves differently according to the presence or absence of that feature” (AMi, p. 260). The neutral monism that William James put forth in his 1904 “Does Consciousness Exist?” rejected the Cartesian notion of mind (and the subject) in anticipation of what nowadays we call “functionalism.” Theory of Knowledge had rejected James’s view because it seemed inadequate to accommodate selective attention to a this presented. Russell’s neutral monism adopts a sympathy for behaviorism which he employed to argue that reactiveness can replace the notion of “selective attention of a subject.” In rejecting the subject, he obviously rejected the existence of those relations whose exemplification require the subject.

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Intentionality is a rich resource for metaphysicians conjuring indispensability arguments in favor of intensional entities acting as meaning propositions that are shared in propositional attitudes.8 The quest for intensional entities to explain objects of belief and the so-called propositional attitudes lends itself to the conflation of meaning propositions with early Russellian propositions. Russell held Brentano’s principle in his Principles era. But the aboutness of Intentionality is not, and cannot be, analyzed in terms of the logical aboutness of the Russellian propositions of the Principles era. The notion of logical aboutness is the notion of an entity occurring as term in a proposition. That is simply the notion of being one among the constituents of a proposition that do not ground its unity. To be sure, Principles accepted a theory of denoting concepts. But again, the logical aboutness conducted by a denoting concept is not the work of intentionality. Rather, the mind’s intentionality has only the role of directing itself to the denoting concept (and to the proposition in which it occurs). The appeal to intensional entities to do the work of intentionality is not found in Russell’s work. Brentano’s principle concerns the mind’s selective attention and is not committed to its being a state of mind (if there is such an entity). As such his principle is not at all challenged by the modern claim that a mental state of pain (or, say, an episodic state of emotion) is not itself a representation about anything.9 In Brentano’s view, the selective attention to a pain in my tooth is of something, namely, the distinctive pain in my tooth. But Brentano wisely demurred over whether the de re “There is this pain I now attend to” can be validly inferred from the de dicto “I now attend to this pain.” Intentionality (selective attention) points, but its pointing does not itself entail any result as to whether there is or is not an entity to which it points. In Principles, Russell embraced denoting concepts in virtue of which one can deny being as well as existence. Russell separated being from existence because he held that genuine ordinary proper names such as “Aristotle” and “Apollo” and so on, get meaning solely from reference. Thus, since he took “Apollo” to be a genuinely ordinary proper name, he concluded that existence is the prerogative of some among beings. Where A is a genuine ordinary proper name, “‘A is not’ must be false or meaningless” (PoM, p. 449). Observe, however, that the form “A is not” does not apply to “the null class is not.” That is because “the null class” is not a genuine ordinary proper name. Thus, Principles quite consistently affirms that the null class is not (PoM, p.  75). In the 1904 “On the Existential

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Import of Propositions,” Russell decided that, although we don’t any longer know the original definite description, “Apollo” is not after all a genuine ordinary proper name. The theory of denoting concepts applies, and thus there is no longer a need for a distinction between being and existence (unless just to indicate a distinction between abstract and concrete). In 1905, however, he altogether abandoned denoting concepts. The only genuine terms in the language of logic are bindable variables. Vehemently rejecting Meinong’s principle, he remarked dryly in “On Denoting” that it is “apt to infringe the law of contradiction.” The existent golden mountains would, on Meinong’s view, have the Sosein of being existent! One may, of course, endeavor to gloss the Sosein principle to avoid contradictions. For example, following Meinong’s student Ernst Mally, one may distinguish the extra-nuclear property ‘existing’ from the nuclear property of ‘existing.’ But however it is to be done, to be a Meinongian is to hold that at least in some cherished cases, the de re conclusion validly follows from the de dicto premise. Russellians hold that it is never valid. Once we allow binding predicate variables, the de re quantification into belief contexts is not so very difficult to imagine. It is important to understand, however, that universals are not immediate objects of beliefs. Together with quantificational apparatus, they are subsumed into the scaffolding of believing. The properties with which one is acquainted become components of the apparatus that enables the directedness of intentionality. Care must be taken, therefore, in addressing the misleading philosophical question: “What is the thought about?” For example, Merricks proclaims that on the correspondence theory of truth the truth-maker for a statement is tied to what the statement is about. He correctly observes that the thought that (x) ~ Hx, that is, that everything fails to be a Hobbit, is not about the property H of being a hobbit. It cannot be a disguised form of the statement that property H is not exemplified, which is about H. Yet we need to know what it is about to address the difficult task of giving what is to be the truth-maker(s) for the statement (Merricks 2007, p. 44). There is a sleight of hand here. The question “What is it about?” may mean to ask quite different questions. Clearly, in one sense, the thought is about everything and thus every entity in every fact contributes jointly to its truth. (Every fact fails to contain an entity with property being a Hobbit unifying it.) But perhaps one means to ask: What property would be exemplified if the determinate intentional pointing were to be successful? That is a question that asks about the property H of being a Hobbit. Alternatively, the “aboutness” question might mean to ask: What is the

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cognitive apparatus that enables determinate intentional pointing to Hobbits? The answer to that question also concerns how the universal H is involved in the quantificational apparatus of intentionality that enables determinate pointing. Finally, one may ask: What kind of thought is the thought that (x) ~ Hx? The answer to that question is simply that is an everything-thought. Crane (2013) gives excellent examples to illustrate the philosophical mistake of thinking properties are immediate objects of thought. He argues that any appeal to properties to avoid commitments to intentionally inexistent objects of thought is “phenomenologically speaking quite unrealistic.” Of course, to “speak phenomenologically” is, by definition, to give an act-object structure of awareness and thus to speak as if there are objects. In this matter, perhaps Meinong’s notion of Aussersein and Husserl’s notion of epoché are helpful. In speaking phenomenologically, we are supposed to bracket all concerns about ontology. If one can’t do this, then one should simply stop speaking phenomenologically. It commits the Meinongian fallacy to infer from “I fear an event of my drowning” to the conclusion that “there is an intentionally inexistent event of my drowning such that I fear it.” Crane gives in to this fallacy. I agree with him when he observes that if I fear death by drowning I’m not fearing the property drowning. What I fear is an event. One’s intentionality is directed to a kind of event by the quantificational apparatus of thinking. The property of being an event of my drowning can certainly be playing a role in directing my thinking without the slightest commitment to there being any such event of my drowning, intentionally inexistent or otherwise. The property is subsumed into part of the apparatus of intentionality and is not an object (immediate or otherwise) of that apparatus. Once we see that it is perfectly legitimate to appeal to properties playing a role in the apparatus of thinking, many cases can be treated in a similar way. Crane (2013) worries about the following: Some Biblical personae did not exist.

He thus distinguishes “exists” as a predicate from the quantifier “some.” The Russellian has: (∃F)(∃x)(x is Biblical predicate characterization of a person • x denotesBible F • ~ (∃x)(Px • Fx)).

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This says that some property F is referred to by a Biblical predicate characterization of a person and it is not exemplified by any person. Binding predicate variables dispatches the Meinongian argument. Similar techniques dispatch arguments such as that of Schiffer (1994) who thinks the multiple-relation theory is unable to recover inferences of shared content. He offers the following example: s believes [ιxYx][Ax] s* disbelieves [ιxYx][Ax] Therefore, (∃p)(s believes p • s* disbelieves p). The above inference, however, is ill-formed for a Russellian. A proper conclusion is this: (∃F)(Fx ≡x Yx.•. s believes [ιyFy][Ay] • s* disbelieves [ιyFy][Ay]). This is perfectly mundane. Consider also the following, where there is a unique entity that is Y about which a speaker S* quite consistently ascribes de re to the person s as having both a belief and also a disbelief: [ιzYz][ (∃F, G)(Fx ≡x x = z .•. Gx ≡x x = z :•: s believes [ιyFy][Ay] • s disbelieves [ιyGy][Ay])]. There is no inconsistency in S* having ascribed such beliefs to s. The quantificational apparatus and a universal are what is shared. The key to the solution is simply to bind predicate variables de re. There is yet another puzzle by Schiffer (2006) which aims to indict any multiple-relation theory. Karol Wojtyla is identical to Pope John Paul II.  Schiffer would hold that the following presents Francesco as having inconsistent beliefs about Stefania’s belief/disbelief: Wojyla is such that Francesco believes that Stefania believes him to be Polish. John Paul II is such that Francesco believes that Stefania disbelieves him to be Polish.

Surely, Francesco does not think that Stefania believes and disbelieves the same thing! The error is easily diagnosed by our demands on proper de re ascriptions. We have:

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[ιxWx][ (∃G)(Gy ≡y y = x .•. Francesco believes [ιzSz][ z believes [ιyGy][y is Polish]])] [ιxJx][ (∃G)(Gy ≡y y = x .•. Francesco believes [ιzSz][z disbelieves [ιyGy][y is Polish]])]. Schiffer’s puzzle assumed that the multiple-relation theory cannot capture ascriptions of general compositional belief. In Chap. 6 on compositionality, we shall see that it can since it can capture the above. The lesson, however, is that all legitimate de re quantification into the context of a propositional attitude must respect the fact that predicational thought is scaffolded by quantification. This makes it illicit to bind an individual variable de re in the scope of a propositional attitude. Indeed, the Meinongian fallacy itself arises from engaging in such an illicit form of de re quantification. The following is ill-formed: [ιxKx][M believes that ~Gx]. A proper form of de re quantification binds a predicate variable, as in the following: [ιxKx][ (∃F)(Fz ≡z z = x .•. M believes that [ιyFy][~Gy])]. In quantifying de re, we must always respect that quantificational apparatus. Of course, since ‘belief’ is not a dyadic relation on the multiple-­ relation theory, the above does not yet reflect the incorporation of the multiple-relation theory. We will see how to do that in Chap. 6. We have to reveal how the multiple-relation theory accommodates an account of such quantification de re. I reject singular thought and work with syntactic depictions of quantificationally scaffolded intentionality. Hawthorne and Manley (2012) accept singular thought and, following current trends, endeavor to find semantic/pragmatic escapes from puzzles about belief ascription. An acquaintance constraint governs the traditional notion of singular thought—namely that in ascribing to a person a singular thought about an object, the speaker is attributing to that person an acquaintance with that object. My approach is rather orthogonal to that dispute, since I reject singular thought altogether.10 All predicational thought has a quantificational scaffold. Admittedly, the semantics/pragmatics of English belief ascriptions does often give the appearance that they render ascriptions of singular thought. Consider the following celebrated cases:

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Ralph believes some person to be a spy. i.e., Ralph believes (∃y)(Py • y is a spy). There is some person whom Ralph believes to be a spy. i.e., (∃y)(Py • Ralph believes y to be a spy). The second and not the first, it seems to Hawthorne and Manley, ascribe de re a singular thought to Ralph. The acquaintance constraint demands that it can only be true if Ralph is acquainted with the person in question. On my view, the second does not ascribe a singular thought. Rather it is this: (∃x)(Px • (∃F)(Fz ≡z z = x .•. (∃x) Ralph believes [ιyFy][y is a spy] )). The acceptable Russellian acquaintance constraint only requires that Ralph be acquainted with a property F. The speaker needn’t know what property it is. Similarly, consider this case: The shortest spy is such that Ralph believes he is a spy. [ιxSx][(∃F)(Fz ≡z z = x .•. Ralph believes [ιyFy][y is a spy])]. Again this is not a de re ascription of a singular thought. The speaker ascribes to Ralph acquaintance with a property F that the speaker takes Ralph to use to single out the shortest spy. Though again the speaker does not know what property it is, the acquaintance constraint demands that the ascription is true only if Ralph is acquainted with the property F. On the view I advocate, binding an individual variable in the context of a propositional attitude is illicit because it obliterates the quantificational apparatus underlying all discursive thinking. It is important, however, to recognize that though a quantificational apparatus is always involved, one can make belief ascriptions without knowing what quantificational apparatus is involved and without demanding that it be that of definite description. Consider this: Le Verrier had a belief about Vulcan.

If one understands the background astronomy according to which Le Verrier hoped to explain the orbital precession of Mercury by hypothesizing that there is a hidden body exerting a gravitational influence, then it seems clear that the speaker uses “Vulcan” as a disguised description—perhaps “the body whose gravitation produces the precession of Mercury.”

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Astronomers eventually concluded that there is no such body. The philosophical issues, in this case, are interesting because of the lack of a sentential context for the use of “Vulcan.” This makes it difficult to apply Russell’s theory of definite descriptions. Moreover, the speaker may well not intend to attribute to Le Verrier the use of “Vulcan” or any definite description. Le Verrier’s intentionality involves some quantificational scaffold and a speaker’s de re ascription must respect this. At the same time, the speaker needn’t know what apparatus it is. In order to accommodate this, I rely on Fregean variables Ωxφx and ΣP(ΩxPx) and so on that are quantificationally structured. For example, there are: s believes Ωxφx s believes ΣP(ΩxPx).11 These indicate the quantificational cognitive apparatus involved. Fregean structured variables help to address the many cases where a speaker ascribing a belief does not know what quantificational apparatus is involved. Fregean structured variables solve our problem: [ιzLz](∃Ω)(ΩxPx ≡P [ιyVy][Py] .•. (∃F)(z believes ΩxFx))]. In this way, Le Verrier’s intentionality was directed so that, were it to have been successful, the property V would be exemplified. Here the speaker is using a definite description involving the property V which purports to be uniquely satisfied by a body responsible for perturbations in Mercury’s orbit. In reporting on Le Verrier, the speaker is saying that some quantificational apparatus Ω is involved in Le Verrier’s intentionality whose use is successful if and only if there is a unique entity that exemplifies V. The speaker need not know what that quantificational apparatus is. The Fregean structured variables are very useful, and soon we shall see that they play an important role in representing quantification de se and in communication with indexicals. First notice that they are helpful in addressing the concern of Church (1951, p. 111, n. 14) who offered this sort of case Meinong thinks about the golden mountain.

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Obviously, one cannot put: [(ιx)(Gx • Mx)][Meinong thinks about x] All the same, to think about the golden mountain one must predicate about it. Hence, one can put: (∃H)(Meinong thinks [(ιx)(Gx • Mx)][Hx]) Admittedly, the quantificational apparatus scaffolding Meinong’s thinking might be not via definite description. This can be captured using a Fregean variable as follows: (∃Ω) (ΩxPx ≡P [(ιx)(Gx • Mx)][Px] .•. (∃H)(Meinong thinks ΩxHx)). Some quantificational apparatus is employed by Meinong in thinking, and it is directed quantificationally by the property of ‘being uniquely a golden mountain.’ In this way we can capture: Meinong believes everything that Brentano believes (Ω)(G)(Meinong believes ΩxGx .⊃. (∃ξ)(∃H)(ΩxPx ≡P ξxPx .•. Gx ≡x Hx .•. Brentano believes ξxHx). Something weaker may be preferred. Perhaps this: (y)(Ω)(G)(Meinong believes ΩxGx .•. ΩxPx ≡P (ιz)(z = y)[Px] :⊃: (∃ξ) (MxPx ≡P ΩxPx.•. (∃H)(Gx ≡x Hx .•. Brentano believes ξxHx))). This says that for every entity y, and every quantificational apparatus Ω, if Meinong has a quantificational belief about y using that apparatus, then there is some equivalent quantificational apparatus Brentano uses to think about y. The fact that we make ascriptions to people of shared belief does not present good evidence for the indispensability of intentionally inexistent objects. Quine (1956) offered concerns about verbs such as “strives to find,” “searches for,” “finds” and the like. One can, for example, search for the fountain of youth. No one will find a fountain of youth because there is no such thing. But as Quine illustrates, the issue is not about whether the target exists. For example, there is the case of “Ctesias strives to find a lion.” There are, after all, lions. But obviously, it does not follow that there is some lion that Ctesias strives to find. Just as obviously it won’t work to say that there are a, b,… each of which is a lion and such that either Ctesias strives to find a or Ctesias strives to find b or… Let us adopt: striveP{ΩzPz}.

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The idea is that striving to find (searching) implements a particular quantificational apparatus. We have: Ctesias strives to find a lion. [ιxCx][ (∃Σ) i.e. (Σ xp {Ωz Pz}.≡. striveP{ΩzPz} :•: ΩzLz)] This says that there is in the quantificational apparatus of Ctesias, an apparatus Ω functioning as an apparatus for striving to find, and together with the property L of being a lion, this apparatus is engaged. As we can see, it by no means follows from the above that there is a lion Ctesias strives to find. Intentionality can target an object without there being (de re) any object targeted. The approach I take to the problem is to regard “searches for” and “finds” (i.e., “successfully searches for”) as also indicating that a cognitive quantificational apparatus is involved. We don’t know, and don’t need to know, what quantificational apparatus Σ composes the mind of Ctesias in saying that Ω is involved. Neither do we know exactly what quantificational apparatus Ω is involved. It might be a quantificational apparatus of definite description such that for some universal H: ΩzPz ≡P Mx[ιyHy][Py • Px]. But the ascription of an apparatus Ω to Ctesias makes no specific commitment. Again, we must be on guard not to fall into the Meinongian fallacy. Thus, similarly, Ctesias finds (successfully strives to find) a lion. (∃y)(Ly • [ιxCx](∃Σ) (Σ xp {Ωz Pz}.≡. striveP{ΩzPz} :•: Ωz(z = y)]). If one attributes to Ctesias a cognitive apparatus involved in his searching for a lion, then it by no means follows validly that there is a lion. But if one finds12 (successfully strives to find) a lion, then it validly follows that there is a lion found. These cases are particularly difficult, but only because we don’t know what quantificational apparatus scaffolds the intentionality of striving to find. The solution to the situation of Ctesias is similar to the solution to cases where we have S* is obligated to pay some money owed to S S* pays some money owed to S. It does not follow validly from the former that there is some money such that S is obligated to pay it to S*. This does follow validly from the latter. Fregean variables capture such differences.

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What I Believe Intentionality embodies an impredicative quantificational scaffold. No one knows how, and as we can see, no one needs to know how in order to make ascriptions of propositional attitudes. In ascribing beliefs to others, we have no acquaintance with the mind M of the person to whom we are attributing the belief, and neither do we have acquaintance with the subject m that is a constituent of a belief-fact generated when M is engaged in believing. This same situation arises as well for ascriptions de se and the general issues arising with the use of indexicals. Acquaintance with the subject is never required for the ordinary practice of using of indexicals in communication. The use of indexicals in communication is well explained by the Russell/Reichenbach token-reflexive theory.13 Russell set it out in his Theory of Knowledge and the theory remains his considered view in his Enquiry into Meaning and Truth (1940). Kapitan (1999, 2001, 2006) points out, however, that the token-­ reflexive approach, while appropriate for the interpretation of indexicals in communication, is inappropriate as a theory of the production of indexicals (in soliloquy). He writes (Kapitan 1999, p. 5): the token-reflexive analysis is appropriate only to the interpretation of indexicals. It is dependent upon the antecedent production of indexical tokens, and very likely cannot even begin without the interpreter’s indexical identifications of the relevant tokens. To understand quasi-indexical attributions, consequently, we must turn to their source in indexical thinking.

Kapitan is correct that the cognitive apparatus that accounts for the production of indexical tokens (sounds and gestures etc.) presupposes an apparatus of determinate intentionality. The presentation of a noticed object comes before any utterance tokening of “this” in an act of communication. For example, Kapitan considers a case of Mary uttering Be careful, this is a snake.

She has already noticed the snake independently of any tokening utterance. Kapitan holds that that noticing is itself an indexical identification and is prior to the production of any indexical token.14 This original indexical identification is what Kapitan calls the “executive meaning” of the indexical, while the token-reflexive meaning is called its “interpretative meaning.” An account of the production of indexical sounds and gestures

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for use in communication relies on a philosophy of the nature of the mind’s determinate intentionality. No adequate account exists today. But Kapitan believes that the development of a philosophy of the intentionality of mind should begin from considerations of the “executive” use of indexicals in first-person thinking. In Theory of Knowledge, Russell argued that his accepting the selective attention to a ‘this’ presented to a subject is incompatible with his accepting neutral monism. But this position is consistent with rejecting Kapitan’s thesis about the direction in which to lead a theory of the mind’s intentionality. In my view, the use of indexicals is a fundamentally social phenomenon of communication which works by agreed-upon conventions governing the learned use of indexical words. The conventions presuppose intentionality. Thus, the use of indexicals in communication cannot guide a theory of intentionality. I am committed to the thesis that there is no first person thinking that is essentially indexical thinking. I follow Theory of Knowledge in holding that the selective attention (communicated by the use of the indexical ‘this’) is conclusive evidence of the existence of the subject. It is thereby conclusive evidence against neutral monism, since the existence of the subject is incompatible with neutral monism. Commitment to the subject does not require that there be singular indexical thinking. A token-reflexive account of the communicative use of indexicals is all we need, and we do well to base it on an indexical-free apparatus of intentionality. Certainly, nothing in successful communication with indexicals offers good reason to hold that there is singular indexical thinking. Admittedly, the apparatus of communication is so prevalent that soliloquy sometimes seems to parallel it—as when we seem phenomenally to speak to ourselves or hear ourselves thinking. The thesis that there is singular thought undoubtedly gets a large impetus from this feature of consciousness. But it is not reliable data for building a theory of the nature of intentionality. On the view I adopt, indexicals only occur in communication. They are not involved in the quantificational apparatus of selective attention. They are linguistic artifacts produced by that apparatus. The striking feature of the use of indexicals in communication is that they never introduce scope distinctions. Otherwise put, indexicals used in communication never take, if you will, a secondary scope. The natural source of this feature is precisely that making indexical sounds invite the hearers of the speech act to use their own quantificational apparatus of intentionality to find the reference in the pragmatic context of the utterance of the speaker. This is nicely illustrated when a speaker asserts, for example,

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Mary believes that I am here now. This use of the indexical “I” sound invites the hearer to find the speaker, the sound “here” invites the hearer to find the location of the speech act, and “now” invites the hearer to find the time of the speaker’s utterance. The thesis that the use of an indexical cannot have a secondary scope has, however, had its challenges by philosophers of language—challenges originating, for the most part, from the pioneering early work of Castañeda (1966, 1967) on quasi-indicators and later explored by many others including Kapitan (1998, 1999, 2001) and Orilia (2010). Consider the following example from Orilia concerning a female stranger knocking at the door of a man (Tomis) expecting his mother to visit. In answering his door, Tomis remarks: I thought you were my mother.

Orilia holds that this occurrence of “you” connotes a secondary scope as would be given by “the person I was expecting to see.” In my view, the hearer parses the assertion of Tomis (the speaker S) as follows: [ιxSx][ (∃F)(Fz ≡z z = the referent of x’s token “you”.•. (∃P)(Pz ≡z z = the mother of x  .•. x thought [ιyFy][Py]))]. Tomis instructs the hearer to apply her quantificational apparatus of intentionality and in the above she is construed as using the quantificational apparatus of definite description with the property F. Nothing here reveals an example of a use of an indexical with a secondary scope. Another candidate for a case of a secondary scope of an indexical comes from the use of what Castañeda calls a quasi-indicator. Consider this Mary believes she herself to be happy.

The phrase “she herself” is a quasi-indicator. It is used to indicate that a first-person perspective is being ascribed by the speaker to Mary. The case is strikingly contrasted with the following: Mary believes that I am happy.

As expected, speaker’s use of the indexical “I” here refers to the speaker. The hearer’s parsing is this: [ιxSx][ιyMy][ (∃F)(Fz ≡z z = x.•. y believes [ιvFv][Hv])].

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Again S is the token-reflexive property ‘speaker uttering the token “I”.’ The hearer understands that the speaker ascribes to Mary the use of a definite description involving a property F known perhaps only to Mary. The quantificational apparatus of Mary’s believing is via the definite description ιyFy that refers to the speaker. That Mary uses a definite description seems innocuous—since Mary’s intentionality is pointing to objects other than herself. If we use a quasi-indicator she herself, however, then the communicative situation has changed. Clearly, this does not intend to say that Mary uses a definite description to think about herself. Thus, the following is clearly inadequate: [ιyMy][ (∃F)(Fz ≡z z = y .•. y believes [ιvFv][Hv])]. How, then, can we accommodate quasi-indicators into our approach? The apparatus that enables first-person thinking is unknown and awaits a viable philosophy of mind that can illuminate the nature of intentionality. Communication gets along quite successfully without anyone knowing what that apparatus is. Castañeda is surely right that successful communication with quasi-indicators cannot be captured by replacing them with definite descriptions and determining scope. But just as surely, nothing in communication demands that the successful use of the quasi-indicator “she herself” ascribes to Mary a singular thought. If, as I believe, all predicational thinking has a quantificational scaffold, then thinking about oneself involves a quantificational scaffold as well. But nothing demands that it involve the quantificational apparatus of definite descriptions. Using Fregean structured variables for quantificational structures, we have the following which escapes the difficulty: Mary believes she (herself) to be happy. [ιyMy][ (∃Ω)(ΩvPv ≡P Py  .•. y believes ΩvHv)]. In ascribing this to Mary, the speaker uses a quasi-indicator without knowing the apparatus that enables Mary to think about herself, and thus the ascription is made without a commitment to Mary’s first-person thought being singular. The quantificational apparatus that enables Mary to think about herself is left entirely unknown—and this is as it should be. Cases involving quasi-indicators and anaphora can be more complicated especially when they occur in multiply embedded contexts of propositional attitudes. For instance: Mary knows that I believe myself to be a Cartesian.

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Such embedding poses no new problems for the hearer who parses the utterance thus: [ιxSx][ιyMy][ (∃F)(Fz ≡z z = x .•. (∃Ω)(ΩvPv ≡P Px .•. y knows [ιzFz][z believes ΩvCv]))]. The hearer takes the speaker’s use of the token indexical “I” as an invitation using token-reflexiveness to find the speaker via a definite description ιxSx. The speaker also indicated to the hearer that Mary has descriptively referred to the speaker and the speaker may well not know which definite description Mary used to do so. Moreover, to parse the assertion, the hearer does not need to have an understanding of the nature of the firstperson cognitive apparatus that enables the speaker to think self-referentially. The hearer need only understand that there is some such apparatus involved. This is as it should be since such communication is successful independently of anyone’s philosophy of mind. The same point applies to embeddings of quasi-indicators within temporal contexts that naturally raise difficult philosophical issues about the identity of an object over time—the personal identity of a ‘self’ over time being particularly vexing. Communication is successful without supposing any solutions to such philosophical questions. Kapitan considers the following case: Yesterday I believed that I-myself was in danger.

Philosophers anxious about personal identity through time may construe the use of “yesterday” as influencing the scope of the quasi-indicator “I-myself” so that it refers to a subject existing only yesterday, which is different from the subject the use of the indexical “I” refers to. But as I see it, the everyday use of indexicals and quasi-indicators in communication carries with it no such anxiety. The phrase “yesterday” cannot trap the scope of the indexical “I,” and neither can it trap the scope of the embedded quasi-indicator “I-myself.” How then does the quasi-indexical succeed? If we use S for the property of being the speaker who uttered the token “I,” then we can render the hearer’s parsing in the now familiar way: [ιxSx] (∃Ω)(ΩyPy ≡P Px .•. Yesterday x believed ΩyDy). Singular thoughts involving subjects are nowhere presupposed by ordinary communication with indexicals and quasi-indicators. On the token-­ reflexive account of the use of indexicals in communication, the pragmatic context of the speaker’s utterance of the indexical “I” and the

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quasi-­indicator “I-myself” constitutes an invitation to the hearers to use their quantificational apparatus to single out the speaker and situate the speaker’s belief in the proper time (yesterday). This explains why indexicals have to have, as it were, a primary occurrence. It is for the hearer’s benefit that the speaker uses an indexical. The speaker certainly does not need to use any indexical apparatus to reflect upon himself yesterday having had a first-person concern about danger. Temporal indexicals are particularly difficult and the use of quasi-­ indicators may well introduce unique problems. The various philosophies of time may well deal with them, with varying degrees of success. Quentin Smith (1993, p. 122) offers the following case: It is now noon and Alice now knows that David is now asleep.

Smith concludes that Castañeda is mistaken that the use of a quasi-­ indicator never expresses the indexical reference of the speaker. He thinks, so long as the utterance is indeed at noon, that in the above example the speaker’s use of “now” does both! If one assumes that the speaker is reporting on singular indexical thoughts, then this appearance is reinforced. If the speaker utters this at noon, then it may seem that the clause “David is now asleep” presents “now” as both an indexical and a quasi-­ indicator, for it seems also to present Alice’s use of the indexical “now.” Smith’s case is puzzling only if one has supposed that there are indexical now-thoughts. There is no good reason to hold that the speaker intends to be attributing to Alice a singular indexical “now-thought” of David’s being now asleep. Abandoning singular now-thoughts, there is nothing new in Smith’s case—except to emphasize that quantificational thinking is tenseless. Nothing more than Alice’s cognitive apparatus of intentionality is needed to direct her to temporal passage. (Chapter 7 will take up some of the problems that arise for the elimination of abstract particulars from the empirical science of time.) Reichenbach’s token reflexive theory is perfectly adequate as an account of communication with indexicals. Kapitan is right that it cannot account for first-person indexical thinking. But since I reject singular thought, there is nothing more to trouble over. It is, however, of utmost importance to resist the temptation to think that the communicative use of temporal indexicals (especially “now”) is conclusively indicative of the existence of first-person singular thinking. Indeed, it is only by assuming from the onset that there exists temporally indexical singular thought that the use

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of indexicals seems straightforwardly incompatible with attempts at rendering truth-conditions of such use—as is advocated by the (so-­called) new tenseless theory of time. The original (so-called) tenseless theory of time notoriously failed to adequately render the meaning of such uses. The arguments are rehearsed and discussed at length by Smith (1993). Suppose in Iowa City on 24 August 1989 at 2 a.m., Austin uttered It is raining now.

Hearing this, Ansel cannot simply swap out the indexical “now” with a place and date signature, say, “in Iowa City on 24 August 1989 at 2 am,” and expect to capture that import of the utterance as It is raining on 24 August at 2 am in Iowa City.

Communication with indexicals does not work by such a naïve replacement rule. There are other important cases as well where quasi-indicators are used. Indexicals seem to demand primary scope (or better no scope), but with temporal quasi-indicators it may seem as though there are cases where one can trap them inside a propositional attitude as in the following said by Lulu about her sister: Yesterday Alice thought she-herself to now be a trombonist.

The speaker, Lulu, intends to capture a thought Alice formed yesterday. The indexical “now” seems used by Lulu but nonetheless indicates Alice’s use of “now” and not the speaker’s. Ultimately, however, these cases are not convincing. They involve Kripke-style and Perry- and Kaplan-style arguments concerning de se ascriptions which simply reveal that the use of indexicals in communication cannot be captured by a translation strategy that replaces them with definite descriptions and adjusts for the scope. I am not endorsing any such replacement strategy. Communication with indexicals and quasi-indicators is ineliminable. But it by no means follows that there is essentially indexical singular thought. In my view, indexicals are not part of thinking since they require thinkers to establish pragmatic conventions for communicating thoughts to others. The apparatus of indexicals is essential only as a vehicle for communication.15 My demands on de re quantifying-in do not clash with the uses of indexicals and quasi-indicators in communication.16 Quite the

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contrary, they enhance them and do not interfere with the orthodox thesis that indexicals never take a secondary scope. One may worry, however, that there is de se belief that is essentially indexical and thus demands singular thoughts (singular propositions). The de se data that supposedly favors the doctrine of the essential indexical, first promoted by Castañeda and later by Perry, Lewis, and Kaplan et al.,17 does not, in fact, support the conclusion that there is singular (indexical) propositional thinking. As we have already seen, self-reflective awareness is not rendered by the quantificational apparatus of definite descriptions. But that does not entail that it does not have some, as yet unknown, impredicatively quantificational scaffold. Suppose S, the speaker, communicates to another S* by saying, I believe that I am happy.

This case is not formally more demanding from the perspective of the hearer S* than the case of “Mary believes that she herself is happy.” The hearer has been instructed by the speaker to find the speaker uttering the token “I.” The hearer S* may well find the speaker descriptively using the property S and thus the hearer forms a belief with the following content: [ιxSx](∃Ω)(ΩyPy ≡P Px .•. x believes ΩyHy). The hearer S* has no idea whatever what quantificational apparatus is involved in the speaker’s first-person thinking. In communication using the indexical “I,” nothing demands that the speaker be regarded as thinking about himself by means of a definite description. The demand on de re quantification only maintains that the apparatus of first-person thinking be quantificational. Having been following a trail of sugar to find the culprit spilling it, a person discovers that the culprit was he himself, and this de se realization is alone what awakens him to the moral obligation that he, in particular, ought to clean up the sugar mess. Perry (1979) concludes that what motivates his cleaning is precisely his de se attitude to the singular meaning proposition I am the person making the sugar mess.

No descriptivist replacement account can capture this. For example, replacing “I” by “the person who is spilling sugar is making a sugar mess” yields the result that one can no longer explain why the person, having made the discovery, feels morally obligated to clean it up himself. Perry is

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correct in rejecting a descriptivist communicative method that would enjoin us to replace indexicals by definite descriptions. But in accepting Perry’s point we are not compelled to introduce an attitude to a singular meaning proposition in attempting to explain the person’s discovery of his obligation to clean up the mess. His discovery, undoubtedly, is his thinking that he himself has made the sugar mess. Nothing in the story entails that the apparatus of his thinking is singular indexical thinking. Perry’s story simply shows that it cannot be captured with a definite description. It is quite obvious that successful communication with indexicals places no demand whatsoever on anyone knowing what that quantificational apparatus is. We get along perfectly well in natural language without any such philosophy of mind. It is communication that essentially involves the use of an indexical. The apparatus of his first-person thinking remains unknown even to himself. Kaplan (1968, 1989) offers modal arguments designed to demonstrate the inadequacy of any descriptivist replacement rule according to which an indexical is to be replaced with an appropriate definite description. Again they offer no good ground for concluding that there is indexical first-­ person propositional thinking. Suppose the communication rule governing “I” were that it is to be replaced by “the one uttering the token ‘I’” (or more simply let “the T” abbreviate “the one thinking”) and we have a speaker and she utters: If I exist, then I am thinking.

Replacement yields this: If the one thinking exists, then the one thinking is thinking. i.e., E!(ιyTy) ⊃ [ιyTy][Ty]. In this kind of case, Kaplan realizes, the replacement cannot be correct since the result is logically necessary. Kaplan imagines the speaker thinking indexically in soliloquy and thereby imposed a secondary scope of “I” in violation of the thesis that indexicals have only (as it were) a primary scope. On the view I adopt, indexicals are never used in soliloquy. When a speaker utters “If I exist then I am speaking,” the speaker employs the indexical “I” to communicate the instruction to a listener S* to engage in tracking the token reflexive utterance of “I” to find the person speaking and to form the appropriate belief about what the person speaking is saying. The listener uses the property T to single out the speaker, and thus the content of the belief ascribed by the hearer S* is this:

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[ιxTx](∃F)(Fz ≡z z = x .•. x believes (E!(ιyFy) ⊃ [ιyFy][Ty])). This, of course, is contingent. Admittedly, on this rendition the hearer S* ascribes to the speaker the use of the quantificational apparatus of definite descriptions in thinking about herself. The definite description seems forced by the Russellian for whom “… exists” does not indicate a property. But using Fregean structured variables, we can avoid this. [ιxTx](∃Ω)(ΩyPy ≡P (ιz)(z = x)[Pz] .•. x believes ΩyTy). The hearer ascribes to the speaker a first person self-awareness of thinking, but makes no commitment to this self-awareness being a singular indexical thought. A Fregean structured variable is used for the unknown quantificational structure enabling self-reflective awareness. This removes the untoward antecedent “I exist” altogether. Similarly, consider someone uttering “I exist.”18 Russellians can avoid: [ιxSx](∃F)(Fz ≡z z = x .•. x thinks E!(ιyFy)). The speaker’s communicative use of the indexical “I” is simply inviting the hearer to use the hearer’s quantificational apparatus of intentionality to find the speaker. Thus, the hearer may have: [ιxSx](∃Ω)(ΩyPy ≡P (ιz)(z=x)[Pz] .•. (∃F)(x thinks ΩyFy)). From these examples, we can see that my demand (that de re and de se quantification into propositional attitude contexts respect the quantificational apparatus of thinking) does not require that a hearer attributes to the speaker a way of thinking about herself that involves a definite description. All that is required is that the ascription be quantificational. It is worth noting, in this regard, that even Kaplan’s infamous “Castor/ Pollux” thought experiment poses no threat to the thesis that the apparatus of first-person thinking is quantificational. Castor and Pollux are cognitive doppelgangers who, according to Kaplan, assert the following: My brother was born before I was born.

Once we accept that they don’t both share a content that is indexical and first person, there is no longer a puzzle concerning the hearer’s understanding when Castor and Pollux simultaneously speak. Castor is a speaker and the hearer S* of his utterance tracks Castor’s token reflexive “I” sound by thinking descriptively using a property, say cS, and thereby singles out

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the speaker making the sound. The hearer may well know nothing about the cognitive apparatus by means of which the speaker, Castor, is thinking about himself and his brother. The same holds when the hearer S* hears Pollux’s utterance, finding him by his distinct token reflexive “I” sound by using a property, say pS. The truth-conditions for the belief-facts generated by the hearer S* pose no difficulty at all. Two distinct beliefs are formed by S* in these cases, namely: [ιx cSx][ (∃Ω)(ΩyPy ≡P (ιz)(z = x)[Pz] .•. x believes[(ιw)(ΩyBro(w, y)][Byw]) [ιx pSx][ (∃Ω)(ΩyPy ≡P (ιz)(z = x)[Pz] .•. x believes[(ιw)(ΩyBro(w, y)][Byw]). Since cS is different from pS, the contents of what is heard by S* when Castor and Pollux simultaneously speak are quite different. What then about the belief-facts formed by Castor and Pollux themselves? Castor and Pollux are psychological doppelgangers in Kaplan’s thought experiment, but that only assures that in each respective belief-­fact generated, the relata to which these two distinct subjects, namely c and p, are related are precisely the same. It certainly does not require that the belief-facts be the same. When Castor thinks, he engages his impredicative quantificational apparatus for self-awareness, and it generates the artifact which is the belieffact containing the subject c standing in a multiple-­relation of belief to universals. We don’t know, and in communication we certainly don’t need to know, what this impredicative quantificational apparatus is. Simultaneously, the belief-fact formed by Pollux contains the subject p standing in a multiple relation of belief to the same universals. These subjects are different, since Castor and Pollux are different, and thus the belief-facts containing them are different. Now it may be that two people employ exactly functionally “the same” quantificational apparatus in thinking about themselves. But, if so, there is still no great puzzle about the constituents of the belief-facts generated. Each belief-fact contains a distinct subject. This is itself sufficient to individuate the belief-facts. Perry and Kaplan et al. are correct that indexicals do not work in natural language communication by a rule which instructs us to replace the indexical by a definite description with appropriate scope. But this does not give good evidence that there are first-person singular meaning propositions and a special use of indexicals in the soliloquy of first-person thinking. Indexicals are needed only in communication. The token-reflexive approach of Reichenbach adequately explains the way they work. We do

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well to be rid of singular meaning propositions. The relation of ‘acquaintance’ of Russell’s Principia era requires a subject. But it does not require singular first-person thinking.

Acquaintance with ‘Acquaintance’ The Principle of Acquaintance is the foundational epistemic idea of both Russell’s Principles era and his Principia era. ‘Acquaintance’ is a special relation whose exemplification requires the existence of a subject (if not also a Mind) as one of its relata. Any universal standing in the relation of ‘acquaintance’ is ipso-facto, simple-type free. The converse of ‘acquaintance’ is the relation of ‘presentation’ which emphasizes the subject’s selective attention to a this presented. The same selective attention holds when a subject is related to an object by the relation of ‘sensation,’ which is a relation also requiring a subject for its exemplification. It is selective attention that grounds the aboutness or pointing that Brentano took in his Principle of Intentionality to be the distinctive mark of the mental. Obviously, significant changes are involved when in 1918 Russell developed his brand of neutral monism and its natural alliance with behavioristic ideas which enable thermometers to be said to have “selective attention.” Indeed, Russell says behaviorism “belongs logically with neutral monism” (LLA, p. 279) and, in fact, it is his behavioristic sympathies that do the lion’s share of the work. Happily, these changes are irrelevant to his 1914 book Scientific Method in Philosophy, which remains squarely in the Principia era. Indeed, in spite of Russell’s 1926 attempt to bowdlerize the book to make it seem consonant with neutral monism, the constructions were built around the subject standing in a relation of ‘sensation’ to a transient physical particular (sense-datum). Russell sometimes kept the expression “principle of acquaintance” after he adopted neutral monism. In the same work in which he explains that the behavioristic noticing replaced his relation of ‘sensation,’ he wrote (MPD, p. 125): 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 must be on guard not to be misled by such comments. Indeed, we can see that this passage is out of sorts with Theory of Knowledge which was

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committed to acquaintance with logical forms (that ground our understanding of the logical particles). Relations like ‘acquaintance’ and ‘sensation’ whose exemplification requires that one of the relata be a subject are important and unique. They will be seen to be instrumental in our understanding the very notion of a subject itself. According to Theory of Knowledge, we are never acquainted with any subject. Indeed, one is never acquainted with any fact in which a subject occurs. All the same, we understand the notion of a subject and thereby the notion of a Mind. I shall, therefore, hold that the notion of a subject is understood only by means of an innate acquaintance with the relation ‘acquaintance’ itself and it is this that enables our understanding of the notion of a subject. Innate acquaintance with a relation is required if the relation’s exemplification requires one of its relata to be a subject. The Principle of Acquaintance is very important for the Principia era. Characterizations of it from the Principles era, however, cannot carry over to the Principia era. It is not surprising that in the 1905 “On Denoting” the principle is expressed in terms of propositions. It was not until 1908 that Russell finally abandoned propositions. (In 1906, Russell abandoned general propositions, keeping non-general ones. In 1908 he entertained without endorsing an ontology of orders of general propositions.) Let’s look at the characterization from “On Denoting” (OD p. 56): in every proposition what 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.

With propositions abandoned, logical aboutness is also abandoned. The only notion of aboutness is psychological, and, with the agenda of scientific philosophy, this obviously cannot be a metaphysics of abstract particulars. A proper characterization of acquaintance for the Principia era must make it wholly independent of any commitment to abstract particulars. That includes meaning propositions indicated by declarative sentences (or utterances). What then is a proper formulation of the principle that would be appropriate to the Principia era? Problems itself, though squarely in the Principia era, still stated the principle of acquaintance using the word “proposition.” But likely the word “proposition” used in the following passage intends to indicate a declarative sentence (statement). Russell writes (PoP, p. 58):

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Any proposition which we can understand must be composed wholly of constituents with which we are acquainted.

In Theory of Knowledge, this is corroborated when Russell writes: “Let us, to begin with, define a proposition as any complete sentence capable of expressing a statement” (TK, p. 105). (See also TK, pp. 109, 112, 117). In “Knowledge by Acquaintance and Knowledge by Description,” Russell puts the principle more carefully, writing (KAKD, p. 219): The fundamental epistemological principle in the analysis of propositions containing descriptions is this: Every proposition which we can understand must be composed wholly of constituents with which we are acquainted.

Happily, nothing in this passage suggests that he embraces entities that are meaning propositions. Russell makes this clear when he gives a “restatement” of his principle (KAKD, p. 221): Whenever a relation of supposing or judging occurs, the terms to which the supposing or judging mind is related by the relation of supposing or judging must be terms with which the mind in question is acquainted.

This seems to be the best statement of the Principle of Acquaintance for the Principia era. It should also be noted that Russell does not regard acquaintance as creating a mediation. The aboutness of thinking does not require that universals and sense-data are the immediate objects of thought, while thoughts about other entities are mediate objects of thought. On such an interpretation, thinking descriptively, for example, about the tallest mountain on earth, we are thinking immediately about the universal ‘being tallest mountain on earth’ and we are thinking indirectly about the mountain. The thesis of mediation is mistaken. Russell holds that the universal plays a role in directing one’s thinking without thereby becoming an immediate object of one’s thinking. All predicational thinking depends on acquaintance with universals which thereby become a part of the apparatus of intentionality, not immediate objects of that intentionality. Indeed, acquaintance is not itself a kind of conceptual knowing and it does not admit of truth or falsehood. Russell explains (OKEW, p. 144): Acquaintance, which is what we derive from sense, does not, theoretically at least, imply even the smallest ‘knowledge about,’ i.e., it does not imply

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knowledge of any proposition concerning the object with which we are acquainted. It is a mistake to speak as if acquaintance had degrees: there is merely acquaintance and non-acquaintance.

Russell is famous for the distinction between knowledge by acquaintance and knowledge by description, first indicated in “On Denoting” and set out in Problems. But he is clear that acquaintance is not a sort of propositional knowing. Moreover, Russell writes (KAD, p. 108): I say that I am acquainted with an object when I have a direct cognitive relation to that object, i.e., when I am directly aware of the object itself. When I speak of a cognitive relation here, I do not mean the sort of relation which constitutes judgments, but the sort which constitutes presentation. In fact, I think that the relation of subject and object which I call acquaintance is simply the converse of the relation of object and subject which constitutes presentation. That is to say that S has acquaintance with O is essentially the same thing as to say that O is presented to S.

