Extensionalism: The Revolution in Logic

Extensionalism: The Revolution in Logic

Nimrod Bar-Am

Extensionalism: The Revolution in Logic

N. Bar-Am Head, Rhetoric and Philosophy of Communication Unit Communication Department Sapir College of the Negev M.P. Hof Ashkelon 79165 Israel

ISBN: 978-1-4020-8167-5

e-ISBN: 978-1-4020-8168-2

Library of Congress Control Number: 2007941591 All Rights Reserved © 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To Gali,

Motto: “… logic … since Aristotle … has been unable to advance a step, and, thus, to all appearances has reached its completion” Immanuel Kant, The Critique of Pure Reason Preface to the 2nd ed., 1787 “Pure mathematics was discovered by Boole … the fact being that Boole was too modest to suppose his book the first ever written on mathematics.… His book was in fact concerned with formal logic, and this is the same thing as mathematics” Bertrand Russell Recent Work in the Philosophy of Mathematics, 1901

Acknowledgements

I am grateful to the Dibner family and their delightful Dibner Institute for the History of Science and Technology at MIT, for their generous support of this project and their invitation to stay there for unforgettable 2 years. I am grateful to Joseph Agassi, old-style philosophy teacher, colleague, close friend, and the most generous person I know. He granted me his time and his wisdom for ridiculously long hours, expecting nothing in return. Most of what I say in this book is but a summary of the many detailed and happy discussions that we have had over the years. I likewise wish to express my thanks to my colleagues Noam Ben-Yishay, Robert S. Cohen, Haim Gaifman, Stefano Gattei, Malachi Hacohen, Amir Meital, and Michael Segre, and to my brother Ahuvia Kahane, for their friendship, valuable criticism and constant, concerned encouragement. I am grateful also to Mark Jago, Springer reader, for unusually perceptive and helpful comments. I am as ever grateful to my parents for their relentless support. I am forever in awe of the miraculously good fortune I had as Gali and I became life companions. She has shown me that my doubts regarding the future of the closed society that we call the academy should not reduce the pleasure of recording my thoughts in this book. She also boosted my hope for it to find some kind readers. This book is for her, then.

ix

Abstract

For a very long time, Aristotelian logic was accepted as a tool (Organon) for the generation of scientific theory. Yet, science did not quite take advantage of this tool: syllogistic terminology rarely appears in the scientific literature, and even Aristotle’s own scientific theories seem to show almost no traces of his own formal tools. Does this discrepancy matter? Why was it so often claimed that Aristotle’s logic was essential to science? And what, then, was the purpose of logic? These are the questions Extensionalism: The Revolution in Logic attempts to answer. The book argues that the discrepancy is indeed of great significance. It suggests that Aristotle’s logic was, in fact, meant to serve as a means, not for the generation of scientific theory, but for judging it. This allowed, crucially, for the conflation of methodology, epistemology, and science. For philosophers, right up to Leibniz and Kant, this offered a means of overcoming major and seemingly insoluble philosophical problems – most notably of the skeptical sort. Logic, sustained by this conflation, functioned as a unifying principle in the attempt to make sense of the world. The problem, however, was that hence advancements in empirical science seemed to threaten the very consistency of traditional logic. This book expands on the history of logic as the story of the undoing of the classic conflation of methodology, epistemology and science. As I attempt to show, the first stage in this undoing was the recognition of the conflation. This was achieved as early as Leibniz, who spelled it out so as to provide an explicit justification for it. Kant attempted to replace the conflation with his own system of so called transcendental logic. Others, such as Bolzano and De Morgan, achieved partial separation of logic and epistemology. But, as I will argue, it was Boole who finally managed to undo the conflation, thus setting in motion a process that reached its culmination a century later with Tarski’s formal semantics. This process of recognizing the conflation and the successful attempt to undo it is the extensional revolution in logic. My aim in this study was not the writing of history. Rather, I have tried to describe the intellectual background to the extensional revolution in logic and to understand some of its major turning points. I hope to explain how methodology, epistemology, and science were linked in a knot commonly known today by the name of Aristotelian essentialism. And I hope to analyze its overwhelming effect on modern philosophy and early modern logic. Consequently, this book attempts to xi

xii

Abstract

revisit one particular feature of George Boole’s mathematical logic: its deliberately unsophisticated extensionalism. It was by reducing logic to the study of extensions that Boole was able to transcend, by default almost, some of Aristotle’s most stubborn essentialist presuppositions, thereby inaugurating a new era in the study of formal logic. More specifically, Boole succeeded in separating the study of valid inferences from the Aristotelian endeavor to provide the complete taxonomy of all things (and of all things known) that he conflated as both logic and science. By and large, Boole’s extensionalism is taken for granted in current literature, which thus fails to acknowledge the novelty of this idea, and prefers to discuss his path-breaking technical contributions instead. My aim is to do the opposite. Without entering into the technicalities of Boole’s theory, I want to focus on extensionalism and to analyze its intellectual background. The study of the technical, mathematical background to the rise of modern logic is undoubtedly an important part of the study of this rise. Yet, technical innovation does not occur in the void. It is thus surprising that the epistemological background to the rise of modern logic is hardly ever acknowledged. And it is, I suggest, precisely epistemological ideals, and above all the union of logic and empirical science, that held back the development of logic after Aristotle, right up to the early modern period. Gradual renunciation of this epistemological ideal allowed logicians the freedom to develop ever more formal logical systems, and thus to discover that this epistemological ideal is unattainable, and finally to denounce it. By discussing the philosophical impact of extensionalism and contrasting its effects with those of Aristotelian essentialism, I intend to explain the former’s role within that extraordinary process of disillusionment that constitutes modern logic. My study thus delimits the role of essentialism in the history of Western thought – positive and negative alike. I discuss the reluctance to render logic extensional and the difficulty and importance of doing so successfully.

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction: In Praise of Shallowness: A Methodological Credo . . . . . .

xvii

Part I Preliminary Notes Outline of Preliminary Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Controversies as Contests: The Birth of Intellectual Responsibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Methodology is not Epistemology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III Conflation is not Confusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV Proof is not Sound Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5 6 10

Part II Setting the Scene: Some Notes on the Pre-history of Logic 1 2 3

The Mother of All Conflations: Parmenides’ Proof. . . . . . . . . . . . . . . . . Early Disagreements Concerning the Power of Proofs: The Uses and Misuses of Dialogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sophists’ Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 23 31

Part III Aristotle’s Logic: The Rise of Essentialism 4 5 6 7 8 9 10 11

The Beginning is the Term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chimera in the Dusk: Essentialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semantics is not Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Conflation of the Source with the True, Good and Beautiful (Source) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Induction as Spell-Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Birth of Induction from Sea Foam . . . . . . . . . . . . . . . . . . . . . . . . . . Taxonomy of Reality by Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 49 61 65 71 77 83 91

xiii

xiv

Contents

Part IV

Essentialism Besieged

12 Ad Hominem Logic: Logic between Aristotle and Boole . . . . . . . . . . . . 13 The Neglect of Judgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Leibniz as Aristotle and Boole Conflated . . . . . . . . . . . . . . . . . . . . . . . . 15 Why Transcendental Logic is no Logic at All . . . . . . . . . . . . . . . . . . . . . Part V 16 17 18

The Fall of Essentialism

Extensionalism as Exorcism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Logic: An Oxymoron . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Last Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Epilogue

97 103 109 115

123 129 137

Extensionalism in a New Context . . . . . . . . . . . . . . . . . . . . . . .

143

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

Name Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

Introduction

In Praise of Shallowness: A Methodological Credo

Every abstract, Borges has suggested, is a mystical wonder: it shimmers with the possibility of condensing a large text into a brief paragraph, covering much ground, with little space and few words. Is it the one genuine marvel of all intellectual endeavors, or is it illusory? Philosophy is a craft, art and science. It is an attempt to know it all – at least a little. Ancient philosophers were polymaths: they studied nearly every conceivable problem, examined every observable fact, and attempted to understand it all, explain it all, describe it all, criticize it all and bring it to a unity. They read everything that could possibly be read, and then they wrote it all down concisely. And they did know it all, a little, to the extent that anything at all could be known, of course, even if a little – given the obvious human limits and the limits of the dawn of time. Some say, the ancient polymaths had it easy: their task was simpler than ours. There was, so they say, much less to know back in the golden childhood days of western reflection, when natural science was little more than commonsense, observed with a critical eye. And mathematics was then hardly more than basic geometry and elementary arithmetic. The information explosion and the over-specialization of our day have put an end, they say, to the ancient philosophical spirit, to the ideal of knowing it all, even if a little. Consequently, they conclude, we can no longer practice these ancient philosophical ideals; we can no longer emulate the ancient polymaths with any reasonable degree of success. This is a mistake. It is a misunderstanding of what it means to know it all a little. It is, thus, a misunderstanding of what philosophy is, of what a Weltanschauung is and of what the aim of philosophers is. Let me offer an illustration. Imagine a terrain that we seek to familiarize ourselves with and to map out. Knowing it all a little simply means having a bird’s-eye view of it: seeing the land as a bird or a pilot would, as an elaborate quilt of fields, cities, forests, rivers and lakes. The higher up, the greater the terrain covered by the bird’s-eye view. But also, the fewer the fine points perceived. This is the philosophical outlook and these are its limits. Like any general and abstract view, the philosophical outlook is flat: it lacks minute details. This is due to a conscious and deliberate choice, of course: it offers flatness as an intellectual service. There are only so many items that one can observe and account for at a single broad glance, a single map, a single book, xvii

xviii

Introduction

a single life-span. Philosophers, then, do not see more or know more, and they do not see less or know less. They aim to see less detail and more of the abstract. Their details, if you like, are abstractions. Walking on God’s earth as a pedestrian, as a farmer working his fields or as a passer-by, one’s picture of one’s surroundings is every bit as intelligent as that of the pilot riding the sky. The views of the field are radically different, however. One sees only a specific field and in all lively detail: the exact pattern of the land, or even the exact outline of a given leaf, grasshopper, grain of sand even. Acquaintance with minute detail is not without its price: details may stand in the way of conjuring the big picture. It may be difficult to compare whichever field one happens to be in with far off fields, with respect to their size or shape or any other quality. One may wish to inquire if far off fields were already planted, harvested, or even if they exist. A pedestrian may find it hard or even impossible to do so. The pedestrian view contains fine points that the pilot’s map never would, but it does not necessarily contain more information, for it lacks the general context. After all, there are only so many items that one can observe and account for at a single glance, a single map, a single book, a single life-span. As philosophy is no geography, how do we translate the metaphor? This is a classic difficult question. What is the bird’s-eye view of things? How is it possible to conjure it? The first question has hardly ever been explicitly asked, yet miraculously the second was answered – unsatisfactorily but answered. The answers were the classic theories of method (methodologies): Inductivism, Deductivism, Hypothetic-deductivism, and even Instrumentalism. Today we know that these classic answers are a bit naive just because the first question was traditionally dodged: the very division of things into abstract and concrete is context-dependant, conjectural, and, of course, delightfully metaphysical. The very distinction between pilots (philosophers) and farmers (specialists) depends wholly on philosophy, on Weltanschauung, no less. When accepting a hierarchy of things, we normally accept it with prudence and with a critical eye, perhaps even irony. And yet, hostility frequently subsists between the specialist and the philosopher. Why? Specialists often call philosophers charlatans who presume to provide general contexts in ignorance of vital details. On their part, philosophers observe that specialists pretend to have direct acquaintance with details while ignoring their own use of some general context – of necessity philosophical. The very idea of building blocks of the world, or of its perception, they note, is already an abstraction, suffused with philosophy. Ignoring it, they observe, is sheer dogmatism. It breeds irrationalism. How then can we avoid both presumption and dogma? We can avoid both presumption and dogma by noting that the two views are symmetrical: every minute detail presupposes some big picture which accommodates it, just as every big picture presupposes minute details that dwell in it. The farmer cannot observe even a single tree or a single beetle without the aid of some general abstract notion of Tree and of Beetle. How else will the farmer avoid viewing the one as the other? And the pilot who observes a forest has some notion of particular trees. Why else would he call it Forest? This is so, even without admitting the confusing fact that all of us, pilots and farmers alike, have a rather

Introduction

xix

fragile understanding of what knowledge is, what an abstract fact is, and what, in the abstract, a concrete particular detail is. It is for a reason that I burden the reader with these semi-trivial philosophical remarks at an introduction to a book about the development of logic. It is to stress that whatever your philosophy, I hope you and I agree that only through cooperation can the farmer (specialist) and the pilot (philosopher) achieve anything. If you do not agree with this observation, you are, perhaps, misusing your time reading this book. The difference between ancient times and our own that matters here, I believe, is not that there is more to be known today than ever before (though this is the case), but rather that we are far more developed as a community of seekers of knowledge than ever before, and a bit more aware of our limits as individuals, aware of the need to cut up and share the responsibility of our search for knowledge. Thus, the more we are developed as a community of knowers, the more knowledge becomes a property not of any one individual but of all of us as a community. We are increasingly in need of the knowledge and expertise of other members of our community, pilots and farmers alike. Mapping the infinite land of historical data is a delicate task. It requires the delicate and complex cooperation of philosophers and specialists alike. Pure data are nowhere to be found: there are no facts totally free of context, ideology and partiality. Historians study reports and reports are history filtered through the contextual sieves of this or that observer. It is, hence, futile to attempt to properly appreciate any minute historical detail without some reconstruction of a bigger picture, a general history, a context, within which these details were originally set and without which they evade all appreciation. The opposite also applies of course: the very meaning of a general context is that it accommodates an understanding of some details. A good general history, then, explains existing details successfully and provides the context which allows new ones to be ushered in and properly appreciated. I am embarrassed to admit: I have yet to read an outstanding general history of logic. I have yet to read even a good one. Please do not take this as a conceit or as intellectual snobbery. Rather, please take it for what it is: a concerned observation regarding a sad state of affairs. Needless to say, there are many talented researchers around. More than a few are personal acquaintances. And many of them are as concerned and distressed as I am. Why then does such a sad state of affairs prevail? I think it does, because there is only one formula or recipe for a general history of logic, a tacit and a terribly wanting one. Before I can tell you about it I am afraid I have to raise the following questions that seem to me basic for any decent and explicit general history of logic: what has provoked the rise of logic in the first place? What caused its development as such? What role was it supposed to have and what role did it play in the history of Western thought? What brought about the periods of its relative stagnation, even at such times as other fields experienced unusual leaps forward? What, finally, has provoked the meteoric rise of modern logic in recent centuries? And what provoked its present spectacular development? The standard recipe for a general history of logic dictates that its narration should present its growth as beyond reason thus: it was formulated about two and

xx

Introduction

a half millennia ago by one individual in a somewhat inexplicable stroke of genius; it then underwent a long period of dogmatic slumber; finally, in a brief outburst of ingenuity, relatively recently, its new version emerged by the addition of some clever technical gear. In oversight of the original aims of the old logical system and of the change of aims displayed by modern logic, there is little or no mention of the old system having been in dire need of replacement. Consequently, there is little or no room for any discussion of the dire need to replace the old system. Indeed there is little or no room for the observation that the old system was replaced. Thus, the standard recipe for a general history of logic repeatedly generates presentations of the rise of modern logic as a rather technical expansion of traditional logic. The resultant standard presentation of modern logic, then, is as an inevitable natural outcome of the ancient system meeting the modern inclination for the formal. In this book I explain why I think that the prevalent recipe for a general history of logic is so wanting. And I offer preliminary suggestions as to how to replace it. I did not follow my own recipe and write a full-fledged history of logic addressing in detail the questions posed above. I only outline my answers. Little can be accomplished in a single book. Here, then, I merely accentuate the outline of a new general history, within which, I hope, decent histories of logic will be written one day. In one sentence, I suggest, that the history of logic from Aristotle to Tarski is the history of the collapse of the classic conflation of logic and empirical science. In one paragraph, my recipe for a general history of logic is this. Aristotle generated logic for it to serve the impossible demand to provide a guaranty for the provability of empirical science. He designed it as an answer to the sophist challenge to rational thinking. Remarkably enough, he succeeded. Yet he (and all of us who followed him) paid a very high cost: that of conflating empirical science with logic. The status of empirical science was thus saved at the cost of making it an inherent part of logic. This had a very undesirable trivial effect: any advancement in empirical science seemed to threaten the very consistency of traditional logic. Stagnation within both logic and science was then established. Analyzing the intellectual background that enabled this classic conflation is the task of the greater part of this book. The rest of it is dedicated to an analysis of the earliest moves that together comprise a breakthrough that ushered early modern logic and their intellectual background. I deem modern logic as the result of the gradual process of (1) recognizing the classic conflation, (2) undoing it, and (3) disengaging from it. The most important step in this process was of the extensionalist revolution in logic, which had occurred around the middle of the 19th century. As I show here, the first persistent representative of extensionalism is George Boole. Boole’s extensionalism had divorced Aristotle’s essences from logic proper by rendering legitimate classes within logic all conceivable – and even inconceivable – collections of particular objects. He introduced such basic clean-cut modern concepts as the universal class (the universe of discourse) and the complement class. Unlike others around him who were toying with similar ideas (most notably De Morgan, of course) Boole’s extensionalism was undaunted. He, thus, also discovered the empty class without which modern logic cannot be conceived, of course. By doing all this Boole provided logicians with the first logical apparatus

Introduction

xxi

within which discussion of the laws of logic was clearly distinct from all knowledge of the cosmos. This is the collapse of the traditional conflation of logic and empirical science. Boole’s extensionalist revolution in logic was silent: while it occurred, and even later, it was not noticed. When it was completed its triumph was too obvious to be denied, and so everybody took it as trivial and inevitable, indeed as too trivial and inevitable to be considered a revolution. In other words, Boole’s extensionalist revolution in logic was silenced by the standard recipe for a general history of logic that did not tolerate acknowledgment of the fact that a revolution in logic had occurred (it presented modern logic as, essentially, formalized traditional logic, thus forbidding open admission of the fact that a fundamental change of the aims and scope of logic had at all been suggested). Indeed, the name for modern logic, namely “mathematical logic”, comes to hint that the modern is an addition to the classical, not the rebellion against it that it was. Here an attempt is made to correct the philosophical injustice (to both Aristotle and Boole) that is the inevitable outcome of this oversight of rather apparent historical facts. I hope you will find what I have to say interesting and helpful despite its being somewhat odd in its display of my aversion, I am afraid, to grinding minutiae. Indeed, I did not intend this book for the lovers of detail for its own sake. Should you be one I regret to say it might very well irritate you. I avoid almost entirely exegetical controversies with colleagues, and in the footnotes along the whole book I mention the names of certain contemporary scholars only when deeming it inescapable: whether because I cite other peoples ideas and I am happy to acknowledge this, or because I wish to draw the readers attention to the canonic literature, or because I wish to correct certain prevalent ideas which might otherwise get in my way. I confess that I have often found the canonic literature at odds with some of the most basic observations that I make and that, I regret to admit, I find quite uncontroversial and so I do not know why I have to make them. (Let me mention but one example: the logical distinction between sound inference, namely a valid inference from putatively true premises, and proof, namely establishing a theorem, seems to me obvious and of immense historical importance and yet I find even admirable leading scholars, such as William Kneale, blurring it.) In this respect I have modeled my book upon the classic Gilbert Murray’s Five Stages of Greek Religion and Carl L. Becker’s The Heavenly City of the Eighteenth-Century Philosophers. I do not boast of standing in the same place as these illustrious scholars. I do follow their obvious, radiant penchant for the abstract. Dibner Institute for the History of Science, MIT Cambridge, MA, USA December 2003 Sapir College of the Negev Sderot, Israel July 2006

Part I

Preliminary Notes

Outline of Preliminary Notes

My apology: here a terminological terrain is cleared out, without which the task of writing and reading this book would be too laborious to undertake. In these preliminary notes I introduce the following eleven basic terms: Methodology-inthe-loose-sense (with no promise for success), Methodology-in-the-strict-sense (with such a promise), Epistemology (theory of knowledge), Conflation, Confusion, Inference, Valid inference, Sound inference, Refutation, and finally Proof, direct and indirect. My excuse is that these terms are not clear enough as they do not have a canonic version. Significant points in the literature in which they play a role are unclear. I am afraid I found it important to clarify them somewhat.

I

Controversies as Contests: The Birth of Intellectual Responsibility

Greece is the cradle of western philosophy. This philosophy comes with a few myths of creation. Let me narrate only one of them, the one that will appear repeatedly in my narrative. The Greeks were the first to admit disputes and controversies as legitimate, and even as desirable. Our world, so the myth goes, contains a manifold of cultures worshipping various gods, endorsing diverse theories about the cosmos, following diverse leaders, obeying different rules and customs, and communicating in a variety of languages. There is much to disagree about. And, so continues the myth, the Greeks first attempted a satisfactory delineation of the universal and its separation from the local, between that which is true by nature (universal truths) and that which is only true in virtue of local agreement (true by convention).1 They were the first to legitimize disputes and controversies without allowing conventional, patriarchal, wisdom to nip them in their bud. Two items were introduced in the previous paragraph: the dichotomy and controversy. The reason for their linkage is obvious yet still unstated. It is this: as such, disputes and controversies are ubiquitous; but without legitimation they are expressions of conflicts and so they are violent. The legitimacy of peaceful intellectual disputes and controversies grants, of course, intellectual freedom, and in philosophy this freedom is grounded in doubt. Befittingly, then, Greece is the cradle N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

3

4

Outline of Preliminary Notes

of systematic skepticism. The legitimacy of disputes and controversies regarding the desirable ruler is democracy. Aptly, then, Greece is the cradle of democracy. Controversies concerning the nature of the world belong to natural philosophy; when concerning the nature of virtues they belong to moral philosophy. Both fields were developed and cultivated in Greece. The institution of controversy was not central to archaic Greek culture, which, at least at the beginning, preferred to it cruder forms of contest (agon). It was not the love of controversy that made the ancient Greeks notice diverse cultures and compare them so fervently, but, rather, initially it was their competitive spirit, the love of winning honorable competitions. “Nothing defines the quality of Greek culture more neatly”, says Sir Moses Finley, “than the way in which the idea of competition was extended from physical prowess to the realm of the intellect, to feats of poetry and dramatic composition”.2 Thus, the Greeks were unique in their having gradually developed a combination of an enormous sense of cultural narcissism and a glorification of cultural superiority as well as individual excellence. Finley has shown that competition between Homeric heroes bordered on violence and yet was contained, since competitors shared a most remarkable code of honor.3 It was the code of excellence – of manifesting prowess and skill – a set of conventions regarding criteria of excellence, criteria that govern the contest and that the participants, the spectators and Homer’s audience all endorse as a matter of course. Notable among these rules concerns the indivisibility of excellence, the painful (or stimulating) realization that it cannot be shared. In each competition excellence is won by the one and lost by the rest: winning is impossible without loss. The Greek love and admiration of contest, of its generation of individual excellence, bore the rules of contest that were in turn applied to disagreement and bore the rules of controversy. Disputes in ancient Greece were therefore initially games which perpetuated a tradition of admiration for excellence – their rules being very much like those of ancient taboos that were steadily turning into (and recognized as) conventional institutions, such as law courts, – except that as things developed all this gradually changed. Initially controversy was nothing more than a sort of contest, a game of skill, but slowly and gradually it became a contest of truth. The truth that contestants aspire to cannot possibly be truth by convention, as truth by convention is (by definition) arbitrary and, hence, not contested (we are each entitled to follow our own conventions). So it must be about the truth by nature. This recognition transpired slowly. Philotimia (the love of honor) turned into philosophy (the love of wisdom). The task, then, gradually became that of separating truth from falsehood, by observing common opinions concerning the nature of things, opinions that are merely local and hence possibly false, and finding among them general ones that are universally true and hence fit to constitute a foundation to science. Can this be done? How? How can truth be sought and found? Homer’s Achilles is said to have established himself as more virtuous than his enemy in battle Homer’s Hector whom he killed in a more or less fair contest. But who is to decide the winners of the race for truth? How does this decision come about? It seems that additional rules for debate are called for, rules for determining a winner in the race for truth.

II Methodology is not Epistemology

5

What are these rules? What are the rules for resolving justly court disputes or controversies in the natural sciences? How can one distinguish truth from falsehood? Such questions invite the development of intellectual and moral responsibility. They have led the Greeks to the creation of Dialectics and Rhetoric, and later on to the formation of Logic, which is the subject matter of the present study. This study is not a part of logic but the story of the emancipation of logic from a variety of limitations imposed on it (consciously, semi-consciously and unconsciously) in fear of futility: of controversies with no winners and no viable results. The fear of endless controversies is a remnant of a world in which all great matters were decided by the gods and their emissaries. It is the fear of intellectual and moral responsibility in a possibly godless world. It is as frightening as the infinite has frightened many a serious thinkers. It constitutes a crucial background to western philosophy and in particular to the classic view of logic. As a result both logic and the views of it traditionally imposed a variety of limitations and directions designed for the specific end of securing final positive results in all debates properly conducted. This is the source of what I call here the (traditional) conflation of epistemology and methodology. It has shaped the history of logic, and the development of modern logic. In fact, modern logic begins when the tendency to conflate epistemology and methodology ends. This, then, is a study of the emancipation of our thinking from the limitations it imposed on itself in fear of its emancipation.

II

Methodology is not Epistemology

The aim of this preliminary is to distinguish between ways of pursuing definite goals: they can be with or without guaranties for success. This difference is of extreme significance when we discuss ways of pursuing the truth. Throughout this study ways of pursuing the truth without guaranties for success will be kept apart as far as possible from those with (alleged) guarantees for success. “Going there” and “being there” are different, of course. Surely conflating them is often reasonable. This is not the case, however, when your destination is “out-of-the-way”. Any study of the function of logic invites the above distinction, since logic is known first and foremost, justly or otherwise, as the theory of the method of generating science, or in short as the methodology of science, and there is a need to make clear the sense of “method” thereby intended. Let us call a method with a guarantee “strict”, or “rigid”, and the other “loose”. The literature, both ancient and modern, uses the term “method” (and equivalent ones) in these two senses, often without a clear-cut distinction between them. At times authors, especially ancient ones, use it, or more often any of its equivalents, even when it is not clear whether or not they promise a guarantee. In this manner methodologies in the loose and in the strict sense are often conflated. When it is not clear whether guarantee is expected or not, it is not clear whether the theory of method (i.e. logic) should incorporate an explanation of the grounds for the promised success. Thus, conflating the two senses of method may breed

6

Outline of Preliminary Notes

ambiguity and error about the ground. For example, the claim that the method of science is inductive may be taken to mean that a guarantee for finding of truth is procured through the theory of inductive inference. The task of the logician, then, would be taken to be that of explaining and describing how this is done. Or, to take another example, rational thinking (whatever it is) may be taken to lead to the discovery of truth (as was silently assumed by almost all those who held that man is rational). The task of the logician may then be taken to be that of explaining and describing how rational inference guaranties truth. One damage caused by the conflation of the two senses of “method” is a further conflation that is less conspicuous: that of method in the rigid sense with episteme, i.e. of means to an end with that end itself. Consequently, the subject of epistemology is historically an odd mixture of promises and their alleged fulfillments: it includes sets of criteria for identifying any truth, for guaranties for its attainment, and sometimes even a straightforward enumeration of general truths allegedly already attained, science. When way to pursue truth is conflated with guarantee to find it, and when guarantee is then conflated with truth itself, then theory of loose method, theory of rigid method and episteme, or science, are all conflated. Logic and science are conflated. Literally “epistemology” should mean the theory of knowledge, and answer the question what knowledge is and methodology should answer the question what method is, that is, how knowledge is to be sought (sought, not necessarily attained). But this terminological strictness is seldom strictly adhered to perhaps because classical methodologies and epistemologies were often presented as unquestionable knowledge, as episteme. Thus, ways, promises, guarantees, and results are often conflated in the history of philosophy in general, and were so in the history of logic in particular. All this may seem like semantical hair splitting. However, as we shall soon see, it is anything but terminological untidiness that we discuss and more than terminological tidiness that we offer. Curiously, often terminological tidiness and deeprooted conflations co-exist in an intriguing harmony. On other times the terminological untidiness is a symptom, an indicative of deep-rooted conflations of philosophical ideas and domains that had an immense influence over the development of logic.

III

Conflation is not Confusion

Let me introduce now the notion of conflation and the distinction between it and confusion. Consider Tom and Dick as different people. Identifying them is a mistake. Identifying them on occasion is confusion. Not noticing the question whether they are identical or different is a conflation. The opposite of conflation is what Plato called Diaeresis. The process of Diaeresis is that of rising the question whether items are identical and deciding that they are not (and perhaps noting and explaining the difference between them). Using the word diaeresis this way,

III Conflation is not Confusion

7

I present this book as a story of a process of diaeresis that took ages to conclude. The diaeresis in question is the undoing of the conflation of empirical science and logic and its influence on the history of logic. Modern logic was used to define the basic logical notions (“validity”, “proof”, “soundness”, “refutation” etc.) in unprecedented accuracy. Of course, as is usual in such cases, an immense gap was thus created between the modern definitions and their looser, vaguer, more elastic historical origins. This gap is an interesting challenge to historians: it is reasonable to wish to employ the modern refined notions in an attempt to understand the ancient text. Yet doing so impetuously, as every good historian knows, may result in crude anachronisms and risk of losing historical understanding. (This is, of course, not to object to anachronism but to warn against the rather obvious confusion that it may introduce.) Consider, for example, the following traditional thorny questions: What is the subject matter of Aristotle’s Logic? Is Aristotle’s logic a theory of proof? Is it a theory of sound inference, or, perhaps, of valid inference? Does Aristotle’s logic deal with inference at all, or, perhaps, with truth conditions of conditionals of a special type? Perhaps it deals with none of the above but with some rules of the modern technique called “natural deduction”? Historians and philosophers of logic often dispute these matters, each arguing for the correctness of their own pet interpretation of the Aristotelian corpus, all of them with this or that degree of justice, of course, while making it difficulty to see their reading as irreplaceable.4 In hindsight (and in hindsight only) we may observe that Aristotle, incredibly ahead of his time as he was, naturally and even unavoidably, conflated what we now call proofs with what we now call sound inferences from mere empirical conjectures. Naturally and even unavoidably he conflated what we now call sound inferences with what we now call valid ones. Naturally and even unavoidably it is impossible to identify exact modern parallel of the items discussed by him. We should not forget that he had no access to the 20th century tool kit. Naturally and even unavoidably then, Aristotle subtly conflated what we know call empirical science, what we know call metaphysics and what we know call logic. Aristotle, of course, was not confused when he conflated what we nowadays call “proof” and “sound inference”. Certainly he was not confused when he seems at times to conflate what we now call “logic”, what we now call “empirical science”, what we now call “epistemology”, what we now call “methodology” and what we know call “metaphysics”. Many aspects of these fields could hardly be meaningfully distinguished from his point of view. (Famously there is not even a single Aristotelian term that designates the entire field that we now call “logic” – the subject matter of the collection of works we now call “Aristotle’s Organon”.) The term “conflation” (rather then “confusion”) aptly describes this delicate situation. For confusion presupposes that at some conscious level there is a more or less clear knowledge of the distinctness of the confused objects. Indeed, confusion is the replacement of one distinct object with another, by mistake and by neglect. Conflation, by contradistinction, is not having made the distinction that later generations made, or even took for granted. And, to repeat, Aristotle could not have made a sharp distinction between logical concepts he could not divine. Conflations,

8

Outline of Preliminary Notes

therefore, are far more elusive than confusions. They presuppose a somewhat amorphous setting which is far more subtle. To explain a given conflation one must first explain the intricate background information that we have at our disposal and that our predecessors lacked, and the background information, conceptions and misconceptions, which prevented them from making the sharper distinctions which we now make. This task often amounts to reconstructing a comprehensive (and in hindsight) elusive world-view. This reconstruction must include theories, known and forgotten, accounted for and not-accounted for, overt and latent, conscious, semi-conscious and unconscious, public and private. A comprehensive nonanachronistic intellectual historiography of any field should pertain to the study of these subtle settings. It ought to reconstruct conflations as what they were, not hide them, nor present them as the confusions that they are today (or in hindsight may seem to have been). We cannot ignore that there exists an enormous gap between modern conceptions of logic and ancient ones. This would render our historiography anachronistic, or apologetic, or both. Allow me then to repeat my point from a different angle and to advance it a little further. Our knowledge is often presented as cleaner than it is. It is presented as a structure of some sort, or at least as an edifice under construction. Descartes has made this metaphor so common, that we hardly ever think of it critically. Science is a sky-scraper, or a tower of Babylon. One day it will reach all the way up to heaven. These metaphors are misleading: real clear-cut foundations (or axioms), building blocks (or theorems), and engineering rules (or derivation rules), rarely exist in the realm of knowledge. We sometimes hope that our knowledge would be arranged deductively, as Descartes hoped it would be, but this, for all we know, is a mere yearning. The various edifice metaphors are a means for expressing an ideal, then, and they sometimes make us forget that this ideal is just that: a yearning for the unreachable. In this sense they are misleading metaphors. My aim is to emphasize the importance for all intellectual historiography of the fact that in so many cases we are ever short of achieving the deductive (sometimes dubious and often obstructive) ideal. Descartes’ metaphor is too sharp to be the right one. I prefer to talk about something less clearly contoured, something like a gradually dispersing cloud. I think that it is more instructive, slightly less misleading. The picture that I portray here will be very partial, of course. I am not trying to replace one myth with another, but rather to make a bit clearer the point of this section, namely, that our knowledge, to the extent that we have any, has no foundations. Instead it has fringes. It can be likened to a land surrounded by a terra incognita, covered in mists. Conflations dwell in these mist covered fringes. They are not real entities but rather they are the vague shapes that we recognize, or seem to recognize, in the mist from afar. Conflations do not support our knowledge (as foundations are supposed to do) but rather softly cradle it, as a set of myths cradles the moral and existential makeup of a society. Conflations are vague intellectual drives and penchants beyond our intellectual horizons, engulfing them silently. They faithfully do so until our horizons expand. We habitually explore our intellectual horizons, perhaps in an attempt to inquire if the

III Conflation is not Confusion

9

vague shapes we thought we recognized in the mist are really there, perhaps in an attempt to insure that they can sustain our intellectual horizons. But one cannot explore these vague shapes. The morning sun drives them away. As soon as one approaches the vague shapes in the fog they dissipate. Any attempt to turn a conflation into a clear and distinct axiom, ends in its evanescence. If the above paragraph contains some truth regarding certain aspects of our intellectual development (personal and social), then historiography of these developments should consist of an attempt to re-create, to the extent that this is possible at all, the subtle conditions that enabled past conflations. This task, by and large, can never be completed even if we have all required information at hand, since intellectual mists are even more difficult to reproduce than real ones: we should reconstruct not only lost theories, lost drives and lost penchants, but those which we grew out of and can no longer sustain. Previous conflations, like childhood conceptions, are something we normally grow out of. Reproducing them means virtually the turning back of time: re-creating and living a lost childhood. We can certainly study our childhood as adults but we cannot return to it. Ignoring conflations results in inferior historiography. Accounting for them in the full is by and large impossible. Inferior historiography presents conflations as naïve confusions, or it ignores them completely, or it explain them away apologetically, thus presenting our historical subjects as wiser and greater than they actually were. It presents our historical subjects as wiser and greater than anyone can be expected to be: as transcending their own intellectual backgrounds. Clearly, those resorting to such readings are the ones who truly belittle their historical subjects, for they do not deem them great enough to do without such readings. In modern introductions to logic discussion of Aristotle’s logic occupies but a few pages. It is very important to note that it contains all that we can get out of Aristotle, whereas tradition got out of him much, much more. Aristotle’s logic is part of an extremely subtle system that is at once outstandingly ingenious and problematic. Those parts in it that can be roughly called methodology are secured by an appeal to a metaphysical framework, scientific theories and a fair amount of commonsense that should have been its results rather than its foundations, but that somehow infiltrated into it by various means. Thus, methodology, epistemology and science are subtly conflated in Aristotle’s system: at crucial points he does not seem to have noticed the importance of making a sharp distinction between them, an importance that we so much take for granted that we may easily ascribe it to him by mistake. Had Aristotle conflated epistemology and methodology clearly and openly, he would not have done so at all (as there would have been no conflation to begin with). The move is obviously circular. Aristotle was a master observer and critic of cases of circular justifications. (Few ever matched him in this respect.) So, circularity leaked into his system silently and subtly and indirectly and unnoticed by means extremely difficult to pin down. We will observe some of the more influential ones, and some of their background, as we go along. We will then see how they shaped logic and its development.

10

Outline of Preliminary Notes

IV Proof is not Sound Inference It is not easy for the lay mind to realize the importance of symbolism.… The fact is that symbolism is useful because it makes things difficult.… … since people have tried to prove obvious propositions they have found that many of them are false. Bertrand Russell Recent Work in the Philosophy of Mathematics, 1901

This section is devoted to presenting some basic logical terminology. It is intended to bridge between faithfulness to history and accuracy by up-to-date standards. My aim then is to emphasize the distinctness of epistemology and methodology while explaining how easy it was to conflate them. In particular, my aim is to stress the distinctness of the empirically sound as very different from the logically sound (at the very least because we admit with ease that our forefathers took as sound what we at times are happy to reject but have trouble doing the same about the logically sound). And this is the source of the historical difficulty in explaining the nature of indirect proof. An inference is a non-empty finite set of sentences one of which is a conclusion (today, some logics study infinite inferences, but discussing them here is unnecessary). The rest of the sentences are its premises. A valid inference is an inference that has no counter-example. A counter-example to an inference is an inference with the same logical form, true premises and a false conclusion. Logical forms will not be discussed here. An inference that has a counter example is invalid. A refutation is a valid inference with a putatively false conclusion. (Hence, at least one of its premises must be recognized as putatively false, or, in other words, the conjunction of its premises must be recognized as putatively false). A sound inference is a valid inference with premises that are all putatively true (hence its conclusion must also be recognized as putatively true). The use of the ambiguous term ‘putatively true’ is deliberate. The term covers a range of historical phenomena that ooze with conflations of the extra-logical with the logical. Even in highly formal logical systems it regularly envelops at least two distinct senses: valid inferences from theorems and from empirical truths. Historically, the situation is much more elusive. For the search for the nature of sound premises takes us straight to the story of self-evident judgments. Clearly, some but only some of the propositions historically deemed self-evident judgments later won the status of logical truths (tautologies). Others were demoted to the status of merely empirically sound propositions, and at times even false ones. Consider the syllogism. Tradition views it as, more or less, valid inference from self-evident judgments (essential definitions). But what are these? Are they empirical, informative, conjectures? Are they vacuously true nominal definitions? Tradition conflates them, which is the source of the trouble. A sharp distinction between the logical and the empirical would have aided us greatly here, but it is not always the case that such sharp distinction existed and/or was fully recognized and/or was fully taken account of. I shall argue that it is the merit of Kant and later of Bolzano and Boole that they tried to be careful about it. Many putatively sound inferences had hidden

IV Proof is not Sound Inference

11

premises, whose status was not even noticed. These hidden premises were discovered only after standards of logical rigor greatly improved. This is an important point: the standards of rigor change whenever it is suspected that certain self-evident judgments are not sufficiently self-evident. Thus, many a putatively sound inference was exposed as a conflation of the logically and the empirically sound. Examples will occupy our attention later on. Inference (valid, sound or refuting) is not to be confused with proof. A proof is a certain collection of inferences taken together. A proven sentence can be placed as a conclusion to any set of premises (even to an empty set of premises, or to a false premise or premises) and the result will always be the same: a valid inference. Such an inference will be recognized as sound if and only if its premises are also putatively true. But, and this is a crucial technical and historical point, the inference does not have to be sound. For example, a proof by reductio ad absurdum, also known as indirect proof, is never based on the putative truth of its premises. (Example: A; if A then B; if A then not B. Therefore: not A.) Thus, strictly speaking, sound inferences and proofs are distinct logical entities and must not be confused. In the empirical study of nature presumed sound inferences may be discovered to be unsound, when their premises are refuted. In a similar manner refutations are never a final matter, in those contexts, since they can also be refuted. And they frequently are refuted (some view this as “progress” nonetheless, and the debate about this point is still open).5 In contrast, it is not contested that if a given inference is valid it will remain so even if we provide different valuations (assigned truthvalues) to our sentences. And a proven sentence, by definition, cannot be refuted. Within modern formal systems the distinction between sound inference and proof is usually inconsequential. When we study the foundation of mathematics, for example, it is reasonable to assume that any sound inference (from axioms) is also a proof (and it can be re-constructed with ease as a refutation, a reductio ad absurdum type of proof, with minor variations). Ultimately, no logic is without certain presuppositions and the point of studying these presuppositions by means of that logic is usually that we take them to be (at least) putatively true. The word “ultimately” in the last sentence is crucial: some systems, natural deduction systems in particular, have no axioms at all, of course, but this is only because the inference schemata that they use are expressions of meta-theoretical presuppositions about truth preservation). Thus, for modern logicians it is often unimportant to distinguish sound inferences and proofs (although some, like the intuitionists, deem it crucial to do so for reasons that will not be discussed here). Significantly, the historical discussion took place long before formal systems (in the modern strict sense of the word) were known. Thus, logical and extra-logical notions were historically conflated, and then proof by reductio ad absurdum was very confusing. Aristotle himself gives wonderful examples to indirect proofs, but he could not fit them neatly into his logic because syllogisms are sound inferences and indirect proofs are not. And yet, although it is difficult for us to imagine, he seems not to have noticed this. In this limited sense, his logic is, by his own standards, incomplete and perhaps even initially outdated. As premises of an indirect proof we

12

Outline of Preliminary Notes

choose ones that we suspect that they are inconsistent, of course. A contradiction is no essential definition, of course. And so, the modern tendency to ignore the distinctness of sound inferences and proofs dulls the bewilderment that this last point generated. So much so that the old bewilderment bewilders many a new historian. This is fascinating and even alarming. I hope I am pardoned for troubling the reader with such minute, intentionally trivial, observations. My reason is simple: before they were noted conflating methodology and epistemology came naturally. Historically, proofs, especially indirect proofs, shimmered with an amazing promise: they seemed to have established certain truths once and for all, “no matter what”, without even admitting other propositions as true. They thus seemed irrefutable in principle even when their conclusions were empirical conjectures. Some of the best studies of Aristotle’s logic play down the distinction between sound inferences and proofs.6 They do so in an attempt to present an accurate historical description of Aristotle’s logic, yet without even warning the reader that from the purely historical point of view, what they do is sacrilegious: their modern apparatuses cover up for ancient conflations. I hope, then, that the reader’s awareness to the fallacy implied here will simplify my work. It is clear that the proponents of ancient logic were largely unaware of the above distinctions, as trivial as they may seem in hindsight. Thus, ancient logicians had the (sometimes dim) impression that they were actually proving (“no matter what”) informative empirical conjectures, when in fact they were at most securing them on the condition that other empirical conjectures are true. Even the great Leibniz made this mistake and as an essential part of his metaphysics, as we will see later on. The more logic became a distinct study, the more it became crucial for its students to demonstrate in a clear manner their claim that it secures final knowledge of the universe. Yet no empirical informative hypothesis is ever logically proven. Is the history of logic the history of an attempt to attain an impossible ideal, then? This is what I will attempt to show here, in a sense at least until Leibniz, in another sense at least until Kant, in another sense at least until De Morgan and Boole. The list can continue, for this impossible ideal had persisted in ways that are less directly relevant to this present study in a sense at least until Frege, and in yet another sense at least until Russell. Let me stop there. (As Picasso once said “when one paints a portrait one must stop somewhere”). No empirical hypothesis is ever logically proven. Admitting that this trivial truth is the trivial truth that it is, is a rather late development in philosophy and an even later development in the study of logic: it was first stated clearly perhaps by Hume (early 18th century), and Perhaps by Kant (late 18th century). It was first incorporated into logic proper perhaps by Boole (mid-19th century), perhaps by Frege (late 19th century) and perhaps only by Russell (early 20th century). The reader may suspect that I ignore ancient skepticism. Not so as I will explain soon (sections 2 and 3). Let me stop here. The extensionalist revolution in logic, this is the theme of this book, has affected the discovery that epistemology and methodology are utterly distinct, and that they are better kept apart. Until the 20th century the best defenders of reason against

IV Proof is not Sound Inference

13

darkness among philosophers battled with it, for they wished to maintain that their methodology (whatever it was) secures their epistemology (whatever it was). It is, for example, a curious fact that no logician expressed an explicit deep interest in proving tautologies as such, until Boole and De Morgan entered the picture, and no logician succeed in proving many of them, until Frege and Russell entered the picture, and no logician provided us with an adequate understanding of what proof is, until Hilbert and Gödel, Tarski and Gentzen entered it. The extensionalist revolution in logic, then, is a requisite of the rise of modern logic.

Part II

Setting the Scene: Some Notes on the Pre-history of Logic

Chapter 1

The Mother of All Conflations: Parmenides’ Proof

In this study three general claims are made. (1) Aristotle’s logic conflates methodology and epistemology. (2) Aristotle’s conflation of methodology and epistemology has had a tremendous influence upon the development of logic. (3) George Boole’s extensionalism has undone the Aristotelian conflation. It, thus, cleared the road for modern logic, for the study of logic as a methodology-in-the-weak-sense. We should begin by a brief inspection of the background to Aristotle’s conflation of methodology and epistemology: the pre-history of logic. For the conflation clearly prevails in the most rudimentary conceptions of logical proof that we know of. It is appropriate to begin, then, with a few notes on Parmenides poem,7 for Parmenides has been called ‘the father of logical proof’ and even ‘the father of logical indirect proof.8 A great deal has been written and said of Parmenides and his “proof”. Little is known about the man and his theory, and little of this little relates to our present context. We will explain here the distinguished titles that he received yet bear in mind that there is at least one crucial difference between Parmenides’ “proof” and modern proofs: he supposedly proved a theory of the universe, whereas today we do not think that such theories can be logically proven. This curious difference should immediately draw our attention for it is an indication of a conflation: a conflation of that which in hindsight will be distinguished as logical with that which in hindsight will be distinguished as extra-logical. Parmenides’ proof, then, provides us with an early example to a conflation of (what we now call) logic with natural philosophy, or (what we know call) analytic reasoning with cosmology. What we nowadays call Parmenides’ proof is a fragment of the fragment that is Parmenides’ poem. In the part called “way of truth” Parmenides formulates his cosmological theory, and names it “piston logon” (which Szabo translates as “proven assertion”). The gist of the proof is an enigmatic seemingly a priori argument, which nevertheless yields a paradoxical description of the cosmos. Its formulation is of the highest simplicity. It can be formulated in one sentence: only that which is, is; and, that which is not, is not. There are a few formulations of the proof, within the poem, and, of course, quite a few differing readings of them and even some incompatible translations that support the differing readings.9 Some of these readings simply seem to stress different aspects of the proof while others seem to uncompromisingly clash. But it is generally N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

17

18

1 The Mother of All Conflations: Parmenides’ Proof

agreed upon that the main line of the argument is that it is self-contradictory to assume the existence of ‘that which is not’. Parmenides seems to have identified ‘that which is not’ with empty space (later called ‘the void’).10 This helps us stress the informative impetus of the proof: the cosmos, containing no void, must be completely dense, and so it cannot tolerate any motion (as there is not room no ‘empty space’ to move to). Similarly, it cannot accommodate plurality or diversity of any sort. It is motionless, timeless and indivisible. And, as Parmenides’ follower, Melissus, has added (correcting some of Parmenides’ religious idiosyncrasies about the infinite)11 it is an infinite-indefinite oneness in a sense of this term that defies imagination as any imaginable unity is imaginably devisable and likewise imaginably contains some diversity (as Boole noted) though perhaps not in reality (as Descartes insisted, when he said that there is only one real substance, namely God).12 Parmenides’ conclusion – that the universe is dense and indivisible – was, perhaps, not as unusual as it might seem at first sight to the modern reader, at least against the background of Greek thought: Greek philosophers who preceded Parmenides seem to have held this view, one way or another, or at least be committed to it and this is exactly what Parmenides had insisted upon.13 Already Thales seems to have held it (or rather to have been committed to it) when maintaining that the whole universe is made of one substance, namely water. Indeed, the first to explicitly and clearly maintain that the void must exist were the atomists: Leucippus and Democritus. Parmenides’ proof, then, is highly unusual and original not in its primary conclusion, but first and foremost in its structure. It is also of immeasurable importance in its unconditional embracing of its paradoxical secondary conclusions, such as the negation of (real) movement, which some of his predecessors seem to have been committed to, perhaps without being fully aware of, and, as far as we know, without even slightly admitting to it. Let us observe the structure of the proof, then. Its compelling power stems from its seemingly tautological, structure. It looks analytic and uninformative (to use a few anachronistic modern qualifiers). However, the crux of the proof is that the proposition expressed by this sentence is anything but uninformative: it yields a description of the universe. Some note that different meanings of “is” are conflated here (and correspondingly two different meanings of “is not”). In particular, they say, Parmenides does not seem to distinguish clearly between “is” in the sense of “predicated of” and in the sense of “exists”.14 In fact Parmenides quite explicitly identifies “is” in the abovementioned, conflated, sense as “that which can be thought of”, “that which can be expressed meaningfully” and, even as “non-empty space”. This point is repeated in the poem and is stressed by most of its standard translations.15 Consequently, Parmenides does not seem to distinguish clearly between “is-not” in the sense of “Non-being”, or “impossible-being” or “meaningless”, or “nothing” and, in the sense of “empty space”, or “the void”. The result is overwhelming in its implications: empty space is declared impossible because it is nothing, because it is an impossible concept. It is declared meaningless because of its meaning. The proof has been viewed as a crude form of reductio ad absurdum, since it shows that the very notion of an existing void is a “contradiction in terms”, it is an absurd proposition (absurd literally means “that which cannot be heard”): the terms

1 The Mother of All Conflations: Parmenides’ Proof

19

“void” and “existing” prove to be mutually exclusive. This is why Szabo regards Parmenides as not only the father of logical proof but also the father of logical indirect proof. Today the two sentences “The void exist” and “The void does not exist” make perfect sense to us. If, however, we allow the conflation of “the void” with “non-existence” the absurdity of assuming an existing void trivially arises. In modern, Boolean, terms we say that the intersection of any two mutually exclusive terms designates the empty class. The story of the recognition of the legitimacy of such an “empty” class, and of the terms that refer to it, is the gist of the history of logic as told here. (This monograph ends with Boole’s discovery of the empty class.) To Parmenides, however, the notion of an existing void does not merely designate an impossible entity: it cannot even be said to express a comprehensible idea. In the most general modern terms, then, we can say that Parmenides’ proof hosts a most subtle conflation of logical thinking and putative knowledge of reality (or, a conflation of semantical considerations and cosmological considerations). Of course, every indirect proof seems to make a “leap” from the study of a language to the ontology that it presupposes. If we wish to totally avoid the conflation of semantics and ontology, then, we must simply deny that such indirect proofs apply to the real world. (We sometimes say instead that they apply to any consistent model of it, which satisfies our suppositions). But the early philosophers were less interested in (and less aware of) such seemingly over-prudent restrictions: when they asserted (or implied, or assumed) that contradiction in terms is impossible16 they meant that it is impossible for contrary terms to designate a real entity, or for a contradiction to designate a real state of affairs (the two distinct readings of this famous maxim where not clearly distinguished until Boole). Indeed this minimal “leap” from semantics to ontology makes much sense to us even today. But the “leap” has cruder forms that are of great importance to the history of logic. We should now consider them briefly. The conflation of language and reality was dominant in all primitive societies, especially those imbued with magic and witchcraft (including ancient Greek society, of course). There, words were taken to have their hooks into things. Important referents had “real” or “true” names. These names were never (or rarely) uttered. Instead, conventional names were used to refer to those objects without the danger of breaking a taboo. The “true” name was considered a taboo because it had its hooks into a possibly dangerous referent. The “true” name was believed to have provided those who uttered it control over its referent because it had its hooks into it. And so the “true” name of revered deities, revered forces of nature, diseases, and their like were kept discrete. Many cultures today still deem the uttering of curses dangerous and the uttering of the true name of a deity calumny. It is a remnant of the faith that words promise control over reality by the very ability to cry them out, as the cries of a baby seem to command and bring food and comfort. Parmenides’ proof is a crucial point in the growing out of this magic imbued world view: it reveals its paradoxical nature by embracing its extreme conclusions. It makes it explicit that the conflation of semantics and ontology generates the view that there are no separate entities at all. And, because there are no separate entities language is useless and misleading. It is a part of the chimerical world of phenomena.

