Essays on Definition (Terminology and Lexicography Research and Practice)

ESSAYS ON DEFINITION

TERMINOLOGY AND LEXICOGRAPHY RESEARCH AND PRACTICE Terminology and Lexicography Research and Practice aims to provide in-depth studies and background information pertaining to Lexicography and Terminology. General works will include philosophical, historical, theoretical, computational and cognitive approaches. Other works will focus on structures for purpose- and domain-specific compilation (LSP), dictionary design, and training. The series will include monographs, state-of-the-art volumes and course books in the English language.

Series Editors Helmi Sonneveld Sue Ellen Wright Consulting Editor Juan C. Sager

Volume 4 Juan C. Sager Essays on Definition

ESSAYS ON DEFINITION

Selected and edited by

JUAN C. SAGER UMIST, Manchester

Introduction by

ALAIN REY

JOHN BENJAMINS PUBLISHING COMPANY AMSTERDAM/PHILADELPHIA

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TM

The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences — Permanence of Paper for Printed Library Materials, ANSI Z39.48-1984.

Library of Congress Cataloging-in-Publication Data Sager, Juan C. Essays on definition / Juan C. Sager. With an Introduction by Alain Rey. p. cm. -- (Terminology and lexicography research and practice, ISSN 1388-8455; v. 4) Includes bibliographical references and index. 1. Definition (Logic). 2. Terms and phrases. 3. Lexicography. I. Title. II. Series. BC199.D4 S34 2000 160--dc21 ISBN 90 272 2327 0 (Eur.) / 1 55619 773 X (US) (Hb; alk. paper)

00-036100

© 2000 – John Benjamins B.V. No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher. John Benjamins Publishing Co. • P.O.Box 75577 • 1070 AN Amsterdam • The Netherlands John Benjamins North America • P.O.Box 27519 • Philadelphia, PA 19118 • USA

Table of Contents

Preface Introduction Alain Rey Defining Definition

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1

Texts PLATO From Theaetus From Laws (Book X)

15 23

ARISTOTLE From Analytica Posteriora (Books I + II) From Topics (Books I, VI + VII) From Physics (Book I) From Metaphysics (Ch. Z + H)

25 44 81 81

ISIDORO OF SEVILLE From Etymologiae (c. 633) The Devision of Definitions Abbreviated from the Book of Marius Victorinus (Book II, Ch. IX)

91

BLAISE PASCAL The Spirit of Geometry (1657) The Art of Persuasion (1657)

95 108

BENEDICT DE SPINOZA From On the Improvement of the Understanding (1677) From Correspondence with Simon de Vries (1633)

119 121

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TABLE OF CONTENTS

JOHN LOCKE From An Essay Concerning Human Understanding (1690) Of General Terms (Book III, Ch. III) On the Names of Simple Ideas (Book III, Ch. IV) On the Remedies of the Foregoing Imperfections and Abuses (Book III, Ch. XI) Of Trifling Propositions (Book IV, Ch. VIII) GOTTFRIED WILHELM LEIBNIZ From Correspondence Leibniz to Hermann Conring (1678) Leibniz to Walter von Schirnhaus (1678) Leibniz to Antoine Arnaud (1686) Leibniz to De Volder (1699) From Preface to an Edition of Nizolius (1670) From On Universal Synthesis and Analysis, or the Art of Discovery and Judgement (1679) From Discourse on Metaphysics (1686) From New Essays on Understanding (1695–1708)

125 125 127 132 138

145 146 147 147 148 149 151 152

GEORGE BERKELEY From A Treatise Concerning the Principles of Human Knowledge (1710) Introduction 159 IMMANUEL KANT From The Only Possible Argument in Support of a Demonstration of the Existence of God (1763) (Section I, First Reflection) From Inquiry Concerning the Distinctness of Natural Theology and Morality (1764) (Reflections)

163 165

JOHN STUART MILL From A System of Logic, Ratiocinative and Inductive (1843) Of Definition (Book I, Ch. VIII)