The relation of ‘acquaintance’ is not akin to the multiple-relations of ‘belief’ and ‘judgment’ which involve the application of concepts and are thus susceptible to misapplication and falsehood. The analysis of falsehood must involve a multiple-relation. Russell distinguishes understanding from believing and, of course, both are distinct from acquaintance. He admits that he had earlier thought that the understanding relation is just the presentation relation—the converse of acquaintance (KAD, p. 159). But having abandoned his early theory of propositions, he came to hold that a fact of understanding must involve a multiple relation in just the way belief-facts require multiple relations. We can, after all, understand “Pegasus exists,” even though there is no Pegasus. Unfortunately, he goes on to maintain that believing presupposes understanding (TK, p. 108): when I speak of ‘understanding a proposition,’ I am speaking of a state of mind from which both affirmation and negation are wholly absent. …Understanding, if I am not mistaken, is presupposed in belief, and can itself be discussed without introducing belief.19

A mind-first view which makes belief-facts truth-apt artifacts of the mind’s intentionality will have no difficulty with the view that a mind understands what it believes. But if it were a logical matter—so that the existence of a

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fact of a subject understanding is logically entailed by the existence of a fact of that subject’s believing—one would have to define belief-facts in such a way that they have understanding facts as constituents. Otherwise, the view would be in violation of the fundamental thesis of the Principia era. (Recall that the thesis is that where facts that are truth-makers are concerned, and where such a fact does not contain other facts as their constituents, the existence (or non-existence) of a fact never logically entails the existence (or non-existence) of any other fact.) I prefer, therefore, to hold that the connection between belief and understanding is a feature of contingent psychology. In contrast, belief does logically entail acquaintance. It is not difficult to imagine that acquaintance facts are constituents of a belief-fact. Consider the structural name:  R2    s—B4—  a  .  b    The lines should be viewed as indicating the acquaintance facts that are its constituents, namely: s—A—R2, s—A—a, and s—A—b. The existence of a belief-fact logically entails the existence of an acquaintance fact. That doesn’t violate the fundamental thesis of the Principia era. Acquaintance facts are constituents of belief-facts. This exempts them. In discussing his Principle of Acquaintance in “Knowledge by Acquaintance and Knowledge by Description,” Russell speaks of a mind standing in the relation of acquaintance. In Theory of Knowledge he speaks of a subject standing in that relation. This difference has to be rectified. In Theory of Knowledge, Russell relied on his four-dimensionalist view of time to distinguish a mind from a subject. He wrote (TK, p. 35): It will be observed that we do not identify a mind with a subject. A mind is something that persists through a certain period of time, but it must not be assumed that the subject persists. So far as our arguments have hitherto carried us, they give no evidence as to whether the subject of one experience is the same as the subject of another experience or not. For the present, ­nothing is to be assumed as to the identity of the subjects of different experiences belonging to the same person.

The passage makes it clear that Russell understood a mind as a temporal series of facts involving subjects as constituents. Russell’s position in

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Problems was likely the same. He says that a “mind” is a constituent of a belief-fact, but perhaps he meant “subject.” Russell does speak of the “Self,” but it is difficult to discern whether he meant a mind or the subject. He wrote (PoP, 19): some care is needed in using Descartes’ argument. ‘I think therefore I am’ says rather more than is strictly certain. It might seem as though we were quite sure of being the same person today as we were yesterday… But the real Self is as hard to arrive at as the real table.

Later he says “… it is probable, though not certain, that we have acquaintance with Self, as that which is aware of things or has desires toward things” (PoP, p. 51). Again, perhaps in this passage he meant to be speaking of the subject. In his amusing 1913 review of Jevon’s book Personality, Russell’s remarks (CW, Vol. 6, p. 375): The second chapter discusses the attacks on the Self by Hume and James. These it dismisses by means of the verbal inconsistences which grammar forces on those who deny the Self. “When I enter most intimately into what I call myself,” says Hume, “I always stumble on some particular perception or other.” To this, Dr. Jevons retorts by asking what is meant when it is said, “I enter” or “I stumble,” thus arriving at the conclusion that the Self is assumed by the very words in which Hume intends to deny it. Such short and easy arguments, however, never really succeed… Whether he is right or wrong remains a very doubtful question; but he was certainly not so stupid as Dr. Jevons’ refutation implies.

Once again it is unclear what he meant by Self (mind or subject) except that it is to be the referent of the use of the indexical “I.” This passage is of a piece with Problems. In Theory of Knowledge, however, Russell doesn’t speak of the Self, but explicitly accepts Hume’s concern that we are never acquainted with the subject. I shall follow him in this, though I must admit that I feel a significant sympathy for Dr. Jevon’s concern. That we are not acquainted with the subject, however, certainly does not establish that one cannot be acquainted with it. Russell’s deference to Hume raises something of a puzzle, for he claims to know that the relation of ‘acquaintance’ requires that one of its relata be a subject. How? If we are never acquainted with any subject, how is it that we have the concept of the subject? On Russell’s behalf, I think that the best answer to hold is that it is our acquaintance with the relation

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‘acquaintance’ itself that renders our understanding of the notion of a subject. In being acquainted with any universal, one understands the nature of the fact that would exist if that universal were exemplified. Hence, it is by being acquainted with the relation of ‘acquaintance’ that we understand the notion of a subject without ever being acquainted with any subject. Perhaps this is the kernel of truth in the Cartesian cogito. Certain universals are such that acquaintance with them assures an understanding that they are occurrently exemplified. The relation of ‘acquaintance’ is one of them. Acquaintance with ‘acquaintance’ would then assure that we understand the occurring existence of a subject even though we are not acquainted with any subject. Russell’s notions of “subject” and “object” are technical. He uses the term “object” for whatever is presented and he includes universals, sense-­ data and logical forms (TK, p. 35). But the expression “s is acquainted with y” and the expression “y is presented to s” are well-formed for any term in the position of “y.” This includes terms for facts and terms for universals. This is very important and it reveals that there are no distinctions in type* when it comes to standing in the relation of acquaintance.20 Now one may have hoped that both the expression “x is acquainted with y” and “y is presented to x” should be ill-formed and not merely the expression of something false when x is not a subject. That is, one may have hoped that a deeper analysis of the relation of ‘acquaintance’ should be possible—an analysis of what a subject is would reveal why acquaintance requires a subject. But Russell has no deep analysis of the relation of ‘acquaintance’ to offer. No one does even to this day. For any p-relation with which we are acquainted, it is quite natural to hold that we are acquainted with all its converses and all the position relations it determines. Lebens (2017, p. 199) finds the view untenable that cognitively we make such permutations. But once we make the connection to the converse of a relation, such permutations are not at all untoward. Indeed, when we have a relation with which we are acquainted we do seem readily able to form converses by active-passive transformations. For example, let R3 be the relation ‘x gives gift y to z.’ We have: R3xyz i.e., x gives gift y to z ⋉R3xzy i.e., x gives to z gift y ↷R3yzx i.e., gift y is given to z by x ⋊R3yxz i.e., gift y is given by x to z § R3zyx i.e., z receives gift y from x ↶R3zxy i.e., z receives from x gift y.

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The transformations giving us the needed rearrangement of variables of converses are, in fact, quite familiar. It is not hard to imagine the same holds for the identification of like variables. We have: R3xyx i.e., x gives gift y to x j↶R3xy i.e., x self-gives y. Such names of converse transformations are obviously unfamiliar. But these transitions are plausible features of ordinary acquaintance with universals. Generalizing we have: (↶Rn)(xn, x1, …, xn − 1) ⇋ Rn(x1, …, xn − 1, xn) (⋉Rn)(x1, …, xn, xn − 1) ⇋ Rn(x1, …, xn − 1, xn) (iRn)(x1, …, xn − 1) ⇋ Rn(x1, …, xn − 1, xn − 1). ↷Rn( x2, …, xn − 1, x1) ⇋ Rn(x1, x2, …, xn − 1) ⋊Rn(x2, x1, …, xn) ⇋ Rn(x1, x2, …, xn) jRn( x1, x2, …, xn − 1) ⇋ Rn(x1, x1, x2, …, xn − 1) ⋈Rn(x2 x1, …, xn, xn − 1) ⇋ Rn (x1, x2, …, xn − 1, xn) §Rn(xn, x2, …, xn − 1, x1) ⇋ Rn(x1, x2, …, xn − 1, xn) These are not, however, the same as the functor’s of Quine’s “Variables Explained Away.” There are no functors (operations) for Russell. There are only relations, and the relations that do the work of Quine’s functors await us in Chap. 6. For the present, we see that the above pose no difficulties.21 In Theory of Knowledge, Russell entertains the striking thesis that a relation is to be identified with its converse(s). The reason, I surmise, is to preserve the fundamental thesis of the Principia era. For example, the existence ofa fact that a R b seems to logically entail the existence of the fact that b R a, where R is the converse of R. With the identification in place, these are identical facts and the fundamental thesis of the Principia era is preserved. In cases where the relation is logically symmetric, the relation does not have a domain and a range. Thereby it cannot be said to be many-one, one-­one, etc. In these cases, the identification of a relation with its converse(s) seems innocuous. But especially in cases of asymmetric relations, where the domain is important, there are serious challenges for any such identification. (Russell knew this. He said it explicitly in PoM, p. 96). Russell’s Theory of Knowledge never quite made good on the identification for relations in general. In Chap. 5, on the problem of direction, we shall see that we can repair Russell’s identification of a relation with its converse in cases where the relation is a p-relation—that is, where it determines

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position relations. In such cases, we can redefine a relation’s domain and range and so forth in terms of the position relations it determines. Russell’s identification of a relation with its converse raises a new difficulty, however. It would require the dyadic relation of ‘acquaintance’ to be identified with its converse ‘presentation.’ And yet there is clearly a difference between the concepts of domain and range for the relation of ‘acquaintance’ since only a subject can be in its domain. Russell’s identification, given the repair I’m proposing, would require that ‘acquaintance’ be regarded as a p-relation. It is only in the case of p-relations (permutative relations) that we can appeal to the position relations they determine in defining the notion of domain and range—thereby avoiding difficulties with the fundamental thesis of the Principia era in identifying the relation with its converse. And yet ‘acquaintance’ is not a p-relation unless the subject can be an object of acquaintance. The only escape I can see is to return to the position of Problems and maintain, Russell’s endorsement of Hume in Theory of Knowledge notwithstanding, that one can be acquainted with a subject (even if this, in fact, never happens.) That is, nothing in pure logic prevents there being two subjects s* and s, both generated as artifacts of the same mind M, such that there is a permutative fact of s being acquainted with s*. In fact, some of Russell’s own comments suggest that this can happen. Using P for the relation of ‘presentation’ and A for ‘acquaintance,’ he writes (TK, p. 38): What is psychologically involved in our acquaintance with the present experience? We require an experience which might be symbolized by s*-P-(s-A-o). When such an experience occurs, we may say that we have an instance of “self-consciousness,” or “experience of a present experience.” It is to be observed that there is no good reason why the two subjects s and s* should be numerically the same: the one “self” or “mind” which embraces both may be a construction, and need not so far as the logical necessities of our problems are concerned, involve the identity of the two subjects.

We can extrapolate from this and accept, after all, that ‘acquaintance’ is a p-relation. In deference to Russell’s concession to Hume, however, we can also accept that s and s* are never the same. In order to successfully repair Theory of Knowledge, one must keep the problem of nature of intentionality separate from the problems of direction and compositionality. I adopt a mind-first approach that makes subjects artifacts of the predicational thinking of minds. Similarly, belief-facts and understanding-facts are artifacts of minds. They are artifacts of believing

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and understanding (respectfully), and they do not contain minds as constituents; they contain subjects. This requires that both a mind and a subject can stand in the acquaintance relation. It is a mind thinking that gives rise to belief-facts containing subjects that are artifacts of what Cartesians distinguished as the mind’s essential feature—thinking. This presupposes that a mind (not just a subject) stands in the relation of acquaintance. This mind-first approach rules out there being a construction of mind as a series of events containing subjects. But nothing in the Principia era suggests otherwise. Material continuants, Russell hoped, would be constructs from series of transient physical particular events. Mind is left intact. Russell’s Principle of Acquaintance demands that we cannot understand a statement unless we are acquainted with the entities indicated by its meaningful constituents. Does an analogous principle also apply to facts? In his summary of “atomic thought,” he writes (TK, p. 177): The question of analysis, we found, is complicated by the doubt as to whether we can be acquainted with a complex without being acquainted with its constituents. What seemed empirically certain was, that we may be acquainted with a complex without being able to discover that we are acquainted with its constituents; but it is not possible to assert positively that we are ever not acquainted with its constituents.

Russell’s view on the matter is not easy to determine. If we are acquainted with a fact, must we be acquainted with each of its constituents? I think we must. Accepting that being acquainted with a complex requires that we be acquainted with the constituents does not, however, demand that we have an acquaintance with the constituents that is temporally prior to our acquaintance with the fact. Moreover, in being acquainted with a fact, we may not always be able to discover that we are acquainted with its constituents. Russell explains (TK, p. 120): There is, however, another proposition [i.e., fundamental thesis], less obvious, and not immediately deducible from the difficulties of analysis: this is the proposition [thesis] that we may be acquainted with a complex without being acquainted with its constituents.

In an effort to find empirical evidence for this, Russell restates it as follows (TK, p. 121):

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We may be acquainted with a complex without being able to discover, by an introspective effort, that we are acquainted with the objects which are in fact its constituents.

This general principle seems quite reasonable. We may be acquainted with objects without being able to discover by introspection that we are, in fact, acquainted with them. Acquaintance with a universal requires that the universal be simple-type free. In Problems, Russell noted that becoming newly acquainted with universals is the foundation of a priori knowledge of mathematical logic. This is compatible with holding that we are not acquainted with any simple-­ type stratified universal comprehended by Principia’s *12.1.11 and so on. This offers a compromise between a Realist interpretation of Principia’s bindable predicate variables and the failed intended nominalistic interpretation set out in the informal introduction of the first edition. We cannot be acquainted with a simple-type universal comprehended by *12.1.11, but we do understand the universal by understanding the wff giving its exemplification conditions. Understanding simple-typed universals through its exemplification conditions is enough for the epistemology of mathematical logic. This approach allows that we can come to be acquainted with ever new (type-free) universals. Russell’s unique Rationalism allows that acquaintance with some universals may be innate and acquaintance with other universals may be acquired. He tells us in Problems that “…there seems to be no principle by which we can decide which can be known by acquaintance, but it is clear that among those that can be so known are sensible qualities, relations of space and time, similarity, and certain abstract logical universals” (PoP, p. 101). But can one acquire an acquaintance with new universals through the senses? Russell writes (PoP, p. 107): Two opposite points are to be observed concerning a priori general propositions. The first is that, if many particular instances are known, our general proposition may be arrived at in the first instance by induction, and the ­connexion of universals may only be subsequently perceived. For example, it is known that if we draw perpendiculars to the sides of a triangle from the opposite angles, all three perpendiculars meet in a point. It would be quite possible to be first led to this proposition by actually drawing perpendiculars in many cases, and finding that they always met in a point; this experience might lead us to look for the general proof and find it. Such cases are common in the experience of every mathematician.

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Russell was surely aware, as Plato was in earnest to point out, that no drawing ever makes the lines meet at a mathematical point. A point is not a sensible entity that can be found among the physical marks of a drawing. Russell was not unaware of this. I think his example was meant to reveal that by experiencing certain facts, one may become acquainted with a universal not itself a constituent of any of the facts experienced. Russell’s example reveals something weak but important—namely, it reveals a case where the empirical study of facts, none of which contains a given new universal as a constituent, leads one to come to be acquainted with that new universal. Thus, I adopt: (A) In order to be acquainted with fact, one must be acquainted with every one of its constituents.

Note that the principle does not require one to know that one is acquainted with every one of its constituents. Russell rejects the Platonic rationalist thesis that no mind can ever acquire an understanding of any universal by means of the senses. I agree with him in this. This leaves us with the question as to whether, in particular, acquaintance with the relation of ‘acquaintance’ is innate or acquired. Now given principle (A), it is not possible to acquire acquaintance with the relation ‘acquaintance’ by being acquainted with a fact whose relating relation is ‘acquaintance’ itself. Consider a fact such as, s* being acquainted with o. Principle (A) says that in order to be acquainted with a fact of s* being acquainted with o, we have to be acquainted with s* and with the universal ‘acquaintance,’ and with o. So one cannot acquire an acquaintance with ‘acquaintance’ this way. Moreover, Principle (A) assures that we cannot acquire acquaintance with ‘acquaintance’ by being acquainted with a fact such as, ‘acquaintance’ is a component of ‘belief’.

So the question of how we could acquire an acquaintance with ‘acquaintance’ cannot be answered by appealing to our being acquainted with a fact in which the relation ‘acquaintance’ occurs as one among its constituents. It also remains very hard to see how there could be such a study of facts, none of which have the relation ‘acquaintance’ as a constituent, that could bring about the existence of fact of one’s acquaintance with the

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universal ‘acquaintance.’ Thus, it seems there is just no means of acquiring acquaintance with the relation ‘acquaintance.’ Though Russell did not speak to the matter, I shall maintain that acquaintance with the relation ‘acquaintance’ is innate. A benefit of accepting that we have an innate acquaintance with the relation ‘acquaintance’ is that there is no need for an account of how we have an understanding of subjectivity (the subject). An understanding of the notion of a subject is given by our acquaintance with the relation of ‘acquaintance.’ Indeed, our understanding of the notion of subjectivity is given by our acquaintance with any universal (e.g., ‘sense,’ ‘presentation,’ ‘belief,’ ‘understand,’ ‘know,’ etc.) whose exemplification requires that at least one among its relata is a subject. In deference to Descartes, however, I will also accept the following strangely looped feature of relations whose exemplification requires at least one of its relata be a subject: (D) Any universal whose exemplification requires that least one of its relata to be a subject is such that occurrent acquaintance with it entails and is entailed by its occurrent exemplification in oneself.

Of course, ‘acquaintance’ itself satisfies principle (D). It is incoherent, however, to appeal to principle (D) to acquire acquaintance with the relation of ‘acquaintance.’ It is surely a precondition for acquiring an acquaintance with a universal, that one already is acquainted with ‘acquaintance.’ Acquaintance is involved in every mental act (in Brentano’s sense) and is the foundation of Intentionality itself.22 Principle (D) enables us to acquire acquaintance with relations whose exemplification requires that one of its relata be a subject. It enables us to avoid the thesis that we are innately acquainted with relations of ‘belief’ of different adicities. The relation ‘belief’ has the feature that if it is exemplified one of its relata is a subject. Thus, an occurrent acquaintance with ‘belief’ entails and is entailed by an occurrent exemplification of ‘belief.’ That is, an occurrent exemplification of an n-adic belief-relation is enough for there to be such an acquaintance. In this way, we acquire an acquaintance with an n-adic belief-relation simply by believing. One may worry that it is cognitively impossible for humans to have acquaintance with any relation whose adicity is rather very large and thus there will be belief-­relations with which we are not acquainted. Interestingly, this does not rule out forming definite descriptions that purport to refer to the belief-­ facts formed by other minds engaged in belief-relations of very high adicity. To

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form such a definite description, one does not have to be acquainted with the belief-relation that that other mind engages. We can describe the belief-relation—so long as we are able to understand that belief-­relations, each of different adicity, all form a family. The strange loop established by principle (D) does not hold for all universals with which we are acquainted. It holds only for universals whose exemplification requires that one or more of its relata be a subject. The property ‘… is in pain’ has this feature. Thus, acquaintance with the property ‘is in pain’ can be acquired simply by being in pain. Without principle (D), we couldn’t have a foundation for a judgment that the sensory universal ‘being in pain’ is exemplified. One would have to be acquainted with a particular fact s-in-pain. But a mind cannot become acquainted with a particular fact s-in-pain without being acquainted with the subject. Hence, a subject s is never acquainted with the fact s-in-pain. Neither can a subject s be acquainted with the fact s*-in-pain because a subject s* is not an object with which s is ever acquainted. The same applies to ‘sensation.’ A subject cannot stand in the relation of ‘sensation’ to the particular fact s-in-pain. How then can one arrive at the notion of pain or become aware that one is in pain? Principle (D) answers all such questions. Although according to Russell we can neither be acquainted with a subject nor with any particular fact in which a subject is a constituent, being occurrently in pain is precisely what assures our acquaintance with the universal ‘pain.’ We become acquainted with the universal ‘being in pain’ simply by being occurrently in pain. (Indeed, it is plausible to maintain as well that it is only by exemplifying the property ‘… is in pain’ that one comes to be acquainted with the property.) A mind S acquainted with a sensory universal ‘being in pain’ generates an immediate understanding that there is a subject s and a particular ‘s in pain.’ ‘Sensation’ is a dyadic relation whose exemplification requires one of the relata to be a subject. It would seem that no subject ever stands in the relation of ‘sensation’ to a subject. Nonetheless, this difference in its relata forces a distinction of its domain and range, and thereby makes it a p-relation. The situation is like that of ‘acquaintance.’ Thus, ‘sensation’ is a relation that has features similar to the relation of ‘acquaintance.’ Unlike ‘acquaintance,’ however, there is no need to maintain that acquaintance with the relation of ‘sensation’ is innate. In accordance with principle (D), we acquire an acquaintance with ‘sensation’ simply by sensing. Interestingly, principle (D) may well enable one to avoid acquaintance with particulars altogether. To understand how, observe that aboutness

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always concerns the intentional activity of determinate pointing. In the intentionality of pointing, there is no difference between veridical cases and non-veridical cases. To explain the nature of pointing, there is no need to postulate an immediate object, qualitatively the same in veridical and non-veridical cases, to which one successfully points. In seeing (and even in sensing) the sun, for example, we do not immediately see (or sense) a sensory particular, nor do we see (or sense) series of them, nor a universal. To see (and to sense) is to point successfully with one’s intentionality. It is because success is presumed in using success verbs such as “see” and “sense” that we speak of a particular that is the direct object of the relation of ‘seeing’ or ‘sensing.’ But there are no dyadic facts of seeing or sensing. And quantificational pointing determinately is always independent of success. Of course, we believe there is a sun and thus we believe that we have successfully pointed when we say we see it. And some of the transient physical particulars produced by the sun impinge upon the sense organs. Still, they are no more accessible as objects of thought of sensation than is the sun itself. Seeing and sensing are kinds of intentionality. Emphasizing that intentionality is at work makes it viable to hold, contrary to Russell, that we are never acquainted with any particulars at all. The idea is that acquaintance with universals, together with a quantificational apparatus that enables the determinate directedness of intentionality, can apply equally well to sensation and experience (introspective or otherwise). If this is viable, it is an attractive way to avoid problems that arise with the view that sense-data are particular physical entities. Banishing sense-data, it becomes viable to hold that the apparatus of sensation always involves (sensory) universals whose exemplification produces events involving the activation of sensory quantificationally scaffolded faculties. The question for further research is to work this out. As with intentionality generally, the theory does not make universals immediate objects of thought or perception.23 Admittedly, the case of seeing is easier to tie to intentionality than is the case of sensing. But Gestalt psychology offers convincing reasons to hold that the alleged mind-independent physical data of sense (e.g., Russell’s physical transitory particulars standing contingently in a relation of sensation) cannot plausibly be passively sensed objects. On the Gestalt view, the “objects” of a subject’s phenomenal field have features actively produced by the mind’s intentionality itself—features including depth, rotational direction, motion, color constancy, neon color spread, and the like. That is, the determinate nature of these

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so-­called qualia (“objects” of experience) is dependent upon the activity of the perceptual apparatus of intentionality. In this respect, the “objects” of a phenomenal field may be as paradigmatically intentionally inexistent as any of the usual favorites (e.g., the golden mountain, the round-square). The proper analysis of phenomenal character (qualia) lies in unraveling the mystery of a special sort of intentional pointing, not in the postulation of special particulars such as qualia at which one points.24 The apparatus of intentionality is at work in all cases of predicational thinking (imagining, entertaining, hallucinating, judging, or what have you). And in all cases, the apparatus of intentionality points independently of whether there is anything at which it points. Intentionality can turn inward, directing itself to sense-data as ‘objects’ given in experience. Meinongianism notwithstanding, the individuation of such intentional states is not determined by special (intentionally inexistent) objects of those states. The individuation lies in the differences in the apparatus of intentionality involved and the universals (sensory or otherwise) it engages with. The qualitative indistinguishability of veridical and non-veridical phenomenal experience does not compel us to introduce phenomenal objects. Acquaintance with sensory universals is a part of the quantificational apparatus of pointing. That is enough to explain it. Russell’s construction (1914) of material continuants in time, of course, relies on the existence of transitory particular events. Abandoning acquaintance with sense-data does not undermine the construction. It is transient particulars that impinge on the sense organs, which are themselves composed of transient particulars. Sensation is still a dyadic relation between a mind and a host of transient particulars sufficient to trigger a neural response. But synthetic a posteriori knowledge does not cognitively employ transient particulars that are sense-data. Principle (D) enables a theory of perception that appeals to acquaintance with sensory universals alone. The well-known Sellarsian dilemma sets up a seemingly insuperable problem for an acquaintance epistemology based on the assumption that perception derives from sense-data as concrete particular objects of sensation.25 Sellars’s concern applies to Russell’s conception of sense-data as physical objects. Thought transient, they have determinate physical properties and thus, conceptual judgments about those properties might well be false. Such judgments would require epistemic justification and undermine their providing a ‘given’ for a Foundationalist epistemology. To take the classic problematic case, if there is a sense-datum of a speckled hen that I experience, then the physical entity that is the sense-datum has a determinate

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number of speckles. I am clearly unaware of the number of its speckles even if the determinate number is what is triggering my neural response. But then, the sense-datum cannot play the fully foundational role it was meant to play in acquiring empirical knowledge.26 For these reasons, it is worth trying to develop a theory that dispenses with sense-data as playing the key role in perceptual cognition. The Meinongian fallacy is to be resisted at every turn. The question “What is the object of sensation?” is on a par with the question “What is the object of thinking?” Thinking about a gray elephant in hallucination or a gray elephant in veridical perception both involve the active mental process of intentionality. The difference lies in the universals, sensory and otherwise, that are part of the quantificational apparatus of pointing. Universals are subsumed into the apparatus of pointing and play a central role whether the objects to which we point be the sun, a transient particular produced by the sun, a mind of another person, the present king of France, the Russell class, a pain quale, Desdemona’s loving Othello, or Desdemona’s loving Cassio.

Adicity As we can see, it is acquaintance with universals that is the keystone. And it brings quite a lot with it. According to Russell’s multiple-relation theory, there are a great many quite distinct belief-relations, each with markedly different adicity. Some have found this untoward. Indeed, MacBride (2005) apparently finds it so utterly untoward that he interprets Russell as maintaining that there is one relation of belief that is multi-grade—that is, its adicity varies upon its instantiation. Whatever one might make of MacBride’s notion of a multi-grade universal, it is quite clear that no such notion is appropriate for Russell’s multiple relation of ‘belief.’ Indeed, the fact that there are quite different ‘belief’ relations is absolutely essential to the viability of Russell’s solution to the problem of giving truth-conditions when permutative facts are involved. The belief-fact whose existence is the truth-maker for “M believes that Rab” is itself a permutative fact. This belief-fact is different from the belief-fact whose existence is the truthmaker for “M believes that Rba.” The difference of such belief-facts lies in the different structure of the constituents; this is due to the adicity of the belief-relation that generates the unity of the belief-fact when it inheres in it as its relating relation.

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MacBride (2005) imagines that the notion of the adicity of a universal can be divorced from the notion of a dyadic, a triadic, and so on, fact in which it inheres as exemplified. He speaks as if the same universal R is the relation providing the ground of the unity in, say, both a dyadic fact and a triadic fact. This is unintelligible from a Russellian perspective (of the Principia era). The Russellian notions of fact and universal are interdependent. MacBride’s idea, therefore, employs the notion of a “relation” in a new sense—as an entity no longer posited to ground the unity of a fact. In turn, this supposes a new sense of “fact” as an entity that can be monadic, dyadic, triadic, and so on, exemplifying a relation without fixed adicity. For Russellians, the notion of a “particular” is the notion of an entity that does not have adicity and cannot occur in fact in such a way that relates the constituents and produces the unity of the fact. Universals (properties and relations), Russellians maintain, are just those entities that account for the unity of a fact. The very notion that there is a dyadic fact of a’s bearing R to b is unintelligible to those of us who, like Russell, maintain that relations are just those entities that, because of their dyadic, triadic, and so on, adicity, provide the ground of the unity of facts that are dyadic, triadic, and so on. This is precisely what it is to be a universal. In short, n-adicity (for some fixed n) is not a property that a universal has (of metaphysical necessity) since it is not a property of a universal at all. Adicity is part of the very theory of what a universal is as an entity that is postulated to ground the unity of fact with n-many positions. The Russellian theory being proposed is that there are universals and particulars that are of this nature. Proposing the theory doesn’t beg any questions. Whoever doesn’t like the theory is free to use the word “universal” and “fact” in a new way. In being acquainted with a relation, we understand its adicity and thereby understand what sort of a fact would exist if the universal were exemplified. (Russell sometimes speaks of this as understanding the form of such a fact.) Indeed, when we understand the adicity of a relation Rn that is not a p-relation, we thereby understand that any fact in which it is exemplified consists of the relation Rn exemplified with exactly n-many positions filled by entities. Thus, the notion of a fact is understood by being acquainted with any universal. More exactly, the notion of an Rnfact is understood because of our acquaintance with an Rn universal. Therefore, we need not be acquainted with any fact to understand the notion of a fact (or if the universal involves time, an event). It is the notion of a fact (or an event) that gives us the paradigm of a particular, though

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it remains an open question (and perhaps one which will never be settled) whether all particulars are facts/events. As we shall see, the notion of a fact is definable only by appeal to the notion of an Rn-fact. Thus, both the notion of a subject and the notion of a fact (particular) are grounded in acquaintance with universals. The notions of universal and fact are inseparably connected—the notion of a universal being that entity whose exemplification grounds the unity of a fact, and the notion of a fact being that which occurs when a universal is exemplified. These notions are not definable by reference to whether a particular is concrete—that is, in space-time (having beginning and ending temporal boundaries). Perhaps, some facts/ events are not in space-time. Acquaintance with a p-relation Rn assures that we understand its adicity and the position relations it determines. If we have doubts about a universal’s adicity, it can only mean that we are not acquainted with it. I hold that the number nof positions in an Rn-fact is exactly the number i of position relations CiR determined by Rn. I hold that the number of position relations that a p-relation determines must coincide with the adicity of the universal. All the same, the adicity of a universal is not a property it has contingently. Thus, it is best to maintain that there is no property of having n-adicity. One might be concerned that this brings too much information. The notion of n-adicity is a numerical notion and thus in being innately acquainted with a relation (such as ‘acquaintance’) we are innately acquainted with number. I accept this. Indeed, I accept that for any universal with which one is acquainted, one understands its adicity. This helps to explain how we are acquainted with ‘belief’ relations of certain adicity and how we understand that there is a family of multiple relations of ‘belief’ of different adicities, many with which we are not acquainted. If we are acquainted with a given ‘belief’ relation, then we understand its adicity. This may at first seem to generate something of a problem. Russell certainly admits that there has been a long history of confusion about the nature of belief and some philosophers mistakenly held that ‘belief’ is a dyadic relation—a confusion which he thinks is responsible for making it so difficult to give an account of false belief without imagining there are entities that are objective falsehoods.27 Clearly, I need a new error theory to explain that history. Since obviously we do recognize that there can be false belief, and since we are acquainted with relations of ‘belief’ (for otherwise we couldn’t ascribe belief to others), I hold that there was never any doubt that it is not always the case that a belief involves a dyadic relation. The philosophical

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puzzle of belief has always been about what sort of many-placed relation of ‘belief’ is involved, not whether a many-placed relation is involved. The situation is akin to the famous Tuned-Deck card illusion which, in fact, consists in a family of different “pick-a-card-any-card” tricks. What made it so difficult to unravel the sleight of hand was that when a given trick was excluded as not being conducted in a given performance, the inquisitor thought that it would not be used in a subsequent performance either. In short, it was a mistake to imagine that exactly one trick was involved.28 The philosophical puzzle of false belief arises from metaphysicians demanding that all cases of belief be dyadic cases and in inventing theories of abstract particulars as meaning propositions to achieve this. Russell’s diagnosis, in offering his multiple-relation theory, is that there is family of ‘belief’ relations of different adicity involved. Different ones (including dyadic ones) are involved in different cases. A similar error infests our understanding of the adicity of ‘acquaintance’ itself. In Theory of Knowledge, Russell admits that he is torn in quite different directions, and at times, he has conflicting intuitions about the nature of his relation of ‘acquaintance.’ One direction pulls him toward maintaining that there is only one kind of acquaintance relation and its object may be any entity whatsoever, universal or particular. This is the pristine view of acquaintance that I accept. The other direction that captivates him is to abandon the relation of acquaintance and replace it with several distinct acquaintance-like relations whose logical analyses may make them of different adicity in one’s quest to somehow embody restrictions on the relata they may have (particular, universal, logic form, etc.). Russell writes (TK, p. 97): It should be said, to begin with, that “acquaintance” has, perhaps, a somewhat different meaning, where logical objects are concerned, from that which it has when particulars are concerned. Whether this is the case or not, it is impossible to decide without more knowledge concerning the nature of logical objects than I possess. It would seem, that logical objects cannot be regarded as “entities”, and that, therefore, what we shall call “acquaintance” with them cannot really be a dual relation. The difficulties which result are formidable, but their solution must be sought in logic. For the present, I am content to point out that there certainly is such a thing as “logical experience”, by which I mean that kind of immediate knowledge, other than judgment, which is what enables us to understand logical terms.

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Quite clearly, Russell is concerned that adopting different kinds of acquaintance will open a Pandora’s box. He has no further analysis to render and nothing further can be rendered since it would require naturalizing intentionality itself. Russell wisely opts for the pristine view that we are acquainted with one unanalyzable dyadic relation of ‘acquaintance.’ I add that acquaintance with ‘acquaintance’ is innate. In sum, acquaintance with a universal yields an understanding of its two-fold capacity—its unique capacity to ground the unity of a fact. It yields an understanding of the sort of fact that would exist if the universal were to be exemplified. Acquaintance with a universal gives an understanding of its adicity, revealing whether a monadic, dual, etc., fact would exist if the universal were exemplified. For any universal with which we are acquainted, we cannot fail to understand its adicity and, if it is a p-relation, we are acquainted with all the position relations it determines. Any genuine dispute about adicity is sufficient to show a lack of acquaintance with the universal in question.

Identity, Poetic License, and the Russell Property All determinate intentional thought is scaffolded by quantificational thinking. We shall see in Chap. 6 how the scaffold enables us to understand the simple-type theory of Principia. For the present it is important to see that the quantificational scaffold of intentionality is not the same as a simpletype scaffold. We can illustrate this by pointing out how the poetic license of intentionality enables thought about the Russell class, and indeed, more amazingly still, the Russell property. We can direct our thoughts to all manner of entities, including logical impossibilities, even though there may well be no such entities to which our thoughts are directed. We think about golden mountains, round-squares, sets that violate simple types, Desdemona’s loving Cassio, and even the Russell property. Poetic license allows us to direct our intentionality determinately. It is wholly unconstrained because it is the working of an unbridled quantificational apparatus. We can thereby think about the Russell class and conjure all manner of seeming paradoxes and similar riddles about what is not. Tractatus (TLP 6.5) maintained that the riddle does not exist—that showing prevents the very articulation of philosophical riddles. To the contrary, the riddle does exist and Russell’s paradox of classes is but one of many examples. Quite obviously, we do think about the Russell class (of all classes not members of themselves). How? Quantificationally. Note, however,

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that formal grammar of Principia’s simple types defines the sign “∈” in such a way that makes both “x ∈ x” and “~(x ∈ x)” meaningless. Principia has: *20.02 x ∈ φ! = df φ!x. The grammar does not allow the expressions “φ!(φ!)” and “~φ!(φ!)” to be well-formed. That in no way constrains the intentionality of thinking. We cannot be acquainted with classes (sets) since, according to Russell, there are no such entities. We cannot be acquainted with the naïve relation ‘∈’ of class membership since there is no such relation. Nonetheless, Russell’s conviction that the technical expressions “x ∈ x” and “x ∉ x” are meaningless in the grammar of Principia does not cause any difficulties for the cognitive apparatus that enables thinking about a Russell class.29 That one can believe (falsely) that there is a simple-type free contradictory Russell class is certainly no great puzzle of intentionality. We straightforwardly have the following: M believes (∃R)(∈(R) • (∃y)(x R y ≡x ~(x R x))). That is, M can naïvely believe that there is a membership relation R, that is, a relation meeting conditions ∈(R) defining a conception of membership,30 and believe that there is an entity y of all and only those entities x that do not bear R to themselves. The Poetic License of intentionality enables pointing toward such illogical “objects” of thought—so long as we resist the Meinongian fallacy of concluding that there are illogical objects about which we think. The quantificational scaffolding of the apparatus of thinking enables thinking naïvely about the Russell class. Similarly, in dealing with beliefs about Zermelo sets obtained by his Aussonderung axiom, we think about a membership relation ∈Z(R) and put: M believes (∃R)(∈Z(R) • (a)(∃y)(x R y. ≡x. x ∈ a • ~(x R x))). This enables M to believe that there is a Zermelo set of all members of a that are not members of themselves. Followers of Zermelo will thereby believe that there is no Zermelo set that is universal. But again following the pattern, one can just as readily think about Quine sets via ∈Q(R) and believe in a universal Quine set. The important point is that while Russell’s cognitive apparatus is quantificationally scaffolded, it by no means prevents one from forming thoughts and beliefs about all manner of entities. The quantification scaffolding of thinking has no deleterious impact on the poetic license of intentionality. One can think about the Russell class even though there is no such entity and no such relation ‘∈’ of

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naïve membership. Intentionality enables us to think “about” type-free sets/classes even if there are no such objects about which we think. Poetic license is realized even though thought is quantificationally scaffolded. The case of thinking about the Russell property, however, is more complicated. Cocchiarella was first to fully appreciate that any approach to propositional attitudes that introduces predictable and referential concepts to do the work of predication requires that the apparatus of thinking be type-­ free. All too frequently, this point is overlooked. For example, Hanks (2015, p.  88) imagines that he can accommodate predicational mental acts by appeal to concepts and kinds of mental act events. But he does not stop to worry that in embracing such concepts for every open wff the paradox of the Russell property will arise. We can, after all, think about the Russell property. Cocchiarella faced this issue head-on. It requires nothing short of a solution to the paradox of the Russell property! Paralleling Quine’s systems of NFU-sets and ML-sets versus ML-classes, Cocchiarella developed a type-free conceptualist theory of unsaturated referential and predicable concepts as evolved cognitive capacities. Mutual saturation of referential and predicable concepts (a process governed by rules of compositionality) is a sort of mental synthesis that results in a mental act of predication. Though completely unavailable to Russell’s Principia era, Cocchiarella’s work establishes the paradigm for such approaches where concepts do the work. Cocchiarella (2007, 2015) accepts that there are singular thoughts and maintains that poetic license leads one to hold that thought involves a type-free holistic conceptualism. On this view, thought consists of synthesizing agencies acting on referential and predicational concepts (as evolved biological capacities) which mutually saturate to form predications. The conceptualism is “holistic” because it allows impredicative concept formation to be a spandrel that is riding on an armature of the ramified concept formation that is wrought by natural selection. To see his point about type-freedom, consider an attempt at capturing a belief about the Russell property as follows: M believes (∃R)(e(R) • (∃y)(x R y ≡x ~ (x R x))). Here e(R) says that R is a relation of exemplification in analogy with ∈(R) which says that R is a relation of membership. The trouble is that the expression “x exemplifies y” is meaningless unless y is a property, and we have decided that there is no property ‘y is a property.’ Cocchiarella’s solution is to introduce the Russell concept, [λx (∃G)(x = G • ~Gx]

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Cocchiarella has type-freedom and singular thought, and thus he has: M believes [λx (∃G)(x = G • ~Gx]([λx (∃G)(x = G • ~Gx]) Russell’s Theory of Knowledge cannot parallel this. According to the theory, I adopt on its behalf, all predicational thinking is quantificationally scaffolded. There is no singular thought. How then do we think about the Russell property? Happily, a quantificational scaffold binding predicate variables is not a simple-type scaffold. This is the key. Thinking about the Russell property is made viable precisely by our acquaintance with the relation of ‘identity.’31 We have: M believes that (∃P)(Px ≡x (∃G)(x = G.•. ~Gx)). By holding that we are acquainted with the relation of ‘identity,’ a quantificational scaffold comes apart from a simple-type scaffold. One can believe (confusedly) that there is a property P, that an entity x has iff x is identical to a property G that does not have itself. The thought is quantificationally scaffolded but not simple-type scaffolded. Of course, in the simple-type regimented grammar of Principia, the expression “x = ψ!” is ungrammatical. But the defined sign “=” of Principia is not a sign for the relation of ‘identity’ with which we are acquainted. There is no simple-type scaffolding involved with any universal with which we are acquainted. That is, any property or relation with which we are acquainted is, ipso-facto, type-free. Once we find the quantificational apparatus that eliminates bound predicate variables, we have a solution to the problem of thinking about the Russell property. The quantificational scaffold of predicational thinking allows us to think about entities such as the Russell property. Without acquaintance with ‘identity,’ we couldn’t do this. The riddle does exist. It exists because intentionality realizes poetic license. Wittgenstein’s Tractarian agenda of showing hoped to make thinking about the Russell property impossible! The internal limits precluding illogical thought are demanded by the notion that the structure of the thought shows its would-be truth-maker. Wittgenstein’s Tractatus notwithstanding, we are able to think about the Russell property. And it is precisely our acquaintance with the relation of ‘identity’ that enables it. We are innately acquainted with the relation of ‘identity.’ One might, at first, seem to have reason to doubt this. Indeed, Russell admitted that early on he was himself attracted to Wittgenstein’s ideas for the elimination of a relation of ‘identity.’ He then thought Wittgenstein’s approach

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might offer a friendly modification consonant with scientific method in philosophy. But as we noted in Chap. 2, he soon changed his mind (MPD, pp.  115–116). Wittgenstein’s elimination of ‘identity’ undermines the Cantorian notion of cardinal number which is a centerpiece of the Whitehead-Russell’s logicist agenda against abstract particulars in the ontology of mathematics. On Russell’s behalf, I accept ‘identity’ as a genuine dyadic relation with which we are innately acquainted. Of course, in Principia we find the sign “=” defined as follows: *13.01 x = y = df (φ)(φ!x ⊃ φ!y). I cannot take this approach. Such a definition would make acquaintance with ‘identity’ ruled out since our understanding of a relation of ‘identity’ would be through comprehension: (∃R)(x,y)(R!xy ≡ (φ)(φ!x ⊃ φ!y)). As we will see in Chap. 6, since we have to identify like variables our understanding of the wff in the above instance of comprehension requires prior independent acquaintance with the relation of ‘identity.’ Understanding a universal by understanding a wff giving its exemplification conditions is never sufficient for acquaintance with the universal. In any case, it is clear that cognitively the notion of identity is not a notion of indiscernibility. Like the case of ‘acquaintance’ I cannot fathom how, if there indeed is such a relation, one could become acquainted with ‘identity.’ Hence, I take it that we are innately acquainted with this relation. The existence of a genuine dyadic relation of ‘identity’ with which we are acquainted may seem at first to be in tension with the fundamental thesis of the Principia era. It is not, as long as we hold that the ‘identity’ relation does not come with hook-eyes: () = [ ]

[ ] = ().

If there were hook-eyes, there would be two distinct identity facts and ‘identity’ would be a p-relation. As long as ‘identity’ is not a p-relation, identity facts are not permutative. The fundamental thesis of the Principia era is consistent with this. The existence of the fact that x = y does not logically entail the existence of another fact that y = x. They are the same fact. Now recall that we have adopted the thesis that acquaintance with a relation yields an understanding of its adicity. If we are acquainted with a relation of ‘identity,’ then we wouldn’t feel uneasy about it being dyadic. Do we feel uneasy? One may worry that Fregean puzzles about informative identity

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shows that we are uneasy about there being a dyadic relation of ‘identity.’ But quite to the contrary, it is because we feel sure that there is a dyadic ‘identity’ relation that the Fregean puzzles about informative identity arise. There is no denying that Wittgenstein took himself to feel uneasy about accepting a relation of ‘identity.’ Since on my view he was innately acquainted with ‘identity,’ his uneasiness must be explained away. Wittgenstein thought that identity is part of pure logic and not given by a theory added on to logic. That seems quite right. The kernel of truth that Wittgenstein intuited is that our understanding of ‘identity’ is inseparable from our understanding of quantification. Where Wittgenstein went astray was with his N-operator account of quantification—an account that hoped to emulate ‘all’ by appeal to an arbitrarily bounded notion of ‘any’ using exclusive free variables. This improperly divorces ‘identity’ from the wholly unrestricted bound variables involved with our conception of ‘all.’ There are no singular identity thoughts. Doubts about ‘identity’ being a genuine relation are produced by missing this. All thought has a quantificational scaffold. When seen in this way, we come to realize that it is Russell’s quantificational theory of definite descriptions, not Wittgenstein’s exclusive quantifiers, that reveals the intimate indivisible connection that ‘identity’ has with the apparatus of quantification theory. In Chap. 6, repairs to Theory of Knowledge to capture compositionality will make use of a Quinean method for eliminating variables bound and free. Wittgenstein’s elimination of identity is not a friendly ally of that method. It would impose serious, perhaps insuperable, complications. All things considered, it is best to accept that we are innately acquainted with ‘identity.’ And by accepting acquaintance with ‘identity,’ we have the foundation of an understanding of ‘difference.’ We are not acquainted with a relation of ‘difference.’ We understand it by means of understanding the fusion of quantification, negation, identity, and all the logical particles. It is this apparatus, including our innate acquaintance with the relation of ‘identity,’ that realizes the poetic license characteristic of intentionality—with a vengeance.

Acquaintance and the Verb On the one hand, the question of how intentionality has determinate directedness is solved by appeal to the quantificational scaffold of predicational (discursive) thought. The nature of intentionality, on the other hand, has been left as an unresolved mystery. This approach entirely skirts the so-called problem of predication made famous by Davidson. I fear that Davidson’s problem of predication is generated from his accepting some

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form of the Biblical or Rylean Fido-Fido assumption that the foundational cognitive apparatus is the apparatus of referring (naming). In the beginning was the name.32 If one begins from this assumption, then explaining asserting, predicating, saying, and so forth becomes a philosophical conundrum. Asserting, say, that Socrates is wise, does not consist of a consecutive series of acts of naming such as the following: Socrates⁀the relation of exemplification⁀wisdom.