20

1 The Mother of All Conflations: Parmenides’ Proof

Parmenides’ proof, then, turns the conflation of semantics and ontology against itself: it shows that it entails its annulment. This is the important tradition according to which there is only one object to be named, and its extreme paradoxical conclusion: that language is a useless tool for coming to grasp with it. It is the gist of all mysticism, of course. Jewish tradition has it that Adam’s language, the first language, had one name (and one name only) for each and every object: the “true” name. It may seem at a first glance that this mystic tradition is incompatible with Parmenides’ annulment of language. The Jewish myth, however, cannot to be taken literally, of course: it is deliberately incompatible with our understanding of language, as in any languages one can (at least in principle) produce infinitely many synonyms to any name (for instance, by replacing it with a definite description and expanding the latter indefinitely). Thus the Jewish myth is an expression of a nostalgic sense of loss, loss of the union of the subject and its object in thought. It expresses a yearning for the paradise-like, womb-like state that had this union (paradoxically) within reach. (Borges ironically suggests that Adam’s language comprised a single word17; perhaps it was Parmenides “is”.) It is no surprise, then, that the paradise of the union of subject and object in thought was lost when curiosity led to the tasting of the fruit of the tree of knowledge. It was lost again, according to the biblical narrative when vanity led man to attempt erecting the tower of Babel (which, of course, is a metaphor for science, for the hubris of science). Alas, scientific knowledge and mystic knowledge do not fuse. Both are forms of hubris that seem mutually incompatible. And so we are left with the paradox that is Parmenides’ proof. To our own context the following fact is crucial: the proof virtually created the central problem of epistemology: the problem of bridging the gap between phenomena and the one reality underlying it. On the one hand, ever since Thales and his school, the search for hidden reality underlying all phenomena was the search for an explanation of phenomena. Phenomena were to be explained by (i.e. derived from) our knowledge of what the underlying reality is. On the other hand, Parmenides declared phenomena to be false and misleading. Phenomena were identified with multiplicity, with movement and diversity that do not exist. How can the false (phenomena) be derived from the true (reality)? Clearly, however little formal logic Parmenides had at his disposal, he recognized that the false could never be validly derived from the true. The very notion that phenomena can be explained by underlying reality was thus under a serious challenge. Conflicting incentives seem to clash in Parmenides’ proof. They do not seem to be resolved. (This is why Parmenides’ poem has so many conflicting interpretations: he is a rationalist, materialist mystic). Conflating semantical considerations with ontological ones leads astray from science to mysticism. Abandoning the conflation altogether leads to the view that we have lost our grip on reality, that we are trapped in the world of phenomena merely guessing what reality really is, and traditionally this too seemed to lead away from science (and into stale skepticism). (The view that science and skepticism are not incompatible belongs essentially to the 20th century.) A refusal to endorse either option to its extreme conclusions is the heart of the ancient fright of terms that have no reference.

1 The Mother of All Conflations: Parmenides’ Proof

21

The dread of terms that do not designate real objects will occupy our attention quite a lot. We must note already here in passing, however, that it manifests itself clearly in Plato’s problematic theory of knowledge, in his problematic explanation of the ontological status of the reference of false propositions, in his dismissal of the world of phenomena as not wholly real, and in his insistence that meaningful terms must denote real objects and that, therefore, abstract terms denote abstract objects (ideas), that are, hence, the only truly real objects (they are the only truly real objects because particular objects are not wholly real). More important to our limited context is the fact that the fright of terms that have no reference culminated in Aristotle’s logic. It is the crux of his notorious “existential import” (to use an anachronism). This will be shown here in some detail. It will be shown that Aristotle’s “existential import” is fundamental to his conception of logic, its tasks and its scope. It will be maintained that the ancient dread of terms without (real) reference is central to Aristotle’s conflation of methodology and epistemology. Aristotle’s “existential import” is explained and criticized later in this study. Here we should stress merely an important rational underlying it, which springs to mind when thinking of Parmenides’ Proof and its paradoxes. Unlike Parmenides and Plato, who seem to be (and are sometimes criticized for) condescending towards the world of phenomena, Aristotle boldly insisted that explaining phenomena is at the heart of science. Deriving appearances from underlying reality is what science is all about, he stressed. But one cannot explain phenomena and concede Parmenides’ way of truth according to which there is only one indivisible reality. This would mean (as Parmenides is understood to have argued) that empirical science is the discussion of the Nothing. There must be more than one underlying indivisible essence to it all, Aristotle concluded, if we are to explain phenomena. Thus, Aristotle resourcefully and ingeniously tried to import plurality into the world of reality: this is his grand matrix of essences. Importing plurality into reality is the essence of Aristotle’s “existential import”. I will return to this point in greater derail later on. Let us sum up. Parmenides is the father of logical proof by virtue of the structure of his unique argument and his unyielding endorsement of the paradoxical conclusions of the conflation of language and reality, a fundamental feature of scientific thought in its early stages. Of course, he too seems to have failed to neatly distinguish between non-reference, meaninglessness, inconceivability, and non-existence. (No one can be expected to do so properly before Russell resolved his famous paradox; not even Frege). However, Parmenides’ proof is justly taken today to be the first known case in which something that (perhaps, in hindsight) resembles natural philosophy is proven by something that (perhaps, in hindsight) resembles logical a priori reasoning. Today we easily distinguish the province of natural philosophy from that of logic. The extensionalist revolution in logic had insured that modern logic will no longer conflate the empirically informative and the logical. But in a world where this could not have been properly done an important tradition gradually appeared: the tradition according to which informative knowledge can be logically proven. This myth is but a refinement of the magic-suffused conflation of language and reality. When this myth was sharpened enough and stated clearly it became the

22

1 The Mother of All Conflations: Parmenides’ Proof

awe-inspiring theory that logic is the theory of the way of producing science, that it can be used to guarantee that universal laws are revealed to those who pursue them properly. This theory was to hold sway of logic and determine its growth, until the modern extensionalist revolution in logic replaced it.

Chapter 2

Early Disagreements Concerning the Power of Proofs: The Uses and Misuses of Dialogues

As Aristotle tells us, and as standard histories of logic agree with him, its inception is in his reflections over a peculiar Platonic procedure known as diaeresis. Diaeresis is a refinement of a common argumentation procedure known as dialektike. Let me tell you briefly my opinion of what dialektike and diaeresis are and why refining them was so important. Dialectics had sprung from reflections over the proper way of conducting the search for truth. Truth was sought by means of disputations, and already Plato’s early dialogues are reports of such disputations and of the search for rules for conducting them fairly. Many interesting (then prevalent) procedures are mentioned there, tested, criticized and improved upon. A classic example is the Protagoras, where Socrates goes as far as to walk out on the search (335c–d) when the rules of conduct are bent by Protagoras who “sails to the ocean of speech” instead of restricting himself to a strict ping-pong of clear questions and brief and precise answers. Later on, when the offer is made to assign a moderator (338a) that would decide the results of the search he again refuses to play along until crowd and interlocutors alike agree to let the conclusion be decided naturally, jointly (338b–e). When attempting to describe dialektike we should remember that its exact character in Plato is understandably somewhat unfixed and perhaps even unclear as he could not formalize it.18 There is little doubt that he identified the highest form of Philosophizing with it, but he lacked variables, the need for which is apparent in his dialogues, where Socrates repeats a sentence about artisans with the change of the art thus saying the same thing over and over again, once about carpenters and once about shoemakers and once about doctors. Aristotle, as far as we know, invented the variable. This revolutionary tool enabled him to easily do away with such repetitions, to generalize them, and, for the first time in history, to study them as such. It is the explicit use of variables that made it possible to study not only particular examples of dialektike but also what is known today as its logical form. Since Aristotle’s formulation of dialectics is the first known attempt at presenting its structure, all reconstructions of previous examples of the dialectical method are speculative. We do not even know if dialectics was a distinct method before Aristotle had formalized it (though Plato often refers to it as such). Speculations as to the nature of dialektike do cluster around a certain debate procedure, and this debate procedure, though rarely sanitized of what we see in hindsight as irrelevant N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

23

24

2 Early Disagreements Concerning the Power of Proofs

riders, repeatedly occurs in the discussions of Parmenides, Zeno, Socrates, Plato and the Sophists. The meaning of “dialektike” is, more or less, ‘the skill of thinking together’, or ‘the ‘know-how’ of discoursing in two’. It seems to have sprung gradually as a distinct procedure of interrogation, a proper conduct of a debate, up to, and including, its very end. It was a prevalent debate procedure. Apparently it was taken to be very effective. It was employed not only in private, as a game or as part of oratory contests, but also, and more seriously so, in public debates and in law-courts. It constituted a primary tool for conducting them and concluding them. The more or less formal procedure that is most commonly identified as the core of dialektike goes roughly so: suppose alpha and beta are debating over some legal matter, and suppose alpha claims that P. Beta (accepting P for the sake of the game) leads alpha to admit that if P is true, then Q must also be true, and then to the admission that Q is false. They then conclude that P must inevitably be false since it leads to plain absurdities. Beta, thus, led Alpha to admit that the initial assumption P was an error, by dialektike. In mathematics, as we have noted (preliminary iv), the refinement of dialektike into a rigorous procedure is known as indirect proof or a reductio ad absurdum. But the reductio is recognized today as distinct from ordinary refutation: it is a rigorous proof procedure practiced in formal systems – one in which a contradiction is formally derived from an inconsistent premise thereby proving its negation. And so we allow the identification of refutation with proof by reductio ad absurdum today only in restricted formal contexts. In other more loose contexts the identification is recognized as far from trivial, and in the empirical study of nature it is rightly deemed invalid since to refute a hypothesis is no proof (final and un-revisable) of its alternative. Strictly speaking, it is not even proof of its negation. We must stress this point since otherwise refutations might be taken as final, and they are certainly not: they are revisable, they can be refuted, and quite often they are. How much of this was known to Plato and his contemporaries? Has it influenced their view of the methodological power of dialektike? Before answering these questions, let us complicate things a little more by stressing yet again a point that we already stressed in the preliminaries. Let us remember that there are intermediate levels of formality and that the less formal the context, the more problematic the coupling of refutation to proof. And we are not always clear today where we should draw the line. Intuitions can go both ways. For example: the books of Euclid are written in Greek, which is far from being a formal language, yet many would maintain that they contain obvious cases of reductio ad absurdum (and they certainly comprise a semi-successful, and possibly conscious, attempt to slice out parts of the Greek language and to turn them into a formal language). There are, to be sure, some formal representations of Euclid’s geometry and also empirical interpretations of those representations, but the distinction between the two stems from developments remote from the world of Euclid. (What we do in non-formal and semi-formal contexts is to be judged, from a strictly logical point of view, as somewhat nebulous, even though mathematics often, and perhaps even

2 Early Disagreements Concerning the Power of Proofs

25

normally, is to be found in such contexts, as Lakatos maintained, since it is hardly ever quite formal.)19 We are thus in a delicate spot: in hindsight we deem almost trivial difficulties that the advocates of ancient logic may not even have realized properly. Plato’s examples of dialectics are typically more confusing than Euclid’s semi-formal proofs by reductio ad absurdum, because they concern metaphysics and ethics and politics, all of which notoriously less yielding to formality than geometry. Bearing this in mind we should nevertheless try to examine the following major question: assuming dialektike was recognized more or less as a fixed debate procedure akin to the procedure described above, what knowledge can it bring? It seems that it was considered the chief method for pursuing the truth and so it seems that deciding its scope is crucial for deciding the status of science. In the previous chapter we considered Parmenides’ proof as a case of dialektike, a kind of reductio ad absurdum of the possibility of void. Presumably this was noticed by Parmenides’ disciple, Zeno of Elea, who is commonly regarded as the father of dialectics. It is, of course, reasonable to assume that at least some Pythagoreans have used it before him, as many have noted. (Popper has added that the followers of Thales did so too a generation or two earlier, when they refuted each other, though it seems that their practice of it was not as explicit, and perhaps it was not explicit at all.20) In any case, already Plato’s Socrates calls Zeno the father of the method of indirect proof (Parmenides 128a) and of disproof (Phaedrus 261d) and Aristotle confirms this verdict.21 To be sure Zeno’s work includes astounding examples of refutations in the mode later to be generally recognized as reductio ad absurdum: he not only refuted the critics of his mentor (who claimed that motion and diversity exist); he clearly alleged that by doing so he had proved his mentor’s theory. He studied the premises of his mentor’s critics and derived absurd assertions from them. Since the distinction between refutation and reductio ad absurdum was not born yet, it is likely that he was taken to have proved his master’s theory (that the universe is a motionless dense unity) and to have refuted the claims of his critics (that the universe is plurality in flux) by the same argument. Socrates, however, had a radically different view of what can be achieved by means of dialektike: in his practice, he said, dialectics served him as a tool for the search of truth, yet almost never as proof techne. This point is crucial. Unlike Parmenides and Zeno, he repeatedly denied that his dialectical quests are final and non-revisable. This fact is well known as a philosophical anecdote and as a Weltanschauung but is rarely noted from the strictly logical point of view. It is well known that Socrates self-proclaimed aim was to facilitate doubt in those deeming themselves in the know. But it is less noted that this entailed a crucial difference of opinion between him and the Eleatics regarding the power of (what would later become) Logic: while the Eleatics seem to conflate refutation with proof of the negation of the refuted proposition, Socrates never views refutation as more than just that, a mere refutation. Therefore he has claimed to have no knowledge but that of his own ignorance. Zeno, then, saw dialectics as methodology-in-the-rigid-sense (that leads to knowledge of the universe) and as part of epistemology, whereas Socrates considered it as methodology-in-the-loose-sense (that does not): he saw

26

2 Early Disagreements Concerning the Power of Proofs

dialectics as a way of pursuing the truth, perhaps as a distinct procedure for pursuing the truth, but as one that does not guarantee final positive results. Sometimes, Socrates fascinatingly emphasizes that his dialectic refutations are the result of an impersonal force. He describes them as a logos with an independent will that frequently takes over him, disregarding his own will as well as that of the person being refuted. This rational infatuation (a deliberate oxymoron) is an obvious reverse allusion to his forefathers ate (divine infatuation).22 Tactless and indifferent to anything but the rational of argument, blind as fate herself, Socratic logos shatters the claims it encounters. Tactless and indifferent, blind as fate herself, it draws absurd conclusions from seemingly down-to-earth observations. Richard Robinson, who notes this in his famous study of Socratic elenchus,23 claims it to be a mere manifestation of Socratic irony, for, so he contends, how could an indifferent (and hence impartial) skill continuously lead to refutations, never to positive results? This question is very sensible, of course, and yet, it should be observed that, it is guided by Robinson’s own conflation of refutations with indirect proofs. Clearly, if refutations are not proofs, direct or indirect, it is not at all odd that Socrates is continuously refuting propositions without (or hardly) ever proving one. Indeed, this is perfectly coherent with Socrates’ overall philosophical outlook (as well as with the modern understanding concerning the limits of logic, and the limits of knowledge in empirical science). There is certainly an irony here, but much of it is self-irony for it is clear that not even the great Socrates (and not even the great Socrates with the aid of his never-erring-whispering-daemon), could bend his dialectical skill so as to yield a proof of a comprehensible definition of some virtue, as some sophists claimed to have done. He seems to have had a solid philosophical ground to believing that this was impossible. No less important, is the fact that Socrates’ ironical confession constitutes a rare testimony to dialektike being recognized as a more or less definite technique. It is recognized as a distinct tool, separate from the will and aims of its users, and from given cases that exemplify it. This is a ground-breaking declaration, for Aristotle had yet to introduced his logical variables and consequently the abstract consideration of dialektike, apart from given cases that exemplify it, was still very difficult. Socrates’ insistence that his technique is an impersonal procedure is, then, the closest he gets to deeming it an abstract logical technique. Some of Plato’s early dialogues clearly achieve results that are not entirely negative despite being perplexing, of no real practical value or even downright paradoxical. Such, for example, is the final definition of courage achieved in Laches according to which courage is knowledge of all goods and evils (199c), and the final formulations of the definition of temperance in Charmides as knowledge of knowledge (166c), knowledge of ones’ self (167a), knowledge of what one knows and does not know (167a), and knowledge that one knows and does not know (170d). In these fascinating peak moments, Socratic dialectics is subtly confronted with its own (logical) limits, as it yields what seem to be dead-ends: useless and even paradoxical conclusions (169b–c). What are we to do when our logic yields illogical conclusions? The Socrates that we meet in these moments is acutely selfaware of the graveness, helplessness and humor of the situation. He is clearly careful

2 Early Disagreements Concerning the Power of Proofs

27

not to derive any nihilistic conclusions of the sort that the sophists embraced as the grounds for their relativism. Even his divine daemon, he repeatedly stresses, has never offered him “positive” definitions, knowledge, only refutations. Of a different category, perhaps, is Socrates’ famous geometrical inquiry with the slave in Meno. It seems to be an exception, for they certainly achieve some form of proof of a very limited case of Pythagoras’ theorem. However it is generally agreed that this part of the dialogue (as well as the discussion of the theory of Ideas which it serves) may be less genuinely Socratic than other parts of it, and was possibly a result of a later revision and or insertion (to the extent that such subtle matters can be decided). Self-irony aside, it is clear that Socrates shared with Zeno the conviction that dialectics is the best means we have for the pursuit of truth. At least some sophists seem to have confronted both and laid down the first important challenge to rational speculative philosophy: they claimed (and demonstrated) that it was possible to refute any sentence as well as its negation. They boasted of the ability to refute any thesis whatsoever, regardless of whether it was true. They thus undermined the very notion of dialectics as a means of pursuing truth (with or without guarantees for success). If it is possible to refute any sentence as well as its negation, then dialectics is completely worthless as a tool of searching for the truth and it certainly isn’t a tool that guarantees final positive results. How, then, can a rational debate be conducted and decided? Is it at all possible? Is science possible? Is the use of reason as a tool for the pursuing of ultimate truth a mere misuse? Gorgias, for example, aimed his attacks directly against Parmenides’ philosophy. He ridiculed him sarcastically in his “On that which is not, or on Nature”.24 Assuming that his listeners where acquainted with Parmenides’ Philosophy, he eloquently demonstrated its negation. He argued that nothing exists, that if something does exist, then it cannot be known, and that even if it can be known, its knowledge cannot be communicated. Nature, then, is the Nothing, he showed. It is easy to mistake this semblance of a proof for a mere poignant parody. However, when taken, as it should be, together with the actual Parmenidean counterpart that proves the exact opposite thesis, it is one of the most profound philosophical attacks ever produced against rational debate as a means for pursuing the truth. For it is a demonstration that proofs are useless, that anything can be proved.25 Gorgias’ anti-philosophical dialectic was generalized and systematized by his followers prominent among whom was his student Isocrates, who may have started the tradition of viewing dialectics as intellectual gymnastics, a tool guaranteeing superiority in public debates, regardless of the truth of the matter at hand and in frank contempt for the arrogance of philosophical quests as opposed to commonsense. (It is not unlikely that some such sophistic attack on the very logic of Parmenides’ proof and its compelling power was the license that the atomists needed to invert it. For where Parmenides assumes that the void is nothing (and hence) that it does not exist (and hence) that the universe is dense (and hence) that movement is impossible, the atomists freely take license to start by negating his conclusion. Thus they freely assume that movement exists and therefore the universe cannot be dense, from which it follows that the void, must exist.)

28

2 Early Disagreements Concerning the Power of Proofs

Plato justly found this startling: if the sophists are right and everything can be proved, then perhaps the love of wisdom is a mere fantasy. He launched a propaganda campaign against them. It was one of the most successful propaganda campaigns in history. He is largely responsible for the derogatory connotations of the very word “sophist”.26 He dismissed them as troublemakers, spreading corruption and confusion by empty rhetoric (another word he may have either invented or redefined as derogatory in the course of his attacks).27 He portrayed them all as moral relativists, political pessimists, and even shameless opportunists motivated by sheer selfish objectives. Sometimes he went as far as portraying them as hopelessly arrogant nihilistic fools (as in the Euthydemus). He refused to acknowledge that their relativism, however morally deplorable, was grounded in a serious challenge to rational philosophy. And yet he accepted their challenge all the same, for it was, to repeat, a serious challenge. He was clearly dissatisfied even with Socrates’ solution – the unending quest for truth by refutations of the common opinions – and hoped to reinstate refutations as indirect proofs, in Eleatic fashion. Accordingly, in some of his late dialogues, a cautious attempt to formulate Eleatic dialektike takes over. A striking example is the dialogue Sophist, where a stranger (the “stranger from Elea”) proves axioms of Plato’s theory of ideas, by dialectical means, while Socrates silently steps aside and pays heed. This brings us to the idea of diaeresis (literally: splitting in two). The very idea of successful proof by dialektike, explains the mysterious stranger, assumes it. Proof requires the imposition of a strict dilemma (a strict ‘either or’ choice) on one’s opponent, so as to clinch the identification of refutations with indirect proofs. If we agree, for example, that Socrates is an animal and furthermore, that all animals should be divided exclusively into rational and non-rational, then the refutation of the assertion that Socrates is non-rational is also proof that he must be rational. This form of argumentation, Aristotle tells us, was a fundamental practice in Plato’s academy. Indeed, when Aristotle defines dialektike he incorporates Plato’s refinement into his definition by demanding that all dialectical quest begin with a reply to an ‘either-or’ question (An. Pr. 24a22–24a25). In hindsight, it is easy to see that at most, Plato was reflecting and specifying in a clear and rigorous manner an assumption that Zeno has already made. We do not know whether or not Zeno had also assumed it explicitly but it is certainly reasonable that he was well aware of it (and Gorgias clearly is aware of it). And since this seems to be all that Plato’s response to the sophists challenge amounts to, the challenge remained. Plato’s dismissal of the sophists was, thus, more an expression of limitation, and personal disgust, than a suitable philosophical response (it bears some resemblance to his alleged dismissal of Antisthenes as slow-minded, and of the atomists, whom he never even mentions by name and always refers to by derogatory descriptions). He could not undo the sophists’ demonstration of the possibility to refute both thesis and anti-thesis, even when a strict dilemma is imposed. Their demonstration was that the imposition of a strict dilemma is useless because both thesis and anti-thesis can be refuted. Indeed the explicit imposition of the dilemma only strengthens their claim that proofs are impossible. To the best of our

2 Early Disagreements Concerning the Power of Proofs

29

knowledge, Plato never answered this challenge in a satisfying manner, indeed (in most contexts) no one has. Let us sum up then. Searching for early conflations of the logical and the extralogical, we have given a brief account of the scope of the argumentation technique known as dialektike before the inception of logic with Aristotle. We noted that the controversy regarding the scope of logical techniques was central to the development of philosophy even before logic was explicitly formulated, and before logical techniques were cleared of irrelevant riders. For the development of logic was the heart of attempts to answer the following questions: Can truth be discovered by reason? Is dialektike a mere loose method of refutation or is it the rigid proof method guaranteeing that truth will be found? Perhaps it is neither, since it is both, as anything can be proved and, or, disproved by its use? The arguments of Parmenides, Zeno and Plato, exhibits a tacit conflation of early notions refutation and indirect proof. They thus exhibit a tacit conflation of the two senses of methodology, the loose (refutations) and the rigid (indirect proofs) and even a conflation of both with science per se (by diaeresis) and hence, even with epistemology.

Chapter 3

The Sophists’ Challenge

Parmenides and Zeno both tried to prove the truth of Parmenides’ philosophy by means of (roughly) isomorphic logical arguments. Let us set aside for a moment the difficulties implied by the very notion that logical considerations may lead to absolute knowledge of the cosmos. We are then immediately taken by this incredible discovery. What is so marvelous and apparent is that Parmenides, Zeno and Plato alike share awareness to the isomorphism which today we know constitutes the heart of logic: sound inferences and refutations share a logical form. Sound inferences and refutations are distinct uses of the same inferential form, the same logical structure. The same valid inference is at once sound when taken to have true premises and is a refutation when taken to have a false conclusion. The isomorphism was first formulated in a revolutionary, though somewhat inelegant, manner by Aristotle, in the opening to his Topics and again in the opening to his Prior Analytics.28 But it was already noticed and celebrated in the most epic philosophical episodes of all times: the meeting of Parmenides, Zeno and the young Socrates in the famous opening of Plato’s Parmenides. There, the young Socrates observes (somewhat sarcastically), that Zeno is at most an innovator of style: his claim for fame is merely his reformulating the Eleatic philosophy (giving it the form of a refutation rather than that of a sound inference, or of indirect proof rather than a straightforward one). Zeno, by the way, is quick to admit this (in the dialogue) in a manner that is perhaps humble and perhaps averse: he claims never to have pretended to offer more than mere structural variations over the Parmenidean theme. Then occurs (in the dialogue) something which has puzzled scholars right up to our own time. First, Parmenides repays Socrates for irritating his young bosomfriend by producing certain criticisms of Plato’s theory of forms. Socrates is compelled to admit that they are both valid and alarming. Then Parmenides offers to initiate the confused Socrates into the subtleties of an expanded form of dialectics (135d–136a). In a shielding, fatherly tone, he recommends to Socrates to practice that method thoroughly before taking upon himself any serious philosophical pursuits. The method he recommends is distinctly sophistic and self-destructive: Parmenides offers Socrates to strive and reduce to absurdity not merely every thesis under consideration but, also, its negation. This is what readers of the dialogue find so hard to digest. The rest of the Parmenides is an elaborate demonstration that such N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

31

32

3 The Sophists’ Challenge

intellectual suicide is possible: Plato’s Parmenides shows that contradictions follow logically from his own theory (in the form of the proposition ‘unity exists’) and from its negation (the proposition ‘unity does not exist’). The great Gorgias himself could not have formulated a mock criticism of the possibility of rational philosophy better. Plato’s Parmenides has been made into a sophistic ridiculer of the historical Parmenides. Parmenides is a difficult, enigmatic text whose message is notoriously controversial: it is the most complete testimony we have of Parmenides philosophy, and we are not even sure that it is a serious one. Plato’s presentation is certainly not humorless, yet it is very hard to tell where humor stops and sober philosophy begins (or if and when they are mutually exclusive in Plato). The dialogue (mainly a monologue really) features what seems to be a proof of the hopelessly problematic character the very method that is supposed to secure rational thinking from the sophist challenge, demolishing in its way Parmenides’ own philosophy as well as key concepts in Plato’s own theory of forms. Then the dialogue ends, abruptly, in silence, with no “positive” results, no explanations, and no concluding remarks to lessen our confusion. Could the dialogue be Plato’s attempt to defend what he recognized as the paradoxical (mystical) theory of the historical Parmenides?29 Is it his honest attempt to criticize certain Eleatic idiosyncrasies and ambiguities?30 Is it his honest attempt to criticize his own earlier misconceptions about Ideas and Proof?31 Is it a subtle scoff at the historical Parmenides?32 Could it be simply a straightforward proof that all proofs are useless? My task here is not to determine once and for all the subtle meaning of the dialogue but rather to note what is visible and unmistakable about it. The dialogue constitutes a direct attack on the aspirations – shared by Parmenides, Zeno and Plato – to achieve knowledge by means of dialectics. The father of indirect proof has been made to disprove his own theory and then to do the same to its negation, thus casting into doubt the very process of indirect proof. The Parmenides, then, seems like the best and most eloquent criticism ever directed against dialektike as methodology-in-the-rigid-sense. Indeed Parmenides’ monologue is suspiciously similar to Gorgias’ on Nature or that which is not (it makes use of similar weaknesses in the Eleatic notions of Being and Unity). It thus seems to be a straightforward attack on the very possibility of attaining knowledge by rational means. It is little wonder that many have deemed such self-criticism too harsh to be serious. Already in antiquity many concluded that the dialogue must be an expression of Plato’s deep esoteric philosophy, a systematic layout of his mystic Parmenidean inclinations. The Neo-Platonists in particular have embraced the contradictions that Plato puts in the mouth of Parmenides as expressions of profound truths that transcend the ‘yes’ and ‘no’ of ‘all-too-human’ human language and its limited grasp on reality). Others have marveled mainly at Plato’s incredible ability to attack key notions of his own philosophy while not being sure how to correct them. Yet others have concluded that the Parmenides must be a parody. But even as a parody it seems to state difficulties to which we find no satisfying answer in the later Plato. (The question is not whether or not Plato had taken his self-criticism seriously but whether or not it is valid and whether or not he had any good answer

3 The Sophists’ Challenge

33

to it.) Some parts of the (alleged) proof are defective and this does not make decision easier. We do not know if (and to what extent) Plato realized (and/or intended) that they would be defective. We are not even sure to what extent Plato would seriously endorse some such defects. It is certainly not unreasonable to assume that some fallacies are deliberate ironical teases by Plato to his audience. (Socrates himself commits such fallacies in earlier dialogues and at least some of them seem intentional.) It is thus interesting to note that these fallacies do not and cannot undermine the self critical thrust of the Parmenides. They can, perhaps, to the extent that they are intentional, undermine the mystical interpretations to it. (Mystics rarely posses self-irony, yet arguably Plato is the most self-ironical of all mystics). If refusing to endorse the contradictions inferred by Plato’s Parmenides as an expression of neither some deep mystic wisdom nor of Plato’s bent towards anarchic sophism, then we must admit that Parmenides is simply an ingenious summing up of the bewilderment of Greek Philosophers, including, perhaps, the author himself, at the force and extent of dialectics – especially since it can so easily be directed against itself. This leaves us perplexed, suspecting that perhaps the sophists where right all along and episteme cannot be mastered. Socrates of the Republic warns against the demoralizing dangerous of allowing the young to engage in dialectical pursuits. But what is the philosopher-king’s answer to the enigma of the Parmenides? We do not know. The enigma, then, is an inseparable part of the marvelous enigma that is Plato. Perhaps all that can be said here in passing is that it is almost inconceivable that Plato would have written it had he not believed either that it is an outline of the truth, or that he holds a solid refutation of the argument propounded in it. Otherwise, it seems unclear why he would perform in such minute detail what seems like his own intellectual execution. Possibly, Plato’s suggested refutation of the message of Parmenides had something to do with his professed improvements of the dialectic method, his diaeresis. After all, diaeresis features the Sophist and Politician the dialogues that immediately ensue the Parmenides and which no one has suggested to interpret as parodies. Possibly Plato allowed himself to toy with certain limitations of Eleatic dialectics because he believed himself to have dramatically improved it by his diaeresis. Yet in hindsight we should also stress that, if this is the case, it was an error on his part. Diaeresis, as far as we know, constitutes neither a dramatic improvement nor a satisfactory answer to the sophist challenge. The Parmenides, thus, gives off that challenge instead of answering it. Jaeger assures us that Aristotle has entered the Academy approximately at the time of the writing of the Parmenides and that he was profoundly influenced by its theme.33 Aristotle followed Plato closely in portraying the sophists as charlatans, who endeavor to make profit from fallacies (e.g. 164a 20–22) and from apparent unreal wisdom (e.g. 165a 18–20). He too devised an answer to the sophist challenge. In his writings, however, the acceptance of the challenge is explicit. To his own testimony (in the epilogue to his On Sophistical Refutations), his first attempts to formulate logic (in the Topics) were the result of his inquiries into the very idea of dialectical confrontations, his attempts to criticize diaeresis, to improve it.

34

3 The Sophists’ Challenge

Aristotle clearly wished to formulate explicit rules of the dialectical confrontation in order to understand it better, so as to prepare his students to outrival the sophists, and (later on) the students of the academy, in public disputations and oratory contests. In the Topics, a manual for dialectical disputes, Aristotle clearly assumes that his addressees are quite familiar with the practice of diaeresis, and perhaps even with some crude formulations of intuitive rules of it. His greatest merit here, we should add, is his outstanding frankness, his laconic lack of irony in his admission of the difficulties at hand and the process of his own mental development. Openly he acknowledges that the sophists did produce a worthy challenge: to tell the apparent from the genuine. Openly he admits his controversy with Plato and exposes the latter’s failure to answer the sophists’ challenge satisfactorily: he demonstrates that even diaeresis provides no guarantees of finding truth.34 And openly he attempts to produce his own improved answer to the challenge. This is the gist of the Organon35: it is the open acceptance of the greatest challenge philosophy has known, the challenge of answering the sophist’s criticism of the possibility of knowledge, it is an attempt to demonstrate the attainability of knowledge by rational thinking.36 Answering the sophist’s challenge remains the central aim of Aristotle’s later logical work, his Prior Analytics and especially the Posterior Analytics. Aristotle’s study of refutation techniques has led him to construct a full-fledged, abstract theory of it. The most striking product of all this is several of the rules of deduction, described in his theory of the syllogism, in the Prior Analytics. Its use of linguistic variables as a substitute for concrete terms is, as I stressed earlier, revolutionary. It is the use of variables as substitutes for concrete terms, and the formulation of strict rules of deduction by means of them, that makes Aristotle discussion of inference, an abstract discussion of logical form rather than a mere pointing out of similarities between specific examples. This is why his own formulation of the isomorphism that is the heart of logic is superior to that of Plato in his Parmenides. Aristotle, then, was the first to suggest a systematization of rational thinking as opposed to mere scattered examples to it, or mere noticing and noting of similarities, even of symmetries, between such examples. The theory that resulted thereby is, perhaps, the most distinctive case in history of an entire theoretical field of science single-handedly created almost ex nihilo. Having made all of this clear, I wish to stress that my aim in this study would be to attenuate it. My aim is to explain why it is misleading to regard the syllogism as a purely formal study of inference. I have no intention of belittling the achievements of one of the greatest minds that ever lived or of producing an anachronistic criticism of what is arguably his greatest achievement – the theory of the syllogism. My intention is rather to delineate those features within Aristotle’s theory of the syllogism that actually hindered the rise of modern formal logic. Such features do exist. It is not by accident that 18th century logic is essentially the same as Aristotle’s logic (with a few minor stylistic variations, most of which are actually signs of corrosion, not improvement). Modern logic, we should quickly add, is very, very different from it. Even those parts of modern logic that are sometimes anachronistically presented as the equivalents of Aristotle’s logic, do not

3 The Sophists’ Challenge

35

remotely resemble it. The dissimilarities between these logics will be dealt here soon. At the moment we should merely note that historians of logic sometimes tend to ignore them or to politely avoid noting them. They do so for various reasons that I will not discuss here. The result is not merely a distortion of our understanding of Aristotle’s logic, but also of the rise of modern logic, especially of the philosophical difficulties that hampered its growth. Presentations of Aristotle’s logic as a modern logical system are obstructive to our historical understanding of the development of logic in that they blur the important fact that the undoing of Aristotle’s conflation of logic and science was both central and crucial to the launching of modern logic. To put it more concisely: modern logic could not have been developed without the straightforward annulling of Aristotelian essentialism, which is inseparable from his logic. I do not know if Aristotle had ever considered a purely abstract study of logical forms. I am convinced that if he had considered it, he, most likely, considered it useless. This is because Aristotle’s logic is, first and foremost, a tool for answering the sophists’ challenge, and because empty logical forms, clearly, do not answer this challenge. Aristotle did not admit into his logic so many logical truths, tautologies, which he undoubtedly recognized as trivially-true, a fact that repeatedly puzzles his anachronistic admirers. They formalize these tautologies in elaborate systems in an attempt to expose the “true” Aristotle. Sometime they go as far as to formulate and formalize for him rules that he clearly used only subconsciously, as if these were legitimate conscious parts of his logic. But the fact of the matter seems to me to be that, until modern logic came and changed this, the very term “tautology” (and its many historical equivalents) designated, for Aristotle and his scholastic followers, a useless truth: useless because it is empty of content. Aristotle’s logic was designed as the method to guarantee the attainment of scientific knowledge, and explain its attainability. As we will soon see, a loosely pre-determined choice of the range of his variables and of the terms that are “legitimate” within his logic, played a crucial role here. (The odd term “a logically legitimate term” is mine, and will be soon explained in detail.) My claim will be that by rendering certain terms “legitimate” and certain terms “non-legitimate” Aristotle attempted to secure the status of logic (methodology-in-the-weak-sense) as epistemology. This was the chief manner by which logic and science were conflated in his system. Today we know that the sophists’ challenge (which was to become the skeptics’ challenge) cannot be answered without some circularity. Aristotle, then, boldly produced a magnificent system in an attempt to answer an impossible challenge. Hidden circularities had to occur (at least in the form of subtle conflations). (In other words, what we now, in hindsight, recognize as “circular justifications” filtered into his magnificent system by means of subtle conflations.) Ultimately, then, Aristotle’s logic explains and guarantees the truth of what he regarded scientific knowledge, and yet the latter is presupposed by the former. The subtle problematic relationship between logic, epistemology and science in Aristotle, became an inseparable part of the tradition of logic for over two millennia. Even those who opposed it found it extremely difficult to sift a purely formal system of logic from his overall system. They saw little use for a purely formal

36

3 The Sophists’ Challenge

system of logic. The very idea that such sifting is desirable, that a purely formal system of logic is desirable, belongs in effect to modern logic and the giving up of the idea that logic should arm us with an answer to the classic problems of epistemology. And so the modern view of logic as a (rather partial) methodology-in-theloose-sense was inhibited by a grand and bold attempt to answer an impossible challenge to epistemology.

Part III

Aristotle’s Logic: The Rise of Essentialism

Chapter 4

The Beginning is the Term

Any discussion of Aristotle’s logic should begin with an explanation of his extremely subtle notion of a term. Aristotle’s logic is a logic of terms. Terms, not predicates, or sentences, or propositions (not even judgments) are its building blocks. I intend to show here that Aristotle’s notion of a term is the source of his conflation of methodology and epistemology. It is the gist of his essentialism. Explaining this claim will be the focus of the next five chapters. Literally “term” is “an end”, a sentence end (horos, in Greek, terminus in Latin, means an edge, a limit or an end). This literal meaning suggests a structural definition: it suggests that the only thing necessary for identifying terms is pinpointing a position in a sentence. It tells us that a sentence is the composition of two ‘ends’ and a middle (the middle being a special binder, the copula). This definition is highly misleading, however, exactly because it seems to suggest that anything that can grammatically occupy a sentence’s beginning or end is a legitimate term. This is not the case in Aristotle’s logic. Aristotle is the first that we know of to introduce variables explicitly. What is a variable, then? A variable is a sign replaceable by a term. The list of terms that can replace it is usually pre-determined. For example, in arithmetic the variable “x” in the formula “2 + x = 5” is replaceable by any numeral (representing a number, in this or that mathematical field; and it is true only if that number is 3). In Boyle’s (original) law “pressure is proportional to density” the words “pressure” and “density” are variables and both have to apply to some given quantity of air. Even the term “someone” in the sentence “Someone pinched my grapes” can be viewed as a variable, especially if it is replaceable by any of the names of my possible suspects.37 Variables do not have to be replaced by any of the terms that they stand for and this is their most cherished feature: they constitute an apparatus which grants us the freedom to discuss such things as sets of numbers, volumes of air and suspects. We can do so in a loose, abstract setting committing to neither specific terms, nor to abstract ontology regarding sets and other non-material objects. It is sometimes mentioned that a common practice of Greek mathematicians preceded and heralded Aristotle’s groundbreaking discovery: when writing proofs of various sorts they habitually used shorthand; they wrote only the first letter of some terms and/or used the letters of the alphabet for numerals. (One dominant such method had alpha standing for 1, beta for 2, and so on). The practice, so it is N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

39

40

4 The Beginning is the Term

sometimes suggested, heralds Aristotle’s discovery. This claim is off the point, and it is misleading. Using the letter S, as shorthand for “Socrates”, or for “Square”, or for “Seven” does not increase the level of abstraction of our discussion. Shorthand may lead to abstraction but not by itself, as is made evident by the very fact that something like it is almost universal without eliciting the use of variables. Thus, to take one notable example, we are told that, strictly speaking, only a few specific instances of Pythagoras’ theorem were proved in Plato’s days, at his Academy, not the general formula which, though certainly apprehended, implied the existence of irrational numbers and thus could neither be proved nor admitted. The value of a variable, then, is not in it being shorthand for a given term, but rather in it being an apparatus for handling abstraction, replaceable in principle by any of a list of terms, a possibly infinite list of terms (such as in the case of Pythagoras theorem). Aristotle’s aim was far and above that of enabling us to write shorter computations, of course. It was, at least in part, to examine the observed similarities, symmetries and isomorphism between inferences that made them all valid or invalid. Consider, for example, the rubber stamp example of an inferences “All Greeks are human, All humans are mortal, Therefore all Greeks are mortal”. Let me repeat myself and say that Aristotle was not the first to observe such an inference, and possibly he was not the first to produce a shorthand version of it. (Since writing was so cumbersome those days, shorthand seems to have been almost inevitable.) He was also not the first to notice that other inferences are structurally similar to it. But he was the first that we know of to have constituted an apparatus which enabled to formulate and study these shared structures (and thus to declare them isomorphic, as opposed to merely similar). Thus, when Socrates defends the validity of an inference, he intuitively invents similar ones. Aristotle does better: he formulates their shared structure and proves it to be valid. In this manner, strictly speaking, he is not dependent on intuition when providing similar ones and can rest assured that no counter-examples ever arise. The logical form expressed by “All A’s are B’s, All B’s are C’s, therefore All A’s are C’s” is clearly shared by the above inference and by the following: “All unicorns are white animals, All white animals are good-natured, therefore All unicorns are good-natured”. It is the formulation and discussion of the laws that apply to such symmetries that truly signify the birth of logic. However, the above presentation is a considerable idealization of Aristotle’s logic. The question before us is this: what terms can replace variables in Aristotle’s syllogisms? The answer to it is short ant yet devious and subtle: in Aristotle’s logic the only legitimate terms are names of essences (and the genera and species that are required for defining them). How can we to tell which terms are names of essences (and which are genera or species)? How can we depict, out of the pool of possible candidate terms, those that are genuine names of essences? (How can we tell genuine genera from apparent genera, and genuine species from apparent species?) Modern logicians explicitly ignore such questions: they declare that, if they are meaningful and answerable at all, they are certainly extralogical. This is exactly why we say, today, that logic is methodology-in-the-loosesense. How did Aristotle handle these questions? Did he deem the answer to them as extra-logical too?

4 The Beginning is the Term

41

Aristotle does make it very clear that his logic is not a formal study of all possible terms. Despite his admirably free use of variables (most notably in his Prior Analytics) his logic was not intended as a content-insensitive theory of inference. It is certainly not a predicate calculus without quantifiers, as it is sometimes presented in anachronistic contexts. In order to demonstrate this claim we must first note a few clear-cut restrictions which he makes explicitly. He divides terms roughly into three groups: proper names, such as “Socrates”, abstract names, such as “Men”, and super-abstract names, or categories, such as “substance”, “quantity” and “quality”.38 In an attempt to delineate agreeable terms, Aristotle accepts into his logic only terms of the second group: abstract terms that do not designate categories. So, as many have noted, following Lukasiewicz, even the classical inference “All men are mortal, Socrates is a man, therefore Socrates is a mortal” is, strictly speaking, not a proper subject matter of Aristotle’s logic, since it contains the proper name “Socrates”. Similarly, the inference “Quantity is a category, categories are exceptionally abstract names, therefore quantity is an exceptionally abstract name” has no place in the realm of Aristotle’s logic: it refers to levels of abstraction that, according to Aristotle, are beyond the province of logic.39 As a matter of fact this level of abstraction seems to be beyond the realm of Aristotle’s metaphysics as well: for he declares that the 10 categories are the most abstract of terms, and in the above syllogism the term “Category” is potentially more abstract than any of these 10 terms (for it includes all of them, all particular categories; thus, the observation that there are exactly 10 categories is an Aristotelian oxymoron) and the term “an exceptionally abstract name” is potentially even more problematic than the term “Category” (for it implies that some categories are more abstract than others, an implication that is unacceptable in Aristotle). More importantly, however, as many have noted (e.g. G. Patzig 1968, p.7), the above restrictions imply Aristotle’s refusal to admit into the province of logic all terms that represent the Universe of discourse (the Universal class), which is forbidden as it is more abstract than any of the categories, as well as all those terms that depict the empty-class (the complement class of the Universe of discourse). Thus, even the abovementioned syllogism about unicorns (“All unicorns are white animals, All white animals are good-natured, therefore All unicorns are good-natured”) was not admitted into Aristotle’s logic. “All unicorns” is neither a name of an existing genus nor a name of an existing species and so it is not a name for an essence: there are no unicorns.40 In Aristotle’s logic, terms that depict impossible entities, non-existent entities, hypothetical entities, and even arbitrarily grouped together entities, are, thus, considered highly problematic and deemed logically undesirable. Aristotle’s restrictions upon terms are not all as explicit as the ones that we just discussed. Here the fine (modern) line between epistemology and logic is blurred. It is useful to distinguish here between terms that he admitted into the province of his logic, and tag them collectively “legitimate terms”, and contrast them, with the terms that he barred, for various subtle reasons, problematic terms, that we will tag here “non-legitimate terms”. The reason for introducing here the new technical term – “legitimate/non-legitimate term” (which will accompany us throughout this

42

4 The Beginning is the Term

book) – is due to my wish to lump together several groups of barred terms, while avoiding exegetical discussions of Aristotle’s “true” intensions as far as possible, resorting instead to a more general analysis of the logical province that he was studying, thus arriving, in as far as this is possible at all, at a fair description of the options he faced and the rationale of his system. The criteria by which Aristotle distinguished between what I call here legitimate and non-legitimate terms are, to repeat, diverse, and are not always explicit. Often they come as a scientific or metaphysical afterthought. Full-fledged formal rules do not come close to comprising them. This is a highly problematical situation. We do not have strict Aristotelian rules for determining whether or not a term depicts an essence and whether or not it is admitted into his province of logic. (We do not know exactly what an essence is.) Is it the case, then, that in Aristotle we can only pursue our logical inquiries once our scientific inquiries have ended and all that exists was found and listed in proper taxonomies (whatever “proper” means here exactly)? Must the essences that comprise our cosmos be fully inventoried before we can make our first steps into logic? Should it not be the opposite, that logic is valid irrespective of our knowledge of the cosmos? Should it not be the case that logic is an inseparable part of the tool kit we use in our search for knowledge of the cosmos, including (if we so wish) knowledge of essences? It is not easy to extract a single unequivocal answer to these questions, in Aristotle. Sometimes knowledge of essences seems to be, for him, a prerequisite for the very possibility of logical enquiries. On other times, it seems that it is not a prerequisite at all (since this would ultimately mean that the sophist was right all along, that their challenge is unanswerable) but rather an inevitable result of logical inquiries. The conflation of prerequisites and results is, in hindsight, the gist of all circular justification. Important hints as to Aristotle’s plan do exist, however. They have been studied since antiquity and will be inspected here. (The reference to science and metaphysics is unhelpful, of course, but it does complicate things in a manner that had a tremendous influence upon the development of logic.) According to Aristotle, the terms ‘man” and “goat’ are legitimate since they denote species, natural kinds, that is, groupings of particulars that share an essence; not so the terms “goat-man”, “man or goat”, “man and goat” “not-man”, “not-goatman”, “not-not goat”, etc., Aristotle’s metaphysics attempts to make the division hold. The reason for the illegitimacy of all complement classes is that there is no essence common to all of them. For example, the term “not-man” is problematic because there is no essence common to all those (stones, plants, and brutes) which are not-man.41 Likewise goat-men do not exist, and hence their essence does not exist. Consequently “goat-man” as a term designating a species is meaningless, and the term is rendered illegitimate. Not-goat-men are in an even worse predicament since they are a complement class of an empty existent class. The problems with Aristotle’s notion of a term do not end here. In fact they just begin: for it is simple to show that an infinite list of illegitimate terms can always be produced from any list of legitimate terms (that allegedly refer to genuine essences) and the logical operations we now call elementary set operations. (This is precisely what was demonstrated above by means of the terms “man” and “goat”.) Aristotle’s logic is

4 The Beginning is the Term

43

usually presented as some formal theory of deductive derivations that is limited to a closed list of pre-chosen terms. However, it is impossible to provide that list, the list of the “legitimate” terms that are allowed in it, both because some of the criteria for “legitimacy” are unclear (as the very notion of essence is unclear) and because it is impossible to keep any list of allegedly legitimate terms closed to non-legitimate terms without thereby disallowing the most basic logical operations between these terms.42 To make things even more complicated, sometimes Aristotle further divided the second group of terms into two distinct groups: he distinguished plural names such as “Men” on the one hand, and still more abstract terms such as “Man” – meaning the idea of a human being – on the other hand. To a large extent, these had already been identified by him as extension and intension, respectively. (His understanding of extension and intension is discussed in chapter 10, in which his theory of induction is explained.) In general, terms can be read as extensional (i.e. as designating an object or a collection of objects) and as intensional (i.e. as designating a property or an idea). Yet there is no explicit differentiation between extensional and, intensional syllogisms in Aristotle since, as we will see (in chapter 10), the fusion of extension and intension by induction is a pillar of his system. (It plays a crucial role in his epistemology which is practically founded on the claim that logic is limited to intensional relations, expressed by definitions, and that these are established by observation of particulars and the comparison of the extensional relations that observable classes of particulars form.) This subtle view created a fundamental problem in Aristotle’s logic. The inference “Humans are primates, Greeks are human, therefore Greeks are primates” is valid. Yet the inference “Humans are numerous, Greeks are human, therefore Greeks are numerous” seems invalid. The inference “Human is a form of behavior, Greeks are human, therefore Greeks are a form of behavior” too seems to be invalid. But all three inferences, on the face of it, seem to share a logical form, as long as extension and intension are not explicitly differentiated. Aristotle’s logic, then, rests not only on a putatively intuitive choice of admitted and barred terms, it rests on a putatively intuitive choice of admitted and barred senses of these terms. The problem, however, is that Aristotle could not make such a division fully explicit: he could not simply dismiss the last two inferences as extra logical, or as invalid, on the ground that they illegitimately fuse intension and extension. This is because, to repeat, he resorted to that very fusion in his theory of inductive inferences (which he regards as a special type of syllogisms), and which justifies, for him, the feasibility of the split between legitimate and non-legitimate terms. The fusion, then, is both a threat to the validity of his logic and a manner of securing it. A strong inner conflict in Aristotle’s theory is present here, one that neatly exemplifies its problematic character. It is, to the best of my judgment, nowhere to be resolved in his known texts. Rather, Aristotle disregards it when context allows for disregard. He seems to declare logic the study of intensions only when context allows, but then, clearly, this is not always the case, and if it is so, it seems that his claim to have secured the status of science by induction collapses, as we will observe later on. (With it, let me stress again, the very distinction between legitimate and non-legitimate terms collapses, and so the whole

44

4 The Beginning is the Term

of the magnificent Aristotelian system.) The point to observe, however, is not that Aristotle’s epistemology may be problematic. It is that his logic presupposes a strict choice of terms (and of senses of terms) that his epistemology cannot allow. The distinction between legitimate and non-legitimate terms is basic to classical logic and is ignored in modern logic. Therefore, readers of this book who are somewhat familiar with modern logic or mathematics or computer science may fail to see the historical impact of this classic and important distinction, whereas readers who are trained in classical biology or in the arts and humanities may fail to see the importance of consciously ignoring the distinction in modern logic. Allow me, then, a few explanatory remarks. The curious and influential idea that successful discourse presupposes a list of all proper taxonomical terms is, essentially, Plato’s. In such dialogues as Sophist and Statesman the stranger (who comes from Elea) explains the division of terms which underlies the search for definitions by diaeresis. It is, he says, a serious metaphysical affair, subtle and delicate. Improper splits give rise to unnatural terms leading to false classifications. And then the validity of diaeresis which relies on these splits, and the truth of the definition obtained by it are threatened. Socrates of the Phaedrus (265d) explicates this perplexing warning: dialecticians, he tells us, dissect the cosmos into forms as a carver of meat, a butcher, dissects the body of an animal: a perfunctory cut, made carelessly, not along the natural joints of the animal results in a useless piece. (The Greeks, alas, must have disliked T-bones). It is as if the world of forms is a giant goat, or –to be less carnivorous– a giant cake that can be cut in slices by means of our terms.43 Some cuts produce authentic parts, slices, glorious slices corresponding to real platonic forms. Others do not. Some terms designate carefully delineated eternal ideas. Others do not. The former will serve us well in our dialectical pursuit of definitions; they will not threaten the validity of diaeresis. The latter will spoil our pursuits for true knowledge and mislead us. We must, then, use only terms that depict real, authentic, slices of reality. Let us observe an example. “Animal” can be properly split into “Brute” and “Man”, says the stranger, yet “Man” is improperly split into “Greeks” and “Barbarians”. Why? Modern philosophers say that Man is a natural kind and Barbarian is not. Why? Perhaps the term Man seems well defined and sufficiently uncontroversial. Some may wish to define “Barbarian” so as to make it as clear and uncontroversial as Man. This, Plato seems to caution us, is dangerous. It is possibly doomed to fail. Barbarians are humans whose discourse makes no sense to us (to Greeks). This is its definition. It is a bad definition: clearly some Greeks are mute, some are deaf, some are poor speakers of Greek and some were born in foreign lands and do not speak Greek at all. Some non-Greeks may speak Greek (even if poorly), and so on. The borders of Greece are conventional and changing (and pureblooded nations are fictitious illusions anyway). So the split of all men into Greeks and Barbarians is an artificial (not a natural) one: it allows for confusing borderline cases (Greek speaking Persians, for example). Borderline case cannot mirror the heavenly world of forms, for they do not represent authentic slice of reality.