173

HEINRICH RICKERT The Theory of Definitions (1888)

191

References

251

Name Index

255

Preface

Definition occupies a central place in all sciences and is a fundamental tool in logic, philosophy of ideas and semantics; each of these areas view the activity of defining from a different angle. In everyday communication, the result of this activity, represented in the various forms of definition considered appropriate for their objective, is also essential for establishing relationships between things and ideas and their names for which purpose they are collected in glossaries, dictionaries and other reference tools. Given these various viewpoints on what is involved in defining and the wide range of uses of definitions, there is, understandably, a great diversity of interpretations of what is meant by defining and its product, the definition. There is, thus, no general agreement about what a definition is, what knowledge it represents and conveys and what quality criteria it must satisfy. Defining and definitions play an essential role in terminology and lexicography because they are the conventional means for establishing the meaning of lexical items, or, expressed differently, for connecting the concept with the word or term that represents it. This crucial role, which lies at the root of language as a symbolic system, is, however, viewed very differently by lexicographers and terminologists. To judge by the many treatises on the subject, not only are there different methods of defining and different resulting types of definition, but even the limited objectives of lexicographical and terminological definitions themselves seem far from clear. It would therefore appear to be useful to step back from the specific concerns of lexicography and terminology and consider the nature and functions of defining and definitions in those sciences which have discussed this topic for the last two millennia. The present collection of essays on definitions from Plato and Aristotle to modern times, assembles interesting, sometimes less widely known and, perhaps, unexpected texts. They examine the subject from the point of view of philosophy, which is essential for a theory of terminology seeking to establish the relationship between concepts and terms. These essays deal mainly with theoretical

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issues but they also consider the practice of defining and therefore serve as background to all manner of studies in terminology. In addition they form a useful complement to the better known discussions of definitions in lexicography.

Acknowledgements “The only possible argument in support of a demonstration of the existence of God” Section I, First reflection (2:70 and 2:71) Translated by David Walford, is reprinted from: Immanuel Kant, Theoretical Philosophy, 1755–1770, by kind permission of Cambridge University Press. The following texts have been translated by J.C. Sager: Isidoro of Seville Etymologiae, Leibniz Letter to Antoine Arnauld & New Essay on Understanding, Pascal The Spirit of Geometry & The Art of Persuasion, Rey Defining ‘Definition’, Rickert The Theory of Definitions.

Introduction Defining Definition Alain Rey

“ce qui ne ressemble à rien n’existe pas” Paul Valéry

Definition can be defined in various ways. In an ontological frame of mind, one can, for example, try to describe it as the essence of a certain logical-linguistic operation which is necessary for the restricted availability of linguistic signs. This is the position of Aristotle who uses a “discourse of limitations”, oÏ}isti~oÈ| loÈgo| abbreviated to oÏ}ismoÈ|, which has been translated as ‘definition’. More modestly, one may endeavour to describe it as the meaning of a set of words in a natural language considered to be equivalent: Latin definitio or finitio, French définition, German Wortbestimmung, etc. Such a description would have to be wide enough to fit all the uses of the words in question at a particular time, and to permit them to be distinguished from all other words of the same language and especially any other semantically related word. Such a description may, but need not, be guided by a morphological analysis of the root; thus English definition goes back to the Latin de-finitio, which refers to finitio, which refers to finire and then to finis. Sometimes, when faced with a fairly precise concept or notion, and when one disagrees with previous usages of the word, it is also permissible to present an analogous description in an authoritative and prescriptive manner, a formulation which is generally preceded by an expression like “by definition I here mean … etc.” The results of these different procedures, which we globally call definition, are, of course, quite different. The first type, which is philosophical, can lead to a metaphysical discourse, a fragment of a set of assertions which simultaneously claim to account for the