It won’t help to add that each act is accompanied by an understanding of the entities named. Thus, if one begins from the view that naming is the foundational cognitive act, then how is predication possible? Where is the unity—the synthesis that is essential to predication in thought and language? Davidson ties predication to a special sentence unity that makes it into an assertion. He writes (Davidson 2005, p. 77): if we do not understand predication, we do not understand how any sentence works, nor can we account for the structure of the simplest thought that is expressible in language. At one time there was much discussion of what was called the ‘unity of the proposition’; it is just this unity that a theory of predication must explain. The philosophy of language lacks its most important chapter without such a theory; the philosophy of mind is missing a crucial first step if it cannot describe the nature of judgment; and it is woeful if metaphysics cannot say how a substance is related to its attributes.

In a follow-up, articulating the problem even more starkly, he writes: (Davidson 2005, p. 85): The problem is easier to state in semantical terms, and Plato gave us what we need to recognize it as a problem when he said that a sentence [statement] could not consist of a string of names or a string of verbs. The sentence “Theaetetus sits” has a word that refers to, or names, Theaetetus, and a word whose function is somehow explained by mentioning the property (or form or universal) of Sitting. But the sentence says that Theaetetus has this property. If the semantics of the sentence were exhausted by referring to the two entities Theaetetus and the property sitting, it would be just a string of names; we would ask what the verb was. The verb, we understand, expresses the relation of instantiation. Our policy, however, is to explain verbs by relating them to properties and relations. But this cannot be the end of the matter, since we now have three entities, a person, a property, and a relation,

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but no verb. When we supply the appropriate verb, we will be forced to the next step, and so on.

The first lesson Davidson hopes to draw is that “a satisfactory account of predication depends on relating it to the truth of sentences” (Op. Cit. p, 141). Davidson holds that whatever our intention may be in a given speech act, the act will not be understood by someone who does not know the truth-conditions of the background assumptions that set the stage for that speech act. He continues: “The second lesson, which the history of attempts to provide a semantics for predicates teaches us, is that since the meaning of a predicate lies in its naming universals (properties, relations or sets), it will not solve the problem because it will always lead to an infinite regress” (Op Cit. p. 143). “Truth,” Davidson argues, “is the key to the unity of the acts we perform by uttering sentences, whether we are interested in giving information, giving a command, or asking a question” (Op Cit, p.  141). Tarski’s recursive definition of “true-satisfaction” will establish the rules governing the admissible ways which one kind of predicate expression φξ engages with its occurrence in another structured complex. This, he hopes, gives us the compositionality and the systematicity of thought and language. In Davidson’s view, at the base case, it is not unity that explains truth and predication; it is rather truth that explains unity and predication. Davidson’s thesis is that facts can only be understood as truth-makers of those expressions that are declarative sentences which make predications. Accordingly, he holds that the notion of a relating relation comes from the linguistic notion of a predicate. Similarly, he holds that the notion of a fact comes derivatively from the notion of statement. Thus, he concludes that the central features of a fact are extrapolated from features of the declarative sentence for which the notion of a fact is invented as truthmaker. Consider the sentence: The meteorite hit the ground with a thud.

The phrase “the meteorite” plays the thematic role of the agent, and “the earth” the recipient and “hitting with a thud” is the relationship. We are then invited to imagine that the linguistic thematic roles of agent and recipient and predicate have ontological counterparts in the fact. This reinforces the mistaken thesis that a relation relating is understood in terms of predicating (attributing, saying).

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Davidson’s “lessons” are astonishing and one can’t help but feel that the train has come off the rails. But as with any genuine philosophical conundrum, we are only in a position to understand what happened after we have found the flaw. I hold that Davidson’s problem of predication is produced by his assumption that the foundational intentional activity is the activity of referring (naming). It is this that drives one into the fly bottle (to borrow the colorful notion from Wittgenstein) of needing a unity that would ground the use of a declarative sentence in an act of assertion (predication). One must reject the assumption. Why begin with naming? The intentionality involved in naming is no less mysterious than the intentionality involved in predicating (asserting). By rejecting this assumption of the foundational nature of naming, one can thereby embrace different artifactual uses of words—uses that make different kinds of words into different tools for communicating—for example, a tool for the activity of naming, a tool for the activity of predicating, and so forth. It is our acquaintance with relations (and properties) that is the foundation of our artifactual use of something as a verb in communicating that an action (an event) has, is, or will occur. Davidson has the cart before the horse. The Russellian notion of a relation that is exemplified in a fact (a relation “relating,” as Russell sometimes put it) is not modeled on the notion of saying or predicating that entities exemplify a relation. It is acquaintance with the relation that comes first. The notions of “universal” and “fact” and “exemplification” are not dependent, as Davidson would have it, on the notions of “verb” and “statement” and “predication.” We are acquainted with universals. The notions of “universal” and “fact” and “exemplification” arrive from cognitive faculties of acquaintance. This is the source of the notion of the “verb,” the “statement,” and “predication.” To repeat, it is incorrect to imagine, as Davidson does, that the special way a property or relation inheres in a fact that grounds the unity of the fact is modeled on predicating and that our notion of relating is a shadow of our notion of predicating. Quite the contrary, it is the understanding that comes from acquaintance with relations that lead us to invent special words to use as predicates in language and thought. A relation exemplified in a fact is not “relating” in the sense of asserting or saying the constituents of the fact are related thus and so. The notion of a relation unifying a fact is not the notion of it occurring in the fact in a way that predicates something. The conflation of relating with predicating must be carefully avoided. The conflation is most apt to occur in the case of properties. Properties inhere in certain facts, but they don’t occur in them predicationally. Unfortunately, there is no ready expression for the special way in which properties inhere

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in facts. So it is easy to fall into the dangerously misleading linguistic practice of speaking as if a property inhering in a fact to ground its unity were a case of the property predicating. It is also quite incorrect, and dangerously misleading, to speak of the special way of inhering as if it were this that brings about the existence of a given fact. Obviously, facts (if they are brought about at all) are not brought about by universals. Orilia (2014) has recognized the importance for a theory of linguistic understanding of the notion of thematic roles associated with the use of verbs—“O-­roles” (such as agent, patient, beneficiary, instrument, and the like).33 We can embrace this. We come to understand how O-roles governing verbs can arise from acquaintance with relations. For example, we have already seen that acquaintance with the relation ‘acquaintance’ is the foundation of the notions of subject and object (particular). Acquaintance with a universal yields an understanding of the sort of fact that would occur if that universal were exemplified. To understand some verbs, one must acquire the O-role of agency. The notion of agent comes from the notion of a subject which we get by acquaintance with ‘acquaintance.’ Our acquisition of an acquaintance with the universal ‘loves’ becomes possible whereby we understand the O-role of agent. O-roles can thereby be embraced as long as one keeps in mind that predicating presupposes a mind’s acquaintance with a universal. It is this acquaintance that explains linguistic O-roles in virtue of acquaintance with relations. Interestingly, complications with thematic O-roles can arise for linguistic theories embracing them. I suspect that the complications are due to our general ignorance of logical forms and our incomplete understanding of the universals in question. If there is ever confusion about an O-role accompanying the use of a verb, that is itself good grounds for holding that it is due to the lack of acquaintance with the relation involved. We must be on the lookout as well for equivocations when objections are raised to the existence of O-roles involved with verbs. Imagine worrying that the verb “to meet at” does not itself come with determinate roles of actor and patient. Suppose the worry comes from realizing that while it can happen that both Russell and Wittgenstein brought about the event of their having met at The Hague in 1919, it can equally as well have been Russell alone or Wittgenstein alone that brought about the event of their meeting. Indeed, it might have been an accident altogether, and that no one brought it about. It is because we recognize the universals involved that we can so readily diagnose this kind of objection to O-roles of the verb “meet” as based on equivocation. Obviously, equivocations do not count as evidence against the existence of O-roles understood by

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acquaintance with relations. Intentional verbs are notorious offenders because they enable ambiguities. If we say that Whitehead traveled to Cambridge, then we may mean that he initiated the process that brought about the event in question. Clearly, this sort of “traveling” is different from the “traveling” his body underwent when it was returned from America to England for burial at Cambridge. Equivocations are revealed because we understand and are acquainted with universals. Orilia’s emphasis on O-roles is important. Accepting O-roles, however, does not confuse issues of thought and predication pertaining to intentionality with issues pertaining to the structure and unity of a fact. To be sure, in Theory of Knowledge, Russell worries that relations don’t have a spatially “first” term and a “second” term etc., as do the expressions of statements of a relation holding. For this reason, he was skeptical of the metaphor of hook and an eye. Accepting O-roles does, to some extent, allow cases where the hook-eye metaphor is particularly apt. It helps to keep the role of the verb front and center in understanding predication in thought and language. Acquaintance with relations and understanding their hook-­eye structures seems to be a precondition essential for inventing and understanding the verbs of natural language and for communication by their means. All the same, we must emphasize that relating (unifying, being exemplified) is not predicating. Intentionality has nothing to do with unity. Relations don’t predicate anything and neither do properties. The understanding of universals gets its foundation in acquaintance. It cannot arise from a cognitive act of referring and naming which are no less mysterious a feature of intentionality than predicating. Our understanding of universals is prior to our adoption of conventions governing language and prior to our manufacture of artifacts suitable to facilitate communication. The conventions settle on verbs with lexical inflections carrying tense. These are used with conventions for recognizing acts of assertion (predication). The linguistic notions of “verb,” “predicate,” and “predication” have their foundation in our acquaintance with universals. When a given particular is an artifact which is typically employed by minds to make a specific assertion, we naturally look to certain parts of the artifact as indicative of the nature of its process of creation. Front and center is always the verb. Our acquaintance with universals provides our understanding of the notion of a fact which would exist were the universals to be exemplified. Universals with which we are acquainted ground the notion of an event which would be occurring if the universal were exemplified. Russell did well in Theory of Knowledge to include a chapter “On the Acquaintance involved in our Knowledge of Relations.” From acquaintance with

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universals, together with a great deal of cognitive stage setting, is born the artifactual use of an expression as a verb of a natural language. Indeed, Jackendoff (1994) suggests that it is not untoward to hold that grammar is, fundamentally, built around the verb.34 The beginning (of language) is acquaintance with relations and, thereby, in the beginning was the verb!

Notes 1. The point is the same whether or not the entities named are private objects of introspection. 2. See Quine (1973). 3. My thesis is in important ways in tension with Quine’s The Roots of Reference. It strongly disagrees with any behaviorist thesis that the origins of the quantificational apparatus lie in the habituation to patterns of stimulation which produce behavior protocols. I agree with Chomsky’s view that transformational grammar is innate. No behaviorist account of the ontogenesis of reference is viable. The quantificational apparatus cannot come in degrees and cannot evolve by natural selection in stages—improving in a process of adaptation to environments. 4. See Landini (2017) for a discussion of Kripke’s Nixon-Jones argument. 5. See Eldridge-Smith and Eldridge-Smith (2010). 6. Let’s find a no-propositions version of Prior’s wff: θ(p)(θp Ͻ ~p) .Ͻ. (∃p)(θp • p) • (∃p)(θp • ~p) Start from θP which is to mean “S thinks of P” and thus: θP =df φF ≡F F = P . •. S thinks [ιFφF][ψF], for some wffs φ and ψ. To recover θ(p)(θp Ͻ ~p), we’ll need a property H such that Hx.≡x. (P)(θP ⊃ ~Px). i.e., Hx ≡x (P)((φF ≡F F = P .•. S thinks [ιFφF][ψF]) ⊃ ~Px). Thus we get: (∃H)(Hx.≡x. (P)(θP ⊃ ~Px) :•: θ(H)).⊃. (∃P)(θP • Py) • (∃P)(θP • ~Py)). But again there is no reason to think that there is a wff φ for which the following is true: (∃H)(Hx .≡x . (P)(θP ⊃ ~Px) :•: θ(H)) i.e. (∃H)(Hx .≡x . (P)((φF ≡F F = P .•. S thinks [ιFφF][ψF]) ⊃ ~Px) :•: φF ≡F F = H .•. S thinks [ιFφF][ψF]), 7. Followers include Parsons (1980) and Zalta (1982, 1988). For an in-depth discussion of Meinong’s Principle of Independence, see Lambert (1983).

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8. Nowadays appeal to intensional entities is out of fashion since, as abstract particulars, they militate against the research program of the naturalization of mind. 9. See Crane (2001). 10. An important discussion is found in Wahl (2007). 11. The use of these Fregean structured variables cannot be treated by the techniques of Chap. 6. But happily, that will not jeopardize their use. 12. The notion of “find,” which means “bumping into” without searching or striving to find, is not relevant. 13. For an important discussion of the Reichenbach view and its capacity to resist the objections of Kaplan et al., see Manuel Garcia-Carpintero (1998). 14. This example is adapted from Orilia (2010, p. 121). 15. Interestingly, Perry (2012) has disavowed the thesis often attributed to him according to which, as he puts it, “…having a self-belief requires the word ‘I’ or some other first-person expression.” Perry explains: “I did not claim, did not believe, and do not believe that first-person pronouns are essentially involved in having self-beliefs” (p. 98). He goes on to say that “Self-notions are about their possessors because of their causal and informational role, not because of convention, as is the case with “I.” “I” is a basic tool for communication, but the self-notion is a tool for organizing information and guiding action” (p. 99). 16. For an excellent detailed discussion of indexicals and a descriptivist approach to them, see Orilia (2010). 17. See John Perry (1979). See also Castañeda (1966, 1967) and Lewis (1979). 18. A Cartesian accepting that indexicals are involved in soliloquy might best avoid making existence into a property by taking “I exist” to abbreviate “I am now thinking this.” 19. See also footnote 2 of “Knowledge by Acquaintance and Knowledge by Description” (KAKD, p. 220). 20. I use the expression “type*” here in a different sense than that which occurs in a Realist (objectual) interpretation of Principia’s simple-type theory as a theory of attributes (properties and relations) in intension. The distinction between a universal and particular, for example, is a type* distinction. It is not a distinction in simple type. 21. Van Inwagen (2014, p. 120) imagines some further difficulty that requires using numerals marking the positions of the variables. See Chap. 6. 22. If intentionality includes something akin to what Sartre called “reflexion,” which is an implicitly self-reflective activity, then Russellians would have to find it in the nature of relation of ‘acquaintance.’ 23. See Forrest (2005) for an attempt at a theory of perception which avoids sense-data in favor of making “structured” sensory universals into immediate objects of perception. I fear that this introduces as many new problems as it avoids. It makes universals immediate objects of perception. In my view, they are components of the cognitive apparatus of perceiving.

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24. To be sure, we cannot easily find the quantificational apparatus at work in thinking, as it were, about qualia. To this date, the logical forms involved in the quantificational scaffolding of introspective thinking are unknown. I maintain only that they involve the formation of impredicative quantificational concepts—concepts that are “looped” insofar as they may be said to quantify over a realm that includes themselves. 25. See Hasan & Fumerton (2015) on how acquaintance with particulars addresses the Sellarsian dilemma. 26. See Fales (1996) for important arguments in favor of the given. 27. See, for example, Russell’s Scientific Method in Philosophy. 28. See Dennett (2005) for a discussion of the Tuned-Deck and its implication for the philosophy of mind. 29. This is an important issue that was raised by Nicholas Griffin in conversation. 30. For example, ∈(R) might stand in for clauses that would include, (a, b)((x)(x R a ≡ x R b) ⊃ a = b). 31. Cocchiarella (1973) was perhaps first to recover the role that identity plays in the characterization of the Russell property. Russell himself identified this role in one of his many early systems. See Landini (1992). 32. Adam is said to have named the animals (Genesis 1: 24–27). 33. Orilia (2014, pp. 283–303). 34. Jackendoff (1994, p. 95) observes that some current natural languages have a generative transformational grammar centered on the verb. Semitic languages such as Hebrew and Arabic offer very good examples of verboriented languages where different forms of the verb preserve the order of the consonants but change the vowels and length of the consonants: kataba (he wrote), kaataba (he corresponded), kutib (was written), kattaba (he caused to write). A single verb incorporates markers for the topic and/or other particulars plus various other markers that would be expressed in English by auxiliary verbs or adverbs. This is an exciting finding for any Russellian for whom the foundation of language is acquaintance with universals.

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Landini, Gregory. 1992. Russell to Frege 24 May 1903: “I believe I have Discovered that Classes are Entirely Superfluous”. Russell 12: 160–185. ———. 2017. Meinong and Russell: Some Lessons on Quantification. Austrian Philosophy Conference Proceedings. University of Texas, Arlington, Axiomathes 2017. Lebens, Samuel. 2017. Bertrand Russell and the Nature of Propositions: A History and Defense of the Multiple-Relation Theory of Judgment. New York: Routledge. Lewis, David. 1979. Attitudes De Dicto and De Se. Philosophical Review 88: 513–543. MacBride, Fraser. 2005. The Particular-Universal Distinction: A Dogma of Metaphysics? Mind 114: 565–614. Meinong, Alexius. 1904. The Theory of Objects. In Realism and the Background of Phenomenology, ed. Roderick Chisholm, 76–117. Atascadero: Ridgeview Publishing Co. Merricks, Trenton. 2007. Truth and Ontology. Oxford: Clarendon Press. Orilia, Francesco. 2010. Singular Reference: A Descriptivist Perspective. Dordrecht: Springer. ———. 2014. Positions, Ordering Relations and O-Roles. Dialectica 68: 283–303. Parsons, Terence. 1980. Nonexistent Objects. New Haven: Yale University Press. Perry, John. 1979. The Problem of the Essential Indexical. Noûs 13: 3–21. ———. 2012. Thinking about the Self. In Consciousness and the Self: New Essays, ed. J. Liu and J. Perry. Oxford: Oxford University Press. Prior, A.N. 1961. On a Family of Paradoxes. Notre Dame Journal of Formal Logic 3: 16–32. Quine, W.V.O. 1956. Quantification and Propositional Attitudes. The Journal of Philosophy 53: 177–187. ———. 1966. The Ways of Paradox. In The Ways of Paradox and Other Essays by W. V. O. Quine (New York: Random House). First published as “Paradox,” Scientific American 206, 1962. ———. 1973. The Roots of Reference. La Salle: Open Court. Schiffer, Stephen. 1994. A paradox of Meaning. Noûs 28: 279–324. ———. 2006. A Problem for a Direct Reference Theory of Belief Reports. Noûs 40: 361–368. Smith, Quentin. 1993. Language and Time. Oxford: Oxford University Press. Soames, Scott. 2014. The Analytic Tradition in Philosophy: Volume 1, The Founding Giants. Princeton: Princeton University Press. Van Inwagen, Peter. 2014. Existence: Essays in Ontology. Cambridge: Cambridge University Press. Wahl, Russell. 2007. ‘On Denoting’ and the Principle of Acquaintance. Russell: The Journal of Bertrand Russell Studies 7: 7–23. Zalta, Edward. 1982. Abstract Objects. Dordrecht: D. Reidel Publishing Co. ———. 1988. Intensional Logic and the Metaphysics of Intentionality. London: Bradford Book.

CHAPTER 5

Direction

Theory of Knowledge solved the problem of direction. That is, it solved the problem of how, working only from those universals with which we have acquaintance, to form a definite description that would be satisfied by a permutative fact. The key is that in being acquainted with a permutative relation (p-relation), we are acquainted with its adicity and all the position relations it determines. We then form a definite description by appealing to the position relations. Russell explains as follows (TK, p. 145): It is obvious that the question of whether a belief is true depends only upon its objects. … The belief is true when the objects are related as the belief asserts that they are. Thus the belief is true when there is a certain complex which must be a definable function of the belief, and which we call the corresponding complex, or the corresponding fact. Our problem, therefore, is to define the correspondence. If our complex is one which is completely determined by its constituents, the problem is simple. A non-permutative belief1 is said to be true when there is a complex consisting of its objects; otherwise it is said to be false. We have now to extend this definition to permutative beliefs. In linguistic terms, our problem is: When several complexes can be formed of the same constituents, to find a method of distinguishing between them by means of words or other symbols. In logical terms, our problem is: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8_5

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When several complexes can be formed of the same constituents, to find associated complexes unambiguously determined by their constituents. By “associated complexes,” here, I mean complexes which exist whenever the original complexes exist, and not otherwise.

Russell goes on to clarify his solution thus (TK, p. 147): C1,…, Cn are a system of relations, determinate when R is given, but requiring to be simply recognized in each case, and not describable in general terms. When C1,…, Cn are given, conversely, R is determinate, thus our complex γ can be described unambiguously without mentioning R, as simply “the complex γ in which x1C1 γ and  x2C2 γ,…, and xnCn γ.” If we have decided, once and for all that when the one with the C1 –position is to be mentioned first, then the one with the C2 –position, and so on, we can denote the complex γ by the symbol R(x1, x2,…, xn). But this symbol, though it has a certain notational convenience, is not sufficiently explicit for philosophical purposes. For philosophical purposes, the symbol (ιγ)(x1C1 γ • x2C2 γ,…, • xnCn γ). is preferable, because it does not make a more or less concealed use of the spatial order of x1, x2,…, xn.

And next Russell summarizes his position (TK, p. 146): We may now generalize this solution, without any essential change. Let γ be a complex whose constituents are x1, x2,…, xn and a relating relation R. Then each of these constituents has a certain relation to the complex. We may omit the consideration of R, which obviously has a peculiar position. The relations of x1, x2,…, xn to γ are their “positions” in the complex; let us call them C1, C2,…, Cn. ….Thus our complex γ can be described, unambiguously without mentioning R, as simply “the complex γ in which x1C1 γ, x2C2 γ,…, and xnCn γ.”

As we can see to solve the problem of direction, Russell explicitly offers the definite description: (ιγ)(x1C1 γ • x2C2 γ,…, • xnCn γ). This is a definite description that purports to refer to a permutative fact γ. It offers a definite description by appeal to position relations C1,…, Cn

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each of which would, if there exists a permutative fact that R(x1, x2,…, xn), occur unifying the following facts: x1C1 γ x2C2 γ ⋮ xnCn γ. Mind you, there may well be no such facts at all. The solution to the problem of finding a definite description for a permutative fact is to be found by having the description concern position relations. It succeeds only if the associated facts are not themselves permutative. It may succeed contingently. It is a purely logical matter that each p-relation determines position relations. The permutative relation and the position relations it determines exist independently of their being exemplified. In virtue of this purely logical matter, the exemplification of a permutative relation, say, R2 yields a fact x-R2-y that is logically associated with certain other atomic facts that exist if and only if it exists. The logical relationship between such associated facts, though their existence may well be contingent, does not violate the fundamental thesis of the Principia era since such facts contain the original fact x-R2-y as constituents. For example, the existence of the permutative fact x-R2-y assures the existence of the facts 2

x- C1R2 -(x-R2-y) x- C2R - (x-R2-y). Of course, if there is no permutative fact x-R2-y, that is, if there is no fact that R2(x, y), then there are no such associated facts either. We describe the permutative fact by appealing to the position relations that its would-be unifying relation determines. In forming the definite description, the believer typically assumes that such would-be associated facts are not themselves permutative. If the believer didn’t assume this, likely she wouldn’t form the definite descriptions in the first place. Russell explains (TK, p. 148): The new complex is molecular, and is non-permutative as regards its atomic constituents x1C1 γ • x2C2 γ,…, • xnCn γ; also each of these atomic constituents is non-permutative… 2

It is important to note that it may well be an entirely contingent matter whether the definite description

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(ιγ)(x1C1 γ • x2C2 γ,…, • xnCn γ). is satisfied. It may well be a contingent matter that we are able to form such a definite description. In forming the definite description, the believer has acquaintance with a p-relation and the position relations it determines. The definite description is contingently satisfied only if the position relations employed in forming the definite description do not, themselves, determine further position relations. That is quite enough. In stating his solution to the problem of direction, Russell is explicit in Theory of Knowledge that the problem concerns finding a proper definite description for a permutative fact. It is quite significant that not a word occurs about belief-facts being representations or about their embodying some intrinsic unity that transforms them into truth-apt representations that are necessarily about their would-be truth-makers. I maintain, on Russell’s behalf, that one’s acquaintance with the position relations determined by a given p-relation is immediate from one’s acquaintance with the relation itself. One might naturally ask, however, whether there are cases where position relations determine further position relations. One might naturally ask whether this is a purely logical issue, so that it is logically impossible for a position relation to determine further position relations. I maintain that it is not, at least not always, a logical matter whether a given position relation determines yet further position relations. Contingency can often be quite sufficient for forming definite descriptions of permutative facts. It is enough for a correspondence definition of “truth” that many permutative relations with which we are acquainted determine position relations that do not determine further position relations. The exemplification of the truth-as-correspondence relation is, at times, doubly contingent in this way.3 It is natural to be concerned that there may be facts x-C-y whose position relation C is itself a permutative relation. But nothing in logic is required to rule this out. To think otherwise creates a fly-bottle from which there is no escape. Wittgenstein himself is guilty when he demands that Parity be a logical matter whereby a belief-fact as truth-bearer shows the logical form of its would-be truth-maker. Those interpreting belief-facts as logically representing their truth-makers do the same, since the representation is assumed to be necessarily tied to what it represents. On Russell’s theory, the existence of such logical relations is rejected and they have no role to play.

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Relations Have Sense (Direction) Intuitively, to say that a fact is permutative seems to be to say that it is possible for there to be a fact composed of the very same constituents, with changes to the positions that are occupied by the constituents. That is, where it is possible for “Rn(x1,…, xi, …,xj,…, xn)” to be made true by an exemplification of the relation Rn, so also for some relations Rn it is possible that there exists at least one fact that makes true “Rn(x1,…, xj, …,xi,…, xn),” where 1 ≤ i ≤ n and also 1 ≤ j ≤ n. This characterization of a permutative fact, however, would land Russell in an untenable ontology of possible entities of one sort or another. To chase such a definition of the notion of a permutation fact in terms of the notion of (logical possibility) is to chase a phantom.4 Long ago, Pears (1967) noticed this. We find (Pears 1972, 216): When a person judges that something is so, he formulates a proposition in his mind. But according to Russell, the formulation is not done with any kind of psychological tokens. It is done by arranging the actual things themselves in thought. Of course, this does not mean that the things which the thinker arranges are in his mind. …But the arranging is mental. … If Russell had not been so averse from possibilities, he could have said that what the person who makes the judgment does is to construct a possibility and judge that it is realised.

Pears thinks that in forming a belief about a permutative fact existing, we are “constructing a possibility.” The notion of “possibility” here seems to be that of logical possibility. But is it? It is far from clear that there is anything structural to suggest that a notion of logical possibility is at work. But is a causal (or metaphysical) possibility at work? Russell rejects such notions. And in any sense of “possibility,” the notion of an entity that is a possible fact is not something that Russell can accept. Obviously, he can no more embrace an ontology of possible facts than he can embrace possible winged horses and golden mountains. Pears thought that Wittgenstein had the answer. After all, his Tractatus spoke of a difference between a fact (Tatsache) and what he called a Sachverhalt. The latter speaks of a fact as a realization of combinational possibilities that are shown by the symbols

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themselves. But of course, Wittgenstein no more embraces possibilia than does Russell. He adopted a Doctrine of Showing according to which a statement’s logical form shows the relevant possibilities of truth-making. The would-be truth-making fact has (logically) to share the logical form of the truth-bearer. Without showing and his picture theory, Wittgenstein’s philosophy has nothing to offer Russell by way of help. Happily, the solution of the difficulty of defining what is meant by a “permutative fact” is found by simply appealing to the notion of a permutative relation (p-relation). A permutative fact is simply understood as a fact wherein the relation providing its unity is a permutative relation. The permutativeness of a relation is not a contingent feature of it and it is given by acquaintance with the relation itself. By understanding a relation we understand the sort of fact that would exist were it to be exemplified. Hence, the notion of a permutative fact is understood simply by being acquainted with a permutative relation. There is no more to it. In being acquainted with a p-relation, we are acquainted with the position relations (if any) that it determines. It thereby becomes viable to hold that one can form a definite description of a would-be permutative fact by appeal to the position relations determined by the permutative relation that would ground the unity of that fact. Acquaintance with a relation renders an understanding of its being a p-relation—if it is a p-relation. Acquaintance with ‘loves,’ for example, yields an understanding of its being a p-relation. It yields this understanding independently of the exemplification of the relation. It may contingently happen that (for some interval) the relation is exemplified in such a way that no one loves anyone but themselves. But that contingent configuration of its exemplification has no bearing on whether the relation ‘loves’ is a p-relation. The same holds generally for the sense of a relation. The sense of a relation (e.g., the notion of it being many-one, one-one, etc.) is something that it has independently of contingencies of its exemplification. We know a priori that the relation of the arithmetic addition of natural numbers is symmetric. It cannot lose this nature when studied abstractly before the mind. That would make mathematical logic impossible. Russell was very clear on this point in Problems. In discussing the multiple-relation theory of belief, he explicitly mentions that mathematical logic depends upon the sense of relation being a metaphysical feature that it has independently of whether the relation happens to be exemplified (PoP, p. 127). For example, Principia’s account of cardinal numbers studies the relation of ‘similarity’ which is logically an equivalence relation (transitive, symmetric, and reflexive). It provides an in-depth a priori study of the Progression property independently of its exemplification. As noted, the sense of a

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relation is central to the relation’s being symmetric, transitive, many-one (functional), one-one, and so on. These features of the sense of a relation require the notions of the relation’s domain and range. But they do not require any knowledge of what entities are in its domain or range. This is precisely how it is that synthetic acquaintance with such universals of mathematical logic enables the relational structures they determine to be studied a priori. The same point carries over to non-mathematical relations with which we are acquainted. We may come by experience to be acquainted with the relation ‘loves’ and thereby discern that it is not symmetric. It remains non-symmetric even if it should be a happy accident that at some interval of time everyone loves one another. To understand that a relation, say, ‘loves,’ is by nature not many-one, not symmetric, and so on does require one to understand the notion of its domain and range (i.e., the notions of agent and recipient). These notions are given by acquaintance with the relation ‘loves’ quite independently of the contingencies of its exemplification. The sense, understood this way, is thus perfectly intact when the relation is abstractly before the mind. In Chap. 1, we saw that Stout, among many other interpreters, was confused about this and took Russell to be agreeing that the sense of a relation is a feature of its contingent exemplification—a feature lost when the relation is considered abstractly before the mind. That is mistaken. This is central to the solution in Theory of Knowledge of the problem of direction. It was also central to Russell’s structural realism of the 1920s, well after he abandoned the multiple-relation theory. Consider a passage from The Analysis of Matter (1927) written in the neutral monist era (AMa, p. 255): It is obvious as a matter of logic that, if our correlating relation S is many-­ one, not one-one, logical inference in the sense in which S goes is just as feasible as before, but logical inference in the opposite sense is more difficult. That is why we assume that differing percepts have differing stimuli, but indistinguishable percepts need not have exactly similar stimuli.

Russell is speaking about the sense of the relation S from transient particular causes of percepts to percepts. He says that the sense of the relation S is many-one since a veritable sea of transient particulars may collectively be the stimuli causing a single percept. The sense of this relation’s converse, Russell tells us, is not many-one and that makes inference about the causes of percepts more difficult—though not impossible. Russell understands these features of the relation of ‘perception’ independently of how it

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happens to be exemplified. There are, undoubtedly, accidents of the way the perceptual relation is exemplified. Perhaps at times it is even exemplified in a way that is one-to-one. But the nature of the relation of perception can, nonetheless, be understood by acquaintance not to be logically one-to-one. We saw in Chap. 1 that in Problems, Russell explained that, acquaintance with universals such as ‘human’ and ‘mortal’ cannot render, a priori, information about the kinds of structure that would be realized if they were to be exemplified. That all humans are mortal cannot be known a priori by acquaintance with the universals ‘human’ and ‘mortal.’ That is because the structure arises contingently from exemplification. Thus, empirical information gained a posteriori is central to the study in the natural sciences of such structures and that can only arise from the exemplification of universals. All the same, the sense of any universal with which we are acquainted is understood quite independently of whether the universal is exemplified. In order to form a belief-fact as a truth-bearer, one must be acquainted with the universal that would unify the would-be corresponding fact. Thus, if there are any contingent facts as truth-makers, some contingently exemplified universals must be objects of acquaintance. Because one can also describe universals via our acquaintance with other universals, a great many physical facts whose unifying universals may never be objects of acquaintance are truth-makers for belief-facts we can form.

The Family of ‘consist-Rn of’ Relations In Principia and in Problems, the multiple-relation theory shows up only in discussion of the correspondence definition of truth for the basic atomic case, where quantifiers and logical particles are not involved. In the simplest kind of base case, where Rn is not a permutative relation, Russell’s plan seems to have been to simply form a definite description of a fact by naming a belief-fact in which an n+2 placed belief-relation Bn + 2 unifies a subject m and the objects x1,…, xn and Rn. The expression  Rn  x  m- Bn + 2-  1   x   n is a convenient name for a belief-fact. Russell’s theory of definite descriptions replaces all names in favor of definite descriptions, and hence ultimately this name will have to be removed too. But for the present, it is

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useful. The thesis is that when one asserts that a mind M believes truly that Rn(x1,…, xn), the wff “Rn(x1,…, xn)” in this expression serves as a disguised definite description. Thus, M believes truly that Rn(x1,…, xn) =df  Rn  x  E!(ιf )(m- Bn + 2-  1  corresponds to f ).  x   n (It is convenient to use “f” here to remind us that we have a definite description of a fact. This will be explained away later when “fact(x)” is defined.) This says that there is a unique fact f to which the belief-fact in question corresponds. If there is no such corresponding fact, then the belief is false. Of course, “correspondence” in the simplest case simply means that the would-be fact f consists of precisely the entities x1, …, xn and Rn with the relation Rn providing its unity.5 That is, in the simplest case where permutation doesn’t arise, the order of x1,…, xn doesn’t matter. Using an alternative description in this case, we can eliminate appeal to a mind engaged in believing. We have: [Rn(x1,…, xn)] is true =df E!(ιf )(f consists of x1,…, xn unified by Rn). One might be nervous about the notion of being “unified by Rn” because it supposes that we understand that Rn is a relation. But acquaintace with an entity is, according to Russell, sufficient by itself to support one’s understanding of whether the object is universal. Thus, in forming a definite n description, it is better to appeal directly to C R to indicate the family of ‘consists-Rn of’ multiple relations. In the monadic case, we can write, x CFf. This says that fact f consists of x unified by the universal F. The following holds: [Fx] is true =df E!(ιf )( x CFf ). Similarly, we can write,

 x1 ,,xn  C R

n

f.

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This says that f consists of x1, …, xn with the relation Rn providing unity. The following holds: [Rn(x1, …, xn)] is true =df E!(ιf)(  x1 ,,xn  C R f ). n

These cases pose no problem because permutation doesn’t arise. These cases are unproblematic since in such cases “consists-Rn of” together with the constituents is sufficient to definitely describe the fact. There is no need to worry about a case where one or more among the x1,…, xn are themselves universals. No problem arises. For example, take the expression that abstractness is a property which we can write as “P(F).” We have the following definite description of the fact: (ιf )(F CPf ). This definite description requires only the ‘consist-P of’ relation. Compare, (ιf )(P CFf ). This is a definite description of the fact that being a property is abstract. The ‘consists-Rn-of’ relation designates what universal is to be the unifier of the would-be fact described. The problem of forming a proper definite description of a fact does not arise here. Note well: the relations CF and CP are not themselves permutative relations.

Position Relations and Adicity All structure comes from relations. The “order type” (or better “structure type”) can very often be depicted by a map. In Introduction to Mathematical Philosophy, Russell gives an example (IMP, p. 60): Given any relation we can, if it is a sufficiently simple one, construct a map of it. For the sake of definiteness, let us take a relation of which the extension is the following couples: ab, ac, ad, bc, ce, dc, de, a

b

d

c e

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where a, b, c, d, e are five terms, no matter what. We may make a “map” of this relation by taking five points on a plane and connecting them by arrows, as in the accompanying figure. What is revealed by the map is what we call the ‘structure’ of the relation. It is clear that the structure of the relation does not depend upon the particular terms that make up the field of the relation. The field may be changed without changing the structure, and the structure may be changed without changing the field.

All structured entities get their structure precisely from universals (properties and relations in intension) being exemplified in them. Interestingly, we shall see that in Theory of Knowledge Russell came to hold that although the understanding relation structures an understanding-fact, there is no way to draw a general map of an understanding-fact. There will be different maps depending on the subordinate relation involved. In considering relations and the structures they exact in the facts in which they relate entities, one must focus on position relations and not be misled by grammar and linguistic or spatial relations involved with conventions of representation. Let us assume that ‘loves’ is a relation that can inhere in a fact relating the constituents. Desdemona loves Othello, taking Shakespeare’s play as our paradigm, and so also does Desdemona love Desdemona. These facts reflect the contingent way ‘loves’ is exemplified. They give rise to several interesting questions. It may seem odd that a dyadic relation ‘loves’ can occur as a relating relation unifying a fact of Desdemona’s loving Desdemona. How can Desdemona occupy both positions in this fact? The answer, obviously, is that we should not imagine that all facts are spatial. The fact of Desdemona’s loving Othello is clearly not spatial. There is no spatially “first” term of this fact and no “second.” It is very important not to conflate the notion of structure with the notion of a fact having its constituents in ordinal positions (first position, second position, etc.) in the fact. Neither is there a “left” term nor a “right.” There are, of course, some spatial relations. Care must be taken in considering this. Insofar as geometry is a part of pure mathematics, with its mathematical ‘points’ and ‘lines’ and ‘planes,’ these relations may well not inhere in any physical facts at all. Logical/mathematical relations must not be conflated with physical spatial relations inhering in physical facts. In modern relativistic physics, physical spatial relations are not dyadic but field relations permeating the universe—with mass and acceleration producing perturbations in the field in accordance with the invariance of electromagnetism. In such a case, the notion of adicity is a complicated matter.

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We are not acquainted with the spatial relations of relativistic physics. Any relation with which we are acquainted is one whose adicity is given. Russell’s paradigm of a pR2-relation (dyadic permutative relation ) is the relation ‘loves’ and in the spirit of his celebrated Shakespearian example we can take the fact of Desdemona’s loving Othello as a permutative fact since nothing prevents there being a fact of Othello’s loving Desdemona, even if (as I interpret the play) he doesn’t and there is no2 such fact. The dyadic 2 relation ‘loves’ determines two position relations C1L and C2L and in being acquainted with the relation ‘loves’ we are acquainted with these position relations. Observe that ‘loves’ is not symmetric by its nature even if it should happen by some happy accident that one day everyone loves everyone. Acquaintance with a relation yields an understanding of its adicity and an understanding of its having a domain, a range, whether it is symmetric, and so forth. Acquaintance with a p-­relation is itself sufficient to yield understanding that it is a permutative relation. Moreover, since p-relations determine position relations, it is intuitive to hold that acquaintance with any pRn-relation itself yields an acquaintance with all the position relations it determines. This is quite important. To form definite descriptions of permutative facts, Russell relies heavily on acquaintance with the position relations that a given p-relation determines. It comes as something of a surprise to find that Russell holds that while acquaintance with a relation yields understanding its adicity and an acquaintance with the position relations it determines, he holds that the adicity of the relation does not itself determine the number of distinct position relations it determines. In Problems, Russell had not yet introduced the notion of position relations and he says nothing about adicity. He takes a simple case of a belief-fact to be this:  R m-B4-  a  b   In Theory of Knowledge, he takes the belief-fact to be this:  R    m-B5-  R( x, y)  a  b    The new component R(x,y), which is sometimes written as xRy, is what Russell calls the “logical form.” It is acquaintance with this entity that

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grounds our understanding of the adicity of the relation and thereby our understanding that, were it to be exemplified, the relation would inhere as the unifying ground of a dual complex (fact). Russell is explicit about the separation, though no one seems to have noticed it until Perović (2016). But I do not think it wise to follow him in this view that adicity does not fix the number of position relations determined by a p-relation. He writes (TK, p. 146): It is to be observed that the relations C1,…Cn are not determined by the general form, but only by the relation R. So far as the general form ‘xRy’ is concerned, the position of A is the same in ‘A-before-B’ as in ‘A-after-B.’ It is only after the relation R has been assigned that the positions can be distinguished. Although, therefore, the various possible positions are determinate when R is given, they are not functions of R.

Russell felt that he had to introduce a separate logical form R(x,y) of a pRn-relation because he held that the adicity may come apart from the number of position relations the p-relation determines. We shall witness a concrete case with the relation of ‘similarity’ below. It is for this reason that Russell holds that one cannot draw a general map of an understanding complex. The map will depend not just on the adicity of the p-relation in question but also on the number of distinct position relations it determines. Ultimately, I will reject Russell’s separation. I maintain that the adicity of a p-relation concurs exactly with the number of position relations that the p-relation determines. Thus, there is no need to introduce a special logical form for adicity. On my view, a p-relation pRn determines n n n exactly n-many relations C1R , C2R ,…, CnR . (There is no need to write these as C1pRn , C pRn ,…, CnpRn since it is clear.) 2 Nothing Russell is doing with position relations violates the fundamental thesis of the Principia era—the thesis that where facts that are truth-­ makers are concerned and where facts containing other facts are exempted, the existence of one fact never logically entails the existence (or non-­ existence) of any other. Consider the following which has to be regarded as logically true: 2

2

(f )( (∃x)(x C1R f ) ⊃ (∃y)(y C2R f )). According to Russell, an entity has a position in a fact. Now the fact

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2

a- C1R -(a-R2-b) 2

is the truth-maker for the statement “a stands in the relation C1R to a-R2-b.” And likewise 2

b- C2R -(a-R2-b) is the fact that is the truth-maker for the statement “b stands in the rela2 tion C2R to a-R2-b.” The existence of the one logically entails the existence of the other. All is well since these facts contain the fact a-R2-b as a constituent. Hence, they are 2exempted from the fundamental thesis. R2 R The position relations C1 and C2 determined by a p-relation R2 are as Platonically existing as is the relation itself. They all exist unexemplified. In the case of an n-placed pRn-relation, Rn that determines n-many position relations, we have the following: n

n

n

pRn(x1,…,xn)  x1 ,, xn E!(ιf )( x1 C1R f • x2 C2R f •….• xn CnR f ). Note that this statement relies on the linguistic convention that the linguistic order in say “R2xy” tracks the positions that occur in the relational fact. These Russellian theses of Theory of Knowledge come together when he considers the relation of ‘similarity.’ It is logically symmetric and not merely contingently symmetric. Suppose then that we are acquainted with ‘similarity.’ Russell felt he must hold that in spite of its being a dyadic relation, it involves only one position relation! He is quite explicit. In a similarity fact, there is just one position—one position that can be occupied by two entities. He writes (TK, p. 122): In general, in a complex of n terms, there are various “positions” in the complex, corresponding to different relations (generally each of them functions of the relating relation) which the constituents have to the complex. A complex may be called “symmetrical” with respect to two of its constituents if they occupy the same position in the complex. Thus in “A and B are similar”, A and B occupy the same position.

Russell, it seems, was struggling with Principia’s fundamental thesis. To avoid conflicting with the fundamental thesis, the fact A-similar-B must be identical with the fact B-similar-A and thus, Russell concluded, it must determine one position.