4 The Beginning is the Term

45

Borderline cases seemed to threaten the validity of diaeresis then: if two terms (A and B) do not sharply split our universe, if x can be a borderline case between them, then we cannot derive “x is B” from “x is either A or B” and, “x is not A”. A Greek speaking Persian seems neither Greek nor Barbarian, and then showing that he is not a Greek is not the same as showing that he is a Barbarian. This is the source of the stranger’s warning against the use of careless divisions: if they are allowed, he suggests, diaeresis cannot guarantee for the attainment of the truth. This point is extremely subtle and confusing, but it is also highly important. I regard it as the corner stone of Aristotle’s logic. If terms are regarded as some kind of ontological hooks on proper parts of the world, then using the wrong terms may seem to result in unsuccessful fishing. This is clearly undesirable. The attainment of the truth, suggests the stranger, is possible only if discussion is limited to “true” terms, i.e. correct splits (that we called “legitimate terms” above).44 Socrates is no brute; therefore, he is a man. This inference is valid, on the understanding that (in the universe of discourse) everything is either man or brute and nothing, absolutely nothing, is both. We should emphasize, however, a somewhat obvious yet crucial point: Plato’s dialectics was not really threatened by arbitrary splits. Barbarians and Greeks can be mutually exclusive with or without mirroring some authentic slice of reality. Many commentators and logician throughout history came to acknowledge this point. But not until Boole was it fully internalized as part of logic proper, by means of neglecting altogether the notion of essence and the search, by means of logic, of essential definitions. Certainly Plato and Aristotle do not seem to fully appreciate it. Paradoxically (and most interestingly) for the sake of illustration, it is best to forget Plato’s world of Forms and Aristotle’s world of essences for a brief moment, and use a Democritian model of the universe to visualize the conception of reality implied by the ancient view of terms.45 Let us assume that reality is a gigantic Lego structure, containing pieces of diverse lengths and colors. Assume that the only parameters for comparing these pieces are their size and color. Then, some splits of that world are clearly legitimate and some not. An authentic slice of reality is a distinct Lego piece, or a distinct set of Lego pieces of equal length and/or color. Furthermore, we may wish that terms of our language would designate only those sets of elements that are chosen by legitimate splits. A term is unnatural, illegitimate, or false, if it designates none of the above but rather designates a part of a piece or a partial set of pieces, or a mixture of such partial sets. A term is unnatural, illegitimate, or false, if and only if it makes us break the true structure of reality in search for its reference. That such improper breaking up of the real world is possible, and that it is to be feared, is made very clear by both Plato and Aristotle. This, by the way, is the rational behind Aristotle’s famous definition of truth. “To say of that which is, that it is, and of that which is not, that it is not, is to say the truth”. To read his texts with Tarski’s modern alternative to it in mind is problematic and misleading, for Aristotle seems to have in mind the act of pairing attributes and objects, tagging terms to entities, not the act of tagging truth conditions to propositions.46

46

4 The Beginning is the Term

Aristotle – who had considerable criticism of the Platonic practice of dialectics – followed Plato’s main assumption closely: he took the legitimacy of certain terms, as well as the non-legitimacy of certain other terms, as both intuitive and crucial for the very possibility of logic. Humans beget Humans, he repeatedly reminds us (and brutes beget brutes). No fear of borderline cases can arise here. We can single out all Humans neatly and sharply, we can even imagine the world without them, but if we try to single out all barbarians, or to imagine a world without them, we immediately get entangled in fuzzy borderline cases depicted by dubious terms, resulting in pseudo-definitions. A successful term for Aristotle is a legitimate term applied correctly to the right entity, and this presupposes knowledge of the right entities (whatever that means). Terms are anchored in reality, they catch a substance: either a “primary substance” (a real authentic slice, a genuine particular, whatever that means), or a “secondary substance” (an essence, or a genuine attribute that resides in that genuine particular, whatever that means). An unsuccessful anchor term catches nothing, or (which is more or less the same, for Aristotle) it catches an alleged attribute that does not actually reside within an existing object, or it catches an artificial attribute, an arbitrary set of objects that do not constitute a genuine species and cannot feature in a genuine definition. According to Aristotle, then, to give a primary substance (its proper) proper name or to give a species, a (real) natural kind (its proper) universal name is to use legitimate terms correctly. Other than legitimate terms, used rightly or wrongly, other terms are illegitimate or false. In Plato it often seems that the role of diaeresis is not only to assume correct splits, so as to formulate correct axioms, but also to discover correct splits (and thereby correct definitions). This is a classic conflation of epistemology and methodology: episteme (knowledge of correct splits) dictates the tools (correct splits) for the method of searching for correct splits. On chapter 10 I show that Aristotle’s theory of induction and intuition commits a more elaborate but similar conflation. In hindsight we may observe that Plato and Aristotle alike wish both to assume and discover intimate acquaintance of the structure of reality and that this intimate acquaintance both dictates and facilitates discovery of correct definitions. This is the bootstrap effect of the conflation of methodology and epistemology. Famously, a crucial difference exists here between Plato and Aristotle: Plato’s heaven is beyond space and time (and quite possibly beyond the reach of our intellects) whereas Aristotle’s essences exist here and now, within the particulars of this world. Thus, in Aristotle, legitimate terms are also existence-pointers. They indicate not only essences, but also the existence of distinct particular objects that exhibit these essences. This fact will be studied in detail in the next chapter. To sum up, following Plato, and as a direct result of reflections on Platonic diaeresis Aristotle assumed that certain terms are more legitimate than others and that they (and only they) should be found and sorted out, from the pool of all possible terms even before proper (logical) investigation can begin. Only legitimate terms are allowed to partake in the dialectical game (in Plato) and in the syllogism proper (in Aristotle). Legitimate terms, for Plato, are those that have their hooks into a Platonic idea. For Aristotle the legitimate terms are those that have their hooks into an essence. Thus, terms that depict the universal class, the empty class,

4 The Beginning is the Term

47

hypothetical entities, non-existent entities, or even arbitrarily grouped together entities, are to be avoided, in Aristotle’s logic, as a matter of course, for using them will (so it was assumed) result in problematic and useless pseudo-definitions. However, as we stressed, undesirable non-legitimate terms are impossible to prevent. The problem is not merely that the notion of an essence is not adequately defined, though this surely is the case. It is that any list of allegedly legitimate terms turns into an infinite list of manifestly illegitimate terms by the most elementary logical operations that we can conceive of. Thus, commonsense knowledge of kinds plays a crucial role in Aristotle’s logic: it is the keeper of the border-gate to the province of logic, which becomes an ideal realm by the very insistence that such a border-gate can exist. The ancient view of terms clearly enmeshes our semantic knowledge with our ontological knowledge. Sometimes it bluntly assumes (as Aristotle had done) that a rough ontological theory, suggested by our language, is already presupposed by our very perceptions of reality, and that because of this, it is true. (For example that, observably, Humans beget Humans, and that, observably, they are rational animals, and that therefore rational animals must exist). This is circular justification, of course. At other times a dialectical detour is needed, and such matters are decided, once and for all, by careful dialectical quest. This too, as we noted earlier, is circular: dialectics is founded on correct splits of terms and establishes them. Either way, the assumption is made that episteme awaits us nearby on the crossroad of commonsense, perception, and careful rational thinking. Noting the conflation of semantics and ontology that cradles the ancient view of terms is, I maintain, indispensable for a proper historical understanding of Aristotle’s logic because the latter is founded on the former. Thus, the ancient view of terms cradles the conflation of epistemology and methodology that is Aristotle’s logic. Plato’s ironical and critical style may blur the fact that, like Aristotle, he conflated semantics and ontology, and consequently methodology and epistemology, in the hope to squeeze final positive results out of his diaeresis. Though he neither presented nor even attempted to present a rigorous logic the problems of the ancient view of terms are inevitably present in his dialogues, especially when he occasionally transcends his unmatched ability to smoothen difficulties through his unmatched poetical skills, and tries to form an explicit theory. Aristotle, on the other hand, was the first thinker we know of to explicitly provide a rigorous theory of logic. It is because of this great endeavor that the problems of the ancient view of terms loom large in his system. He was, in this respect, bolder and more direct than Plato. Alas, because of these merits, the deficiencies of his system are more apparent to us today. Plato elegantly and elusively conflates semantics and ontology while Aristotle attempts to rationalize and formulate this conflation as a more or less explicit theory.

Chapter 5

Chimera in the Dusk: Essentialism

The most influential result of the conflation of semantics and ontology in Aristotle’s logic is his assumption that “legitimate” terms can be sorted out and that they are legitimate by virtue of their depicting essences directly and particular objects of our world indirectly. This result has sometimes gained the name “Aristotle’s existential import”. We will now inspect its philosophical background closely. The aim of this chapter is to describe very briefly the classical problem of change and to explain that Aristotle’s notion of a term in general, and his “existential import” in particular, reflect an intended solution to it. This solution, I will maintain, is a compromise between concretism (also known in as “nominalism”) and abstractism (also known as “realism” or “Platonism”). Ontologically, concretism is the theory that only individual or concrete entities exist, and abstractism is the theory that abstract entities exist as well (or, as Plato seems to suggest, that only abstract entities really exist). Semantically, they are theories about names. Traditional conceptions of meaning suggested that names mean by having their hooks into things. So, nominalists maintained, meaningful names must name concrete things (like Socrates). And all other names, they said (like, Pegasus and Humanity) are meaningless. Platonists, however, allowed meaningful names to name abstract entities (like ‘Man’ or ‘Humanity’). The two theories are, thus, incompatible (ontologically and semantically). Of necessity, then, Aristotle’s compromise between them is of the highest logical import (for strictly speaking it is suspected of being incoherent). It is known today as (Aristotelian) essentialism. For over 2,000 years, in various philosophical variants, Aristotelian essentialism was responsible for the perpetuation of the conflation of ontology, logic, epistemology and methodology. The diverse variants shared a characteristic that inhibited the development of a purely extensional system of logic, prior to the time of Boole. The terminating of their influence upon logic comprises the extensionalist revolution in logic. Thus, we cannot possibly understand the significance of the extensionalist revolution in logic without understanding the compelling force of essentialism. And we cannot hope to understand the tenacity of Aristotelian logic without noticing its immense philosophical appeal: the crucial problems that it purported to solve, those it had solved, and those it seemed to have solved. We begin, then, with the ontological problem: What is there? Today when the vastness and complexity of the universe are realized better than ever before, we may N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

49

50

5 Chimera in the Dusk: Essentialism

readily admit that the ontological problem is too “big” for us. Its dimensions are monstrous. But even in older times it was felt that the monster must be tamed a little before it is approached safely. The early Greek thinkers had a powerful aid: the division of all that can be known into truths by nature on the one hand, and truths by convention on the other. This dichotomy has many known versions. A few of them are: reason vs. arbitrariness/ essence vs. appearance/ real vs. illusionary/ the constant vs. the changing/ the permanent vs. the ephemeral and, of course (as Parmenides taught us) the true vs. the false. Generally it was assumed that knowledge of the real would explain the changing, the diverse, the apparent. And if the real explains the changing, then the real is unchanging. If the real explains diversity, then it must be one. If the real explains the complexity of the apparent then the real must be simple. Once the ontological problem is tamed by means of these dichotomies, it paraphrases into the following problem: “What is the true unchanging reality? What is the underlying essence of it all? Even in this last, somewhat narrower formulation, the problem remains vast almost beyond imagination. And few but the earliest of Greek thinkers had dared to face it directly. A view of reality as a whole seems both too much and too little. That it is too much is obvious. That it is too little transpires when we see what we want the theory to do for us: we want it to explain you and me, and the differences and similarities between us, and the trees in the woods and the birds that reside in them. We want to explain different aspects of reality, like ships sailing the sea and like dreams. We regularly recognize the existence of certain entities that we deem particulars or individuals, such as you and me, a specific tree or a specific bird. We do not know if they are “authentic slices of reality” (in the sense explained in the previous chapters) since we do not know what being an ontologically authentic slice of reality amounts to. But we still deem them particular enough for them to get our attention as particulars: we wish to study them. We think that they are compounds of smaller particulars, such as specific cells, specific molecules specific atoms. And we gather some of these specific objects into groups or types that we call species or genera. Logic contrasts proper names and universal terms (properties and relations). This reflects an ontological contrast between particulars and universal objects. Universals get our attention at times, but differently: we attend to them as we wonder about the nature of things, about the essence of types, about the common to particular members of a given species. We do not know whether universals are objects or not, whether they are authentic slices of reality or not, but we study them too, sometimes as means for the study of (what we normally take to be) less abstract objects, such as the ones we deem particular objects, and sometimes due to interests we have in them, in themselves. Some of us want all our answers to be included somehow in our one big answer to the ontological problem and be explained by it. This is the ideal of explaining it all by reduction to a level of reality that we deem more fundamental and (somehow) more real than other such levels. Some of us consider this ideal misleading. But we all accept commonsensical divisions of reality, criticize them and move on to better divisions, and this seems to tame a little the ontological problem: things seem simpler, at least at first sight, once we discuss you and me and the trees and the birds (as particulars or as universals)

5 Chimera in the Dusk: Essentialism

51

and not merely the cosmos as a whole. At least at first sight we leave the amorphous question of explaining the totality of it all, that seems to defy our intellectual capacities, and settle for the more considerate, down to earth set of questions about the nature and character of each and every particular and each and every universal. To make this point sharp we can view our changes of opinion about types. Our taxonomy is not the same as that of Aristotle’s. (He declared that whales are fishes, for example). Nor do all taxonomists today are of one mind, and there is a dazzling number of methods for establishing what a species or a natural kind is, or should be. What is this disagreement about? Nominalists should say, it is about how to name real things. Platonists should say, it is about what taxa are real. What do essentialists say about it? Allowing all this as necessary preliminary for the taming of the ontological problem, we may have finally managed to slice it down to the size of our human capacities. We have reached the level from which the problems of change will emerge (and the problem of identity is but a facet of the problem of change). The problems are those of finding the unchanging nature, or identity, or essence of a given object of inquiry, particular or universal, and of separating this essence from the erratic changing appearances of that object of inquiry. There are numerous problems of change, since we regularly assume many distinct entities, but we can divide them all into two groups, adopting the commonsensical division (into particulars and universals) that was explained above: 1. What about a particular item (say, Socrates) is a true unchanging essence? And what of that item is mere erratic appearance? Socrates may be young or old, short or tall, single or married, yet he remains Socrates nonetheless. What is it that makes one a Socrates? We may also note that Socrates is a man and in order to understand him we may want to study the essence of Man. This leads us to our second concern. 2. What is it about human particulars that makes them members of the same kind? Some humans are young and some are old, some are tall and others are short, some are Greek and some are Barbarians, yet they are all parts of the human kind. By what virtue? What is the essence of humanity as such? What is the essence of the universal Man? The division of the problem of identity into these two layers – particulars and universals – is traditional and it is usually considered natural. We do not have the final blueprint of reality, of course. We do not know if such a blueprint exists, or if it makes sense at all to assume that it does. Therefore we cannot know for certain that Human is more general than Socrates in some deep ontological sense. This really depends on our perspective. And, to repeat, today we do admit that the very idea of a blueprint of reality is somewhat nebulous to us – at least in the sense that we now readily admit that we are very far from understanding what it amounts to. What we take to be a particular is usually simply a system within a system (a sub-system). It does not have anything ontologically special about it. It is simply considered for one purpose or another as a distinct item. (Of course, some such purposes may be important for our survival, for example they may aid us in sorting out poisonous

52

5 Chimera in the Dusk: Essentialism

and nutritious foods, but this does not mean that their pursuit leads to an understanding of the structure of the universe, whatever this means.) The quest for knowledge has knowledge as its end (this incidentally is why pragmatism is wrong). As Aristotle has put it, Thales is the father of philosophy because he was the first to declare a hidden essence to all matter, which lies beyond its appearance and explains it. Clearly he did not deem essential the difference between different humans, or even between them and other animals or minerals. The difference between things, Thales most likely taught, is one of condensation or intensity. All is water, he said. His followers disagreed with him. They proposed that other essential substances underlay all matter. However, none of them disputed his claim about the very existence of an underlying essence to all matter. A tradition of philosophy emerged: explain all change as diverse modifications of the manifestations of the one original, essential, substance. Parmenides may very well have noticed that it is absurd to speak of various degrees of condensation or of intensity of one and the same unchanging reality. Unless we admit some other matter that functions as a substratum for the changes of condensation, added Democritus later. The assumption that the void exists allows attempts to explain changes of condensation. But, Parmenides argued, this assumption is absurd. He concluded that no change is real, and so there is no (real) plurality that awaits explanation. The differences between you and me and between humans and birds are illusory, he said. He, thus, undermined the very assumption over which the problems of change rests: he employed the dichotomy between truth by nature and truth by convention, between reality and appearances, to show that only the truth by nature is true, that truth by convention is false, that only the real exists, that appearances are mere illusions. Heracleitus employed the dichotomy in the opposite direction. Some scholars maintain that he did this on the ground that only change exists. Others maintain that he noted an underlying fixed and unchanging law of change, which is a kind of an essence of all change. Possibly this controversy is merely semantical: either way he denied the view of the world as composed of a fixed reality, of fixed building blocks of any kind, whether particulars or universals. Both Parmenides and Heracleitus, then, have rejected the very admission of items on both sides of the dichotomy, the admission that enabled the ancient Greek thinkers to confront the ontological question. Aside from these two remarkable exceptions and their followers, the tradition went on, though with some modifications. Democritus continued it by confronting both Parmenides and Heracleitus. He asserted that the void exists. Assuming its existence, along side fixed elementary building blocks, the unchanging atoms, permits explaining plurality and change by reference to the unchanging. Plurality and change are explained as variations of shape, size and position in space and time of the atoms and of their aggregates. The atoms themselves are of various shapes and size, but cannot change (aside from their position in space and time). But the structures that they form may change. The forms, which they create when combined into larger complexes, may change. These forms define the nature of the complexes.

5 Chimera in the Dusk: Essentialism

53

Thus, Socrates is Socrates either as an atom or as a set of atoms, and then he is what he is by virtue of the particular atoms of which he consists, and then he endures some minor structural changes that do not effect his general constitution. He is human by virtue of the form his atoms are arranged by, which resembles that of other humans. All humans share the same form to a large extent, yet each is composed of different atoms. Democritus maintained that particular atoms determine the particularity of the particular objects that they constitute, and the form shared by many such particular objects defines universals (universals, then, are for him geometrical forms). It is, thus, possibly Democritus rather than Plato who is the father of the theory of forms, according to which the nature of an entity is its (geometrical) form or shape. If so, then Plato added two improvements to Democritus’ theory: first, shapes delineate parts of space, so that the matter that they shape according to Democritus is redundant; and second, shapes – geometrical abstract shapes – are real and eternal, though they do not occupy parts of space. So he replaced the atoms and the void with abstract shapes and the eternal realm (which is a sort of an intellectual stratum, a void). Plato’s theory, then, was that the (geometrical) form in which things partake is the real: he declared that universals – general types – are full-fledged objects, universal objects. Strictly speaking then, for Democritus, the geometrical form shared by all humans is neither an entity nor an object. It is a similarity between things. Unlike an atom or an aggregate of atoms it is an abstraction. It arises when we compare aggregates that share it. This is the scientific background of Concretism (“Nominalism”), the view that only particular things really exist. Whatever names it goes by, nominalism seems to have been a part of our commonsense since times immemorial. Humans think of themselves as concrete individuals, as particular objects (like “atom”, “individual” literally means “indivisible”). We even used to view mountains and rivers as individuals, but let us leave that for the past. We recognize particular objects like the circle that we draw on a piece of paper. We do not think of Humanity or of the perfect Circle in that same manner, presumably because we consider them to be abstractions. This is so because the simplest reading of nominalism tends towards materialism: we presume that reality is material, and abstractions are not. Instead, we deem abstractions products of intellectual activity. We compare particulars and find similarities and differences between them. These we call universals. When we are asked where are universals to be found, we often say that they reside in the intellect, beyond space and time. We understand this in the trivial sense that the contents of our thoughts are not material objects, not in space and time. This is often taken to be some version of Platonism. It is a remarkable and most significant fact that our commonsense freely mixes up materialism and Platonism with no regard for consistency. The reasons for this remarkable fact will soon be explained. Platonism is anti-materialist and anti-concretist. This is why later ages contrasted Platonism with nominalism: nominalism is a refusal to view abstract concepts as representations of real, abstract objects. This is because they refuse to admit universal objects. It is interesting, therefore (and in my view also significant) that Democritus is never mentioned by name in Plato’s entire corpus, not even as a

54

5 Chimera in the Dusk: Essentialism

target for criticism. He is the “present-absentee” in Plato’s dialogues. An anecdote in Diogenes Laertius suggests that Plato had much to do with Democritus’ anonymity: he was so envious of his achievements that he hoped to erase his name off the record. This testimony may be false, of course, but it agrees with a well-known fragment by Democritus that reports his visit to Athens: no one seemed to recognize him, it says. Possibly it expresses his humility and his faith in equality. Perhaps it is a hidden complaint. This report, if true, is certainly a sad one. However it is revealing even as a mere rumor, since the mere existence of such a rumor suggests intense undercurrents. Democritus and Plato shared much and differed significantly. Sharply distinguishing between their philosophies may very well have been a difficult task.47 Plato had an interesting criticism of, and an improvement over Democritus’ theory: he maintained that since the actual forms of things were subject to defacement and corruption, it had to be the abstract geometrical forms of things and not their concrete manifestations that characterized them. That which makes a human human cannot be a perishable shape: it must be fixed in some manner. So only the abstract shape manifest in a particular can contain the eternal truth about the nature of that particular. Plato improved the theory that identity is characterized by form, by shape. But since he deemed only eternal shapes as the true reality, his improvement carried a high price: the perishable, material universe – you and me and the birds and the trees – was no longer a proper part of ontology. A Platonist may successfully explain the nature of abstract entities such as a planetary orbit, or Goodness, and perhaps even of Man and Bird. But it is difficult for him to admit the existence of concrete things, and explain them. Socrates, he may say, partakes in the idea of Man. This is clearly unsatisfactory since it is true of you and me as well. Whatever we may derive about Socrates from the idea of Man, we may derive about all men. What makes us distinct, then? And what makes Socrates the child, the same as Socrates the old man? Can the problem of change be solved for individuals? Perhaps Plato thought that neglecting this aspect of the ontological problem was a small price to pay, since all perishable things were negligible. Perhaps he did not consider them as negligible as we sometimes take him to have done. After all, existence is hard to deny. Perhaps this explains, at least partially, Plato’s hostility to Democritus: it seems impossibly difficulty to distinguish the common and the divergent between Platonism and atomism, without loosing important aspects of ontology. But, since Plato’s main quest was after the nature of virtue in the abstract, a virtue that can never fully manifest itself in perishable particulars, he might have considered the loss negligible. As an aside we may notice that the neo-Platonists, who really took over the Platonic school for a long while, made a virtue of necessity, and advocated the idea that individuals were second-class citizens in the Kingdom of God. This is a compromise between Parmenidean mysticism and Democritian realism. It is a compromise that much resembles the one that is Aristotle’s essentialism. This may explain the fact that the neo-Platonists habitually deemed Aristotle one of them.

5 Chimera in the Dusk: Essentialism

55

You and me are not negligible, if only because we want explanations about us. Nor is Socrates, admirable and virtuous as he surely was. There were others, even among Socrates’ followers, who disagreed with Plato. Most notable among them was Antisthenes, who is said to be Socrates’ most noteworthy follower prior to the arrival of Plato. He is still a puzzling figure: very little is known about him, and the little that is known is inevitably distorted. Antisthenes denied that abstract objects existed and said he could not agree to pursue (either by thought or by action) the non-existent. He, thus denied any form of definition of terms that was not (a result of) a simple pointing out onto an object. And he denied the possibility of contradictions on the basis that they were unreal. Consequently, he saw little point in Plato’s diaeresis. This was dismissed by Aristotle, as simplemindedness. The verdict was perpetuated by Hegel who influenced generations of scholars to present Antisthenes as slow minded, as one who was unable to follow Plato’s dazzling intellect. The truth may very well be different: Antisthenes, perhaps, simply did not share the latter’s penchant for the meshing of Abstractism and ontology: his “simplicity” may have been an expression of anti-Platonism. Like Democritus he seems to have dismissed that-which-is-not-in-space-and-time as unreal (and like Parmenides he insisted that whatever exists must be consistent) Still, Antisthenes was no materialist. He was simply a concretist. Another such non-materialist, concretist, was Aristippus the Elder, the father of the Cyrenaic school. He too was a notable follower of Socrates. Some vulgar interpreters (influenced by Stoic-Christen traditions) have dismissed him as a shameless hedonist, because he identified Goodness with the sum of concrete pleasures. This is as offensively inaccurate as calling Epicurus a hedonist for the same reason. Like Antisthenes, Aristippus saw little point in Plato’s hunt for the abstract. It may very well be the case that in trying to be a consistent Anti-Platonist, that is, in trying to be a consistent concretist, he was led away from materialism and into sensationalism. This point is crucial for the understanding of western philosophy as a whole. It is devious. I hope I am excused for slowing down a little. Please pay attention. Materialism leads to concretism (since the denial of the abstract realm leaves us with the concrete, and since matter is viewed by commonsense as concrete). But concretism often leads away from materialism. Indeed often concretism leads to the complete denial of matter. This is because often the things that materialists consider as material (trees and stones etc.) are, on second thought, aggregated abstractions made of many concrete sensations. What we encounter seems to be not objects in themselves but sensations of objects. Thus (and this is a historical observation not a logical argument) often concretism lead to sensationalism (the view that the only objects we encounter are sensations). Sensationalism is (usually taken to be) the opposite of materialism since sensations are (usually taken to be) non-material. They are usually taken to be part of the mental realm. (This incidentally is why our commonsense allows materialism and idealism to co-exist with no sense of inconsistency.) Sensationalism, then, leads – however reluctantly – to idealism, perhaps even to solipsism, namely to the denial of the existence of matter, one way or the other. In sum, materialism leads to concretism; concretism leads to sensationalism;

56

5 Chimera in the Dusk: Essentialism

sensationalism leads to idealism; idealism is as far from materialism as one could possibly get. Those who linked scientific curiosity with realism and materialism sought ways to justify and strengthen this link. They took realism and materialism to be the denial of the existence of anything except concrete objects. So they became concretists. Then many of them discovered a scandal: material objects are abstractions. The only concrete things we encounter seem to be sensations. We have no immediate encounter with objects, rather we construct them, we stipulate their existence. (George Berkeley’s aim was to expose the fact that science is based on stipulations, just as Christen faith is. David Hume’s aim was to defend science, and wipe out all stipulations. Alas, tradition presents both of them as skeptics! This is so, simply because they stress that material objects are abstractions, stipulations. Kant taught – or led us to see – that even sensations are not concrete objects. They usually involve (among other things) general expectations, which are a special type of abstract stipulations. This, however, takes us far-off from our current discussion, so it seems best that we stop now.) I do not suggest, of course, that Antisthenes or Aristippus knew all of the above. Far from it: it is at any rate hard to tell what they knew, since so little about them has survived.48 In particular, it is hard to tell if they realized that their concretism had driven one away from materialism and into some form of idealism, just as Platonism did. It is very likely that they (and Plato) did not have the clean-cut concepts that we use today. But these concepts help us understand why Aristippus identifies Goodness with the sum of all that is good (concretism), and then identifies what is good with what is pleasurable (sensationalism). It also helps us understand why this move from concretism to sensationalism was feared by generations of ambivalent Stoic-Christen interpreters, who feared sensationalism as the doctrine that lead to promiscuity. They sensed that if Goodness is the sum of pleasurable sensations, than the pursuit of Goodness leads to hedonism and thus to promiscuity. They did not care that both Aristippus and Epicurus repeatedly insisted that temperance and virtues are the secrets of the maximization of pleasure, that both of them rejected and even despised “hedonistic” ways of life. They were both brushed off as hedonistic simply because of their anti-Platonism and the idea that anti-Platonism encouraged immoral conduct. Antisthenes is reported to have refuted the definition of man as a featherless biped, not with an abstract argument but with a concrete example: he waved a plucked chicken in front of Plato’s face. His follower, Diogenes of Sinope, continued this tradition of ridiculing Platonism. He is said to have deliberately visited Plato’s residence with muddy feet, staining the former spotless carpets. (This should be understood as a metaphor for his rejecting the sterilized platonic world of forms.) Plato, we are told, had called him “The Socrates gone mad”. His most famous criticism of Platonism was his claim to be able to see horses, not Horseness. Seen from the concretist anti-abstractist perspective, we can understand all this easily, as well as the famous myth about his “defacing of the coinage”, which puzzled some scholars already in antiquity. They surmised he had conducted a (fairly common) criminal act against the currency. I take it to be a misunderstood allegory, a

5 Chimera in the Dusk: Essentialism

57

metaphor taken too literally. Diogenes defamed Platonic ideas, i.e. he tainted the heavenly mint where all perishable coins are molded. The meaning of the metaphor is simple then: he was a shrewd anti-Platonist, he criticized platonic forms. There was a price to concretism, however, one that even subtle nominalists of later ages, like Berkeley and Hume, failed to reduce. It had much to do with the conflation discussed above, of semantics and ontology in the form of the idea that it makes no sense to speak of the non-existent. Since concretists denied abstract objects, and since it makes no sense to speak of the non-existent, universal terms (terms designating properties and relations) were deemed devoid of meaning. How can general laws be formulated without universal terms? How is science possible without admitting abstractions? Since Goodness, Humanity and Horseness were not admitted as objects, concretists had no adequate words with which to speak of them. How can we identify particulars as members of the same universal type if universal terms have no real reference and hence no meaning at all? How can we say that Socrates, you and me, are all human, and how can we study such universal propositions as ‘All humans are mortal’? Hume famously tried to answer these questions. An ardent nominalist, he applied Ockham’s razor and admitted no universal objects and no universal concepts. In an attempt to save universal terms – without which science is impossible – he said that a set of concrete concepts, representing particular sensations, might function as an abstract concept (by the help of a universal term which designating them). Universal terms, he said, designated all concrete concepts of a certain type. This is unsatisfactory, since it is no more than renaming the problem: it is no solution to it. Clearly, in order for the adequate set of concrete concepts to be grouped under one universal term, an adequate abstract analogy between them must be given. For example, if all and none but concrete concepts of particular men are to be grouped under the universal term “human”, it must be so because we find that all of them satisfy some criterion, some analogy, some common feature. This something, this common feature, is an abstract concept, of course. It enables us to decide the question whether a given new object is human or not. Hume said we acquire such (abstract) criteria by habit, and by trail and error, and by a priori rules of association. This mixture of a priorism and nominalism is a silent admission of defeat. Traditionally, nominalism is associated with sensationalism and with the attempt to avoid the admission of a priori knowledge, including that which Hume appealed to. As Jaeger noted, when Aristotle had entered Plato’s academy the most burning problem of the time was that of resolving the problematic relation between particular objects and the abstract Ideas.49 It should not surprise us, then, that Aristotle attempted to make peace between Platonism and Concretism (and indeed, to synthesize the theories of Plato, Antisthenes, and Aristippus). It was Aristotle’s peacemaking that shaped logic until the time of Boole. Aristotle was justly unhappy with the split between Concretism and Platonism. Neither view constitutes a satisfactory basis for science. And it was science that he cared about. Platonists belittle the material world, and concretists reject universals and, thus, general laws. Both their views, consistently followed to extreme conclusions, amount to the abolishment of

58

5 Chimera in the Dusk: Essentialism

the natural sciences, and yet endorsing them both is inconsistent.50 The conflation of semantics and ontology played a critical role here. It inhibited the discussion of nonexistent objects by declaring worthless names without references. (Note, however, that Plato’s criteria for existence had nothing in common with Aristippus’: Plato’s was imperishability, whereas Aristippus’ was tangibility.) Aristotle accepted that science should provide general laws and recognize the (material) objects that obey them. But this seemed to entail a dilemma, since the conflation of semantics and ontology forbade the admission of meaning of universal terms without prior admission of existence of properties and relations. He therefore concluded that certain properties and relations must exist. And the same goes for particular (material) objects: they too must exist. The conclusion seems inevitable: Plato’s Heaven must somehow be admitted alongside particulars. Even though Humanity and Horseness are not tangible objects, they must somehow be out there, somehow in the real world (since – as Aristotle repeatedly reminds us – Humans beget humans, not horses). Aristotle’s attempt to resolve the controversy between Platonism and concretism was an ingenious and vital foundation to science. But, we should also note that against the background of the conflation of semantics and ontology it amounts to reconciling two contradictory views by declaring them both true. Indeed he assembled them into one subtle theory. He declared that particulars exist and that universals also must exist. (It is the same as attempting to resolve the famous dispute in Solomon’s judgment by duplicating the child under dispute so as to satisfy both parties.) In an attempt to moderate the obvious tension, intensified by the conflation of semantics and ontology, Aristotle emphasized, that by admitting into his ontology universals, he was not thereby admitting them as objects. In his Categories he distinguished different modes of existence. He said that certain abstract objects existed, but not as objects, not in themselves, but rather that they existed as essences of particulars that reside in them, in a subtle sense. From the strictly logical point of view it is only fair to observe that, in principle, any contradiction can be circumvented by distinguishing different senses by which each of the contradicting propositions is true. (Even Antisthenes, who denied the possibility of real contradictions, would not object to this). This is so regardless of what issue is at hand. Because of this trite fact, it is rarely an acceptable move and hardly ever a constructive one. (It is the same as giving each of the two contestants in Solomon’s judgment the same child, but in different senses of “giving”.) Aristotle often resorted to this strategy, which seems to originate in common practices of dialectics at the Academy. When faced with a dilemma, he often refined his terminology and his theory so as to endorse both lemmas. In our case (and as part and parcel of his solution to the Parmenidean predicament about being) he distinguished different levels of existence and different kinds of objects (of substances). He gave primary existence to particulars and secondary existence to certain kinds. He said that in one sense both particulars and universal objects existed, but that in another sense, only particulars did. He thus resolved the conflict, since now abstractism and concretism are not really mutually exclusive.

5 Chimera in the Dusk: Essentialism

59

In order to support and maintain the crucial distinction between primary and secondary existence, Aristotle erected one of the most elaborated and subtle intellectual apparatuses known. Today, it is generally referred to as Aristotelian essentialism. The heart of it is an enigmatic claim: only concrete objects really exist, and yet abstract objects are out there too: in some subtle sense, they too exist. They are either general characteristics of particulars or their very essences (or both).51 They decide the character and nature of particular objects and they define the particular’s identity, in some subtle sense. We are reaching the main point of this chapter. Aristotle’s criticism was right: Platonism is insufficient a basis for science, because it belittles the material world and concretism too is insufficient a basis for science as it blocks all possibility to explain. Justly he wanted to have it both ways. But admission of universal terms as meaningful without giving a place of prominence to universals in ontology was a prohibited move. To justify it Aristotle resorted to the conflation of semantics and ontology. His solution then looks simple: Universal terms are allowed, not because universal objects exist, but because they designate essences (and their attributes). Essences (and their attributes) are universals objects, but they are not really objects. They are entities whose existence implies the existence of objects, of particular objects. Thus, some (and only some) universal terms are allowed because they (and only they) imply the existence of particular objects. This is Aristotle’s “existential import”. The reader should note that the trouble we had earlier with nominalism, namely that we do not know how to decide which object to call a human and which to call a horse, now reappears: we need to explain how we came to obtain the ideas of humanity and of horseness, and their definitions. We will study Aristotle’s answer soon (chapter 10). But before we do that let us go back to variables and attempt to sum up the last two chapters. Aristotle’s reconciliation between Platonism and concretism imposed limits over the possible range of his logical variables: the list of terms that a variable was allowed to be replaced with was limited to terms that designated essences and their attributes. Since essences do not really exist and since terms that do not refer to existing objects are meaningless, these terms had to carry an existential import, they had to imply the existence of particular objects, objects that are known to exist. Without this “existential import” Aristotle’s metaphysical structure collapses. With it, logic, ontology, epistemology and methodology become one inseparable mash. Allow me to add one more note before we move on. One of the most curious results of Aristotle’s essentialism is this: it solves the problem of change only at the abstract level, the level of universals and essences (by means of his famous theory of definitions). Yet his ontology asserts that only concrete things really exist. It admits only particular objects as real substances. This means that Aristotle suggested a solution to the problem of change for entities that his ontology denies real existence (essences). And he left unanswered the problem of change for particulars, though he asserted that only they really exist. It was an inevitable result of his essentialism, since in its core lies a chimera, a compound of concretist ontology and (a subtle variation of) Platonist epistemology. The glue that held these two

60

5 Chimera in the Dusk: Essentialism

incongruent parts together was “existential import”. Thus, it is a fundamental mistake to view Aristotle’s “existential import” as a formal rule, a matter of logical choice or convenience, to be endorsed or avoided at will. It was the only glue available: his philosophical apparatus would disintegrate back in incongruent parts if he did not endorse it. His essences would ascend to Plato’s Heaven, and his primary substances would stay here with us (and with Diogenes) ever in search of general laws to explain them.

Chapter 6

Semantics is not Ontology

Aristotelian essentialism, as we have just seen, is a most subtle metaphysical fusion of Platonism and concretism. A conflation of semantics and ontology determines success conditions for this fusion: it dictates that the reference of universal terms must exist if they are to be meaningful and logically usable. Yet, says Aristotle, the reference of these terms cannot really exist: universal objects cannot really exist: they are not objects. And so, Aristotle concludes, every use of (some) universal terms presupposes the existence of particular objects that it designates. This is known today as “Aristotle’s existential import”. It is the metaphysical “super-glue” that prevents the incongruent parts of the chimera – Platonism and concretism – from disintegrating. Its merit is obvious: it was the first glimpse of a theory of meaning that allows universal terms to be meaningful even without admitting that universal objects exist. Let us now discuss some of the difficulties entangled within. It is the habit of historians of logic to minimize the drawbacks of Aristotle’s existential import, and to portray it as a minor idiosyncrasy of an ingenious system. That the system is ingenious is not contested. But its drawbacks are inherent and significant and minimizing them is both defensive and misleading. Modern apologetics of existential import often begins by an offhanded explanation of it as (possibly erroneous) license to derive existential sentences out of universal ones, and no more than that. A partial truth here veils the whole truth. It is true that in a formal presentation of Aristotle’s logic from the sentence “All green goblins have wings” it is permitted to derive “some green goblins exist”. It is also possible to derive “some winged green goblins exist”, and, I dare say, Aristotle would have also granted us permission to derive “some winged things exist” from it. Then it is noted that the choice to endorse or reject the permission for such derivations is a contingent matter; Aristotle chose to permit them, while modern logicians – with their well-defined notion of truth in a model – may freely prefer to go the other way. A crucial fact is veiled by this explanation, however: Aristotle could not have seen it as a matter of choice, since his solution to the problem of change, the corner stone of his metaphysics, rests on it, and logic is there subordinated to it. Giving it up is tantamount to rejecting his entire system. And there is more: the only existence that modern “existential import” imposes is in its denial that the universe of discourse is empty, not in any prohibition of the use of terms that do not designate an essence. N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

61

62

6 Semantics is not Ontology

Aristotle’s main point, here, is radically different, it is that terms without substance as reference are to be banned from the realm of logic. Hence all sentences within logic must discuss entities known to exist. That is why modern “existential import” is an anachronistic tag-name for Aristotle’s conflation of science and logic.52 Aristotle’s existential import, then, is not about the validity of this or that inferential form. It is about allowing extra-logical knowledge (from simple commonsense observations to highly abstract ontological theories) to leak into logic and determine its scope. The crux of existential import is the conflation of semantics and ontology that manifested itself in the ancient view of terms: the view that some terms are legitimate (like “Man”), some (like “Green goblin”) are not, and that those terms are legitimate have their hooks into a substance. Thus, it is not the case that in Aristotle’s logic one can derive “Some green goblins exist” from “All green goblins have wings”. Both sentences are, strictly speaking, outside the province of Aristotle’s logic, as he did not believe in the existence of Green goblins. They are precluded by existential import. The crux of existential import, then, is the view that some terms are legitimate while others are not, and that legitimate terms can be sharply distinguished from non-legitimate ones, and finally, that, somehow, this distinction is a part of our logical intuitive apparatus. It is the thesis that logic is useless unless, somehow, episteme is given, intuitively, in advance. In chapter 10 we will see that episteme also seems to be guaranteed by logic, in Aristotle. Aristotle’s Existential import, then, at least in retrospect, seems like a subtle circular move. A few words about the common apologetics for “existential import”. All apologetics is defensive and hence undesirable. Apologetic history is no exception, of course. The challenge of history is to tell of past events (and to analyze their impact on present ones). Historians, with the exception of outspoken hagiographers, have no interest in belittling or even erasing the mistakes of favorite heroes. Even if they are their own favorite heroes. It is distressing to report how often Aristotle’s existential import has been defended exegetically. Its commonest modern apologists present it as a legitimate part of commonsense. It is observed that we regularly presuppose the existence of the objects that we discuss, or else we see little point in discussing them.53 Talking of non-existent things is generally seen as a waste of time, if not madness (as Descartes famously noted). This defense of the conflation of semantics and ontology deserves some attention. It is not serious, but discussing it may have great importance from a historical point of view, since it uncovers irrational motives and their role in any intellectual development. The success of any non-serious defense may be due to its effect: the shifting of attention from an irrational motive into a trite controversy. A classic case is the use of verbal violence instead of arguments in order to prevent serious discussion of some criticism. The portrayal of skepticism as a form of madness is a classic example: both Augustine and Descartes declare all skeptics mad, just before affirming the existence of a deity whose existence (we now readily admit) cannot be proved (and may easily be doubted). Plato’s reported calling Diogenes “The Socrates gone mad” perhaps set the precedent and standard for such moves, down to the very usage of the word “mad”. A similar move is made by both Plato and

6 Semantics is not Ontology

63

Aristotle when they dismiss the sophists as charlatans just before answering their challenge, which, we now readily admit, cannot be answered. The standard defense of Aristotle’s existential import discussed here, by reference to commonsense as a criterion for validity, is a very refined and subdued example to the same move: it is clearly not the task of logic to celebrate commonsense. Let me address the argument anyway, however. It is not true: commonsense thrives on fictitious entities that are deemed a proper part of it; we usually don’t take these objects too seriously. We need not mention here Aristotle’s famous goatstag, which, incidentally, became a unicorn, that is a crossbreed of a rhinoceros and a horse (not of a goat and a stag): commonsense had conflated the two cross-breeds because it seemed useless to distinguish between two non-existing things. The use of terms that refer to nothing is inevitable, as Aristotle well knew. Of particular interest here is the fact that any controversy regarding the existence of anything is clearly a controversy regarding whether or not the terms denoting it denote anything. Any controversy, any proof or disproof of the existence of any x assumes “x” is meaningful regardless of whether it denotes anything. Otherwise all proofs by reductio ad absurdum are impossible. Thus, endorsing existential import amounts at least to excluding all controversies regarding existence and all proofs by reductio ad absurdum. It makes the question “Does the term x refers to an essence?” meaningless and so even variables become suspicious. A heavy price, one might add, one that Aristotle would have been the last to accept.54 Aristotle clearly endorsed both proofs by reductio ad absurdum and existential import. Dialectical disputes too are forms of reductio ad absurdum. Thus, admitting dialectical disputes alone already amounts to admitting the legitimacy in principle of terms that are shown (by the dialectical procedure itself) to have no reference. Here, then, lies the oddity: dialectics, the original subject matter of Aristotle’s logic, seems incompatible with his clear endorsement of (what is known today as) existential import. Is an explanation of this oddity possible? (At what price?) Something like a weak and incomplete explanation will be provided here when we discuss Aristotle’s theory of induction. But we should also stress that there is no real need for such an attempt. Abandoning apologetics turns our historiographic challenge into a fascinating one, and, possibly, also into a significant one. The challenge is not to render an incoherent corpus coherent by exegesis, which is always possible and also undesirable unless it is very brief and significant. The challenge, as Joseph Agassi has shown, is rather to expose the discrepancies and the complex background leading to them, the background that makes them almost invisible to the eyes of the historical figures and traditions in question.55 Discrepancies (including contradictions) are often observed only in hindsight. There is nothing odd about this: it is the reason for the use of dialectics (and indirect proofs). Often, then, a subtle conflation smoothes down an irresolvable conflict between the excitement of an ideal and its unattainability. The incoherency that we observe in hindsight is just an expression of our inability to view it as the conflation that it originally was: it is our very realization that the historical aspiration underlying the conflation is unattainable, and that the conflict between the aspiration and the limited means to attain it is irresolvable. (Thus, today we are less sympathetic with

64

6 Semantics is not Ontology

Descartes dismissal of all skeptics as mad, simply because today we are closer to skepticism than he seems to have assumed possible for sane people.) Aristotle’s logic is the grandest attempt ever erected to answer the sophist challenge (only Kant’s transcendental logic is, perhaps, commensurate with it). But, to repeat: today we almost take it for granted that the classic challenge is unanswerable: informative theoretical propositions cannot be proven by logical means, they cannot constitute knowledge in the traditional sense, they cannot be ascertained as justified true beliefs. Aristotle’s “existential import”, then, attests to his acceptance of a challenge that could not be answered. The oddities of existential import are easily observed today as incoherencies in the Aristotelian text. This is not to say that Aristotle, or his followers, did not at times expresses incredibly clear awareness to problems directly related to it. They do express clear awareness to it (just as Descartes did, to skepticism), but (again, as is the case with Descartes) only at times: their awareness dissipates at critical points. Some of the best criticisms of Aristotle’s philosophy that I am aware of are his own. Without realizing and admitting this, one cannot expect to have a comprehensive understanding of the subtle character of Aristotelian corpus, its incredible complexity and astonishing ingenuity. Why, then, were the conflations not simply exposed as the confusion that we now recognize them to be? There are many partial answers to this interesting question, but we can never have a satisfying answer to it. However, in order to answer it, if only partially, we must describe the service that the conflation provided, or seemed to have provided, those that resorted to it. And we must compare it with the possible service that exposing it as the confusion that it is would have been. The conflation of semantics and ontology sustained the view that the skeptic challenge could be answered. The grand aim of this exercise was to enable science (by assuming that the scientific project is complete). The aim seemed so lofty, so sublime, and perhaps also so evidently within reach, that the weakness of the means to attain it, seemed to encourage endless tinkering with it, rather than despairing if it. We should now study how Aristotle’s solution to the problems of knowledge had shaped his theory of the syllogism.