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sense of words and terms, for the nature of the corresponding general ideas and finally for the nature of ‘things’ — that is, of objects, phenomena or operations. Other modes of expressions would only cover a part of this ambitious undertaking. The second type, which is linguistic and philological in the proper meaning of the word, respects the variations in linguistic form, generally of a lexical item, produced by actual usage, and leads to something which may be comparable to what is found under the heading definition, oÏ}ismoÈ| or Wortbestimmung in language dictionaries, namely a combination of definitions and notes supported by examples. The third type, by contrast, and like the first, can produce a unique expression. It is motivated by the intention of limiting the notion and prohibiting any other usage. In theoretical discourse, it frequently serves not only to establish scientific theories, but also to lay down its terminology, especially in Law, and in purist discourse. Such a creative or prescriptive definition is, by its nature, opposed to polysemy and is contrary to the precepts of linguistics in its function as the scientific observation of usage. Only the second method of definition seems to be compatible with what today is understood by dictionary definition. These basic differentiations already show the great variety of methods of defining and hence the degree of ambiguity of the word definition, and its equivalents in other languages. This is true regardless of the angle from which it is tackled, and not only because there are so many types of definition. Even without going into details about the nature of this semantic and semiotic method, the objective of the act of defining itself is far from clear. The classical opposition between definitions of words and definitions of things, discussed especially in the 17th and 18th centuries, is hardly satisfactory; its resolution into a single category which Richard Robinson (1950) calls ‘word-thing definition’ is necessary but there, too, ‘word’ and ‘thing’ are too brutally contrasted. Replacing ‘word’ by ‘stable linguistic sign’ and ‘thing’ by ‘referent of such a sign’ are useful amplifications and refinements, but this does not solve the problem. Defining a word or a stable linguistic sign — natural or otherwise — means establishing a contact between this sign and others, either at the same semiotic level (in which case we can speak of periphrastic synonymy, which is useful for dictionary definitions), or at the level of a constructed metalanguage, which can occur in the natural sciences and, sometimes, in philosophy or any other theory which is concerned about its terminology. This is a first ambiguity. The second ambiguity relates to the very objective

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of any definition. This objective can vary considerably according to the domain of knowledge, (e.g. philosophy, logic, lexical semantics, lexicography, terminology) and, in each domain, according to its theories and methodologies. For example, the objective may be to reveal, by means of rational language (logos), the being, the essence, the quiddity of an object of thought, namely of a state (rather than a being), external to the knowing subject. This was Aristotle’s method, which was later crystallised by the medieval realists and frequently taken up by metaphysicians. Today we rather refer to the method used by the Stoics which, from the 17th century onwards, was revived by modern philosophy. Or, much more modestly, the objective may be to establish a body of synonyms by means of periphrases which can clarify either the meaning or the uses of lexical units for a more or less specific user, as Wittgenstein proposed. All the intermediate positions between these two poles are possible and many have been attempted. It is also possible to exploit the pragmatic opposition between (a) the descriptive definition of a factual state which respects the normal usage by the majority of language users, as sanctioned by the social norm, of the signs to be defined; and (b) the concept-creating definition (in the Kantian sense), one objective of which is to construct ‘a well-formed language’ according to scientific principles, which serves as a metalanguage for natural language, or, at the aesthetic level, in the words of the French poet Mallarmé, ‘donner un sens plus pur aux mots de la tribu’ (to give a purer sense to the words of the tribe). In the second case, that of the creative, imposed, stipulative definition, it may be a matter of re-naming an object of thought, which had been considered to be clear and which had previously been designated by a natural language word, by relying on signs supposedly corresponding to previous but unambiguous objects of thought. In the constructed theories, this is the purpose of certain terminological definitions and there are admirable historical examples of this process. One of the most remarkable of these is undoubtedly Lavoisier’s and Guyton de Morveau’s elaboration of modern chemistry in and by means of a new terminology. Leaving aside, for the moment, the situation of the applied sciences and institutions, it has also to be noted that the nature and the purpose of such definitions differ according to whether they occur in formal or in hypotheticaldeductive theories, whether in sciences of observation or in sciences of induction (where one may have to form nomenclatures) or in non-formal theories which are, however, partly subjected to observation and induction (the so-called ‘soft’ sciences. The requirements of the applied sciences and institutions are practical or arbitrary, and are, therefore, fundamentally different, but they, too, have a need of terminologies, and with them for definitions. The concept-creating and/or prescriptive definitions rely on a well-known