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Russell’s thesis that the dyadic relation of ‘similarity’ determines one position fillable by two distinct entities is strikingly difficult to accept. (This is different from accepting that a dyadic relation can ground the unity of a fact in which a single entity occupies its relata.) The difficulty can be avoided by maintaining that there is no dyadic relation of ‘similarity’ that we are acquainted with. The logically symmetric relation of comparative similarity is quite different. It is expressed with the wff “x is similar to y in respect F.” We do understand this relation, but as we have explained, our understanding of the wff does not assure acquaintance with any relation whose exemplification is given by it. Consider, Lewis’s counterpart relation of comparative similarity. It is surely not a relation with which we are acquainted. We understand it by means of the wff “no object is more similar to x in relevant respects F1,…, Fn than y.” This relation is not logically symmetric.6 We can, therefore, protect Principia’s fundamental thesis by simply denying that we are acquainted with a dyadic relation of ‘similarity.’ In this way, we can avoid Russell’s strange notion that a dyadic relation can determine one position relation that may be exemplified by two distinct entities. Interpreting Russell as in a struggle to adhere to the fundamental thesis of the Principia era helps us to understand why he found himself with such an untoward thesis concerning the relation of ‘similarity’ and how he came to separate the adicity of a permutative relation from the number of position relations it determines. Since we reject the thesis that we are acquainted with ‘similarity,’ we can reject Russell’s thesis of separation altogether. In Theory of Knowledge, Russell devises maps in an effort to depict the multiple relation of understanding and in them he tried to represent his separation of the adicity of a relation from the number of distinct position relations it determines. Russell spoke of maps of an understanding-­fact that is involved with what he calls “understanding a proposition.” Of course, these notions are made particularly difficult because there are no propositions on Russell’s view. Russell’s maps are interesting and worthy of a detailed examination. Not surprisingly, there were significant changes in his thinking on how to render a proper map of an understanding-fact that exists when a person S understands that A is similar to B. The maps of Theory of Knowledge involve the relation ‘similarity.’ Russell offers three maps (which we tag as A, B, and C) attempting to render a structural representation of the understanding-fact involved. The

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three maps reveal important new insights. Let us look at the following maps, A and B, that he drew side by side (TK, p 200): A A

R(x,y) R(x,y)

Similarity

Similarity

B

B

S

S

Map A

Map B

The earliest of these seems to be Map A.  They reveal that Russell was developing the view that the adicity of a relation does not coincide with the number of distinct position relations that it determines. Presumably representing the understanding relation with arrows, one of Russell’s experimental maps shows the striking complexity of a map of an understanding-fact capturing a person S’s understanding A’s being similar with B. We see in the maps that acquaintance with the universal yields an understanding of the position relations it determines but does not itself give an understanding of its adicity. That is why he needs to add acquaintance with the logical form R(x,y) which, according to Russell, renders an understanding of the adicity of the relation. Admittedly, it is far from obvious how these maps are to be interpreted. Quite clearly, the first thing to notice in distinguishing Map A from Map B is that there is a different subcomponent (below) in Map B. Here we find a double arrow depicted as such: R(x,y)

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This seems minor. But the second thing in Map B to notice is quite major. We find Russell attempting to depict (below) his notion of there being only one position. It is given saliently in the conspicuously emboldened arrow that comes to a point before dividing. Russell has: A Similarity B This, I believe, reflects his thesis that a similarity fact determines one position relation occupied by the two entities A and B. Map A does not have this feature. The first difference which seemed minor can now be understood as very likely to have been related to the second. That is, to represent the one position involved in similarity facts, the logical form R(x,y) is depicted in Map B with a double arrow. This becomes particularly clear because Russell offers an alternative depiction as well. That is, he has: R(x,y)

This suggests that Russell was agonizing over his view that distinct entities can occupy one and the same position in a similarity fact. Indeed, Russell even considers an alternative depiction. It may well be that Map A was drawn at a time before Russell was engaged with the idea that both A and B occupy the same position in the fact of A’s being similar to B. The relation ‘similarity’ offers an example, he thinks, of a case where the adicity does not fix the number of distinct position relations—there being only one position in a similarity fact in spite of its being a dyadic relation. In Map B, this manifests itself in the arrow that is not present in Map A. Map B attempts to depict the one position relation involved in the dyadic relation of ‘similarity.’ In this map, we see that the subject s is acquainted with

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the logical form R(x,y) and this is independent of the subject s being acquainted with the relation ‘similarity.’ If this is correct, then there is a mistake editors made in their reconstruction of Theory of Knowledge: The 1913 Manuscript. Map C (below) was inserted into Part 2 chapter I: The Understanding of Propositions (p. 118). A R(x,y)

Similarity

B

S That is the wrong map. Map C is clearly a version of Map B because it shares the feature depicting one position determined by the similarity relation. Still, Map C is different from Map B because it does not contain the double-arrow of the logical form R(x,y). Map C depicts Russell’s thesis (first introduced on p. 122 of his Part 2, Chapter II Analysis and Synthesis) that a dyadic relation such as ‘similarity’ nonetheless has only one position. He doesn’t speak about ‘similarity’ having only one position until after p. 118. In reaching p. 118, Russell had not yet come to separate the adicity of a relation from the number of distinct position relations it determines. I conclude that Map A was originally to appear on p. 118 and Map C was drawn after Russell settled on separating adicity from position. Once separated, we find that unlike Map B, in Map C the arrows accompanying the logical form R(x,y) do not flow toward A and B, but first come to a point and then divide.7 Russell speaks of the dyadic form R(x,y) and not the adicity of the relation directly, but his point is the same. The adicity of a relation does not determine the number of distinct position relations determined by it. A dyadic relation R2 such as ‘similarity,’ that is logically symmetric, only determines one position relation. Because the adicity of the relation does not itself determine the number of distinct position relations, one has to be acquainted not only with the adicity (i.e., the dyadic logical form R(x,y)

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but also with the given relation R in question to be able to know how many position relations it determines. Thus, one cannot draw a single map of an understanding-fact for arbitrary R. If we tag the position relation in Map C, we get the following: A C

R(x,y)

Similarity Similarity

B

S On this interpretation, the arrow tagged with the form R(x,y) is pointing not merely to the relation ‘similarity’ but also to the one position relation, namely CSimilarity that is determined by ‘similarity.’ Following this line of development, consider the relation ‘loves’ which is dyadic and which determines two distinct position relations. In Russell’s view, the logical form is for the adicity being dyadic. It does not reflect the existence of the two position relations, C1Loves and C2Loves . They are independently given, as is the adicity, by S’s acquaintance with the relation ‘loves.’ In this case, a map of the understanding complex would include the two position relations together with the logical form. Thus: Loves

C1

A

R(x,y)

Loves

Loves

B

C2

S The map depicts the two distinct position relations that one understands. One must keep in mind that the map does not require that the entities A and B bear the position relations they are understood to have. There may well be no fact of A’s loving B. Obviously, the understanding-fact is not a composite of other facts including a fact of A’s loving B and a fact of A’s standing in a position relation to that fact.

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Understanding the adicity of a relation tells us what sort of fact (complex), dyadic, triadic, and so forth, would exist if the relation were exemplified. Russell holds that the object (logical form) we are acquainted with in understanding the adicity comes apart from the number of the position relations determined. This thesis entails that the solution to the problem of direction is caught up in his introduction of abstract particulars just as is his solution to the problem of compositionality. In Russell’s view, acquaintance with a p-relation yields acquaintance with its adicity and with all the position relations it determines. This is all quite independent of whether the relation is exemplified. Russell came to think that the adicity of a relation does not coincide with the number of position relations (if any) it determines. For this reason, he thinks for adicity he needs to have acquaintance with a separate logical form. I reject Russell’s thesis, maintaining that the adicity of any p-relation always coincides with the number of distinct position relations it determines. For this reason I write a dyadic relation as 2 R2 and the2 exactly two position relations it determines are written as C1R and as C2R . There is no need for acquaintance with a separate logical form to ground acquaintance with its adicity. The issues Russell was facing occur only because of his feeling compelled to hold that the adicity of a relation (which determines the adicity of the complex that would exist if the relation were to be exemplified) has come apart from the number of position relations it determines. Unfortunately, it is easy to run together the issues Russell was addressing in forming a definite description that would be satisfied by a permutative fact with the issue involved in attempting to explain the nature of the intentionality involved in predication itself. The conflation makes it appear as though an understanding-fact requires the introduction of a constituent entity that serves as a propositional content. The impetus for introduction of a propositional content is what makes one feel as though one’s understanding that, for example, A is similar to B requires that the relation ‘similarity’ is predicationally uniting A to B. Russell says (TK, p. 116): in order to understand ‘A and B are similar’, we must know what is supposed to be done with A and B and similarity, i.e., what it is for two terms to have a relation; that is, we must understand the form of the complex which must exist if the proposition is true.

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In speaking of our knowing “what is to be done with A and B,” Russell introduces what he calls “the problem of understanding of the proposition.” This is infelicitously put since, after all, he holds that there are no propositions. But it is natural to imagine that a synthesis is involved. He writes (TK, p. 116): It is essential that our thought should, as is said, “unite” or “synthesize” the two terms and the relation; but we cannot actually “unite” them, since either A and B are similar, in which case they are already united, or they are dissimilar, in which case no amount of thinking can force them to become united. The process of “uniting” which we can effect in thought is the process of bringing them into relation with the general form of dual complexes. The form being ‘something and something have a certain relation,’ our understanding the proposition might be expressed in the words ‘something, namely A, and something, namely B, have a certain relation, namely ‘similarity.’ I do not mean that this is a full analysis, but only as suggesting the way in which the form is relevant.

We must not be misled by this passage. The issues confronting Russell do not themselves suggest that he was concerned to present a theory of the nature of intentionality and predication in thought and language. The problem of predication, of synthesis, is not to be addressed by appeals to facts. Predication, understanding, believing, judging, thinking, and so forth do not consist in the existence of any facts. And we must always remember that no relation predicates anything at all. An exemplified relation relates but it does not predicate. The unity of a fact is not constituted by a predicational synthesis involved in intentionality. It is better to say that some would naturally turn to features of the intentionality involved in predication in their attempt to address the issues Russell was attempting to address. I have skirted that issue entirely. Forming any definite description presumes the intentionality involved in predication. But intentionality and predication certainly do not consist in the existence of a belief-fact or an understanding-fact. All such facts are artifacts produced by intentionality.

Converses of p-Relations In Theory of Knowledge, Russell offers the remarkable thesis that a relation is to be identified with its converse(s). I think his motivation was that the thesis protects the fundamental thesis of the Principia era. Interestingly,

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Russell’s thesis is viable! But it requires, in the problematic cases, that one define domain, range, and so on, by appeal to the position relations determined by a p-relation. I think Russell had just this point in mind. Relations that are p-relations determine position relations whether or not they are exemplified, and we are acquainted with the position relations by being acquainted with the relation that determines them. Acquaintance with a relation renders an understanding of both its adicity and the number of position relations (if any) it determines. Consider the following passage (TK, p. 88): we require the notion of position in a complex with respect to the relating relation. …if we are given a relation R, such that if x and y are terms in a dual complex [a-R-b] whose relating relation is R, x will have one of these [position] relations to the complex, while y will have the other. The other complex [b-R-a] with the same constituents reverses these relations. … Thus the sense of a relation is derived from the two different relations which the terms of a dual complex have to the complex. Sense is not in the relation alone, or in the complex alone, but in the relation of the constituents to the complex which constitute “position” in the complex. But these relations do not essentially put one term before another, as though the relation went ‘from one term to another’ this only appears to be the case owing to the misleading suggestions of the order of words in speech or writing.

It is important not to draw2 the conclusion that Russell thinks that the posi2 tion relations C1R and C2R that are determined by the dyadic p-relation R2 arise only in virtue of R2 being exemplified. That would make it impossible to form definite descriptions that purport to refer to permutative facts by appeal to position relations. The passage should be interpreted to mean that position relations are essential to recovering the notions of domain, range, and so on of a p-relation. It is in this context that Russell is endeavoring to identify a relation with its converse(s). 2 Observe that “x C1R z” does not say that x has the first position in the fact z. In the expression “R2xyz” the sign “x” comes first. There is no “first” position and no “second position” in a fact. In his discussion of the sense of a relation, Russell makes this clear. He endeavors to avoid the many confusions that naturally arise. He writes (TK, p. 87): It might be supposed that every relation has one proper sense, i.e., that it goes essentially from one term to another. In the case of time relations, it might be thought that it is more proper to go from the earlier to the later

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term than from the later to the earlier. And in many relations it might be thought that one term is active while the other is passive; thus “A loves B” seems more natural than “B is loved by A”. But this is a peculiarity of certain relations, of which others show no trace. Right and left, up and down, greater and less, for example have obviously no peculiarly “natural” direction. And in the cases where there seemed to be a “natural” direction, this will be found to have no logical foundation. In a dual complex, there is no essential order as between the terms. The order is introduced by the words or symbols used in naming the complex, and does not exist in the complex itself. Our [permutation] problem arises from the fact that, although this is the case, a different complex results from interchanging the terms, and that such interchanges looks like a change of order.

In a dual complex, there are no essential ordinal positions (i.e., no ordinally first entity in the complex, second entity in the complex, etc.). Russell’s notion of “order” (ordinality) here must be distinguished from the notion of “order” which means structure. Relations that inhere in facts produce structure in them. According to Russell, all structure is due to relations. Russell’s position on this matter—a position held in Principia—has not changed. All structure is due to relations and Principia’s logic is the study of all the kinds of structures by studying relations themselves (whether the relations are exemplified or not). Russell’s point in Theory of Knowledge does not conflict with Principia. His point is that the structure that a relation produces when it is relating in a fact is not properly to be construed in terms of an entity occurring first in the complex, an entity occurring second, and so on. In Theory of Knowledge, Russell held that dyadic relations do not come before the mind with a hook at one end and an eye at the other (TK, p, 86). To be sure, the hook and eye cannot be understood as if they were ordinal positions (“first,” as opposed to “second” etc.) in a complex because structural features of complexes are not ordinal features. Russell tended to associate ordinal features with the hook-eye metaphor. There is no reason to make such an association. I maintain that relations do have hook-eye features and that acquaintance with a relation yields acquaintance with its adicity and its hook(s) and eye(s), if any. Different relations may well present different sorts of hooks and eyes. For example, if we are acquainted with the relation ‘loves,’ we are acquainted with its being dyadic and there being places for an agent and a recipient. If we are acquainted with the relation ‘being left of with respect to,’ then we are acquainted with

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its being triadic, and there being no agent or recipient. Now the existence of the hooks and eyes are precisely what gives rise to position relations appropriate to positions in facts. I agree with Russell, however, that there are no hooks and eyes that render ordinal notions of a “first” and a “second,” occurrence in a fact. Moreover, the ‘from-to’ notion can be misleading—that is, a relation exemplified in a fact of a-R-b does not proceed “from” a “to” b. That is, I agree with Russell that domain and range (which give the ‘from-to’ feature) are not genuine properties of p-relations simpliciter. Instead, they are properties that are properly articulated with respect to the position relations that a p-relation determines. The comprehension axiom schemata of the Principia era assure that it is a purely logical matter that every relation has at least one converse. For example, comprehension *12.11 yields: (R)(∃S)( S2xy ≡x, y R2yx). It is convenient to denote the logical converse of a dyadic relation R2 by the expression R 2 . Recall, however, that nothing in comprehension assures that we are acquainted with the relations comprehended. All the same, the existence of logical converses immediately raises the question as to how the existence of logical converse relations can be made consonant with the fundamental thesis of the Principia era. The answer is that to protect the fundamental thesis, one must hold that no two relations R and its logical  converse R , both of which we are acquainted, can be distinct. In such a  case, R = R . An analogous situation arises if we were acquainted with a relation S that is logically symmetric. In such a case, where distinct entities stand in the relation, the difficulty arises that it seems clear that the fact that a-S-b is not identical to the fact that b-S-a. In Theory of Knowledge, Russell tried to handle this situation by maintaining, appearances to the contrary, that these facts are identical. I cannot agree, and so I hold that the only logically symmetric relation with which we are acquainted is the relation of ‘identity.’ In The Principles of Mathematics (1903), Russell maintained that in the case of a relation that is logically symmetric, the relation might well be regarded as identical with its converse. He was certainly not motivated to make the identification since at that time he did not hold any analog of the fundamental thesis of the Principia era. After all, clearly propositions readily stand in logical entailment relations. It is illuminating that Russell went

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on to note that where we don’t have logical symmetry—for example, where p-relations are concerned—serious problems arise for any attempt to identify a relation with its converse. In a dyadic relation, the domain may well not be the same as the range. To say that x is in the domain of a dyadic function R2, we have: 2

Dom R (x) =df (∃y)(R2xy). And similarly, we have: 2

Rng R (y) =df (∃x)(R2xy). The usual definition of a dyadic relation being many-one (i.e., functional) is this: funct (R2) =df (x, y, z)( R2xy • R2xz .⊃. y = z). Some relations are many-one (such as the trigonometric relation of the sine of an angle) and yet their converses are not many-one. These concerns were couched in the Principles era of propositions. The same concerns carry over to Principia era facts involving relating relations. Yet in order to maintain the fundamental thesis of the Principia era, one has to identify such relations with their converses—for any case where we are acquainted with the relations in question. Happily, where p-relations are concerned, Theory of Knowledge paves the way for a solution which preserves the fundamental thesis of the Principia era. The problems are resolved by appeal to Russell’s notion that every p-relation determines position relations. This readily enables us to identify any p-relation with its converse and to define domain, range, many-one, and so forth, in terms of the position relations the relation R 2 of a p-relation R2 determines two position determines. 2The converse  R R2 relations C1 and C2 . In accepting that R2 = R 2 where R2 is a p-­relation, we accept the following logical truths:  R2xy ≡x, y R 2 yx 2

2

2

2

C1R = C2R .•. C2R = C1R . Accordingly, this yields the following logical truth: 2

2

E!(ιf )( x C1R f • y C2R f ) ≡x, y E!(ιf )( x C2R f • y C1R f ) . 2

2

2

2

This follows because the fact that a C1R f is identical to the fact that a C2R R2 f . And likewise, we accept that the fact that b C2 f is identical to the fact

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2

2

R R 2 that R2 =  2 b C1 f. Be advised that we cannotR2vary RR 2 in C1 . The identity 2 R does not license the2 thesis that C1 = C1 because “R ” is not an independent part of “ C1R ” and is connected to the subscripted “1.” What 2

2

2

2

follows is that C1R = C2R and C2R = C1R . Once we see this, we arrive at the thesis that the notions of domain, range, and whether p-relation is many-one are all properly defined in terms of the position relations determined by the p-relation. This gives Russell a reply to Davidson’s concerns in “True to the Facts,” which argues that converse relations threaten the very notion of a fact as a truth-maker. He writes (Davidson 1969, pp. 41-42): The statement that Naples is farther north than Red Bluff corresponds to the fact that Naples is farther north than Red Bluff, but also, it would seem, to the fact that Red Bluff is farther south than Naples (perhaps these are the same fact). Also to the fact that Red Bluff is farther south than the largest Italian city within thirty miles of Ischia. When we reflect that Naples is the city that satisfies the following description: it is the largest city within thirty miles of Ischia, and such that London is in England, then we begin to suspect that if a statement corresponds to one fact, it corresponds to all.

Davidson engages in equivocations of the scope of a definite description and “London is in England” is an empty truism given the definition of “England” as, say, “the region of the United Kingdom that contains London as its capital”. By appeal to position relations, Russell can readily solve the concern Davidson raises, identifying a relation with its converse(s). Davidson’s conclusion becomes a non-sequitur. Such converses do not threaten the integrity of facts as structured entities. Using “wrt” to abbreviate “with-­respect-­to,” the relation ‘x is north of y wrt z’ simply has ‘y is south of x wrt z’ as its converse, which one might as well have been expressed by ‘y has x north of it wrt z.’ The apparent problem arising from a difference between a relation and its converse turns on conflating position relations. Where p-relations are concerned, the problems of identifying a relation with its converse(s) is solved by definitions of domain, range, and so forth, in term of position relations. We have: R2

2

2

DomC1 (x) =df (∃f )(∃y)(x C1R f • y C2R f ) R2

2

2

RngC1 (y) =df (∃f )(∃x)(x C1R f • y C2R f ).

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From these we readily get:   R2 R2 R2xy ≡x, y R 2 yx :⊃: DomC1 (x) ≡ x RngC2 (x) 2  R2 CR R2xy ≡x, y R 2 yx :⊃: RngC1 (y) ≡ y Dom 1 (y) There is no difficulty that arises from a relation being many-one when it comes to identifying a dyadic relation with its converse. For what Russell calls a “many-one” relation, we have: 2

2

2

2

funct ( C1R ) =df (x, y, z)(f )( x C1R f • y C2R f • z C2R f .⊃. y = z). 2

We can now see that for a given p-relation R2 relation C1R is many-one R2 and yet C2 is not. But this in no way jeopardizes the identity of a relation with its converse. We have: 2  2 R2xy ≡x, y R 2 yx .⊃. funct ( C1R ) ≡ funct ( C2R ). Russell’s position relations resolve the matter entirely. Let us apply this to a specific case. Consider the case of the relation in intension ‘loves.’ The relation ‘loves’ is not logically many-one but it may well be that happenstance features of its exemplification one day make it such that its converse is exemplified in a many-one way. Notice that the problem doesn’t show itself when entertaining the question of whether  the fact that Lab is the same as the fact that Lba. The issue concerns all the facts that make  up the global way the relation L happens to be exemplified. Thus, L = L seems impossible to maintain. The solution to this problem is to realize that the notion of the features of the contingent exemplification of a p-relation like ‘loves’ have been wrongly characterized. A proper characterization uses the position relations determined by L. Thus, in our 2 2 case the contingent exemplification of C1L is not many-­ one and yet C2L is 2 2 many-one. This is no great shock. The relation C1L that is identical to C2L 2 2 is many-one and the relation C2L , which is identical to C1L , is not manyone. The fact of Desdemona’s loving Othello is the same as the fact of Othello’s being loved by Desdemona. The position relations determined by the relation ‘loves’ are, however, quite different. Wherever we have an expression for a particular relation, the active-­ passive transformations of English readily enable us to concoct all the various converse expressions. So actually it is quite natural to explain this by maintaining that acquaintance with a relation gives acquaintance with its converse(s). And with position relations in place, we have a solution of

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many of the difficulties that seem to arise in identifying a relation with its converse. New definitions for domain, range and functionality, and so forth are needed for many-placed p-relations. If we are given a specific triadic relation such as ‘x gives gift y to z,’ then it is easy to discern the domain and range. These pose no difficulties. But what is to be regarded as the domain and what is to be its range of a triadic p-relation in general? The notions of domain, range, and so forth must not depend upon speaker intentions or subjective aspects of a perspective one can take on a given fact—as when we notice that being spatially left of requires a frame of reference for its intelligibility. None of the following are more natural than any other: 3

Dom R (x) =df (∃y,z)(R3xyz) 3

Dom R (y) =df (∃x,z)(R3xyz) 3

Dom R (z) =df (∃x,y)(R3xyz). The point is not that a triadic relation has multiple domains. The point is that what is well-defined is the notion3 of the domain of a3 position relation 3 determined by R3. Now we have C1R and C2R and C3R as our position relations. Thus, we have: R3

3

3

R3

3

3

3

R3

3

3

3

3

DomC1 (x) =df (∃f  )(∃y, z)(x C1R f • y C2R f • z C3R f ) DomC2 (y) =df (∃f  )(∃x,z)( x C1R f • y C2R f • z C3R f ) DomC3 (z) =df (∃f  )(∃x,y)( x C1R f • y C2R f • z C3R f ). 3

3

Thus, since C1R = C2R , we have: R3

 R3

DomC1 (x) ≡ x DomC2 (x). All is well. The definition of the domain defined in terms of a position relation determined by a relation R2 enables the identification of the relation with its converse. The same holds for the notion of the range of a relation R3, once we see that it is properly defined only with respect to a given position relation determined by R3. And again, our solution resolves the similar problem of defining functionality of a relation R3. It is a position relation determined by R3 that is functional. For example, for an n-adic relation, we have: n

n

n

n

funct( C1R ) =df (x, y, z)(f )( x C1R f • y C jR f • z C jR f .⊃. y = z).

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These are properties of position relations. With Russell’s position relations, we can identify a p-relation with its converse(s) and comply with the fundamental thesis of the Principia era.

Position Relations (often) Don’t Permute In Theory of Knowledge, Russell solves the problem of direction by revealing how to form proper definite descriptions that purport to refer to permutative facts. The solution relies on position facts that are not permutative. The notion that a relation Rn is a pRn-relation (permutative relation) has to be taken as primitive and unanalyzable. Russell has to accept that one is acquainted with permutativeness. Now when non-permutative relation is involved, it is enough to describe the nfact as that which consists of the constituents so and so. We are using C R for the family of “consists-Rn- of” multiple relations. We can write x CFf  to say that fact f consists of x and the property F, with F unifying the fact. Thus: [Fx] is true =df E!(ιf )( x CFf ). Similarly let us use

 x1 ,,xn  C R f n

to say that fact f consists of x1, …, xn and the relation Rn. The following holds: [Rn(x1, …, xn)] is true =df E!(ιf  )(  x1 ,, xn  C R f ). n

These cases are unproblematic since p-relations are not involved and “consists-Rn- of” together with a list of the constituents is, therefore, sufficient to purport to refer to a unique fact. For example, letting S be a property uniquely exemplified by a subject, we have: [ιxSx][ E!(ιf  )( (x, F) CAcq f  ]. This says that there is a unique entity x that exemplifies the property S and there is a unique fact f consisting of it and the universal F, structured by the relation Acq of ‘acquaintance’ as its unifier. The list of constituents x and F is sufficient because these constituents can only go together in one way. The relation “consists of” is obviously not itself a p-relation. There is no notion of a Russellian fact (in the Principia era) that is not joined at the chest to the notion of a universal. The notion of a dyadic fact is incomprehensible without the notion of a dyadic universal. The notion

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of a permutative fact is unintelligible without the notion of the adicity of a relation. Hence, it is natural to maintain that we are acquainted with universals and immediately thereby their adicity and any position relations they determine. Acquaintance with a relation, and any position relations it determines, obviously does not offer a source for knowledge about what entities are constituents of a fact (if any) in which it may inhere. The notion of a position in a fact would be unintelligible without the notion of a relation’s adicity—and it is the adicity of a relation that grounds the structure of the facts in which the relation inheres as relating relation. It is precisely because all structure is due to relations, and because relations and position relations exist quite independently of their being exemplified, that we can study structures independently of knowing whether any complexes (facts) are structured by the relations. In order to fully address the problem of direction, some definitions are useful. We can define the notion that z is a fact whose relating relation is pRn as follows: n

n

pRn-fact(z) =df (∃x1, …, xn)( x1 C1R z .•.,…, .•. xn C2R z ). Moreover, we can define the notion that z is a fact whose relating relation ν is p R n (i.e., not a p-relation) as follows: p R n -fact(z) =df (∃x1, …, xn)(  x1 ,,xn  C R z ). ν

n

In virtue of these definitions, we have: ν

fact(z) =df (∃Rn)( pRn-fact(z) ∨ p R n -fact(z)). We can then go on to define (f  )φf =df (z)( fact(z) ⊃ φz) (∃f  )φf =df (∃z)( fact(z) • φz). These definitions enable the convenience of using letters “f” and “g” for facts. With our new variables for relations, we can define the notion of a fact being permutative without appealing to any notion of possibility. We have: ν

perm-fact(z) =df fact(z) • ~(∃Rn) p R n -fact(z). This will be important later. It should be observed that we do not have to adopt special axioms governing the notion of a fact that we have just defined. For instance, we don’t require an axiom (whose status as a piece of logic would be worrisome) such as the following:

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n

(z)(x)( x CiR z ⊃ fact(z)), where 1 ≤ i ≤ n. This is clearly a theorem that follows from our definitions. 2 (Note that an expression such as “x C3R z” is not ill-formed, but every instance is simply false.) It is quite important to understand that the wff “fact(z)” is not on par with the expression “z exists” which of course is unacceptable to Russell. According to Principia, there is no property indicated by “exists.” This is compatible with allowing the wff “(∃y)(x = y).” Wittgenstein, no doubt, would have objected, demanding that “fact(z)” be a pseudo-predicate that must be shown and not said. We now see that there is no reason for Russell to agree with him on this matter. We are not acquainted with any such property, but we can understand the property because we are acquainted with universals and the position relations some of them determine. The special case of finding proper definite descriptions for permutative facts—that is, where different facts can consist of the very same constituents and unified by the same relation—is solved by appeal to position relations. Russell gives a general explanation of the situation in the following passage from Theory of Knowledge which is the most important, and often most neglected, passage in the entire book. He writes. (TK, p. 147): By means of the above account of “positions” in a complex we can give a non-permutative complex associated with the complex γ, namely: “there is a complex α in which x1C1 α, x2C2 α… xnCn α.” Here “α” is an apparent [bound] variable. Instead of the one relation R, we now have the n relations C1, C2…Cn. The new complex is molecular, and is non-permutative as regards its atomic constituents x1C1 α, x2C2 α… xnCn α; also each of these atomic constituents is non-permutative because it is heterogeneous. Whether any difficulties arise from the fact that the molecular complex is still permutative with respect to the constituents of its atomic constituents, is a question which must be left until we come to deal with molecular thought. But it seems fairly evident that no difficulties can arise from this fact.

Russell’s use of the word “complex” here is infelicitous. A definite description is not a complex in the sense in which a fact is a complex. In any case, it is quite clear from this passage that Russell’s theory of definite descriptions was directly involved in his solution to the problem of permutation. To see how the theory of definite descriptions was intended to be applied to the problem, let us use D as a property unique to Desdemona and O as a property unique to Othello. We can give the truth-condition by means of the following which uses a definite description:

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[Desdemona loves Othello] is true =df 2

2

[ιxDx][ιyOy][E!(ιf  )(x C1L f • y C2L f  )] 2 2 i.e., [ιxDx][ιyOy][ (∃f )( x C1L g • y C2L g .≡g. g = f )]. It should be noted that the passive transform concerning the converse involves a different definite description, but no distinct fact is described. With the help of position relations, the permutative relation ‘loves’ is identified with its converse. We have: [Othello is loved by Desdemona] is true =df 2

2

[ιxDx][ιyOy][E!(ιf  )(x C2L f • y C1L f  )] 2

2

i.e., [ιxDx][ιyOy][ (∃f  )( x C2L g • x C2L g .≡g. g = f )]. All is well. Russell’s position relations are a powerful tool in solving problems arising with the correspondence theory of truth. We can see, however, that it is quite important that acquaintance with a relation yields an understanding of its adicity and an acquaintance with the position relations that it determines. 2 Russell realizes that one might be worried about the conjuncts “x C1L 2 f ” and “y C2L f.” The Russellian account of the truth-conditions for “Desdemona loves Othello” presupposes that he can give truth-conditions for these conjuncts without encountering the very same problem of permutation that invoking position relations was designed to resolve. Russell’s answer to this concern is that the problem of permutation does not arise in such cases. Since Desdemona loves Othello, there are associated complexes (facts), namely, 2

d - C1L –(d-L2-o) 2

o- C2L –(d-L2-o). Here we are using “d” and “o” for convenience for the subjects in question. These facts are2 not themselves permutative. Desdemona bears the position relation C1L to the fact of her loving Othello. Desdemona’s lov2 ing Othello certainly does not bear the position relation C1L to Desdemona. The same holds for the other associated complex. Desdemona’s loving 2 Othello does not bear the position relation C2L to Othello. It is essential to Russell’s solution to the problem of forming a definite description for a permutative fact that these associated facts are not themselves permutative and thus do not upturn the ability of the definite description to purport to single out a unique fact. Since permutation does not arise for these

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position facts, ordinary definite descriptions not involving position relations handle their truth-conditions. That is, we have: 2

L2

[d C1L d-L2-o] is true =df   E!(ιf  )( (d, d-L2-o) C C1 f ) L2 2 [o C2L d-L2-o] is true =df   E!(ιf  )( (o, d-L2-o) C C2 f ). Since permutation does not arise, these definite descriptions are proper. Russell’s solution to the problem of direction of finding definite descriptions for permutative facts requires the appeal to facts which are not themselves permutative. Not all relations are p-relations—that is, not all relations determine position relations. Position relations often do not themselves determine position relations. Russell’s solution to the problem of direction relies on it being the case that one can appeal position relations that are not themselves permutative relations. Russell can regard the problem of direction solved if he can assure that the position facts referred to in forming definite descriptions of permutative facts are non-permutative. This raises the questions, however, as to whether Russell’s solution to the problem of direction requires that it be a logical matter that position facts be non-permutative and whether it is incumbent upon Russell to define what it is for a fact to be permutative. The answers are a resounding no. Acquaintance with any universal renders an understanding of its adicity and any position relations it determines. Accordingly, this is the source of our understanding of what sorts of facts would exist were the relation to be exemplified. Acquaintance itself is the foundation of our understanding of whether a given relation is a p-relation or not and whether a purported fact is permutative or not. In order to form a definite description that purports to refer to a fact, one must be acquainted with the universal that would unify such a fact. If the universal is a property or if it is a relation that is not a p-relation, then one can readily form a definite description in virtue of our acquaintance with the relation ‘consists’ which is not a p-relation. In other cases, position relations play the role. When, for example, we come to be acquainted with the p-relation ‘love,’ our acquaintance yields an understanding of its adicity and the position relations it determines. Acquaintance with a p-relation ‘love’ thereby explains how it is that we understand the permutativeness of any fact in which this universal is exemplified. To say that the fact of Desdemona’s loving Othello is permutative is not to say something about the structure of the fact. It is to say something about the relation ‘loves’ itself. Hence, appearances to the contrary, the question of whether a given fact is permutative concerns the nature of its

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relating relation and is not to be found by looking at the structure of the fact. Indeed, it is precisely because it is tied to the universal’s nature that it may seem that the question of what it is for a fact to be permutative is a logical matter. But the proper thing to say is that it is given by acquaintance with the relation in question. In this regard, it must be understood that there is no property of being a p-relation—that is, being the sort of relation that determines position relations that delimit what sort of facts would exist were the relation to be exemplified. Thus, there is no structure for logic to investigate. The nature of a relation as a p-relation is given in acquaintance with that relation. The same holds for relations with which we are related that are not p-relations. Chapter 4 explained that we are innately acquainted with the relation ‘acquaintance’. We thereby understand that relation’s being a p-relation. We may, however, also hold that no fact of a subject’s being acquainted with something is permutative. That is to say, no object is acquainted with a subject. Since we have taken ‘acquaintance’ as a primitive relation we have decided that no logical structure is to be uncovered. There is, therefore, no impetus to define what it is to be a p-relation in terms of more primitive notions involving the notion of possibility. All the same, it is important not to trivialize what is required for the permutativeness of a fact. It is obviously not logic that is involved since the existence of the fact in question may well be contingent. The well-formedness of both the expression “R2xy” and the expression “R2yx” is certainly not sufficient to understand permutativeness—let alone the existence of any fact. Quite the opposite, it is our acquaintance with universals that enables us to understand such expressions. Acquaintance with any fact relies on acquaintance with universals, and not vice versa. Of course, a given p-relation with which we are acquainted may well not be exemplified at all. In the case of Desdemona and Cassio, there is no fact of Desdemona’s loving Cassio and neither is there a fact of Cassio’s loving Desdemona. Nonetheless, ‘loves’ remains a p-relation independently of whether and how it may be exemplified. When there is a fact, for example, that Desdemona loves Othello, we can say that the fact is permutative in virtue of its containing ‘loves’ as its relating relation. The puzzle of defining the permutativeness of a fact can now be addressed without invoking possibility. The puzzle is solved by realizing that there is no such property to define.8 Given a p-relation, however, we

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were able (see above) to define what it is for an entity to be a pRn-fact. The definition of “pR2-fact (z)” permits us to write: 2

2

pR2-fact ((ιf  )(a C1R f • b C2R f )). The theory of definite descriptions yields, 2

2

(∃f  )( (g)( a C1R g • b C2R g .≡. g = f  ) • pR2-fact (f  )). All is well. Observe that if z is not a fact, or if it is not a fact in which the p-relation R2 is exemplified, then it is not a pR2 fact. Let us apply the definition to the specific fact of Desdemona’s loving Othello. According to Shakespeare’s play, Desdemona loves Othello. But Othello does not genuinely love Desdemona (or so we shall interpret the play since, after all, he murders her from jealousy and the tragedy is that his jealousy prevents him from what otherwise would have been a genuine love). Thus, let us suppose that there is a fact of Desdemona’s loving Othello but no fact of Othello’s loving Desdemona. We have: 2

2

E!(ιf  )(d C1L f • o C2L f ) 2

2

~E!(ιf  )(o C1L f • d C2L f ). For convenience, let’s use the name “d-L2-o” for the fact of Desdemona’s loving Othello. Now our definition readily yields the following: pL2-fact (d-L2-o) 2 2 i.e., (∃x, y)( x C1L d-L2-o • y C2 L d-L2-o ). The following axiom is very intuitive: 2

2

(Rn)(w)(x, y)( x CiR w • y CiR w .⊃. x = y), where 1 ≤ i ≤ n. The next results are theorems. We have: 2

2

2

2

(R2)(w)(x, y)( x C1R w • y C2R w .⊃. w = (ιf  )(x C1R f • y C2R f )) 2

2

2

(R2)(a, b)(x, y)( x C1R (ιf  )(a C1R f • b C2R f )) • 2

2

2

y C2R (ιf  )(a C1R f • b C2R f ) .⊃. x = a • y = b). In our present case, we have: 2

x C1L d-L2-o .

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Using our above results, it follows that we know that x = d and since we have 2

y C2 L d-L2-o we know that y = o. The permutativeness of the fact d-L2-o is immediate from the nature of the p-relation L2, that is, ‘loves.’ It does not rely on any notion of the logical possibility of there being a possible fact exactly like d-L2-o but involving a permutation of its d and o constituents. When a mind is acquainted with a relation, say R2, it understands its adicity. More than that, in being acquainted, the mind must understand immediately whether it is a p-relation. And its being a p-relation is not a property of it that we discover empirically by examining cases of its exemplification in facts. This lends itself to the feeling that it must be a logical matter. But this feeling comes from acquaintance with the relation R2 and not from the discovery of a logical structural relation of the facts in which the p-relation R2 is instantiated. Thus, the quest for a general definition of a fact being permutative is misguided. At first, it may seem that in order for Russell to be able to establish that the fact 2

d- C1L -(d-L2-o) is not permutative he must show that it2 is impossible (logically) that a fact be formed so that d-L2-o bears the C1L relation to d. This is mistaken. What has to be shown is the modest result, namely that 2

~( (d-L2-o) C1L d ). Even if Desdemona were a series of facts/events, it doesn’t happen. If Desdemona is a series of events/facts, 2 then facts will bear position relations to her. But they won’t bear the C1L relation to her. That seems manifestly obvious because Desdemona is not composed of facts one among which contains Desdemona as a constituent. As we can see, for a great many p-relations, we know clearly enough that the position relations it determines are not themselves p-relations. We know this, for instance, for the position relations determined by the relation ‘loves.’ That is enough to assure that we can safely make a definite description that invokes position relations—a definite description which, were it to be satisfied, would single out a permutative fact. Even if we feel we know that it never happens, it doesn’t follow that it is logically impossible. If it is logically impossible, there must be some further logical analysis yet to be uncovered that reveals that there is no such structure. To assess whether it is logically impossible, one would need a complete

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analysis, and it is always difficult to know whether an analysis is complete. This is a difficult matter to decide. Much depends on the relation in question. There is an interesting argument that, perhaps, suggests that the result is general, that is, that for every R2, we have: 2

(a, b)( ~( (a-R2-b) C1R a ). Consider what happens with this: 2

(a-R2-b) C1R a. If this were the case, then for some x and y, we have a = x-R2-y. Hence, we have 2

(a-R2-b) C1R (x-R2-y). But this implies (a-R2-b) = x. Thus, for some x and y, we have: a = (a-R2-b)-R2-y ((x-R2-y)-R2-b) = x. We have an entity a that is equal to a fact that has a as one among its distinct constituents. We have arrived at an untoward consequence that, in a great many cases, we know does not happen. But a proper definite description of a permutative fact doesn’t require that it never happens. We have not arrived at a logical impossibility. Perhaps an entity can be equal to a fact that has itself as one among its distinct constituents. The only relation for which I can imagine this untoward situation happening is the ‘acquaintance’ relation itself. My imagination is driven by the fact that no one has a viable analysis of the relation of ‘acquaintance’ because no one has a viable philosophy of mind that explains the nature of mind. Consider, for the moment, that a Mind (‘Self’) may be a complex that is a proper part of a proper part of itself. That is, if a mind is a complex fact (event) then, when it is acquainted with its being acquainted with something, we may have a viable case where a ‘self’ is a fact that is a proper part of a proper part of itself. There is no indication whatever that Russell entertained that there are such strange metaphysical complexes. In Theory of Knowledge, Russell tries to model “self-consciousness” by appealing to

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facts in which different subjects occur, temporal series of which constitute a Mind, He allows s-Ҏ-(s′-A-o) where Ҏ is a presentation relation of experiencing in which objects of sensation are presented but objects of memory are not present (TK, p.  38). Russell is careful to note that the subject s is not the same as the subject s′. For Russell, a Mind, not the subject, is a continuant—a series of states (facts). This series might be said to be itself a very complex fact. But as long as the series is composed of quite distinct subjects, there is no chance of a whole series (a fact) which is a mind ever having itself as a proper part of a proper part of itself. Likely, such an untoward situation just never happens. There seems, however, to be nothing in the structure of the case, insofar as Principia’s logical forms are concerned, to establish that this untoward situation is logically impossible. The lesson here is simply that applying a program of logical analysis is difficult and may involve a great deal of a posteriori scientific knowledge in an effort to assign logical forms and discover what is, or is not, logically possible. Nothing suggests there is a non-well-founded structure that Principia’s general study cannot capture. Russell’s solution of the problem of direction does not stand or fall on finding a new logical form that reveals that it is logically impossible for a position fact to be permutative. Forming a definite description that would uniquely be satisfied by a permutative fact involving the relation ‘loves’ relies on our understanding that this situation does not arise with the position relations determined by the ‘loves’ relation. We know that: 2

~((a-L2-b) C1L a ). Forming the definite description does not require it being logically impossible for there to be such a fact. It simply relies on there not being such a fact. In short, I see no good reason that Russell must reject the thesis that for some p-relations these matters are contingent. We are, after all, considering what it takes to form adequate definite descriptions which would be contingently satisfied. The adequacy of such a definite description relies upon contingent features of the world. That a relation is a p-relation is a logical matter. That when exemplified, the position relations that a p-relation determines form facts that are not permutative may well not be a matter of logic.

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Notes 1. Better put, it is a belief whose truth-maker is a would-be permutative fact. 2. Russell oddly repeats the example using α instead of γ. I have omitted this feature here. 3. God’s belief-facts, if there should be any, do not stand (or fail to stand) in Russell’s correspondence relation. Since falsehood cannot arise for them, they do not involve belief as a multiple relation. In short, Russell’s correspondence definition of “truth” and “falsehood” cannot apply to God. 4. I chased it to no good end in Landini (2015). 5. This is an important point made by Hochberg (2000). 6. See David Lewis (1983). 7. That is not the only new feature Perović (2016) noticed. She observed that Russell’s annotation of Map C speaks as though there are distinct relations each depicted by its arrow instead of one multiple relation of understanding (depicted by all the arrows taken together). 8. In this matter I must reject the attempt I made in my “Types* of Russellian Facts” which endeavored to work with Russell’s analysis of logical possibility to define the notion of a permutative fact. See Landini (2015).

Bibliography Works

by

Other Authors

Davidson, Donald. 1969. True to the Facts. In Inquiries into Truth and Interpretation. Oxford: Clarendon Press. Hochberg, Herbert. 2000. Propositions, Truth and Belief: The Wittgenstein-­ Russell Dispute. Theoria 66: 3–40. Landini, Gregory. 2015. Types* and Russellian Facts. In Acquaintance, Knowledge, and Logic: New Essays on Bertrand Russell’s The Problems of Philosophy, ed. B. Linsky and D. Wishon. Stanford: CSLI Publications. Lewis, David. 1983. Counterpart Theory and Quantified Modal Logic. In Collected Papers, ed. David Lewis, vol. I, 26–38. Oxford: Oxford University Press. Pears, D.F. 1967. Bertrand Russell and the British Tradition in Philosophy. New York: Random House.

CHAPTER 6

Compositionality

Theory of Knowledge all but solved the problem of compositionality. Variable binding introduces complicated syntactic constructions, and without compositionality, cognition of quantification would be impossible. Compositionality and systematicity are features of determinate intentionality that must be embraced by any viable philosophy of mind. They must somehow have vestiges that are manifested in the belief-facts that are artifacts of intentionality and which are prevalent in deductive inference. What then are the constituents of the belief-facts that are the truth-bearers for quantificational believing? Given the complications, one may at first worry that compositionality has to be captured by templates which fit together in fixed syntax-like operational ways comprising a sort of algebra of thought. That approach is, however, entirely closed off. It faces the problem of direction, since different templates can be filled in different ways and one would be back to the problem of having to say which entity goes where in the template. Worse still, it is completely inadequate to impredicative quantificational thinking which enables an unlimited understanding of new universals whose exemplification conditions are comprehended by wffs. Impredicative quantificational thinking is not constrained by fixed operations on fixed units. It is ever at the ready to discover new universals,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8_6

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reference to which then feeds back into the process to form an endlessly bountiful spring. Russell’s approach to compositionality was to introduce structureless logical forms as objects of acquaintance. The structureless feature of Russell’s logical forms is centrally important. In Theory of Knowledge, the theory of logical forms as objects of acquaintance emerges first in connection with the need for a logical form for the adicity of a relation. We saw in our last chapter that Russell’s held that the number of position relations determined by a p-relation may not coincide with its adicity. Hence, a separate logical form for adicity is needed and Russell identifies it as an abstract fact. For example, our understanding of the dyadic adicity of a relation lies in our acquaintance with the structureless abstract fact ‘something has some relation to something.’ Let us recall that he writes (TK, p. 98): In order to understand the phrase ‘dual complex’ or the phrase ‘dual relation’, we must be capable of the degree of abstraction involved in reaching the pure form. It is not at all clear what is the right logical account of ‘form,’ but whatever this account may be, it is clear that we have acquaintance (possibly in an extended sense of the word “acquaintance”) with something as abstract as the pure form, since otherwise we could not use intelligently such a word as ‘relation’.

Russell was quite clear that his logical forms are structureless (TK, p. 114): the logical nature of this fact is very peculiar. For ‘something has the relation R to something’ contains no constituents except R; and ‘something has some relation to something’ contains no constituents at all.

Happily, we found that we were able to avoid having a logical form for adicity by making the number of position relations determined by a p-relation coincide exactly with its adicity. Acquaintance with structureless logical forms was invoked by Russell to do much more than just to facilitate our understanding of the adicity of a relation (“dyadic,” “triadic” etc.). They are required for our acquaintance with logical notions such as “all,” “some,” “and,” “or,” and “not.” He generalizes his plan of appealing to logical forms. Recall that he wrote (TK, p. 98):

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Such words as or, not, all, some, plainly involve logical notions; and since we can use such words intelligently, we must be acquainted with the logical objects involved. But the difficulty of isolation here is great, and I do not know what the logical objects involved really are.

Like the logical forms for adicity, these logical forms are structureless (abstract facts with no constituents). We mustn’t forget, of course, that there was an impasse that Russell reached. Eventually, he had to evade logical forms as abstract particulars so that his solution to the problem of compositionality is not in conflict with the agenda of his scientific method in philosophy. The identification of his logical forms with structureless abstract facts was tentative and we have seen evidence that originally he had hoped that he might overcome his impasse. One immediate way out would be to try to identity logical forms with contingently existing structureless concrete particulars. Unfortunately, it is hard to imagine any concrete particulars that could play the needed role. Logical forms have to be particulars and yet shareable by believers. They (at least the foundational ones) are not learned. Acquaintance with them is innate. The importance of the structureless nature of Russell’s logical forms has long been missed. Of course, interpretations naturally and legitimately objected to the very notion of a structureless fact. Such a criticism is not, however, where the focus should be. It is precisely the structureless feature of Russell’s logical forms that is revealing. It illuminates the trajectory of his approach to the problem of compositionality. He hoped to find a consecutively ordered syntax for quantification theory. A consecutive sequence of logical forms, held in an order by a ‘belief ’ relation, would then parallel the syntax. Now a consecutive order for quantifier-free syntax, I shall argue, was very plausibly known to Russell in 1913. It is rather immediate from Sheffer’s paper, which we know Russell had then read, on the adequacy, relative to Principia logical particles, of the stroke and dagger expressions. Sheffer would later extend his propositional dagger (and stroke) to quantification with the variable binding operator ↓ . But a consecutive ordering for quanx tificational syntax was certainly unknown in 1913. It awaited the technical development in logic of the elimination of variables. I surmise that Russell would have found the technical logical apparatus had he not

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given up. We are reminded that in a letter to Ottoline Morrell he writes (A, p. 66): My impulse was shattered, like a wave dashed to pieces against a breakwater. I became filled with utter despair, … 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.