Chapter 7

The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms

We have seen that Aristotle’s notion of a term is cradled by a conflation of semantics and ontology. We will now observe how this basic conflation turns into a fullfledged conflation of methodology and epistemology, of the study of valid inferences and the study of the empirical informative knowledge that valid inferences (sometimes) convey. Aristotle portrayed science as the taxonomy of reality by syllogisms. Before we can continue we must reflect upon this intriguing portrayal. It is certainly not trivial, though generations of habit may have led us to believe that it is. Its outstanding consequences will now be explained and examined. The view that science is taxonomical is based on the idea that classification constitutes an explanation of the things classified. Please think of this idea afresh, if you can. Is it not peculiar? How can the assemblages of objects into groups constitute an explanation of their nature? Interestingly Aristotle ascribes this idea to Socrates. Socrates, he says, had fixed thought in definitions. (In other words, Socratic dialectics had a profound impact on Greek thought.) This observation is revealing because it accentuates the fact that Socratic dialectics is a quest for classifications, for taxonomies, by definitions. Yet Socrates’ attitude towards the possibility of discovering true definitions seems to have been crucially different from Aristotle’s: he relentlessly refuted the putative definitions that he and his adversaries had suggested, and he had no irrefutable, alternate ones to offer. In obvious contrast to this, Aristotle’s science begins with supposedly undisputed definitions (called “first principles”, or “starting points”). Socrates, then, “had fixed thought” in the mode of an endless search for definitions (which I tagged here methodology-in-the-weak-sense). But it was Aristotle who had fixed it in the mode of a report of supposedly undisputed definitions and the taxonomies that are (trivially) entailed by them (episteme). Note that the Socratic notion of wisdom renders the conflation of knowledge and taxonomy reasonable: for both the knowledge of definitions and the ability to explain the manner and nature of its subjects are repeatedly found to be unattainable. Thus, the failure to provide a good definition is manifested by the failure to defend it as hypothesis in the dialectical game. In contrast to this, it would seem that Aristotle’s notion of knowledge clearly allows one to know a true taxonomy statement (be it a first principle or a statement derived from it) without knowing why it is true.56 And so our question is repeated: How is the assemblage of objects N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

65

66

7 The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms

into groups relevant to explaining their behavior? An example might help: we may know that all men are mortal without knowing why this is the case. By observing that all men are mortal we already implicitly classified Man as Mortal. Would an explicit classification change our knowledge then? Aristotle overcame this obstacle in an interesting and highly influential fashion. First principles, he said, are beyond explanation because they are self-evident, and, all other true taxonomy statements are in fact explanations (or parts of explanations). Thus, men are mortal because they are (classified as) animals (by definition) and because all animals are (classified as) mortal (by definition). In this way taxonomy and the knowledge that is required to compile it and to explain it were united. Taxonomy, Syllogism and scientific explanation turned one. In a very profound way. Allow me to elaborate a bit this seemingly trivial point for it will lead us to an interesting and non-trivial understanding of the structure of Aristotle’s logic and of its weaknesses. Taxonomy, Aristotle noted, is based on the recognition of similarities and (thus) of diversities. Recognizing a similarity between items establishes a class (such as the class of all animals). Recognizing a particular difference between items of the same class establishes a sub-class (as men are a sub-class of all animals). The list of similarities and diversities that single out a subclass uniquely – its place among the sum of all objects and its uniqueness – is all that definition should convey and traditionally it is exactly what definition is supposed to do. (It is the classical definition of “definition”). For example, the famous definition “Man is a rational animal” supposedly singles out men as unique among all the animals by virtue of them being the only rational ones around, that is, the only ones that are not brutes. According to Aristotle, then, the catalogue that is science is a compilation of definitions: scientists notice (essential) similarities and diversities and form a conclusive set of definitions. These comprise the catalogue of nature, a unique cosmic tablature which is God’s very own blueprint of Creation. Like a list of all the animals in Noah’s arc and their god given distinctive attributes, Aristotle’s arc, his science, lists all essences, their definitions, and the taxonomies which can follow from them. It is The Grand Matrix of Being. There is a cluster of difficulties here that we must leave for a later moment if we are to continue with our present course: Aristotle declared that some and only some true taxonomy statement are first principles. What are the criteria by which such a distinction can be made and maintained? Also, some true taxonomy statements are said to deal with essences (for example, Man is a rational animal) and others merely with accidents (for example, “Man is a Featherless biped”). This is so despite the fact that the accidental definitions too may single out all men uniquely (by some successful accident). What are the criteria by which such a distinction can be made and maintained? Instead of attempting to answer these questions now let us simply note here that the difficulty that underlies them closely corresponds to the problematic division, made earlier, between legitimate and non-legitimate terms. Indeed, as I will soon explain in detail, they are the same: in as much as Aristotle’s terms are embryonic sentences. Thus, the view that “Man is a rational animal” is a real definition whereas “Man is a featherless biped” is not, since it is merely an accidental definition, corresponds to the view that “Rational animal” is a legitimate term (say, because

7 The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms

67

it slices an authentic slice out of reality: the essence Man), and that “Featherless biped” is not a legitimate term for it designates an arbitrary slice of reality, not along reality’s natural joints, thereby creating problematic borderline cases (such as Ikarus and Antisthenes’ plucked chicken). Here is the main point of this chapter: Aristotle’s terms are in fact embryonic taxonomy statements and definitions, and even embryonic syllogisms. As such, they are embryonic descriptions of reality and even embryonic explanations of it. This point has a trivial, formal side to it, but also extremely subtle, and historically influential, epistemological dimension. In this chapter I explain its formal side. In the following chapters I discuss its epistemological dimension. Here is the formal side of the point, then: according to Aristotle valid syllogisms are extractions of (our knowledge of) the (genuine) matrix of (legitimate) terms, which thus perceived, constitutes the entirety of science. To put it in other words: the matrix of legitimate terms spawns all definitions and all syllogisms. That this matrix is hopelessly ideal should not concern us at the moment. Suffice that we agree to assume with Aristotle, for the time being, that it is within reach. We will have time enough to inspect its viability. The point to notice at the moment, then, is simply that should such an incredible and hopelessly ideal matrix of legitimate terms exist, then Aristotelian science, in its entirety, would have been given by it. Our point can (and should) be expanded to the psychological realm because often it is there that the act of spawning of syllogisms is performed. Also, it is here that our point is most intuitive: we should thus assume that the matrix of legitimate terms is mirrored (and quite possibly produced by) a matrix of “legitimate” concepts. Having agreed upon this expansion we should expand the central point of this chapter as follows: in Aristotle, the matrix of legitimate concepts spawns all (and only) sound propositions, or judgments, and all and only sound syllogistic reasoning, and thus science in its entirety. Let me illustrate this point. Assume (however problematic the notion of it may seem to readers who are familiar with modern logic) that the matrix of legitimate concepts is given to us. This means that we know which collection of items constitutes a “legitimate” sector of the grand matrix, an essence, or a proper attribute of such an essence (e.g. the collection of objects: Human, Animal, Rational etc.) and which not (e.g. Rational-goat, Green Goblins etc.). Assume that our incredible matrix includes the following concepts: Animal, Human, and even Humananimal. And, assume that it does not include the concept Non-animal-human (which is, thus, implicitly and indirectly declared “non-legitimate”). Then the taxonomical fact that all humans are animals is given in this matrix of legitimate concepts. This is to say that the (truth of the) judgment that all humans are animals is given in it. Note that the conjunction of any two concepts creates a new concept, just as the conjunction of any two terms creates a new term. If the two original concepts are given to us as mutually exclusive (like say, Goat and Rational, or Non-animal and Human) then the new conjoined concept (Rational-goat, or Non-animal-human) would be non-legitimate (and the error of considering it legitimate would be dubbed

68

7 The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms

“a contradiction in terms”). And so knowledge of all true judgments regarding essences, which, when spelled out, become definitions, of course, is clearly given as part and parcel of the matrix of legitimate concepts. Also: a matrix of legitimate concepts implies a hierarchy of concepts. It tells us which concepts are complex, which are less complex, and which are simple, by implying which concepts contain other concepts and which concepts do not. To continue our illustration, assume that Animal is a legitimate concept, and that Human-animal (Men) and Non-human-animal (Brute) are legitimate concepts. Further assume that Non-animal-human is not a legitimate concept (as it is dubbed a “contradiction is terms”). Knowing all this is the same as knowing that all humans are animals, but not all animals are human. It is the same as to know that the concept Animal is contained in the concept Human, but not vice versa. To put things in formal terminology: we have learned – merely by observing concepts – that the extension of Animal includes that of Human and, that the intension of Animal is contained in Human (so that the former is larger from the extensional point of view but smaller from the intensional point of view). Now, if a given legitimate concept is complex, then the proposition that describe its relationship to any of its sub-concepts is a logical truth, of course (we say that it expresses an analytic truth). The point I am getting at is this: Aristotle’s syllogism is a simple outcome of this trivial, formal fact: a given complex legitimate concept that includes another complex legitimate concept comprises a reasoning that, when put into words is called a syllogism. The result is quite overwhelming: the whole of science is given to us by our very concepts, it is given to us by our knowledge of the list of legitimate terms. Imagine, for example, observing the sector Animal in the matrix of our concepts and finding that the (extension of the) concept Mammal is included in it, and then observing the sector Mammal and finding that the (extension of the) concept Man is included in it. This act is a case of the syllogism Barbara (as it was called in the middle ages) the mother of all syllogisms (as Aristotle had believed). Similarly, it follows that all men are mortal, from the legitimacy of the concept Human-animal, and the non-legitimacy of the concepts Immortal-animal and Non-animal-human. The result, I think, is as overwhelming as it is trivial: Aristotle’s theory of the syllogism is simply a literal expression of his incredible assumption that we have gained knowledge of the proper matrix of concepts (and, the corresponding matrix of terms). The result is overwhelming because, by existential import, these syllogisms are not empty deductions: God’s very own matrix of being is spelled out by them. To conclude: by the union of syllogism and explanation (of the objects that are classified by these syllogisms), the cosmos in its entirety, is explained by the mere assumption that we have obtained the matrix of legitimate concepts. This curious fact is famously reflected by Aristotle’s own terminology: he uses the term horos sometimes to designate a term and sometimes to designate a taxonomy statement without worrying about the fact that terms and taxonomy statements are two very distinct linguistic entities.57 This is also why Aristotle sometimes speaks of “primary terms” in the sense of “true taxonomy statements”, or even in the sense of “first principles” or of “real definitions” (e.g. An. Post. I.3. 72b23–25, where it

7 The Mother of All Matrices, or, How Terms Spawn Definitions and Syllogisms

69

is clear that arche epistemes is embryonic definition). As Guthrie have noted, this renders real definitions “potted” syllogisms.58 So much for the formal point. Now let us move into a discussion of its metaphysical and epistemological dimensions.

Chapter 8

The Conflation of the Source with the True, Good and Beautiful (Source)

This chapter is an explanatory digression. We observed that all complex concepts spawn taxonomy statements and syllogisms. Furthermore, we observed that legitimate, complex, concepts spawn science: they spawn all first principles and, thus, all syllogisms. Science is the (true) taxonomy of reality and it is given, in its entirety, by the very list of legitimate concepts. How is this list given to us, then? We have mentioned that taxonomy rests on the act of recognition of similarities. But whence is the source of this observed similarity? In fact what is similarity? How is it established? How are concepts generated? And how do we come to choose some concepts (supposedly “legitimate” ones) and avoid others (supposedly non-legitimate ones) in the first place? Locke famously contended that concepts evolve naturally as results of experience. Leibniz famously disagreed: he contended that concepts are inborn, a priori, given. What was Aristotle view of the matter? Let us first note that Lock’s and Leibniz’s answers are both unsatisfactory in a manner that is often ignored and is not easy to notice. What they have in common is that they obfuscate the problem they come to solve: the problem to which these views answer is: “How do we generate concepts?” The problem their authors were trying to solve is “How do we generate legitimate concepts?” Locke and Leibniz were well aware of the fact that our conceptions of reality might be wrong, of course: their explicit aim was to avoid error. Nevertheless they avoided the second, tougher question, not admitting the seriousness of the possibility that there is no answer to it. For to assume that we can at all generate legitimate concepts and only them, let alone generate them at the first effort naturally, or to be born with them, is to assume quite a lot. Let me explain. Consider the following serious difficulties. To be able to generate only the legitimate concepts amounts, as we have seen in the previous chapter, to the knowledge of their legitimacy. How do we know this? When are we justified in viewing a particular case as demonstrating a general one? How do we choose the generalization (or abstraction) that is illustrated by the case at hand so as to demonstrate its adequacy? When is one general concept justly considered as illustrating another (even more general) concept? How do we decide of two different concepts, which is the more general? (What is generality?) Obviously a case deemed particular might be regarded as illustrating different abstract concepts; the same particular case may even be N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

71

72

8 The Conflation of the Source with the True, Good and Beautiful (Source)

regarded as illustrating opposing abstractions. How, then, are we to decide which particular stands for which general/abstract case? The choice here has to be founded on a criterion that separates the correct or legitimate from the incorrect or illegitimate. But how are we to choose such a criterion? Observe, for example, the concept Human (a concept Human, to be exact, for there are clearly many mutually-exclusive conceptualizations of Human). Locke contended that this concept is a natural outcome of our encountering humans. This begs our question: for the question is on what ground, to begin with, do we determine that an object of perception is a human? Does this concept allow for mistakes in identifying objects as Human? Does it admit replacement (If so is it not illegitimate?) For example: are women human? Are slaves human? (Aristotle, we should remember, deemed both groups not fully rational, and hence not fully human, and he was, of course, not alone in this.) Are retarded humans human? Are psychotic humans human? Are all races human? Are our enemies Human? Disagreements (no matter how despicable) clearly thrive here. Why do we choose the concept Human rather than Northern-Hemisphere Humans, or White Humans, or Sane Humans or Featherless Bipeds, or Animals that Laugh, or Users of tools, or “Incitatus and/or a Human? Or, simply, Us, or even This? Are some concepts really more legitimate than others? More Natural? If so, Why? On whose authority? Does the fact that several humans are rational justify our view of the concept Rational as an inseparable (essential) part of the concept Human? Leibniz contended that the concept Human is simply identical with a list of a priori attributes (including that of being a rational animal). But how do we determine which concept necessarily includes which attributes and which it includes merely by accident? This is the central problem of epistemology: it is the problem of justifying our accounts of our experiences (our professed knowledge of reality). Its most familiar version is the classical problem of the justification of induction, which will be dealt soon extensively. We should now note that conflating the source of our concepts with the Good (infallible) source of our concepts (i.e. their justification), as both Locke and Leibniz did when they obfuscated the differences between the two questions, is the archetype of conflating methodology and epistemology. Take Induction, for instance. It is a method for obtaining general accounts of reality, not a method of their justification. But Locke and Leibniz were searching for the sources of our accounts of reality with the intention of finding explicit justification of them. (In doing this, as we will soon see they both followed Aristotle.) The problem of justifying our accounts of reality, however, is concerned not with the source of our assumed knowledge of reality but with its justification, unless source and justification are conflated. The conflation of the source of our knowledge with its justification is natural and universal. Indeed it can be said to precede philosophy. Even universal mythological representations of the source of all life reflect it, as do all projections of infantile reality: it is easy and natural to conflate the life-giving womb, the life-providing breast, the sheltering bosom, the Good, and the Just. Plato famously made extensive use of an abstract version of this myth: he said that the source of all knowledge is

8 The Conflation of the Source with the True, Good and Beautiful (Source)

73

the True, Good and Beautiful, and that knowledge is justified by virtue of this reliable source. (Descartes and Leibniz called this reliable source “God”). However, since all three recognized the existence of both good and bad generalizations, and since they did not agree on what source is reliable, the problem of the justification of our accounts of reality threatens to lift its ugly head. The problem has two ugly heads (and both heads seem a bit too much to handle). The first head asks: how do we choose concepts among the infinite set of possible concepts? The second head asks: what is the distinction between good and bad correlations of such concepts? In particular, which correlations between concepts constitute true taxonomies and which constitute false taxonomies? However, in the previous chapter we noted that the ancient view of terms presents these two heads as one and the same head viewed from two different angles: the (impossibly) ideal matrix of legitimate concepts already includes the answer to the problem of deciding true correlations between concepts, since correlations between concepts are concepts in themselves (and as such, their status has already been decided by the ideal matrix). Once we allow a conflation of the source of our concepts with their justification (and thus with justification of the taxonomies and syllogisms that they include) epistemology becomes merely the procedure of extraction, or of articulation of the taxonomies and syllogisms that are inherent in these concepts. At most these taxonomies and syllogisms can be said to be tacit and/or latent in us, until they are properly extracted and articulated. Dialectics has been often suggested as such a method of extraction and articulation. The task of the epistemologist, then, becomes akin to that of Socrates leading the slave into discovering/remembering a case of the Pythagorean lemma. Consequently methodology hangs in between these two highly distinct but easily conflated actions: discovering and remembering. The problem of justifying our accounts of reality dissipates against the background of a description of the process of acquiring some account of reality (just as the problem of justifying a case of the Pythagorean lemma dissipates against the background of a series of questions provoked by a drawing in the sand). This point may seem subtle but only because it is so trivial. I wish to state it clearly because I regard it as an underrated key force in the history of Western philosophy. For assume, for the sake of illustration, that the (impossibly) ideal matrix of essences, of legitimate concepts, is somehow given to us (and thus guaranteed). If the matrix is given (and thus guaranteed) a priori (as such diverse thinkers as Plato, Descartes and Leibniz have maintained), even if it is known merely implicitly and vaguely (or merely as a transcendental expectation à-la Kant), then all knowledge of correlations of these concepts is known in advance, as a given (as Plato and Descartes and Leibniz have maintained), even if that knowledge is known merely implicit and vague as they all admitted (or even if it is merely a transcendental expectation as Kant has maintained). The central problem of knowledge for all these great thinkers is the problem of our knowledge of our knowledge: the problem of explicating and articulating whatever knowledge we (allegedly) already posses, if only tacitly and implicitly, by means of some infallible good source. Or, assume that the ideal matrix of legitimate concepts is formed (or, somehow, inferred) naturally, by induction, as a result of our experiences (as Bacon and Locke

74

8 The Conflation of the Source with the True, Good and Beautiful (Source)

and Mill have maintained). Then all factual knowledge that can at all be known has been somehow inferred by induction as a natural result of our experiences (as Bacon and Locke and Mill have maintained, at least as an ideal). Thus, if it is granted that some method secures our matrix of concepts, if it is granted that this method secures and justifies their status as legitimate concepts, then epistemology in its entirety becomes merely the act of extracting taxonomies out of our (experience approved, experience certified, experience guaranteed) concepts. Concepts, as we have seen, in as much as they are complex concepts, embrace taxonomy statements and syllogisms. We can now appreciate the fact that this formal, trivial, truth was a lifeline for classical epistemology, especially when hopes for attaining episteme seemed faint (or when their faintness was not comprehended completely). Concepts were then said to be the natural building blocks of our thoughts. And they were said to be the immediate (and hence genuine) results of our experience. We often doubt some general sweeping judgment concerning the world (say, whether, or not, all men are mortal, or whether or not all swans are white) but we do not often notice that such a judgment is the result of a seemingly innocent concept, which we rarely hesitated to endorse (such as our commonsense concepts Man, and Swan). Justification of our concepts was rarely sought in the same critical zeal as justification of the taxonomies that comprise science: concepts seemed a more solid ground than the taxonomies which they contain. For, properly understood, justification is not a procedure that applies to concepts because they are not sentences or theories, but rather abstractions of mental activities. Yet they do spawn taxonomies. And since their credulity was less questioned, they helped certify the definitions and syllogisms that they spawned. Concepts, however, are just as dangerous as taxonomies and syllogisms exactly because they are pregnant with them. In particular, when we deem a certain concept atomic, we find ourselves implying a hierarchy: we imply that some other concepts, complex ones, contain it. When we accept some complex concept as a natural result of our experience, we imply that we have learnt some informative taxonomical fact, perhaps even a law-like regularity, from experience. We may then be tempted to call these taxonomies first principles and regard them as self-evident. When, finally, we observe that one complex concept contains another complex concept we imply the validity of a particular syllogism, an inference. And, if this embedment relation is deemed a natural result of our experience, then we imply that the valid inference is a sound one. This, I contend is, in very general terms, Aristotle’s theory of terms and (thus) of syllogisms. It comprises his conflation of epistemology and methodology. The conclusion of such a sound inference is then easily (mis)taken to comprise our knowledge of the universe. And the totality of legitimate concepts is (mis)taken to yield the totality of true judgments and sound syllogisms – Aristotle’s science. Finally, as we have explained in the previous chapter according to Aristotle inferences are explanations (they explain the less general taxonomies that participate in them, also called “conclusions”). Traditionally, then, concepts function as epistemological mediators: they mediate between our conjectures regarding our world and the grounds for deeming them true and explained judgments. The means for generating concepts turned into the means for attaining and explaining truth

8 The Conflation of the Source with the True, Good and Beautiful (Source)

75

itself. The sources of our thoughts and theories and the Good, Beautiful, Truthful source of those thoughts and theories were conflated. To sum up: experience is often allowed a dual and highly deceptive role in epistemology: it is not only the source of our scientific riddles, a tag name for all that we try to explain by means of our scientific methods, but also the silent dictator of unequivocal true answers to those riddles. It induces or impresses concepts upon us, and these concepts, properly explicated, induce (the true) science upon us. When such a dual role is allowed, the task of the epistemologist becomes merely that of extracting the taxonomies and syllogisms that comprise our science, from our concepts. This, in very general terms, is the gist of Aristotle’s view of logic. Logic is explication of legitimate concepts (of notions of essences). Since it is just about the most influential epistemology in the history of Western thought, it is also the gist of a good number of the theories of knowledge that followed it. According to Aristotle, the tool for extracting and explicating the judgments and syllogisms that comprise our science from our concepts is logic, then. His logic, then, is the logic of scientific discovery and formal logic in one. The view that our experiences induce upon us the true explanation of these experiences is traditional Inductivism. Our next task is discussing the manner by which such induction process occurs, then, according to Aristotle. From a purely epistemological point of view it is the one pillar which supports the whole Aristotelian edifice. Should it stand, the edifice would stand. Should it crumble and fall, the edifice would crumble and fall. In the latter case, it would be our task to observe which of the edifice’s parts were taken and re-used by later builders of systems, and with what effect.

Chapter 9

Induction as Spell-Casting

According to Aristotle experience induces or impresses (legitimate/correct) concepts upon us. These concepts, in their turn, contain knowledge of the taxonomies and syllogisms which science comprises. The extraction of such syllogisms out of the concepts and taxonomies that contain them is Aristotelian logic. Syllogisms justify and explain the very taxonomies which they extract. (Strictly speaking, first principles need no justification and no explanation, and syllogisms rely on them to justify and explain other taxonomies, namely the conclusions of syllogisms). As seen in previous chapters, this deductive picture is an elaborated explication of the (highly problematic) ancient notion of terms and its dubious conflation of semantics and ontology, that is its implicit presupposition that we have somehow obtained the legitimate concepts and terms, which properly slice up our world into objects (whatever that means exactly) and which spawn self-evident definitions and syllogisms. The validity of science, then, depends upon the validity of the process by which experience allegedly impresses (legitimate) concepts upon us, and the validity of the process of extracting taxonomies out of such concepts. In Aristotle there seems to be a cluster of closely connected such validation and extraction processes. Their exact function and character is under debate ever since they were described by him. The most notable among them is induction. Other such process, closely connected to it and partially overlapping it, are dialectics and intuition. They will be discussed now. A few preliminary reservations are in order, however. Throughout this book (and in the following chapters in particular) I discuss Aristotle’s theories in the abstract, accentuating the intellectual apparatus and rationale which, I believe, sustains them. It is impossible to achieve this task without a deliberate disregard for many interesting, subtle details (including technical glitches and straightforward inconsistencies) which are an inseparable part of the magnificent Aristotelian text. This task inevitably involves a huge selection process, the complexities of which have been spared from the reader. I dare say this study is not one which Aristotle’s scholars will like, since for them Aristotle is found in those minute details. Perhaps they are right. I mention all this here in order to stress that the Aristotelian corpus contains no complete theory of induction, but rather scattered hints, highly problematic examples which often replace a discussion of what they are supposed to exemplify, and many, many, enigmatic loose ends, which only by great effort can be woven into a philosophical fabric. These hints evoke more than one theory of induction and they are N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

77

78

9 Induction as Spell-Casting

the source of many interesting exegetical controversies. Some of the theories that they evoke complement each other in interesting ways while others either partially overlap, or are incompatible. In fact, like our general subject matter (logic) our present subject matter (induction) has been created, by and large, by later readings of the text that we are about to analyze, and so is itself an idealization of the Aristotelian corpus. It is impossible for us to view Aristotle’s theory of induction afresh, independently of such later traditions. However, it is also highly misleading to do so. For every scattered hint and every statement that Aristotle wrote of induction has been developed by generations of scholars into some partial or full-fledged theory of induction which in turn, has silently sunk into our tacit knowledge of general philosophy. Any such partial or full-fledged theory could then be anachronistically read back into the rougher, essentially less detailed, and sometimes vague Aristotelian text. This deserves our attention, and merits our caution: the parameters that we use in order to identify and delineate our subject matter in the Aristotelian corpus, have been largely determined by later readings of that corpus. Remembering this, we should also seek the role that the Aristotelian corpus played in shaping the traditional need for a detailed theory of induction. Obviously, such a need would not crop up if a full-fledged clear and explicit theory of induction existed in the original corpus. General features of the Aristotelian corpus (which have nothing to do with induction per se) play crucial part here, features that cannot be discussed here, such as its draft-like, somewhat hurried style, which often results in many terminological glitches and exegetical difficulties. Aristotle’s use of examples is particularly confusing: even when his theoretical point seems clear, he often attempts to clarify it by means of an example which complicates things, sometimes beyond repair, by emphasizing minor aspects of the general point, or even by making a borderline case into a major paradigm. Part of the traditional need for a theory of induction, however, is unique to Aristotle’s discussion of induction. For Aristotle is responsible for the view that induction provides a service, which in fact, we know now, it does not and cannot provide: justifying our theories of reality. This fact is somewhat boosted by the counter-effective rhetoric of the Posterior Analytics: a detailed theory of induction is repeatedly promised throughout its pages, but it is not to be found there, at least not explicitly, although it is repeatedly stressed that without it science would have no foundation, and the skeptics challenge would remain unanswered. Consequently Aristotle significantly amplified the need for a theory of induction as a prop to science, beyond anything that was felt by his contemporaries and predecessors. This is in part also the result of the fact that he was quite possibly the first to portray all the sciences as deductive systems (modeled roughly upon geometry, as it was later immortalized by Euclid, and which already the Platonists seems to have considered to be central to science). The deductive structure distills the problem of knowledge, perhaps even to a point of exaggeration: it sets it in the cleaner-than-life vocabulary of justifying the foundations (the axioms) of science. The problem of knowledge, of course, was well recognized before Aristotle. His own formulation of it, in his Posterior Analytics, resorts both to Plato’s classic formulation of it in his Meno, and to the skeptics doubts concerning the possibility of securing some knowledge beyond any doubt. His formulation

9 Induction as Spell-Casting

79

clearly implies that the skeptic’s challenge was a central concern in his time, and one that he did not dismiss offhandedly, as much as he may have wanted to. But it seems illuminating to me to observe that it was Aristotle who first formulated the skeptic’s challenge distinctly in terms of justifying the first principles of science. It was he who repeatedly stressed that without an explicit justification science is useless. And, to repeat, such justification is constantly promised and masterfully postponed throughout the Posterior Analytics. Then it is (perhaps) presented by a few hurried and notoriously obscure sentences. The result is an alarming sensation of a promised land not reached: a chasm between Aristotle’s unprecedented clear and didactic presentation of the problem of knowledge in the first two chapters of this magnificent work, and the puzzling solution to it that he may (or may not) have offered, in its closing sentences. Thus, the father of deductive logic provoked a mood which slowly became dominant in the history of epistemology: science is the greatest of human achievements, but without induction it is as worthless and baseless as the ravings of a madman. Enough apologies. Let us review whatever little we have. Induction is the literal meaning of the Greek term epagoge. Epagoge means “leading into”, and it suggests shades of the magical act of controlling an adversary by charm, i.e. leading adversaries into admitting whatever speakers wish them to admit. The term was used to designate the act of persuasion, of eliciting an opinion, primarily in the context of the dialectical duel. As is often the case with Aristotle he has adopted here a loose, rather common word for his own theoretical use: refining its meaning, jargonizing it and finally redefining. Thus, the birth of induction is not in methodology but rather in the hazier intersection of rhetoric and dialectics, at a stage when the epistemological power of both of these practices was still debated and not fully understood. It seems reasonable to guess that induction slowly matured into its role as the method for securing the first principles of science in a similar manner to the maturation of dialectics from a game of skill into a scientific method. The catalyst may have been the gradual maturation of Aristotle’s own thought: his growing awareness of the need for a sound and secure method of justification of the first principles of science is a result of his growing awareness of Plato’s failure to do so. For originally induction was a persuasion method not a scientific method. As such, it was not necessarily a “clean” method: it was a mode of argumentation aimed at eliciting consent, regardless of the truth of the matter. And it was not a single well-defined procedure at all. Rather it was a family of procedures. Let us examine a few by means of an example. Recall the context of the dialectical duel. Assume that our adversary point of departure is an observation P. Our role, as his opponents, is to lead him to admitting that 1. If P is true then another statement, Q, is also true. And that 2. Q is evidently false. This should induce his admission that his initial observation, P, is false. How can induction aid this process? Let P be “It is now raining hard in Athens”. Then we may wish to argue along the following lines: If P (it is now raining hard in Athens) then Q (Socrates would not be out in the marketplace). (For he is never out in the marketplace, when it is raining hard). However (not-Q) Socrates was just now spotted in the marketplace. Thus (not-P) it is not raining hard in Athens at the moment.

80

9 Induction as Spell-Casting

Now, any of the statements that we made in our argument can in principle be supported by any of the argumentation modes that comprise the family of induction: the a fortiori argument, the example (which Aristotle also called “rhetorical induction”), the analogy, and the inductive syllogism proper. Consider the basis to our claim Q: “Socrates is never out when it is raining hard”. If we argue for it by noting that “Socrates is never out even when it is raining softly, let alone when it is raining hard”, we have provided an argument a fortiori. It is a form of induction, according to Aristotle, because “hard rain” and “soft rain” are instances, occurrences, of the same general case “rain”. It is, Aristotle maintains, legitimate to argue from a stronger occurrence to the weaker one, as long as they both instantiate the same continuum, the same general regularity. We may also attempt the following mode of argument: “Plato is never out even when it is raining hard, Aristippus is never out even when it is raining hard, Antisthenes is never out even when it is raining hard, And Socrates too is never out when it is raining hard”. This, according to Aristotle, would be an argument by examples. Note the underlying general statement “All men are never out when it is raining hard”: though it is certainly crucial for the success of our argument – for identifying all the particular cases as exemplifying the same general case – it was not made explicit. It plays only the implicit role in an argument by example: making us see that all named cases are similar to the one we wish to argue for. An argument by analogy is essentially the same as an argument by example. (It seems that the crucial difference between analogies and examples, in Aristotle, is simply that in analogies the particulars discussed may well be far apart in the grand matrix of legitimate concepts: possibly, analogies need not even share the same genus in order to convince us.) Thus, the following may be an argument by analogy: “Socrates’ dog, and cat, and even Socrates’ coat and sword are never out when it is raining hard, and Socrates too is never out when it is raining hard”. Induction, in the service of rhetoric, is an argument by examples that spells out explicitly not just the examples but also the general case that is supposedly exemplified by them. Often it is a straightforward attempt to establish the general case by enumerating particular cases that exemplify it. In the case presented above induction may be an attempt to establish the general fact that all humans stay at home when it is raining hard, by pointing at the habits of particular men (such as Plato, Aristippus, etc.). Having established the general case we can apply it by deduction to our QED, which is the particular case of Socrates. Let me stress that the chief goal of induction at this stage, at this specific context, is persuasion, not the finding of truth. Perhaps because at this stage, or under this specific context, the persuasive effect of induction is more important than its logical structure, the criteria for induction are not very strict. To be sure, Aristotle includes under its title a host of interesting complementary procedures, formal and non-formal, that facilitates its desired effect. For example, the dialectic rules impose upon the participants of the dialectic duel an obligation: to note a counter-example to any statements that they refuse to admit. Should they fail to produce such a counter example they are obliged to admit it, regardless of their judgment of it, indeed regardless of its truth-value. This procedure too belongs, according to Aristotle, to

9 Induction as Spell-Casting

81

the family of induction. Indeed, all not refuted (and irrefutable) general claims seem to belong to the family of induction (and they are used in a host of interesting manners to create the rhetorical illusion required for persuasion). A simple straightforward observation sentence, like “Look, Socrates returns from the market place”, is a case at hand (so long as the context excludes “ostensive definitions” being refuted). Famous quotes, accepted truisms, popular beliefs, even properly executed geometrical drawings serve the same purpose, and are all considered forms of induction, simply because their truth is regularly taken for granted. Even the prosodic melody of the inductive argument seems to be an inseparable part of the appeal of induction. Aligning in a single file and enumerating similar cases has the monotonous appeal of a lullaby. Like counting sheep. Monotony entices anticipation for regularity. We should not be surprised, then, that Aristotle chose for induction a word that resonates with magical control over subjects, nor should we be surprised that he often stressed that this magical control is much more effective in eliciting consent when addressing the young, or the multitudes, or the less experienced adversaries in the dialectical game. Magic has always been less effective when practiced over the critically minded. So much for induction as a rhetorical-dialectical mode of argumentation. We must move to methodology proper, however. For induction as rhetoric merely sets some of the background context of the birth of induction aright. With induction as scientific method however, things get less clean, for clearly here the stakes are considerably higher: bewitching an inexperienced audience by the skillful use of monotonous examples is one thing, but explaining and specifying the true method for establishing the truths of science is quite another.

Chapter 10

The Birth of Induction from Sea Foam

Dialectics in Aristotle is no mere display of rhetorical supremacy. It is a logical method burdened with a heavy epistemological cargo: searching, testing and possibly even establishing definitions.59 In this sense, it echoes Zeno’s method of semi-formal indirect proof and Plato’s diaeresis, though it is not exactly identical with either. It may seem only natural, then, that induction, initially a tag name for a family of dialectical modes of argumentation, should share with dialectics that heavy epistemological overload.60 In the Posterior Analytics induction takes over this very overload. Here is the problem: Aristotle (justly) considers dialectics an inseparable part of logic (being a method of refutation, or of a semi-formal indirect proof). Induction, however, is not an inseparable part of logic. Nevertheless, Aristotle sometimes presents it as if it were, as if it is a peculiar aspect of the syllogism. This situation confuses readers, especially modern readers, since it seems to suggest that Aristotle’s understanding of the situation does not resemble our own modern understanding of induction, and the syllogism. As we understand it today, induction is no deduction and no aspect of the syllogism. Aristotle, it seems, at times had it differently: at times he seems to stretch the notion of the syllogism so as to fit into it the inductive syllogism and at other times he seems to shrink the notion of induction so as to fit it in as an aspect of the syllogism. (Since he practically invented these terms, and as technical terms, this was not as difficult then as it may sound.) Let us now examine his view closely, since it is directly responsible for the fixating of logic as a part of unreachable epistemological ideals for over two millennia. Allow me to slow down a little. Aristotle had opened his Topics as well as his Prior Analytics with a staggering observation: the syllogism, he said, is the form that dialectical syllogisms and apodeictic syllogisms share.61 We have observed already that this is one of the most decisive moments in the history of logic. Aristotle clearly recognizes here that his logic is a description of the isomorphism of dialectics and of apodeictic inferences. The observation not only makes perfect sense to us; it also justly demands to be regarded as the very moment of the inception of logic. For, we should understand it to mean that dialectical refutations and sound inferences share a logical form (the valid inference, the syllogism). I have stressed this point repeatedly in preliminary iv: valid inferences exhibit the isomorphism shared by sound inferences and N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

83

84

10 The Birth of Induction from Sea Foam

refutations, since they transmit truth from premises to conclusion (thus, rendering themselves sound inferences) and they likewise transmit falsity from conclusion to premises (thus, rendering themselves refutations). However, Aristotle’s staggering and profound observation is not as neat and clean as it may seem at first sight. For Aristotle does not show interest in the neat logical isomorphism that we have just worded for him. For dialectics sometimes seems to be aimed at searching, testing, and possibly even finding and establishing definitions. And, as such, it often overlaps with induction within the Aristotelian corpus. We will soon observe the rationale behind this overlap. Before this we must notice the questions that this overlap invites: Can a purely deductive method lead to the discovery of definitions? Can it establish them? Is induction such a method? Is dialectics such a method? Is inductive inference isomorphic to sound inference? Is it isomorphic to dialectics? What then is the logical form of induction? Is it a valid form of inference? If yes, what is it? If no, how can it be isomorphic either to sound inferences or to dialectic refutations? Clearly, today we deny flatly that there exists an isomorphism between induction and any valid inference. But it looks as if Aristotle thought differently, at least some of the time. And then our task is to reconstruct his outlook and the rationale behind it. Let me give a classic example to the complexity of the situation that we discuss now. Zeller’s famous rewording of Aristotle’s staggering isomorphism observation discussed above is the following clear-cut remark: “Aristotelian logic (in the “Second Analytics”) deals with induction as well as proof; but both are preceded by the doctrine of the syllogism, which is the form common to both”.62 We will now try to reconstruct the intellectual setting that made this extremely influential and problematic view possible. The overlap between induction and dialectics in Aristotle has its roots in his understanding of Socratic dialectics. For he often made clear that Socratic dialectics somehow anticipated and even involved induction. Aristotle regularly presented Socrates as his one bona fide predecessor, who discovered (partially, and within the limited field of ethics) the method of induction, of tracing the universal within the particular, and formulating the finding by means of definitions.63 A good Platonic source for this view seems to be the short excerpt from the Phaedrus which we have already mentioned with reference to the distinction between legitimate and non-legitimate terms; in it, Socrates states clearly the two principles that dialectics comprises of: “That of perceiving and bringing together in one idea the scattered particulars, that one may make clear by definition the particular thing which he wishes to explain…[and]… That of dividing things again by classes, where the natural joints are, and not trying to break any part, after the manner of a bad carver” (265d–e).64 We have already explained in detail diaeresis but have identified it as Plato’s version of Socratic dialectics. Here, however, diaeresis appears as only half-of-the-dialectical-story: its other half, its inversed mirror reflection, is the very method “of perceiving and bringing together in one idea the scattered particulars”, namely, the method of induction. This makes induction and division isomorphic, of course, for induction is the inverse of division!

10 The Birth of Induction from Sea Foam

85

The isomorphism that we find here is not exactly the isomorphism that we seek in Aristotle. As far as Socrates is concerned, both dialectics and induction (division and integration) are pure methodology-in-the-weak-sense: for Socrates was not in the habit of offering definitions which he endorsed and defended. On the contrary, he observed the isomorphism of induction and division because both methods offer speculations, hypotheses, refutable conjectures. Induction for Socrates is an attempt to integrate in one idea scattered particulars, so as to explain them, and division is an attempt to explain scattered particulars by naming the ideas which (by the same hypothesis) they fall under. Aristotle tells a different story, which complicates things somewhat. He viewed the method of induction as establishing irrefutable definitions, not conjectures. Socratic dialectics, then, may very well be the source of Aristotle’s view that induction and division go up and down the same ladder of abstraction, but we should not regard Socratic dialectics as the source of the view that we have found such a heavenly ladder by which to climb to legitimate abstractions. Aristotle presents the heavenly ladder as his logic. If we are to reconstruct his view, we must take a closer look at the structure of induction. I am afraid that I cannot forego here introducing briefly yet another terminological distinction. In explaining the rationale of induction it is crucial to make significant use of the division of logical techniques into the analytic and the synthetic.65 Please let me explain. Synthesis is a common Greek word denoting putting together. In the present context, it is the putting together of truths with increased complexity, as in geometry, by deriving complex theorems from simpler ones, and ultimately from axioms. Analysis is the contrary movement: it is the decomposition of the complex to the simple. When the structure in question is a complex statement, its elements may be elementary truths or elementary errors. The need for the analysis of error rests on the supposition that to recognize elementary errors is easier than to recognize complex ones, that common ones are usually complex, and that this process takes place in a dialectical game (and in indirect proofs). Similar to the analysis of error is the analysis of truth: it is hard to see that an advanced theorem is true, but its analysis brings about conviction: in the very reduction of it to elementary, easily recognizable truths, it becomes obvious. The purported service of analysis then is this: it makes the complex into a complex of simples, so as to make its status evident. What is induction, then? Is it analysis or synthesis? The answer is not as simple as it may seem. According to one traditional view, which will now be inspected, induction is synthetic. According to another traditional view, which will also be inspected now it is analytic. And, odd as it is, both views are right. For the answer depends on one’s ontology and epistemology: it depends on what one deems particular, simple objects, and what one deems elementary, simple truths. Consequently, it depends on what one deems complex objects and truths. Aristotle had his own astounding answer to this question which reveals the astounding depth of his vision and which is closely connected to his compromise between the admission of the commonsense-concrete and the Platonist-abstract (as explained in detail in chapter 5). His answer, which actually sums up his conflation of semantics, ontology, methodology and epistemology, is that induction is both synthetic and analytic.

86

10 The Birth of Induction from Sea Foam

In order to explain this amazing contention let me first put it in Aristotle’s favorite compromise formula: Induction, he says, is in one sense (which he calls the “prior in relation to us”) a method of synthesis and in another sense (which he calls the “absolutely prior”, or the “prior by nature”) a method of analysis.66 In Aristotle the two senses not only shift so as to accommodate different epistemological and ontological difficulties, they also justify one another. This, then, is perhaps the most clear-cut and intriguing case of circularity in Aristotle. Allow me to explain. There are at least two very different notions of induction which complement each other, in Aristotle as well as in general. Together they form a kind of bootstrap theory of induction: they allegedly turn it from a fallible loose method for formulating conjectures into a seemingly rigid method which seemingly delivers infallible definitions. The two different notions correspond to the two distinct ontologies which Aristotle united by means of his existential import (as we know call it): the ontology of the commonsense-concrete and that of the Platonist-abstract. Let us recall: the two ontologies put forward opposite “things” as simples, as building blocks of the cosmos. According to the view of the cosmos as commonsense-concrete, its elementary building blocks are you and me and particular sticks and stones. Abstract objects like Swan or White are then complex assortments or collections of particular objects (particular swans and white objects). Indeed, as we explained earlier, they are so complex that materialists and nominalists alike deny that they are genuine objects. These complexes, then, are the result of synthesis, of the putting together of many scattered simples under one roof. Induction, according to this view, is such a putting together of scattered particulars into complex assortments. (More accurately, induction is the synthesis of singular propositions lumped together in one universal proposition; but, we may remember, the difference between a universal concept and a universal proposition is blurred within the ancient view of terms; indeed, even for Hume the so called “problem of induction” and the so called “problem of universals” are parallel if not identical, although he does not seem to be fully aware of this interesting fact.) We can call this type of induction, then, induction by enumeration or simply induction by synthesis.67 One particular (say a swan), and another, and another, together make up a general concept (Swan), so it is claimed. Nature’s lullaby puts a spell on us: we observe that the first swan was white, and the second, and the third and suddenly, after this or that critical point, critical reason itself is lulled, and we fall into a dogmatic slumber, reckoning that all swans must be white. We already noted that Induction by synthesis is no effective rigid methodology: it conflates the source of our concepts with the good, infallible source of our concepts, and it does not explain why we choose certain general concepts while ignoring others that are just as general and no less adequate. Already Bacon called that type of induction childish. We all tend to rely on such inductive generalizations, of course. (It was assumed, for example, as a matter of course, that all swans are white – until James Cook discovered Australia.) But habits are irrelevant from the rigid epistemological perspective, as Hume noted: we all have our good and bad habits, of course, but it does not follow that we should make up a philosophy out

10 The Birth of Induction from Sea Foam

87

of them. However, we certainly do not need to resort to Hume’s famous critique of induction in order to stress this point: Aristotle’s square of oppositions would do just fine. For it states clearly that all judgments of the form “some x are y” can be true, alongside “some x are not y”; the wrong conclusion that “all x are y” from “some x are y” would be sanctified if all induction by enumeration were allowed. And if not, which ones should be allowed? And why? Aristotle, then, was well aware of the epistemological limits of induction by enumeration, by synthesis. This is why he also resorted to a second, complementary, type of induction. It is born out of the Abstract-Platonist aspect of his ontology. Recall that Platonism recognizes only abstract forms as genuine building blocks of the cosmos. Concrete objects, such as particular sticks and stones, then, are not real (not genuinely real). They are only real to the extent that they partake in this or that abstract Form. And they partake in many such abstract Forms at once, so that they are, in this sense, complex. According to Platonism, then, abstract ideas are the simple parts of our thought, as well as simple parts of the cosmos. And note this: Aristotle stresses that there is a strict hierarchy of such abstract universals: the more abstract a universal is, the simpler it is, for other ideas may participate in it, but not vice versa. For example, every white patch is also colored but not vice versa, and so Colored, according to Aristotle, is simpler, more general, than White. (The sketch of a single and unique hierarchy of abstractions is unworkable, of course, and is one of the main reasons that Aristotle’s Grand matrix of essences was never assembled, but it must be allowed here as a part of the presentation of the rationale behind his theory of induction.) Aristotle, then, presented his categories as the simplest, most abstract, building blocks of the cosmos. As he has put it, they are simpler and more basic than all else in the “absolutely prior” sense, for nothing is contained in them and everything else contains at least some of them. (You may recall that this is exactly why they are not allowed into his logic: they are too simple to take the place of a subject of a sentence). Slightly less simple objects are second in order: predicates such as (possibly) Colored or (possibly) Animal. (These are guesses, of course, for, to repeat, The Grand Matrix was never assembled.) Next in order are complex predicates such as White, or Mammal or Fish. (These, again, are guesses.) Next are even more complex predicates such as Human or Whale (and perhaps Greek). Finally, the most complex objects around are objects which cannot be predicated at all of anything else, but contain the most predicates in themselves. These are objects such as you and me, a certain bird or a certain tree. Each one of us individuals, then, is the most complex object in the universe from the “absolutely prior” perspective, exactly because each one of us is the simplest and most basic particular around from the point of view that considers the cosmos as comprising commonsenseconcrete things. A similar (though not identical) idea exists in classic sensationalism and idealism: the complex idea of a particular apple, as Berkeley noted, is decomposable into the simple conceptual atoms of the idea of its shape, the idea of its color and the idea of its scent. By the Idealist, then, induction has nothing to do with synthesis, for it accents simplification and abstraction rather than collection and enumeration. It breaks up

88

10 The Birth of Induction from Sea Foam

a single apple into its basic components: Redness and Roundness. And it breaks up these components to simpler ones: that of being colored and that of being a closed curve. Thus, induction is, strictly speaking, analysis of a particular (such as an apple) into the universals which comprise it. To analyze particular (immediate) observations into the universals and universal regularities which explain them is science, if idealism is true. Similarly with Platonists: they seek to analyze the universal ideas (eternal forms) that are implicit within each and every particular. They seek the universals that make the very existence of common particulars possible. They too, then, view induction as analysis. The difference between induction as analysis and induction as synthesis may seem structural. Induction as analysis may seem merely a new way of formulating the process of induction by synthesis, by enumeration. This is true, of course, from the epistemological point of view, but not from the ontological or the methodological points of view. For the ontological disagreement between concretists and abstractists implies a methodological disagreement about the direction of induction. What one group views as premises of induction the other group views as conclusions, and vice versa. Commonsense-concretists enumerate observation statements as immediate premises and universal regularities as conclusions resulting from them. Abstractists, by contrast, stress that universal regularities are logically prior to observation statements: they are their presuppositions. Consequently, whereas concretists deny the existence of universal forms, abstractists observe that observation statements were impossible unless universal forms existed. It should not surprise us that this last formulation of induction as analysis sounded like a transcendental argument: it is. For abstractists stress that we would be unable to conceive of a single individual had we not implicitly assumed, beforehand, that it participates in some universal idea. Thus what seems like an invalid attempt to reach a sound generalization by enumeration (for the concretist) is really a transcendental argument (according to the abstractist). This is the historical transition from Hume to Kant. It should not surprise us to find, then, that commonsense-concretists stress that they enumerate as many particular cases as they can in the hope of making their universal generalization stronger, better founded. They try to exhaust by enumeration the extension of a universal in an attempt to achieve a good grasp over its intension, and they try to exhaust the extensions of different universals in order to achieve a good grasp over their intensional relations. Abstractists, who perform induction as analysis, however, typically have no need of such an exhausting exercise: they need only one (perhaps two) concrete objects in order to dismantle the universals which these concrete objects (allegedly) pre-suppose. For example, idealists strip the universal Red out of a single observation of a given red apple or, at most they compare a single red apple to a single non-red apple (as Hume had done), and then infer Redness by comparison.68 Thus, induction by analysis is not about exhausting extension so as to get a better grip of intension. Rather, it is about extracting intensions that are presupposed by the very existence of extensions. What may, perhaps, surprise some modern readers is how well Aristotle noted all this, and how often, and how central he deemed it to his system. We need no go

10 The Birth of Induction from Sea Foam

89

as far as his grand opening chapter to his Physics. The grand opening chapter to his Posterior Analytics should do. Having established there that, “…inductive arguments…prove the universal from the self-evident nature of the particular” (71a 4–9), he explains that there are two senses in which previous knowledge is necessary (71a11–28): “Recognition of a fact may sometimes entail both previous knowledge and knowledge acquired in the act of recognition; viz., knowledge of the particulars which actually fall under the universal, which is known to us… that this figure inscribed in the semicircle is a triangle we recognize only as we are led to relate the particular to the universal (for some things, viz., such as are ultimate particulars not predictable of anything else as subject, are only learnt in this way …)”. Ignoring the geometrical example, which complicates things by sending us to the overly neat field of geometry, we may conclude and say that, according to Aristotle, the very recognition of ultimate particulars already presupposes knowledge of the universals within which they participate. Otherwise, we would not have recognized them as what they are. For example, when we perceive Socrates, by necessity we also perceive a human being, and likewise an animal. This, according to Aristotle, is part and parcel of our very ability to perceive Socrates (So that syllogisms are presupposed by observation statements!) A bit later on Aristotle resolves the priority controversy between concretists and abstractists as follows: “There are two senses in which things are prior and more knowable. That which is prior in nature is not the same as that which is prior in relation to us, and that which is < naturally > more knowable is not the same as that which is more knowable by us. By ‘prior’ or ‘more knowable’ in relation to us I mean that which is nearer to our perception and by ‘prior’ or ‘more knowable’ in the absolute sense I mean that which is further from it. The most universals concepts are furthest from our perception, and particulars are nearest to it. And these are opposite to one another.” (71b32–72a5) According to Aristotle, then, we are able to analyze the perception of an individual into the universals which make it up, and we are also able to do the exact opposite, to collect many scattered particulars into a single universal term. He calls the faculty which enables us to perform the first task, analysis, Nous. (Nous is normally translated as intuition, and, whether or not this translation is accurate, it is clearly the source of Kant’s concept of the transcendental intuition; they share an epistemological function and structure: justifying and explaining universal propositions as presuppositions or prerequisites of our immediate perceptions of a particular fact). Nous, then, is Aristotle’s name for the ability to perform induction-as-analysis: it is the faculty which enables us to perceive particulars, by presupposing the universals which comprise them. Two things remain to be noted. The first is that by elucidating induction-asanalysis we have explained why Aristotle thought that it somehow overlaps with dialectics. For both methods are methods of analysis, of decomposing and simplifying the immediate (be it a proposition or an observation) into the more basic which it presupposes. The only important difference between these two methods seems to be that dialectics aims (mainly) at exposing these presuppositions as inconsistent, whereas induction as analysis aims at showing that the presuppositions are necessary, if the immediate proposition is to be admitted as true.