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property of language, namely the arbitrariness of the sign. This arbitrariness can go to extremes as certain formalist philosophies have shown; their limitations, indeed their communicative absurdity, can easily be demonstrated, as Lewis Carroll has done in his famous chapter of Through the Looking Glass. This text which retells the conversation between Alice and Humpty-Dumpty is often quoted to illustrate the impossibility of a completely free definition. (Compare what Robinson (1950: 75) states with respect to stipulative definition). We recall that this fantastic character, the human egg, Humpty-Dumpty uses the word “glory” and gives it an unforeseen meaning in “There’s glory for you”, which is supposed to mean “There’s a nice knock-down argument for you”. It has rarely been observed that Humpty-Dumpty respects semantic rules, confining himself to adding specific semes full of a fairly flexible notion. We remember his aggressive statement: “When I use a word it means just what I choose it to mean, neither more nor less.” To which Alice replies “The question is, whether you can make words mean so many different things” which, in its turn, is countered by the famous rejoinder “The question is, which is to be the master — that’s all”. This exchange can be seen as defying the all-powerful usage and collective arbitrariness: violence and cant are indeed inside the “social shell”. This dialogue also reflects the amused sadness of the reverend Dodgson, logician and mathematician, who, like his famous predecessors Hobbes and Locke, regretted the steadily growing ambiguities of natural languages. But we must not forget the rest of the chapter. Just like Plato’s Cratylus, Humpty Dumpty wants all names, even proper names, to have sense and only one sense. Indeed, his own name, as he explains, is indicative of his form, hump — humpback, dump — dumpy, which can describe a short and fat person. In revenge he says “Alice, it’s a stupid name enough! … With a name like yours, you might be any shape, almost”, invoking the extreme case of motivation and absence of arbitrariness. Humpty-Dumpty is an extremist full of contradictions. Moreover, having declared himself to be the master of meanings — and of definitions — he demonstrates this skill in a highly responsible and didactic manner when he explains for Alice the obscure words of the difficult poem ‘Jabberwocky’, incidentally, creating and exemplifying the concept of ‘portmanteauword’ — in French ‘mot-valise’ — now in common usage among lexicologists. This extremely rich text covers the full scope of definition, evoking both its relation to effective usage and the creation and transmission of new meanings, each of which implies different objectives: to account for sense in usage (and Humpty-Dumpty becomes a proper philologist since he attempts to explicate a text); to attribute a precise conceptual value to signs, either by creating them (as,

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for example, in neologisms and the jargon of the sciences) or by using existing signs. Carroll’s character, we must admit, does this in an antisocial and comical way.