What is revealing in this passage is the emphasis on the need to find a new construction in logic. Russell likely had reached an awareness that he needed a technical way to produce consecutiveness and eliminate variables in his effort to solve the problem of compositionality Thus, we can praise him for his idea that structureless logical forms would, if developed, capture compositionality. He was on track for the solution. Indeed, if we pursue Russell’s solution far enough, we in fact find a general solution not only to the problem of compositionality but a way to transcend the impasse he had reached. He gave up just as he was about to succeed. My repair will be to introduce what I call “L-forms” that are not abstract particulars with which we are acquainted. I shall argue that we notice them as vestiges modifying the belief-relation that unifies the belief-fact that is the artifact of the intentionality of a mind engaged in believing. Moving them to the belief-relation does threaten to reintroduce the problem of direction. Once binding predicate variables is treated, however, this concern vanishes.

Punctuation Marks At first blush, Russell’s introduction of logical forms as structureless particulars seems untenable as a theory of compositionality. Suppose one is acquainted with logical forms such as And, Not, and Or, each tied (respectively) to the logical particles signs “•,” “~,” and “∨.” The logical particle signs enable infinitely many syntactically distinct expressions and the cognitive analogs of logical equivalents must be kept distinct. (For example, a belief that p ∨ ~q is clearly distinct from a belief that ~q ∨ p.) It is not viable to require a yet new logical form Or-not-and to capture p ∨ (~q • r), and a new logical form not-and-Or for (~q • r) ∨ p, and so on. That launches us on

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an infinity of logical forms and the systematicity of compositionality is lost. It is of no help to imagine a relation of ‘dis-believing,’ a relation of ‘orbelieving,’ and a relation of ‘and-believing.’ One would also need ‘dis-orbelieving’ and ‘dis-and-believing,’ and ‘dis-or-and-and-believing,’ and so on and on. It has been thought that one needs templates, or at least to emulate the use of templates by invoking universals or universal-concepts (as unsaturated or gappy referential capacities for thought and reference) and even to transform thought into subject-predicate.1 Thus, for example, using lamba abstracts to form predicable concepts terms, one might put: (x)(~Hx ∨ Mx) ⇆ [λF (x)(~Hx ∨ Fx)](M) (∃x)(~Hx) ∨ (x)Mx ⇆ [λF (∃x)(~Hx) ∨ (x)Fx](M). We noted in Chap. 4 that this avenue requires a solution to Russell’s paradox of predicates, and in any case, it is entirely blocked to Russell who makes no blanket assumption that we are acquainted with such universals and/or referential and predicable concepts. The first step in the right direction is to realize that intentionality has an impredicative quantificational scaffolding. There is no singular, subject-predicate thought. The lesson I draw from Russell’s “On Denoting” is that cognition of the impredicative “all” scaffolds all thought. Therein lies the source of our ability to recognize patterns and to understand “any,” “recursive definition,” “ancestral,” “number,” the and so on of consecutive a series, and so forth. The thesis that quantificational thinking must be built upon a more fundamental and conceptually prior singular thinking is, unfortunately, a traditional empiricist dogma. For all his innovative thinking about the coeval status of quantification with the singular term, Quine’s Roots of Reference didn’t quite transcend this empiricist dogma. We can witness the deleterious implication of the dogma in the recent work of Lebens (2017) who assumes (mistakenly) that it follows straightforwardly from the thesis that a recursive definition of the wffs of a formal syntax begins from independently characterized atomic wffs at its base.2 He says that one must adhere to the maxim: “…you build your general

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claims up out of substitution instances that you already understand” (p.  206). “Propositions involving generality,” he proclaims, “are really molecular. They really are constructed out of atomic propositions in successive stages” (p. 204). Lebens tries to extend Russell’s multiple-relation theory to quantification by imagining that an existential judgment-fact, say, that M judges that Peter envies someone, is recovered as the outcome of M performing an existential abstraction on an understanding fact, say, of understanding that Peter envies John. Targeting John, he takes this case of abstraction to consist in a relation ∃j of existential judgment unifying the following belief-fact (Lebens, p. 205): ∃j (m, U(m, E, p, j), j). To make it clear that this is not a wff but names a judgment-fact, let’s depict it thusly:  E    m U   p  m– ∃j—   j  .     j    Lebens imagines a similar analysis using ∀j to catpure M judging that Peter envies everyone. This endeavors to formulate a multiple-­relation theory in which quantification is recovered by a process of abstraction from singular thinking (Lebens 2017, p. 205): Molecular propositions arise from cognitive acts performed upon previous stages of understanding. This seems right: how could one assert a molecular proposition unless one already understands its atomic constituents? How can one assert a general proposition if one understands none of its substitution instances? Writing the constructional history of molecular propositions into our theory of judgment will also help us to guard against ambiguities of scope…

Unfortunately, the intentionality of quantificational thinking, understanding, and judging cannot be recovered this way. The intentionality involved in a cognitive act of abstraction cannot be constituted by a relation unifying a fact. The targeting of John is supposedly accomplished by the several

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occurrences of John himself in the fact. I don’t see how occurrences of a particular in a fact can serve as if it were to give rise to an existentially bound variable. And an existential judgment-fact, whose unity lies with a relation, cannot embody the cognitive activity of existential abstraction. Imagining a special relation ∃j does not help. No amount of unity performed by a relation can constitute thinking existentially. Far from helping in a quest to explain compositionality, the empiricist dogma is precisely what inhibits progress. Quantificational scaffolding must be the starting place for a theory compositionality. There is no way to reach up to such lofty heights from below by appeal to mysteries of cognitive acts of abstraction. Empiricists resist this and insist that some faculty of abstraction enables one get us from singular thinking to the concept ‘all.’ Emphasis on abstraction suggest that the empiricism is entitled to the concept ‘any’ as a foundation. But it is not. To illustrate, the most striking effort at using ‘any’ to capture ‘all’ is found with Hilbert’s epsilon term, εx φx. The following are definitions: (∃x) φx =df φ(εxφx) (x) φx =df φ(εx~ φx). A first it seems remarkable that one can arrive at the following theorem: (x) (p ⊃ ψx) .⊃. p ⊃ (x)ψx i.e. p ⊃ ψ(εx~(p⊃ψx) .⊃. p ⊃ ψ(εx~ ψx). But together with Modus Ponens, the result comes from the following axiom schema and rule: Axiom schema: φx ⊃ φ(εxφx) Rule: From φx, infer φt, where t is a term. A little reflection reveals that to understand the axiom schema and rule one must be able to understand the variable binding occurring in the term “εx φx” itself, and one must have the ‘all’ concept to recognize generality in applying the axiom schema. Wittgenstein’s Tractatus is not properly classified as in the camp with those accepting the dogma, but that is because we saw that he was willing to simply sacrifice features of mathematical logic if required to realize his intuition of showing. The Tractatus attempted to avoid ‘all’ in favor of the

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singular ‘any’ which amounts to an ‘and so on’ notion of consecutive sequence that is rendered by a recursive recipe for going on in a patterned way. Interestingly, by using recursive functions and a combinatorial logic (and arithmetic), Wittgenstein tried to realize a unification of all logical particles (and, or, not, all, some, identity). Their inseparability comes from his combinatorial algebraic practice of calculating outcomes of operations. What logic and arithmetic have in common, he thought is their source in the practice of calculating tautologhood and the correctness of arithmetic equations. He colorfully says: TLP 5.4611 Signs for logical operations are punctuation marks.

His early work to try to achieve this was his ab-Notation (and tf-Notation) of his 1913 Notes on Logic and 1914 Notes Dictated to Moore. Ultimately, he realized that it couldn’t be extended, as he had hoped, to identity and quantification. We have already seen that his Tractarian idea, which traces back (at least) to May 1915, was that operations are given by recipes for their repetition (… and so on). This is found in his notebooks from 1914 to 1916 and, finally, in his Tractatus itself. Wittgenstein offered his N-operator N( ξ) , where ( ξ) renders the recursive recipe for the notion of “and so on.” Wittgenstein held that all and only logical equivalents have one and the same representation. In this way, logical propositions are not truths that can be said, but are shown in the correctness of tautological calculations of the N-operator equations. Similarly, arithmetical propositions are not truths that can be said, but are shown in the correctness of the outcomes of operations (functions) characterized recursively with numeral exponents. It cannot succeed. Wittgenstein came in the 1930s to witness the discovery by Church that his goal of representational system in which all and only logical equivalents have the same representation is impossible. Quantification theory is not decidable. The concept ‘any’ just cannot do justice to the concept ‘all.’ Cognition of (unbridled impredicative) ‘all’ is fundamental. The kernel of truth of the Tractarian aphorism that logical particles are “punctuation marks” is simply that one cannot understand any logical particle without understanding its intimate interconnectedness to every another and also to the apparatus of quantification. The logical particles are intimately tied together. But not in the way Wittgenstein imagined, for ‘all’ cannot be built bottom-up from the notion of ‘any.’ All the logical notions are

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innately given at once, all together, as a cognitive package. It is never a cognitive discovery, for example, that ‘all’ distributes equivalently over ‘and.’ Unfortunately, the impredicative ‘all’ of quantification with its bindable predicate variables is often considered a bridge too far. Quine rejected it and Chomsky seems oddly not to emphasize its centrality to a viable transformational grammar. Happily, we have found a synthesis of the valuable features of their respective views by accepting, not resisting, a quantificational cognition that is essentially impredicative. Quine’s demand that first-order quantification be the sole apparatus of determinate thought and reference has very untoward implications. Indeed, it is the cause of his failure to accept that there is introspective determinacy in translation. Quine is right to emphasize that in translation a linguist must adopt “analytic hypotheses” concerning what quantificational apparatus holds for a linguistic community. But demanding analytic hypotheses be limited to a first-order apparatus of quantification is not viable. Behavior certainly does underdetermine what first-order quantificational apparatus the linguistic community holds. But it is impredicative quantification with its distinctively bindable predicate variables that scaffolds the cognitive faculty of intentionality. I don’t know whether behavior underdetermines analytic hypotheses of this sort. But even so, every impredicative quantificational apparatus grounds the same determinate aboutness characteristic of intentionality. Excluding impredicative quantificational apparatus as Quine does throws out the baby with the bathwater. I accept the fusion of all the logical particles into a unitary impredicative quantificational scaffold. The point against separation is embedded in Russell’s own employment of the quantificational apparatus of definite descriptions and it is also suggested in Principia’s *9. Predicational thought about a particular, a, does not have the form Fa, but is a cognitive analog of [ιxAx][Fx]. We must avoid the misguided thesis that some thoughts are singular thoughts and some are molecular. This is not due simply to Russell’s theory of definite descriptions. It is due to the fact that all thinking is quantificational. There is no singular and no molecular predicational thinking. The separation of distinct cognitive “parts,” such as a “not” component and an “all” component and so on is a mirage produced by looking that the vestiges of quantification found in the belieffacts and other artifacts of the mind’s activity of thinking. All the same, compositionality remains an essential part of that cognitive activity of determinate thinking. This is compatible with holding that no logical particle can be understood without understanding the whole of

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quantification. The thesis that the logical particles are cognitively inseparable shows the way toward finding a complete solution to the problem of compositionality.

Compositionality Not Predication In repairing Russell’s Theory of Knowledge, compositionality is not to be recovered by imagining a belief-fact to be a structured entity whose special kind of magical unity provided by the ‘belief’ relation transforms it into a representation that is about something or a predication that says something. By taking a mind-first approach which regards both subjects and belief-facts as artifacts of intentionality, one can simply say that a belief-fact is truth-apt precisely because it is an artifact of the determinate intentionality of a mind engaged in believing. The compositionality of the belieffact is a product of the Mind’s intentionality. In this way, the problem of compositionality can be solved while keeping questions of the nature of predication at arm’s length—and entirely unresolved. In Chap. 4 on Acquaintance, we noted that Davidson focused modern philosophy of language on his “predication problem”—the problem of how declarative sentences (or other semantic particulars) have aboutness. This leads him directly to the assumption that some very special kind of unity of such entities is what explains their aboutness. Gaskin (2008) agrees. He writes: “The problem of unity is this: What distinguishes a declarative sentence from a mere list of words?” Declarative sentences say something, predicate something, while lists do not. Gaskin’s concern is that the unity of a semantic particular must ground its aboutness. This misinterpretation is quite widespread; it also infests a great many works, including Candlish (1990), Newman (2002), and most recently Lebens (2017). Such traditional formulations of the predication problem characterize it in terms of the assumption that predicating, saying, representing, and aboutness concerns the special unity of a mental semantic particular. The wheels are off the rails. Russell’s view, pure and simple, is that there are no semantic meaning particulars. It is, therefore, a serious interpretative misstep to attack Russell’s multiple-relation theory by construing it as if it requires that the unity provided by the ‘belief’ relation in a belief-fact transforms that fact into a semantic particular that represents (is about) its would-be truth-makers. In contrast, Collins (2011) is quite concerned to point out that there are quite different notions of “unity” at work in the characterization of the predication problem. In The Unity of Linguistic Meaning, he holds that

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there are pseudo-problems afoot. He focuses his discussion on what he calls “linguistic meaning,” and he takes pains to avoid conflating this issue with issues pertaining to unity of any particular. Collins agrees with Soames (2010) in objecting to the widespread characterization of predication (as it relates to the nature of the aboutness of intentionality) with issues pertaining to the nature of the unity of a complex. This a giant step in the right direction. He writes (Soames 2010, p. 7): I have come to think that propositions in this traditional sense do not exist. If I am right, we face a dilemma. Either we must learn to conceptualize our philosophical, linguistic, and cognitive problems and theories without appealing to propositions, or we must conceive of them in a fundamentally different way. I will argue for the second horn of this dilemma by offering a new account of propositions that reverses traditional explanatory priorities. Propositions, as I understand them, can play the roles for which they are needed in semantics, pragmatics, and other areas of philosophy. However, they are not the source of that which is representational in mind and language. Sentences, utterances, and mental states are not representational because of the relations they bear to inherently representational propositions. Rather, propositions are representational because of the relations they bear to inherently representational mental states and cognitive acts of agents.

Soames maintains that certain special interpretable particulars get semantic properties derivatively from the minds that are engaged with them in mental acts. His thesis is that the philosophy of mind should embrace structured particulars which are interpretable because coupled with a mental act offering a predication attitude toward it. He writes (Soames 2010, p. 29): One might ask what we mean by “predication”—what, in effect, the analysis of predication is. Although it is unclear that an informative answer can be given to this question, it is equally unclear that there is anything to worry about. Some logical and semantic notions—like negation—are primitive. Since this elementary point typically doesn’t provoke hand-wringing, it is hard to see why the primitiveness of predication should.

Soames’s holds that mental acts of predication are primitive (fundamental). He maintains, however, that “…an adequate account of meaning and propositional attitudes requires structure-encoding propositions” (Soames 2010, p. 55). Collins (2011) endeavors to provide such an account using some ideas from the modern followers of Chomsky linguistic studies. Now

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Russell’s Theory of Knowledge did not embrace and does not require the existence of any particulars whose semantic properties of representation are grounded in their unity. Belief-facts don’t represent anything. But the important question is this: Does the solution of the problem of compositionality require the structured entities that are the belief-facts of Russell’s multiple-relation theory to be “interpretable particulars” in the sense that Collins uses the expression? The answer is that it does not. Although they are structured entities, the belief-facts of Russell’s multiple-relation theory are not “interpretable particulars” as Collins (2011) uses the expression. They are not structured entities that are “interpretable” because they are composed of separable units. There are no units whose combinatorial interactions are governed by fixed rules of a transformational process such as that which the followers of Chomsky call “Merge.” They are not governed by a “language of thought” as Fodor and Pylyshyn once imagined.3 Thinking is not a product of any recursive-like process of forming structured entities. Belief-facts are structured entities that are products of thinking. Collins accepts the “minimalist” research program of the followers of Chomsky which appeals to the cognitive operation Merge. This operation was developed by followers in hope that there exists some single operation, one that could be imagined to evolve by natural selection, for building the hierarchical structure involved in human language syntax—an operation that takes any two syntactic units and combines them into a structure ready for recursive computational manipulation that finds the so-called head of the combination which is typically verb phrase oriented. The recursion is thought to be essential because our linguistic capacity enables the construction of a potential infinity of interpretable syntactic complexes, each compositional and finitely structured. Language, on this view, requires recursion, taking items from a lexical stock of perhaps 40,000 units with Merge producing hierarchically interpretable combinations. Chomsky holds that there is an innate universal generative transformational grammar, independent of semantic information bearing states, without which natural languages could never arise. These syntactic generative grammar and its transformation rules are not learned, and they are universal—somehow coded into the neural activity patterns in the brains of users of any natural language. Chomsky’s thesis implies that there are preconscious syntactic formations at work generating and structuring the transformations. Collins agrees and he has precisely this in mind when he speaks of certain strings being “interpretable.” He writes (op. cit., p. 156):

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…we want to ask why prediction patterns one way as opposed to another, and if the answer is not too arbitrary or merely historical, then it will be structural, grounded in a conception of the agent as realizing a system that admits and prohibits certain forms of types realizable as token predications. According to this conception, the unity problem bears on the nature of the cognitive system humans realize, not on what they do, which is a constrained effect of the cognitive system … Of course, we can always ask why a cognitive system is designed one way as opposed to another, but we know how to answer those questions by way of biology, physics…

These rules are not guided by meanings (semantics)—as is witnessed by Chomsky’s famous: “Green ideas sleep furiously.” Indeed, transformational phrase structures often out run any semantic considerations that drive the meanings of the words involved. Collins himself notices the trouble when one becomes concerned that without a “synthesizing” agency or system, one winds up with the untenable position that any sequential string might be an equal candidate for something “interpretable.” Thus, in following Chomsky in accepting a generative transformational grammar, he attempts to appeal to a synthesizing aspect of the generative process in the hope of solving the problem of the existence of phrase structures that are interpretable. Accepting Chomsky’s idea of an innate generative transformational grammar, however, does not commit one to the thesis that some operation which evolved by natural selection (be it Merge or another) can do the work of intentionality. One can certainly embrace Chomsky’s empirical evidence that there exists an innate universal generative transformational grammar underlying all natural languages while rejecting the philosophical thesis that thinking (predicating/asserting) consist, in part or whole, in the existence of series of structured states.4 On a mind-first approach, the artifacts of natural language are interpretable precisely because they are deemed suitable to be used artifactually for the purposes assigned to them by the minds that made them. They are governed by the conventions of the speakers of the natural language—conventions which are malleable and can and do change. There is nothing obscure about that. Being interpretable is something that a particular has in virtue of its being suitable to be used artifactually the way it is, and its being a linguistic artifact lies in there being minds that use it in their activity of communicating. I fear that Collins has the cart before the horse. The Mind’s intentionality comes first, not particulars whose “interpretability” lies in their being appropriately structured—where what is “appropriate” is the result of natural selection.

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Collins takes the Merge operation to produce “interpretable” strings of separable units and then endeavors to construct semantics (if not also Mind with determinate intentionality) as something that arises out of such allegedly foundational processes such as a merging operation. His project remains in the camp with those who endeavor to explain (explain away?) the intentionality of Mind, bottom-up, with a healthy number of handwaiving appeals to evolution by natural selection. If it is to be repaired, Russell’s Principia era scientific method in philosophy can have no patience for any such empirical theory of Mind which would make the epistemology of Principia’s mathematical logic up to its neck in commitments to contingent empirical, and also heavily metaphysical, theories of particulars. Its epistemology is intended to be above the fray of metaphysicians engaged in civil wars over abstract particulars. On a mind-first view, no particulars situated in time are “interpretable.” It won’t help to proclaim that they emerge from a Merge operation well-honed by a process of evolution by natural selection that happens (so far) to have kept a stable equilibrium of species-environments. The Mind-first approach is appealing in a way that the notion of an “interpretable particulars” is not. A version of a Mind-first view may be found in the work of Hanks (2015). Though it would be too much of a distraction to go into details, it is important to investigate some of the engaging ideas Hanks offers for the treatment of compositionality. He focuses on predicating and believing as mental activities. Hanks construes such mental activities as complex universals that are types of actions and that function, in spite of their being complex universals, as sharable propositional contents. Hanks says that “… speakers perform different types of predicative actions in saying ‘Clinton is eloquent’ and ‘It is not the case that Clinton is eloquent’” (p. 100). The latter is distinguished from “Clinton is not eloquent.” He explains as follows (p. 101): Here ‘not’ functions as a predicate modifier that expresses a function from properties to properties … Like acts of quantifier expression, e.g., EVERY, NOT is an act of expressing a function … This commits me to negative properties.

This introduces ontological analogs of the logical particles as functors that modify properties (and relations) to form new properties and relations. Quine’s functors enable him to eliminate variables altogether, which is a valuable step in the right direction. But an ontology of functors would, for

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a Russellian, have to be many-one relations and this is incompatible with the fundamental thesis of the Principia era. If they were relations they could inhere in facts which would thereby violate the fundamental thesis. Hanks rejects the fundamental thesis and he readily accepts that a person’s tokening of a mental act type of conjunction, disjunction, and so forth commits him to general and molecular mental act events. The introduction of such functors to accommodate his theory of compositionality is nonetheless worrisome. Such an approach would seem to commit him to a myriad of properties (and relations) for every distinct syntactic nuance of the wffs he accepts. And what wffs does he accept? The problem of how a person can be thinking of Russell’s paradox might dangerously entangle Hanks in a commitment to the Russell property (of non-self-predication) since it seems needed for the mental act type of asserting (predicating) the property of non-self-­predication to itself.5 Hanks introduces mental act types (properties) such as ‘asserting (predicating) eloquence of Clinton,’ which exists (as it were) in Plato’s heaven awaiting tokening by an astute mind, and which is to be his ersatz for the meaning proposition ‘Clinton is eloquent.’ Hanks therefore offers a boldly engaging denial of the traditional separation of force (asserting, judging, entertaining, hoping, inquiring etc.) from what constitutes propositional content. He explains that “propositions are assertive in the sense that they are types of actions whose tokens are judgmental or assertoric in character” (Hanks 2015, p. 91). It should be noted that the mental act type of ‘asserting (predicating) eloquence of Clinton’ is intended to allow cases in which it involves a referential act using “Clinton” as a rigid designator (Hanks 2015, p. 114). Of course, he realizes that he has to confront the difficulty of what happens with, for example, ‘asserting (predicating) eloquence of King Lear’—there being no such king to appeal to in helping to characterize the type. (See Hanks 2015, p. 134). In any case, one immediately wonders whether he imagines there to be ontological analog of the rigid designator “Clinton” occurring in the referential act type that he construes as a property. A related question arises as to whether there is a structural assay of the property (action type) into components that are its constituents? Is the act type ‘asserting (predicating) eloquence of Clinton’ the same as the act type of asserting (predicating) the property ‘being a property Clinton has’ of the property of ‘being eloquent’? If they are different, then what are the components (if any) of a given action type? Perhaps the notion that it (the action type) has fixed components is misguided—so that it is only the event token of the type that can be said to

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have compositional components. I feel sympathetic to such a view. But in his example of Obama’s assertion that some politician is ineloquent, Hanks offers just such a parsing. He takes it as asserting (predicating) the property ‘being a negative property some politician has’ of the property of ‘being eloquent’ (p. 103). Such parsings conflict with the intuition that neither ‘Clinton is eloquent’ nor ‘Some politician is eloquent’ is about the property of ‘being eloquent.’ Since some mental act types (properties and relations) are themselves assertive, Hanks is drawn to the thesis that there are composite mental act types of canceling assertiveness. In favor of his view, Hanks assumes it to be clear that because of the “speech act effects built into the semantics of the expressions” the mental act type of the assertion (predication) of conjunction (of p and q) makes two assertions, one for each conjunct (p. 104). I’m not convinced. In any case, this assumption leads him to the attending notion that there is a mental act type asserting “p or q” which involves a cancelation of assertions constitutive of the subordinate mental act types p and q involved (Hanks 2015, p. 103). It is an engaging idea, but certainly doesn’t fit well with the role that compositionality was intended to play in inference as Russell imagined it. For example, Hanks holds that in spite of the familiar De Morgan’s laws, in asserting that ~(p ∨ q), one does not make two assertions (op cit., p. 103ff ). Why not hold that negation cancels the “or-cancelation”? Hanks finds himself distinguishing between predicating a negative property and predicating denial. While there are interesting lessons to be drawn from the treatment of compositionality in terms of theory of mental act types (as properties and relations) offered by Hanks, neither his rejection of the force-content distinction nor his approach to compositionality involved with the logical particles is acceptable to Russellians working in the Principia era. A Russellian theory of cognitive compositionality is designed to facilitate (not inhibit) the understanding of basic cognitive inferential transformations such as De Morgan’s—transformations which seem to Chomsky and his followers to be innate. On the Russellian view, no one who understands negation and conjunction can fail to understand disjunction and can fail to engage De Morgan’s laws. Part of the wonderful mystery of our cognitive grasp of logic is that understanding the logical particles comes all together in one compositional cognitive package. Indeed, according to Quine and Chomsky, Russell and Wittgenstein alike (albeit in quite different ways), unzipping our cognitive endowment reveals that it comes with quantification and identity as well. Perhaps it was Wittgenstein who

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noticed this first, daring to propose in his Tractatus that the entirety of quantification theory with identity (where allowed) is worn on the cognitive sleeve of every proposition. Compositionality entails that no one who understands “all” and “some” can fail to understand that “all” distributes equivalently over “and,” and “some” distributes equivalently over “or.” No one who understand “not” can fail to understand that “not all” is “some not” and “not some” is “all not.” These are innate cognitive primitives of every deep transformational grammar (= a quantification theory with identity), and none requires (contrary, it seems, to Hanks) a grasp of a truth-table or indeed a grasp of a concept of truth or of the notion of a semantic interpretation (model) in a domain. Understanding “p and q” cannot be dependent, without embracing an untenable circularity, on understanding “p is true and q is true.” Compositionality and quantification theory with identity are cognitively prior to every concept of truth. On the mind-first view I’m after, one must address compositionality independently of having any account of the nature of intentionality or truth or semantics. In introducing mental act types, Hanks might have seemed to be moving in this positive direction. In an effort to assure the shareability and repeatability of the propositional contents of minds when different subjects engage in tokening the same type of mental act, I fear, he winds up introducing separate properties for every syntactically nuanced complex predicate expression of language. What is needed for shareability, however, is not an appeal to such properties as mental act types. What is needed is an appeal to the quantificational structures of cognition that compositionally scaffold every predicational thought, be it modal, involving bindable predicate variables, relevant, paraconsistent, intuitionistic, or what have you. And of course, in repairing Russell’s Theory of Knowledge, the quantificational apparatus of descriptions is to be followed to the letter, not rigid designation or the use of indexicals. The pragmatics of communication is not proper data for building theory of mind since the social use of indexicals and rigid designators in communication cannot get off the ground until after there are minds using their determinate intentionality to settle on conventions of engagement with others. The cognitive interconnectiveness of the logical particles of quantification with identity is already embedded in the definite descriptive quantificational scaffolding of intentionality that is involved in thinking that Clinton is eloquent, i.e., in thinking that (∃x)(Cy ≡y y = x .•. Ex). It already involves quantification and identity and negation. Moreover, thinking Clinton is eloquent is no different in its quantificational apparatus from in thinking King Lear is eloquent,

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or thinking that there is no such king or thinking that there is no Russell property. As long as we take care to avoid the Meinongian fallacy, the great insight of Russell’s multiple-­relation theory, at least in its original pristine form, is that the only entities that need be shared are just the universals with which the different subjects are acquainted—and no such universals embody ontological analogs of logical particles and quantifiers.

Structureless Logical Forms Russell’s approach to compositionality was to invoke structureless logical forms. How does it work? The only viable answer is that it works by transforming wffs into consecutive sequences of units each of which is a logical form. What is needed is simply consecutive sequencing. One must find a syntax of consecutively ordered syntactic units and associate each unit with an unstructured particular (logical form) with which we are acquainted. They then can be placed consecutively before the subject in a structured belief-fact paralleling the consecutively ordered syntax. This renders a cognitive analog of the consecutive syntax. The consecutiveness is all one needs. In the case of quantifier-free wffs, this is rather easy. Such a notation is often a said to originate with the 1924 Polish notation of Łukasiewicz. In a parenthesis-free notation, consecutiveness does all the work of compositionality. The Polish parentheses-free notion is illustrated in the chart (below). In offering structureless logical forms, it seems likely that Russell had known that such a consecutive notation is available to him. Evidence that Russell knew may be extrapolated from his familiarity in 1913 with Sheffer’s publication on the stroke and the proof that the stroke (and its dual which is now call the “dagger”) are each individually expressively adequate to the truth-functions, given the expressive adequacy of the primitive signs ~ and ∨ of Principia.6 Russell received Sheffer’s (1912) paper on 15 April 1913. The time-line is viable since Russell would have been aware of this in April before embarking in May on writing for Theory of Knowledge.7 Polish

Principia

Opq Np Apq OpANqr AOpNqr

p∨q ~p p•q p ∨ (~q • r) (p ∨ ~q ) • r

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A consecutive notation of this sort preserves the thesis that logical equivalents have different expressions and thus different cognitive analogs paralleling those expressions are assured. To illustrate, consider what would be the belief-facts if the following were true:  Or   R2     a   b  M believes that R2 ab ∨ ~S3 cde.    m- B10-  Not  .  2   S   c   d     e  This puts the wff R2 ab ∨ ~S3 cde into a consecutive notation: OR2 ab NS3 cde. Compare  Or   Not   3   S  3 2 10  c  M believes that ~S cde ∨ R ab.   m- B -  d .    e   R2   a     b  This puts ~S3 cde ∨ R2 ab into the consecutive notation: ON R2 ab S3 cde. The structural difference is welcome. Such detailed differences are expected from a theory of the nature of the belief-facts involved. The consecutive ordering does all the work. Russell needs to extend this idea of a consecutive ordering to quantification with identity. What are the belief-facts that are the truth-­bearers for general belief? To achieve this, we need a consecutive notation adequate to quantifiers which does away with variable binding altogether. But how? Clearly the problem of finding a consecutive notation whose parts can be associated with logical forms is complicated when it comes to the wffs of quantification theory. Russell requires a recipe for a two-way transcription

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of the wffs of quantification theory into wffs in which all the individual8 variables are consecutively presented at the end and where we can rearrange as needed and identify repeating variables. They are then able to be bound one by one in the consecutive order of their appearance (moving right-to-left). In this way, one arrives at a consecutive order of structureless units that track the syntax of any quantified wff. In simple cases, the quantifiers are not given with subordinate occurrence of logical particles, and one can readily find the needed consecutive ordering by simple appeal to converses of relations with which we are acquainted. To illustrate, consider the question of what belief-fact exists when we have a person such that M believes that (x)(∃y) R2xy. To find the belief-fact, we first eliminate variables as follows: (x)(∃y)R2xy ⇆ (x)(∋R2)x ⇆ ∀∋R2. I use the expression ⇆ to indicate the transformation. These are just notational variants. Next assign logical forms to the logical particles involved. The belief-fact is this:    m- B4-    .  R2    Recall that the use of hyphenated name is an explanatory convenience. The key is that we can find a definite description for this belief-fact even though it is permutative. The belief-relation involved is certainly a p-relation but we appeal to the position relations it determines. We have the following: (ιh)(m C1B h • ∀ C2B h •  ∋ C3B h • R2 C4B h) 4

4

4

4

Where R2 is a p-relation, the belief-fact is true iff 2

2

(x)(∃y)E!(ιf )(x C1R f • y C2R f ).

6 COMPOSITIONALITY 

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This accords with Principia’s recursive definition of “truth.” Truthmaking is conducted by the contribution of several facts; no one of them is sufficient as a truth-maker. To switch the order of the quantifiers, we can appeal to the converse ↶R2 of the relation R2. Consider the belief-fact that exists when we have: M believes that (∃y)(x) R2xy. We first eliminate variables as follows: (∃y)(x) R2 xy ⇆ (∃y)(x) ↶R2 yx ⇆ ∋ ∀↶R2 Here I have used ↶R2 which is a converse of R2. Thus, we have the belief-fact:      m- B4-    .   R2    We can easily form a definite description for this belief-fact: 4

(ιh)(m C1B h •

∋ C24

4

4

h • ∀ C3B h • ↶R2 C4B h).

The belief-fact is true iff 2

2

(∃y)(x) E!(ιf )(x C1R f • y C2R f ). All is well. A similar technique applies, to find the belief-fact that exists when we have: M believes that (y)(∃x)R3xyx. Eliminating the variables, we first arrive at the following: (y)(∃x)R3xyx ⇆ (y)(∃x)(↷R3yxx) ⇆ (y)(∃x)(i ↷R3yx) ⇆ ∀∋ i ↷R3.

300 

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Here I use the converse i↷R3 of the relation R3. Thus, the belief-fact is the following:      m- B -    .  i  R3    4

Again there is no trouble giving the following definite description of this permutative fact, 4

(ιh)(m C1B h •

∋ C24

4

4

h • ∀ C3B h • i↷R3 C4B h).

The belief-fact is true iff 3

3

3

(y)(∃x) E!(ιf )(x C1R f • y C2R f • x C3R f ). There are many facts: no one of which is a truth-maker but each of which contributes. The technique of using converses and identification to rearrange as needed, in cases of relations with which we are acquainted, is not so far away from what Russell explicitly did in Theory of Knowledge that it should seem improbable that he might have had this technique in mind in imagining the logical forms that provide him with a solution to the problem of compositionality. If so, then Russell was in anticipation the fundamentals of Quine’s “Variables Explained Away.”9 Among converses, we have the following: (↶ Rn)(xn, x1, …, xn − 1) ⇆ Rn (x1, …, xn − 1, xn) (⋉Rn) (x1, …, xn, xn − 1) ⇆ Rn (x1, …, xn − 1, xn) (i Rn) (x1, …, xn-1) ⇆ Rn (x1, …, xn-1, xn-1). These are akin Quine’s major inversion, minor inversion, and reflection (respectively). But Quine certainly did not imagine them as converse relations. Likely, we do understand conventions which transform one wff into another because the transformation of expressions has an isomorphic (or at least analogous) connection to the work done by converses. (Recall that we are including self-giving and such relations as itself among the

6 COMPOSITIONALITY 

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converses of a relation.) But this is only the start of the techniques needed to parallel Quine’s “Variables Explained Away.” One must be able to reorder variables in connection with the logical connectives, and then we must identify the same variables on the right so that they can be selected consecutively one after another for quantification. Quine adopts a linear convention with the logical particles • and ∨ and ~ which puts all the variables at the right. I’ll mark the linear convention thus: ~R2 xy ∨ S3 xyz ⇆ ∨ ~ R 2 S 3 (xyxyz) ~R2 xy • S3 xyz ⇆ • ~R2S3(xyxyz). In contrast, the default I adopt takes every other one in sequence (working left to right) with the remainder (if any) on the right. For example, ~R2 xy ∨ S3 xyz ⇆ ∨~R2S3(xxyyz) ~R2 xy • S3 xyz ⇆ • ~R2S3 (xxyyz). If we want the remainder on the left, we can put: ~R2 xy ∨ S3 xyz ⇆ .∨~R2S3(zxxyy) ~R2 xy • S3 xyz ⇆ .• ~R2S3 (zxxyy). It is entirely a matter of adopting conventions and marking their use. Next, observe that we can build Quine’s reflection into the quantifier itself, so that e.g., by repeating the i any finite number of the same variables at the end of a string may all be bound together. Thus, for example, (x, y)(~R2 xy ∨ R2 xy) ⇆ (x, y)(∨~R2R2 (xxyy)) ⇆ ∀ ∀ ∨~R2R2. i

i

If we want to change the order of the bound variables, we put: (y, x)(~R2 xy ∨ R2 xy) ⇆ (y, x)(~↶ R2 yx ∨ ↶R2 yx) ⇆ (y, x)(∨~↶R2 ↶ R2 (yyxx)) ⇆ ∀  ~↶R2 ↶ R2. i

i

For notational convenience, however, I shall use ∀i instead of ∀ which is i cumbruous to write. The same points apply to the existential quanifier.

302 

G. LANDINI

Now consider, for example, the case where the following is true: M believes that (y, x)(~R2 xy ∨ R2 xy). We can now find the belief-fact that is its truth-maker, namely,  i     i  m—B7—    .  ~   2  R    R2    Of course, what is wanted is a definite description of the permutative belief-fact. This poses no problem: 7

7

7

7

7

(ιf ) (m C1B f • ∀ C2B f • ∀ C3B f .•. ∨ C4B f • ~ C5B f 7

i

7

i

• ↶R2 C6B f • ↶R2 C7B f ).

As we can see, a great deal can be done with converses and the use of conventions which stem from our ability to recognize patterns.10 The transformations of wffs rearranging variables over the logical particles is made by various conventions. With the permission to invent and mark new conventions at will, there is no need to fear that we might not effectively rearrange variables pulling them to the right of a wff in any order one wishes. Hence, we may as well adopt the following which is the easiest   way to execute such rearrangements effectively. Use n and n . Returning to our example, take M believes that (y, x)(~R2 xy ∨ R2 xy). We can now find the belief-fact that is its truth-maker, namely  i      i  m— B7—  1, 2   .  ~   2   R   R2   

6 COMPOSITIONALITY 

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We found this belief-fact via the transformation:   (y, x)(~R2 xy ∨ R2 xy) ⇆ (y, x)(∨~R2R2 (xxyy)) ⇆ ∀i∀i12 ∨~R2R2. This avoids the use of ↶R2 entirely and paves the way for adapting the Quinean technique for the binding of predicate variables. In any case, we have a glimpse of the trajectory that Russell was on with the introduction of structureless logical forms.

L-Forms with en Resolves Russell’s Impasse We have lately seen how that structureless logical forms can do precisely what Russell said they can do. They solve both the problem of direction and the problem of compositionality. The real difficulty, of course, is that the logical forms must be eliminated altogether as entities that are relata for the belief-relation. We cannot rest until logical forms are removed entirely from relata of the belief-fact. Only in this way can the acquaintance epistemology of Theory of Knowledge be made consonant with the agenda of its scientific method in philosophy. The way to eliminate Russell’s logical forms as objects of acquaintance becomes clear when we extend the Quinean apparatus to treat quantifiers binding predicate variables. The central new feature that adapts Quine’s technique to enable the binding of predicate variables is the introduction of the transformation en + 1. It assures the following: en + 1(Rn, x1, …, xn) ⇆ Rn (x1, …, xn). Observe that there is, in any atomic wff, one and only one predicate variable occurring in a predicate position—the first position (as it were) of en + 1, which is a p-relation. Thus, there is no need to be concerned with recognizing which among Rn, x1, …, xn is the predicate variable. In the above, it is designated by the special upper-case letter Rn but we shall see in the next section that once Principia’s simple-type regimentation is added, such a special letter is not needed. Here I have kept the convention of putting the predicate variables left most. Observe as well that we know the adicity of the universals R2, S3 involved because they are coded by the adicity of e3 and e4. That is, we know that they are involved because of the syntactic positions they have and the adicity e3 and e4. The work of designating which is the predicate is done by en + 1 itself and not any special style of variable.

304 

G. LANDINI

The adoption of en + 1 preserves the intimate links between logical particles and quantifiers. The familiar De Morgan’s laws and double negation are as follows:  m e n ⇆ • e m e n ∨e • en m e n ⇆ ∨ e m e n e ⇆ en . For the quantifiers, negation reveals the linkages between the particles. Using ♀ as a stand in for any needed rearrangement and/or identification of like variables, the following hold:  m en ∋ ♀∨emen ⇆ ∀♀ ∨e  ♀•emen ⇆ ∋ ♀ • e m e n ∀ ∋ ♀•emen ⇆∀♀ • em en  ♀∨emen ⇆ ∋ ♀ ∨e  m en . ∀ These are all immediate and cognitively natural, and thus I use the sign ⇆. The transformations are cognitively innate. But this feature is not, as Wittgenstein hoped, the result of the elimination of all the logical particles in favor of his multi-adicy N-operation (and its attempt to use the “and so on” of a recipe instead of “all”). Quite to the contrary, we saw that it is “all” and impredicative comprehension that is required for the conception of “any,” “and so on,” the “general term of series,” and so forth. The addition of en both enables binding predicate variables and addresses the problem that Russell’s structureless logical forms cannot be abstract particulars. It enables us to move the logical forms to the beliefrelation without reintroducing the problem of direction. I’ll hereafter call them L-forms. Once we have en, the binding of predicate variables offers no new troubles. Consider what happens when the following is true: M believes (R2)(x, y)(~R2 xy ∨ R2 xy). The belief-fact involved can be named as follows: 3 3

m— iii  e e B1 We found this by the following transformation:

6 COMPOSITIONALITY 

305

(R2)(x, y)(~R2 xy ∨ R2 xy) ⇆ (R2) (x, y)( e 3 ( R 2 , x, y) ∨ e3(R2, x, y)) ⇆ (R2) (x, y)(∨ e 3e3 (R2, R2, x, x, y, y) ) ⇆ (R2)(∀i∀i ∨ e 3 e3 ( R 2 , R2) ⇆ iii  e 3e3 . This is a general belief-fact of pure logic. The sign of its being a logical truth does not, however, lie in there being only the subject as the relatum of the belief-fact. It lies in the main component being ∨ e 3 e3 . We have the sequencing of all the needed variables so that even predicate variables can be bound. Moreover, the Quinean technique enables us to move the logical forms so that they are not objects of acquaintance in the belief-fact. We have “moved” all the logical forms to the belief-relation itself. Let us now apply our solution to the celebrated case of providing the permutative belief-fact involved when Othello believes that Desdemona loves Cassio. Now given we are innately acquainted with the relation ‘identity,’ it is not implausible that given we are acquainted with a property F, we are also be acquainted with the property 1F of being uniquely F. I’ll indicate such a property as 1F. (1F) x ≡ (u) (Fu ≡ u = x). On Russell’s view, neither Desdemona nor Cassio can be objects of such a belief-fact since Othello cannot be acquainted with either person. Othello must think about Desdemona and Cassio by means of definite descriptions. So let’s use 1D and 1C for this. We have: Othello believes that [ιxDx][ιyCy][L2 xy]. ∋∋i•• e2e2e3 4 B

[ιoOo][E!(ιf )(o C1 ∋∋i•• e2e2e3 4 B

.•. 1D C3

∋∋∋i•• e2e2e3 4 B

f • L2 C2 ∋∋i•• e2e2e3 4 B

f • 1C C 4

f

f )].

To find the definite description for the permutative belief-fact, we apply our Quinean technique [ιxDx][ιyCy][L2xy] (∃x)(∃y)(1Dx • 1Cy .•. L2xy)) ⇄ (∃x)(∃y)(e2(1D,x)•e2(1C, y ) .•. e3(L2, x, y) ) ⇄ (∃x)(∃y)(•e2e2(1D, 1C, x, y) .•. e3(L2, x, y) ⇆ (∃x)(∃y)(••e2e2e3(1D, L2, 1C, x, x, y, y) ) ⇄ ∋ i∋ i ••e2e2e3(1D, L2, 1C).

306 

G. LANDINI

To avoid L-forms as abstract particulars, we employ our en relation and move all the many L-forms to the belief-relation. With this in place, the belief-fact involved is just this: o—

∋ i ∋ i •• e e e

2 2 3

 1D    B 4 -  L2  .  1C   

The case of Othello believing that Desdemona loves Cassio is now finally solved. It is very important to realize that we can only move logical forms to the belief-relation in this way because of the addition of en. If we try to move them without it, we would need to appeal to templates marking gaps where the relations are supposed to go. The problem of direction would be resurrected. To see this, consider the following case: M believes that (x)(R2xx .∨. R2xx • S2xx) Transforming without using en, we get: (x)(R2xx .∨. R2xx • S2xx) ⇆ (x)(R2xx .∨.•R2S2(xxxx)) ⇆ (x)((∨R2•R2S2(xxxxxx)) ⇆ ∀iiiii∨R2•R2S2 Imagine trying to move the logical forms using template parameters such as ∆ as follows:  R2    m— ∀iiiii ∨△•▽△B3 –  R 2  .  S2    This template approach faces the problem of which among R2 and S2 goes repeatedly in the ∆ spots and which goes in the ∇ spot. The problem of direction returns. Note that if we adopted a convention, the name of the belief-fact is fully representative of the permutative belief-fact needed— the convention being that repetition of the ∆ requires that the same entity fill it, and the first occurrence of a relata must fill the first occurrence of the ∆. But it is not with names of the belief-facts that the direction problem lies. The problem is that one cannot properly form a definite description

6 COMPOSITIONALITY 

307

of the permutative belief-fact. Position relations are determined by p-relations which ground the unity of a fact. A given position relation is a relation between a constituent of a fact and the fact in which it is a constituent. They are not relations to templates. It cannot be otherwise. With the inclusion of en, the problem is solved. We simply have: (x)(R2xx .∨. R2 xx • S2xx) ⇆ (x)(e3 (R2, x, x).∨. e3 (R2, x, x) • e3 (S2, x, x)) ⇆ (x)(e3(R2, x, x). ∨ . • e3e3 (R2, S2, x, x, x, x) 3 3 3 2 ⇆ (x)(∨•e , R2, x, S2, x, x, x)  e e3 (R 3 3 ⇆ ∀iiiii 3 ∨ •e e e (R2, R2, S2). If the belief is true, then there is the belief-fact m—

 iiiii 3 • e3e3e3

 R2    B4 —  R2  .  S2   

And as always, there is no trouble forming a definite description of this belief-fact. Before we go on to examine the binding of predicate variables in Principia Mathematica with its simple-type theory, let’s further illustrate the new technique. Consider how Russellians address the Meinongian worry that non-existents are involved when, for example, someone believes, say, that the x, such that x is left of itself, does not exist. Of course, there is no such object of belief. To think there is, is just to commit what we’ve called the “Meinongian fallacy.” Russell’s multiple-relation theory, now armed with the account of general belief, affords the complete solution. To illustrate the power of our new technique for eliminating logical forms, let us take the cases of how Russell’s multiple-­relation theory can capture primary and secondary scopes afforded by Russell’s theory of definite descriptions when they occur embedded within the context of an ascription of belief. Consider, then, the following ascription of belief by a speaker S: M believes that [ιxKx][~Px] (i.e., M believes that some unique present king of France is not bald.)