90

10 The Birth of Induction from Sea Foam

The second thing to be noted is that the method of induction-as-analysis and the method of induction-as-synthesis are not only inverse mirror images of the same process, in the philosophy of Aristotle (and elsewhere): They are used to justify one another. Thus, in the closing paragraphs of Posterior Analytics Aristotle explains that general concepts are formed (in our Nous) by repetition of reoccurring similar events. Yet he likewise repeatedly stresses the very opposite: recognition of the very similarity of any two objects already presupposes universal criteria for similarly. (Criteria that are imposed by our Nous on our perceptions). For example, different humans would not be viewed as human unless we previously attained the universal Human and applied it to them, and we would not have attained any such universal concept, Human, unless we were raided steadily with (similar) perceptions humans. The result of this circularity is peculiar: every observation (allegedly) sends epistemological aid to the two far ends of the province of our knowledge: it sets and strengthens the universal concepts which it exemplifies (induction as synthesis) and it performs the exact opposite, it deduces (by induction as analysis) the universal which the very perceptibility of the particular presupposes. This guarantees the logical certitude of empirical science, according to Aristotle: empirical science becomes the logical result of experience. The cost is that of considering the project of science inherently complete and completely given by experience (and logic). To put it differently: the cost is that any advancement in empirical science entails the inconsistency of our logic. Induction-by-analysis is, of course, as much a way of formulating our conjectures as induction-by-synthesis, and no more. The two methods are parallel in a strict manner and hence not complementary: the one cannot be used to justify the other, or vice versa, for both offer conjectures regarding abstraction; both are methodology in-the-loose-sense. To stipulate, for example, that the humans Socrates, Plato, and Aristotle presuppose a single and unique concept Rational, is the same as to observe that these humans add up to determine a single and unique concept Rational. Setting this unachievable ideal of a full-proof bootstrap theory of induction aside, however, the greatness and acuity of Aristotle immediately captures us. Kant, to note the most famous historical case of induction-as-analysis, had argued that Euclidian geometry and even Newtonian mechanics function as presuppositions of our perceptions. He deemed this staggering – later on refuted – conjecture, a novel type of proof, a transcendental proof, and a second Copernican revolution. Not wishing to belittle this enormous achievement, nor its claim for originality, we should note that already Aristotle recognized the gist of that proof and utilized it, at least in a rudimentary form, as the foundation of his epistemology.

Chapter 11

Taxonomy of Reality by Syllogism

This chapter concludes my discussion of the epistemological dimensions of Aristotle’s logic. I offered delineation and discussion of some of the influences of his notion of term over his notion of taxonomy statement (and thus) over his notion of syllogism. My aim was neither to criticize these notions nor the system as a whole, of course, but rather to note the knotty package deal that they comprised. It was a tough, take-it-whole-or-leave-it, package deal. It offered a great lot and demanded too much. It offered the first and best theory of inference for over two millennia and a solution to the sophists challenge, in one neat parcel. It demanded the conflation of these two tasks by means of acceptance into logic of a series of (not always explicit) extra-logical intuitions which became increasingly antiquated and outdated but ever so hard to pinpoint and replace. Aristotle’s corpus includes various suggestions as to why he deemed feasible the separation of legitimate terms – that designate essences – from the class of all terms. It is impossible to do justice to these suggestions because their very persuasiveness is the result of irredeemable conflations that cradle them. I have tried to do so here nevertheless. Thus, I noted that the proper acquisition of “legitimate” terms (and “legitimate” concepts) was declared by Aristotle to be a natural result of experience (a claim that rests on the conflation of the source of our concepts with the Good/infallible source of our concepts), and the result of the dialectical quest of the “good carver” (a claim that rests on the conflation of dialectics and proof), and the result of a subtle interplay between induction-as-analysis and induction-assynthesis (that comprises Aristotle’s bootstrap theory of induction and Nous). Also, Induction, Nous, and dialectics seem to prop up each other, in Aristotle (and perhaps even more so in the Aristotelian tradition). Often Induction, Nous, and dialectics are difficult to tell between. Perhaps they are the same method viewed from different perspectives (or reflecting different phases in Aristotle’s intellectual development). We should not be surprised that the conflations themselves have a tendency to conflate: they are the same epistemological mist viewed from different places. When (supposedly) legitimate terms are admitted as presuppositions to logic by a (seemingly) logical process, the conflation of logic and episteme is complete. For these terms, by means of the concepts that they represent, become embryonic taxonomy statements, embryonic definitions and even syllogisms. Since these terms are N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

91

92

11 Taxonomy of Reality by Syllogism

supposedly logically established their status is deemed incontrovertible. Since they are said to have an existential import, they have empirical content and meaning. Thus, empirical content and logical necessity are fused. I have stressed the role of concepts and terms in the history of logic (a role that has been neglected in the past century as a direct result of the rise of modern logic). For the purposes of the rest of this book it may help to couch these conclusions in a more familiar early modern terminology. Let us reformulate them in terms of definitions then. This, I think, is crucial for an adequate understanding of the history of logic, for the theory of procuring definitions was at the center of attention of traditional logic, especially in the early modern period. Discussing the conflations presented above in terms of definitions would constitute an appropriate preliminary to any discussion of early modern philosophy of logic, since the theories of Locke, Leibniz, and Kant typically stress the role of judgments and definitions rather than mere concepts and terms. We should begin with the definition of definition. Definition in the modem sense of the word, is any equation (or, sometimes, an equivalence statement) expressing nominal, or verbal, identity. One side of the definition (the left one in English) is termed the definiendum and the other the definiens. All modern definitions are traditionally called nominal definitions, yet traditionally not all definitions are deemed merely nominal. (In modern times, let me repeat, it is taken for granted that all definitions are merely nominal definitions, not so in the Aristotelian tradition.) Aristotle says of nominal definitions that they are an explanation of the meaning of a name and that they carry no real existential conviction.69 Both of these features insured that nominal definitions had a highly desirable property: they were true regardless of reality. For example, one could easily show, without any reference to reality, that all Greeks are mortal, that is, that “Mortal” is part of the definition of “Greek”. This is easily done by noting that the nominal definition of “Greek” includes the term “Human”, and that the nominal definition of “Human” includes the term “Mortal”. The inference is, of course, valid, but it is also sound: it is sound by virtue of dictionary alone, regardless of reality (indeed regardless of whether or not there is a reality, and certainly regardless of whether or not Greeks, humans and mortals exist). Traditionally nominal definitions had been viewed as uninteresting just because they are merely verbal, just because they do not carry real existential conviction; this is why the word “tautology” was used derogatorily until very recently. It meant a truth too trivial to be studied by anyone, especially the logician, whereas in modern times, of course, they are the stuff that logic is made of. Accordingly, syllogisms that are sound by virtue of dictionary alone were deemed barren, vacuous and uninteresting. Aristotelian logic was traditionally taken to contain and justify the taxonomy of reality, not merely the spelling out of this or that arbitrary dictionary. Speculative dictionaries did not interest Aristotle, and this is exactly why his theory of the syllogism is not an abstract theory of taxonomical roles. Rather he concentrated on the dictionary that, he believed, comprised the list of essences. Consequently, that dictionary carried an alleged existential import. Thus, tradition valued only what it named the real (or essential) definition. A real

11 Taxonomy of Reality by Syllogism

93

definition, however, is, really, not a definition at all, and this point is crucial. Rather it is an informative law-like proposition which was believed to be true. Aristotle says of real definition, however, that “it explains why a thing exists” and that it is “a quasi-demonstration of the essence”.70 It is supposed to explicate the nature of an entity by listing its essential (unique and irreplaceable) properties. Real, or essential, definitions, then, were distinguished from mere nominal definitions. They were also to sharply distinguished from mere accidental truths. Merely accidental truths purported to do the same service as real definitions, but purportedly failed to do so. They failed to capture the essence of an entity, to explain why a thing exists, to “quasi-demonstrate” it. Thus, from the modern point of view, real definitions are a crossbreed, a hybrid of nominal definitions and accidental truths, for they are supposed to be as logically necessary as nominal definitions, when contrasted with accidental truths, but they are also supposed to be as informative as any accidental proposition, when contrasted with nominal definitions. The classic theory of (seeking and establishing) definitions constituted an inseparable part of logic until very recently. It can be easily traced back to Plato’s early dialogues and was, possibly, first orderly formulated by Aristotle. It was preserved and stylized in the Aristotelian spirit during the middle ages and held sway into modern times. (It still appears, to take one example, almost exactly in its Aristotelian wording in Kant’s logic lectures; Kant explicitly names it as one of the three classical parts of logic). It expresses an aspiration to list all the criteria that guarantee that the search for real definitions will be successfully concluded. In effect it includes five main guidelines: (a) Avoid circularity (such as the definition: “All men are men”); (b) avoid infinite regression, i.e. the definition of terms by means of new and unfamiliar terms, which in turn have to be defined by means of new and unfamiliar terms, and so on; (c) divide the defined term into clear and distinctive parts, and especially avoid a situation where the defining terms are congruent, even in part, for example, “Man is a rational, thinking animal”; (d) avoid defining a term by means of more obscure terms; and (e) avoid expressions whose meaning is equivalent to “etc …”, and in general avoid definitions based on listing that require such expressions (for example: “An animal is a dog or a cat or a horse or a donkey, etc. …”; also, “man is an animal that is not a dog and not a cat and no a horse and not a donkey, etc.”). Let us reconsider the division of all propositions into real (essential) definitions and merely accidental propositions, in the light of this theory. Consider the following two propositions: “Man is a rational animal”, and “Man is a featherless biped”. On the assumption that both are true and informative (that is to say, that they are neither nominal definitions nor false or meaningless). How do they differ? The first is an essential definition and the second is merely an accidental truth, we are usually told. Any classic introduction to logic would tell you that. But what justifies this? How do we know that this is the case? Clearly, both propositions comply with the five traditional guidelines. Thus, the criteria are of no (real) help here. It is impossible to give account of the heaps of inadequate justifications of the above distinction. Aristotle, who is by far its best defender, states that men merely

94

11 Taxonomy of Reality by Syllogism

happen, by accident, to have no feathers but that of necessity they are rational. This is a classic and highly influential detour, it is the traditional union of modal logic and essentialism: the rules for discovering necessary facts are said to be the same as the rules for making and recognizing essential definitions. But what are these rules? Modal logic can formulate our modal intuitions neatly, but it cannot do the work for us, it cannot find justifications for essential definitions for us. Let me repeat this point for it is crucial: both the real definitions and the accidental truths are really informative taxonomies. Today we would simply treat them as empirical conjectures. Let us assume that these conjectures are true. Then out of the class of all such truths an elite class is (allegedly) chosen: the class of necessary, essential truths, the class of real definitions. This class is allegedly special: it is not only informative, as any conjecture is, but also necessary as any nominal definitions is. What is it that makes an informative truth necessary, essential and profound? What guaranty is there that we can render the distinction between essential and accidental truths valid? Here the traditional theory of definition does not reply but rather sends an urgent desperate call for the cavalry: induction, and Nous and the dialectical art of good carving. In short: it resorts to the ancient theory of terms already imbued with the conflation of rational thinking and science. This implicit and subtle theory allegedly guaranties that “featherless biped” is, somehow, illegitimate, that it does not depict a genuine, authentic, chapter of the grand matrix of living things. In short it guaranties that good carvers and bad carvers can always be told apart. From Aristotle to Kant, logicians assumed that there was a solution to the problem of recognizing real definitions from all accidental truths. They expected that slow and careful clarification of mental processes would explain how we come to posses the cosmic tablature, the blueprint of creation. In hindsight it is easy to see that the propositions deemed real definitions are empirical conjectures just as those deemed accidental truths. Only nominal truths are logically necessary. In modern terms, then, we can sum up the above concisely: Aristotle seems to have conflated two distinct senses by which propositions can be true: the logical sense, which renders them nominal definitions, or the real sense which renders them informative truths. Real definitions were presented as necessary as nominal definitions, when contrasted with accidental propositions, and as informative as accidental truths when contrasted with nominal definitions. They allegedly united the empirical and the logical. Thus, Aristotle succeeded to provide an answer to the (unanswerable) sophist’s challenge by an elusive conflation of epistemology and methodology. Once the assumption is made, however vaguely, that it is possible to restrict logic to terms that depict authentic slices of reality, essences, the conflation is already complete, for terms are embryonic definitions and even embryonic syllogisms. Ideal becomes reality, then: informative theoretical knowledge (of the compositions of our concepts) is presented as part and parcel of our theory of valid inferences; mere valid inferences become sound inferences, and mere sound inferences become proofs. Logic and empirical science become one. Let us now set out to see how these conflations are slowly recognized, and how they finally evaporate with the extensionalist revolution in logic.

Part IV

Essentialism Besieged

Chapter 12

Ad Hominem Logic: Logic between Aristotle and Boole

The aim of this study is to contrast Aristotelian essentialism with Boolean extensionalism and to discuss the aftermath. This precludes discussion of much interesting material and of many interesting periods in the history of logic. Megarian and Stoic logic are mentioned now only in passing and to a very limited end.71 Significant works of Aristotle’s followers (Greek, Roman, Arabic and scholastic) are ignored here. We are about to leap forward in time, then: from the time of Aristotle’s to that of Leibniz. The next two chapters should cushion the landing. Some formal limitations of Aristotle’s system of logic were well known at least since the Stoics worded new rules of inference (such as the ones known today as the modus ponens and tollens). This suffices as a rather straightforward refutation of the claim that the Aristotelian system is complete: it does not describe all valid reasoning. As it was well known, since antiquity, how did this knowledge influence the history of logic? I expand a bit on the formal limitations of Aristotle’s logic in this chapter. Between Aristotle’s time and Boole’s time, the major vehicle of essentialism – and thus of the conflation of epistemology and methodology – was the classic theory of judgment that I thus far ignored. It signifies here as it elicited a heated controversy on the nature of analytical judgment, a concept so central to the epistemologies and logical systems of the early-modern philosophers; its final rejection was a major achievement of modern logic and the demise of the framework within which these epistemologies thrived. I expand a bit on judgment in the next chapter. Let me begin then with the limitations, as first noted by the Megarians and the Stoics. (I call the logics of the Megarians and the Stoics MS logic, treating the two logics as one, since both have the same function here and this despite the fact that we do not know if the Megarians worded rules of inference as the Stoics did.) MS logic constitutes the earliest clear challenge to the claim that Aristotle’s system of logic is complete. We do not know how clear this was to the challengers themselves, because they had their own conflation of their logic with their ontology and epistemology. To bring Aristotle’s logic and MS logic on a par is difficult, since they have little in common, exactly because neither system is completely formal. When the scholastics tagged the Stoic innovations to Aristotle’s system of logic, claiming that they are complementary, they thereby added confusion, since the two conflations did not sit well together. N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

97

98

12 Ad Hominem Logic: Logic between Aristotle and Boole

MS logic heralds what we now know as the logic of propositions (in contradistinction to the logic of terms). But the few testimonies that we have about them are too partial and our own notion of proposition was not available to anyone that I am aware of until the twentieth century. Aristotle obviously made use of certain non-syllogistic rules of inference (including, of course, the modus ponens and tollens); we all do; but it is equally obvious that he nowhere formulated these rules explicitly the way he formulated the syllogism. MS logic did. By so doing it refutes the claims of Aristotelian logic for completeness. This was well recognized by Greek, Roman, Muslim and Scholastic philosophers, who all admitted that Aristotle’s logic is complete as a matter of course. How did they do that? I offer no answer to this fascinating question here. I will soon expand on the question, though. Before that let me offer another question, one that is more directly relevant to the present study. Being roughly a part of propositional logic, MS logic seems uninfluenced by essentialistic-style conflations that limited Aristotle’s logic. Modern readers, then, may tend to view it as extensional, especially when its formulae are read out of context, as they then naturally seem to modern readers to be context-free. In that reading (the purely extensional reading) the two systems – The Aristotelian and the MS – are complementary rather than in the conflict. In particular, the rules of inference formulated by the Stoics (as reported by Diogenes Laertius) seem wonderfully abstract and context free. The peculiarly Aristotelian conflations of epistemology and methodology, and the essentialistic limitation that they entailed within his logic, play no role here, or seem to. MS logic then looks distinctly modern – because then it is easy to read it extensionally unawares. More so, since the Megarians as well as the Stoics were directly opposed to Aristotle. Their logic is, to a great extant, an elaborated attack on his philosophy. They seem not merely modern, then, but consciously and deliberately so. For the purpose of the present study, then, the following questions signify. Did the Stoics present a formal, context-free, extensional logic? Did they seek such a logic? Did they demand that logical theory should be independent of empirical knowledge? In other words, is MS logic strict, pure, methodology-in-the-weak sense? Are the Stoics, then, precursors to the extensionalist revolution in logic? Needless to say these questions deserve a separate detailed study. As I am unable to do so here, let me answer them briefly nonetheless. The Stoics did conflate logic ontology epistemology and methodology. However, they did so in their own peculiar, distinctively anti-Aristotelian way. Hence, their logic was not contextfree. It was also not extensional. The Megarians and the Stoics criticized certain aspects of Aristotle’s essentialism and so, in this limited sense, their system was free of his conflations of logic and science. But this is not to say that they thereby produced a logical system free of comparable conflations. Admittedly, at certain points MS logic tends to be more formal than Aristotle’s; yet it certainly is not context-free or extensional, nor was it aimed to be. As an example, let me present the most influential conflation peculiar to MS logic: a conflation of logic and metaphysics. The philosophical background to it is the Megarian doctrine of necessity which was strongly influenced by Eleatic metaphysics. It is a form of monism that, as Aristotle himself tells us, is in discord with

12 Ad Hominem Logic: Logic between Aristotle and Boole

99

his dichotomy of actuality and potentiality. The Megarians seem to have deemed the notion of potentiality illusory on the ground that the world is already fully actualized: it is a motionless Eleatic unity. At this junction the Megarians were unable to avoid conflating logic with ontology and with epistemology. Let me show this in the following paragraphs and ask the reader’s patience. Eleatic monism entails the denial of time and change. It is more at ease with mystic silence than with a constructive, well developed, logical system. The Megarians and the Stoics seem in two minds about this. On the one hand, Eleatic monism justified their determinism, since it portrayed reality as fixed and immutable. On the other hand determinism looses its meaning as time and change are denied. Indeed, any attempt to soften or attenuate Eleatic monism, while retaining its core thesis – that all is one – seems to invite endless paradoxes, since it invites the holding of a double set of standards (as indeed we see already in Parmenides, who split all knowledge into truth and conjecture). One set of standards applies to reality as a permanent unity, the other applies to it as devisable and in flux. And any attempt to bring those two sets of standards into one system must fail. We do not know how the Megarians solved such puzzles. We do know that they took them seriously and that they sharpened their logic against them. It seems that they regarded past, present and future events as necessary and inevitable aspects of the one immutable, unchanging reality. In other words, in some sense they declared time illusory and in another sense real. And they tried to do so within an explicit detailed consistent logico-philosophical theory. Consequently they found themselves dwelling in the midst of logical puzzles and paradoxes, for the distinction between the two senses was not a logical one but rather an epistemological one. They both admitted and denied the existence of events past, present and future. Megarian determinism obviously gives rise to such predicaments, and even more so does Stoic determinism which is a softened version of the same (leaving a narrow range of freedom for free will). (The very notion of a softened determinism seems inconsistent from the strictly logical point of view.) One of the most important conflations in the history of logic seems to have its origin here: the conflation of two kinds of necessity, physical and logical. From a strictly monistic point of view there is no need to separate them: since reality is fully actualized, to speak about mere possibilities is to speak about the impossible, the illogical. Consistent Megarians, then, denying the notion of potentiality should have also denied the notion of possibility, and thus conflate logical and physical impossibility. In their view, all possible events that did not and will not occur are just impossible. Causes and their effects, then, seemingly contingent and seemingly separated by time, are not really contingent and not really separate by time: they are necessary timeless aspects of the immutable reality. And (seemingly) separate (seemingly) physical events become the logical, timeless, results of (seemingly earlier) events. The whole point of Stoic philosophy, however, was not to deny change, but rather to help humans accept it. This, perhaps, is why they were continuously at odds with their Megarian and Eleatic roots. Perhaps this is why they were so preoccupied with logic in the first place. They sought a rational-logical solution to an epistemological-ontological puzzle. Not wishing to exclude all tense-propositions

100

12 Ad Hominem Logic: Logic between Aristotle and Boole

as meaningless, as perhaps they should have done had they been consistent Eleatics, they found themselves dwelling in logical subtleties and controversies concerning the possibility of possibility. Epictetus famously reports and discusses the most famous summary of these controversies – the so called “Master argument”. I will leave all this and note that, interestingly, although resolving the master argument is inevitably a part of his philosophical heritage, Epictetus reports it with a distinctive (and typical) mixture of elegy and (somewhat un-Stoic) scorn.72 It is evident, then, that the Stoics, at least some of the later Stoics, found annoying their inability to resolve such controversies once and for all. This tension leads to many eccentricities of MS logic. Some of them noted that the truth value of propositions about the present is contingent upon the place and time of their utterance, whereas they denied that this is the case with propositions about the (unchangeable) past. Yet, instead of denying the existence of the present, as the Eleatics did, Stoics tried to tackle such difficulties logically, headlong. And since past events were supposed to determine future events by necessity, the Megarians and the Stoics could not agree about conditionals. No wonder that this was the heart and soul of their propositional logic: their controversy about conditionals resulted from their ambivalence towards the nature of necessity (physical/ metaphysical). Debates on conditionals immediately translated themselves into debates about the legitimate range of variables. This is well documented and stands in direct conflict with the view of MS logic as extensional and as context-free. To note but one brief example (discussed in some detail in chapter 3 of Kneale and Kneale 1962), Chrysippus, the greatest of the Stoic logicians and reputedly the greatest logician until Frege, declared that the claim “X is dead”, while X is alive, is self-contradictory. He also declared that the claim “X is dead”, while X is dead, has no meaning, since meaning is reference and the bygone have reference no longer. (The acute reader will note, that Chrysippus was thus obligated to the Eleatic, mystic, view that all meaningful sentences are necessarily true, as all false sentences are either meaningless or self-contradictory). Having explained the attraction of the conflation between logic and science in Aristotle I try to show that this attraction prevented efforts to take the opportunity and revolutionized logic, the way Boole did. Historically we know of two candidates for such opportunity, the Stoics and the Scholastics had it. Neither took advantage of it. As in the case of the Stoics so in the case of the Scholastics I will show that modern historians who ignore the historical conflation may read simply too much into old, pre-Boolean, texts. Let me return then to my first question regarding the effect that the very existence of an alternative logic must have had on the scholastics. It was minor. MS propositional logic was recognized as an important addendum to Aristotle’s logic. But it nowhere seems to elicit awareness of the dire need for an overall re-evaluation of it (and of the metaphysical background that cradled it), let alone revolutionize the field. The Scholastics were well aware of many theoretical holes and loose ends in Aristotle’s theory. They were aware of many examples to its limitations as a complete theory of inference. They usually take note of such examples, classify them and study them. Yet they rarely consider them qua criticisms and they never attempt to formulate a new logic that would

12 Ad Hominem Logic: Logic between Aristotle and Boole

101

include them. The most well known examples to such neglect come from (what we now call) the logic of relations. For modern logic of predicates, which includes Aristotelian logic, stripped of its conflations, rectified, and made a special case, is but a small part of modern logic of relations. Consider for example, the following inference: A is taller than B; B is taller than C; therefore A is taller than C. Clearly we want it to be valid. Yet this is perplexing from the Aristotelian point of view since the classic Aristotelian subject-predicate structure, which is applied to all sentences in all syllogisms, does not allow acknowledgment of relations as real: it acknowledges only the copula (a one-placed relation) as a binder of subjects and predicates. A consistent Aristotelian should view this inference as a syllogism and declare it invalid: viewed as categorical, the premises have four terms between them and no middle term (A, taller than B, B, and taller than C). Or consider an even more basic logical fact: certain relations are reversible, or symmetrical (for example, “as tall as”), whereas other relations are irreversible or asymmetrical (for example, “taller than”). Thus, from “A is as tall as B” viewed as categorical, we should deduce by simple eduction “B is as tall as A”, and similarly from “A is taller than B” we should be able to easily educe “B is not taller than A”. Not admitting relations as logical, however, Aristotle’s theory could not adequately satisfy such elementary desiderata. As the scholastics were well aware of such examples, one may wonder why they did not deem them as refutations of the principles of the view of Aristotle’s logic as comprehensive. The answer is sadly dogmatic: maintaining so seemed blasphemous as Aristotle’s logic was already part and parcel of their Christen dogma and education. Instead, they rendered them irregular. The notion of irregularity is very confusing. What is it? What is its logical status? How is it to be handled? Possibly it is very rare. Yet, as any student of logic can observe, and as they all knew only too well, there are at least as many examples to such “irregularities” as there are to (supposedly) regular cases. How common must a case be before it is deemed regular and explained as such? Answers to these questions were offered yet they were subordinated to the official view that Aristotle had said the last word on the subject of logic. The scholastics, then, collected and classified such refutations of Aristotle’s logic as curiosities, as ornament to his grand corpus, as problems to be solved, even in minor insignificant modification. A common strategy that survived well into Boole’s time was to tag such inferences “materially valid”. Material validity was then distinguished from formal validity: the mark of a true syllogism. This meant that the validity of the above inferences was deemed inferior simply because it did not fit Aristotle’s logic neatly: these inferences, it was thought, merely happened to be valid in our (accidental) world. Open admission of the need to reassess Aristotle’s logic, then, seemed too costly and so tacit, intuition-based modifications were supplemented in oversight of the fundamental questions. This situation is part and parcel of the general puzzle that scholasticism raises in modern scholarship. On the one hand the intellectual vividness of the scholastics is sweeping and alluring. Russell, for example, famously reports to have been stunned by it, despite (and, perhaps, because) his confessed initial expectation to the

102

12 Ad Hominem Logic: Logic between Aristotle and Boole

opposite (having been brought up in an anti-scholastic, classic tradition). On the other hand modern thinking shares little with scholasticism. Indeed, modern logicians are more open to Aristotle’s cannon than they are to the scholastic texts. For example, scholastic logic, though essentially Aristotelian is more deeply imbued with matters extra-logical than Aristotle’s. It is imbued above all with Christian theology, its mysteries of faith, and the subtle disputation theory that was often called to support it. It is a logic with a mission. Modern logicians regularly think of their subject matter as among the most abstract and context-free fields of study. Obviously they do not regularly think of Christian mysteries of faith in the same manner. And so scholastic logic seems to them idiosyncratic and unfeasible conflation of logic with certain (contingent) matters of faith. (Others find amusing the ease with which Muslim theology was transposed into Christen.)

Chapter 13

The Neglect of Judgment

A good way to leap forward in time is to attempt and provide a bird’s-eye-view of the rise and fall of the classical problems of analyticity. The classical problems were: (1) What judgment is analytic? (2) What judgments are analytic? Today we are used to formal logic so much that we easily concentrate on the title “analytic” and forget all about the meaningful, portentous, term “judgment”, and ignore its influence upon the history of logic. Logicians from Aristotle to Frege typically discuss judgments and not propositions or sentences or statements, and this preference is not merely semantic; it is epistemological: judgments were the standard vehicle of the standard conflation of epistemology with methodology; abandoning it – as Boole did, before Russell, and all those who had followed him ever since – was a huge step in the direction of modern logic. To avoid the conflation we should notice that under the title “analytic judgment” two items were traditionally presented: one concerning analytic propositions, and one concerning the knowledge of their contents. Similarly, under the title “judgment” two items were traditionally presented: one concerning propositions as such, and another concerning the conviction (judgment) that they are true. What do convictions have to do with logic? This point has already been explained in detail: Aristotle declared that certain concepts, which we tagged here “legitimate concepts” are the building blocks of our knowledge and that agreement and disagreement of such concepts creates new concepts. All such complex, legitimate, concepts yield judgments. (For example, the observation that red is a color, is a judgment describing the fact that Color is an attribution of Red.) The very recognition of the fact that a certain concept is complex, thus, yields the conviction that a certain judgment, aptly describing its content, is analytic. The classic problems of analyticity are an inseparable part of the story of the development of modern logic since the answers to them reflect clearly the disagreements about the province logic: it was ubiquitously agreed that the province of logic is the province of the analytic. However, different concepts of analyticity have allowed different types of knowledge to be regarded as logical. The problems of analyticity have a somewhat nebulous pre-history: they become a major focus of attention only in the early modern period. Aristotle did not pose them explicitly and some key terms in his corpus overlap only partially and uncomfortably with that of analyticity. If we try to impose the notion of analyticity upon his corpus, the result N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

103

104

13 The Neglect of Judgment

is more puzzling than helpful. He resorted to definitions, or first principles, and the problematic notion of their primacy over all other taxonomies. He said that only definitions can function as suppositions in a syllogism. The status of definitions is unclear and confusing even if we avoid reading later theories of judgment and analyticity into it. It is reasonable to view all definitions as analytic. But it is clear that primacy and analyticity are not identical. For example, Aristotle said, rightly of course, that primacy is non-hereditary (An. Post. I.2. 71b26–29 and see also 72a5–8). Thus, in Aristotle, strictly speaking, the conclusion of a syllogism, since it was derived through definitions by means of a middle term, cannot be a definition in itself. The primacy of definitions, as Aristotle understood it, whatever exactly it is, is, thus, certainly lost even after the first trite logical move. (Incidentally, this blocks all possibility of chains of reasoning in Aristotle: for the conclusion of a syllogism, since it involves a middle term is not a definition, therefore is not primary and, thus, cannot function as a premise in a new syllogism: to repeat, only definitions are allowed as premises in the proper syllogism). However, analyticity clearly is hereditary: a judgment derived from an analytic judgment, is analytic. Consider the sentence “All humans are rational animals”. If it is analytic, then so is “All non-rational animals are non-human”. As Aristotle has taught us, they are logically equivalent. Nevertheless, traditional logicians who followed him took it as a matter of course that the former is somehow primary whereas the latter is not. Alberto Coffa notes the above in the opening chapter of his magnum opus.73 He adds that Arnauld seems to have provided a pretty straightforward “Aristotelian” idea about analyticity: a judgment is analytic, Arnauld said, whenever it expresses a necessary, true assertion accurately describing an essence. This is true but more confusing than helpful, as we have seen, since there is no difference that we know of between analytic truths and necessary truths and since the reference to essences is impractical. And Arnauld still had the initial need to explain why analyticity is hereditary, and primacy is not hereditary. Spinoza crystallized an absurdity within the traditional conflation that comprised the traditional theory of judgment. He said: when all is said and done, all truths are necessary in the eyes of God (from the viewpoint of Eternity). The claim in itself is no novelty: we have already discussed its Stoic roots. It was embraced, however, by traditional Christian theology: from the viewpoint of eternity all truths are logical truths. But it is difficult to exaggerate the importance of Spinoza’s crystallization of this observation upon further developments of logic, and in particular upon Leibniz. For Leibniz quickly understood that Spinoza implied that the attempt to distinguish between the contingent and the necessary is merely a sign of human ignorance. Like other Spinoza readers he sensed an outrage: he rightly understood this to imply Spinoza’s determinism. Determinism, many people suggested, threatens the very foundations of Christian morality: moral choices must be contingent if there is free will. When all is said and done, it was argued, humans must be punished for sinful acts only if these acts are products of choices and not otherwise. And so Leibniz tried to reconcile free will with the idea that all truths are necessary from God’s view point.

13 The Neglect of Judgment

105

This central theological puzzle increasingly turns logical-theological, in Leibniz. Analytic judgments, he says, merely spell out what is already known about their subject. Thus, since every truth is known in advance from the view point of eternity, all truths must be analytic from the view point of eternity. But, he adds, analyticity is, ultimately, a property of judgments not of knowers. Thus, he concludes, all truths must be analytic judgments in principle, be they necessary or contingent truths, and regardless of whether they are known by humans or not. How, then, can we distinguish between necessary and contingent judgments, he asks. Without the formulation of such a criterion free will is an illusion, he adds. So his effort is to distinguish between two types of analyticity: necessary and contingent, and to offer a solid criterion for this distinction. Leibniz claims the we ought to resort to a proof apparatus that he had invented and labeled ‘Characterisitca Universalis’ and that we ought to add to it an axiom he titled ‘the principle of sufficient reason’ in order to prove the analyticity of contingent judgments. Sometimes he admitted that even this would not always suffice, for, he admitted, to prove a contingent analytic judgment might require us to perform an infinite number of deductive transformations, a task beyond the capacity of humans. But, largely, he concluded, the undertaking was possible, if only by God (that can grasp intuitively an infinite inference, he said). In the next chapter I discuss Leibniz’s extraordinary attempts to provide such an extraordinary proof apparatus and its relevance to the main theme of this book. In retrospect, it is easy to point out the source to the difficulties he faced: Aristotelian logic dealt almost exclusively with informative truths that in the Aristotelian tradition were presented as judgments. Some such judgments were deemed necessary (“All humans are Rational”), others were deemed accidental and even contingent (“All humans are featherless bipeds”). But in fact, it was impossible to distinguish the two groups by mere logical means because (logically) both were really sets of informative, empirical, conjectures. This fact portrays the important aspect of the Aristotelian tradition throughout the middle ages: it was gradually crystallizing into an explicit agenda: to prove that which cannot be proved. It, thus, invited the theory of judgment to smooth over the difficulty that (the very possibility of) counter-examples put forth: judgment (allegedly) rubs them out. In retrospect, it is interesting and crucial to observe that Hume already noticed this, if only vaguely. He observed that only non-informative truths were analytic. This, at least, is how Russell presents Hume’s critique of induction, and though this presentation is obviously an anachronistic idealization (I discuss it in chapter 15), it helps to reveal an existing undercurrent: the idea was there, even if it was dimly conceived and vaguely expressed. (Hume clearly spoke in the limited terms of his sensationalistic framework, not in the logical terms of Russell’s framework). Russell stresses that Hume’s observation is probably the greatest single shock modern philosophy has known. We should remember that even Russell was stunned to discover it. I mention this here because it highlights an interesting fact that was hitherto ignored: Russell makes Hume the first grandparent proper of modern logic proper. For, he says, Hume was the first to anticipate the modern scope and limits of this yet-to-emerge new field we now call modern logic; he was the first to observe that the traditional theory of judgment was a conflation of nominal definitions and

106

13 The Neglect of Judgment

informative propositions; the first to remove informative propositions from the realm of the provable. This was the inevitable effect of his critique of induction, of course. Thus, Hume was the first to sense, by a mixture of insight and foresight, that a theory of proof would produce tautologies, not the chimerical real definitions (informative, and yet logically necessary). Dim premonitions and distant images aside, modern logic was still not in existence. Before its birth, Kant would face Hume’s challenge. His aim was to replace induction by a new kind of logic, transcendental logic. He, thus, divided all real definitions into analytic judgments on the one hand and synthetic, informative ones on the other. And he claimed that analytic judgments comprised only a small part of the sum of all provable judgments. He formulated transcendental logic so as to prove all provable universal judgments that were not provable by classic logical means: all synthetic, informative, judgments. These then are the three classical answers to our question (what is analytic judgment?). Alberto Coffa suggested arranging them neatly, a-historically, as a gradual narrowing down of the domain of analyticity: by Leibniz all true judgments are analytic, both necessary and contingent; by Arnauld only necessary judgments are analytic, and not the contingent ones; and by Kant only those necessary judgments that are not proven by transcendental logic, are analytic. Coffa also observes that the very presupposition to the classic question was later on dismissed as untrue. Thus, the classical question (what is an analytic judgment?) was replaced by the modern one (What is an analytic proposition). This was the discovery that analyticity is a property of propositions, not of judgments. It was the end of the theory of judgment and more importantly the end to the conflation that cradled it. Logic, then, was declared to be the discussion of propositions (and the valid inferences that they may form) not of alleged judgments (and the allegedly sound inferences that they form). Who made this remarkable discovery? It seems that, strictly speaking, no one did before Russell. And even he admitted it somewhat reluctantly, though clearly. Frege’s odd refusal to acknowledge valid inferences from false propositions (and even from mere suppositions) attests to this amazing delay in the recognition of the exclusion of judgment from the domain of formal logic proper. (This insistence is so interesting because it is refusal to admit reductio ad absurdum unless it is formulated as a conditional with a false antecedent.74) Coffa has argued that the shift from judgment to proposition was first stimulated by Bolzano’s attempt to decompose the classical problem (what judgment is analytic?) into (1) what proposition is analytic? and (2) what makes the truth of a proposition analytic? Bolzano also distinguished these two questions from the psychological question, what is an act of assertion? Thus psychology (judgments and convictions), epistemology (synthetic propositions) and logic (analytic propositions, tautologies) were set apart, at least in principle. The answers that Bolzano gave to these questions will not be dealt here. Let me say though, that they are unsatisfactory from a strictly modern point of view, exactly because they still express the yearning for the conflation of epistemology and methodology (science and logic) that they helped to disperse. It is Bolzano’s stress on the questions that earns him the honorable place in the

13 The Neglect of Judgment

107

history of logic, a place he so rightly deserves; yet he had little to offer in terms of novel formal, or even quasi-formal, logical apparatus. Coffa’s narrative should be expanded on a little. He was unaware of Boole and his contribution to the rise of modern logic. Against the subtle epistemological complexities discussed here the simplicity of Boole’s algebra of logic is striking. It is a straightforward, no nonsense, theory of terms and propositions, with no place of honor for judgments and no place at all for real definitions. It explicitly suggests a formal division of all truths into logical truths (tautologies) on the one hand, and contingent ones (empirical, informative) on the other – with very little mention of knowledge. Boole was not a skeptic, of course, he easily made the aforementioned distinction, not worrying about securing knowledge, as he believed that he could secure it later on with the aid of his theory of probability, which he regarded a part of logic too. So he did not hesitate to construct a logical apparatus that made the distinction between informative, contingent propositions on the one hand, and tautologies on the other. The distinction became a formal one, then. Only tautologies are true in all universes of discourse and hence are analytic. Contingent propositions, he showed, were true merely in some universes of discourse and were hence synthetic. (Boole did believe that some synthetic truths were so highly probable as to be nearly certain, yet their degree of probability did not alter their fundamental logical nature) This excluded from the province of logic all ‘real definitions’ and all informative-yet-analytic judgments by fiat. Coffa should have noticed this: Boole is the first to have refuted in a clear, explicit and unequivocal manner the classical (presupposition to the) problem of analytic judgments, and to make this the foundation of a new, epistemology free, logic. As an addenda to all this we should include here the following historical note. Frege is the undisputed father of modern logic. He is so by virtue of his insistence that logicians qua logicians should abandon epistemology, at least for the time being, and study the analytic truths of arithmetic instead. It is, thus, extremely interesting that, following Kant, Frege could not give up the Aristotelian view of logic as a theory of sound inferences, not merely of valid ones. He could not endorse the classical view of judgment, but he could not completely abandon it either. He, thus, stuck to judgments nonetheless as indispensable for logic, yet in a new (weaker) sense of the term, which he defined and which was aimed at a new compromise between the nominal and the real senses in which judgments were allegedly true. Indeed, his life’s project was based on the assumption that despite Kant, arithmetical propositions were analytic, yet despite their analyticity (the fact that they all shared the same reference) they were nevertheless informative (in a new sense) since they did not share a sense. Thus, in Frege’s view, arithmetical judgments are necessary and yet informative, but in new and weaker sense of informative, which rests on his classical refutation of the traditional theory of meaning. The final transition from judgment to proposition and sentence, then, clearly adumbrated by Bolzano, was entrenched as part of logic proper in Boole and his followers (most prominent among whom was Russell), despite some Fregean last ditch efforts to hold onto judgments (in a weaker version). Boole’s logic is so amazing

108

13 The Neglect of Judgment

just because it starts logic afresh, completely ignoring judgments, as if they were not the center of attention of almost all who surrounded him and all who preceded and heralded him. As to Russell, clearly, his need to resolve his own paradox by declaring it meaningless was the watershed, as it made clear for the first time that there is a logical need to distinguish between a proposition and a judgment if only in order to recognize contradictions as (meaningful) propositions proper, as an integral part of language. Since no one can uphold a contradiction as a judgment and since contradictions are not meaningless, they had to be admitted as plainly false, yet meaningful, propositions. The reign of judgment was over. Modern logic was in full swing. Back to Leibniz.

Chapter 14

Leibniz as Aristotle and Boole Conflated

Leibniz signifies one of the most difficult and confusing phases within the history of logic. He is the greatest logician of the early modern period as well as one of the greatest thinkers ever, and yet his profoundest achievements as a logician were magnificent outlines of a grand program that was doomed to failure. He was a prophet whose insistence on the immediate actualizing of ideals had forced his followers to notice the impossibility of doing so. He tried to spell out and explicate Aristotelian essentialism by constructing a proof apparatus (a semi-formal language of sorts)75 that would enable the computation of everything known, and everything that could in principle be known, by trivial substitutions of synonyms that lead to identities of the form A = A. He maintained that such proof apparatus is within close reach that a few years of collaborative work of the “republic of letters” would suffice to construct it. It would bring an end to all controversies (including theological disputes), he promised, by reducing them to elementary arithmetic calculations. Let me explain very briefly why his attempts to achieve these ideals were so magnificent and why they were doomed to failure. Leibniz was a Modern who revered his scholastic predecessors. Like many of them he was well aware of many imperfections of the purportedly prefect Aristotelian logic: he studied them with a peculiar mix of critical independence and dogmatic awe. He rejected, for example, the validity of some of the inferences whose validity rests upon existential import, but he did not reject all of them and his reasons for his rejections seem to us today somewhat fanciful, perhaps even capricious. He held that an account of relations is indispensable for any complete logic, and considered the old Aristotelian system too stringent to deal with them properly. Yet he did not try to give relations a place of their own within a new logic, but rather simply to force their reduction to the traditional subject-copula-predicate form. He tried to determine the validity of inferences whose validity depends on relations by their reduction to inferences of the subject-copula-predicate form. This could not succeed, of course. But it did produce his well-known notion of Monad, whose properties are relations with infinitely many arguments, no less. Leibniz’s also tried to generalize all deductive thinking (syllogistic and other) and find its underlying principle, since as he noted clearly in a famous letter to the Countess Elizabeth: “arguments in proper form do not always bear the stamp of Barbara and Celarent”.76 And amazingly he declared that he had found it: it is the N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

109

110

14 Leibniz as Aristotle and Boole Conflated

principle of identity of the indiscernibles, expressed by any identity of the form A = A. Today, following Frege, we recognize that identity is a relation between names, not between entities; however, already Leibniz expressed this clearly by his observation that no two monads can be identical since if they are, they are one and the same thing, one and the same Monad (thus denying in effect the sameness of things the status of identity). Accordingly, his most famous formulation of the principle of identity declares that synonyms (names sharing a meaning) can be replaced salva varitate, that is, with no effect over the truth value of the sentence in which they occur. The most remarkable result of Leibniz’s study of (different versions of) the principle of identify was his realization that very basic, trivial transformations of synonyms are interesting and that they merit serious study. This realization constitutes his novel notion of proof as well as his systematic critique of Descartes’ careless reliance on the ‘clarity and distinctness’ of ideas that should substitute for proof. Almost everything merits proof, said Leibniz, if we are not too proud to attempt to do so. Since Leibniz’s time, and by his direct influence, we nowadays take proof to be a series of elementary replacements of synonyms (salva varitate) on the way to formal identity, identity of the form A = A. This crucial break from the Aristotelian tradition is rarely appreciated as such. Let us stress it then: for Aristotle and his followers A = A, like other tautologies, is somewhat uninteresting from a logical point of view, since it is said to be too trivial to be profound and seems empty of content. Even Descartes clear and distinct judgments were in essence a hidden conflation of the informative (the contingent) and the tautological. For Leibniz all true propositions, profound and trivial alike, are expressions of the formal identity A = A. Formally speaking, trivial and profound become one and the same, in a manner that does not allow their future conflation. This is why Leibniz plan was so magnificent and why its inevitable failure is so important. Allow me to stress this crucial breaking point between tradition and modernity by means of an example. Leibniz is the first mathematician to formulate a proof for 2 + 2 = 4. The proof goes roughly as follows: A. 1 + 1 = 2. (definition of the term 2) B. 1 + 1 + 1 + 1 = 4. (definition of the term 4) C. By A: 2 + 2 = 1 + 1 + 1 + 1. By B and C, 2 + 2 = 4 is reducible to 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 which is a formal identity. End of proof. This is mind-blowing. Descartes famously hesitates to decide whether or not it is possible at all that even an almighty God would be able deceive us regarding that very evident truth. And yet, Leibniz sets out to prove it. Step by step. The proof is not some accidental computational scribbling of an irksome pedant. Here is one of

14 Leibniz as Aristotle and Boole Conflated

111

the busiest and most creative intellectuals of any period, one of the most brilliant mathematicians in history (the co-founder of the infinitesimal calculus), and yet he is troubling himself and his reader with a trivial proof of what until that moment was considered a trivial truth, that is, a truth too trivial to be in need of proof. The notion of proof that underlies it is, perhaps, as significant to the history of logic as the invention of the infinitesimal calculus is to the history of mathematics. It heralds a radical shift of agenda: it heralds the recognition of the profound theoretical value of proofs which were until that moment taken to be mere trivial truths, too trivial to be reflected upon, let alone proven. That Leibniz was well aware of the shift of agenda that he was inaugurating, is undisputed.77 Yet he did not realize that his call for strict and explicit standards of proof was the beginning of the end for Aristotelian essentialism for it was the beginning of the separation of logic and empirical science. The Aristotelian system clearly presupposed much that was informal in principle, implicit and intuitive. Some presuppositions of the kind that Aristotle regarded permissible, indeed, all real definitions, were conjectures disguised as necessary truths. They were informative sentences presented as self-evident by means of subtle conflations. By demanding outmost clarity in proofs Leibniz imposed a demand to make all these would-be-evident- presuppositions explicit identity statements. This had placed an unbearable burden upon advocates of the old system. Chief among them was Leibniz himself, of course, and he ingeniously and relentlessly tried to upgrade the old system so as to satisfy his new standards of clarity in proof. Yet, how can “All men are mortal” be reduced to a formal identity? How can “All swans are white” or “All whales are Fish” be reduced to “A = A”? And if they are reduced, what are we to do with the discovery of black swans, or the discovery that whales are mammals? (Discovery, not the arbitrary taxonomical decision as instrumentalists would have it, since clearly Aristotle’s contention that Whales are fish was no mere nominal definition.) Let me elaborate this point a little. I do not know if Leibniz explicated or expanded the Aristotelian notion of definition. (This clearly depends upon reading of the Aristotelian text which is not sufficiently detailed on this point.) But it is clear that he had the Aristotelian notion of definition in mind when he argued that every subject in every sentence is replaceable by a definite list of predicates, its defining predicates. A subject, he said, is shorthand for the list of all the predicates that are true of it. Hence, in a sense all of them are its defining predicates: the full list of all the predicates that are true of a subject, put in an orderly manner, is its definition. Leibniz spelled out the consequence of this far-reaching move: all contingent truths must be analytic. And so, he added, they are reducible, at least in principle, to identity. Finally, Leibniz argued, a proof apparatus should be constructed that would enable such incredible reduction. The guiding idea was to reduce all proofs into inspections of a list: in order to prove a truth, any truth, we would inspect whether or not a certain item (a predicate) is found in a certain list (the definiens of the subject). Such inspection is easy to perform even if the list is very long, as long as it is given. It can be performed mechanically, that is to say, it can be performed by a

112

14 Leibniz as Aristotle and Boole Conflated

machine, if only the list and the item to look for are provided to it: it requires no imagination and no intuitions and, hence, it allows very little place for error. For, Leibniz realized, the more we rely on our intuitions (our allegedly clear and distinct ideas and allegedly evident judgments) the more prone we are to commit error. In Leibniz’s ideal proof apparatus, proofs manifest themselves. This is done in the following manner: every possible subject of every possible sentence is represented by a complex symbol made of elementary symbols. Elementary symbols stand for the elementary predicates that define it. The compound symbol, then, manifests its defining predicates by its very form. Thus, the subject of any possible sentence in that language and the complete list of predicates that define it become not merely equivalent but one and the same. For example, if we are to prove that all men are rational animals, we can make “Man” represent this truth by its form: we understand “M” to be “Animal” and “an” to be “Rational”. Then “Man is a rational animal” becomes “Man = Man”. And Man is Rational becomes Man = an. All truths regarding that subject become formal identities of the form A = A, or AB = A. (The copula was, allegedly, the only relation; so it was allowed to be both class identity and class inclusion although not explicitly so; and so a traditional conflation of these two forms persisted until Hamilton’s and De Morgan’s quantification of the predicate had put and end to it). Such proofs are intuitive in the original sense of that word, that is, they rely on explicit perception of forms alone (not in Descartes’ subtler sense which relies on God’s grace.) Leibniz outlined numerous fragments of such a proof apparatus.78 They quickly became his grand obsession. They gave a wonderful glimpse to what formal logic, one day, would be like. But they had all failed. And for a basic reason: they were designed to explain how informative knowledge is provable, and thus their full construction depends on the completion of science, as only science can give us the full description of a thing so as to be able to reduce its name to the list of names of its properties. A famous early series of such attempts was modeled upon arithmetic and the fact that any natural number is uniquely decomposable into its prime factors. Leibniz likened concepts to natural numbers, and elementary concepts, which make up complex concepts, to their prime factors. In one such outline Leibniz maintained, that if “animal” is represented by the number 2 and if “rational” is represented by the number 3, then “man” must be 2 times 3, that is, 6. In order to prove that all men are rational, he continued, we simply point out that 6 is uniquely decomposable into its prime factors 2 and 3. This wonderfully simple example easily collapses into chaos, of course, if we try to expand it without the ideal, complete list of all truths. For note this: 3 is a prime factor of infinitely many numbers, yet only men are supposedly rational. How are we to prevent such numbers as 15 = (3 times 5), or 21 (3 times 7) in our apparatus? 5, let us note, is a prime, and so too must represent some elementary concept. Let us assume that it stands for Winged. Then 30 = (2 times 3 times 5) now means “Rational winged animal” which, we suppose not to exist. How are we to avoid such mistakes unless science is given? The greatest merit of Leibniz’s ingenious proof apparatus, then, is that it makes explicit the traditional Aristotelian requirement that we should explicitly know in

14 Leibniz as Aristotle and Boole Conflated

113

advance all truths, all natural, proper taxonomies of all things and all ideas, before we set out to formulate and outline even a single proof. This is like demanding of babies to speak before they prattle or mumble or stutter (as Hegel said of Kant). Aristotle had demanded this somewhat implicitly by means of his subtle theory of induction/intuition. Leibniz recognized that this is an explicit construction requirement of his proof apparatus: he viewed logic as the tool for expressing the achievements of science and their presentation as final and complete. This was too much to ask, of course: we really do know very little. Assuming that science is complete, an outline of its skeleton would be advantageous indeed, but Leibniz wanted more. Let me stress this last point from a different angle. Leibniz was in two minds about whether his intended proof apparatus should be a mere tool for spelling out what we already know or a tool which opens the door to new fascinating discoveries, and offers resolutions of controversies. He was thus genuinely in two minds about whether or not his proof apparatus is an expression of God’s knowledge or a ladder. Clearly if we know everything in advance, as God assuredly does, no discoveries are possible and no genuine controversies. But Leibniz insisted that his proof apparatus allows genuine controversies to be resolved, and discoveries to be made. In some sense truths have to be unknown or at least only implicitly known, and then, and only then, be discovered and established as the formal identities that they allegedly are. (This is how Plato has reconciled his views on apriori knowledge and discovery.) Yet, in order to formulate explicitly Leibniz’s proof apparatus one has to be God, for these same truths have to be given in advance as explicit, not merely as implicit, if we are to construct the proof apparatus that shows that they are explicit. If these discoveries are given in advance as explicit knowledge, how could they be discoveries? How could genuine controversies exist? And what use do we have of a proof apparatus? If genuine controversies exist they immediately express themselves as disagreement about formalization, about how to construct the proof apparatus; they would not be resolvable by it, for they are all pre-logical. How are we to decide pre-logical controversies then? (Students of modern logic don’t have this problem, but Leibniz clearly did). What, for example, is the procedure to decide between those who deny that all men are rational and those who avow it? We can try and point out that the symbol “an” in “Man” stands for “Rational”. But then those who deny that man is rational would deny this too. And, more fundamentally, how are we to know that Rational is the predicate of Man and not vice versa? This difficulty throws us back (as it had thrown Leibniz and Aristotle and Plato before him) to the metaphysical depths of the unsolvable problem of finding genuinely elementary predicates, and in general, a genuine hierarchy of concepts, terms and of things. For, the very idea of the completion of the project of science was assumed to include and imply such a hierarchy. We must decide what particulars are and what abstracts are, so as to find the elements and the compounds of our alleged proof apparatus. (It is, perhaps, not surprising, then, to find that Leibniz regarded as particulars the monads which are the most abstract and complex entities in the history of philosophy.) How are we to agree upon such highly metaphysical matters before a perfect proof apparatus

114

14 Leibniz as Aristotle and Boole Conflated

is provided to us? And if it is not provided to us in advance by what logic are we to discover it and formulate it? Thus, Leibniz rationalized and simplified the traditional conflation of the theory of valid inferences and the logic of discovery. He did it with such rigorous strictness and unprecedented clarity that it had virtually forced his followers (Kant specifically) to admit that the conflation is not promising, that when it is rationalized and simplified it clearly yields an impossible project. A wonderful example of a proof by Leibniz will close this brief chapter. He had reported to have proved the impossibility of the concept of the motion having the greatest speed. He says: “The motion having the greatest speed is impossible in any body whatsoever, because, for example, if we assumed it in a circle, then another circle concentric to the former circle, surrounding it and firmly attached to it, would move with a speed still greater than the former, which, consequently, would not be of the greatest degree, in contradiction to what we had assumed.” He concludes: such motion “has no idea, since it is impossible.”79 Of course today we flatly deny the conclusion of this simple and ingenious proof. But the proof is incontrovertible. What we deny today is not its logic but rather the conflation of logic, Euclidean geometry and physics that its originator makes, unawares. We deny the conflation of logical necessity and physical necessity that it makes unawares. Leibniz’s firm belief that empirical truths are analytic has led him to the view that logic must allow the discovery and establishment of such truths. This is why his grand failure cleared the road for Kant’s transcendental philosophy: it was the last grand attempt to sustain the conflation of science and traditional logic.