Definition, logic and language Many philosophers have asked these questions long before Carroll. For example, in his famous essay De l’esprit géométrique — the title really refers to the spirit of ‘mathematics’ — Pascal distinguished two types of definition: a ‘definition of the name’, consisting (so he claims) of freely giving a name to the ‘things one has clearly outlined in perfectly well-known terms’; and a ‘definition of a thing’ which leaves words their ordinary meanings, and is ‘intended to make the thing correspond with this formulation’. So, regarding the concept of ‘time’, Pascal shows that if one intends to define the idea, while respecting the usage of the word, one arrives at a proposition which is ‘not at all free but subject to contradictions’. For him, only the first one is a true definition, and he characterises it as ‘free’ (i.e. arbitrary), ‘permitted’ and ‘Geometrical’(i.e. logicomathematical). This definition consists of assimilating the concept of a known ‘thing’ and attributing a designation to it. It has its absolute limits; namely in that all fundamental concepts (represented by ‘primitive words’) are indefinable. Pascal inserts definition into the thinking process, into mathematical proofs, and into value and truth systems. Beside these formalisable nominalisms, which can create permitted formulae and which are the ‘true’ imposed definitions, Pascal states that the so-called definition of a thing is only an arguable proposition which is not at all ‘free’, because it is linked to the ‘common meaning’ of the word. He adds that this type of definition is one of the scourges of reasoning when it gets mixed up with permitted definitions. But in his essay, Pascal is only interested in definitions corresponding to concepts which can figure in a demonstration. Moreover, in the name of selfevident and intuitive knowledge, which is at the basis of any system, the fundamental concepts of this category, for example, time, space, being, admit of no definition without the risk of becoming ridiculous tautologies. As a result, Pascal does not only not envisage the description of common meanings but the rules for definition, which he gives in the second part of his essay, do not apply to the definition of lexicographers. This, in turn, does not correspond to either of the two types of definitions used by Pascal. The common lexicographic definition is totally alien to a system where definitions, axioms and demonstrations are interdependent, and where the reform of logic is subjected to the rigours of

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mathematics and is also subordinated to the intuition which supplies our reasoning with the indefinable ‘perfectly well-known terms’ to which it is useful to add ‘explained terms’ — those which are defined by the provision of a definition for each term and do not belong to either of these two categories, and which, for this reason remain ‘a little obscure’ or ‘ambiguous’. In this way Pascal’s definition explains terms by dispelling their obscurity and eliminating their ambiguity. It ensures the designation of the necessary concepts for our reasoning; it is never a substitute for the intuition of fundamental concepts, and, finally, it is independent of the common meaning of words. Thus he presents a terminological methodology applicable to the formalisable reasoning represented by mathematics, which carefully avoids the problem of the relationship between the semantics of natural language and the construction of meaningful tools required for thinking. Nevertheless, some of Pascal’s rules also concern the semantic explicitation of linguistic units. The rule whereby the ‘definition of things’ is not free without risking ‘inexplicable embarrassment’ is equally valid for terminology and dictionaries. But, for Pascal, this liberty, admitted for ‘definitions of words’ and prohibited for ‘definitions of things’, only applies to scientists, researchers and mathematicians. For the scientist, Pascal states that, beside the primitive words ‘which the world understands by itself’, the terms used in geometry — by which he means mathematics — ‘are so clear and precise that there is no need of a dictionary to understand them’. In this way, the dictionary starts where mathematical thinking finishes with its definitions. By a strange perversity of designation, only Pascal’s ‘definition of things’, which is always questionable and completely limited, may be expressed in dictionaries, because his definitio nominis never affects the working of everyday language, nor the system of meanings, but only the terminological process of designation. It is never concerned with semasiology, but only with a very special onomasiology appropriate for the needs of scientific thinking. Even more explicitly than Arnaud and Nicole, the authors of the Port Royal Logic, Pascal and after him several logicians, prove to what extent the scientific definition, with its two methods, differs from the definition of the semanticians of the lexicon of natural language; he also demonstrates the close relationship of the two contrasting types, definitio nominis and definitio rei with the definition used by terminologists.