308 

G. LANDINI

Our Quinean technique enables us to transform as follows: [ιxKx][~Px] ⇄ (∃x)(1Kx • ~Px) ⇄ (∃x)(e2(1K, x) • ~e2 (P, x)) ⇄ (∃x)( •e2 e 2 ( 1K , P, x, x)) ⇆ ∋i • e2 e 2 ( 1K , P). If the ascription to M is true, then there is the following belief-fact, namely:  1K  2 2 m— ∋ i • e e B3 —  .  P  It is in virtue of this that we can find the definite description for the belief-fact. [ιmMm][E!(ιf )(m C1

∋ i•e2e2 B3

∋ i•e2e2 B3

f • 1K C 2

∋ i•e2e2 B3

f • P C3

f )].

That is, we are using a definite description that would pick out the appropriate belief-fact were the ascriptions made by a speaker S to be true. Observe that this belief-fact is true (i.e., M believes truly) iff (∃x)(E!(ιf ) (x C

1

K

f ) • ~E!(ιf ) (x CP f )).

Many facts conspire together at truth-making. Now let us compare the quite different cognitive situation involved in the following belief ascription by a speaker S: M believes that ~[ιxKx][Px] (i.e., M believes that no unique present king of France is bald.) ∋ i•e2e2 3

[ιmMm][E!(ιf )(m C1

B

∋ i•e2e2 3 B

f • 1K C 2

∋ i•e2e2 3 B

f • P C3

f )].

In this way, we have found the belief-fact that exists if the ascription is true, namely this: m—

∋ i • e2 e2

1  B3 —  K  .  P 

6 COMPOSITIONALITY 

309

To see how we arrived at the relevant definite description, translate using our Quinean technique. This yields the following: ~ [ιx Kx][Px] ⇆

∋

i •e2e2 (1K, P).

Observe that this belief-fact is true (i.e., M believes truly) iff ~ (∃x)(E!(ιf ) (x C

1

K

f ) • E!(ιf ) (x CP f )).

There are no general facts as truth-makers. No general facts, negative facts, or molecular facts play any role as a truth-maker. At first it seemed impossible to capture compositionality in the multiple-relation theory following Russell’s idea of appealing to a consecutive sequence of structureless logical forms construed as abstract particulars that are relata of the belief-relation. Structureless particulars don’t combine, and clearly it would be impossible to have a distinct particular for each distinct syntactic difference in the wffs of the language of quantification with identity. Systematicity seemed lost. Quine’s technique saves the day. The elimination of variables enables the systematicity involved with compositionality to be completely captured by the unique consecutiveness of the associated series of logical forms. They need not be combinable at all. Russell was on the right track—Wittgenstein’s concerns notwithstanding. But the problem of compositionality in connection with the elimination of logical forms as particulars was the impasse Russell was facing—his “real difficulty” as he called it. We have finally answered the fundamental question and solved the paralyzing difficulty arising in Theory of Knowledge concerning compositionality and the elimination of logical forms. It is time to indicate how the accounts in Chap. 4 on Acquaintance and de re and de se ascriptions of belief integrate with Russell’s multiple-relation theory. Let us assume that we are acquainted with a property M that is uniquely satisfied by a subject. Thus, we want to capture: [ιxKx][(∃F)(Fz ≡z z = x .•. [ιmMm][m believes that [ιyFy][~Gy]]). We do this as follows: [ιxKx][ (∃F)(Fz ≡z z = x .•. [ιmMm][E! (ιg)(m C1

∋ i•e2e2 B3

∋ i•e2e2 B3

g • 1F C 2

∋ i•e2e2 B3

g • G C3

g)])].

310 

G. LANDINI

The process of finding the needed definite description of M’s belief-fact is now clear. We first get: [ιmMm][m believes that [ιyFy][~Gy]]. Having enabled the multiple-relation theory to accommodate compositionality, we have the technique for doing this—that is, we use the Quinean technique to transform the clause [ιyFy][~Gy] to eliminate all the variables in it. This yields the following: [ιyFy][~Gy] ⇄ (∃y)(e2 (1F, y) • e 2 ( G, y)) ⇄ (∃y)(i•e2 e 2 )(1F, G, y) ⇄

∋i•e2

e 2 (1F, G).

The use of en enables a nice characterization for the would-be belief-fact that would exist iff M believes that [ιyFy][~Gy]). The definite description of this belief-fact is this: ∋ i•e2e2 B3

(ιg)(m C1

∋ i•e2e2 B3

g • 1F C 2

∋ i•e2e2 B3

g • G C3

g).

As we can see, Russell’s thesis that belief is a multiple relation does not interfere with the intelligibility of forming beliefs about beliefs held by others. Thus, the account of de re belief ascription of Chap. 4 is perfectly consonant with our solution of the problem of composition. The ability to treat primary versus secondary scopes readily handles the concern Prior (1971) raises about the multiple-relation theory of belief when paired with Russell’s thesis that perception, being veridical, consists in a dyadic relation to a fact. Prior takes the case of Felix believing that Joe perceives the fact a-R-b. Now the speaker S says: Felix believes that Joe perceives the fact that a R b.

Where F is exemplified uniquely by Felix, both primary and secondary scope can be readily accommodated by Russell’s theory since we have found the needed quantificational belief-facts. A primary scope is this: 2

2

[ιuFu][(∃f )(a C1R f • b C2R f • [ιxJx][ (∃H) (Hz ≡z z = x .•. u believes that [ιyHy] [y perceives f])].

6 COMPOSITIONALITY 

311

It is captured by the repaired multiple-relation theory as follows: 2

2

[ιuFu][(∃f )(a C1R f • b C2R f • [ιxJx][(∃H) (Hz ≡z z = x.•. 







E!(ιg)(u ∋i 3,4•e e C1B g • 1H ∋i 3,4•e e C2B g • P 2 ∋i 3,4•e e C3B f • f ∋i 3,4•e e C4B g))]] 2 3

4

2 3

4

2 3

4

2 3

4

The belief-fact g that Felix has, if indeed the ascription is true of Felix, is this:  1H   2 3   2 u— ∋i 3,4•e e B 4 —  P .  a - R2 - b    This is quite different from the secondary scope reading which is this: 2

2

[ιuFu][ u believes that (∃f )(a C1R f • b C2R f .•. [ιxJx][ x perceives f])]. It is captured by the repaired multiple-relation theory as this: [ιuFu][ E!(ιg)(u • C2R • 1J

2





∋i 5•e3e3 ∋i 3,4•e2 e3

  ∋i 5•e3e3 ∋i 3,4•e2 e3





∋i 5•e3e3 ∋i 3,4•e2 e3 7

C3B g • a 5

C5B g • P

7

C1B g • C1R 



∋i 5•e3e3 ∋i 3,4•e2 e3

 3 3  2 3 2 ∋i 5•e e ∋i 3,4•e e





∋i 5•e3e3 ∋i 3,4•e2 e3

2

5

C5B g • b



7

C2B g 

∋i 5•e3e3 ∋i 3,4•e2 e3

5

C5B g )].

If this is true, the belief-fact containing the Felix-subject u, is this  C1R   2  C2R       a  ∋i 5•e3e3 ∋i 3,4•e2 e3 7 u – B — .  b   1   J   P2    2

5

C5B g

312 

G. LANDINI

The case is fairly complicated. But we are able to find the needed definite description of the would-be belief-fact using our Quinean technique. We just had to transform as follows: 2

2

(∃f )(a C1R f • b C2R f .•. [ιxJx][ x perceives f])   2 2 ⇆ ∋ i 5 • e3e3 ∋ i3, 4 • e2 e3 ( C1R , C2R , a, b, 1J, P 2) The transformation gives us the constituents of the would-be belief-fact asserted by Felix to exist. These cases are engaging as Prior knew, but we see that they are not insuperable for the multiple-relation theory once its account of compositionality has been repaired. Now let us apply the account of compositionality to cases where we have an ascription of doubly embedded belief—that is, where speaker S attributes to a person the belief that she believes something. There are various kinds of cases to consider. But the issues are often not different in kind from what we have already seen. Of course, the speaker cannot be acquainted with another mind and cannot be acquainted with any subject that another mind may generate as an artifact of believing. Moreover, we must respect that fact that belief is, on Russell’s theory, a multiple relation. For convenience, let M and M* be properties that a speaker S is using to uniquely pick out the subjects m and m* (respectively). Consider the following: [ιxM*x][ιmMm][x believes that m believes that (∃z) R2zz]. This illicitly binds an individual variable in the scope of a belief context. Instead, we have: [ιxM*x][ιmMm][(∃F)(Fz ≡z z = m .•. x believes that [ιyFy][ y believes that (∃z) R2zz )])]. This is captured in the repaired multiple-relation theory as follows: 

∋∋ i 3 ,5i•e2•e3e3 B4

[ιxM*x][ E!(ιg)(x C1 ∋e3

• C1

B2

  ∋∋ i 35i•e2•e3e3

C2

B4

∋e3

g • C2

B2



∋∋ i 3 ,5i•e2•e3e3 B4

g • 1F C1

 ∋∋ i 3 ,5i•e2•e3e3

C3

B4

g 

∋∋ i 3 ,5i•e2•e3e3 B4

g • iR2 C1

g )].

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To see how the appropriate transformation is found, let’s take it piecemeal. Start by realizing that we need a definite description for the wouldbe belief-fact y—

∋ e3

B2 —iR2

This would name the belief-fact that would exist iff y believes (∃z)R2 zz. Thus our definite description is this: ∋e3

(ιf )(y C1

B2

∋e3 B2

f • iR2 C2

f ).

Hence, we get the following transformation: [ιyFy][y believes that (∃z) R2zz )] ∋e3

i.e., [ιyFy][ E!(ιf )(y C1

∋e3 B2

B2

f • iR2 C2

f )].

But there is no need to trouble over the definite description. A description is sufficient and uniqueness may, as it happens, not be part of M’s belief. So for ease of exposition, let’s take the following: ∋e3

[ιyFy][ (∃f )(y C1

∋e3 B2

B2

f • iR2 C2

f ).

To find the belief-fact would exist if this is true, we use the Quinean technique as follows: ∋e3

(∃y)(1F y • (∃f )(y C1

B2

∋e3 B2

f • iR2 C2 ∋e3

(∃y)(∃f )(e2(1F, y) •e3e3 ( C1 ∋e3

B2

B2

∋e3 B2

, C2

f )) ⇄

, y, iR2, f, f ) ⇄

∋e3 B2

(∃y)(∃f )(•e2 •e3e3 (1F, C1 , y, C2 , y, iR2, f, f ) ⇆  ∋e3 2 ∋e3 2 ∋ ∋ i i3, 5i •e2 •e3e3 (1F, C1 B , C2 B , iR2) Thus, if the original belief ascription is true, there would exist the following belief-fact, namely:  1F   e3 2   2 3 3 C B  x— ∋∋ i 3,5i•e •e e B5 —  1e3 2  .  C2 B     iR 2   

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Of course, the names of belief-facts we used to help us were only a heuristic aid helping us to find the needed description. There are no names of belief-facts in the theory. We must describe them. There may well be no such belief-facts. The speaker S must describe them and of course the speaker is in no position whatever to be acquainted with a mind or a subject. We see, however, that this in no way interferes with the applicability of Russell’s multiple-relation theory of belief as applied to doubly embedded contexts. This leaves yet to be addressed the case where a speaker S uses what Castañeda calls quasi-indicators in making ascriptions of first-person beliefs to others. We shall want an account of belief ascriptions de se. The Russellian multiple-relation analysis must handle this situation while adhering to its thesis that one is not acquainted with the subject. This is handled by the use of bound Fregean structured variables such as MxFx and ΣF(Mx Fx), and so on. For example, consider the following where a person S says: M believes she herself to be an x such that (∃y) R2xy)]. This is captured as follows: [ιxMx][ (∃Ω)(ΩzPz ≡P Px .•. x believes Ωz (∃y) R2zy)]. In this way, we shall see that one can use quasi-indicators in communication to ascribe a first-person belief to another without knowing what quantificational scaffolding such a first-person’s reflexive self-reference has. One’s commitment is only to the thesis that such a first-person reflexive self-reference involves a quantificational apparatus (as yet unknown). In virtue of this, we addressed Perry’s concern about essentially indexical de se belief. The thesis that all belief is quantificational does not require that in communication one must know what quantificational structure enables first-person belief. Unfortunately, the general Fregean use of structured variables cannot, so far as I can discern, be accommodated by Quine’s technique. Happily, for the use of quasi-indicators, we don’t need a general treatment of Fregean structured variables. We need only capture one sort of Fregean structured variable, namely, ΩzFz, which will then perform the task of quasi-indication required for the attribution of first-person reflective awareness. This can be accommodated by our Quinean treatment that

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eliminates variables. The case we are considering requires that we apply our Quinean analysis to this: Ωz(∃y)R2zy ⇄ Ωz(∃y) e3(R2, z, y) ⇄ Ωz( ∋ e3)(R2, z) ⇄ (Ω ∋ e3)R2. Next, we have to accommodate the following: m believes Ωz (∃y) R2 zy. We form a definite description for the would-be fact that is its truth-maker: e3

(ιf )(m C1

B2

e3

f • R2 C1

B2

f ).

Hence, we can take the case where we have a person S making the following belief attribution: M believes she herself to be an x such that (∃y) R2xy)]. This is captured by the following: [ιxMx][ (∃Ω)(ΩzPz ≡P (ιz)(z=x)[Pz] .•. x believes Ωz (∃y) R2 zy)]. Here the quasi-indicator (she herself ) alerts us that the speaker S is attributing to M a first-person generation of a self-referentially reflexive belief-fact. This is captured as follows: e3

[ιxMx][ (∃Ω)(ΩzPz ≡P (ιz)(z=x)[Pz] .•. E!(ιf )(m C1

B2

e3

f • R2 C 2

B2

f )].

The first-person self-referentially reflexive belief-fact that exists if this is true is this: m—

e3

B2 —R2.

In this way, we can accommodate the use of quasi-indicators in attributions de se.

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We can also accommodate the case where a speaker S makes the following yet more complicated belief attribution. Here the speaker S attributes to M a first-person belief that she herself believes in a self-reflectively aware way that she loves herself. Thus:  believes she herself to be an x who believes she herself to bear L2 to M herself. The quasi-indicator is captured as follows: [ιxMx][ (∃Ω)(ΩzPz ≡P (ιz)(z=x)[Pz] .•. x believes Ωu (u believes ΩyL2 yy))] The repaired multiple-relation theory can accommodate this: [ιxMx][ (∃Ω)(ΩzPz ≡P (ιz)(z=x)[Pz] .•. Ωe2

E!(ιf )(m C1

B3

f•

Ωie3

Ωe3

B 2 C2

Ωe3

B3

f • L2 C3

B3

f )].

The belief-fact that would exist were this to be true is the following:  ie B2  B2 –  2  .  L    3

m—

Ωe3

As we can see, such cases do not pose insuperable difficulties for the multiple-relation theory. Moore had an engaging case of a speaker S who attributes to M, say, the belief that both (∃y)L2yy and that he himself believes ~(∃y)L2yy. We can put this in the form: [ιxMx][ x believes both that (∃y)L2yy and that he himself believes ~(∃y)L2yy]. Thus, we have: [ιxMx][ (∃Ω)(ΩzPz ≡P (ιz)(z=x)[Pz] .•. ∋ i • e3e3 B3

E!(ιf )(L2 C1

f•

ie3

∋ i • e3e3 B3

B 2 C2

∋∋ i • e3e3 B3

f • x C3

f )].

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To see how we captured this, start with the clause (∃y) L2 yy •

ie3

B2 (m, L2).

We next apply our Quinean technique to this to arrive at the following: (∃y) L2 yy •

ie3

B2 (m, L2) ⇄

(∃y)(L2 yy •

ie3

B2 (m, L2)) ⇄ 3

(∃y)(e3(L2, y, y) • e3 ( ie B2 , m, L2)) ⇄ (∃y)(• e3e3(L2,

ie3

B2 , L2, y, y, m) ⇄

(∃ • e3e3 ↷i↶ ) (L2,

ie3

B2 , m)

So far as I can tell, nothing in Russell’s multiple-relation theory prohibits or resolves Moore’s questions about whether such belief attributions are “paradoxical.” Further assumptions would be needed to establish that.11 The multiple-relation theory is nonpartisan. This is a welcome outcome. Questions in the philosophy of mind concerning the cognitive apparatus for what I (myself ) believe in first-person reflective awareness remain as perplexingly mysterious as ever.

Understanding Principia’s Simple Types Only a Realist semantic interpretation of Principia seems capable of making valid every instance of its impredicative comprehension axioms *12.1.11, and so on. We have to admit that the nominalistic semantic informally sketched by Russell in the introduction to its first edition was a failure. Nonetheless, it had a positive contribution in leading Russell to look askance at any epistemic theory according to which a mind is acquainted with the simple-type scaffolded universals comprehended through *12.1.11. We understand such universals only by understanding the wffs of comprehension that give their exemplification conditions. All universals with which we are acquainted are, ipso facto, type-free. They are, thus, values of Principia’s individual variables (of lowest simple type). Whitehead certainly interpreted Principia in this Realist way.12 The evidence comes from his opening for vol. 2  in which he notes that the

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number of individuals of a simple type (t) may be any number (even an infinite number) greater than the number of universals of the simple type t. Principia’s failed nominalistic semantics also played a role in Russell’s philosophical justification of simple-type regimentation. This happened because in “ψ(t)(φt)” the simple-type index “t” marks (excepting when t = o) predicate occurrences in the wffs of the nominalistic semantics that are substitutable for “ψ(t). ” This correctly likened our understanding of simple types to our understanding of wffs and their truth-conditions. The Realist compromise with Whitehead and Russell we reached is that while any universal we are acquainted with is, ipso facto, simple type free, we are not acquainted with any universal grasped only by means of our understanding of the wff of its comprehension. Our access to simple types of universals is only through our understanding of the simple-type regimented wffs that render their exemplification conditions. Our task is now to explain what this understanding of simple-type regimented wffs amounts to in terms of L-forms. What is the belief-fact that holds when, for example, we take a wff of Principia and a mind M who believes that (φ)(∃x)(φ!xx)? Note that in Principia’s typical ambiguity, the shriek ! serves only to indicate that we are dealing with a bindable predicate variable φ! as opposed to a schematic letter φ for a wff. (Ramification is a figment of the imagination of interpreters such as Church.) Thus, it is merely a notational variant to write (R) (∃x)R2xx. We have already had occasion to employ en in such a situation and we saw that the apparatus for binding predicate variables is easy. By proceeding in this way, we find that the language of Principia’s simple types poses no special troubles for applying our Quinean techniques. The acceptance of L-forms that scaffold the cognitive universals that are belief and understanding relations enables our account to address what entities are before the mind of a person who understands a wff of cp Logic2, i.e., so called second-order logic where impredicative comprehension is allowed but bindable predicate variables may only occupy predicate positions. But we require an account of what is before the mind when a person engages a wff of Principia’s simple-type theory in its primitive notation (where typical ambiguity is removed). In Principia we have cp Logicn. We must extend our Quinean methods to accommodate it. In order to employ Quine’s techniques to develop a theory of cognitive understanding of simple-type regimentation in Principia, we must introduce a general cognitive apparatus that effects the positioning of all the variables so that they are ready to be rearranged and selected one by one

6 COMPOSITIONALITY 

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by a consecutive repetition of quantificational operators. Let us continue writing upper-case letters for predicate variables (usually R, S, T, F, G, H). One can readily use lower-case letters only, adopting a convention as to what marks a predicate position. Thus, instead of using a capital letter for a predicate variable and writing, Rn (x1, …, xn) one can simply write y(x1, …, xn). The position of the letter y and the brackets can be regarded as sufficient to mark y as a predicate variable of appropriate adicity. But in order to bind predicate variables, we need an apparatus to put y into a subject position so that we can rearrange in anticipation of Quine’s technique of sequential quantification. This is what led to our use of the L-form en. That is, we have the following transformation: (en + 1)(y, x1, …, xn) ⇆ y (x1, …, xn) . This idea has to be extended to simple-type regimentation. But we can see that the adicity of the relation y is given already by the superscript attached to the sign en + 1. Thus, an adicity superscript for the sign y can be dropped in (en + 1)(y, x1, …, xn) since it is clearly one less than the adicity superscript for en + 1. Moreover, when we are presented with (en + 1)(y, x1, …, xn), we need not worry about marking y among y, x1, …, xn as the predicate since y is clearly designated by its left-most position in the expression. At first blush, it might seem that for this reason we could dispense with en altogether. But at once we realize that we cannot dispense with en since the Quinean technique of binding variables requires that we be able to rearrange the variables and a rearrangement would hide from us which is the predicate. The question before us is how to apply our Quinean technique for ordering and eliminating variables in the context of simple-type regimentation. In cp Logicn, i.e., the simple-type regimentation of the language of Principia, the atomic wffs look like this: y 1

t ,,tn 

x

t1 1



,,xntn .

In cp Logicn where predicate variables may occur in subject positions as well as predicate positions there may well be several predicate variables in a given atomic wff. But this is no problem because simple-type indices do

320 

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all the work of conveying which letters are predicate variables of a relative simple-type variables and which are individual variables of a relative simple type. The individual variables of lowest type always have a simple-type index of o. Now if we adopt Principia’s simple typical ambiguity, an expression such as y(x1, x2, x3) gives us information both on adicity and on simple type. Simple typical ambiguity restores indices so that we have information on relative type as follows: y 1

t ,t2 ,t3 

( x1t2 , x2t2 , x3t3 ).

Thus, we are in good shape for applying the Quinean techniques as long as we are only interested in conveying simple types in a relative way through the convention of typical ambiguity. Indeed, we now see that we need no new apparatus for capturing the cognitive components for the relative simple types cp Logicn than we had for cpLogic1. An important question immediately arises as to whether there is any significant cognitive difference (concerning L-forms) in beginning from Logic1, i.e., so-called first-order quantification theory where only individual variables can be bound, and transitioning to Logic2, i.e., so-­called second-order quantification theory, where predicate variables may be bound in predicate positions. Once one has the Quinean L-form en, we can see that there is no significant cognitive difference. Likewise, we may ask whether there is a significant cognitive difference as we make the transition from cp Logic2 to cp Logicn, which is the full simple-type regimentation with impredicative comprehension. (Recall that cp Logic2 is standard second-order quantification theory with impredicative comprehension.) The answer, of course, is the same. There is no cognitive difference in terms of L-forms. This is precisely Quine’s own view and it shows up if one tries to read in English a wff such as “(∃F)Fx.” One finds oneself saying: There is some universal and x exemplifies it. Unfortunately, Quine tries to use this feature of cognition to force firstorder quantification theory as the only legitimate quantification. Adopting pseudo-predicates “U” and “E,” he offers the expression: (∃y)(U(y) • x E y). This makes it appear as though cp Logic2 is a naïve set/class theory in disguise. But contrary to Quine, the proper cognitive analog of Logic1 is Logic2, not cpLogic2 with its impredicative comprehension. Given the

6 COMPOSITIONALITY 

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cognitive transition operation en, there is no difference in cognitive significance (in terms of L-forms) between Logic1 and Logic2. But this certainly does not render any privileged status to Logic1. Cognitively nothing more is required to understand cpLogic2, but the cognitive operation en has no import against an ontology of universals. In fact, cpLogic2 is consistent while naïve set/class theory is inconsistent. It cannot be set/class theory in disguise. Moreover, Logic2 is not semantically complete in its standard semantics while Logic1 is semantically complete. The lack of a cognitive difference is irrelevant to the formal semantic and ontological issues involved. To be sure, cpLogic2 has ontological commitments that are quite a bit more modest than those of cp Logicn. One might feel more easy with the former than the latter even if cognitively nothing more is required for understanding the latter. But it is not the ontological question that is concerning us when we speak of an extension of the Quinean methods to capture what is before the mind in understanding the wffs of cp Logicn. We can agree with Quine that in the presence of operation en, cognitively, there is no difference in transitioning from cp Logic1 straight to cpLogicn (simple-type theory). However, we do need a slight modification of our Quinean apparatus if we want to capture the detailed information of relative simple types afforded by simple-type indices, and not merely exploit the typical relativity which leaves all simple-type indices suppressed under conventions of restoration. In order to accomplish this, we need to only modify the L-forms for quantification. We shall need, ∀tRn and ∃tRn. Let us illustrate this Quinean transformation with a few examples. Consider the following typically ambiguous wff in simple-type theory: (ψ)((y)(Gy ⊃ ψ!y) ⊃ ψ!x). Let us transform it using our analog of Quine’s apparatus: (ψ)(~(y)(~Gy ∨ ψ!y) ∨ ψ!x) ⇆ (ψ)(~(y)( e 2 (G, y) ∨ e2(ψ!, y)) ∨ e2(ψ!, x)) ⇆ (ψ)(~(y)(∨ e 2 e2 (G, ψ!, y, y)) ∨ e2(ψ!, x)) ⇆  ∨ e 2 e2 e2 (G, ψ!, ψ!, x)) ⇆ (ψ)(∨ "i   ∨ e 2 e2 e2 (G, x)) "i 3, 2 ∨ ∀i

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These exploit Principia’s simple-type ambiguity. It should be noted, however, that we cannot use our Quinean technique to transcribe directly from defined signs of Principia. We must proceed from wffs in its primitive notations. From these examples, we can see that nothing prevents the application of our technique to find the cognitive apparatus for understanding the wffs of simple-type theory. Any wff regimented by simple types, and thus any instance of impredicative comprehension axiom schemas, is captured; thus we can find the belief-facts and understanding-facts that are the truthbearers involved in general belief. Russell’s Theory of Knowledge, buttressed by an innovation within logic that its research program itself anticipates, supplies the acquaintance epistemology for Principia Mathematica.

Notes 1. See, for example, Cocchiarella (1989, 2007) and Hossack (2007). 2. Lebens overlooks that wffs and also complex predicate expressions (composed from wffs) can be co-recursively defined. 3. See J. Fodor and Z. Pylyshyn (1988). 4. A similar objection applies to Hanna (2006), p. xiii. Instead of a Merge operation, Hanna embraces an innate “minimal logic” which he thinks underlies all quantification theory and all non-classical logics (relevant, intuitionist, paraconsistent, modal, etc.). That is, they are to have the minimal logic in common. This is exceedingly hard to fathom since, for example, relevant entailment wants to abandon even disjunctive syllogism. In Chap. 7, I shall argue that Principia Mathematica captures all such systems as studies of different kinds of structures given by relations. 5. In contrast, we saw in Chap. 4 on Acquaintance that Cocchiarella addresses this problem head on, offering something of a solution to Russell’s paradox. 6. Post (1920) proved the expressive adequacy of Principia’s logical particles. 7. Recognizing that the stroke sign is dyadic, one need only write /pq instead of p/q and one immediately can see the viability of a parentheses-free notation. For instance, p ∨ (q • r) would be //pp/qr. 8. We shall see how to do this for predicate variables and for Principia’s individual variables of higher simple type that are its bindable predicate variables. 9. W. V. O. Quine, “Variables Explained Away,” Proceedings of the American Philosophical Society, vol. 104 (1960), pp. 343–347.

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10. This, it seems to me, goes a long way toward addressing the criticism (e.g., Lebens 2017, p. 198) that such rearrangements of variables are cognitively implausible. It should be noted that every computational philosophy of mind must rely on such rearrangements involving the recursive functions they take to emulate (or constitute) the mind’s intentionality. 11. See Landini (2022c), forthcoming. 12. This evidence was first pointed out to me by James Levine in 2013. Whitehead rejected both the introduction to Principia’s first edition every bit as surely as he rejected Russell’s introduction to the 1925 second edition. Neither introduction was meant to be part of the formal work.

Bibliography Works

by

Other Authors

Candlish, Stewart. 1990. The Unity of the Proposition and Russell’s Theory of Belief. In Ray Monk & Anthony Palmer (eds.), Bertrand Russell and the Origins of Analytic Philosophy. Bristol: Thoemmes Press. Cocchiarella, Nino B. 1989. Conceptualism, Realism, and Intensional Logic. Topoi 8: 15–34. ———. 2007. Formal Ontology and Conceptual Realism. Dordrecht: Springer. Collins, John. 2011. The Unity of Linguistic Meaning. Oxford: Oxford University Press). Fodor, J., and Z. Pylyshyn. 1988. Connectionism and Cognitive Architecture: A Critical Analysis. Cognition 28: 3–71. Gaskin, Richard. 2008. The Unity of the Proposition. Oxford: Oxford University Press. Hanks, Peter. 2015. Propositional Content. Oxford: Oxford University Press. Hanna, Robert. 2006. Rationality and Logic. MIT Press. Hossack, Keith. 2007. The Metaphysics of Knowledge. Oxford: Clarendon Press. Landini, Gregory. 2022c. Good Assumptions for Paradox in Moore’s Paradox(es). Lebens, Samuel. 2017. Bertrand Russell and the Nature of Propositions: A History and Defense of the Multiple-Relation Theory of Judgment. New York: Routledge. Newman, Andrew. 2002. The Correspondence Theory of Truth: An Essay on the Metaphysics of Predication. Cambridge: Cambridge University Press. Post, Emil. 1920. Introduction to a general theory of elementary propositions. American Journal of Mathematics 43: 165–183. Prior, A.N. 1971. Objects of Thought. Oxford: Oxford University Press. Sheffer, Henry. 1912. A Set of Five Independent Postulates for Boolean Algebras, with Application to Logical Constants. Transactions of the American Mathematical Society 14: 481–488. The paper was read before the society in 1912. Soames, Scott. 2010. What is Meaning? Princeton: Princeton University Press.

CHAPTER 7

Scientific Philosophy’s Necessity

Scientific Method in Philosophy was a book illustrating avenues for solving (dissolving) various traditional philosophical problems. Conspicuous by its absence is any direct assault in the book on the notion of necessary truth itself. Its theory of truth had been put on hold. Nevertheless, its aim was to reveal that logical necessity is the only necessity. All non-logical kinds of necessary are frauds. Happily, there are various papers and manuscripts that are particularly revealing when it comes to what were Russell’s views on the subject of necessity. He argues that all notions of necessary truth (and possible truth), including the notions of logically necessary truth (and logically possible truth), are confused and ought to be reformulated as invariance notions. In an early manuscript “Necessity and Possibility,” which may date as early as 1905, he hoped to explain away various de dicto invariance notions of necessary truth in terms of universal quantification. The summum genius is the notion of logically necessary truth (for first-­ order wffs), and it is defined as full universal generality in the language of Principia coupled with truth. In “On the Notion of Cause” (1912–1913), he also speaks of necessary truth and possible truth as confused notions. He undermines both the Laplacian determinist (where laws are thought of in terms of a deterministic causal necessity that brings about subsequent states in time) and Libertarian agency (where a person can bring about events in time). In sorting things out, he appeals to degrees of generality

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8_7

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of form that can be expressed by universal quantification. In his Lectures on Logical Atomism of 1918, he returned to his quantificational treatment of invariance notions of necessity given in his Principia era. Unfortunately, Russell’s analyses have been widely misunderstood. His thesis was that all invariance notions of “necessary truth,” including the invariance notion of “logically necessary truth,” are to be given quantificational analyses. Logical necessity, however, is not an invariance notion. It is grounded in logical facts all of whose constituents are universals. The long history of misrepresentation seems to have begun with Wittgenstein who, as we recall in Chap. 2, intuited that showing elucidates the nature of truth. He held that notions of necessary truth and possible truth are shown. We find: TLP 5.525 It is not correct to represent the proposition “(∃x)fx” in the words “fx is possible” as Russell does. The certainty, possibility or impossibility of a situation is not expressed by a proposition, but by an expression being a tautology, a proposition with sense [truth-conditions], or a contradiction. TLP 6.1231 The mark of logical positions is not their general validity. To be general is only to be accidentally valid for all things. An ungeneralized proposition can be tautologous just as well as a generalized one.

Wittgenstein offers an improper representation of invariance analyses. Russell never held that a true instance of the general existential “(∃x)φx” is sufficient for its being a logically possible truth. Russell did not say that truth of an instance of the general universal “(x)φx” is sufficient for it being a logically necessary truth. Full generality requires binding of all variables after a completed analysis which removes all predicate constants, function constants, and individual constants in favor of appropriately bindable variables. Even with this correction, the Tractarian passages in criticism of Russell seem to convict Wittgenstein of the fallacy of strawman. In fact, there is more agreement than it appears. They both hold that logical necessity and logical possibility are not invariance notions. Russell’s quantificational analyses of invariance notions of necessary truth and possible truth don’t apply. Wittgenstein sought to find a marker by which one could discern whether or not a statement belongs to logic. Since logic is not decidable, there is no such marker even for first-order quantification theory. Unaware, Wittgenstein intuited that all and only statements belonging to logic have

7  SCIENTIFIC PHILOSOPHY’S NECESSITY 

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a logical form that is shown (in N-notation). Though the question of decidability was not settled until the 1930s, Russell never had any patience for Wittgenstein’s intuition of showing. Moreover, he never held that, for a fully universally general wff φ in the language of Principia, knowing that φ is true is sufficient for knowing that φ belongs to logic. Principia’s statement Infin ax and its statement Mult ax are examples. They are fully general in the language of Principia, and they seem (epistemically) to Whitehead to be logical truths, and yet to Russell they seem not to be. Wittgenstein’s concerns aside, the central impediment today facing a revival of Russell’s scientific method in philosophy, with Principia’s cpLogic as its essence, has been its apparent hostility to the many so-called rival logics (relevant, intuitionist, alethic modal, etc.) and to non-logical de re notions of “necessity.” Metaphysicians today remain in intractable civil wars. Their respective abstract particulars and favored kinds of necessity exclude one another. Each is committed to rejecting the Russellian thesis that Principia offers a genuinely universal logic that is the essence of scientific method in philosophy. Happily, this is misguided. The synthetic a priori science of Principia’s cpLogic is the study of all the kinds of relational structures that there are. Principia’s cpLogic is not itself a kind of structure and certainly is not one among the kinds of relational structures it studies. In contrast, we shall see that all so-called rival logics are just that. They are studies of invariance features of relational structures and can all be studied within Principia. Scientific method in philosophy maintains that invariance notions of necessity and the “rival” logics are, one and all, fully compatible with Principia’s cpLogic and its logical necessity being uniquely genuine. Any invariance notion of necessity can be accepted and studied within Principia as a kind of structure. Such rivalry between “logics” is no more genuine than the rivalry between the various geometries (projective, descriptive, metrical, Euclidean, or otherwise). The Russell’s scientific method in philosophy treats invariance notions of necessity and “rival logics” in a way that is akin to the treatment of the faux necessities of non-Euclidean geometries. In short, they are not rivals. They are simply important studies within mathematics of different relational structures. In what follows, I point the way for Principia to accommodate so-called rival logics and different invariance conceptions of necessity. Quantification is the key and thus the same L-forms that enable understanding quantification give everything needed for our synthetic a priori cognition of such notions.

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Logical Necessity Is Not an Invariance In his book Possibility, Buchanan (1927, p. 5) sounded the warning of the dangers of equivocation on the notion of “possibility,” writing that “the philologist could, no doubt, tell an interesting story of the verbal alchemy that lies in back of this term possibility.” In modern modal systems, “◽p” is used to say p is necessary and “⬦p” is used to say p is possible, leaving it open what kinds are intended and leaving it to the axioms and their semantics to sort it all out. (If both are adopted, it is axiomatic that ~◽~p ≡ ⬦p, otherwise one is defined in terms of the other as in: ⬦p =df ~◽~ p.) The sorting continues today and philosophical argumentation, as ever, uses a jumble of different modalities awaiting clarification. Buchanan echoes Russell’s concerns, but he went on to warn us at length that to capture “absolute possibility” is a pursuit in vein to find ultimate reality, the totality of self-completion. He wrote: “Thus the mystic is tempted to deny the adequacy of systematic order and assert that ultimate universality and completeness of totality alone.” He goes on to say: “The identity condition of a given parameter is determinate; that is, it is a differentiation within the field of variability of a higher parameter. This means that there can be no highest parameter in any absolute sense” (Buchanan 1927, 79). Buchanan would be right if he meant that no invariance notion of “necessity,” however inclusive it may try to be, can catch its own tail. But Principia’s logical necessity does not deserve his indictment. It is not an invariance notion. Logical necessity is not a notion of invariant truth. As understood in Russell’s Principia era, it is quite distinct from all invariance notions of necessity (= necessary truth), including the notion of logically necessary truth (logical truth) itself. All notions of necessary truth are invariance notions. The concept of “logically necessary truth” is no exception. According to our repairs to Russell’s Theory of Knowledge, the study of the ontological ground of cpLogic cannot be conducted as a study of what is true, not even of what is logically necessarily true—that is, of what truth-­ makers there are for the truth-bearers that there are for beliefs concerning logic. The ontological ground of cpLogic lies in abstract particular facts composed solely of universals. These are neither the subject matter of cpLogic nor the truth-makers for true beliefs about it. cpLogic finds its ontological ground in abstract particular facts, but logical truths (aka logically necessary truths) do not. Logical truths, as truths, require belief-facts as truth-bearers and because of the contingencies of what belief-facts there are, they are not made true by the abstract logical facts that ground logic. Consider, for example,

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(R)(x)( R2xx ∨ ~R2xx). The truth-conditions are just as recursive as ever. We have: 2

2

2

2

(R)(x)( E!(ιf)(x C1R f • x C2R f ) ∨ ~E!(ιf )(x C1R f • x C2R f )). All this accords with the recursive definition of “truth” and “falsehood” of Principia. The ground of cpLogic lies in the existence of abstract facts all of whose constituents are universals. The ground of cpLogic comes apart from issues pertaining to truth. The ontological ground of Principia with its impredicative comprehension is not found by examining truth at all. Given Principia’s comprehension principles, Russell’s invariance analysis of the de dicto notion of logically necessary truth remains fully viable and it parallels the Tarski definition of “logical truth” in terms of invariant truth in every interpretation. Consider the following: ◽ {E!(ιxBx) ⊃ [ιxBx][Bx]} (ψ)(E!(ιxψ!x) ⊃ [ιxψ!x][ψ!x]). It is a logically necessary truth that if the inventor of bifocals exists, then he invented bifocals. This agrees with Tarski’s semantic idea that logically necessary truth is invariant truth in every admissible interpretation the predicate “B” over any domain of any non-empty cardinality. Logically necessary truth is a fundamentally de dicto notion of invariant truth with respect to the given logical form involved. One must, of course, find the logical form before we apply the Russellian analysis and that is rarely an easy matter. The ordinary language form may be extremely misleading when it comes to logical form—and yet we are not in any position to judge whether a given wff is logically possibly true until after we have found its logical form. Some examples will make this more clear. Assume for convenience there are properties M and Y (where Mx: x is married; Yx: x male) with which we are acquainted, and consider this: ◽ (x)( ~Mx • Yx . ⊃. ~Mx) (φ, ψ)(x)( ~φ!x • ψ!x .⊃. ~φ!x).1 It is a logically necessary truth that all unmarried males are unmarried. Since the second of the above is true and expressed with full generality in the language of Principia, “All unmarried males are unmarried” is,

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according to Russell, a logically necessary truth. This case is easy. Consider next the following which involves ordinary proper names. Assume that the expressions D and C give us properties and L expresses a relation ‘loves.’ Russell has: ⬦ [ιxDx][ιzCz][Lxz] (∃φ)(∃ψ)(∃θ) [ιxφ!x][ιzψ!z][ θ!xz] It is a logically possible truth that Desdemona loves Cassio. One must eliminate all names and constants before applying the analysis. Russell’s quantificational theory of definite descriptions plays an important role as does the elimination of “exists” as a logical predicate and the acceptance of a relation of identity as a part of the logical form. This is less easy but straightforward. If one wants to know whether it is logically possibly true that parallel lines meet, then one will need an analysis of ‘lines’ and the meaning of ‘parallel.’ The Whitehead-Russell logicist embracing nonEuclidean geometries will say Yes. If one wants to know whether it is logically possibly true that 2 + 3 = 3, then one needs an a priori analysis of the nature of finite cardinals, of cardinal addition and identity—and as we have seen, if we don’t raise the simple type the Whitehead-Russell Logicist analysis would say Yes. If one wants to know the structures involved in evaluating whether it is logically possibly true that lead might transform into gold, then one has to know a great deal about the physics of lead and gold. Only thereafter can one know the logical forms involved to answer. Similarly, if one wonders whether it is logically possibly true that a human mind survives the death of its body, the discovery of logical forms involved may be centuries away. The partly a priori and partly a posteriori quest for logical form can be a herculean matter to discover. But this is all as expected.2 When it comes to a de re notion of necessary truth (and possible truth), matters are importantly different. A de re notion logically necessary truth seems unintelligible because there is no structural form over which to define the invariance. On this point, Russell and Quine are in agreement.3 All de re expressions such as the following should be regarded as ill-formed: (x) ◽ (x = x) (∃x) ◽ Fx [ιxBx][~ ◽Bx ]. Such expressions obliterate structure and all invariance notions of logical necessary truth must be defined over structures. Remarkably, it turns out that one can rectify matters, permitting the syntax but making it semantically innocuous. Cocchiarella (1975) demonstrates the semantic

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equivalence of de re and special de dicto wffs in a system in which every model is an anti-essentialist model. The technique requires not just that there are anti-essentialist models, but that one rig the semantics so that every model is what Parsons (1969) calls an anti-essentialist model.4 In this way, the spirit of Russell’s structuralist unification (for first-order wffs) of logically necessary truth and logical truth (in Tarski’s semantic sense) is realized. Russellian analyses cannot rest here. This by no means exhausts what one wants from a scientific method in philosophy. We are after invariance analyses of the kinds of necessary truth and possible truth that allow de re claims about invariances over possible worlds. The following has become paradigmatic: There is a unique inventor of bifocals such that it is not necessary that he invented bifocals.

The de re ascription intended here is surely intended as a physical or causal or perhaps metaphysical notion of necessity. It remains, however, an invariance notion concerning physical, causal, or metaphysically possible worlds. As such, I want to reveal that a Russellian scientific philosopher can take a good many steps toward capturing it, not dismissing it. Of course, Russell’s scientific method in philosophy must reject all invariance notions of necessary truth—logical necessity being the only genuine notion. But Principia’s monolithic cpLogic, in fact, is capable of studying any kind of structure and, as we shall see, that includes the notions of non-logical necessary truth as studies of invariance over (physical, causal, or metaphysically) possible worlds. The task ahead is to show how a Russellian can reveal a fully quantificational analysis of such possible world invariance notions of necessary truth and thereby reveal that L-forms fully accommodate our understanding of them.