Chapter 15

Why Transcendental Logic is no Logic at All

Consider Kant’s famous acknowledgment in the opening remarks to his Prolegomena: Hume’s critique of induction, he famously notes, had awakened him of his dogmatic slumber. Is it not a little puzzling? Hume explicitly criticized Locke’s philosophy of science: he portrayed the limits of Locke’s empiricism. Kant (even in his deepest of dogmatic slumbers) was no advocate of Locke’s philosophy of science. He was certainly no empiricist. Rather, to the extant that such matters can be determined at all, he was a Leibnizian (or, perhaps more accurately, a Wolffian). Leibniz, let me stress, was the greatest critic of Locke until Hume came around. So what exactly had impressed Kant in Hume’s criticism of a philosophy he did not advocate, to the point of shattering his old (completely opposite) philosophy? Importantly, Kant deserves here credit which he does not take, and one for which he is hardly ever credited: his distinctive understanding of Hume’s critique is one of his great contributions to philosophy. He understood Hume to have implied the bankruptcy of Leibniz’s project, the project of rationalizing the conflation of science and traditional logic (by means of the explicit founding of the former on the latter). Hume, let me stress, simply did not speak in those terms. Our commonsense history of modern philosophy is initially Hegelian. Despite its many merits it is still suffused with Hegel’s misleading intellectual determinism: it renders all intellectual developments inevitable. In following Hegel’s standard portrayal of the development of modern philosophy we take it for granted that Kant was, somehow, destined to unite empiricism and intellectualism. Kant’s great originality is, thus, muted by the sound of the march of inevitable (dialectical) progress. But it is Russell’s famous re-formulation of Hume’s critique of induction that is the main source for the prevailing attitude to Kant’s modest confession. Russell was simply not interested in historical accuracy when he named his “problem” of induction “Hume’s problem”. His formulation, however, is much more elegant and general than Hume’s original claim, and so it hardly does historical justice to Hume or to Kant – the one who first understood it in the broadest context of his day. Hume noted that we have no impression of necessity, when we observe an event which we deem a cause and subsequently an event which we deem its effect. According to Russell, however, Hume had demonstrated that induction is not a deductive inference and that only deductive inference, which provides no informative content, can be necessary. Russell, then, sets Hume’s claim in a Leibnizian setting – logic oriented N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

115

116

15 Why Transcendental Logic is no Logic at All

one– as if this is trivial, thereby obfuscating the fact that it was Kant’s achievement to do so clearly for the first time. Reading Hume’s critique of induction as the declaration of bankruptcy of Leibniz’s project is, however, the starting point of Kant’s original philosophy.80 He understood Hume to have shown that some of the most meaningful parts of science comprise not analytic judgments, but rather informative, synthetic ones. He agreed with Leibniz that synthetic judgments cannot be proven by classical logic alone. He thus concluded that a new logic is needed, so as to prove them. And he bravely set out to formulate it. He titled it “Transcendental logic” and argued that it stands in direct contrast to classical logic, since it can secure even informative theoretical knowledge, synthetic judgments, just as Leibniz had envisaged the securing of science by logic. What, then, is this new transcendental logic? What are its axioms? What are its rules of derivation? Where does its novelty lie? How does it stand with respect to classical logic? (By what means are the two logics to be compared?) Transcendental logic, said Kant, is based on transcendental proof. What then is a transcendental proof and what transformation rules render it valid? There is an obvious sense in which these questions are somewhat unjust: formal logic was not yet born and the explicit need for an explicit formulation of logic, any logic, as a system of straightforward derivation rules and axioms was only dimly realized. Leibniz’s attempt to describe logic as formal was an early exception, and since he left no complete cannon describing complete standards of proof the need for such standards was, at best, sensed but not fully realized. The text-book of Kant’s logic lectures, for example (Georg Friedrich Meier’s Excerpt from the Doctrine of Reason) makes no interesting innovation in this respect. Any look at the fascinating notes prepared by Kant’s students of his logic lecture-course attest to the fact that he sensed the need for such standards of proof but did not go as far as to provide them. Thus, Kant admirably attempts to avoid conflating the definitions of concepts, judgments and inferences, but in practice he inevitably does so. And then we are not surprised, nor should we be. Psychology and logic were not fully distinct doctrines at the time. Alberto Coffa has forcefully argued that this had an immense effect on Kant’s logic in general and on his notion of proof in particular: Kant’s notion of analyticity, and hence of proof, is a conflation of two incompatible notions: analyticity in the formal sense (roughly equivalent to what later on will be identified as proof) and analyticity in the psychological sense (which is the act of clarifying a vague concept, and making it into a distinct judgment). As we have seen in Leibniz this is the conflation of a discovery mechanism with a proof mechanism. As we have seen in Aristotle, this is the essence of essentialism: it is the standard way to smuggle epistemology, disguised as psychology, into early notions of proof and of logic. Despite great difficulties, and in deference to the historical limitations that confine even the greatest minds, let me attempt to give a summery of Kant’s transcendental logic. As I have noted (chapter 10), Aristotle’s nous and Kant’s transcendental proof play similar roles: indeed the latter is a refinement and an elaboration of the former. The two roughly share the following structure: the existence of perceptual

15 Why Transcendental Logic is no Logic at All

117

experience is taken as evident; then some (supposedly) universal knowledge is declared its inevitable presuppositions. This is all there is to it. The (supposedly) universal knowledge is declared a necessary prerequisite to the very possibility of perceptual experience. We then conclude that it must be true. Aristotle used a rough version of this line of argument as a justification of theories that he deemed scientific. Kant refined it, developed it, and used it for the very same purpose. (Obviously, they were not referring to the same theories.) Aristotle also used this method to justify induction-as-synthesis. Kant did so too. This is most interesting: Hume, to repeat, had argued that we observe no impression of necessity when we have an impression of (what we deem) a cause and subsequently an impression of (what we deem) its effect. In general, he added, assuming causality and regularity in nature is the result of habit. Kant argued that the very possibility of any perceptual experience presupposes that events would naturally fall into causal chains. And he said the same of expectation for regularity in nature: the expectation that future causes would elicit the same effects, as similar causes in the past is a priori, he said. He thus made Hume’s ‘habit’ into an a priori requirement, into an inevitable presupposition to the very possibility of any perceptual experience. His a priori principle of induction, then, justifies expectations of causality in nature in the same manner that Aristotle’s induction-as-analysis justifies expectation that induction-as-synthesis would not fail us. In general, wherever Hume had found a limit to the provability of empirical knowledge, Kant found a transcendental proof of it as an a priori presupposition to the very possibility of such knowledge. Ingenious as all this is, from the strictly methodological point of view it is confusing. What is the logic of transcendental logic? Is it any different from classical logic? By which rules? The classic standard answer to our question is this: transcendental proofs are deductive. They are variations of modus tollens (or of modus ponens, as it all depends on how you formalize them, which no one did at the time anyway). They have roughly the following structure: q is impossible unless p; yet q is observed (and hence certainly possible); therefore p. For example, to simplify one of Kant’s illustrious transcendental proofs, perceptual experience is impossible unless it obeys the axioms of Euclidian geometry; perceptual experience is undeniable; therefore, the axioms of Euclidian geometry are (a priori) true. All this is confusing from the strictly logical point of view since, to put it bluntly, modus tollens is just about the oldest rule of logic. Indeed, since it is the model for Socratic dialectics, it is older even than logic itself. Why, then, was the title “transcendental” appended to such arguments, no matter how ingenious they were? What exactly is extra-logical about them? A standard answer is this: from the strictly logical point of view transcendental logic is classical logic plus the empirical supposition-turned-axiom: “we have perceptual experiences”. This is what renders it transcendental, then. It is a set of classic inferences that spell out logical presuppositions to the (perhaps, undeniable) fact that we have perceptual experiences. This, to repeat, is the standard answer to the question regarding the logic of transcendental logic: it is classical logic plus the claim that we have experiences. But it was flatly refuted, however, already in Kant’s time. Showing that it was refuted – as we will do presently – will leave us with a confusing conclusion: that

118

15 Why Transcendental Logic is no Logic at All

transcendental logic, no matter how ingenious it is, is no logic at all, not even classical logic with a newly added axiom, or a set of valid inferences with an interesting, empirical supposition. Let me briefly explain this point. Although many readers are familiar with it, they may not have considered it from a strictly logical point of view. Most of Kant’s transcendental proofs were later shown to be not merely unsound inferences, as some prefer to think of them, but downright invalid. This is crucial from any strictly logical point of view: Kant’s allegedly sound transcendental inferences were invalidated. For a complete transcendental proof needs more then the claim that our ability to have perceptual experiences requires this or that presupposition; it needs this claim as a conclusion and more: it needs showing that the only thing that renders perceptual experience possible is this presupposition. Kant did nothing of this sort. He simply insisted that no experience is possible without his presuppositions. (And to excuse him by reference to the backwardness of logic, then, is to say that nowadays we can do better. We cannot.) When this was realized, Kant’s transcendental proofs were not neglected of course. They were taken as challenges rather than as the end-to-epistemology that they purported to be. Take, for example, the proof mentioned above that the axioms of Euclidian geometry are presuppositions necessary for the very possibility of perceptual experience. This was taken as the following challenge: describe a nonEuclidian geometry (as Gauss did in direct reaction to Kant’s alleged proof) and then give it empirical meaning, so that it can become possible perceptual experience and so a possible part of empirical science (as Einstein did). Achieving this was a huge development, of course, and one which Kant, overwhelmed by the unprecedented success of Newtonian science, could hardly anticipate. The crucial logical point, however, is not whether repudiation of Kant’s transcendental proofs could, or could not, have been anticipated by anyone at the time. Rather, the crucial logical point is that once Kant’s transcendental deductions were repudiated they were thereby shown to be no deductions at all; they were shown to be merely claims for deductions, that is, simply invalid inferences. Kant, then, has left us with a well planned set of ingenious conjectures which cannot, in principle, comprise a logic: they comprise invalid inferences. They could not even manifest novel uses of an old logic. Already in Kant’s time it was noted that Kant had also made an odd use of his newly added transcendental axiom (that perceptual experience is manifest). For, he assumed that experience imposes the true science upon us. And so he conflated, as a matter of course, the fact that we have perceptual experience, with the far-reaching supposition that the successful completion of the project of science is, somehow, guaranteed by our perceptual experience. Incidentally, this conflation too is basic to Aristotle’s induction-as-analysis just as it is basic to Kant’s transcendental logic. The result was that explaining the possibility of experience was often indistinguishable in their respective systems from explaining the possibility of perfect empirical science. The brilliant Shlomo Maimon was the first to observe and criticize this tendency in Kant. He did not doubt that Kant’s transcendental proofs are valid. Rather, Maimon observed, Kant assumes that science is complete and final, and

15 Why Transcendental Logic is no Logic at All

119

then sets out to show that there are presuppositions to this fantastic assumption. Of course he rejected it. What Kant really needed for his transcendental proofs, explained Maimon, is first and foremost a proof that science is, indeed, complete and final. Otherwise all we have is an allegedly sound inference, with an empirical (and, he added, objectionable) supposition. Kant’s reaction to Maimon’s acute criticism is worth telling here for it communicates the crucial symptom of modern philosophy to which this chapter is dedicated. His first reaction was gallant appraisal: he hailed Maimon as his greatest critic.81 However, he offered little in reply to Maimon’s criticism. And soon he forgot about the whole affair. Maimon was quick to observe this, and persistently asked for an answer. Tension between the two mounted. Kant ignored Maimon’s repeated, respectful request for additions or clarifications. Than Kant denied that he had ever read the criticism in question.82 One would dismiss this as natural absentmindedness of an aged overstrained Professor were it not for the fact that only eight months earlier he had discussed the criticism in considerable detail, showering Maimon with superlatives as he never did before or since. His final reaction to Maimon, however, is flabbergasting: He writes “…as regards the “improvement” of the critical philosophy by Maimon (Jews always like to do that sort of thing, to gain an air of importance for themselves at someone else’s expense), I have never really understood what he is after and must leave the reproof to others”.83 Since Kant is the greatest spokesperson for the Enlightenment movement, and one of the most liberal, honest and sensitive philosophers in history, most of his admirers do not know what to make of this ugly expression of contempt; all the more since it was expressed towards the person whom he also hailed as his greatest critic. Many feel the need to dismiss it. Even the standard Cambridge edition of Kant’s works (which, by the way, is one of the finest standard editions ever) suggests that “…the remark, ‘Isn’t it just like a Jew to try to make a reputation for himself at someone’s else’s expense’ should be overlooked, on the grounds that Kant was always hypersensitive to criticism and, at that point of his life, concerned about the apostasy of his followers.”84 Now there is little doubt that Kant’s comment on Maimon is indicative of pressure, yet anti-Semitic expressions are intolerable regardless of any pressure that might elicit them, and attempting to dismiss them as the result of pressure is flimsy excuse. A recommendation to overlook a racist expression as a grand old man’s slip of the tongue may be reasonable; the standard edition’s recommendation to overlook that slip on the ground that he was hypersensitive to criticism is not. The pressure that Kant suffered is real and is at the center of our interest. In hindsight, it was the result of the fact that, he boldly undertook an incredible task without even realizing the exact conditions for its success: he set out to prove the foundations of natural science knowing too little about proof. This, incidentally, is true of Kant as it is true of Leibniz or of Descartes. It is, in hindsight, the major symptom of early modern philosophy. The more modern philosophers learned about proof the more they realized that their task was less simple than their predecessors had deemed it. They thus, paradoxically, augmented their own sense of unease. They came to rescue epistemology from the shortcomings of their

120

15 Why Transcendental Logic is no Logic at All

predecessors, and then, having clarified that task better than their predecessors, left it in a graver state then they have found it: its shortcomings were now only more apparent, more apparent than they had been before they set out to rectify them. Descartes, having done that, has placed the burden of securing proofs on the goodness of God Almighty. Already in Descartes’ time, this was taken by many to be an admission of defeat. Leibniz, who tried to improve on Descartes’ program in this point, contributed more than any other modern philosopher to clarification of proof conditions. But, paradoxically again, this only made his task more difficult, since he attempted to prove informative theories. He reacted to the pressure by continually postponing the completion of the construction of his grand apparatus for generating proofs of the theories of natural science. Kant, who realized that Leibniz project cannot be completed successfully, undertook the incredible task of formulating an alternative proof apparatus. He had been so preoccupied with this project that he virtually lost interest in his critics, dismissing them as pests. This is, perhaps, understandable, and if this is all that the editors of his works meant to say, then so be it. Kant’s philosophy is the pinnacle and end of early modern philosophy. He had summed up and completed his own system so well, and it summed up and completed all that preceded it so well, that little was left for ardent original followers to develop. Unique among them was William Whewell who was the first to realize that if experience imposes the correct theories upon us, as Kant argued, then the real epistemological challenge is not to explain how science is possible, since this becomes trivial (as it becomes inevitable), but rather how come it takes so much time and effort to attain it. Maimon’s criticism of Kant was misused as license to distort Kant’s teaching. Kant’s major concepts were then gradually turned into metaphors: Hegel’s logic is as unrelated to logic proper as one can possibly conceive. Indeed it amplifies confusion by deliberately fusing the historical and the logical. Schopenhauer deemed even Kant’s concept of the real, of the thing-initself, a kind of a metaphor for our innermost being, for the Will. Others, with less metaphorical aspirations, tried to revive the Leibnizian project in some version. Chief among them was the ingenious Bolzano, who as Alberto Coffa has shown, is the predecessor of both Frege and Wittgenstein. What follows, I hope will not be taken as an attempt to belittle Bolzano. Rather, let me now complement Coffa’s story by explaining the important contribution of George Boole, the person whom Russell regarded his only true predecessor, and whom Coffa was little aware of. It is a curious fact that Boole’s most distinctive contribution to logic – undaunted extensionalism – is yet to receive its proper place in the history of logic. Let me attempt and do so now.

Part V

The Fall of Essentialism

Chapter 16

Extensionalism as Exorcism

This discussion is devoted to explaining the revolutionary character of extensionalism. We are thus slowly zeroing in on Boole: you should soon feel the domino effect that he impelled, the rapid unstoppable toppling of one traditional dogma after another. Our aim is to emphasize the radical alternative to Aristotelian essentialism that Boole’s extensionalism comprises; that it thus exorcises Aristotle’s essences, thereby setting logic free of its ancient, traditional constraints. Let me first offer in this chapter a description of extensionalism in the abstract. Then, in the following chapters, I will enrich that description with a discussion of a few decisive moments in that historical development. Revolutionary ideas rarely mature at the moment of their inception. Extensionalism is no exception, though it did mature at an astonishingly high rate. Since things will happen fast once they are set into motion it helps to take a good look at the fully fledged idea before surveying its chronicles so as to find its earliest sources and announce its precise moment of inception. Extensionalism, like essentialism, is not a theory. It is an attitude toward logic. It is a logical point-of-view with radical, swiping, and distinctive implications. The fundamental concept at issue is that of a class. Whether classes come with their essences or not was, under various terminologies and theoretical disguises, the hottest traditional dispute about traditional logic. Classes in modern logic clearly and indisputably replace essences. A class is then any arbitrary collection of particular objects (whatever an object may be). Extensionalism stresses this and this is its gist: a class is nothing but any arbitrary collection of particular objects. I contend that Boole was the first to have embraced completely this seemingly paradoxical and somewhat simplistic view of logic and that he thereby inaugurated a new era in the history of logic. Since a class is nothing but any arbitrary collection of objects, as a matter of course, within logic all classes are indisputably equally legitimate. This is all that there is to it. Extensionalism asserts that within logic the very distinction between legitimate/correct and non-legitimate/incorrect terms is obsolete and useless. And since this distinction silently sustained the traditional conflation of logic and science, in its various forms, this conflation too is gone. Extensionalism stands in contrast of sorts to traditional Aristotelian essentialism, the dominant meta-logical view which dominated the field ever since its inception. Extensionalism is a contrast of sorts in the sense that it need not be anti-essentialist; N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

123

124

16 Extensionalism as Exorcism

suffice it that it postpones discussion of essences to a later stage of any inquiry. Essentialism portrays logic as the (true) taxonomy of reality. It thus obviously conflates taxonomy and taxonomy rules, and by the sheer insistence that only essences participate in syllogisms. By rendering this insistence logical it achieves, by a vicious circle, self-justification. Paradoxically, it achieves so much by the seemingly innocent act of limiting logic to terms that name essences. These terms are said to designate directly essences and indirectly particular, tangible, objects. It thus presupposes that an entire given ontology of entities exists, allegedly organized by “correct” groupings into “legitimate” or “natural” kinds or essences. By that stage the conflation of logic and science is already complete. Traditional logic, then, contains the semi-explicit assumption that all traditional problems of epistemology (and indeed of philosophy) have been solved by its very saturation with essentialism. By contrast, extensionalism has no ontological doctrine. It does not deny the existence of essences. It has, however, an explicit agenda with an astounding effect: to exorcise the essences out of logic proper, perhaps in order to reintroduce them later in a more appropriate way (as Boole had certainly hoped). Thus, extensionalism tears logic and ontology apart. And it also tears logic and epistemology apart, and in the very same manner. It takes essentialism to be an undesirable conflation of the extra-logical and the logical, of science and its method. All criteria – at least temporarily – lose their import for the determination of the legitimacy of terms that name classes: all terms are declared equally legitimate within logic and all classes are declared equally legitimate there. In other words, extensionalism suggests that from the strictly logical point of view, any criterion for assembling any collection of objects is as good as any other. The question of the legitimacy of classes or union of classes is, within logic, rejected as extra-logical. Hence, questions of legitimacy –epistemological, metaphysical, etc. – become irrelevant to the construction of any logical system. In modern logic three classes, or three fundamentally or essentially different class types, are new and of the outmost importance. They find their immediate way into logic as a result of Boole’s radical new and permissive attitude: the universe of discourse (or the universal class), the complement class of any class, and the complement class of the universal class, the empty class (Boole called it “The class Nothing”). The extensionalist revolution in logic contains little more than these three classes, or class types. Historically, these three classes, or class types, were deemed problematic and undesirable. Consequently they were exiled from the realm of logic. They designated no essence, and hence had no place there. Tradition still allowed a universe of discourse: the entire cosmos, the sum of all things in the world. Since it was a sum, however, it could not have an essence of its own; it was too vast, too abstract an aggregate to be pinned down by a single term, or a single definition. It was supposedly too abstract to be the subject of predication, too abstract to be a substance and consequently too abstract to participate in syllogisms. Similar considerations, we should recall, caused the complement class to be rejected as unacceptable. For, consider the term Not-Goat under a traditional logical apparatus: it includes every single entity that is not a goat. Since the only admissible universe of discourse is the whole cosmos, Not-Goat indistinguishably includes

16 Extensionalism as Exorcism

125

Humans, Trees, Minerals, Platonic ideas, Cities, Laws, Feelings, even Dreams and Reflections. Not a very useful term, from the traditional point of view, of course. For, the traditional point of view demanded that each and every term should designate an essence. And most important: traditional logicians refused to admit the empty class as legitimate. Ever since Parmenides, they cast a taboo on terms that had no reference. Indeed, traditionally the notion of a term with no reference was so problematic that traditional logicians regularly conflated (under this tag), meaningless terms, contradictions (“in terms”, as traditional terminology goes) and even terms that depicted entities that merely happened not to exist (whether golden mountains or mermaids, not to mention centaurs). What truly strikes us, from the historical point of view, however, is not even the aforementioned novelties but rather the uninhibited manner by which they were suddenly introduced. Extensionalists are not disrespectful of these ancient taboos. They simply have no sense of veneration for essences. They fail to comprehend the ancient sense of dread that their essentialistic predecessors were coping with by means of their taboos. Every class is legitimate, extensionalists argue; therefore, clearly, even a class whose objects have been chosen by inconsistent criteria is legitimate. It is not very special, it is just empty. The empty class is legitimate, then, as are all the terms that name it (such as “square circle” or “largest prime number”). If every term is legitimate, then terms formed by the addition of the word “not” (or any other sign for negation) to any term are legitimate too, and the class of all objects that are not members of any other class is legitimate as well. It is its complement class, of course. If every class is legitimate, then the complement to the empty class is also legitimate, and this is the universal class or the universe of discourse. Indeed, for any two terms that an essentialist would not hesitate to endorse (‘man” and “goat’, for example) the extensionalist immediately produces by means of elementary class operations an infinite list of terms that the essentialist would reject (such as “goat-man”, “man or goat”, “man and goat” “not-man”, “not-goat-man”, “not-not goat”, etc.) These changes are not cosmetic. They lead to a metaphysically neutral logic. And this new openness to any metaphysics, the ability to suspend all metaphysical judgment while doing logic, resonates in every corner of the logical cosmos: it is the demand to replace judgments by propositions and sentences, and it is the demand to replace classic syllogisms by modern formal inferences. For, in traditional logic the ancient division (or aspiration for division) of terms into legitimate and non-legitimate has created the division (or call for division) of judgments into true and false and as this doesn’t work the true is further divided into the essential and the accidental: traditional essentialists, then, must divide judgments into essential or necessary (“Man is a rational animal”) on the one hand, and accidental (“Man is a featherless biped”) on the other. The task then forced itself to differentiate between these allegedly distinct sets of judgments. This task weighed heavily upon traditional logic. The reasons for this burden is that the propositions deemed necessary were empirical conjectures (some of them in error) just as those deemed accidental. The reason for the burden, then, was that a distinction was called for when there was no distinction to be made. Insisting upon making it (and further stressing that

126

16 Extensionalism as Exorcism

it is logical, and hence necessary, and even trivially so) merely put logicians in an even more uncomfortable position when they failed to specify this distinction or at least criteria for making it. The distinction, then, was problematic first and foremost because there is no logical differentiation between the propositions traditionally described as necessary and those traditionally described as accidental: both are informative, and hence contingent, from the strictly logical point of view. These two groups should have been placed together in direct opposition to logically necessary propositions (tautologies), but tautologies were traditionally dismissed – as negligible and uninteresting, exactly because they are empty of content, because they offered no taxonomy of essences and thus of things, because they did not express anything like an essential definition. (Thus, the traditional underrating of tautologies was a disguised effort to avoid admission of the fact that generating informative tautologies is impossible. Only this way can we understand the titanic efforts of Leibniz and of Kant, and their inevitable doom.) Extensionalism provided the basis on which to construct a new algebra. By means of that algebra the contents of any two propositions are comparable – in the sense that they do or do not express the same content. This was a tremendous advance over classical logic. Consider the following propositions: “All citizens are loyal” and “All traitors are foreigners”. Both express the same content and are therefore expressed by the same formula. However, within classical logic the issue is difficult to decide: intuitively, these propositions are obviously equivalent. Aristotelian metaphysics makes things awkward by its division of all expressions into positive and negative. The expression “loyal” is positive, whereas the expression “traitor” is negative. It is therefore considered inferior and therefore unsuitable for designating an essence. Thus, Aristotelian metaphysics ties the hands of traditional logic in order to ensure that only propositions of a certain type may be included in the results of Aristotelian epistemology. Indeed, it is very hard (perhaps impossible) to determine who really has the upper hand here, since traditional logic is saturated with Aristotelian metaphysics. Therefore the question –are the above propositions interchangeable? – gets out of hand. Inevitably, extensionalism changes also our conception of inference. It is, as matter of course, permissive in the choice of conclusions derivable from any given class of premises, as long as it is valid, of course. Traditional logic determines for the user one conclusion for each single premise (eduction) or each pair of premises (syllogisms) by means of fixed schemes. It is not a neutral system of inference, but rather an instructional system of taxonomies. Ordinary logical intuition is not free to operate within such a pre-fixed taxonomy. All this is obliterated (offhandedly) by the concept of logic as the study of extensions: suddenly an inference can include any number of premises, and a set of premises can have any number of logical conclusions. Logic, then, turns into a garden of forking paths along which one may wander at will, as long as one takes care not to stray. Traditional logic, by contrast, is a strict tour guide who purports to lead one towards a sublime destination, when in truth this destination lies beyond the guide’s ken. Extensionalism must not be confused with formalization, though the two are closely linked. Extensionalism invites formalization yet it is not identical with it.

16 Extensionalism as Exorcism

127

(Essentialism does not challenge the very idea of formalization; it simply gets in its way by resorting to informal intuitions, informal judgments, that comprise extralogical agenda.) Extensionalism and essentialism are mere attitudes to logic, whereas the question whether a system is formal or informal concerns its properties. A formal system is one open to manipulations in complete disregard for the meanings of its signs. Success in doing this is due to the presence of the complete list of permissible manipulations within it. Extensionalism, then, invites formalization by insuring us that such manipulation of signs is warranted since it allows all classes as legitimate. For example, Boole’s system of logic is extensional without being formal (as yet). To conclude, extensionalism amounts to a revolution within logic: it radically changed its most basic concepts, its most basic methods and its aims and scope, thus opening the road to modern, mathematical logic. It gave the notion of term a new meaning which in turn gave the notion of a proposition a new meaning which in turn gave the notion of inference a new meaning, more formal and more mathematical: extensionalism is permissive: all classes allowed by essentialists are allowed by extensionalists but not vice versa. And this completely erases the need to establish by logical means alone the theories of empirical science, since it permits empirical hypotheses such as that one class is a sub-class of another, while insisting that such empirical hypotheses are extra-logical. All this happened by fiat: by the mere declaration that a class is any arbitrary collection of particular objects, and that any collection of objects is as much a class as any other, at least within the province of logic.

Chapter 17

Mathematical Logic: An Oxymoron

Extensionalism revolutionized logic. Revolutions seem somewhat implausible when first conceived and plainly inevitable in hindsight, and usually the hindsight comes naturally: the antecedents require reconstruction. Extensionalism is no exception. Fascinating, detailed studies of the hindsight perspective comprise investigations into 18th and early 19th century mathematics. They emphasize every precursor to extensionalism. They often show that in hindsight attitudes clearly identifiable as extensional slowly crept into the agenda of mathematics raising rigor. To this end mathematicians found it fruitful to consider certain mathematical objects and operations in the abstract, disregarding their particular features, concentrating, instead, on formal characteristics shared by these objects and operations. This cleared the road to the study of certain formal mathematical laws in the abstract. These were the early steps in the study of formal (“universal”, or “symbolical” as it was labeled then) Algebra. Studies of the history of the formalization of algebra are thus of the background to the history of the extensionalist revolution. In hindsight they are clearly right: indeed, we now know that the very notion of a formal law anticipates extensionalism. Such studies, then, are vital, interesting, and rarely contestable. Rather than summing them up here,85 however, I have focused instead on the obstacles on the way to the revolution, for these are rarely discussed. The extensionalist revolution was not a mere outcome of the advancement in modern mathematics. Though many of the problems that led to a development of a purely extensional view were mathematical, as was the apparatus that enabled and encouraged it, the development we discuss here was within logic. More importantly: there were major theoretical obstacles that inhibited this development. These were not mathematical and strictly speaking not even logical: they were philosophical, epistemological. They made any extensionalist view seem somewhat dubious and suspicious. This was the case even for the most advanced, open-minded, extensionoriented mathematicians of the time. It holds already for Leibniz, Euler and Lambert, who preceded De Morgan, whose hesitations about undaunted extensionalism are discussed in the next chapter. None of these great Mathematicians have had the audacity to present logic as a mere study of extensions. Why? What they lacked is not the mathematical apparatus and certainly not the mathematical acumen. Even while advocating something that in hindsight looked like extensionalism, they did not do it. Boole has had the audacity to do it. To the extant that he fell short of N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

129

130

17 Mathematical Logic: An Oxymoron

completing the extensionalist revolution this was due to technical obstacles and mathematical biases, not to philosophical inhibitions; this is why I regard him as the proper starting point of the extensionalist revolution. And indeed those who have completed the revolution that he had started (Jevons, Peirce, Schroeder, Huntington) added nothing in terms of philosophical theory, they simply offered superior technical solutions to technical problems that he had solved unsatisfactorily or had left open.86 In order to appreciate the difficulties on the way to extensionalism we must first notice that the first, and in a sense the only, obstacle to extensionalism is the overwhelming incentive not to be one: extensionalism seems to lead to skepticism, since it is an invitation to suspend judgment, at least within the province of logic. Skepticism was universally considered as the termination of rationality. Since rationality is proof, we want proof: proof of knowledge of reality, of mathematics and of logic. This is where conflations come into play. The conflation of semantics and ontology promises us a grasp on reality, and the conflation of epistemology and methodology promises us a grasp on all knowledge. The subtle interplay between these two conflations is the traditional conflation of logic and science. All traditional theories of knowledge, thus, make use of these conflations. They are intellectual means to avoid extensionalism, to avoid skepticism. The removal of conflations is disintegrative and thus disturbing. Of course afterwards comes conscious reintegration and somewhat restores our confidence. The integration of logic and mathematics is peculiar, however: it did not precede a conflation of logic and mathematics. The integration of logic and mathematics was successful only when serving a deliberate attempt to disintegrate the subtle conflation of logic and epistemology. This point is crucial to our story. In semi-historical presentations of modern logic it is often noted that Aristotelian logic is insufficient for formulating basic mathematical proofs that he himself described. The point to observe, however, is that the very task of reducing mathematical proofs to logical ones was not central to Aristotle’s agenda. The reason for this is that traditionally both logic and mathematics were already, separately, conflated with separate aspects of knowledge of reality. Only after the disintegration of these original conflations could these fields be identified as co-extensive in a meaningful manner. The integration of logic and mathematics, then, is distinctly modern and quite revolutionary, though we have grown too accustomed to it to suspect that this is the case. (It is among the greatest achievements of modern logic that we tend to forget this.) Aristotle’s logic, though greatly indebted to reflections over the idea of mathematical proof, was not offered as a study of the structure of all mathematical proofs, not even of all geometrical ones. Though Aristotle often uses mathematical examples, in various contexts, he does not seem to have deemed it desirable to reduce all mathematical proofs to syllogisms.87 Proofs and syllogisms exist side by side in his corpus, sometimes as if existing in parallel universes. The reason for this seems to be that Aristotle’s official view of the syllogism was that it dealt with essences, and that he frequently endeavored to keep essences apart from mathematics, that is, apart from the shadow of Plato’s forms. For, of course, the very ideas of quantity and magnitude are achieved by stripping off objects from their Aristotelian essential features, ignoring

17 Mathematical Logic: An Oxymoron

131

their essence, and by studying them as mere ordinals, or as mere shapes etc. Aristotelian essences, then, whatever they may be, are certainly not mere quantities and magnitudes. Plato’s axioms and Aristotle’s definitions must be kept apart. Consequently, within the Aristotelian tradition, Mathematics and logic make up an unhappy match.88 The view of logic and mathematics as fundamentally distinct prevailed fairly undisturbed throughout the Middle-Ages (the former but not the latter is among the trivium). The prevailing view even gained a certain momentum with the Scholastics subtle fusion of logic and (Christian) theology. For mathematics is certainly no theology. And the first criticisms of this tradition came simply as general criticisms of the Aristotelian system. Indeed, the first signs of increasing intimacy between logic and mathematics came simply as reflections over the signs of the loss of status of logic and the rise of status of mathematics as the foundation of science. (To name the most obvious of these signs: Galileo famously declared that the book of nature is written in mathematical signs; Descartes suggested the revolutionary program to reduce all knowledge to arithmetic, and Hobbes contented that all thinking is combinatory.) This coming together in hindsight looks like a move towards extensionalism. In fact it was, at the beginning, almost the opposite. For, mathematics ran into epistemological troubles similar to the ones that logic had run into, such as unwarranted use of judgments, and downright paradoxes that could not be solved until the 19th century at the earliest. Until then, mathematicians, like the early logicians, used good sense to sidestep these difficulties, a fact that defenders of tradition, such as Berkeley, were quick to grasp and expose. This “good sense” invited the logicians’ theory of judgment which, as we have seen, rested upon the conflation of good sense with truth. The road to extensionalism was found blocked. When one theoretical foundation gradually comes to replace another, it is reasonable to ask at some point – what do they have in common? The suspicion that mathematics and logic may indeed have something in common raised a further difficulty: as the two shared with classical epistemology the idea of proof, they sharpened the question – What can be proven? These, as we have seen, were Leibniz’s questions, and in a move towards an answer to them he offered an outline of a calculus whose task was to mediate between essences and some formal laws that apply to the manipulation of magnitudes, between taxonomy and abstract arithmetic, as he understood it. He could not do it: as Russell was first to note, Leibniz could not succeed since he did not let go of Aristotle89: it is impossible to arithmetize traditional epistemology unless it is given in its entirety. Of course Leibniz was well aware of this for he had assumed that epistemology is given in its entirety. But assuming and proving this are two very different things, and the inevitable resort to “good sense” made sure that he would not succeed. Whenever he tried to apply his logic to the actual proof of an informative truth, then, the weakness of his apparatus loomed large. This, as we have seen, invited Kant to separate Arithmetic from Logic sharply. Arithmetic is synthetic, he said, whereas logic is not. In fact, he added, logic, unless it is transcendental, has nothing to do with the proof of informative knowledge. It was the reaction to Kant that launched the modern study of the shared foundations of logic and mathematics. This study was the opposite of

132

17 Mathematical Logic: An Oxymoron

what Leibniz had anticipated: it put essentialism aside. Mathematics and Formal Logic became “Analytic”. Yet many of Boole’s contemporaries still refused to put essences aside. They thus looked at the idea that logic is a formal study of extensions, or that it can be formulated by means of mathematical formulas, as blaspheme. Even those who opposed Kant and regarded mathematics as a study of analytic truths deemed ‘Mathematical Logic’ an oxymoron. This point is central to understanding the obstacles on the way to extensionalism. It is a subtle point though: even when abstract algebra was budding in England (in the innovative works of the Rev. Robert Woodhouse of Cambridge and his younger Cambridge admirers, the analytical society), the prevailing view amongst logicians was that algebra and logic are essentially different sciences with essentially different provinces. It was impossible to change that view until the completion of the disintegration of the traditional conflations of logic and essentialism. Let me give an example. The most important traditional introduction to logic that precedes Boole’s Mathematical Analysis of Logic (1847), it is agreed, was Richard Whately’s Elements of Logic (1826). Boole often uses it for reference. It is likewise agreed that Whately was one of the most influential writers on logic in his time (the early 19th century), but not why. His Elements of logic, though a remarkably clear introduction, is largely traditional. Why is it considered such a milestone then? This question is curious and requires some background. The book replaced Henri Aldrich’s Artis Logice (1690) as the standard textbook on logic at Oxford. Over a hundred years separate the two books yet they agree on almost everything. This is not entirely surprising: about two millennia separate Aldrich and Aristotle yet they too seem to agree on almost everything. But between Aldrich and Whately something did happen, as we noted: Hume had delivered his critique of induction, and Kant had accepted it – and even as the trivial yet profound observation that it is. Hume, then, made it harder to conflate logic and informative knowledge. And he had made it impossible to regard induction and deduction as isomorphic. This was the new situation that caused a radical change from Aldrich to Whately: defenders of traditional logic and its import for science now had to admit that logic and epistemology have too little in common: they were compelled to recognize and renounce the conflation, at least the way their predecessors effected it. Even Bacon – the greatest critic of scholastic logic – did not do that, much as he wished he did: even he regarded induction and deduction as isomorphic; he merely added that as a methodology deductive logic is dogmatic and defective since it puts theory before observation. Whately could no longer make do with Bacon: he realized that his predecessors in logic were in error since informative knowledge is extra-logical. The study of valid inferences, he admitted, can never yield informative knowledge. How then can logic be of any value for science? This was his question, and this, I submit, is his claim for fame. Whately’s answer to it is interesting and complex. It is complex, we now know, because it openly attempts to reconstruct and return to a lost innocent childhood. He split induction into two parts, one purely deductive and outside the province of epistemology, the other informative and outside the province of logic. Indeed, he

17 Mathematical Logic: An Oxymoron

133

was compelled to stress openly and with unprecedented clarity – and this is how he enters our story, even though marginally – that the conflation of logic and epistemology is a serious error that it leads to the devaluation of logic. But alas! He did not make it. There was a slippery slope that he could not avoid even though avoiding it was his chief aim. Rendering logic a mere theory of deductive reasoning meant for his readers that the study of logic and that of mathematics are both studies in deductive reasoning and to that end they are interchangeable. Please note that until now we did not discuss studies at all, yet I am afraid discussing it now, if only in two paragraphs, is unavoidable. Whately and his readers (Boole too, and we will come to this soon) were concerned with the proposal that was then in the air to replace mathematics with logic in the university curriculum. Abstract algebra which was increasingly gaining popularity at Cambridge had already started this process of erosion of the status of logic in the university curriculum. And there was Euclidian Geometry too. What separates these fields from logic? If nothing separates them, could not the cumbersome Logic be replaced by the highly practical Mathematics. The pressure on the traditionalist patrons of traditional logic was thus enormous, as it was both theoretical and practical: they were to either prove the distinctive value of logic and to successfully define its distinctive province, or else to reduce its weight in the standard curriculum and allow its replacement with other deductive sciences, possibly more useful to students. The standard reaction to this pressure was to claim that logic and mathematics discuss altogether different entities, different objects: mathematics discusses magnitudes and quantities whereas logic discusses essences. Yes, essences again. Whatever essences were, they had one chief role: to let in informative knowledge through the back door. We have analyzed the mechanism which enabled this throughout this book. Speaking of essences warranted logicians to use whatever they regarded to be trivial knowledge as if it was logical. The very distinction between essential and accidental, genus and differentia, concern information, as classic theories of definition guaranteed. This was not an easy dilemma then. On the one hand, logic includes methodologycum-epistemology (because it is a study of essences). Then it is successfully differentiated from mathematics. But its status is challenged by Hume’s critique. On the other hand, logic is a mere study of merely verbal definitions. Then Hume’s critique is virtually irrelevant to it. Yet, abstract algebra, then, is a better substitute for it. It is to relieve these pressures that Whately had written his Elements of Logic. Conflicting views of logic reflected conflicting social pressures: the idea that mathematics can replace logic as an elementary study in universities had fascinated radicals keen on the exact sciences and angered conservatives keen on the humanities. The social dispositions of university students were rapidly changing and this could not fail to have an effect over the curriculum. The practical value of the study of syllogisms was never obvious and many students deemed it superfluous. (Even Aristotle, you will remember, did not employ syllogisms in his scientific writings; what was their use then?) Whately added a book-size appendix to his book about the applicability of logical analysis to the study of Economics. He did this merely to exemplify the supposed practical value of logic to non-exact sciences which seemed to rest upon mathematics. He tried to make Aristotelian logic useful.

134

17 Mathematical Logic: An Oxymoron

The appendix, interesting as it is, was not to the point. The problem was that a pure theory of deduction that permits his analysis is easily replaceable by symbolical algebra. Otherwise, logic is more than such a pure deductive theory and this access requires explicit reflection and consideration. Whately could not win. Hindsight wisdom aside, we should mention that, at his time, Whately seemed successful. He looked able to dissipate these difficulties with an air of assurance and clarity. This, I submit, is the secret of his remarkable popularity. He was the best available defense of an impossible, yet much desired, project. Alas, he could neither evade the dilemma nor openly admit it: well-hidden yet unavoidable inconsistencies were unavoidable then. Indeed his book is read as if written by two writers. Whenever he discussed the attacks on logic-cum-epistemology he dismissed them as misunderstandings of the province of logic. Yet when the issue was the usefulness of logical reasoning he insisted that verbal, namely, nominal and thus empty-yetnecessary definitions of logical terms are also real, namely, essential and thus informative-scientific judgments “implying every attribute that can belong to the thing signified”. (pp. 158–159). And when he discussed the possibility of replacing the study of logic by the study of mathematics he said, this is impossible since: “Mathematical reasoning, as it calls for no exercise of judgment respecting probabilities, is the best kind of introductory exercise; and from the same cause, is apt, when too exclusively pursued to make men incorrect moral-reasoners” (p. 288). This split approach permeates Whatley’s book. It forces us to ask two questions which he did not, and could not, answer: What are the logical terms whose nominal definitions are also real? And why do these terms turn logic from a mere theory of deductive reasoning into a profound study of essences and probabilities? Whately’s significance to the history of logic, then, is that, much against his will, he helped bring the conflation of epistemology and methodology to the surface, and thus to its belated disintegration. For the mere bringing of a conflation to the surface, even if halfheartedly, is a major step towards its disintegration. For example, putting inductive inferences deductively renders their premises hypothetical and leaves their soundness for another time. The validity of the inference is then a part of logic, but the truth of its premises, becomes an extra-logical matter. The gain is in clarity and explicitness. Observe Whately’s example for an inductive inference that is deductively valid: “a property which belongs to the ox, sheep, deer, goat, and antelope, belongs to all horned animals; rumination belongs to these; therefore to all” (ibid., p. 128). Clearly, it is a valid inference, and equally clearly, its premise is a conjecture, not a definition, real or accidental! But since Whately could not admit that scientific theories are conjectures, the problem of the justification of science lifts its ugly head, thus bringing back real definitions, judgments and induction. Whately was in two minds, then. This produced inconsistencies in his text. These mislead historians, even eminent ones. One of these is Theodore Hailperin. Oddly, he asserts that Whately’s book is distinctly extensional and that this had influenced Boole and his milieu (Gasser 2000 p. 133). Few claims by this distinguished scholar have been more careless. (Perhaps it is worse than carelessness, as it also features his “Boole” entry in the distinguished Routledge encyclopedia for Philosophy.) He grounds this odd assertion in the following false observation of his:

17 Mathematical Logic: An Oxymoron

135

“Throughout Whately’s book syllogistic principles are being explained solely in extensional terms, that is, in terms of classes, no mention being made of ‘ideas’, or ‘concepts’ ” (ibid., 133). Let me quote here Whately’s definition of judgment: “the comparing together in the mind of two notions or ideas which are the objects of apprehension …”(p. 61). This definition opens the synthetic part – the major part – of Whately’s book. The notion of ‘idea’ then is central here. Nevertheless, Hailperin does have a point: admittedly, Whately also uses the term “class” (pp. 35–37) to formulate syllogistic principles, especially in the short “analytic” presentation of the subject, opening his book (p. 42). Read out-of-context, this expression (“class”) may appear as extensional, and rightly at times, but only at times. In context, its only aim is to stress that abstract objects are abstractions, not objects (p. 55). This point is elaborated in Whately’s detailed, traditional, discussion of definitions. He says there (p. 157) “logical definitions consists of the “Genus” and “Difference”;” and he stresses that these do not designate “real parts into which an individual object can … be actually divided, but only different views taken [notions formed] of a class of objects, by one mind” (italics and square brackets in the original text). Thus, in Whatley’s sense classes are neither real nor arbitrary: they are the nominalist way of referring to abstract objects, which his ontology forbids. This is not extensionalism. An extensionalist is not a mere nominalist, for nominalism is, after all, ontology and semantics conflated. An extensionalist regards the very terms “Genus” and “Difference” obsolete. Whately’s occasional use of the term “class” is only seemingly modern rather than extensional, since what makes a system extensional is the wish to avoid the conflation of logic with the extra-logical and the readiness to admit any class, however artificial and arbitrary. Whately does not even consider such an option. Were Whately an extensionalist, he would not have insisted that logic is essentially different from mathematics and even morally superior to it. (See the quotation from him above. ibid., p. 288) Likewise, he would have avoided all traditional essentialistic distinctions such as Genus and Differentia. Thus, we see what role Whately played as a junction, a crossroad where the prevailing views of logic met and were crystallizing into a paradox: logic must be a purely deductive, formal, system, yet it cannot be that because its alleged value is in its very discussion of essences that usher informative theoretical knowledge into the province of logic depriving it of its status as a purely formal, deductive system. What is so marvelous about Boole’s work is that it had effectively deleted this very dilemma. Boole flatly proved that logic and mathematics are studies of the same abstract laws. This gives his work its uniqueness and import. There are but few cases like it in history: a long standing metaphysical controversy has been rendered solvable by a novel technical apparatus, and then solved by it, once and for all. Allegedly empty mathematical formulas and allegedly profound real definitions were united. “Mathematical logic” has turned from an oxymoron into a commonplace platitude.