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Incompatible points of view As Pascal observed, the entire defining tradition developed by logicians and philosophers lies outside that of dictionaries. And I believe that much confusion has arisen as a result of the efforts of many later thinkers, especially Leibniz, to apply this tradition to dictionaries. This is the case of the concept of ‘dictionnaire raisonné’ which emerged from Locke’s philosophy of knowledge. Instead of opposing itself to Locke’s view ( as did Leibniz). Diderot and d’Alembert’s Encyclopédie in particular is a hybrid product of an epistemology based on the lexicon and the terminologies — without a clear analysis of these two complementary points of view — and of a didactic methodology, adumbrated by Furetière, and implemented by Chambers in the first Encyclopaedia Britannica, before it was finally taken over by Diderot and d’Alembert. This method of an alphabetical encyclopedia was considered to be complementary and apparently similar to both the mono- and multilingual language dictionaries compiled in the Western World during the 16th and 17th centuries. These two poorly distinguished methods do, however, correspond to two different points of view, the one focusing on lexical semantemes and their meanings in common usage, the other focusing on concepts and their instantiation by terms, which unfortunately were confused with the words and expressions representing them. All discussions about a general concept of ‘definition’ suffer from this confusion which can neither be blamed on scientists, epistemologists and logicians, nor on lexicographers, but results from the ambiguities inherent in the popularisation of knowledge, and an inadequate perception of the originality of each process. The point of articulation, in matters of definition as in others, is then terminology, itself victim of an inadequate theoretical basis, just like lexicography. The intrusion of linguistics and semantics into the debate about definition has permitted us to clarify this question with regard to dictionaries. Contrary to appearances, this intrusion has been felt since Roman times. It is chiefly exemplified by Varro; Cicero, by contrast, pursuing a pragmatic purpose, applied an Aristotelian epistemology to his Rhetoric. Linguistics comes to the fore in the writings of the Port Royal group whose semantics is ambiguous because their universalising ‘grammar’ is constantly attracted by a logic which is itself strongly language-oriented. In Diderot’s contributions to the Encyclopédie, the influence of linguistics produced a unified concept of definition, especially when it came to technical vocabulary, whereas d’Alembert’s approach to definition, like Pascal’s, did not permit those factors, which were later to be called

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sociolinguistics and pragmatics, to interfere with the epistemological methods of a mathematician striving for ‘true’ knowledge; he actually wrote: Il est un grand nombre de sciences où il suffit, pour arriver à la vérité, de savoir faire usage des notions les plus communes. Cet usage consiste à développer les idées simples que ces notions renferment, et c’est ce qu’on appelle définir. (Elements of Philosophy, IV) (In order to arrive at the truth, in a large number of sciences it suffices to be able to use the most common concepts. This use consists of developing the simple ideas contained in these concepts, and this process is called ‘definition’.

A summarising study of definition, like Richard Robinson’s well-known manual (1950), is an impossible undertaking because it can only list and try to relate incompatible points of view. Beside this attempt by a historian of philosophy and logic, one can imagine other studies from the viewpoint of the epistemology of science, from that of the theory of terminology, from that of linguistic semantics, indeed, today, from the viewpoint of variously automated cognitive science. They would all have their particular merit, but none, I believe, would end up with a broad enough concept encompassing all defining methods nor all theoretical viewpoints of the subject. By contrast, the mathematical or logical definition, the terminological definition and the lexicographical definition have each been the subject of cogent descriptions which have sometimes yielded important theoretical insights. The best of these acknowledge the existence of other defining methods; they admit other meanings of the term definition than the one they are dealing with, and do not confuse them. They also abstain from judging the one with reference to the needs of the other. (I am particularly thinking of the irrelevant criticisms by semanticians on the subject of the traditional dictionary definition) These descriptions more or less clearly admit other meanings of definition, but marginalise them: we have seen how Pascal proceeded in order to dispose of the problem of the definitio rei and the limitations of the definitio nominis. Others went further, like Kant, according to whom only mathematics knows true definitions (nur Mathematik hat Definitionen), a viewpoint which imposes a new and very restrictive definition upon the concept. By contrast, Gerolamo Sacchieri, one of the successors of the logic of the Stoics, follows the tradition of Port Royal. In his Logica demonstrativa (1697), he expands on the definitio quid rei, at the same time both interfering with the dictionary definition — the famous definition of things of the encyclopedists — and stressing the difficulties he experiences in his position as epistemological