L-forms of Necessity In virtue of our thesis that all predicational thinking is impredicatively quantificational, I dismiss the question as to whether human cognition involves, for example, a “classical” notion of ‘not’ as opposed, say, to an Intuitionist notion of ‘not’ or a Relevant notion of ‘not.’ The question is misguided. There are no relations ‘not,’ ‘or,’ ‘all,’ ‘some,’ and so on. They do not introduce a kind of structure over which an invariance can be built.5 The very same quantificational cognitive scaffolding of

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intentionality that enables our understanding of Principia enables our understanding of “rival” modalities and logics. Impredicative concept formation enables the mind to recognize any pattern. A mind with determinate intentionality can study relations giving inference patterns of various kinds. To proceed on behalf of Russell’s scientific method in philosophy, the first point is to realize that to study a “logic” is to study a kind of invariance over a relational structure of a special sort—the sort which one designates as patterns of transition to be called “formal inference.” We can find the relational structures involved in the various “rival logics” simply by appealing to the canonical invariance semantics that have been offered for them.6 The invariance is, of course, expressed by quantifiers and thereby we have a way to capture it within the L-forms of quantification theory. This is how we capture a priori, and in one fell swoop, an understanding of the various “rival logics.” No mind whose intentionality has a quantificational scaffold can fail to understand such systems because they are invariance notions over a relational structure captured by quantification. To glimpse how, in general, L-forms for quantification enable epistemic access to any would-be rival logical system of invariance, we shall begin with an example of alethic modal systems. The first step in subsuming non-logical conceptions of necessity (and possibility) is to view them in terms of their canonical semantic interpretations, and thereby translate them using quantifiers and relations of accessibility. This is straightforward since there is a close connection between the operator ◽ and universal quantification and the operator ⬦ and existential quantification. The translation into quantification yields the following: (◽p)w =df (α)(wRα ⊃ pα) (⬦p)w =df (∃α)(wRα • pα) ((x)φx)w =df (x)(xIw ⊃ (φx)w) (p ∨ q)w =df pw ∨ qw (~p)w =df ~(pw ) (α)φα =df (x)(W(x) ⊃ φx) (∃α)φα =df (∃x)(W(x) • φx). These translations are well-known. But they leave entirely open how to render a Russellian analysis for “pw” which is unacceptable as it stands. That will be addressed ahead. Once we translate modal operators into quantifiers, we can see that the Rule of Necessitation is akin to the rule of universal generalization in

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quantification theory and thus readily knowable a priori. The Rule of Necessitation for a system S is this: (Nec) ├S φx → ├S ◽ φx. In quantification theory, for a given theory S, we expect the following: ├S φx → ├S (x)φx. Recall that where S stands for the closed wffs that are proper axioms for a (non-logical) theory, the following holds: ├S φx =df ├ S ⊃ φx. Thus, we have: ├ S ⊃ φx → ├ S ⊃ (x)φx, where x is not free in S. This then explains the quantification rule for the theory S. The purely logical status of the Rule of Necessitation now becomes clear. Notice, however, that universal generalization in quantification theory requires that we have ├ ψx, where ψx is a logical truth (e.g., a tautologous form or instance of the axiom schemas of quantification theory). And of course, if ψx is a logical truth, then so also is S ⊃ ψx. Consider the analogous justification of a Rule of Necessitation for a modal theory S. The rule for S is this: ├S αRw ⊃ (φx)w → ├S (w)(αRw ⊃ (φx)w). Similarly, we have ├ S .⊃. αRw ⊃ (φx)w → ├ S .⊃. (w)(αRw ⊃ (φx)w). Of course, every Russellian in the Principia era will be concerned about whether any instance of S .⊃. αRw ⊃ (φx)w is a logical truth. The expression (φx)w must be further analyzed away. We will come to this shortly. For the present, it becomes clear that when “◽p” is interpreted as a universal quantification, the Rule of modal Necessitation is forced upon us as a piece of pure logic knowable a priori. It is simply a form of universal generalization in quantification theory.7 The same logical status belongs to the following ubiquitous axiom K. That is, after translation, K follows simply from quantification theory alone. We have:

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K: ◽ (p ⊃ q) .⊃. ◽ p ⊃ ◽ q The translation of K is just this: (α)(wRα ⊃ (p ⊃ q)α) .⊃. (α)(wRα ⊃ pα) ⊃ (α) (wRα ⊃ qα). This follows from pure quantification theory. Thus, we see that both Rule (Nec) and K are disguised features of pure quantification theory. Different propositional systems of modal logic are next built up from the propositional calculus (pc) together with Rule (Nec) and K, and various axiom schemas such as the following: D: ◽ p ⊃ ⬦p T: ◽ p ⊃ p S4: ◽ p ⊃ ◽◽ p S5: ⬦p ⊃ ◽⬦p B: p ⊃ ◽⬦p . Many other systems can also be formulated. Consider translation of D which yields this: (α)(wRα ⊃ pα) ⊃ (∃α)(wRα • pα). Observe that this is not an instance of the form ‘(α)φα ⊃ (∃α)φα’ of classical quantification theory. Similarly, translation of T yields: (α)(wRα ⊃ pα) ⊃ pw. This is not an instance of the form (α)φα ⊃ φα of quantification theory. It requires that R be reflexive. From this perspective, we can see that different systems make different assumptions governing the accessibility relation R. Thus, for example, we get: reflex(R) ├ (◽p ⊃ p)w trans(R) ├ (◽p ⊃ ◽ ◽ p)w trans(R), symm(R) ├ (⬦p ⊃ ◽ ⬦ p)w symm(R) ├ (p ⊃ ◽ ⬦ p)w All these are knowable a priori precisely because quantification theory is knowable a priori. There is no special kind of epistemic access to such non-logical alethic modalities. Access to quantification theory is sufficient for the whole of it. With this in place, we see that the L-forms that enable the understanding of quantification enable their understanding too. Russell’s Theory of Knowledge as repaired by L-forms offers an account which presupposes the intentionality of mind and its ability to reason. It

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offers no pretense of explaining the mind’s rationality. It does not claim that rationality consists in the transitions of any system of rules that are valid (i.e., invariant over a preferred structure). It claims only that intentionality and reason have an impredicative quantificational scaffold which provides our synthetic a priori cognitive access to Principia’s revolutionary (non-Fregean) Logicism. We can thereby study any kind of relational structure there is, and that includes all the modal logics, paraconsistent and relevant logics as well. Compare Hossack (2007) who maintains that a Rationalist must model a priori knowledge as an S4 modal logic. That would be akin to the Kantian claim that the mind is scaffolded by Euclidean geometric intuitions! It is quite clearly false since we study all manner of non-Euclidean geometric systems a priori. And S4 is, in fact, not knowable a priori. What is knowable a priori (once translated into quantifiers) is that S4 is a theorem from assumptions that the accessibility relation R is both reflexive and transitive. (This yields the model-theoretic knowledge that S4 is invariantly true in every reflexive+transitive frame.) Hossack wants to scaffold the mind with Intuitionist proof theory. (If true, it feels self-undermining since that very thought would seem to be free of such a scaffolding.) He holds that there is a “correct modal logic” governing a priori knowing and it is KT4 [i.e., S4 built from axiom K, T, and S4]. A “proof,” as Hossack defines it, is a sequence of facts (Hossack 2007, p. 132): on the rationalist theory’s conception, a proof is not language dependent; so we define a proof as a sequence of facts, each of which is either primitive or a logical consequence of earlier facts in the proof. (This notion of proof is idealized; it is not suggested that a human being can follow every such ‘proof’.) Every a priori fact has a proof. For we defined the a priori as the primitive facts and their logical consequences, so every a priori fact is a descendent under the logical consequence relation of some primitive facts. If it is a descendent, there is a family tree of its descent; this ‘tree’, itself a complex fact with facts as constituents, is a ‘proof’ of the fact.

With these notions of proof and fact in place, Hossack needs the Quinean apparatus for the elimination of variables so that he can make intelligible the metaphysical constituents of general facts and molecular facts as truth-­ makers. He hopes that this is knowable a priori and in this way he buttresses his hope that intuitionistic arithmetic is a priori as well (op. cit., p. 138). He favors a Peano Arithmetic couched in a first-order quantification theory embellished by an omega-rule (controlling for non-standard

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models). Reason, on this view, consists in S4 invariance transition behavior protocols, and thus he fetters the a priori to Intuitionist mathematics. Hossack’s conception of ‘proof’ is, of course, completely inadequate to Principia’s cpLogic where we find an apparatus of impredicative comprehension that makes mathematical logic a genuinely synthetic a priori science. cpLogic is not so much as mentioned by Hossack and the entire cognitive apparatus of impredicative quantificational concept formation is lost. The essentially impredicative notions, which makes cpLogic informative and enables a Logicist understanding of number, ancestral, and every Cantorian diagonal argument, are out of Hossack’s reach. So also far out of reach is our obvious ability to understand modal systems other than S4 and relevant and paraconsistent and many other rival logics. Intentionality and reason by no means consists in “inference patterns.” Intentionality involves the application of relational structures to different fields to achieve different goals. Our Minds may study relational structures of S4 and determine their usefulness in some fields, and it may study relevant entailment and perfectly well conclude that in dealing with fields such as “information” (however it is to be defined), relevant entailment transition patterns better guide one’s goals. It is meaningless to speak of a “correct” relational structure. That is as misguided as speaking of a “correct” geometry. Our intentionality studies all relational structures. The Russellian account of our epistemic grasp of the synthetic a priori finds its foundation in acquaintance with universals together with the quantificational L-forms that scaffold impredicative quantificational thinking. This approach permits a priori knowledge about all kinds of systems of invariance, including those for non-­logical necessity, be it a Kripkian metaphysical necessity or otherwise. But this is knowledge about logical necessities concerning the invariances with respect to various kinds of relational structures. Thus, for example, we saw that the Rule of Necessitation and K are none other than quantification theory itself. Only logical necessity is knowable a priori. Non-logical necessity notions are not genuine necessity. They are invariance notions. The cognitive apparatus of quantification is the key. But so far our translations have not yet included the treatment of de re expressions such as {(x) ◽ Gx}w and their rival semantic interpretations that are involved in rival quantified modal logics. In extending them to a treatment of such de re expressions, we do well to observe that nothing in the pure logic of quantification theory militates for or against various semantic intuitions about what objects are in what worlds. In sorting out the issues, one certainly seeks semantic completeness so that all and only those the wffs regarded as universally valid under a given

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semantic intuition are captured as theses (axioms or theorems) of a quantified modal logic tailored to such semantic intuitions. However, the privileged status of Principia’s logic over various modal invariance “logics” demands that one’s quantificational apparatus does not change with changes in the rival semantic intuitions and changes in the deductive systems that capture them. One semantic intuition is that all worlds have the same objects in them, and another is that there should be an “inclusion requirement” according to which if a world is accessible to another, so that wRα, then the objects that are in world w are in world α. Yet a third semantic intuition is that there should be no restrictions whatever on what objects are in what worlds. Given that Principia’s logic must lord over all such semantic intuitions, one must find a stable quantification theory independent of them all. Principia had adopted this: *10.1 (x)φx ⊃ φy, where y is a variable free for x in φ.8 This wants modification to comport with different semantic intuitions about what objects are in what worlds. In a non-logical modal system with a Rule of Necessitation, *10.1 is not appropriate. It would allow the theorem ├ (∃y)(x = y), which by necessitation and universal generalization and necessitation, would yield the following: ├ ◽ (x) ◽ (∃y)(x = y). This cannot be allowed as a theorem of a stable quantification theory that is independent of the semantic intuitions governing objects in worlds. The task of finding a stable quantification theory independent of the rival semantic intuitions is readily solved by replacing *10.1 by the following: **10.1 (y)( (x)φx ⊃ φy), where y is a variable free for x in φ. The revised system retains the usual classical9 theorems: ├ (x)φx ⊃ (∃x)φx ├ (∃x)(x = x). Accepting **10.1 instead of *10.1 stabilizes the quantification theory making it independent of the various semantic intuitions governing objects in worlds. We only get ├ (x)(∃y)(x = y) and thus by the Rule of Necessitation we get ├ ◽ (x)(∃y)(x = y). All is well. With this settled, there is no longer a difficulty with accommodating rival semantic intuitions. Consider, for

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example, the following Carnap-Barcan wff which is universally valid in some semantics and not in others: (x) ◽ φx ⊃ ◽ (x)φx Different semantic intuitions of what is universally valid are captured by different axiomatizations, but none have any bearing on the stable quantification theory that holds independently of all such semantic intuitions. The same holds for the converse of the Carnap-Barcan wff which is this: ◽(x)φx ⊃ (x) ◽ φx. The Carnap-Barcan wff is universally valid in those semantics in which all worlds have the same objects and thus the wff needs to be added to the deductive systems which endeavor to capture all and only the wffs universally valid in such a semantics. Similarly, the converse of the Carnap-Barcan wff is universally valid in a semantics with the inclusion requirement, and thus it must be added to those deductive systems endeavoring to capture what is valid under the intuitions of that semantics. Some semantic interpretations of “◽p” are appropriate to the Carnap-­ Barcan wff and some are not. There are no “correct” ones and “incorrect” ones. These are intuitions governing rival invariance structures and objections concern only which is appropriate in applications.10 None of this has any bearing on logical necessity and quantification. The metaphysical controversies stem from turning a blind eye to the fact that “◽p” has many different meanings. When it means “logically necessary truth” it concerns logical structures, and their logical forms are entirely different from those involved when “◽p” concerns non-logical alethic modality (physical, biological, etc.), with entities at worlds which may or may not be accessible to one another. I fear that equivocating on the notion of “necessity” is one of the primary causes of disagreements among metaphysicians over these matters. Russell’s scientific philosophy is again the arbiter. One system may wish to adopt the converse of the Carnap-Barcan wff in the form: (∃x1),…(∃xn) ⬦Φ(x1,…,xn) ⊃ ⬦(∃x1),…(∃xn)Φ(x1,…,xn) alternatively, ◽ (x1),…(xn)Φ(x1,…,xn) ⊃ (x1),…(xn) ◽ Φ(x1,…,xn), where Φ(ξ1,…,ξn), is any wff of the language of physics that comprehends a physical configuration. It would be confused to try to offer the following as if it were a counter-example: (∃x)⬦~(∃y)(x = y) ⊃ ⬦ (∃x)~(∃y)(x = y).

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Quite clearly, there is no physical or biological configuration ~(∃y)(ξ = y). Identity is part of cpLogic and cpLogical necessity is not an invariance notion concerning possible worlds. Like mathematics, cpLogic transcends the modal metaphysics of invariance altogether. With different non-logical alethic notions of “◽p” come different wffs of disambiguation. Whether or not the Carnap-Barcan wff is appropriate depends on the notion of “◽p” involved. Suppose one is interested in physical “necessity.” Now suppose one is a combinatorialist who requires that accessibility between physically possible worlds be constrained by all physical complex entities being arrangements of the same fundamental particles (quarks, electrons, etc.). One may then endorse the Carnap-­ Barcan wff as appropriate: ⬦ (∃x1),…(∃xn) Φ(x1,…,xn) ⊃ (∃x1),…(∃xn) ⬦Φ(x1,…,xn) alternatively, (x1),…(xn) ◽ Φ(x1,…,xn) ⊃ ◽ (x1),…(xn)Φ(x1,…,xn), where Φ(ξ1,…,ξn) is any open wff of the language of physics that comprehends a physical configuration. For example, we may observe that it is physically possible that some quarks, electrons, and so on are configured so that, say, Venus has a natural satellite. It follows that some quarks, electrons, and so on are such that it is physically possible that they be so configured. There is nothing untoward about that—when couched in its stipulated physics. Observe that it engages in the fallacy of equivocation to try to criticize such a combinatorialist definition of “physical possibility” by noting that it is surely possible that there be more, less, or different physical atoms than there are. That introduces a different notion of “possible” (logical, epistemic, etc.). The same points apply to the case where a biologist interprets “◽p” as involving a biological notion of “necessity,” and her combinatorialism demands, say, that all living biological organisms be composed of cells. Given the stipulated biological theory at hand, it is not biologically possible for a living organism to fail to be composed of cells, but it may yet be (physically, epistemically, even metaphysically) possible. As we can see, the first step toward accommodating the use of our L-forms for understanding non-logical alethic modality is to realize that, unlike logical necessity, they are notions of invariance which are expressed by translating the modal operators into quantifiers. The first step, however, is not sufficient since it does not provide a full translation. Russellians rejecting propositions obviously require that the notation “pw” be eliminated. How then can a Russellian in the Principia era proceed? Lewis (1968) has shown the way—for non-logical alethic modalities. Lewis’s

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counterpart theory offers appropriate logical forms to analyze away “pw” altogether. Lewis does this by realizing that the key feature of non-logical alethic modality is invariance relativized to comparative similarity. Thus, Lewis rejects the notion that there are propositions obtaining in worlds. Instead, Lewis has a relation yCx which says that y is a counterpart of x and means that no other object is more similar to x than y. Of course, similarity is always with respect to a feature that sets the parameters of the reference class of similarity. There are different and each quite legitimate ways of setting the parameters. To assure the plurality of the needed reference classes, Lewis imagines worlds, writing Iyw to say that y is at world w. Lewis places counterparts in worlds, not states of affairs or other abstract particulars. In writing yCx • Iyw Lewis is saying that y is in reference class w and no other object in that reference class is more similar to x than y. Only counterparts are in “worlds” and no counterpart is in more than one world. There are no propositions. Lewis’s worlds are not properly understood as parallel universes.11 We have (◽Fx)w =df Ixw • (α)(wRα .⊃. (y)(yCx • Iyα .⊃. Fy)) (⬦Fx)w =df Ixw • (∃α)(wRα • (∃y)(yCx • Iyα .•. Fy)). Lewis’s translation procedure does not create difficulties for the rule of necessitation understood in terms of universal generalization. Notice that if φx is a thesis then so is φx′. That is, we have: ├ φx → ├ φx′. Thus, we have: ├S αRw • Ix′w • Cx′x .⊃. φx′ → ├S (w)( αRw • Ix′w • Cx′x .⊃. φx′). With Lewis’s translation procedure, we are closer to the quantificational logical forms demanded by a Russellian analysis of non-logical alethic modality. Lewis’s techniques require Russell’s theory of definite descriptions. To give an example, consider the following de re modal statement and its transcription into Lewisian counterparts: The inventor of Bifocals might not have invented bifocals ([ιxBx][⬦~Bx])w

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Transcribing this into Lewis’s counterpart theory, we have the following: (∃x)(xIw • Bx • (y)( Iyw • By. ⊃ . y = x)• (∃α)( wRα • (∃y)(Iyα • yCx • ~By))). Objections to Lewisian translations sometimes fail to pay attention to scope ambiguities that can arise with definite descriptions. For example, Plantinga (1974) famously argued that Lewis’s translation is problematic because it fails to capture the falsehood of: Socrates is not necessarily identical to Socrates. Plantinga didn’t stop to notice that this statement is ambiguous and thus the adequacy of Lewis’s translation is adequate if some one or another translation is false. He certainly does not need to make them all false. Using Russell’s theory of definite descriptions, we get the following primary scopes: [ιxSx][ ~◽ (x = x)] [ιxSx] [ιySy][ ~◽ (x = y)]. They are logically equivalent on the standard Kripke-style modal semantics, but their translations in Lewis’s theory of counterparts are not logically equivalent because a person Socrates may have two distinct counterparts in a world. But so be it! Lewis’s counterpart language has an expressive power (concerning counterparts) that is not available without it. There is no good reason to think that Lewis’s counterparts are inadequate to capture nonlogical alethic modalities once we accept the elimination of modal operators in favor of quantification and the notion of comparative similarity. There is an interesting argument by Williamson and Fara who would have us believe that Lewis’s counterpart interpretations are inadequate once we recognize the need for an actuality operator in quantified modal logic.12 The case concerns the question of capturing: All who are rich might have been poor. It won’t do to put: (x)(Rx ⊃ ⬦Px). This says that for each person who is rich there is some world at which that person is poor. The original says something different, namely that all who are rich might have been poor together—that is, in the same possible world. To deal with the problem, Williamson and Fara write: ⬦ (x)( @Rx ⊃ Px).

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This introduces an actuality operator @[p]. This, of course, is incompatible with what a counterpart theorist allows as intelligible since it permits particulars to inhabit more than one world. Happily, the Williamson and Fara objection to Lewisian translation is misguided since we have: (x1, …, xn)( Rx1 • … • Rxn .⊃. ⬦ ( Px1 • … • Pxn)). The above has a straight transformation into Lewis’s counterpart language. Thus, in a finite case, however large, the issue is resolved. There remains the infinite case. The problem here derives not from modality, but from the fact that no first-order language can express the togetherness of infinitely many without appeal to sets. The impetus for an actuality operator in first-order modal language comes from the demand of keeping the expressive limitations of a first-order language itself and thus has no bearing on the issue of modality. If one’s language binds predicate variables, no such problem can occur. The same point applies against Melia (2003, p.  31) who objects that quantified modal logic cannot accommodate a statement such as: There are three ways Joe could win his chess match. A group of non-coexemplifying properties, winning Joe’s chess match doing A, and winning Joe’s chess match doing B, and so on, can be counted. That is, to count the ways of winning Joe’s chess match is to count how many such non-coexemplifying properties there are. No first-order language without set notations can count the ways because it cannot express the counting of properties. Lewis’s counterpart theory judiciously not only rejects the notion of an entity inhabiting more than one world but also analyzes away the expression pw. In counterpart theory, entities have properties and stand in relations simpliciter, not at worlds.13 The transcriptions are an important ally of the Principia era thesis that there are no propositions (and thus that there are no propositions at worlds). Lewis has provided important Russellian steps toward finding the correct logical form for the transcription of non-logical invariance modalities into a quantification theory. The Russellian spirit of Lewis’s theory is important. Merricks lost sight of this when he objected that Lewis does not provide for actual truthmakers for (non-logical) de re modal statements. He writes (Merricks 2007, p. 100): a corollary of Truthmaker is that, in general, actual truths have actual truth-­ makers. But Lewis cannot satisfy this corollary of Truthmaker when it comes to claims of de re modality.

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This imagines that a winged Secretariat counterpart existing at a metaphysically possible world is what Lewis must take to be the truth-maker for “It is possible that Secretariat was a winged horse.” This is mistaken. Moreover, Lewis does not embrace states of affairs residing in possible worlds. It is only entities that are counterparts that are in Lewis’s possible worlds, and according to Lewis, it is a fact (an actual fact, there being no other sort of fact) that the truth-conditions hold concerning there being such counterparts. Observe that a counterpart is at a world, but it is meaningless to speak as if it exists at a world. Lewis fully agrees with the Russellian that the expression (x exists)w is meaningless. Russellian scientific philosophers should embrace Lewis’s logical forms for non-logical alethic modality. Of course, they are also in earnest to reject his interpretation of those logical forms as being about possible worlds. It is Lewis’s ontology that Russell’s scientific method in philosophy must reject, and it is certainly an ongoing research program to find a new interpretation. The notion of a possible world and a counterpart, Lewis reminds us, are notions intimately involved with the notions of relevant reference class and comparative similarity of entities. Since there is already a vast array of reference classes and comparative similarities between entities, one may wonder whether there is any significant reason to worry that they are inadequate to the task of providing all the truth-makers for nonlogical necessity (and possibility) in their various senses. The central question, of course, is which similarities are relevant and this is an ever difficult matter and naturally gives rise to disputes. This is not entirely due to philosophical issues but arises because of disputes in the empirical sciences themselves. Is it biologically possible for Secretariat to have been a winged horse? Perhaps the relevant similarity concerns creatures like the platypus with reptilian and mammalian features. Perhaps the place to look is the complex interactive chemical processes that turn genes on and off and their involvement in the production of phenotypes. The relevant similarity class for such genetic processes has to do with genes and for this we don’t need there to be worlds in which there inhabits a grotesque counterpart of Secretariat that has wings instead of front legs. Relevant counterparts may well be among us, each inhabiting strictly different relevance classes (not Lewisian possible worlds). Typically, it is our ignorance of the empirical facts in question that drives intuitions that worlds and counterparts are needed to explain the truth-conditions for non-logical alethic modality. As empirical science improves, these intuitions will diminish. The lesson is that Russellians can embrace Lewis’s logical forms for non-logical modality. They transform non-logical modalities into quantificational statements.

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Lewis (1973) has also shown the way for the Russellian scientific philosopher to treat were-would subjunctives (and counterfactuals). Lewis has done a great service in having found an acceptable logical form for such statements.14 He offers this: (p ◽→ q)i=df (∃w)( pw is accessible to i) ⊃ (∃w)( (p • q)w • (w′)((p • ~q)w′ ⊃ w ≤i w′). Thus, an analysis of counterfactuals can fit within a Russellian framework if a Lewis-style account of non-logical modalities can. It should be noted that this illustrates the solution of another concern Merrricks raises about the correspondence theory of truth—namely the concern that the correspondence theory must introduce a heavy bit of metaphysics to call out “cheaters” who find easy properties and relations and facts in which they inhere as truth-makers whenever challenged by a difficult case. For example, consider the truth-maker for the belief-fact that exists when it is the case that M believes that (Fa ◽→ q). A cheater’s strategy is simply to regard the truth-maker for the belief-­fact to be a fact of a exemplifying G, where G is such that (x)(Gx .≡. Fx ◽→ q). Merrick’s quite legitimate concern about cheaters cannot arise for Russell’s Theory of Knowledge approach as we have repaired it. By transforming the subjunctive wff Fa ◽→ q into a quantificational expression, we can find the complex Russellian belief-fact involved. But when we do, all the universals involved as relata have to be objects of acquaintance and certainly no such property as G will be admitted.

The Necessity of L-forms Once alethic modality is subsumed into quantification, we can see that any such apparently “rival” logic built from such modal notions is not genuinely a rival at all. They are just studies of different structures and perfectly cognitively accessible by the L-forms that enable our understanding of quantification. The case of Intuitionism is immediately handled by the fact that it is an S4 modal alethic system which we have already treated. Where p and q are atomic wffs not containing any of the intuitionistic signs ⊃i, ~i, &i, ∨i, the transcription is just this: p ⊃i q =df ◽ (◽p ⊃ ◽q) ~i p =df ◽~ ◽ p p ∨iq =df p ∨ q p •iq =df p • q.

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The signs ∨i and •i intuitionism parallel the classical “∨” and “•.” Because of this, there is no need to indicate a change in sign in these cases. Intuitionism accepts p ⊃i ~i ~i p i.e., ◽ (◽p ⊃ ◽ ~ p).15 And we can also understand how it is that Intuitionism does not accept: ~i ~i p ⊃i p i.e., ◽ (◽ ~ ◽ ~ p ⊃ ◽ p). These are just ordinary features of S4. Indeed, the following is not intuitionistically invalid: (p ∨ q .•i. ~ip :⊃i: q i.e., ◽ (◽ (p ∨ q .•. ◽ ~ ◽p) ⊃ ◽ q). By employing L-forms, the mind can recognize and apply intuitionistic structures. They are captured by the L-forms of quantification. Relevant entailment structures, however, take us outside of alethic modal systems. It is hard to imagine either subsumes the other. They seem incommensurable. At first, one might imagine that relevant structures might be able to subsume alethic modal structures. Where the relevant logic expression pa is understood as saying that p is information at state a, one might imagine a possible world being a maximally complete consistent state of information. Many states of information are neither maximally complete nor consistent. But it is easy to see that the notion of “maximal” is not expressible in the object language expressions of the formal language of relevant entailment. Capturing the notion of a possible world seems out of reach. The L-forms of cognition, however, fully capture relevant entailment because it can be expressed quantificationally and Relevance is just a form of invariance. Thus, our strategy applies. We can reconstruct its formal system quantificationally by considering its standard Routley-Meyer semantics. Since it is couched in an ordinary quantification theory, it is fully cognitively available to our L-forms. To see this, note that relevant entailment has a triadic accessibility relation R, and different systems will be built upon what axioms govern it. The following are among the usual fundamental axioms and rules, where “o” designates a basis for access to classical logic: Idempotence: (x)Rxxx Star1: From Rabc infer Rac*b* Identity (x)Roxx Star 2: b** = b

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Monotony: From Roab, Rbcd infer Racd Inheritance: From Roab, Aa infer Ab Commutation: From Rabc infer Rbac Now if one were to add the axiom (x)Roox, the relevant system yields (x)Roxo and is said to “collapse” into the classical inference patterns (replete with explosion). So obviously, this axiom is not desired. In relevant entailment, Roxy can be interpreted to mean that all the information at information state x is at state y, where o is a very special index of transfer of information. As noted, it seems natural to interpret the superscript as in Aa to indicate that A is a bit of information at state a.16 . Of course, a complete transcription of relevant entailment into quantification requires the elimination and analysis of Aa, i.e., of the notion of A being a bit of information at information state a. Unfortunately, since the very notion of information itself is distinctive of the intentionality of mind and since the nature of the mind remains unknown, this cannot yet be further analyzed. But in any case there is no need, Priest’s Dialetheism notwithstanding, to be engaged in a quest to make some notion of an “impossible world” intelligible. Contradictory information certainly does not produce an insurmountable problem for reasoning that uses an inconsistent state of information. How do we do it? The signs ∨r and •r of relevance parallel the classical “∨” and “•.” That is, we have: (A ∨r B)b =df Ab ∨ Bb (A •r B)b =df Ab • Bb. Because of this, there is no need to indicate a change in sign in these cases. But the relevant expression ~r p is quite special. For convenience, I’ll use ¬p instead of ~r. In the standard semantics of relevant entailment + negation, an expression such as (~p)a is meaningless. Instead, “conflicting” (as it were) information at a situation is expressed by the relevant hook. The semantics offers this (¬A)b =df ~(Ab*). This is the infamous Routley star. In this way, relevant entailment blocks “explosion” (i.e., in reasoning relevantly, not everything follows from conflicting information). The relevant entailment sign is also a special sign. It wouldn’t do to write ⊃r because there is no such thing as a

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“classical conditional” sign. There are no conditionals at all in Principia, which has p ⊃ q =df ~p ∨ q. The Relevant logician, therefore, is free to argue that the natural language “if… then” is very often intended as a relevant entailment. Let’s introduce the arrow for it with the usual relevant semantics: (A → B)b =df (x, y)(Rbxy .⊃. Ax ⊃ By). Now the expressions “(p ⊃ q)x” and “(~p)x” are meaningless and what is required are “(p → q)x” and “(¬p)x.” Observe that the sign “→” is a relation sign. The classical sign “⊃” is not a relation sign. To study relevant entailment is to study a structure by studying a triadic relation R and the axioms governing the relation → and its connections with the operation. The relevant-valid wffs are such that (A)o. Various relevant systems can be developed by adopting special axioms needed to recover desired theorems. Consider for example the following: r-(syll) ├R (A → B :  →  : B → C. → . A → C)o. This is not forthcoming unless one embraces an axiom such as the following: (∃t)(Rabt • Rtcd) ⊃ (∃t)(Ratd & Rbct). Of course, one may not want r-Syllogism. It depends on the kind of structure one is endeavoring to capture. If one wants to capture classical inference patterns, one can adopt the axiom (x)Roox which will enable the system to yield all the analogs of the usual tautologies of classical logic, including: (A •  ¬ A . → . B)o (A ∨ B . → . ¬ A → B)o. The relevant logician agenda, of course, is to avoid classical collapse and sift for the entailments appropriate to the goals of one’s applications. The goals of using relevant systems usually require blocking certain undesired classical inference patterns. Consider the following theorem: r-(mp) ├R (A . • . A → B :  →  : B)o i.e., (x, y)( Roxy .⊃. (A . • . (A → B))x ⊃ By ).

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By Inheritance, this yields a derived rule of relevant Modus Ponens. r-(MP) From Ab and (A → B)b infer Bb. But we can see that this rule is entirely unrelated to Modus Ponens MP From A and ~A ∨ B infer B. Modus Ponens is just a version of disjunctive syllogism. Interestingly, it is perfectly acceptable to embrace relevant Disjunctive Addition: r-(Add) From Ab infer (A ∨ B)b. Indeed, there is no expectation that we should have From (A → B)b infer (¬A ∨ B)b. And in relevant entailment systems, we find: ⊬R (¬A . • . A ∨ B :  →  : B)o. Relevant logic categorically rejects the validity of the following: From (¬A)b and (A ∨ B)b infer Bb. This should be of no surprise whatever because it has no relation to Disjunctive Syllogism. Indeed, the relevant logician fully accepts the following instance of Disjunctive Syllogicsm: ~(A)b, (A ∨ B)b ├ Bb. Moreover, it also accepts explosion in this form: ~(A)b, (A)b ├ Bb. This explosion is innocuous because it is not an explosion at an information state. Note, however, that the correct thing to say in response to a relevant logician is not that the rules governing the classical “⊃” are the correct meaning of this relation sign. There is no such relation sign and indeed neither is “∨” or any of the logical particles proper to regard as relation signs. And the notion of a “correct” meaning is misguided. Cognition enables an understanding of the relevant “¬” just as readily as anything. Again the key point is that relevant entailment is an invariance notion that would be impossible to understand without our capacity to understand quantification. Our cognition of the kinds of structures that relevant “logic” brings to the table is no different from our cognition of other

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kinds of structures. They are, one and all, understood by the same quantificational cognitive apparatus of L-forms. There is no conflict between relevant logics and modal logics and any other kinds of relational structures of invariance that we understand and study. Which structure is most suitable to apply may remain, as always, an empirical question. Contrary to Hossack, a priori reason does not consist in the constraints of S4 transition patterns, and it doesn’t consist of relevant entailment transition patterns either. Indeed, it is not constrained by anything at all—or better, the point is that it is meaningless to imagine quantification and the L-forms scaffolding intentionality being a constraint. With a sketch of the Russellian treatment of modal and relevant logic set out, there remain issues concerning rival “logics” of probability. Given the constructions already sketched, mathematical notions of “probability” do not pose any serious threat to the stance against abstract particulars that scientific method in philosophy takes. Notions of “probability” involving physical propensity (or frequency) would be subsumed into theories involving modal logics of counterfactuals and subjunctives. Special difficulties may well arise, however, with the work of Keynes (1921) according to which “probability” is to be interpreted in terms of the credence that ideally rational creatures would recognize as holding between the informational contents of their beliefs. Keynes imagines that there are propositions (as the contents of beliefs) and that between any two such Keynespropositions, h and e, either of which may be molecular, there obtains a relation of conditional probability c(h/e) which measures the degree of rational credibility that h has relative to evidence e. Keynes-­propositions are very like bits of information and are thus caught up in the nature of intentionality. In any case, he held that c(h/e) may not always be measureable in numerical values and that such probabilities are only partially ordered. Carnap (1950) offers a similar notion which distinguishes itself from mathematical and frequency/propensity notions as a notion of “logical probability.” Capturing his notion of “probability” will require that the propositions that stand in Keynes’s relation ‘c(h/e)’ be eliminated; it will be particularly difficult since instances where “h” and “e” are molecular wffs are certainly allowed by Keynes. Perhaps the best approach is to find a way to subsume Keynes’s theory of probability as degrees of rational credence into a relevant entailment structure involving transitions between information states. The development of a reconstruction of Keynes’s theory of probability will have to be left for further work.17 Much remains to be done. But with the L-forms in place, there seems no serious impediment to doing it.

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Time for Scientific Method in Philosophy Principia construes magnitudes, including temporal magnitudes as “vector” relations—relations whose instantiation is repeatable. There are many different physical temporal magnitude relations of periodicity (e.g., pendulum swing, diurnal rotation, electrical oscillation, etc.). The swing of a pendulum from position x to y (i.e., Pxy) and then from y back to x (i.e., Pyx) is a vector relation xP/Px. Physical temporal relationships of comparative periodic magnitude constitute physical time. There are no abstract particulars that are times or events that are assumed in measurement of magnitudes. That is antithetical to Principia’s agenda, and the work is not committed to the existence of events as concrete particulars. Admittedly, in Whitehead’s Introduction to Mathematics, he speaks of periodicity in nature in terms of events, writing (Whitehead 1911, p. 166): we can not only … say A came before B and B before C; but also we can say that the length of time between the occurrences of A and B was twice as long as that between B and C. Now quantity of time is essentially dependent on observing the number of natural recurrences which have intervened. … The determination of the broad general consistency of the more important periodicities was the first step in natural science.

In Principia’s discussion, however, events are never mentioned. We find (PM, vol. 3, p. 338): We conceive of a magnitude as a vector, … i.e., as a descriptive function… Thus for example, we shall so define our terms that 1 gramme would not be a magnitude, but the difference between 2 grammes and 1 gramme would be a magnitude, i.e., the relation “+1 gramme” would be a magnitude. On the other hand a centimeter and a second will both be magnitudes according to our definition, because distances in space and time are vectors. It will be remembered that we defined ratios as relations between relations; hence if ratios are to hold between magnitudes, magnitudes must be taken as relations. …We demand of a vector (1) that is shall be a one-one relation, (2) that it shall be capable of indefinite repetition.

The point is expanded in the following passage (PM, vol. 3, vi). In our theory of ‘vector families,’ which are families of the kind to which some form of measurement is applicable, we have been able to develop a

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very large part of their properties before introducing numbers; thus the theory of measurement results from the combination of two other theories, one a pure arithmetic of ratios and relation numbers with reference to vectors, the other a pure theory of vectors without reference to ratios or real numbers.

Magnitudes are relations and measurement concerns magnitudes. There are mathematical measure relations on magnitudes which can be studied a priori. Physical universals that are magnitudes exist, as do mathematical universals, quite independently of their instantiation. The pure mathematics of measurement tells us nothing about whether or not a given magnitude can be measured in isolation from others. Einstein’s revolutionary ideas cannot impact the mathematics of measurement, but it can demand that the measurement of a physical temporal magnitude not be conducted in isolation from consideration of other magnitudes such as mass and acceleration and electromagnetism. Principia’s general mathematics of the measurement of a magnitude, as indicated in vol. 3, shows no signs of making a special place in the theory for the measurement of temporal magnitudes as opposed to other magnitudes. By 1912 Whitehead was engaged with Einstein’s special relativity and its orientation to physical events. It would not have impacted his thinking about pure mathematics as an a priori study of relational structures. It might, however, have influenced his discussion of the mathematical features belonging to the general theory of the measurement of magnitudes. Unlike the mathematics of space (geometry), time is excluded from the mathematical study of kinds of structures. Temporal relationships, like the contingent relations between universals ‘human’ and ‘mortal,’ arise from the contingencies of the exemplification of physical relations. The structures that emerge cannot be studied a priori since they depend essentially on the nature of their being exemplified. Tooley (1997) gives a telling argument that quantification is tenseless.18 I should like to go further. Scientific method in philosophy is committed to the meaninglessness of tensed quantification. Given our fusion in Chap.  6 of the logical particles with quantification, this tenselessness belongs to the logical particles as well. Quantificational thinking cannot in any intelligible way be said to be “restricted.” Any attempt at restriction has to be structural (syntactic) and, as Russell remarked in 1906, that is incoherent. The quantifier “every” is not more restricted in saying (x)(Hx ⊃ Mx) than in saying (x)Mx. Both speak equally of every. Scientific method

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in philosophy is committed to quantification being tenseless. It is not committed to allowing tenseless predication. It is not committed to the existence of physical events as concrete particulars. And of course its agenda is opposed to a metaphysics of events as abstract particulars. Reichenbach (1947) offers tense diagrams which assume that the interpretation of a tensed statement involves a tripartite structure involving events: E (time of event), R (time of reference), and S (time of speech act): I see  R,E ,S 

 

I saw   E ,R S

I will see    S ,R  E

I will have seen   SER

Additional indications concerning duration can be diagrammatically indicated as follows: I am seeing I had been seeing I will have been seeing          E E RS S E R S ,R The hearer uses the time of the speaker’s speech act event as a temporal reference point for interpretation. Reichenbach notwithstanding, a theory of physical events is surely not forced upon us when it comes to understanding tense inflections. We understand tense independently of any contingent theory of physical events. One may wonder whether understanding of tense requires an a priori metaphysics embracing mental (phenomenal) event states as abstract particulars. But it is worth putting aside such Meinongian considerations in the phenomenology of time. The central point is that scientific method in philosophy requires that we reject events as abstract particulars every bit as much as it requires the rejection of propositions as abstract particulars. It must define away tense operators on propositions such as P(past), F(future), G(ever will be), H(ever was) of Prior (1967), and the since and until relations of Kamp (1968): p since q p until q

Pq • p has been ever since q was Fq • p • p will cease when q

Scientific method in philosophy is committed to tenseless quantification. But nothing in scientific philosophy demands tenseless predication. Nothing excludes it either, and nothing prohibits an appeal to contingent physical events. All the same, tenseless predication is required if one appeals to concrete particular events, especially concerning the

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characterization of the types of events and in speaking of relations between events at times. This makes it desirable to try to get along as far as possible without appeal to the existence of concrete particular events. Moreover, as we shall soon see in detail, any such appeal assures that there can be no analogy between rival logics of time and the rival alethic modal structures that we found can be recovered within scientific method in philosophy. Experimenting with the plan of tenselessness being found only in quantification (and accordingly its logical particles), and avoiding appeal to contingent events and tense operators, it is open to require tense inflec 19 For example, for a present tense, F tions on the predicate expressions.  for simple future, and F for simple past tense. Then one can write: . This says that everything what was, is, or will be human, was, is, or will be mortal. The quantifier is wholly unrestricted. It should be noted that Russell’s theory of definite descriptions should be followed in a universal instantiation. Principia has: *14.18 (x)ψx .⊃. E!(ιxφx) ⊃ [ιxφx][ψx]. To illustrate the use of this schema, consider these instances: (x)ψx .⊃. E!(ιx x) ⊃ [ιx x][ψx].   (x)ψx .⊃. E!(ιxFx) ⊃ [ιxFx][ψx]. The quantifiers and particles thereby alone remain tenseless and predication is always tensed. Certainly, one may go on to ask what a speaker referred to. But this is a semantic question of how to interpret the quantifiers of the speaker’s speech act (statement, etc.,) over a domain. I can ask such a question about my own speech act too. But thinking quantificationally cannot consist in semantically interpreting a speech act! To think quantificationally is not to interpret. This fact by no means gets in the way of giving a Tarski-style semantics that adopts an interpretation function which renders a given quantified wff “(x)φx” an interpretation over a fixed domain. That is perfectly acceptable. And quantificational thinking will be used in setting out the domain of any semantic interpretation. Working experimentally in an austere language which eschews event talk and which demands that all primitive predicate expressions require diacritical tense inflections, let us see whether we can represent some

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features of the main philosophies of time: Presentism, Eternalism, and Growing Block. Let us indicate the various tenses with inflections required on all primitive predicates including relation signs. For instance, we have: , (loved, loves, will love, had loved, will have loved). . In this language, the tenseless One should allow more still such as expression Lxy is not permitted. Now consider the following:   (∃x)( Px • Ax). This says e.g., that someone was a person who authorized the emancipation of slaves. Note that this doesn’t say that something now exists that was a person who authorized the emancipation of slaves. That is meaningless. In scientific method in philosophy, “exists” is not a predicate—and tense inflected “exists now,” “existed,” “will exist” are similarly meaningless. One can write:

This says e.g., that someone was a person who authorized the emancipation of slaves and no one is a person authorizing the emancipation of slaves. One can write:    (∃y)(Cy • (∃x)( A x • Myx)). This says that someone who was a confederate sympathizer murdered someone who had authorized the emancipation of the slaves. All is well. Tense comes always with the predicates. Again, it is important to remind ourselves that saying (thinking quantificationally) is not the same as rendering a semantic interpretation of one’s statement token over a domain. Of course, the question of how best to give a semantic interpretation of, say, the wff

is perfectly legitimate. But to answer it will require one to think quantificationally, speaking about what one may want in the domain of the semantic interpretation and giving an interpretation function that makes   assignments to the predicates P and as well as A and , assignments that may render different subsets of the domain (respectively). For example, a given entity in the domain may readily be a member of both of the different subsets of the domain that form the respective interpretations of the  predicates P and . Nothing requires that the domain contain Lincoln

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(since of course he is deceased), and nothing requires that the predicate letters be interpreted to mean anything but what the interpretation function assigns them to mean. As we can see, doing such a formal semantics is simply not germane to our concerns. In order to allow comparative temporal notions without appeal to propositions or events, we need tensed comparatives to capture notions such as Kamp’s until and since. For example, we can introduce the following, where Rn and Sm are relation signs:  |R | ∇ |S m | (x1,…,xn, y1,…,ym) x1,…,xn had, and ceased to have Rn precisely while y1,…,ym had and ceased to have Sm  n m ∇ R |S (x1,…,xn, y1,…,ym) x1,…,xn had Rn until y1,…,ym had Sm n  |Rm | S (x ,…,x , y ,…,y ) ∇ 1 n 1 m x1,…,xn had, and ceased to have had Rn while y1,…,ym had Sm n

Similarly, we have the following and the like: n

 |Rm| ∇ |S | (x1,…,xn, y1,…,ym) x1,…,xn will have, and will cease to have Rn precisely while y1,…,ym will have and will cease to have Sm. With such tensed comparatives in place, we can write:    (∃x)( T x • (∃y)(  C|L/C| y, y, x))). ∇ This says that someone who authored Tom Sawyer had lived from when Halley’s comet had been at perigee until when Halley’s comet was again at perigee. (Here yCz says that y is the perigee position and z the apogee position of Halley’s comet from the sun.20 The notion of tomorrow is captured also by appeal to a periodicity, in this case diurnal.) A language of tense inflected predicates with tenselessness only in the quantifiers and logical particles does not prevent the expression (in scientific philosophy) of at least some differences between Presentist, Eternalist, and Growing-Block theorists. All adhere to this: .

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Everything is such that there is a property it had, it has, or it will have. The question: “Do past/future entities exist?” is meaningless. Thus, Presentism is not properly expressed as:

It is meaningless to say that everything that existed or will exist, exists (now). One might worry that Presentism can’t otherwise be expressed. But at least some differences are substantive.21 Presentism requires the following: . Everything is such that necessarily, for every F there is some R such that if it had or will have F then some entities are standing in relation R. For example, a Presentist endorses this: . Accordingly, we have: . This says that if some person uniquely emancipated the slaves, then some person who uniquely emancipated the slaves is such that necessarily there is some relation R such that if that person was murdered then some entities are standing in relation R. This is a substantive thesis which every Eternalist would surely reject. However, to assure that the comprehension of universals doesn’t trivialize all philosophies of time, one may well need the following restriction when wffs with tense inflected primitive predicates are involved in the comprehension of universals: 

◽ (∃Rn) ◽ (z1, …, zn) ( R n (z1, …, zn) ≡ φ(z1, …, zn)), 

where R n does not occur free in the wff φ, and φ contains no quantifiers and all primitive predicates have the same tense inflections which concur  with that of R n . This would prevent:   ◽ (∃G) ◽ (z)(Gz . ≡ . Az • Mz). If the universal G were allowed via comprehension, the Presentist could trivially offer the following (in the above case of the Lincoln murder):

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With the qualification on comprehension, a Presentist cannot appeal to such trivialities. There is, however, room for disputes about the comprehension of universals. Now the block notion of the Growing Block metaphysics of time requires this: . Everything necessarily is such that if there is some property it had (or was that it will have), then there is some property it has. But clearly, we need to use tense inflected comparisons if we are to do more to capture the expression of the growing block philosophy of time. The notion of growing, whether a part of the Growing Block or Presentist metaphysics of time, is notoriously difficult to understand, but advocates may want to accept the following:

Everything is necessarily such that if there are no properties that it had, then every property is such that if it doesn’t have it and will have it, then there is some property E that it will have such that no property it will have is such that it will have it after it has E. An Eternalist denies this. As we can see, we can express at least some important difference in the Presentist, Growing Block, and Eternalist ontologies of time using our minimal language for time which has no tense operators and demands all tense inflection be on primitive properties. There is no mathematics of time, but only the physics and phenomenology of time. In Scientific Method in Philosophy, Russell seems to agree that there is no pure mathematical study of temporal magnitudes. But his Eternalism is salient in the following (OKEW, p. 166): Past and future must be acknowledged to be as real as the present, and a certain emancipation from slavery to time is essential to philosophical thought. The importance of time is rather more practical than theoretical, rather in relation to our desires than in relation to truth. A truer image of the world, is obtained by picturing things as entering into the stream from an extra world outside, than from a view which regards time as the devouring tyrant of all that is. Both in thought and in feeling, to realize the unimportance of time is the gate of wisdom.

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The mathematical unimportance of time certainly does not force Eternalism, and neither does it demand that growing is unintelligible. The nature of time is certainly not philosophically unimportant. Indeed, Russell knew that it is of concern to scientific method in philosophy because the philosophy of time is a field rich in providing apparent indispensability arguments for abstract particulars including ontologies of events and times. Whitehead and Russell thought that the clear direction of relativistic and quantum physics was toward events and away from enduring objects. The introduction of empirical theories of events impacts debates about whether there are physical objects enduring through time. The rival hypothesis is that of perdurance. On such a view, no physical object endures through time. What is called a “continuant” is an event that is a composite of temporal event parts, each of which is at a time. The notion of the “continuant at a time” refers to its part which is at that time. In his 1914 book, Scientific Method in Philosophy, Russell held that physical laws traditionally written in terms of enduring physical objects (“things”) should be reformulated in terms of the contingent existence of transient physical particular events. Russell writes (OKEW, p. 110): Now physics has found it empirically possible to collect sense-data into series, each series being regarded as belonging to one ‘thing,’ and behaving, with regard to the laws of physics, in a way in which series not belonging to one thing would in general not behave. … We must include in our definition of a ‘thing’ those of its aspects, if any, which are not observed. Thus we may lay down the following definition: Things are those series of aspects which obey the law of physics.