Chapter 18

The Last Step

Undaunted extensionalism seemed – still seems – scandalous, and not only to traditionalists. Even advanced innovators express reservations about it, if not downright objections. The great Augustus De Morgan an ardent advocate of the union between logic and mathematics, had found undaunted extensionalism disagreeable on a few accounts. Perhaps it was his outstanding acquaintance with the history of logic that inhibited him and prevented his becoming a full blooded extensionalist. This is the last chapter of my survey of the farewell to essentialism. I devote it to presenting De Morgan’s greatest and most interesting objection to extensionalism. To repeat, the traditional preference for traditional logic lied in its practical, and thus seemingly informative, character. It is thus not surprising that the first real advances in the modern study of logic were made by those who viewed mathematics as able to serve such classic practical (and moral) ends. Those who surmised that mathematics can justify science could then contribute. This is where probability theory enters our story. It is also where De Morgan enters it. And Boole too. Nowhere was extensionalism more natural and less conspicuously revolutionary then within the context of probability. At the beginning of the 19th century probability theory was viewed as the proper representation of inductive inference. It was one of the most important catalysts of modern logic. Probability theory seemed to promise to reduce meaningful, essential, inter-relations between concepts to the merest acts of comparison of extensions. Within that frame, then, perhaps paradoxically, extensionalism looked pregnant with epistemology exactly because it served as a means of aiding the rationalization of induction. The undoing of the conflation between logic and science seemed a loss, as was emphasized in our brief discussion of Whately’s Elements, and then probability theory came to restore that loss: it promised to reintegrate the lost part, namely epistemology. It seemed to do so, we now know, by yielding to a new and highly influential conflation: the conflation between probability and statistics. (The one studies the likelihood of certain events within certain mathematical models and the other with the possibility that these models are also faithful representations of the world we live in.) The study of probability had numerous subtle influences on the rise of the new logic. De Morgan, for example, studied such clearly extensional terms as “most men” and even “30% of all men”. He thus significantly expanded the classic notion of the syllogism with such probabilistic syllogisms as, say, “Most humans are N. Bar-Am, Extensionalism: The Revolution in Logic, © Springer Science + Business Media B.V. 2008

137

138

18 The Last Step

rational, most humans are females, therefore some females are rational”. De Morgan insisted that the validity of such inferences is not “accidental” in the traditional Aristotelian sense. The validity of such syllogisms, he insisted, is mathematical, and hence logical, and quite significantly so. The study of logic from the perspective of probability theory has a devastating effect upon some dogmas of essentialism. For example, it undermined the prominence of the classical copula. Traditionally, it was the only relation admitted as real, and hence the only logically admitted relation. From the mathematical point of view things look radically different, however. There, it is difficult to find its exact equivalent. (Kant’s insistence that, within the equation “5 + 7 = 12”, the expression “= 12” is a predicate seems to us today needlessly strenuous, even completely arbitrary). The very claim that there is a single mathematical copula is by now known to be false. (Russell discovered his paradox, we should note here in passing, came as a result of his effort to show just how crucial it is to distinguish between different copulas.) Traditionally there was a tendency to identify, as a matter of course, the copula with the arithmetical “greater/and or equal to”. Even this De Morgan had to show, is problematic. For clearly in certain contexts it is important to distinguish between “greater than” and “equal to”. He was the first to express the importance of doing this for logic as a whole and, returning to traditional logic, he declared that the traditional copula is ambiguous. It must be split in two, he argued, so as to achieve clarity. He offered to split it into something akin to inclusion and equality. Thus, he offered to replace the classical four categorical judgments with eight, ridding himself, more or less, of some but not all copula related ambiguities. (Russell and Tarski did the rest.) Clearly the ancient logicians were skillful enough not to be misled by such ambiguities. But this, as we repeatedly stress, does not suffice: the original system needed overhaul so as to eliminate the ambiguity. Overcoming it on the level of intuitive understanding, on the level of “good sense”, on the judgment level, only entrenched the conflation that harbored many more. De Morgan’s new system came to eliminate the very possibility of ambiguity. It thus replaced some implicit logical knowledge with some explicit logical rule. Doing so all the way is modern formal logic. A sad consequence of this interesting discovery was that De Morgan had entered into a famous quarrel of primacy and originality with the Scottish (Oxford bred) logician William Hamilton. Hamilton had offered De Morgan his notes on the subject containing a misleadingly similar system a little before De Morgan had published his own. Let me note here in passing something that few commentators take note of: Hamilton’s explicit aim was the exact opposite of De Morgan’s: it was that of preserving the prominence of the copula, as well as the primacy of intensional relations over extensional ones.) The quarrel itself is not within the bounds of our narrative, however. What is significant about it is that it led De Morgan to reflect upon the features of relations, to attempt and offer taxonomy for them (e.g. transitive, reflexive and symmetric) and even to make rudimentary suggestions for a (future) logic of relations.90 De Morgan’s most profound contributions to the advancement of extensionalism, however, are even more basic. They are the introduction of the universal class

18 The Last Step

139

(initially the sum of all chances) and the complement class (initially the sum of all chances that an event will not occur). Traditional logic deemed these classes undesirable and even hazardous, for the very reason that they are extensional. De Morgan introduced these concepts so as to extend classical syllogism to the probability calculus. Still, he was no extensionalist. Extensionalism requires also the complement class of the universal class and this class caused the most trouble to classical logicians, you will remember, for it meant admitting the class nothing, or the empty class. (It is no accident that Lukasiewicz’s axiomatization of Aristotelian logic includes the axiom ‘there is no predicate without some referent’, better known as “PiP”.) The empty class was dreaded for admitting its very possibility contradicted the one and only theory of meaning that was ever proposed until Frege had refuted it and replaced it: the theory that the meaning of a term is its referent. Thus, terms with no reference seemed meaningless and so the idea that terms could designate an empty class seemed absurd. De Morgan, then, had good grounds for his hesitation to endorse it. Let me explain this hesitation, as it is the last form of such hesitation when the extensionalist revolution was afoot. When De Morgan introduced rudimentary versions of the universal class and the complement class he was already thinking of how to generalize the theory of deductive inference to include inductive ones (which most everybody identified with probability). De Morgan regarded the urn for drawing lots that serves as the basis for Bayes’ theorem as the universe of his discourse; he called it “Universe”. The idea of a universe of discourse, then, is not new and with respect to probability theory it is even old: it is the sum of all chances. However, De Morgan turned it into a basic concept within logic and relativized it to specific probability cases to be redefined accordingly. This was a great improvement and expansion of the Aristotelian system which admitted only one such universe and forbade all discussion of it. De Morgan also maintained that every Universe is neatly composed of a pair of complementary terms, which he called “contrary names”. “Contrary names”, he said, “with reference to any one universe, are those which cannot both apply at once, but one or other of which always applies”.91 Now the complement class of the Universe is empty, of course. De Morgan did not fail to see this. (It is the sum of all chances that an impossible event will occur.) But like all logicians before him he deliberately refused to acknowledge the legitimacy of terms that have no referent. De Morgan prohibited any of his terms to “fill” the universe so as to prevent their complement to become “empty”. This is his restraint from becoming fully extensionalist. It also renders his notion of “universe” not logical, but rather meta-logical. In order to clinch matters, De Morgan set it as a maxim: “In future” he said, “I always understand some one universe as being that in which all names used are wholly contained: and also (which it is very important to bear in mind) that no one name mentioned in a proposition fills this universe, or applies to everything in it.”92 Obviously, if a term is allowed to “fill” the universe, then, its complement is “empty”. This is precisely what De Morgan ruled out by his maxim. Consequently, he could not resist the traditional conflation of semantics, psychology and ontology in the style that Alexius Meinong rendered explicit soon. De Morgan followed the

140

18 The Last Step

Aristotelian tradition in demanding that all terms should have referents (“PiP”), but since some terms obviously do not refer to real objects, he allowed some terms to refer to mere concepts (as Meinong did later too). De Morgan’s problematic theory of reference is nicely summed up in his Syllabus of a Proposed System of Logic.93 He points there at his greatest difficulty with terms that refer to nothing. This difficulty is of outmost importance to our story, as it is patently Aristotelian. De Morgan endorsed the traditional idea of an inverse relation between extension and intension: the greater the extension of a term, the smaller its intension, and vice versa. And he says: “The name of greatest extension and of least intension, of which we speak, is the universe.” (ibid.) This would suggest that terms that refer to Nothing have the greatest intension. The problem was, then, not simply to admit that a class with no members is at all possible, but rather to admit the possibility that a class with no members has the greatest intension! It seemed absurd. Please recall. The absurdity of the Nothing was the starting point of our journey: it was the crux of Parmenides’ proof, the mother of all proofs, and the source of the conflation of semantics and ontology. To admit that meaningful terms within logic can have no reference was to cut the umbilical chord that ties language to reality. So long as it was not cut, the hope lurked that with enough patience, diligence, humility and ingenuity, some a priori consideration within logic will emerge and suggest some a priori justification of some a priori ontology, and thus of scientific knowledge. As long as the empty class was forbidden, then, the conflation of logic and science persisted. De Morgan, then, is one of the key proponents of the extensionalist revolution in logic, and is without doubt one of the founders of Modern Logic, yet he was no undaunted extensionalist. He still tried to associate intension and extension under the same logical system (still suffused with traditional terminology) and he consequently got entangled in a problematic logic which barred the empty class and the terms that depicted nothing and so also the universe of discourse. They could not participate in his logic for they threatened to let in the dreaded empty class through the back door. One had to use good sense, then, for how is one to know that certain terms (“the greatest prime number” or “black swans) are, or are not, to be admitted into logic? Indeed, De Morgan now added a new version of the conflation (one which seems to have been first presented by Bolzano), as he embedded in his logic a conflation of psychology and semantics, which threatened to become a full-fledged conflation of methodology and epistemology, with those (like Husserl) who regarded psychologism as the new road to epistemology. Despite his great local success in describing probable inferences and describing the beginning of the logic of relations, De Morgan was still (essentially) too traditional. Boole was far more ambitious. Perhaps as a result of this he was also slightly blinded or even slightly simplistic. Hard boiled extensionalism is the secret of all his innovations within logic, and the source of the simplistic clarity of his system of logic. It is truly striking how casually Boole dismissed De Morgan’s hesitation, offhandedly, in the typical manner in which he brushed aside all epistemological difficulties (as extra-logical, not as unproblematic or as undeserving serious

18 The Last Step

141

consideration). This holds for all extensionalists. Boole explicitly allowed terms to “fill” his universe of discourse and, consequently, he allowed their complements to become “empty”. His mere notation suggests a trivial proof that this “empty” class, the class Nothing as he called it, is nothing more than the complement class of the universal class. The proof is as basic as arithmetic can get. It is this: if the universal class = 1, then by definition of the complement class, it is 1–1, that is 0. End of proof. (It should be noted that when Boole first presented it, the proof included a semi-conscious conflation of the class Nothing with the number zero; the problem of admitting the class Nothing thus became no more, nor less, problematic than the problem of admitting zero as a number;94 like zero, the class Nothing is a mere matter of (logical) convenience. The conflation of zero with it, however, is not essential for the success of the proof. Indeed, it stands in the way of such a proof. Thus, the triviality of the proof is truly striking. In his Laws of thought (1854, p. 52) Boole had released himself from this conflation. He then presents the empty class in the following historical words: In accordance with a previous definition we may term Nothing a class. In fact, Nothing and Universe are the two limits of class extensions for they are the limits of the possible interpretations of general names, none of which can relate to fewer individuals than are comprised in Nothing, or to more than are comprised in the Universe. Now, whatever the class y may be, the individuals which are common to it and to the class “Nothing” are identical with those comprised in the class “Nothing” for they are none.

As for De Morgan’s hesitation, suffice it to say that today we readily admit that in various senses the class Nothing has the greatest intension: we admit, for example, that a contradiction entails any conclusion (it has the greatest possible content as all propositions are its consequences), and we similarly admit that the empty set is a sub-set of all sets. In Boole’s system one is led to refer to the class Nothing also as the union of everything and nothing (i.e. as the class of individuals shared by the Universe and its complement class, Nothing). Thus, the union of 1 and (1–1) is the union of 1and 0. It is 0. This is a special case of X*(1 – X) = 0. The latter case is of course the more interesting and more general case. It immediately leads to Boole’s famous, and striking index law, X = X*X. The index law, as is well known, is fundamental to Boolean algebra, and has been regarded by Boole as its core. The first logical system thus came into being that sets ontological and epistemological considerations apart from logical ones. Logic became, at least in principle, fully extensional. The extensionalist revolution in logic had begun. Let me stress that what I find so impressing in this historical moment is the philosophical attitude that let to it. Boole’s crucial achievement in logic, then, is not that of suggesting new concepts or even a new algebra, as hindsight studies of the history of logic may suggest. (W. R. Hamilton did so a few years before him, with his system of quaternions, without thereby becoming a key figure in the history of logic, of course.) Boole’s crucial achievement is not even that of insisting that his algebra is an accurate representation of the rules of valid inference, and that its equations faithfully capture the meaning of the categorical propositions and the validity of the classical syllogisms. His crucial achievement in logic is much more fundamental and basic. It is his emphatically unsophisticated extensionalism, his

142

18 The Last Step

seemingly simpleminded insistence that logic discusses classes and nothing else, and that classes are arbitrary collection of objects and nothing else. By doing so he was a trailblazer: he provided the first apparatus which clearly and unflinchingly separates epistemology from logic, empirical knowledge from reasoning. Let me quickly add that Boole did not complete the extensional revolution. The most obvious example to this is that his Algebra lacks some rules of inference. It is thus not completely formal and in the strict modern sense it is incomplete: it makes use of intuitions (and mathematical judgments) on the way to proofs of syllogisms. Even his formalization and manipulation of the categorical particular (existential) sentence (affirmative and negative) makes use of such a semi-formal appeal to intuition. Only when complete formalization of his system was achieved (Schroeder 1890 and Huntington 1904) did his system have a fully extensional presentation, as Boole had intended, but by then it has been realized that in important respects it is insufficient for the new aims of modern logic. This reservation, however, is technical and not entirely a part of our story: Boole was the first logician to intend his logic to be fully extensional. And he certainly went as far on that track as he could possibly go. By doing so he started a process that completely obliterated the old conflations of logic and epistemology and ended in a totally new logic. This was the extensionalist revolution.

Epilogue: Extensionalism in a New Context

If you feel that this book has ended abruptly, then you are right. It has. And just when things get really interesting: right on the verge of the most exciting period in the history of logic. But my aim in this book was not to share with you my excitement with modern logic. And it was not even to tell you of the immense impact of extensionalism on it. That would certainly be the most deserving task of my next study. My aim was rather to present you with the story of its first (and devastating) impact on the aims, methods and scope of traditional logic, brought upon by a person who remained within tradition much more than his followers. Almost every conceivable significant innovation in modern logic resonates with attempts to either entrench extensionalism or to attenuate it. Alas, modern students of logic have grown too used to extensionalism, and with it to the idea that logic is free of metaphysics. They thus often fail to realize that the very wish to have it free of metaphysics was part and parcel of a revolution that had occurred within logic, almost as an afterthought. The story I have told you, thus far, was rather that of the difficulties standing in its way. Boole regarded arithmetic and logic as two manifestations of the same abstract laws (“the laws of thought”, to use his terminology). He did not call these to doubt. With admirable freedom he says of these laws: “if they were other than they are, the entire mechanism of reasoning…would be vitally changed. A Logic might indeed exist, but it would no longer be the Logic we posses”.95 Hence, he would have been somewhat surprised to learn that his followers spent so much ingenuity and effort in attempts to secure Arithmetic by means of logic (which, to repeat, he regarded as parallel to it). This, in retrospect, is the most important conceptual leap from Boole to Modern logic: Frege was the first to realize the importance of a task that until that point was either not noticed (Aristotle), insufficiently distinguished from the impossible task of securing the whole of science (Leibniz, Kant), or seemed too trivial to be of genuine interest (De Morgan, Boole). The advent of modern logic is usually identified with the contributions of Frege and Russell. They have explored undaunted extensionalism. Both have restrained it in interesting ways which can only be hinted at here. They helped spell out many of its hidden consequences, make them distinct, endorse some of them and criticize others (in truth or in error). The case of Frege is fascinating because there seems to be a neat division between his logical system (which is extensional as a matter of 143

144

Epilogue: Extensionalism in a New Context

course, and even to the point of inconsistency as Russell demonstrated) and his philosophy of logic (which contains some of the most interesting criticism of extensionalism ever produced). Frege’s philosophy of logic results from his improvement on Kant’s theory of judgment as I have already noted (chapter 13). His very wish for arithmetic to be reduced to logic is a defense of Kant’s concept of analyticity as provability by logical means alone. For this, of course, he had to expand these ‘logical means’ in a manner neither Kant nor even Boole imagined. His endorsement of his own peculiar version of Kant’s theory of analytic judgments, then, is responsible for his interesting reservations about undaunted extensionalism. For within Kant’s definition of analyticity he had no reservation of this sort. As I have noted before (chapter 13), he refused to admit proposition as such into the province of his logic, which resulted in refusal to admit reductio ad absurdum unless it is translated (with the aid of what we now call the deduction theorem) into a conditional with a false antecedent, never as a valid inference with false premises. Better known is his fascinating refutation of the classical theory of meaning (as reference) which redefined Platonism and gave rise to so many interesting debates about the ontology of logic and mathematics. Russell’s attitude to undaunted extensionalism is a bit more complex to describe in a nutshell exactly because his paradox employs only the apparatus of logic proper, extensionally understood. He overcame the paradox by his clear cut distinction between the meaningful and the meaningless where the meaningful is, as with Frege, true or false. This way of overcoming the paradox limits extensions to the meaningful. It is, thus, not undaunted extensionalism. It is interesting to note that the source of his Paradox was his attempt to clarify the Aristotelian copula by splitting it into two: the relation of set membership on the one hand and set inclusion on the other. Taking self-inclusion as obviously true for all sets and seeing that self-membership is normally not the case, he tried to show that the normal case (a set not being a member of itself) is the universal case. This attempt ended up with the discovery of his paradox: is the set of all normal sets normal? In the motto to preliminary IV I have quoted Russell’s astute observation that since people have tried to prove obvious propositions they have found that many of them are false. I have told you the story of this discovery with respect to many fundamental propositions of traditional logic. But I suspect that Russell intended to apply it also to his own system, his own discovery that undaunted extensionalism is impossible. Attempts to apply extensionalism all the way, then, exposed first the limitation of traditional judgments and then the limitation of the attempts themselves. By extensional means new intensions are discovered. The future is still open, then, even for the most obvious.

Notes

1. The learned literature has various descriptions of the discovery of the dichotomy between nature and convention as a creation myth of rational philosophy. The classic ones are J. Burnet 1897 pp. 328–333., Sir E. Barker 1918, chapter IV (the section on physis and nomos), and G.C. Field 1930 the end of chapter 6. The boldest (and thus clearest yet most susceptible to minute criticism) is still Sir Karl Popper’s [1945] 1966 Vol., chapter 5. Agassi 1977 pp. 222–255 is a most valuable critical study of these sources. All descriptions point at the fact that the dichotomy replaced traditional taboos by presenting them as mere conflations of customs and laws of nature. This fact, by the way, sits well with the common observation that the first philosophers were exiles: there is no human condition more prone to illicit doubt in the universality of one’s tradition than that of seeing it subdued by another. It is thus, not surprising to find that Herodotus goes as far as to ascribe the very discovery of the dichotomy to the Persians (who are, of course, responsible for turning so many of the first philosophers into exiles). See L. Strauss 1953 p. 85. and Sir K.R. Popper 1994 p. 37. 2. I. M Finley’s [1954] 1967 p. 139. 3. The classic discussion of the Homeric hero’s code of valor, its values and morals is chapter 5 of I. M Finley’s [1954] 1967. 4. J. Lukasiewicz [1951] 1972 started the modern fashion of anachronistic formal studies of Aristotle’s logic. Of course, his (deliberate, conscious) anachronism cannot be accused of spreading confusion. Unfortunately this cannot be said of all those who offer alternative “more correct” versions. Whatever the merit of their anachronism, confusion is never a merit. The very question, “what is the correct modern formalization of Aristotle’s logic?” is misleading and confusing: criticism of any of the alternatives should be directed against the making of the confusion due to the ascription to Aristotle, explicit or tacit, of ideas he could not divine. The following texts are the best known, formal studies of Aristotle Logic: G. Patzig 1968, T. J. Smiley 1973, J. Corcoran (ed.) 1974, J. Lear, 1980, P. Thom 1981. 5. For the controversy as to whether the refutation of a refutation is progress see Sir K.R. Popper 1963 (2002), chapter 10; Lakatos, and A. Musgrave (eds.) [1970] 1999 pp. 91–196; D. Miller 1974 pp. 166–177; and J Agassi 1981 pp. 576–579. Hacohen 2000, pp. 528–535, offers a very short and careful summary. 6. R.D. Mckirahan 1992 p. 24. 7. Fairly recent English translations of Parmenides’ poem that include particularly accessible and exciting comments are J. Barnes 1987, chapter 9 (see also his 1979, chapters 9–11) and D. Gallop 1984. The less recent, classic, Cornford, 1939, is my favorite, though this, perhaps, is a sentimental preference. W. K. C. Guthrie 1962–1981, Vol. 2, includes the standard discussion of the standard secondary literature. 8. A. Szabo 1978 pp. 216–220. See also p. 250 there: “It was Parmenides who proved his theses by refuting their negations. The discovery of indirect proof was perhaps his greatest and most lasting contribution to philosophy.” Popper adds, to this: “The old pre-Aristotelian formal proof

145

146

Notes

was, it seems, mainly the indirect proof, the elenchus (reductio ad absurdum). Parmenides mentions it by name in B7: 5. It is good that there can be no doubt about its meaning, as it derives from elenche (‘to disgrace’, ‘scorn’, ‘dishonor’); in this case to dishonor an assertion” K. R. Popper 1998 p. 77, n 4. This explains in part why it was so natural to indict Socrates, the greatest master of Elenctic disputations, with dishonoring the Gods and corrupting the youth: he had taught them the art of ‘scorn’. 9. A whole literature is devoted to the interpretation of this fragment of Parmenides’ poem. Most of the standard major studies are mentioned in C. H. Kahn’s seminal paper (1988) that divides them by their understanding of Parmenides’ use of the verb ‘to be’. Efforts to resolve the paradox that Parmenides’ proof imposes go as far back as Plato (the Parmenides and the Sophist in particular) and Aristotle (Metaphysics 992b18–24, for example). Already they suggested that the paradox can be resolved by distinguishing between different meanings of the verb ‘to be’. My addition to Kahn’s learned discussion is this: in addition to any interpretation of the proof it is necessary to acknowledge openly that it is invalid. My concern with this invalidity is due to its conflation of the real with the possible, of the cosmological with the logical. 10. One reading that rests distinctly on the claim that Parmenides identified ‘that which is not’ with the void; is K.R. Popper, 1998 (pp. 287–288). I find his interpretation most instructive, since it clearly exemplifies my own: that Parmenides conflated the logical with the cosmological. 11. The first known philosopher to have used a particular word for “that which is not” seems to be Parmenides’ follower Melissus of Samos. He used the Greek word Kenon – meaning emptiness or void. This, however, may have been an inconsistency on his part. The absence of a particular word for it in the text of Parmenides may well be intentional: his theory declares meaningless expression that designates a non-entity. (Is ‘that which is not’ not a meaningful term? For a discussion of this problem see S. Austin 1986, and G.E.M. Anscombe, 1968/9. More on Melissus in the next note. 12. Oddly, and not in accord with the momentum of Parmenides’ argument, the universe according to Parmenides is bounded spatially. Melissus corrected this discrepancy. Barnes, 1979 pp. 180–184 is a most sympathetic brief account of Melissus proof, unyielding to the Aristotelian tradition that looked down on him (Met. 986b26 and Phys 186a9). The passage from Aristotle’s Physics (186a9) is interesting from the logical point of view since it exhibits his having made no distinction (at least here) between sound inference and proof. Aristotle says there of Melissus’ proof that it suffices to “admit one ridiculous proposition, and the rest follows – a simple enough procedure”. Since, in retrospect, Melissus’ argument is an indirect proof not a sound inference, since it purports not to rest on any suppositions, Aristotle’s hasty judgment rests on his conflation of sound inference and proof. 13. The very idea of indirect proof, says Szabo, came to Parmenides as a result of his attempt to refute the cosmogony of Anaximenes; ibid., p. 219 and p. 250. 14. M. Grene 1963 p. 40, clarifies the point: “The fact is that the statement ‘Theaetetus is (exist)’ and the statement ‘Theaetetus is snub-nosed’ or ‘Theaetetus sits (or flies)’, use the concept ‘to be’ in two different senses. Only the second is predicative rather than an existential ‘is’. Now, of course, Parmenides, living long before Aristotle and logic, had no notion of this difference. He thought that to say ‘Theaetetus is snub-nosed’ was the same sort of business as to say ‘Theaetetus is’ ” 15. D. Gallop’s translation of Fragment 2. 7–8 is this: “For you could not know what-is-not (for that is not feasible), Nor could you point it out” (p. 55); and of fragment 6. 1–2: “It must be that what is there for speaking and thinking of is; for [it] is there to be, Whereas nothing is not; that is what I bid you to consider” (p. 61); and, once more, in fragment 7.1: “For never shall this prevail, that things that are not are; But do you restrain your thoughts from this route of inquiry” (p. 63) 16. As I observed in the previous note, the (later called) “principle of non-contradiction” is stated quite clearly already in Parmenides (2.7–8, 7.1 and elsewhere). It is presented there as a revelation. In Plato’s Republic 436b it is already a platitude. Aristotle calls it “the most certain of all principles” (Met. 1005b19–23). Famously (and, of course, ironically) he demonstrates that for those who would violate it “all roads lead to Megara”.

Notes

147

17. “Undr” in J. L. Borges 1998 pp. 455–459. Since a considerable part of Borges’ output deals with that theme (the absurdity and inevitability of the idea that our language is a mirror of nature) it is perhaps ridiculous to refer to a single story here. In “God’s Script” it is not a word but rather a series of fourteen strips on the back of a leopard that codifies the entirety of the cosmos and its secrets. In “The Aleph” it is a ludicrously arbitrary point in space at the cellar of an incompetent poet. In “The library of Babylon”, the library, its index and the story itself have the same referent. See also “Total library”, and “John Wilkins’ Analytical Language” in Borges (1999). 18. The extent is understandably undecided to which Plato has recognized dialektike as a distinct method. See the superb study of this point by G. Ryle (1971), chapters 5 and 6. Following Jaeger, Ryle speculates that Plato was still alive when Aristotle had begun to teach in the Academy contents of what would become his Topics. He adds that Aristotle’s early logical terminology is somewhat influenced by Plato’s own (pp. 114–115). To my knowledge no one has ever doubted Aristotle’s own confession (at the end of his On Sophistical Refutations) that he had to figure out on his own the formal account of dialektike. Indeed, it is this formality that made him the first logician. 19. I. Lakatos 1976 pp. 1–5. 20. K.R. Popper [1963] 1989, chapter 5. 21. Diogenes Laertius reports that Aristotle has made the ascription of the invention of dialectics to Zeno, in his lost dialogue Sophist. Laertius mentions this once in his life of Empedocles viii, 57 and once in his life of Zeno of Elea ix, 25 (R. D Hicks 1925); Sextus Empiricus also reports it in his Against the Logicians, i. 7. (See R. Bett 2005). 22. E.R Dodds (1951) is the classic discussion of the role of divine infatuation, ate, in the Homeric world. Chapter 6 vividly describes the intellectual clash between this form of irrationalism and the rationalism of the 5th BC enlightenment. 23. Dialectics in Plato reflects a gradual attempt to formulate a method out of the cluster of argumentation habits and procedures known as Socratic elenchus (See note 8 for Parmenides’ use of that term, which originally meant simply ‘to defame’ or scott [a proposition]). Elenctic disputation is not discussed here as a separate method, distinct from dialektike, since it seems to have been hardly a distinct method at all. A detailed study of Elenchus is the admirable and seminal R. Robinson 1953, especially its first three chapters. Robinson centers on the study of elenchus under the assumption that it is more-or-less a distinct practice, more genuinely Socratic than its later Platonic refinements: dialektike and diaeresis. 24. R.K. Sprague 1972 pp. 42–46. 25. W. K. C. Guthrie 1962–1981, Vol. III, rightly criticizes the literature that deems Gorgias’ argument a mere parody on Parmenides’ poem (p. 194). Indeed, it is improbable that Gorgias was unaware of, or indifferent to, the philosophical significance of his parody. After all, he was parodying the belief in the ability of rational thinking to achieve Truth, presenting it as barren metaphysical speculation. This is no small matter. It is interesting to compare Gorgias’ didactic, dramatic tone with that of the hilariously irresponsible and silly Socrates in Aristophanes’ Clouds. Aristophanes’ Socrates is clearly a friendly caricature, as indicated by the evident amity of Socrates and Aristophanes in the Symposium. It is hard to imagine Parmenides and Gorgias exchanging compliments and mutual friendly teases on a similar occasion. 26. E. Schiappa 2003 pp. 3–11 is a concise summary of the literature discussing Plato’s reputed responsibility for the prevailing negative connotations of the label ‘sophist’. 27. E. Schiappa 2003 pp. 39–63, especially 40–41. Whether Plato had actually coined the term ‘rhetoric’ or not is, of course, of little import. It is significant that almost single-handedly he made it an insult. 28. Top. 100a29 and also 104a8; and An. Pr. 24a23–24b–13. Aristotle’s notion of syllogism is (roughly) that of a sound inference with categorical judgments. His notion of apodeictic syllogism is (roughly) that of a sound inference from first principles, unlike his notion of dialectical syllogism that is (roughly) that of a (putatively) sound inference from “apparently true and generally accepted” premises. Aristotle surely deserves the credit for being the first to formulate the isomorphism between apodeictic and dialectical syllogisms; yet he did not get the

148

Notes

isomorphism correctly. (For, properly speaking, the isomorphism is between sound inferences and refutations, i.e. valid inferences with admittedly true premises on the one hand and those same valid inferences with admittedly false conclusions on the other hand). Thus, Aristotle did not quite realize that dialectic syllogisms are refutations and not at all sound inferences. Alternatively, a double notion of dialectical inference makes his intentions unclear: for clearly a valid inference from “apparently true and generally accepted” premises is not a refutation. Indeed Aristotle could not find a proper place for refutations in his theory (His Refutations of the Sophists notwithstanding): he cared only for sound inferences. (For a detailed discussion of this point see chapter 10 above.) From the modern point of view Aristotle’s notion of dialectics is somewhat peculiar, as it ignores that which is essential to it (its role as a method of refutation) and stresses that which is accidental to it (its beginning with the acceptance of a certain premise as putatively true). Thus, instead of stressing that dialectics is reasoning by valid inferences from false conclusions to false premises (as Socrates and Popper stress), Aristotle stresses that the inference used in dialectics is only putatively sound. Although aware of refutation being re-transmission of falsity (from conclusion to premises), he does not take explicit account of it: valid inference as such plays no role in his logic! Thus, he found no place in his theory for valid inferences from false premises (despite a fleeting, and admittedly obscure, acknowledgment of their existence in An. Pr. B 2 53b3–10; his acknowledgment is then incompatible with his view of all syllogisms as sound). Patzig 1968 pp. 196–202 has proposed an attempt to defend Aristotle’s fleeting remark about syllogism from false premises. His attack on the traditional interpretation (and straightforward reading) of B2 53b3–10 is surprisingly vigorous. And he presents Ross and Meier as exceptional whereas clearly they follow tradition. Alas, Patzig speaks of ‘the truth (i.e. the validity) of syllogisms’ p. 199, thus fusing validity and truth. Aristotle’s conflation of these, inelegant as it is, is historically understandable. Not so Patzig’s fusion of the two. 29. The mystic interpretation of the Parmenides has been the dominant one until about a century ago. E.R. Dodds 1928 pp. 129–142 is the classic (still irreplaceable) brief study of the sources of this tradition. The first explicit source of this reading of Parmenides is Plotinus’ Enneads V. 1 (translated by S. MacKenna, 1992) and the best known is the extensive Commentary of Proclus on Plato’s Parmenides (translated by G. Morrow and M. Dillon, 1987). 30. F. M. Cornford 1933 maintains that the Parmenides is a sober and on the whole serious criticism of the ambiguous notions Unity and Being, as they were (illegitimately) used by the Eleatics (and almost everybody else at the time). 31. G. Ryle 1971 pp. 1–44 argues that the chief aim of Parmenides is self-inspection. Plato, he claims, is exposing here certain idiosyncrasies in his own use of the terms ‘Being’ and ‘Unity’, thus opening to criticism his own theory of Forms. Possibly, he adds, Plato did not know how to answer satisfactorily all the criticism that he considered. See also ibid., pp. 45–53, which is a careful criticism of Cornford 1933 as well as a defense of his own reading. It has many merits, yet (as he was the first to admit) he could not explain why Plato would chose to put clearly devastating criticism of his own theory into the mouth of Parmenides, all the more since Ryle insists that the dialogue has little to do with Parmenidean Monism (p. 53). 32. A. E. Taylor 1986 [1926] pp. 349–370. 33. W. Jaeger 1962 [1934] pp. 14–16. It is still the standard view that Aristotle entered the Academy at the time Plato wrote Parmenides. (See Guthrie 1962–1981, Vol. VI, pp. 22–23.) For a different view see G. Ryle 1971 (p. 124). Ryle suggests that Plato had composed the second part of Parmenides for the pedagogic benefit of Aristotle’s students in the Academy. Aristotle entered the Academy as a youth (at the age of 17 or 18), around 366–367 BC., and stayed there for about 20 years. It is, of course, unreasonable that he had started teaching at the academy upon entering it. These conjectures are of course highly speculative, as Guthrie stresses (ibid., p. 24 n. 1): the speculation that Aristotle has taught logic at all in the Academy is, indeed, Jaeger’s, since the only fairly reliable evidence that we have is that he had taught a class of rhetoric there, opposing the ideas of Isocrates. 34. For the classic studies of Aristotle’s criticism of Platonic dialectics see J. Stenzel [1940] 1973. Especially 75–85 and of course also H. Cherniss 1944 pp. 1–64.

Notes

149

35. Here are a few incidental remarks for the benefit of the curious but less-informed reader regarding the corpus of Aristotle’s writings on logic, the Organon. It comprises the following works: The Categories, On Interpretation, The Topics, On Sophistical Refutations, Prior Analytics and Posterior Analytics. As far as we know, it was neither compiled by Aristotle nor given by him that title (Organon). The works are edited, and our knowledge of the philosophical agenda of his editors is insufficient. This is a serious deficiency: we do not even know who was (or were) the first to append to it it’s, now ubiquitous, title (Organon). The very word “Logic” is notoriously absent in it (Aristotle uses, rather, the word Analytica). The first to use the word Logic, as far as we know, was Alexander of Aphrodisia, and about 500 years separate these two great thinkers. We do not know if the Organon was designed to be a consistent system or rather reflects distinct, incompatible, phases in Aristotle’s development. Today, following the seminal studies of T. Case (‘Aristotle’, the 11th edition of the Encyclopedia Britannica, Vol. II, pp. 501–522) and W. Jaeger ([1923] 1962), it is generally accepted that the Organon comprises works written over a considerable time span, reflect considerable intellectual development, blurred by Aristotle’s editors in their endeavor to present it as a complete, exhaustive and consistent system. As M. Grene 1963 noted (pp. 15–37) those who maintain that the Organon was designed and written as a system find it difficult to explain its many inconsistencies. However, she adds, those who view the Organon as the result of Aristotle’s gradual intellectual development have produced an interpretative tool that is often too flexible to be helpful. It suffers from “inherent arbitrariness” since “instead of leading the student to do his best to understand the philosopher in question, it makes him try his outmost to dissolve the thinker he is studying into many opposing thinkers and finally to legislate him out of existence altogether” (ibid., p. 29). Aristotle, then, seems to have revised his works repeatedly, especially the works that were not published until late in his life (and even more so the ones not published during his life time). Here is one, well known example to a conjecture about revisions: it is generally believed today that books II-VII of the Topics are considerably earlier than books I and VIII. It is generally agreed that books II-VII were written even before the discovery of the syllogism, whereas books I and VIII were written afterwards. It is stipulated that Aristotle, having discovered the syllogism, felt the need to tinker with his earlier work, revise and update it, adding to it a new introduction and a summary. All this pertains to my present discussion only marginally: my concern is with the impact of Aristotle through the ages, and that Aristotle is composite. 36. The observation that Aristotle’s logic was designed to guarantee the attainment of scientific knowledge, and thus to answer the sophist challenge to the possibility of knowledge is still surprisingly under dispute, perhaps because it amounts to an open and explicit admission that Aristotle could not make a clear-cut distinction between the search for the purely formal (logic) and the search for informative truth (science). Some modern admirers of Aristotle take it for granted that he had made this distinction clearly and endorsed it. Others invent new terms so as to avoid the issue. Notable among these is J. Corcoran (2003 p. 286). By contrast, the terrific M. Grene writes (1963 p. 69): “We may, therefore, legitimately consider Aristotelian logic not as the first adumbration of a formal system but as a discipline enabling the student to acquire scientific knowledge”. Later on (ibid., p. 71) she adds: “Aristotle’s logic is not a pure logic, a system valid for ‘all possible worlds’, like the formal systems envisaged by Leibniz.” In between stands W.K.C. Guthrie, who writes cautiously (1962–1981, Vol. VI, pp. 135–136), observing that Aristotle’s logic “is close to what is meant today by ‘scientific method’ where the word ‘scientific’ is used in its proper, all embracing sense”. What is this “proper, all embracing sense”? It is not easy to answer this question if only because the matter is under dispute even today. Perhaps oddly, Guthrie does not answer the question, but rather cites others who have. He cites G.R.G. Mure 1964 (p. 211, n. 2.) who put it rather bluntly: “Aristotle never teaches a logic of mere validity” and the passage of Grene just cited. 37. “Someone” can also be understood as a quantifier, of course, especially when the context overlooks the question of the existence of a list of suspects, as when the little bear in goldilocks says “someone ate my porridge”.

150

Notes

38. The famous list of Aristotelian categories comprises 10 items. Catt. 1b25–30, and Top. 103b21–23). The list seems to have been composed for reasons best described in An. Pr. 49.7–10, where Aristotle says that “the statement that X is true of Y must be understood in as many different senses as there are distinct categories.” This is a direct allusion to the Eleatic paradoxes of Being and an attempt to solve them by distinguishing the different senses of “is”. Burnet, Ross and Guthrie note that the list may have been “developed in the Academy and only taken over by Aristotle” (Guthrie, 1962–1981, Vol. VI, p. 141 and n. 2). Notoriously (and, if I may say so, typically) the number of categories is not fixed in the Aristotelian corpus. For example, in Phys. 225b5 Aristotle names only eight categories and elsewhere gives (deliberately partial?) lists of six or even five categories. This only amplifies the uncertainty regarding the problem of recognizing and delineating legitimate terms in Aristotle. 39. Aristotle’s ejected singular terms from the province of logic throughout his logical career beginning with his discussion of predicable terms – Catt. (2b28–34, 3a35–36, etc.) – and ending in a concise form in his An. Pr. (i. 27. 43a25–43). Lukasiewicz [1951] 1972 criticizes this as somewhat misleading and capricious, as it is clearly inconsistent with his (Aristotle’s) examples, as he uses singular terms in syllogisms even in the Prior Analytics (p. 6). But Lukasiewicz himself concludes that it is logically sustainable (p. 7), so long as Aristotle could justify his claim to be able to tell the singular from the universal. Since modern logic allows the arbitrary division of sets of terms, he concludes that Aristotle’s distinction is sustainable. This, I contend, is one of the very few anachronistic passages in his outstanding study of Aristotle’s syllogism; for here Lukasiewicz uses the modern (extensional) license to simplify Aristotle’s highly complex, metaphysical problem situation, which allows no arbitrariness. Patzig 1968 (pp. 4–8) too notes that Aristotle uses singular terms within syllogisms (An. Pr. B 27 70a16–20) despite his being “obviously inclined to exclude them” (p. 5). However, unlike Lukasiewicz he attempts to defend Aristotle’s distinction by formulating axioms for him that would render his inclination to include singular terms logically warrantable (p. 7). Ross justifies Aristotle’s outlawing of singular terms within syllogisms by the observation that science ignores the particular. This is not so in astronomy. Indeed, a rarely noted example is Aristotle’s famous explanation of the lunar eclipse in the Posterior Analytics. (Book II, chapter 2). 40. Aristotle’s view that logic does not deal with hypothetical and non-existent entities is confusingly similar but far from being identical to modern questions regarding the theory of meaning as reference. Both Frege and Russell, for example, clearly admitted in their logic only names that have reference. But there are crucial differences here. Aristotle did not even deem all terms with references legitimate (for example, “All things that are not-Socrates”). Rather, he deemed only some terms with reference legitimate, those successfully depicting Aristotelian essences. Other terms cannot make up an essential definition. So, for example, the term “NotSocrates” is certainly legitimate from a Fregean point of view, but highly problematic from an Aristotelian point of view, for there is no essence that is common to all those humans, brutes, oaks, fossils, prime numbers, etc., that are Not-Socrates, it simply does not makes sense to attempt and encompass them all by means of some essential definition. 41. Aristotle’s attempts to determine the exact status of the complementary class are scattered throughout the Organon. For example, Catt. 3b24–30, De Int. 16a30–16b, and again throughout chapter 10 there, especially 20a31–36. His intuition as a grammarian clash with his intuition as a metaphysician: as a grammarian he clearly accepts complementary terms, yet as a metaphysician he clings to his grand plan of describing the grand matrix of being solely by means of terms depicting genera and species and no complementary terms. Thus, what he says of them dawdles between grammatical tolerance to all terms and metaphysical intolerance. 42. Patzig 1968 pp. 193–194 says, “Aristotle’s syllogistics is the theory of the relative products of … binary relations between terms.… Aristotle’s logic is thus a special part of the logic of binary relations.” This is a false and anachronistic statement: no list of terms can satisfy Aristotle’s restrictions without thereby disallowing the most basic binary relations. It is also difficult to square this declaration with Patzig’s own presentation (in the first chapter to his book) of several limitations that Aristotle had imposed on possible terms.

Notes

151

43. Aristotle’s terminology is profoundly indebted to the analogy between grammar and anatomy. As D. J. Allen [1952] 1978 (p. 102) observes, such terms as syndesmos (conjunction) and arthron (article) are deliberate borrowing from anatomy where they mean ‘ligature’ and ‘joint’. 44. To the extent that it can be formulated in our modern terminology, Plato’s and Aristotle’s aim was that of determining truth-conditions for terms. They were trying to separate the right terms from the wrong terms. This point is very confusing since the very notion of truth-conditions for a term is absent from modern logic. Today we deem only sentences (or propositions) true or false. Today we take the notion of a truth-condition for a term as involved with hopeless confusions. Tarski has clarified all that in his famous study of the semantic notion of truth when he sharpened the distinction between the notion of truth and the notion of satisfaction. My assertion, that Aristotle’s aim was to determine truth-conditions for terms, may elicit surprise among some Aristotle experts, since it is well known that he explicitly denied that simple terms (like “man” or “white”) can be true or false (Catt. 2a5–7). But I nevertheless insist that despite this admirable observation Aristotle indirectly violates it repeatedly when he insists that some terms (names of essences) can be rightly predicated of some entities while other terms (just as befitting, from the extensional point of view of modern logic) cannot. It is not at all surprising that a person who is well aware of a certain distinction in some context will be unaware if it in other context. 45. Stenzel [1940] 1973 has shown that this seemingly paradoxical analogy is not odd at all. He says: “In order to grasp the peculiar relation between Plato and Democritus, we should start from the point that they were both Atomists. Plato takes over from Democritus his principle of a division repeated until the ‘atomic form’ is reached.… Everything in Plato’s writings on this subject which remains in the twilight of figurative expression finds its explanation through this parallel” p. 158. See also p. 163 46. This explains Lukasiewicz frustration with Aristotle’s conflation of semantics and ontology. See J. Lukasiewicz [1951] 1972 p. 6. To this W.K.C. Guthrie 1962–1981, Vol. VI adds, “If this means that Aristotle when using words had in mind their meaning rather than treating them as symbols with no more content than x or y, the so-called confusion was essential to his philosophy. His indifference to the distinction appears in his use of the expressions ‘predicated of’ and ‘present in’ a subject. What is predicated, according to Lukasiewicz (p. 6), is a term, but what is in something must be the attribute expressed by the term”. (ibid., p. 139, n. 1). Let me add that by Aristotle “the attribute expressed by the term” is not merely meaning but also substance, not merely a part of semantics proper but a part of ontology proper. M. Grene stresses this (1963 pp. 72–73) Consequently, the “so-called confusion” which Guthrie says “is essential” to Aristotle, is not merely the confusion of terms and their meanings, but also the conflation of terms and things (Lukasiewicz and Grene). 47. Stenzel [1940] 1973 pp. 157–164 is less impressed than myself by the fact that Plato deliberately avoids the mentioning of Democritus. He says: “Plato, far from ‘avoiding a contest with the best of philosophers’ (Diogenes, 1c), was very deeply and clearly conscious that it was his duty to confront his brilliant predecessor before his own interpretation of nature could claim to be established. It will be agreed by those who know the conventions of ancient literature that the avoidance of any overt mention of a man is anything but a sign of pure negation and conflict”. 48. My favorite study of the sources about Antisthenes is L. E. Navia’s touching Antisthenes of Athens: Setting the World Aright 2001. I hereby recommend it to you wholeheartedly. 49. W. Jaeger 1962 p. 17 (n. 1). 50. M. Grene’s formulation of this dilemma is unmatchable: “Knowledge is of the real and universal; but whatever is real is not universal; therefore if knowledge is of the real it is not of the universal, if of the universal, not of the real. Yet Knowledge by definition must be both. Here is a contradiction indeed!” (M. Grene 1963 p. 24) 51. In Aristotle there are not only different senses of existence (that is, existence of particulars on the one hand and existence of universals on the other hand) but also different senses by which a universal can be said to apply to a particular: it can be said to be ‘predicated of’ a particular

152

Notes

and/or be ‘present in’ one. Catt. (1a 20–b8). For the sake of accuracy, then, it should be noted that in Aristotle there are various senses of existence even for universals. 52. Modern logic rarely discusses empty domains, of course, and it allows a derivation of existential sentences from universal ones only through the mediation of a non-empty proper name. This, of course, is not to say that modern logic endorses Aristotle’s existential import. 53. Already William and Martha Kneale (1962 pp. 58–61) have noted this prevalent tendency and criticized it. They note “It has therefore been suggested that for Aristotle, and perhaps also for the ordinary user of a natural language, universal statements always have existential import…this suggestion has the merit of providing a justification for what Aristotle says about subalternation and contrariety, but it does not agree with his account of contradictory opposition” (p. 58). 54. Jonathan Barnes tersely sums up the point: “Aristotle” he says, “urges that being is predicated of everything…It is not clear how Aristotle would reconcile these views with his remark that goat stags do not exist” (1975 p. 206). 55. See the fantastic Agassi 1963 pp. 54–67. 56. As elsewhere in this monograph, I offer an epigrammatically brief account of the Aristotelian views that shaped traditional logic. In this case it is the view of the unity of taxonomy and explanation by syllogism. I deliberately avoid entering the finer (and by no means uninteresting) Aristotelian distinctions regarding explanations, and in particular the distinction between ‘Knowledge why’ and ‘Knowledge that’). This is neither an attempt to belittle the importance of the distinction nor that of the interesting literature discussing it. From the narrow point of view of my study, the distinction is simply of little relevance, for I simply observe here that a thorough explanation, in Aristotle, is identical with proper classification by syllogism. 57. That Aristotle used horos at times in the sense of ‘real definition’ and at times in the sense of ‘term’ is a simple uncontestable textual fact. Some, including myself, argue that it is also a significant fact, as it is indicative of Aristotle’s conflation of logic and science. Lukasiewicz [1951] 1972 attempts to defend Aristotle’s intriguing (and clearly suspicious) equivocal use, by insisting that Aristotle never actually conflated the two senses of horos (pp. 3–4, n. 6). Lukasiewicz notes that distinguished scholars as Carl Prantl have ascribed the conflation of the two senses of horos to Aristotle. He censures them severely for causing a “disastrous confusion” (ibid., p. 4). As I explain above I think Prantl was right on the mark. I also think that Lukasiewicz attack is misdirected. Prantl does not maintain that a mere semantical conflation of two senses of a word (horos) had led Aristotle to a grand scale metaphysical conflation of logic and science. Rather, he claims the exact opposite: that Aristotle’s grand scale conflation of logic and science is demonstrated even by his terminological heedlessness. Lukasiewicz is here conflating an idealization of the Aristotelian text with its description. 58. W.K.C. Guthrie 1962–81, Vol. VI, p. 176. “Real definition is thus a kind of potted apodeixis, which packs in the middle term along with the major and the minor in the same sentence”. 59. Dialectics has a distinct role in the processes of recognizing ‘first principles’. This is made explicitly clear in the opening to the Topics (I. i, 101a35–101b–5). Since it is usually assumed that book I of the Topics is a relatively late addition, an introduction of sorts to the original version (books II–VII), it is not easy to dismiss what Aristotle says here as an early, Platonic, slip of the tongue. Guthrie minimizes the importance of this declaration because it entails Aristotle’s view that induction and dialectics are isomorphic. Since the isomorphism claim is false, Guthrie avoids ascribing it to the great master. I inspect and explain this interpretation in detail in the chapter above. (See also the next three notes). 60. Top. xii, 105a10–12, and elsewhere. It is interesting to observe that within the rhetorical context, the overlap of induction and dialectics is sometimes also manifest by Aristotle’s repeated application of parallel descriptions to them. He uses similar words to characterize both methods. This certainly creates the impression that they are interchangeable. He says, for example, that induction is the method which affects the inexperienced multitudes most, and yet also that dialectics is the method which affects the inexperienced multitudes most. This is no inconsistency on his part. The parallelism as we will see in the above chapter has its roots in Aristotle’s’ understanding of the Socratic method. Some would claim, perhaps, that the overlap between

Notes

153

induction and dialectics in Aristotle is no more than candid terminological untidiness. I do not believe this to be the case, for the reasons provided in the chapter above: it is certainly rare to find terminological untidiness in Aristotle that is genuinely candid, in the sense that it does not reflect some objective, which is not easily achieved by tidier terminology. 61. An. Pr.24a23–24 b 13. A more or less similar observation opens the Topics see Top. 100a25– 100b 24. The point has already been stressed and explained, but let me repeat it here nonetheless. An isomorphism proof between two structures presupposes that they are rigid. Aristotle’s “syllogism” refers only at times to rigid structures, however. One would expect him to have said that dialectic and apodeictic inferences use the same form, one as a refutation and one as a sound argument. Instead he says that they use the same form, one as allegedly sound argument from accepted premises, and one from admittedly sound argument from accepted premises. This makes the only difference between dialectic and apodeictic inferences not formal but circumstantial. Nevertheless, modern scholars frequently repeat that distinction as if it makes sense. It does not. Incidentally, even the Posterior Analytics opens with an observation similar to the two mentioned above: An. Post. 71a4–9. Here, however, the isomorphism observed is between apodeictic inferences and induction, and not between apodeictic and dialectic inferences. This may very well be yet another result of the overlap between induction and dialectics. 62. Zeller 1889 p. 182. I deliberately appeal to the 19th century giant Zeller and not to more upto-date scholars. Modern philosophy and logic have shown us unequivocally that induction has nothing to do with deduction. Zeller, then, demonstrates how natural it was to read Aristotle as one who conflated dialectics and induction, before modern logic had taught us that this is a rather basic error. Since the basic lesson was learnt, it had become unfashionable to ascribe the conflation, now recognized as a basic error, to the greatest logician ever. Zeller was unaware of all this, of course. Consequently, he provides us with a straightforward reading of Aristotle, one that is more truthful, accurate and laudable. 63. See, for example, Met. xiii. 4. 1078b 27 for a clear case of this ascription: “…; for two things may be fairly ascribed to Socrates – inductive arguments and universal definition, both of which are concerned with the starting point of science.” Such ascriptions are scattered throughout the Aristotelian corpus. See also Met i. 6. 987b 1. And PA. i. 1. 642a 28, where Aristotle states his view that Socrates started off on the right track leading to his (Aristotle’s) method of essential definitions by induction and that it was Plato and his disciples who took on a misleading sidetrack, off the Socratic-Aristotelian way to truth, with their unnecessary admittance of the independent existence of essences and their ineffective method of diaeresis. 64. To make clear that this is the heart of his method of philosophical quest, “Socrates” adds a somewhat personal confession: “… and if I think any other man is able to see things that can naturally be collected into one and divided into many, him I follow after and ‘walk in his footsteps as if he were a god’. And whether the name I give to those who can do this is right or wrong, God knows, but I have called them hitherto dialecticians” (266b). Here, typically, Plato confesses his admiration for Socrates by means of Socrates’ own confession. 65. The ideas which I explain here can be easily traced back to Aristotle’s writings as I will demonstrate soon. However, it is should be noted that for the sake of historical accuracy that I use my own refined terminology before discussing his interesting formulations. The most noteworthy classic discussion of Analysis and Synthesis is Pappus’ Mathematical Collection, Book VII. In what follows I do not follow Pappus’ definitions. 66. An. Post. I, iii, 72b 30–32: “for the same things cannot be at once prior and posterior to the same things, except in different senses, – I mean the distinction between “prior to us” and “absolutely prior” – with which we become familiar through induction”. See also 71 b 33–72 a 6. 67. This, generally, is the notion of induction prevailing in the Topics. For example, Top. I, xii, 105a15–18: “Induction is the progress from particulars to universals; for example, if the skilled pilot is the best pilot and the skilled charioteer the best charioteer, then, in general, the skilled man is the best man in any particular sphere.”