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logician. According to Sacchieri, this definition, which no longer corresponds to the simple ‘right to abbreviate’ of the definitio quid nominis, presupposes the existence, or at least the possibility, of what is being defined (later to be called the referent) and must always be justified. Consequently, even in mathematics, a definition of this ‘real’ type creates problems when it does not correspond to a theorem of existence. Sacchieri chooses the example of ‘parallel’; and the revolution in mathematics and physics in consequence of the redefinition of Euclidian geometry in a wider framework is a perfect exemplification of the accuracy of his reasoning which moves towards the construction of an allembracing concept. According to the philosophy of Port Royal, this real definition — of ‘things’ — serves to explain an idea already contained in the word (today we would say in the meaning of the lexical sign) by showing how it is formed by means of simpler ideas. We have already mentioned that this was still d’Alembert’s position with respect to: la nature des êtres envisagés par rapport a nous, (qui) n’est autre que le développement des idées simples renfermées dans la notion que nous nous formons de ces êtres. (id. ibid.IV) (the nature of beings envisaged with reference to us which is merely the development of simple ideas contained in the concept which we have of these beings)

For d’Alembert these definitions are neither ‘of names’, because they do not restrict themselves to explain what Pascal called the abbreviation by means of a term, nor ‘of things’, because they explain the nature of the object, not ‘the way it is, but the way we conceive it’. This movement from ontology to cognitive psychology indicates an important shift in the theory of definition, but does not modify the social function of definitions, any more than other epistemological mutations do. In the meantime, in the 19th century, several logicians and epistemologists, especially John Stuart Mill and Augustin Cournot, have dealt with the whole of this problem, the latter occupying himself with practical considerations no longer relevant only to their structure, but to the transfer of knowledge, which is very useful for lexicography.

Common practices in the of definition In everyday language, the practice of defining and the use of definitions is complex: it is not limited to dictionaries intended to represent the lexicon of a

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language, but covers glossaries, whose purpose is the description of the vocabulary of a text, or a speech, and encyclopedias, terminological collections, treatises and manuals explaining a science or a subject field, and also standards, laws and codes, which attach a particular meaning to the terms of a special language. Though wrongly, definition is also applied to the clues used to make us find a word in games like cross-word puzzles, or even free defining statements so frequent in literature and proverbs. Though it would be perfectly appropriate, the word ‘definition’ is not used to designate translingual correspondences which play an equivalent role in bilingual dictionaries. Nor is it normally applied to the spontaneous formulations meant for the acquisition of lexical competence, in learning, as, for instance, in conversations between adults and children. Even when we limit ourselves to the limited and restricted communicative functions and corresponding methods of dictionaries and encyclopaedias, numerous and well-known ambiguities surround the term ‘definition’. Some are not very troublesome. It is simply a matter of convention that ‘definition’ has been called a semic equation, or the second element of such an equation. Nevertheless, because of its condensed form, the text of the defining statement in dictionaries is ambiguous. For example, the expression ‘word x means …’ is not properly differentiated from a phrase like ‘that which is called x — the referent — is …’. The ambiguity between meaning and being, which occurs both in the metalanguages of semantics and lexicography, has been the subject of detailed study, especially by J. Rey-Debove. Other ambiguities are more serious. Above I have pointed out the one which consisted in borrowing logico-philosophical categories, subject to the evolution of theoretical systems, to account for the methods of defining in everyday use. Though the history of logic and philosophy and even dictionaries of philosophy provide substantial information about this topic, this cannot be applied unreservedly to the analysis of what might be called the ‘common didactic definition’. A reading of the article ‘Définition’ in Lalande’s Vocabulaire technique et critique de la Philosophie (1902–1923) suffices to convince us both of the fact that its content and analyses are hardly appropriate for general language dictionaries, and of the rightness of his warnings against the ambiguity of outdated and superseded concepts, such as ‘definition of word’ and ‘definition of thing’. Does this mean that a theory of lexicographic definition can manage without the tradition of logic and philosophy? Certainly not, any less than it can do without the implicit and explicit theories of terminology. But it must be constantly aware of the differences, especially of the incompatibility of the respective points of view. Diderot and d’Alembert were not the only practitioners of the common