A physical theory of events is certainly not forced upon scientific method in philosophy whose agenda is to remove appeals to abstract particulars from physical science. Russell’s construction endeavors to rewrite the physical laws of continuants (“things”) enduring in time in terms referring only to transient particular physical events (capable of being sensed). Thus, he included sense-data as well as what he calls “unsensed sensibilia.” He respected, though rejected, a “phenomenalistic” standpoint where the rewriting is done in such a way that permits only reference to physical events of sense-data and to no unsensed transient particular events. Indeed, he also respected and rejected the phenomenalistically solipsistic view point where only one’s own sense-data are referred to in the rewriting. In every case, however, it is those transient physical events that are capable of

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being sense-data that are central to Russell’s rewriting of the laws. Transient particular events that are too complex to be capable of being sensed are not part of Russell’s constructive rewriting. Whitehead held that there is good empirical evidence that favors the view that physical relations of overlapping appealed to in measurement are at least as dense as are the Rationals, so that for any two events, there is always an event temporally in between (overlapping with neither). In his 1927 book, The Analysis of Matter, Russell criticized Whitehead’s view of events. He wrote (AMa, p. 292): Starting from events, there are many ways of reaching points. One is the method adopted by Dr. Whitehead, in which we consider “enclosure-­ series.” Speaking roughly, we may say that this method defines a [physical] point as all the volumes that contain the point. (The niceties of the method are required to prevent this definition from being circular; also to distinguish a set of volumes having only a point in common from such as have a line or a surface in common.) As a piece of logic, this method is faultless. But as a method which aims at starting with the actual constituents of the world it seems to me to have certain defects. Dr. Whitehead assumes that every event encloses and is enclosed by other events. There is, therefore, for him, no lower limit or minimum, and no upper limit or maximum, to the size of events. …The events which we can perceive all have certain duration, i.e., they are simultaneous with events which are not simultaneous with each other. Not only are they all, in this sense, finite, but they are all above an assignable limit. … There might be empirical evidence, as in the quantum theory, that events could not have less than a certain minimum spatio-temporal extent. Dr. Whitehead’s assumption, therefore, seems rash.

It is odd that Russell appeals to “the actual constituents of the world,” since his imposition of a lower temporal bound seems motivated only by his epistemological agenda. To be sure, Russell notes that quantum theory may demand minimum temporal bounds for quantum events in time. But if quantum theory requires all quantum events in time to be finite in duration, so be it. It also relies on events of quantum superposition that are not themselves situated in space-time. Indeed, it would appear that special relativity also requires ongoing events of electro-magnetic invariance which, together with privileged events of acceleration, constitute metrical events of space-time. Whitehead seems correct that physics imposes no limitations (maximum or minimum) on events.

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Whitehead and Russell do not imagine that events are abstract particulars. Scientific method in philosophy cannot accept a metaphysics of abstract particulars conjured as a part of a metaphysics of time.22 Indeed, no scientific philosophy is forced to be committed to the existence of physical events. But it can, with trepidation, accept events as empirically well-confirmed concrete particulars of a physics of the measurement of temporal magnitudes. Whitehead and Russell both viewed special and general relativistic physics as committed to concrete contingent particular events.23 Quantifiers for times have to be defined in terms of quantification with respect to events involving temporal relations between parts of events. The notation et (of an event being at a time) and the notation t < t* (of a time preceding another) are not primitives but must be defined by appealing to other events and the temporal magnitude relations between them. Thus, for example, a race event has a beginning event part happening when event e1 (a pendulum was at a given position) and has an ending part happening when e2 (the pendulum was at a position after so many repeated swings). The notation et lends itself to the formation of questions that may well be little more than pseudo-questions such as what is the temporal rate at which events are at times, and whether for a given event e, it is determinate whether e is at time t or e is not at time t. In his 1934 “On Order in Time,” Russell begins from a primitive transitive and irreflexive temporal relation ePf of an event e wholly preceding another event f. A partial overlapping relation S of events is defined thus: eSf =df ~(ePf ) • ~(fPe). This assumes that in the wff ~(ePf ), the letters e and f are special variables and thus that there are such events and that all events are situated temporally. Accordingly, Russell has the following: e(S/P)f        e begins before f i.e., (∃g)( ~(ePg) • ~(gPe ) .•. gPf ). Similarly, Russell has:  e(S/ P)f          e ends after f e(P/S)f        e ends before f  e(P / S )f          e begins after f. Sameness of temporal duration of events overlapping partly with e and those overlapping partly with f is expressed by:   S ‘e = S ‘f .

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The language of events as concrete particulars uses quantifiers, and as before, they must be understood via the tenseless cognitive apparatus of quantification. But a distinctive feature of the language of events is that it embraces not only the tenselessness of quantification but also at least some tenseless predications. And there remains as well a grave danger of falling into the allure of natural language which conjures up expressions for types of events willy-nilly. It is physics, not philosophy of language, that determines what physical events there are. One may wonder what types of events there are and whether tense inflected predicate expressions or only tenseless predications may be employed in characterizing their types. Does an event have a property of being of a type and does it have such a property at a time? Can it, the same event, lose its type so that an event of radiating can, at a later time, no longer be an event of radiating? Advocates for the contingent existence of events as concrete particulars must answer such questions, and it seems reasonable to require that there be no tense inflections on predicates occurring in any allowable wff p characterizing a type of event. Nothing in physics demands, for example, that there be an event f of event e being of type P. In an effort to evade the many confusions that wait for us at every turn, we can invent a special quantifier and write: p

pt =df (∃e)( et), where p is a wff appropriate to postulate as characterizing a type of event. p

This uses a special variable e for an event of a given type p. The reason for this special variable is to avoid the expression Pe which would speak as though event e is of the type P. For atomic p, we have: ├ pt ∨ ~(pt) p

p

i.e., ├ (∃e)( et) ∨ ~(∃e)( et)). The notation still leaves room for disagreement as to what wffs p are appropriate in a physics of time in characterizing the type of an event. 24 In the Principia era, that is a matter for physics itself to decide. There is no mathematical study of time that is in any proper way analogous to pure geometry as a mathematical study of space. There can be no a priori mathematical study of temporal metrics. That is not very surprising. More surprising is the fact that once events as metaphysical abstract particulars are banished from scientific method in philosophy, there is no study of time within Principia analogous to our studies of rival alethic modalities and rival logics. To be sure pw has to be eliminated in alethic modalities no less so than pt. But in the case of the former, a Lewisian

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treatment is available which transforms the alethic necessitation into pure universal quantification. We shall find no analog for the treatment of pt. There are no analogs of the Rule of Necessitation or the axiom schema K that we found central to modal systems and rival logics. It won’t help to appeal to Prior’s tense operators, G and H, writing rules such as: NecG ├ p→├ Gp NecH ├ p→├ Hp. In scientific method in philosophy, such operators have to be translated away without appeal to propositions or events as abstract particulars. For the scientific philosopher who rejects abstract particulars such as propositions, the expression (p)t must be defined away. The following are fine, where p and q indicate atomic or quantifier-free molecular wffs not involving negation: (p ∨ q)t =df pt ∨ qt (p • q)t =df pt • qt. But consider next the following: (Gp)u =df (t)( t > u .⊃. pt) (Hp)u =df (t)( t < u .⊃. pt). These take us back again to the problem of analyzing away pt. The following fare no better: (Fp)u =df (∃t)( t > u .•. pt) (Pp)u =df (∃t)( t < u .•. pt). If one invokes contingent concrete particular events, one can make a start (provided, the tenseless notion of temporal precedence, t > u, is properly defined in terms of the overlapping of parts of events). Where p is atomic one may have: p

~(pt) =df ~(∃e)( et). p

(Fp )u =df (∃e)(∃t)( t > u • et) p

(Pp )u =df (∃e)(∃t)( t < u • et )) p

q

p

q

(F(p ∨ q) )u =df (∃e)(∃ f )(∃t)( t > u .•. et ∨ f t) (P(p • q) )u =df (∃e)(∃ f )(∃t)( u < t .•. et • f t ).

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It is difficult to see how to go much further this way, and in any case, the entire enterprise is clearly outside the parameters governing the development of a logic of time. The above rules NecG and NecH cannot get their foundation in universal generalization the way the alethic rule Nec of modal necessitation does. In his book, The Future, Lucas (1989, p.  109) offers the following as if it were a piece of temporal logic: tNec ├ p →├ pt. But again this relies on the admissibility of pt and thus of an ontology of abstract particular propositions holding at times. And again, when p is not atomic or defined away we get gibberish. Consider this: ├ p ∨ ~p →├ (p ∨ ~p)t. Lucas (1989, p.  111) notwithstanding, the expression (p  ∨  ~p)t is ill-­ formed because (~p)t is ill-formed. Lucas was working in an ontology of propositions as abstract particulars. It won’t help to put: p

├ p ∨ ~p →├ (∃e)( et ∨ ~( et)). This is well-formed, but it is not in the province of any logic acceptable in scientific philosophy which requires that events be contingent concrete particulars. Curiously, Lucas takes (~p)t to be interchangeable with ~(p)t. But scientific philosophy has no patience for negative events—for example, the cat’s not being on the mat. The expression (~e)t is meaningless. Similarly, both (e ∨ f )t and (e • f )t are meaningless. Lucas also allows: ├ pt ⊃ (pt)t. But it is questionable in scientific philosophy that there is an event et of event e being at t. But suppose p

(∃e) (et ⊃ (et)t). It by no means follows that it a logical matter. So long as abstract particulars are rejected, it is never a matter of any rival logic that there is an event et. Similarly, nothing in pure logic demands that one hold: p

(∃e)((et)u ⊃ (et)v), for all v ≥ u. The existence of an event as a concrete particular is always contingent.

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In light of this, note that there is no temporal analog of axiom schema K which we found to be part of pure quantification theory in alethic modal logics. Consider: tK ├ (p ⊃ q)t .⊃. pt ⊃ qt. This is ill-formed, because (p ⊃ q)t, that is, (~p ∨ q)t, is ill-formed. Prior has axioms such as: ├ G(p ⊃ q) .⊃. Gp ⊃ Gq ├ H(p ⊃ q) .⊃. Hp ⊃ Hq. But these fare no better. Such difficulties plague the hope of finding a proper logical study of time on the par with the study of modal and other rival logics of invariance notions of necessity that can be formulated within Russell’s scientific method in philosophy. Recall that scientific method in philosophy required the analysis of pw in modal systems. The partial solution was found by appeal to Lewisian analyses. We had: ((∃x)(Sx • ⬦ Dx)w =df (∃x)(Sx • Ixw • (∃α)(∃y)(αRw • Iyα • Cyx • Dy)). Similarly, one might imagine that a Lewisian analysis could help when it comes to an analysis of temporal ascriptions such as pt especially when couched within Russell’s agenda against continuants. But alas, no. Consider the following, ((∃x)(Sx • F(Dx)))u = df x

(∃x)(Sx • (∃z)(z at u • (∃t)(t > u • (∃y)(y at t • C yz • Dy)))). x

This endeavors to say that the sun is radiating. I write C yz to say that y and z are among x’s temporal counterparts. Now D speaks of irradiance and thus the transcription speaks of the temporal part y as having D (radiating) simpliciter. But it is the sun that is radiating through time, not its temporal counterpart. A scientific philosopher in the Principia era may well maintain that it is events that are at times (as defined relative to the measure of temporal magnitude relations). Of course, in saying that the counterpart y is at t, one need not imagine it being at an instant since t could be a temporal interval. The problem is not here, but with the clause Dy. If there are temporal counterparts with properties, series of which constitute the sun radiating, one has to find those properties and construct the solar radiation event as a series of events involving these parts. Such properties are nowhere to be found in any physical science. Their existence is a postulate

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of a metaphysical analysis that remains irremediably incomplete. It won’t help to try a reformulation as follows: ((∃x)(Sx • F(Dx)))u = df Dx

Dx

(∃x)(Sx • (∃e )( eu • (∃t)(t > u • (∃ f ) (eCf • f t)))). This construes events e and f as both events of the type solar irradiance. Dx But we now see that in characterizing event e as an event of solar irradiance the problem is clear. The analysis presents event e as being of the type Dx and that is in tension with the thesis that the sun is a series of events—a whole defined in terms of its temporal parts. The temporal parts are themselves intelligible only in terms of the whole series which is the sun. The part-whole imagery of space is getting in the way of the account of time even if one holds a perdurance rather than an endurance theory. Whitehead was on to this problem, offering the following objection (Whitehead 1919, p. 23): The essence of an organism is that it is one thing which functions and is spread through space. Now functioning takes time. Thus a biological organism is a unity with a spatio-temporal extension which is the essence of its being. The biological conception is obviously incompatible with the traditional ideas. This argument does not in any way depend on the assumption that biological phenomena belong to a different category to other physical phenomena. … The only reason for the introduction of biology is that in these sciences the same necessity becomes more clear. (PNK, p. 3)

Iron and a biological organism both require time for functioning. There is no such thing as iron at an instant; to be iron is a character of an event. The metaphysics of physical time is not properly construed as involving a part-whole relation. The right relation is that which is tied to the measure of temporal magnitudes itself. It is the determinable-determinate relation. That is the heart of Whitehead’s objection. Solar irradiance is a determinable which is given by the characterization Dx of the event type as e . The determinate events at times are defined by appeal to determinables and measured by appeal to the overlapping of events that are productive of temporal magnitude relations. This is caught up with the nature of the relationships that constitute magnitudes of temporal measurement in modern (relativistic and quantum) physics. It is the sun-event that is radiating (through time), and the determinate events are intelligible only via the determinable ‘radiating.’

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The conflation of the temporal determinate-determinable relation (involved with the measure of temporal magnitudes) with a part-whole relation (so natural to the conception of space) may well be the root cause of the misconception that embracing concrete particular events in physics requires that one adopt temporal parts and constructions of continuants in terms of event series of their temporal parts. Whitehead glimpsed this (Whitehead 1919, p. 65): It is an error to ascribe parts to objects, where “part” here means spatial or temporal part. The erroneousness of such ascription immediately follows from the premise that primarily an object is not in space or in time. The absence of temporal parts of objects is a commonplace of thought. No one thinks that part of a stone is at one time and another part of the stone is at another time. The same stone is at both times, in the sense in which the stone is existing at those times (if it be existing). But spatial parts are in a different category, and it is natural to think of various parts of a stone, simultaneously existing. Such a conception confuses the stone as an object with the event which exhibits the actual relations of the stone within nature. … But the confusion of the object, which is a unity, with the events, which have parts, is always imminent. In biological organisms the character of the organism as an object is more clear.

Ultimately, the collaborations of Whitehead and Russell came apart over Whitehead’s intuitions governing the nature of events, the measurement of temporal magnitudes, and the question of whether physics must embrace a notion of process. Any tensed predication that serves as a characterization of the type of an event presents that event as a process in time. Russell felt that Whitehead, perhaps under Bergson’s influence, hoped that organic processes of life and consciousness might be accommodated into an empirical science by embracing unique natural processes of growth which involve dynamic notions of motion and change. Whitehead’s earlier views hadn’t yet crossed the line. Russell’s concerns properly apply to Whitehead’s later views (e.g., in Process and Reality) which fall into the quagmire of explaining growing/becoming. The physics of special relativity promotes the view that any physical temporal measurement is relational, depending on exemplified temporal relations between events. The event(s) appealed to as the inertial frame relative to which the measurement is physically determinate cannot itself also be measured by that very frame. The same point concerns the notion of an inertial frame of events of uniform motion that is the mainstay of

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special relativity. There is no privileged frame, no absolute (universe) frame from which all temporal measure relations of space-time are made determinate—though all physical laws reached relative to any frame are precisely those reached from any other. This impacts the notion of a temporally bounded event—that is, an event in space-time with beginning and ending temporal bounds. The notion of physically determinate temporal measure is relational and relies (ultimately) on an event frame of reference (inertial frame of uniform motion) which I call the “ongoing events.” The temporal measure of an event is physically determinate ultimately only relative to the ongoing events of the inertial frame. Accordingly, events constitutive of a given frame are not in the very space-time whose physical determinacy is dependent on relations to the ongoing events definitive of that frame. It is only physics that can decide what ongoing events are, in fact, proper in relation to which temporal measure is physically legitimate. Einstein’s Special Relativity focused on ongoing electro-magnetic events. This focus categorically rejects holding that physical length be defined as a relation to molecular ongoing events of an iron bar in Paris under fixed pressure and temperature. Space-time events are, in special relativity, constituted by the overlapping of ongoing events of electro-magnetic propagation that constitute an inertial frame. Events which are regarded as ongoing with respect to relations of temporal measure cannot be said to have temporal parts (in relation to that measure framework). Only events determinate in space-time (with beginning and ending temporal boundaries) may be said to have temporal parts. It is the notion of temporal ‘determinate-determinable’ and not temporal ‘part-whole’ that does the foundational work in understanding the magnitude that is physical time. It is in the hands of empirical physicists to get evidence for what physical events are legitimately candidates for the reference frame of ongoing events in relation to which temporal and spatial and gravitational metrics are made physically intelligible. Electro-magnetic and gravitational fields are the currently empirically successful candidates. When it comes to fitting intentionality into this story, however, one wants to say that no special considerations arise about the nature of temporally bounded mental events. As with all temporally bounded events, the meaning of there being temporal bounds of a mental event (e.g., my thought that the cat is on the mat) is dependent on relations to ongoing events. But a twist rises here. The ongoing events in such a case may be electro-magnetic, gravitational, and, perhaps also, my ongoing thinking. While a given thought may well be

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(on a naturalist view) an event determinately in space-time relative to ongoing events, it seems knowable (perhaps synthetic a priori) that an event of one’s own thinking is always among the ongoing events relative to which a given thought’s temporal bounds have determinacy.25 For example, Descartes’s many mental events of 1637 are temporally bounded. Their temporal bounds are determinate relative to ongoing events— electro-­magnetic as well as Descartes’s intentionality. As Borges (1946) put it: “Time is a river which sweeps me along, but I am the river.” It may seem strange that Descartes’s intentionality is itself among the ongoing events relative to which his mental events of 1637 have temporal determination. The implication, however, is just that Descartes’s intentionality does not consist in a series the “parts” of which are “his” temporally bounded mental events. The part-whole imagery is inapplicable. Whitehead, not Russell, seems better on this point. The determinable ‘Cartesian intentionality’ is involved in the determinacy of the temporal boundaries of any of the determinates which are “his” mental events situated in time. (The same may well not be the case for the sun for which no appeal to a solar ongoing event is mandated.) It by no means follows that Descartes is still thinking. That mistakenly suggests that Descartes’s intentionality was in time and remains so. Note as well that on the thesis of Chap. 4, there is no indexical first-person thought, and thus there is no indexical first-person mental event. We may use a quasi-indexical to say that in 1637 Descartes had the thought that he himself is thinking. This is not committed to there being an event situated in time that is an indexical thought. The apparatus of intentionality is quantificational. The introduction of the notion of an “ongoing event” is something of a compromise between Russell’s Eternalism and certain aspects of Whitehead’s process metaphysics of time. It is much in sympathy with Eternalism in its rejection of the intelligibility of a process of becoming governed by physical law. But since the very physical determinacy of a temporal boundary (space-time measure) of an event requires a coordination of events relative to ongoing events, the thesis is incompatible with traditional philosophical Eternalism. There is no way to conceive, as the traditional Eternalist does, that all events are temporally bounded, i.e., in a space-time metric of a Universe. Any space-time metric is relative to an inertial frame (of ongoing events), but no frame is universal. There is no intelligible notion of a Universe with a space-time metric because there is no privileged inertial frame of absolutely ongoing events. There is no physical meaning assignable to the notion of an absolutely ongoing event. The Growing-Block theory of time incoherently imagines relative to a given time, of a growing temporal boundary after which there are no

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events.26 This is meaningless. Every temporal boundary is constituted by the overlapping of events temporally before and after it relative to the inertial frame of ongoing events. In this way, the notion of an ongoing event is compatible with special/general relativity. The notion of a growing-­block is not. Similarly, the thesis that ongoing events of an inertial frame give physical determinacy to space-time boundaries entails that the Presentist notion of being present is physically unintelligible. There can be no physical notion of a temporally bounded yet present event. A temporal bound is only physically determinate in relation to events temporally before and events after the bound. As we can see from these modest and very programmatic sketches, there is every good reason to hold that a scientific method in philosophy inspired by the Principia era is quite fruitful and viable. The cpLogic of Principia transforms many philosophical issues into issues that can be studied by means of an apparatus of quantification together with acquaintance with universals. This is viable because the cpLogic of Principia alone gives us all the needed tools for the analyses, and that includes analyses of the various sorts of invariance notions of non-logical necessity. We have every good reason to hold that the kinds of objections to the correspondence theory of truth do not impact our repaired form of the acquaintance epistemology of Theory of Knowledge. All “truths” have truth-makers and all truth-makers are actual. That includes logical truths, modal truths, subjunctive conditional truths, and temporal truths. We found that we must guard against the fog that comes from the Meinongian fallacy involving the question of aboutness—worrying over what negative existential truths are about, what tensed truths are about, and what subjunctive truths are about. They are, one and all, approached in the same way—by giving a quantificational account. From Russell’s “On Denoting” and his multiple-relation theory, an important lesson emerged: All discursive thinking is quantificational. We can capture what is shared in thought and communication by referring only to universals with which we are acquainted. In the beginning was acquaintance with universals and the cognitive apparatus of quantification. Reference and predication do not begin from singular thinking, work their way up to molecular thinking and then eventually to quantificational thinking. There is no singular thinking and no molecular thinking at all. There is only quantificational thinking. On the mind-first view, which accepts that intentionality is scaffolded by impredicative quantification, the existence of separate units that Russell imagined as particular logical forms is an appearance produced by looking

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at features of the belief-facts that are the artifacts of the mind believing. The appearance of separate units has the same derivative status as the appearance of the subject. They are, one and all, vestigial remnants in time of the mind’s quantificational thinking. Naturally, they do suggest something about the mind that generated them, but it is mistaken to conclude that quantificational thinking is composed of separable units of a “language of thought” understood as a sort of Boolean algebraic combinatorics. No combinatorial algebra and theory of recursive functions (which capture bounded quantification) can constitute the intentionality of mind and its impredicative quantificational thinking. By keeping the problems of the nature of representation and intentionality entirely separate from the problem of direction and the problem of compositionality, repairs to Russell’s Theory of Knowledge became available. Russell’s multiple-relation theory no longer faces what he called his “real difficulty”—the difficulty of embracing logical forms as abstract particulars. We have at last found the epistemic theory Russell needed to support his Scientific Method in Philosophy in a way that is consonant with the pristine thesis of The Problems of Philosophy that synthetic a priori knowledge lies solely in our acquaintance with universals. The lessons learned from a study of Russell’s Theory of Knowledge are quite instructive. Mathematical logic is synthetic a priori. For it to be the essence of a research program in scientific philosophy, it has to have a privileged status over and above all other fields of inquiry (metaphysical or otherwise), and our knowledge of it has to be every bit as privileged. Russell’s science of philosophy requires the elimination of abstract particulars not only from mathematics but also from Principia’s logic and its epistemology. Although abstract particulars must ground the ontology of mathematics (and of Principia’s mathematical logic), they are not part of its subject matter and play no role whatsoever as truth-makers (in the sense in which truth is correspondence with fact). Looking back, Russell remarked that logicism was attractive because it made an advancement upon Kronecker’s thesis that while God created the natural numbers the mathematicians created the rest, viz., fractions, reals, imaginary, complex, and so on. He wrote (MPD, p. 219): It was comforting to find that they could all be swept into limbo, leaving Divine Creation confined to such purely logical concepts as or and not and all and some. It is true that when this analysis had been effected, philosophical problems remained as regards the residue, but the problems were fewer and more manageable.

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Russell never was quite able to manage. But the acquaintance epistemology he abandoned for his Theory of Knowledge—his Scientific Method in Philosophy—was actually what offered his most promising attempt. The truth-makers for the truths of mathematical logic are not the abstract facts that ground mathematical logic. Russell’s acquaintance epistemology remains above the metaphysical conflicts in ontology (i.e., the study of necessity). This opens the way for the revitalization of Russell’s research program. There remains a great deal of research in the field to be done to undermine arguments for the ineliminability of abstract particulars in the sciences. There is no so-called easy ontology. But only Russell’s scientific method in philosophy is making progress. It is Theory of Knowledge, among all schools of philosophy, that remains the most promising avenue for unraveling the problems of philosophy.

Notes 1. Recall that the exclamation φ! simply indicates that we have a bindable predicate variable instead of a schematic letter φ for a wff. 2. As expected, Russell’s method makes the following de dicto iterations trivially true: ◽ p ≡ ◽◽ p ≡ ◽ ◇p.

◇p

3. See Quine (1966), pp. 156–174. 4. For the definition of an anti-essentialist model, see Parsons (1969). 5. This point leaves open the question of what semantic interpretation of a formal system enabling transition states best explains the behavior of the system. A Boolean algebra can be so implemented to enable the landing of an airplane and so also can an Intuitionist algebra. 6. Compare Hanna (2006). Influenced by Chomsky, Hanna imagines an innate “protologic” faculty that is, he says, “… distinct in structure from all classical and non-classical logical systems, that is used for the construction of all logical systems.” Hanna imagines a cognitive faculty for “representing logic” that is foundational to all rival systems that have claim to being logics. I fail to see how there could be any such foundational system lurking behind such rivals as relevant logic and the alethic modal logics and classical logics, and so on. Hanna’s bottom-up approach is in stark contrast to the mind-first top-down approach I’m after. It is the vast resources of our understanding of impredicative quantification that enables the development of studies of various systems of invariance that parade themselves as if they were rival logics.

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7. In the neighborhood semantics that Cresswell (1995) gives for C.I. Lewis’s S1, (◽p)w is not interpreted as a universal quantification. In this way the semantics evades the unrestricted Rule of Necessitation. Lewis’s systems S1 S2, and S3 are not properly transcribed in terms of quantification over worlds. They equivocate between strict implication which is a relevant entailment and the quite distinct ◽ (~p ∨ q). See Landini (2022b). 8. Now the axiom schema *10.1 is perfectly innocuous in Principia, where individual variables of lowest type may be said to range over universals as well as concrete particulars—there being no type considerations where universals of acquaintance are concerned. (Universals, on this view, are not the sorts of entities that are “at worlds.”) 9. Much of the question here turns on whether one imagines that bound individual variables must range over concrete contingent particulars. The semantics Whitehead and Russell originally intended for Principia’s axiom schema *10.1 included universals in the ranges of its individual variables (even of lowest type). Individual variables are not concrete particular variables. In his 1919 Introduction to Mathematical Philosophy, Russell imagined that the quantification theory of Principia should be altered so as to accommodate his interest in a semantics which has the individual variables of lowest type restricted to concrete particulars. He was never able to find such a formal alteration to its quantification theory. The only accepted change to the 1925 second edition of Principia was the replacement of its *9 with the quantification theory of *8 which conducts quantification without free variables. 10. For a contrasting opinion, see, Hayaki (2006). 11. In criticizing popular physics writers such as Lawrence Krauss’s A Universe from Nothing, many correctly noted that “nothing” is a quantifier phrase and not a name of anything—and obviously it is an equivocation to regard it as a name of the void, the quantum vacuum or any other physical entity or process. It is no less important to note that “everything” is also a quantifier phrase and not a name of something. The notion of the universe as if it were a physical entity, understood as everything or everything taken together, is every bit as untoward as the equivocation that makes “nothing” a name of the void. The point that there is no universe is made by Russell himself in his famous BBC debate with Copleston. 12. See Fara and Williamson (2005). 13. A counterpart theorist can, however, write the following: (∃β)(x)( Iβx • (∃y)( xCy • Ry) .⊃. Px). 14. See Lewis (1973). Stalnaker (1968) offers a rival. Stalnaker’s alternative theory coincides with Lewis’s theory if certain assumptions are made concerning accessibility and the relation ≤i . So we take Lewis’s approach as our exemplar. In particular, Stalnaker accepts CEM: ~( p ◽→ q) .⊃. p ◽→ ~q. Lewis rejects it.

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15. Note that in S4 we have: ◽ ◽p ≡ ◽p. 16. See Mares (2004). 17. In Problems, Russell voiced an enthusiasm for Keynes’s work (part of which he has seen in draft) while at the same time vehemently rejecting propositions. Russell may have been led astray by Keynes’ theory being called a “logical” probability notion. In Problems, Russell regards the fundamentals of probability to be part of mathematical logic. 18. See Tooley (1997) for a nice discussion of the hopelessness of rejecting the tenselessness of quantification. The quantificational nature of intentionality enables reference to entities such as universals which are out of time, even if their exemplification may, in some cases, be through time. This ­reference would be impossible if quantification were essentially tensed. Thinking about particulars that are events situated in time also requires quantification to be tenseless. 19. This approach formulates a language independently of taking a stand between the so-called A-theory and the B-theory of the metaphysics of time. Sullivan (2016) attempted a different experiment searching for an A-theory of time without Tense  operators.   20. Thus x( C/ C )y, that is, (∃z)(xCz • z C y), has taken it from what had been a perigee position to what had been an apogee and then from that apogee to the perigee position. 21. Compare Grazini & Orilia, forthcoming. 22. Recall in the Principia era “ontology” concerns what (if anything) is necessary, not what is. 23. Relativity is mentioned in OKEW, p. 103. 24. The existence of God who is omniscient requires one to accept every instance of “p ⊃ KGp.” It follows that ~p ⊃ KG~p , but contrary to Dummett’s Truth and the Past (2004), one cannot validly arrive at KG{pt} ∨ KG{(~p)t}. Even if one allows: pt ⊃ KG{pt} as well as (~p)t ⊃ KG{(~p)t}, nothing in logic yields: pt ∨ (~p)t. Now we do have ~(pt) ⊃ KG{~(pt)}. But this yields: KG{pt} ∨ KG{~(pt)}. Omniscience does not entail knowledge of the future. There may be nothing to know. 25. See Prior (1959) for an argument that an Eternalist theory of time fails to account for our relief that painful past events are in the past. 26. See Landini (2014b).

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Borges, Jorge Luis. 1946. A New Refutation of Time. In Labyrinths: Selected Stories and Other Writings, ed. Donald Yates and James Irby. New York: New Directions Publishing Co. Buchanan, Scott. 1927. Possibility. London: Kegan Paul, Trench, Trubner & co. Carnap, Rudolf. 1950. Logical Foundations of Probability. Chicago: University of Chicago Press. Cocchiarella, Nino B. 1975. On the Primary and Secondary Semantics of Logical Necessity. Journal of Philosophical Logic 4: 13–27. Cresswell, Max. 1995. S1 is Not So Simple. In Modality, Morality and Belief: Essays in Honor of Ruth Barcan Marcus, ed. Walter Sinnott-Armstrong, Diana Raffman, and Nicholas Asher, 29–40. Cambridge: Cambridge University Press. Fara, Michael, and Timothy Williamson. 2005. Counterparts and Actuality. Mind 114: 1–30. Hanna, Robert. 2006. Rationality and Logic. MIT Press. Hayaki, Reina. 2006. Contingent Objects and the Barcan Formula. Erkenntnis 64: 75–83. Hossack, Keith. 2007. The Metaphysics of Knowledge. Oxford: Clarendon Press. Kamp, J. 1968. Tense Logic and the Theory of Linear Order, PhD Thesis, University of California, Los Angeles. Keynes, John Maynard. 1921. A Treatise on Probability. London: Macmillan. Landini, Gregory. 2014b. Methodological Cartesianism. In Defending Realism: Ontological and Epistemological Investigations, ed. Javier Cumpa, Greg Jesson, and Guido Bonino, 63–98. De Gruyter. Lewis, David. 1973. Counterfactuals. Cambridge: Harvard University Press. Lucas, J.R. 1989. The Future. Oxford: Basis Blackwell. Mares, Edwin. 2004. Relevant Logic. Cambridge: Cambridge University Press. Melia, Joseph. 2003. Modality. Montreal: McGill-Queen’s University Press. Merricks, Trenton. 2007. Truth and Ontology. Oxford: Clarendon Press. Parsons, Terence. 1969. Essentialism and Quantified Modal Logic. Philosophical Review 78: 35–52. Plantinga, Alvin. 1974. The Nature of Necessity. Oxford: Clarendon Press. Prior, A.N. 1959. Thank Goodness That’s Over. Philosophy 34: 12–17. Prior, Arthur. 1967. Papers on Time, Tense. Oxford: Oxford University Press. Quine, W.V.O. 1966. Three Grades of Modal Involvement. In The Ways of Paradox and Other Essays, 156–174. New York: Random House. Reichenbach, Hans. 1947. Elements of Symbolic Logic. New York: Macmillan, Co. Stalnaker, Robert. 1968. A Theory of Conditionals. In Studies in Logical Theory, American Philosophical Quarterly, Monograph Series 2, 98–112. Oxford: Basil Blackwell. Sullivan, Meghan. 2016. An A-Theory of Time Without Tense Operators. Canadian Journal of Philosophy 46: 735–758.

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Tooley, Michael. 1997. Time, Tense & Causation. Oxford: Oxford University Press. Whitehead, A.N. 1911. The Axioms of Geometry. Encyclopedia Britannica, 4th ed. Reprinted in Whitehead (1948), pp. 177–194. ———. 1919. An Enquiry into the Principles of Natural Knowledge. Cambridge: Cambridge University Press.

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———. 2007. The Russell/Bradley Dispute and Its Significance for Twentieth Century Philosophy. New York: Palgrave Macmillan. Cappelen, Herman, and Josh Dever. 2013. The Inessential Indexical; On the Philosophical Significance of Perspective and the First-Person. London: Oxford University Press. Carey, Rosalind. 2007. Russell and Wittgenstein and the Nature of Judgment. London: Continuum Press. Carnap, Rudolf. 1950. Logical Foundations of Probability. Chicago: University of Chicago Press. Castañeda, Hector-Neri. 1966. He: A Study in the Logic of Self-Consciousness. Ratio 8: 130–157. ———. 1967. Indicators and Quasi Indicators. American Philosophical Quarterly 4: 85–100. ———. 1974. Thinking and the Structure of the World. Philosophia 4: 3–40. ———. 1979. Fiction and Reality: Their Basic Connections. Poetica 8: 31–62. ———. 1983. Reply to Burge. In Agent, Language, and the Structure of the World: Essays Presented to Hector-Neri Castañeda with His Replies, ed. J.E. Tomberlin, 355–372. Indianapolis: Hackett Publishing Co. Chellas, Brian, and Krister Segerberg. 1996. Modal Logics in the Vicinity of S1. Notre Dame Journal of Formal Logic 37: 1–24. Chisholm, Roderick. 1957. Perceiving: A Philosophical Study. Ithaca: Cornell University Press. Chomsky, Noam. 1965. Aspects of the Theory of Syntax. Cambridge, MA: MIT Press. ———. 1995. The Minimalist Program. Cambridge, MA: MIT Press. Church, Alonzo. 1951. The Need for Abstract Entities in Semantic Entities. Proceedings of the American Academy of Arts and Sciences 80: 100–114. ———. 1976. A Comparison of Russell’s Resolution of the Semantical Antinomies with that of Tarski. The Journal of Symbolic Logic 41 (4): 747–760. Clark, Romane. 1978. Not Every Object of Thought Has Being: A Paradox in Naïve Predication Theory. Noûs 12: 181–188. ———. 1983. Predication Theory Guised and Disguised. In Agent, Language, and the Structure of the World: Essays Presented to Hector-Neri Castañeda with His Replies, ed. J.E. Tomberlin, 111–130. Indianapolis: Hackett Publishing Co. Cocchiarella, Nino. B 1973. Whither Russell’s Paradox of Predication. In Logic and Ontology, vol. 2 of Studies in Contemporary Philosophy, ed. M. K. Munitz. New York: New York University Press: 133–158. ———. 1975. On the Primary and Secondary Semantics of Logical Necessity. Journal of Philosophical Logic 4: 13–27. ———. 1980. The Development of the Theory of Types and the Notion of a Logical Subject in Russell’s Early Philosophy. Synthese 45: 71–115. ———. 1985. Frege’s Double Correlation Thesis and Quine’s set theories NF and ML. Journal of Philosophical Logic 14: 1–39.

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Index1

A ab-Notation, 45, 78, 79, 81, 85–89, 91, 94, 97–102, 112n13, 113n29, 113n32, 113n33, 286, 327 Adicity, x, 36, 50, 65–67, 136, 146, 215, 216, 219–223, 227, 239, 267–271, 274, 303, 319 and number of position relations, x, 50, 221, 248–262, 280–281 B Blackwell, Kenneth, ix Borges, Jorge, 368 Bradley, F. H., 9, 20, 121 Brentano, Franz intentionality principle of, 8, 108, 203

C Candlish, Stewart, 73n49 Castañeda, Hector-Neri, 194 Chomsky, Noam, 177, 287–291, 294 Church, Alonzo, ix, 123–125, 144, 157, 172n29, 286, 318 Cocchiarella, Nino B., ix, 70n17, 74n57, 130, 225–226, 322n5, 330 Collins, John, 288–292 Compositionality, problem of, x, 34, 39, 44, 50, 63–68, 78, 162, 279, 281, 282, 288, 290, 300, 303, 309, 370 Comprehension principle (impredicative), vii, 2–5, 10, 14, 33, 69n4, 79–82, 101, 111n5, 118–120, 123–125, 135–138, 151, 156, 161, 262, 304, 317–318, 322, 329, 336, 356–357

 Note: Page numbers followed by ‘n’ refer to notes.

1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Landini, Repairing Bertrand Russell’s 1913 Theory of Knowledge, History of Analytic Philosophy, https://doi.org/10.1007/978-3-030-66356-8

393

394 

INDEX

Connelly, James, 73n46 Crane, Timothy, 185, 235n9 D Davidson, Donald, 106, 288 Denoting concepts, 120–121 Descartes, René, 368 Dewey, John, 30 Direction narrow, 46–55 problem of, 39, 43, 46, 174n48, 210–212, 239, 240, 242, 245, 258, 267–271, 276, 303–307 of a relation (see Sense) E Eames, Elizabeth, 42, 70n7 Event, viii, 19–21, 33, 108–109, 212, 217–221, 232–234, 274–276, 293, 325, 350–369, 373n25 F Fact, viii, ix, 8–12, 17–22, 84–86, 161–166, 239–244, 246–253, 255, 257–276, 326–329, 335, 343 belief-fact, 31, 41, 47, 49–62, 66, 67, 160, 192, 201–202, 206–208, 211, 212, 219 conjunctive, 165, 166 definition of permutative, 53, 239, 307 definition of R ^ n − fact, 220 general, 65, 162, 163 impossible to name/describe, 113n36, 114n38 logical, 28, 40, 41 negative, 108–111, 163–169 ongoing, 359, 366–368

picture of, 88 vs. propositions, 30, 121 structureless, 65, 281 Frege (Gottlob), 3, 11–16, 33, 69n3, 148–155, 172n31, 314, 335 Friedman, Michael, 19 Fumerton, Richard, ix Fundamental thesis (of Principia era), ix, 41, 87, 164–165, 207, 210, 211, 227–228, 251–263, 293 G Galaugher, Jolen, x Gandon, Sébastien, 71n22 God, 22, 72n36, 370, 373n24 Gödel, Kurt, 15 Griffin, Nicholas, ix, x, 46–47, 73n41, 73n47, 88, 112n17, 172n34, 236n29 H Hanks, Peter, 16, 56–58, 73n52, 86, 87, 225, 292–296 Hanna, Robert, 322n4 Hasan, Ali, ix Hawthorne, John, 187, 188 Hossack, Keith, 322n1, 335, 336, 349 Hylton, Peter, 31, 170n3, 171n11, 173n42 I Intentionality, 8–11, 33–36, 61, 62, 177–203, 205–207, 211, 216–218, 222–228, 258, 259, 279–292, 295, 323n10, 332, 334–336, 346, 349, 367–368 J Jackendoff, Ray, 234

 INDEX 

K Kapitan, Tomis, 192–194 Kaplan, David, 197–203, 235n13 L Lebens, Samuel, 36, 41, 51–52, 73n49, 73n51, 73n53, 170n1, 170n4, 283–285, 288, 322n2, 323n10 Levine, James, 323n12 Lewis, David, 173n44, 199, 235n17, 339–344, 361, 364, 372n7, 372n14 Logic cpLogic, 2–8, 23–32, 35, 38, 69, 69n3, 118–119, 124, 181 infinity of, 150–156, 171n21 intuitionistic, viii, 331, 335, 336, 344, 345, 371n5 modal, viii, 327, 332, 334–337, 339, 341, 342, 344, 349, 353, 362, 364, 369, 371n6 necessity of, 153, 325–333 probability and, 26–27, 70n9, 349 purity of, 138–139 relevant, viii, 345–349 Logical forms, 7, 8, 12, 23, 24, 32–36, 276 as abstract particulars, ix, 39–46, 63–69, 204, 222, 281, 282, 304, 306, 309 modifying belief, 282 as structureless, 65–68, 280–282, 296–304, 309 Logicism of Frege, vii, 3, 15, 150 of Whitehead-Russell, ix, 1–8, 15, 38, 45, 118, 121, 127, 139, 150 Lucas, J. R., 363

395

M MacBride, Fraser, 9, 54, 70n12, 72n34, 73n48, 219, 220 Maclean, Gülberk Koç, x Manley, David, 187, 188 Meinong, Alexius, 47, 109–111, 177–181, 183–191, 218, 224, 234n7 Meinongian fallacy, 224, 296, 307, 369 Merricks, Trenton, 37, 38, 184, 342 Mind, 49, 54–60, 69, 108, 191–200, 203–212, 235n8, 275, 276, 282, 287–293, 312–317, 346 as ongoing, 359 vs. subject, 29, 47, 58, 111, 207 N Necessitarianism, 37 O Orilia, Francesco, 194, 232–233, 235n14, 235n16, 236n33 O-roles, 232–233 P Paradox of the Barber, 70n10 contingent, 178 Frege’s belief, Russell’s of classes, 12, 15–16, 118, 127, 223 Liar, 129 of predicates, 225 of po/ao, 122, 123, 130–133, 170n3 Prior’s, 178 semantic, 80, 118, 121, 122, 128, 170n3, 171n11 Veronique/Pinocchio, 178 Parity, 68, 88–89, 99, 102–111, 242

396 

INDEX

Pears, David, 31, 243 Perović, Katarina, x, 9, 50, 70n13, 251, 277n7 Position relation, 50, 62–63, 73n48, 209–211, 221, 239–242, 244, 248–260, 262–276 Prior, Arthur, 178, 179, 234n6, 310–312 Proposition image proposition, 108–109 linguistic meaning, 36–38, 51, 57, 180–183, 202, 203 Russellian, ix, 6, 8, 31, 36, 41, 48, 51–52, 56–58, 72n36, 170n4, 178, 334, 339, 342, 349, 352, 355, 362–363, 373n17 Q Quine, W. V. O., 15–16, 30, 38, 67, 72–73n37, 128, 139, 172n33, 177–181, 190, 210, 224–228, 234n3, 283, 287, 292, 294, 300, 301, 303, 309, 314, 318–321, 322n9, 330, 335 R Ramsey, Frank, 79, 81–82, 94, 107, 111n5, 112n23 Reichenbach, Hans, 192, 197, 235n13, 352 Representation, 36–38, 47, 50–59 S Schiffer, Stephen, 186–187 Sense data of, 12, 27, 33, 34, 203, 205–209, 217–219, 235n23 vs. nonsense, x, 56 of a relation, 59–62

Soames, Scott, 180, 288–290 Soutor, Milan, x Stout, G. F., 60–62, 73n51, 245 Substitutional (theory), 113–114n36, 118–133, 142, 156, 170n2, 171n11 T Tarski, Alfred, 129, 158 Time, 5, 16, 19, 20, 22, 25, 33, 38, 39, 70n12, 135, 181, 196–198, 207, 213, 218, 220, 260, 292, 350–370 eternalist theory of, 20, 354, 355, 357, 358, 368, 373n25 invariant, 90, 113n28 no logic of, 10, 361–370 presentist, growing block, 354, 355, 357 truth-aptness, 86 Tooley, Michael, 351 Truth, 36, 37, 41, 42, 47, 51, 156–160 invariant, 90, 113n28, 329 logical, 3, 7, 31, 325–331, 333, 369 necessary, 32, 325, 326, 328–331, 338 recursive theory of, 156–160 science as a body of, 80–84, 102 truth-aptness, 52, 56–58, 86, 87 Truthmaker principle of, 37, 342 U Unity (of Russellian propositions; of Russellian facts), x, 9, 36, 50–59, 107, 160, 170n2, 170–171n10, 223, 228–233, 244–253, 259, 285, 288–291, 307

 INDEX 

Universals Armstrongian, 10, 11, 70n12, 70n14, 165 ramified types of, ix, 81–82, 118, 123–125, 133, 157, 225 robust Russellian, 9–11 simple types of, 38, 46, 72n33, 317–322 W Wahl, Russell, ix, 72n30, 74n56, 74n57 Whitehead, Alfred N., ix, 1–6, 8, 15–21, 35, 38, 45, 70n18, 71n19, 71n20, 71n21, 71n22, 71n25, 85, 112n17, 112n20, 117–127, 135–139, 142, 143,

397

146–156, 165, 171n22, 172n35, 172n38, 317–318, 323n12 Williamson, Timothy, 7 Wittgenstein, Ludwig, ix, x, 9, 12, 21, 29, 42–46, 68, 71n26, 72n29, 77–91, 94–107, 109–111, 111n2, 111n3, 111n8, 112n9, 112n10, 112n13, 112n14, 112n15, 112n21, 113n27, 113n28, 113n30, 113n36, 114n39, 114n40, 114n43, 124, 161, 162, 168, 171n22, 226–228, 242–244, 269, 282, 285–287, 294, 304, 309, 326–327 Z Zermelo, Ernst, 15–16, 149