154

Notes

68. Within the pre-Boolean British tradition, both Whately and Aldrich resort to the curious and misleadingly simple operation of Abstraction and endeavor to distinguish it from that of generalization. See, for example, Whately 1826 [1859] pp. 54–56. The fact merits our attention, and although it properly belongs to the ending chapters of this book, let me mention it here for the sake of clarity. Whately and Aldrich where in two minds about the methodological tradition to which they belonged. On the one hand they denied that induction and deduction are symmetrical, and indeed that induction is at all a legitimate part of logic. However, on the other hand, they endeavored to retain the essentialist world-view which attempted to secure the status of logic as the foundation of empirical science. This meant for them that at least some version of induction still had to remain symmetrical to deduction, or else there would be no logical basis for empirical judgments. Consequently they have found it imperative to distinguish between similar but (allegedly) fundamentally distinct faculties: abstraction on the one hand and generalization on the other. Abstraction, they said, can only occur as a result of studying a single object (e.g. stripping redness off a single rose). It is a legitimate logical process, they assure us, since no generalization is involved in the process. Any observation of a rose, they said, presupposes its redness. Generalization, on the other hand, is inductive, they said. It requires the comparison of several objects. For example, the aggregate ‘red’ may arise as a result of the observation of a bouquet of red roses. The distinction, then, attempts to delineate and expel the extra-logical inductive generalization “all roses are red” out of the province of legitimate epistemological moves, while insuring us that the concept ‘red’ is presupposed by the concept ‘rose’.… The exercise is unsuccessful, of course, since it is un-reflectively metaphysical: unless we decide in advance what to regard as a particular object and what to regard as a complex one, we cannot tell an abstraction from a generalization. Consider, for example, a red patch. Someone may observe that since it is red it is also colored. Is this a case of abstraction or a result of a generalization? If it is an abstraction, then the inference “this is a red patch, hence it is colored” should be logically valid. (As even Wittgenstein seemed to have suggested in his highly intriguing essay on logical form). If it is a generalization, however, it is logically unwarranted and the sentence “All red patches are colored” becomes a contingent conjecture. Clearly the problem is not in the question regarding the logical status of the sentence. It is with the confusion of the empirical and the logical that it allows. To assume that we know which objects are particulars and which are not, and hence, which abstractions are also generalizations and which are not, is, by any standard, to assume quite a lot. Not so our distinction between the logical and the extra-logical, for it is, in most cases, rather trivial. 69. An. Post. II, x, 93b29–36 70. An. Post II, x, 94a1–10. The unique and idiosyncratic term “quasi-demonstration” may seem perplexing but is in fact pretty straightforward and helpful: real definition is merely quasidemonstration of an essence and not full demonstration thereof. This is so because a definition is not a syllogism (and because only syllogisms are (full) demonstrations). The definition expresses a very similar content to that of the syllogism while remaining a mere definition. The problem, of course, is not with the notion of quasi-demonstration but rather with the very claim that essences can at all be demonstrated, even if quasily. 71. A recent brief and authoritative survey of Stoic logic is S. Bobzien (in B. Inwood 2003 pp. 85–123). The classic and, in my mind, still irreplaceable is Mates [1953] 1961. 72. Epictetus, Discourses, Book II, chapter xix. 73. A. Coffa. 1991 pp. 7–21. 74. All relevant references to Frege’s curious and persistent insistence to disallow mere valid inferences within his logic have been ably collected and masterfully commented upon by G. P. Baker and P. M. S. Hacker in their recommendable 1984, pp. 37–39. Of particular interest are Frege’s (in retrospect almost bizarre) letters to H. Dingler and to P.E. B. Jourdain. The crux of the matter is this: Frege acknowledged only judgments into his logic and not mere (possibly false) propositions or sentences. Consequently, he acknowledged only sound inferences into his logic and not mere valid ones. Yet, as Jourdain and Dingler point out, acknowledging valid inferences is crucial for a complete account of logic. For example: proofs by

Notes

155

reductio ad absurdum are never sound inferences, for they are based on the falsity of their premise; they must be valid, of course, but their premises are manifestly false. Frege denies this, however and offers a bypass: he simply denies that inferences with false premises are a part of logic. Instead he acknowledges reductio ad absurdum into his system only as a conditional (with false antecedent) and not as the corresponding valid, yet unsound, inference. 75. Leibniz may have come closest to proposing a system of logic that can be viewed as semi-formal in his 1686 Generales inquisitions de analysi notionum et veritatum (G.H.R. Parkinson 1966 pp. 47–87 is the standard English translation). T. Hailperin has offered an analysis of that attempt and a comparison of it with Boole’s system (J. Gasser 2000 pp. 129–138). This anachronistically clear comparison is unacceptable, even if right. Its conclusion is that Leibniz “carefully constructed abstract formal system” whereas Boole offered “No actual formal system” (ibid., p. 137). Hailperin deliberately ignores the epistemological and metaphysical limitations of Leibniz’s system. As he says “Much of it [Leibniz’s aforementioned text] is concerned with analysis of concepts and of truths, and to matters of a grammatical, philosophical or metaphysical nature. We shall be ignoring these as not being germane to our topic.” (ibid., 130). Let us grant him that. The fact that in-retrospect-germane matters have shaped Leibniz’s logical apparatus, defined its limits and, were deemed by Leibniz the gist of his logic is, thus, completely missed, and quite possibly misunderstood by Hailperin. Let us follow Hailperin nonetheless. What remains for Hailperin to do is to cite a text from Leibniz and explain why he thinks that it is a complete formal system. Remember that for a system to be formal all that is required is the possibility to ignore all possible interpretations of the system. Therefore, when Hailperin ignores all questions of interpretation of Leibniz and rewords his assertion in a modern formal manner, he has already succeeded in accomplishing his mission without any argument. The question is what can be gained from such an easy exercise. Let me give an example. On page136 Hailperin mentions that whereas Boole clearly recognized the universal class and the empty class, Leibniz had great difficulties to do so. Why? Hailperin blocks this question by his dismissal of it as “germane” in the passage quoted above. It is clear that if we have to remember of every class that it is neither empty nor the universe then the system can hardly be a complete formal system. (We can, of course, rectify this situation by adding an appropriate axiom, but, significantly, Hailperin did not take the bother to do so, and for a good reason: he couldn’t find it in Leibniz, of course.) It is more accurate and more appropriate, then, to save the title “formal” to systems that were conceived, designed and successfully crafted as such, and not those that can, in retrospect, be explicated as such. 76. R. Ariew and D. Garber 1989 p. 239. 77. It is rare to find anyone so conscious of the value of his innovations, despite their trite appearance to many who surround him. Let me quote here in full the excerpt that I quoted above (taken from R. Ariew and D. Garber 1989 p. 239): “There is a way of avoiding error which these able men have not condescended to use; it would have been contrary to the greatness of their minds, at least in appearance, and with respect to the common people. All those who wish to appear to be great figures and who set themselves up as leaders of sects have a bit of the acrobat in them. A tightrope walker does not allow himself to be braced in order to avoid falling; if he did so, he would be sure of his act, but he would no longer appear a skillful man. I will be asked, what then is this wonderful way that can prevent us from falling? I am almost afraid to say it – it appears too lowly. But I am speaking to Your Highness who does not judge things by their appearance. In brief, it is to construct arguments only in proper form. I seem to see only people who cry out against me and who send me back to school. But I beg them to be a little patient, for perhaps they do not understand me; arguments in proper form do not always bear the stamp of Barbara Celarent. Any rigorous demonstration that does not omit anything necessary for the force of reasoning is of this kind, and I dare say that the account of an accountant and calculation of analysis are arguments in proper form, since there is nothing missing in them and since the form of the arrangement of the whole reasoning is the cause of their being evident”. 78. For some of the more interesting examples see R. Ariew and D. Garber (eds. and trans.) 1989, pp. 10–22.

156

Notes

79. R. Ariew and D. Garber 1989 p. 238. 80. Understanding Hume’s critique of induction as entailing the bankruptcy of Leibniz’ project involves some injustice to Hume, since it facilitated his not entirely just portrayal as a skeptic. Hume, let me stress an obvious and yet often ignored fact, was as ardent an advocate of Newton’s mechanics as Kant was: he never doubted that it is an expression of the absolute and final scientific truth. Kant should have stressed that this may well be an inconsistency on Hume’s part. Instead he chose to ignore Hume’s clear admiration for science, thereby commemorating him unjustly as the great skeptic he never claimed to be. 81. Here are Kant’s words: “I had half decided to send the manuscript back immediately, with the aforementioned, totally adequate apology. But one glance at the work made me realize its excellence and that not only had none of my critics understood me and the main questions as well as Herr Maimon does but also very few men posses so much acumen for such deep investigations as he …” Letter to Marcus Herz, May 26, 1789 in: The Cambridge Edition of the works of Immanuel Kant: Correspondence, pp. 311–312. 82. Kieswetter’s report of the meeting is found in his letter of December 15, 1789, ibid., pp. 327– 328. Kant’s reply is in his letter of February 9, 1790, ibid., 336. 83. Ibid., p. 476, Letter of March 28, 1794. 84. Ibid., p. 595. 85. There are many fine studies of the history of mathematics in general, and of that part of it which is relevant to the history of early modern logic in particular. There are very few studies of extensionalism. Usually extensionalism is presented, if at all, as identical with formalism. Alas, formalism is a much broader concept and one which glosses over the problems specific to extensionalism, that are more distinctly philosophical than those related to formalism in general, especially if one takes formalism as its father (Hilbert) did, not as a comprehensive philosophy but as a technique, particularly one for checking consistency. As for philosophical studies of extensionalism, let me mention here as antecedents to my own study the marvelous but outdated C.I. Lewis [1918] 1960 and the interesting but muddled C.I. Lewis and C.H. Langford [1932] 1959. I should also mention here the early volumes of M. Bunge’s Treatise on Basic Philosophy (1974 and 1977), which though enlightening are unfortunately not reader-friendly. As for general readings in the history of formal mathematics, let me first say that there is very little of it. I. Grattan-Guinness 1994 is a standard recent source book. I shall only mention here the classic A. Robbinson 1965 (“Formalization 64”), and N. Bourbaki [1984] 1994. Moving into the narrow field of Boole scholarship, it is impossible not to mention D. MacHale, 1985, the only monograph in existence dedicated to Boole’s life and thought. It contains basic summary of the mathematical background to Boole’s discoveries as well as some interesting anecdotes. J. Gasser 2000 is the only anthology of studies on Boole. The articles by L. M. Laita (pp. 45–59) and M. Panteky (pp. 167–212) examine the mathematical background to Boole’s logic. Especially recommendable is M.-J. DurandRichard’s paper there: it discusses some interesting historical background facts to Boole’s achievements which are not discussed here. There is, let me add, no substitute for readings Boole’s original first essay on the topic The Mathematical Analysis of Logic (1847). There are many more rather obvious references which, I am sure, any reader would come across having performed the most basic bibliographic search. Many have listed them before me and I see no point in repeating such lists. Instead, let me express here my grateful debt to some less known studies, which have influenced my understanding of the history of extensionalism and are, sadly, mostly ignored by the current establishment: P.E.B. Jourdain 1910, 1912, and 1913, on the early development of modern logic, J.O. Wisdom 1947 is a difficult but rewarding reading, and let me also mention here the lucid and profound A.E. Musgrave 1972. I am greatly indebted to the logical works of K.R. Popper (all of which are discussed in Lejewski 1974), and to the studies of J. Agassi, especially his brilliant unpublished work on the logical background to the philosophy of Wittgenstein, and his outstanding 1978 (“Logic and Logic of”). 86. There are three noteworthy technical improvements to Boole’s Algebra, all are discussed by C. I. Lewis [1918] 1960 pp. 54–78. Stanley Jevons is responsible for the most significant of improvements: he suggested using the inclusive “either/or” interpretation to Boole’s “+”,

Notes

157

rather than Boole’s “Either or, not both”. This is not merely a technical change: Boole’s “+” was highly problematic on a number of accounts as it left certain terms un-interpretable (most notably “x + x”; Boole refused to admit it to be “=0”, and hence left it as un-interpretable). As a direct result of this Jevons could formulate the well known “index law” for “+”: “x + x = x”, so basic to Boolean algebra ever since. Another important result of this improvement was that it enabled and facilitated the use of De Morgan’s famous substitution rules, which did not apply neatly to Boole’s original system. The Boole-Jevons correspondence, a fascinating chapter in the history of logic, is discussed by P.E.B Jourdain 1913 and I. Grattan-Guinness 1990. Another important improvement to Boole’s system is the elimination of Boole’s problematic notion of division, a mathematical operation prevalent in his system despite having no “logical” interpretation. Finally, the much desired elimination of Boole’s highly problematic treatment of “Some”, a residue of the problems that originated with the Aristotelian copula, was indirectly but effectively achieved by replacing Boole’s equation sign with Peirce’s horseshoe. 87. Let me quote here N. Bourbaki [1984] 1994 p. 4: “Aristotle was still sufficiently informed of the mathematics of his era in order to have seen that schemas of this type [syllogisms] were not sufficient to take account of all the logical operations used by mathematicians, nor all the more, of the other applications of logic”. See also next note. 88. Early neo-Platonists are a well known exception. Since they conflated Aristotle and Plato with no sense of inconsistency, they labored to show that mathematical proofs can be cast into syllogistic form. They did this by means of ultra-subtle exegetical reconstructions of Aristotle’s basic logical notions. In a sense they destroyed the theory of the syllogism by forcing it to achieve what it could never achieve and what Aristotle realized could not be achieved: a satisfactory account of all mathematical proofs. Though the ingenuity of these exercises is hardly contested, it is clear that their results, in hindsight, are of little value, exactly because their aim was apologetic. They succeeded however, in creating a long lasting impression (still dominant amongst traditionalists’ circles at the 20th century) that the gist of Euclidian geometry is somehow faithfully captured by Aristotle’s theory of the syllogism (e.g. McKirahan 1992 pp. 133–163). 89. B. Russell 1917 (1963) p. 62: “Leibniz foresaw the science which Peano has perfected, and endeavored to create it. He was prevented from succeeding by respect for the authority of Aristotle, whom he could not believe guilty of definite, formal fallacies …” 90. A good example is in De Morgan 1966 pp. 50–65. Hamilton’s subtle objections to De Morgan’s system are also mentioned there and answered. 91. A. De Morgan, 1847 [1926] pp. 41–42. The classical concept of a sum of all chances is vague, and it is often stated that it may be different than 1 (See, for example, Bernoulli’s Ars Conjectandi Part four, chapter III). A particularly interesting collection of notes on the history of probability and its state at the first half of the 19th century is found in De Morgan 1830 92. A. De Morgan 1847 [1926] p. 64. 93. A. De Morgan 1860 pp. 37–42. 94. G. Boole 1847 p. 21. Earlier, yet on the same page, the sign “0” appears for the firs time: Boole formalizes the open sentence “All Xs are Ys” as xy = x and then immediately transform it into x(1 – y) = 0 (with no explanation, other than the trivial algebraic knowledge which he seems to assume that his readers share). As Kneale and Laita have noted this “0” seems to be closer to the algebraic zero than to anything like the class Nothing. 95. G. Boole 1847 p. 6

Bibliography

Ackrill, J.L. (trans. and comm.) Aristotle’s Categories and De Interpretatione, Oxford: Clarendon, 1963. Agassi, J. “Towards an Historiography of Science”, History and Theory Beiheft 2, 1963. Agassi, J. Towards a Rational Philosophical Anthropology, The Hague: Martinus Nijhoff, 1977. Agassi, J. “To Save Verisimilitude”, Mind XC, pp. 576–579, 1981. Agassi, J. “Logic and Logic of”, Poznan Studies 4, pp. 4–11, 1978. Agassi, J. “Presuppositions for Logic”, The Monist 65, pp. 465–480, 1982. Aldrich, H. Artis Logicae Rudimenta [1691] Reprint H.L. Mansel (ed.) Oxford: Henry Hammans, 1849. Allan, D.J. The Philosophy of Aristotle, Oxford: Oxford University Press, 1952; Reprint 1978. Anscombe, G.E.M. “Parmenides, Mystery and Contradiction”, Proceedings of the Aristotelian Society 69, pp. 125–132, 1968/9. Ariew, R., Garber, D. (eds. and trans.) Leibniz: Philosophical Essays, Cambridge, MA: Hackett, 1989. Aspray, W., Kitcher, P. (eds.) History and Philosophy of Modern Mathematics, Minneapolis: University of Minnesota Press, 1984. Austin, S. Parmenides. Being, Bounds and Logic, New Haven, CT: Yale University Press, 1986. Baker, G.P., Hacker, P.M.S. Frege: Logical Excavations, Oxford: Blackwell, 1984. Bar-Am, N. “Demarcation Problems in Linguistics”, Conceptus 84, pp. 23–34, 2001. Bar-Am, N. “A Framework for a Critical History of Logic”, Sudhoffs Archiv Band 87, Heft 1, pp. 80–89, 2003. Bar-Am, N. “Extensionalism and Induction in Boole”, Physis XLI, pp. 97–123, 2004. Barker, Sir E. Greek Political Theory, Plato and His Predecessors, London: Methuen, 1918. Barnes, J. (trans. and comm.) Aristotle’s Posterior Analytics, Oxford: Clarendon, 1975. Barnes, J. The Presocratic Philosophers, London: Routledge, 1979. Barnes, J. Early Greek Philosophy, Harmondsworth: Penguin, 1987. Barnes, J., Schofield, M. and Sorabji, R. (eds.) Articles on Aristotle, 4 vols. London: Duckworth, 1975–1979. Bell, E.T. The Development of Mathematics, New York: McGraw-Hill, 1940; Reprint 1945. Bett, R. (trans.) Sextus Empiricus, Against the Logicians, Cambridge: Cambridge University Press, 2005. Bochenski, I.M. A History of Formal Logic, Notre Dame, IN: University of Notre Dame Press, 1961. Bonitz, H. Index Aristotelicus, Berlin: Reimer, 1870. Boole, G. The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Cambridge: Macmillan, Barclay & Macmillan 1847; Reprint Bristol: Theommes Press 1998. Boole, G. An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities, London: Walton and Maberley, 1854.

159

160

Bibliography

Boole, G. Studies in Logic and Probability, R. Rhees (ed.) London: Watts & Co, 1952. Boole, G. George Boole. Selected Manuscripts on Logic and Its Philosophy, I. Grattan-Guinness and G. Bornet (eds.) Basel: Birkhauser, 1997. Borges, J.L. Collected Fictions, New York: Penguin, 1998. Borges, J.L. Selected Non-fictions, New York: Penguin, 1999. Bourbaki, N. Elements of the History of Mathematics, J. Meldrum (trans.) Berlin: Springer, 1994. Boyer, C.B. The History of the Calculus and its Conceptual Development, New York: Springer, 1959. Boyer, C.B. A History of Mathematics, New York: Wiley, 1968. Bunge, M. Treatise on Basic Philosophy, Vols. I and II, Dordrecht: Reidel 1974. Bunge, M. Treatise on Basic Philosophy, Vol. III, Dordrecht: Reidel 1977. Burnet, J. Early Greek Philosophy, London/Edinburgh: Adam and Charles Black, 1892. Burnet, J. “Law and Nature in Greek Ethics”, International Journal of Ethics, Vol. 8, 1897. Cherniss, H. Aristotle’s Criticism of Plato and the Academy, Baltimore MD: Johns HopkinsUniversity Press, 1944. Cherniss, H. The Riddle of the Early Academy, Berkeley LA: University of California Press, 1945. Coffa, J.A. The Semantic Tradition from Kant to Carnap, Cambridge: Cambridge University Press, 1991. Corcoran, J. (ed.) Ancient Logic and Its Modern Interpretations, Dordrecht/Boston, MA: Reidel, 1974. Corcoran, J. “The Founding of Logic”, Ancient Philosophy, 14, pp. 9–24, 1994. Corcoran, J. “Aristotle’s Prior Analytics and Boole’s Laws of Thought”, History and Philosophy of Logic 24, pp. 261–288, 2003. Cornford, F.M. Plato and Parmenides, London: Routledge, 1933. De Morgan, A. On Probability, London: Baldwin and Cardock, 1830. De Morgan, A. Formal Logic, London: Open Court, 1847; Reprint 1926. De Morgan, A. Syllabus of a Proposed System of Logic, London: Walton and Malbery, 1860. De Morgan, A. On the Syllogism and Other Logical Writings, P. Heath (ed.) New Haven, CT: Yale University Press, 1966. Dodds, E.R. “The Parmenides of Plato and the origin of the Neoplatonic One”, Classical Quarterly 22, pp. 129–142, 1928. Dodds, E.R. The Greeks and the Irrational, Berkeley: University of California Press, 1951. Finley, M.I. The Use and Abuse of History, London: Chatto and Windus, 1975; Reprint London: Penguin, 1990. Finley, M.I. The World of Odysseus, London: Chatto and Windus, 1954; Reprint (with revisions) London: Penguin, 1967. Forster, T. “Kant, Boole and Early Metaphysics”, Synthese 113, pp. 43–70, 1997. Forster, T. Logic Induction and Sets, Cambridge: Cambridge 2003. Frege, G. Posthumous Writings, Oxford: Blackwell, 1979. Frege, G. Philosophical and Mathematical Correspondence, Chicago, IL: The University of Chicago Press, 1980. Frege, G. Collected Papers on Mathematics, Logic, and Philosophy, Oxford: Blackwell, 1984. Furley, D. Cosmic Problems, Cambridge: Cambridge University Press, 1989. Gallop, D. Parmenides of Elea, Toronto, ON: University of Toronto Press, 1984. Gasser, J. A Boole Anthology, Dordrecht: Kluwer, 2000. Gottschalk, L. (ed.) Generalization in the Writing of History, Chicago, IL: University of Chicago Press, 1963. Grattan-Guinness, I. (ed.) From the Calculus to Set Theory: 1630–1910, London: Duckworth, 1980; Reprint Princeton, NJ: Princeton University Press, 2000. Grattan-Guinness, I. “Living Together and Living Apart. On the Interactions between Mathematics and Logics from the French Revolution to the First World War”, South African Journal of Philosophy 7(2), pp. 73–82, 1988.

Bibliography

161

Grattan-Guinness, I. “The Correspondence Between George Boole and Stanley Jevons, 1863–4”, History and Philosophy of Logic 12, pp. 15–35, 1990. Grattan-Guinness, I. (ed.) Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 2 vols. London: Routledge 1994. Grene, M. A Portrait of Aristotle, Chicago, IL: University of Chicago Press, 1963. Guthrie, W.K.C. A History of Greek Philosophy, 6 vols. Cambridge: Cambridge University Press, 1962–1981. Hacking, I. The Emergence of Probability: A philosophical study of Early Ideas About Probability, Induction and Statistical Inference, Cambridge: Cambridge University Press, 1975. Hailperin, T. Sentential Probability Logic: Origins, Development Current Status and Technical Applications. Cranbury, NJ: Associated University Press 1996. Hamilton, W. Lectures on Metaphysics and Logic, Boston, MA: Gould and Lincoln 1859. Hicks, R.D. (trans.) Diogenes Laertius, Lives of Eminent Philosophers, 2 vols. Cambridge, MA: Harvard University Press, 1926. Huntington, E.V. “Sets of Independent Postulates for the Algebra of Logic”, Transactions of the American Mathematical Society 5, pp. 288–309, 1904. Huntington, E.V. “New Sets of Independent Postulates for the Algebra of Logic with Special Reference to Whitehead and Russell’s Principia Mathematica”, Transactions of the American Mathematical Society 35, pp. 274–304, 1933. Inwood, B. (ed.) The Cambridge Companion to the Stoics, Cambridge: Cambridge University Press, 2003. Irwin, T.H. Aristotle’s First Principles, Oxford: Oxford University Press, 1988. Jaeger, W. Aristotle: Fundamentals of the History of His Development, Oxford: Oxford University Press, 1934; Reprint with revisions 1962 (First German edition: 1923). Jevons, S.W. Elementary Lessons in Logic: Deductive and Inductive, London: Macmillan 1870. Jevons, S.W. The Principles of Science: a Treatise on Logic and Scientific Method, 1874; Reprint New York: Dover, 1958. Jevons, S.W. Pure Logic and Other Minor Works, R. Adamson (ed.) New York: Franklin, 1971. Jourdain, P.E.B. “The Development of the Theories of Mathematical Logic and the Principles of Mathematics”, The Quarterly Journal of Pure and Applied Mathematics 41, pp. 324–352, 1910; 43, pp. 219–314, 1912; 44, pp. 113–128, 1913. Kahn, C.H. “Being in Parmenides and Plato”, La Parola del Passato 43, pp. 237–261, 1988. Kant, I, The Cambridge Edition of the Works of Immanuel Kant: Lectures on Logic, Vol. 9, J.M. Young (ed. and trans.) Cambridge: Cambridge University Press, 1992. Kant, I, The Cambridge Edition of the Works of Immanuel Kant: Opus postumum, Vol. 12, E. Foerster (ed. and trans.), M. Rosen (trans.) Cambridge: Cambridge University Press 1993. Kant, I, The Cambridge Edition of the Works of Immanuel Kant: Correspondence, Vol. 14, A. Zweig (ed. and trans.) Cambridge: Cambridge University Press, 1999. Kapp, E. Greek Foundations of Traditional Logic, New York: Columbia University Press, 1942. Kitcher P. “Mathematical Rigor – Who Needs it?” Nous 15, pp. 469–493, 1981. Kneale W. “Boole and the Revival of Logic”, Mind 57, pp. 149–175, 1948. Kneale W. and Kneale, M. The development of Logic, Oxford: Oxford University Press, 1962; Reprint (with revision) 1984. Lakatos, I. Proofs and Refutations: The Logic of Mathematic Discovery, Cambridge: Cambridge University Press, 1976; Reprint 1991. Lakatos, I., Musgrave, A. (eds.) Criticism and the Growth of Knowledge, Cambridge: Cambridge University Press, 1970; Reprint 1999. Lear, J. Aristotle and Logical Theory, Cambridge: Cambridge University Press, 1980. Lejewski, C. “Popper’s theory of formal or deductive inference”, in: The Philosophy of Karl Popper, La Salle, IL: Open Court, pp. 632–670, 1974. Leibniz, G.W. Leibnitii opera omnia, 6 vols., L. Dutens (ed.) [1768]; Reprint Hildesheim: Olms, 1989. Leibniz, G.W. Mathematische Schriften, 7 vols., C.I. Gerhardt (ed.) Berlin and Halle: A. Asher and H.W. Schmidt [1849–63]; Reprint Hildesheim: Olms, 1962.

162

Bibliography

Lewis, C.I. A Survey of Symbolic Logic, New York: Dover, 1918; Reprint 1960. Lewis, C.I., Langford, C.H. Symbolic Logic, New York: Dover, 1959. Lewis, F.A. Substance and Predication in Aristotle, Cambridge: Cambridge University Press, 1991. Loemker, L.E. (ed. and trans.) Leibniz: Philosophical Papers and Letters, Dordrecht: Reidel, 1969. Lukasiewicz, J. Aristotle’s Logic from the Standpoint of Modern Formal Logic, Oxford: Clarendon, 1951; Reprint (with revision) 1972. MacHale, D. George Boole: His Life and Work, Dublin: Boole, 1985. MacKenna, S. Plotinus: The Enneads, Burdett: Larson, 1992. Mates, B. Stoic Logic, Berkeley: University of California Press, 1953; Reprint 1973. McKirahan, R.D. Principles and Proofs, Princeton, NJ: Princeton University press, 1992. Miller, D. “Popper’s Qualitative Theory of Verisimilitude”, BJPS 25(2), pp. 166–177, 1974. Morrow, G. and Dillon, M. (trans. and comm.) Proclus’ Commentary on Plato’s Parmenides, Princeton NJ: Princeton University Press, 1987. Mourelatos, A.P.D. The Route of Parmenides, New Haven, CT: Yale University Press, 1970. Mourelatos, A.P.D. The Pres-Socratics, Garden City, NY: Doubleday, 1974; Reprint 1993. Mure, G.R.G. Aristotle, New York: Oxford University Press, 1964. Musgrave, A.E. “George Boole and Psychologism”, Scientia 107, pp. 593–608, 1972. Navia, L.E. Antisthenes of Athens: Setting the World Aright, Westport, CT/London: Greenwood, 2001. Owen, G.E.L. Logic Science and Dialectic, Ithaca, NY: Cornell University Press, 1986. Parkinson, G.H.R. (ed. and trans.) Leibniz: Logical Papers, Oxford: Oxford University Press, 1966. Parkinson, G.H.R. (ed. and trans.) Morris M. (trans.) Leibniz: Philosophical Writings, London: Dent, 1973. Patzig, G. Aristotle’s Theory of the syllogism, Dordrecht/Boston, MA: Reidel, 1968. Popper, K.R. The Open Society and its Enemies, London: Routledge, 1945; Reprint 1996. Popper, K.R. “New Foundations for Logic”, Mind 56, pp. 193–235, 1947. Popper, K.R. “Logic without Assumptions”, Proceedings of the Aristotelian Society 47, pp. 251– 292, 1947. Popper, K.R. “The Trivialization of Mathematical Logic”, Proceedings of the10th International Congress of Philosophy, pp. 722–727, Amsterdam: North Holland, 1949. Popper, K.R. Conjectures and Refutations: the Growth of Scientific Knowledge, London: Routledge, 1963; Reprint (with revision) 2002. Popper, K.R. “A Realist View of Logic, Physics and History”, in: Physics, Logic and History, W. Yourgrau and A.D. Breck (eds.) New York: Plenum, 1972. Popper, K.R. The World of Parmenides, London: Routledge, 1998. Rhees, R. “Boole as Student and Teacher”, Proceedings of the Royal Irish Academy 57, pp. 74–79, 1955. Robinson, A. “Formalism 64”, in: Logic Methodology and Philosophy of Science, Y. Bar-Hillel (ed.) Amsterdam: North-Holland, pp. 228–246, 1965. Robinson, R. Plato’s Earlier Dialectic, Oxford: Clarendon, 1953. Russell, B. A Critical Exposition of the Philosophy of Leibniz, London: Allen & Unwin [1900] 1937. Russell, B. Mysticism and Logic, London: Allen & Unwin [1917] 1963. Russell, B. Introduction to Mathematical Philosophy, London: Allen & Unwin, 1919. Russell, B. The Autobiography of Bertrand Russell, Vol. I: 1872–1914, Boston, MA: Atlantic Monthly Press [1951] 1967. Ryle, G. Collected Papers, Vol. I. London: Hutchinson, 1971. Schiappa, E. Protagoras and Logos, Columbia, SC: University of South Carolina Press, 2003. Schilpp, P.A. (ed.) The Philosophy of Bertrand Russell, Evanston: Northwestern University Press, 1944. Schilpp, P.A. (ed.) The Philosophy of Karl Popper, La Salle, IL: Open Court, 1974.

Bibliography

163

Segre, M. “Peano’s Axioms in their Historical Context”, Archive for the History of Exact Sciences 48, pp. 201–342, 1994. Sheffer, M.“A set of Five Independent Postulates for Boolean Algebra, With Application to Logical Constants”, Transactions of the American Mathematical Society 14, pp. 481–488, 1913. Smiley, T.J. “What is a syllogism?” Journal of Philosophical Logic 2, pp. 136–154, 1973. Smith, G.C. The Boole-De Morgan Correspondence: 1842–1864, Oxford: Clarendon, 1982. Sprague, R.K. (ed.) The Older Sophists, Columbia, SC: University of South Carolina Press, 1972. Stenzel, J. Plato’s Method of Dialectic, Oxford: Clarendon [1940] 1973. Stokes, M. One and Many in Presocratic Philosophy, Washington, DC: Center for Hellenic Studies, 1971. Strauss, L. Natural Right and History, Chicago, IL: University of Chicago Press, 1953. Styazhkin, N.I. History of Mathematical Logic from Leibniz to Peano, Cambridge, MA: MIT, 1969. Szabo, A. The Beginnings of Greek Mathematics, Dordrecht/Boston, MA: Reidel, 1978. Segre, M. “Peano’s Axioms in their Historical Context”, Archives for History of Exact Sciences 48, pp. 201–342, 1994. Tarski, A. Logic, Semantics, Metamathematics. Papers from 1923–1938, Oxford: Clarendon 1959. Taran, L. Parmenides, Princeton, NJ: Princeton University Press, 1965. Taylor, A. E. Plato the Man and His Work, London: Methuen, 1926; Reprint 1986. Taylor, G. “George Boole”, Notes and Records of the Royal Society of London 12, pp. 42–52, 1956–7. Thom, P. The Syllogism, Munich: Philosophia, 1981. Venn, J. Symbolic Logic, London: Macmillan, 1881. Venn, J. “Boole George”, Dictionary of National Biography, Vol. II, pp. 831–832. Whately, R. Elements of Logic, London: Mawman, 1826. Whewell, W. The Philosophy of the Inductive Sciences, 2 vols. London: Routledge [1840] 1996. Wisdom, J.O. “The Analyst Controversy: Berkeley’s Influence on the Development of Mathematics”, Hermathena 54, pp. 3–29, 1939. Wisdom, J.O. “The Analyst Controversy: Berkeley as a Mathematician”, Hermathena 59, pp. 111–128, 1942. Wisdom, J.O. The metamorphosis of Philosophy, Cairo: Al-Maaref, 1947. Wisdom, J.O. “Berkeley’s Criticism of the infinitesimal”, BJPS, pp. 22–25, 1953/4. Whitehead, A.N. and Russell, B. Principia Mathematica, Cambridge: Cambridge University Press [1910], 1964. Witt, C. Substance and Essence in Aristotle, Ithaca, NY: Cornell University Press, 1989. Zeller, E. Greek Philosophy, New York: Henry Holt, 1889.

Name Index

A Achilles, 4 Adam, 20 Agassi, J., ix, 63, 145, 1n, 5n, 152, 55n, 156, 85n Aldrich, H., 132, 154, 68n Anaximenes, 146, 13n Anscombe, J.E.M., 146, 11n Antisthenes, 28, 55–58, 67, 80, 151, 48n Aristippus, 55–58, 80 Aristophanes, 147, 25n Arnauld, A., 104, 106 Augustine, St., 62 Austin, S., 146, 11n

B Bacon, F., 73, 74, 86, 132 Baker, G.P., 154, 76n Barnes, J., 145, 7n, 146, 12n, 152, 54n Becker, C.L., xxi Berkeley, G., 56, 57, 87, 131 Bernoulli, J., 157, 91n Bolzano, B., xi, 10, 106, 107, 120, 140 Borges, J.L., 20, 147, 17n Bourbaki, N., 156, 85n, 157, 87n Boyle, R., 39 Bunge, M., 156, 85n Burnet, J., 145, 1n, 150, 38n

C Charmides, 26 Cherniss, H., 148, 34n Chrysippus, 100 Coffa, A., 104, 106, 107, 116, 120, 154, 73n Corcoran, J., 145, 4n, 149, 36n Cornford, F.M., 145, 7n, 148, 30n, 148, 31n

D Democritus, 18, 52–55, 151, 45n, 47n De Morgan, A., xi, xx, 12, 13, 112, 129, 137–141, 143, 157, 46n, 90n–93n Descartes, R., 8, 18, 62, 64, 73, 110, 112, 119, 120, 131 Dingler, H., 154, 74n Diogenes Laertius, 54, 98, 147, 21n, 151, 47n Diogenes of Sinope, 56, 57, 60, 62 Dodds, R., 147, 22n, 148, 29n

E Einstein, A., 118 Epictetus, 100, 154, 72n Epicurus, 55, 56 Euclid, 24, 25, 78 Euler, L., 129 Euthydemus, 28

F Finley, M.I., 4, 145, 2n, 3n Frege, G., 12, 13, 21, 100, 103, 106, 107, 110, 120, 139, 143, 144, 150, 40n, 154, 74n, 155, 74n

G Galileo, 131 Gasser, J., 134, 155, 75n, 156, 85n Gauss, K.F, 118 Gentzen, G., 13 Gorgias, 27, 28, 32 Grattan-Guinness, I., 156, 85n, 157, 86n Grene, M., 146, 14n, 149, 35n, 36n, 151, 46n, 151, 50n

165

166 Guthrie, W.K.C., 69, 145, 7n, 147, 25n, 148, 33n, 149, 36n, 150, 38n, 151, 46n, 152, 58n, 59n Gödel, K., 13

H Hacker, P.M.S., 154, 74n Hacohen, M., ix, 145, 5n Hailperin, T., 134, 135, 155, 75n Hamilton, W., 112, 138, 157, 90n Hamilton, W, R., 141 Hegel, F., 55, 113, 115, 120 Heracleitus, 52 Herodotus, 145, 1n Hilbert, D., 13, 156, 85n Hobbes, T., 131 Hume, D., 12, 56, 57, 86–88, 105, 106, 115–117, 132, 133, 156, 80n Huntington, E.V., 130, 142 Husserl, E., 140

I Ikarus, 67 Isocrates, 27, 148, 33n

J Jaeger, W., 33, 57, 147, 18n, 148, 33n, 149, 35n, 151, 49n Jevons, S.W., 130, 156, 86n, 157, 86n Jourdain, P.E.B., 154, 74n, 156, 85n, 157, 86n

K Kahn, C.H., 146, 9n Kant, I., 10, 12, 56, 64, 73, 88, 90, 92–94, 106, 107, 113–120, 126, 131, 132, 138, 143, 144, 156, 80n–82n Kneale, M., 100, 152, 53n Kneale, W., 100, 152, 53n, 157, 94n

L Laches, 26 Laita, L.M., 156, 85n, 157, 94n Lakatos, I., 25, 145, 5n, 147, 19n Lambert, H.M., 129 Langford, C.H., 156, 85n Leibniz, G.W., 12, 71–73, 92, 97, 104–106, 108–116, 119, 120, 126, 129, 131, 132, 143, 149, 36n, 155, 75n, 156, 80n, 157, 89n

Name Index Lejewski, C., 156, 85n Lewis, C. I., 156, 85n, 86n Locke, J., 71–74, 92, 115, 131 Lukasiewicz, J., 39, 141, 145, 4n, 150, 39n, 151, 46n, 152, 57n

M MacHale, D., 156, 85n Maimon, S., 118–120, 156, 81n Mates, B., 154, 71n Meinong, A., 139, 140 Melissus of Samos, 18, 146, 11n, 12n Mill, J.S., 74 Musgrave, A., 145, 5n

N Navia, L.E., 151, 48n

P Parkinson, G.H.R., 155, 75n Parmenides, 17–21, 24, 25, 27, 29, 31–34, 50, 52, 55, 99, 125, 140, 145, 7n, 8n, 146, 8n–14n, 16n, 147, 23n, 25n, 148, 29n–31n, 33n Patzig, G., 41, 145, 4n, 148, 28n, 150, 39n, 150, 42n Peirce, C.S., 130, 157, 86n Phaedrus, 25, 44, 84 Plato, 6, 21–24, 28, 29, 31–34, 44–47, 49, 53–57, 62, 72, 73, 80, 90, 113, 146, 9n, 147, 18n, 23n, 27n, 148, 31n, 33n, 151, 45n, 151, 47n, 153, 63n, 64n, 157, 88n Plotinus, 148 Popper, K.R., 25, 145, 1n, 5n, 8n, 146, 8n, 10n, 147, 20n, 148, 28n, 156, 85n Proclus, 148, 29n Protagoras, 23 Pythagoras, 27, 40

R Robinson, R., 26, 147, 23n Russell, B., 10, 12, 13, 21, 101, 103, 106–108, 115, 120, 131, 138, 143, 144, 150, 40n, 157, 89n Ryle, G., 147, 18n, 148, 31n, 33n

Name Index

167

S Schiappa, E., 147, 26n, 27n Sextus Empiricus, 147, 21n Smiley, T.J., 145, 4n Socrates, 23–28, 31, 33, 40, 41, 44, 45, 49, 51, 53–57, 62, 65 Spinoza, 104 Sprague, R.K., 147, 24n Stenzel, J., 148, 34n, 151, 45n, 47n Szabo, A., 17, 19, 145, 8n, 146, 13n

Theaetetus, 146, 14n Thom, P., 145, 4n

T Tarski, A., xi, xx, 13, 45, 138, 151, 44n Thales, 18, 20, 25, 52

Z Zeller, E., 84, 153, 62n Zeno of Elea, 24, 25, 27–29, 31, 32, 83, 147, 21n

W Whately, R., 132–135, 137, 154, 68n Whewell, W., 120 Wilkins, J., 147, 17n Wisdom, J.O., 3, 4, 28, 33, 65, 156, 85n Wittgenstein, L., 120, 154, 68n, 156, 85n Wolff, C., 115

Subject Index

A Abstractism, 49–60. See also Platonism Analyticity; theory of, 97, 103–108, 111, 114, 116, 132, 144. See also Judgment, analytic Apodeixis, apodeictic syllogisms and inferences, 83, 147, 28n Atomism, 52–54

B Barbara, 68, 109, 155, 77n

C Categories, 41, 58, 87, 149, 35n, 150, 38n Change, problem of, 49–60. See also Identity problem of Classes Boole’s notion of. See Extensionalism classes vs. essences, 40–47, 123–127 complement class, 41, 42, 124, 125, 139, 141 empty class, 19, 41, 46, 124–125, 139– 142. See also Nothing, the class universal class, 46, 124, 125, 138, 139, 141 Concretism, 49–60. See also Nominalism Conflation of dialectics and induction 83–90, 147–148, 28n of empirical science and logic, xi, xx, xxi, 7, 10–11, 29, 35, 91, 135 of epistemology and methodology, 5–6, 10, 17, 21, 29, 46–47, 65–69, 73–74, 94, 97, 98, 106, 130, 133–134 of meaningless and absurd, 18–22 of meaningless terms and terms having no reference, 18–22, 125, 139–142

of ontology and semantics, 17, 19–22, 29, 32, 47, 49, 57–65, 97–100, 114, 130, 135, 139–142 of refutation and indirect proof, 11–12, 23–29, 32, 91 of sound inference and proof, 10–12, 146, 12n Contradiction, 12, 18, 19, 24, 32, 33, 55, 58, 63, 68, 108, 114, 125, 141, 146, 16n, 151, 50n Cosmology, 17

D Definition classic theory of, 91–94, 133 essential, 10, 12, 91–94, 104, 106–107, 126, 131, 150, 40n, 153, 63n, modern theory of, 92 nominal, 10, 91–94, 105, 107, 133, 134 real, 65–69, 91–94, 106–107, 111, 134, 135, 152, 57n, 58n, 154, 70n Determinism and indeterminism, 99, 104, 115 Diaeresis, 6, 7, 23, 33, 34, 44–47, 55, 83, 84, 147, 23n, 153, 63n Dialectics, 5, 23, 25–27, 31–33, 45–47, 58, 63, 65, 73, 79, 83–85, 89, 91, 117, 147, 21n, 23n, 148, 28n, 34n, 152, 59n, 60n, 153, 61n, 62n

E Epagoge, 79 Epistemology, 5–6. See also Conflation of epistemology and methodology Essentialism, xi–xii, 41–47, 49–60, 61, 94, 97–98, 109, 116, 123, 127, 132, 137, 138

169

170 Existential import, 21, 49, 59–64, 68, 86, 92, 109, 152, 52n, 53n Explanation, Aristotle’s theory of, 65–68, 74, 150, 39n, 152, 56n Extension vs. intension, 43, 68, 88, 140, 141 Extensionalism, xi–xii, xx, 123–127, 129–132, 135, 137–141, 143, 144, 156, 85n

F Forms, theory of, 53–55, 151, 45n Free will. See Determinism

H Horos, 39, 68, 152, 57n. See also Terms, ancient theory of

I Idealism, 55, 56, 87, 88 Identity, problem of, 51. See also Change, problem of Induction as-analysis, 85–90, 117, 118, 152, 59n, 154, 68n Hume’s problem of, 86, 87, 115–116, 132–134, 137 Kant’s problem of, 94, 115–118 Russell’s problem of, 115 as-synthesis, 85–90, 117, 153, 67n, 154, 68n Inference sound, 7, 10–13, 31, 74, 83, 84, 94, 106, 107, 118, 119, 146–148, 154, 155 valid, 3, 7, 10, 11, 31, 40, 65, 74, 83, 84, 94, 106, 107, 109, 114, 118, 132, 138, 141, 144, 148 Intuition, 77, 89–90, 112, 113, 127. See also Induction-as-analysis Intuitionism, 33

J Judgment analytic, 103–108, 116, 144 synthetic, 106, 116, 134 theory of, 10–11, 68, 74, 97, 103–108, 116, 125, 127, 131, 134, 135, 138

Subject Index M Meaninglessness and contradictions, 18–21, 100, 108, 125 and the false, 18–21, 42, 49, 59, 100, 125 and the void, 18–21, 139, 146, 11n, 12n Megaric Logic, 97 Methodology, 3, 5–7, 9, 10, 12, 13, 17, 21, 25, 29, 32, 35, 36, 39, 40, 46. See also Conflation of epistemology and methodology

N Nominalism, 49, 53, 57, 59, 135. See also Concretism Nothing, the class, 124, 139–141 Nous, 89–91, 94, 116

P Platonism, 49–60, 61, 87, 144 Pragmatism, 52 Probability, 107, 137–139, 157, 91n Progress in Science, 33, 145, 5n Proof; Indirect, 10–12, 17, 19, 24–26, 28, 29, 31, 32, 63, 83, 85, 145, 8n, 146, 12n, 13n. See also Reductio adabsurdum

Q Quantification of the predicate, 112

R Range, 35, 39–47, 59, 100. See also Variables Realism, 49, 54, 56 Refutation, 3, 7, 10, 11, 24, 25, 28, 29, 31, 33, 34, 83, 97, 107, 144, 145, 5n, 148, 28n, 153, 61n Relations, 43, 50, 57, 58, 88, 101, 109, 137, 138, 140, 150, 42n

S Sensationalism, 55–57, 87 Solipsism, 55 Stoic logic, 97–100, 154, 71n Substance, 18, 41, 46, 52, 62, 124, 151, 46n

Subject Index T Terms ancient theory of, 20–21, 41–47, 62, 66, 67, 73, 123–125 conflation of terms and judgements, 66–69, 103. See also Horos Transcendental argument, 116–119. See also induction-as-analysis Truth Aristotle’s theory of, 45 by convention, 3, 4, 50, 52 empirical, 10, 114

171 logical, 10, 35, 68, 104, 107 by nature, 4, 52 Tarski’s theory of, 45

U Universals vs. Particulars, 50–53, 153, 67n

V Variables, 23, 26, 34, 35, 39–41, 59, 63, 100