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didactic definition who attempted to apply concepts from logic and philosophy to it. In the past, many compilers of encyclopedias have tried it and produced typologies which are incomprehensible today, and in any case unusable, even though some of them continue to be of interest. A noteworthy case is that of Isidore of Seville, the most distinguished encyclopedist of the Middle Ages. Isidore’s fifteen types of definition, specifically meant for rhetoric, borrowed from Marius Victorinus via Cassiodorus, can be found in the second volume of his Etymologiae. In the analytical spirit of this rhetoric, and of scholasticism in general, Isidore gathers fairly disparate notions but does not relate them to his own defining method which, as we know, is most frequently ‘etymological’, in the old sense of the word used by Cratylus. In this connection, it is worth noting that the modern accounts and syntheses of definition not only omit reference to all non-western traditions, but, among the western ones, neglect the rhetorical tradition. Even the most distinguished among them, authors of deep reflections on definitions, are left out of consideration. And yet, Cicero and Quintilian, only to mention the major cultural influences, interpret the Aristotelian theory with its application in common usage in mind, which it may be worth contrasting with the much later theory of definition, in dictionaries and elsewhere. Quintilian, in particular, uses Aristotle’s categories (genus, species, differences, properties) and Cicero’s contrast between divisio, an intensional semantic method, and partitio, an extensional method that is useless for definitions. But he shifts and modifies them by inserting them into the methodology of argumentation, very often with the intention of their application to legal texts. In this way he clarifies the discussion, endowing definition with two purposes: Nam tum est certum de nomine, sed quaeritur, quae res ei subjicienda sit tum res est manifesta: et quod nomine constat. Quintilianus, De institutione oratoria, 1. VII, ch.3.) (When the name is known with certainty, look to what thing it applies; if, however, the thing is manifest, look for the word that suits it.)

This statement anticipates the starting points of lexicography and terminology; but it is expressed with a quite different intention, namely the interpretation of the law. But the criterion of applicability he uses is what, in dictionaries, is jointly expressed by ‘example’ and ‘definition’. Quintilian also observes the wide differences in defining statements applicable to the same word. He adds that this finitionum diversitas is not serious, unless it conflicts with another sense (si sensu non pugnant). The divergencies arising from the comprehensio are useful for intellectual disputations, but not for the arguments of the lawyer. In this way,

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an entirely practical theory of definition pushes aside philosophical subtleties. Perhaps lexicography — and logic — needed a Quintilian for limiting the legitimate claims of the theory.

Conclusion Ranging from the ontological definition, which today is no longer acceptable, to the defining methods for pedagogical and practical purposes, there are many concepts which corresponded and still correspond to the general perception of definition. Beside their traditional applications in rhetoric, lexicography and law, their interpretations in theoretical traditions such as philosophy, logic, and now also in philosophy of natural language, pragmatics and cognitive science, while still remaining quite distinct, reveal many intersecting aspects. Among the different traditions which have evolved on the subject of ‘defining’, two other traditions, beside scientific and legal definitions, are worth noting: the dictionary definition, being entirely practice-oriented, and the terminological definition, which rests on both traditions. This double association leads to syntheses which are simultaneously efficient and illusory, as in Diderot’s Encyclopédie. The criticism of one tradition from the point of view of the other is useful but always in danger of misleading. Criticising the obvious logical weaknesses of dictionary definitions from the point of view of scientific definitions, and their linguistic shortcomings from the point of view of a semic or ‘prototypic’ analysis — inherently deceptive by its implications — is as pointless as criticising the arbitrariness of designations in the name of the regularities of the lexicon, or the arbitrariness of scientific definitions from the point of view of general usage, as widely practised by the critics of scientific jargon. Dictionary definitions must be evaluated from the point of view of the semantics of natural languages, as a manipulation of quasi-synonymy, but also from the point of view of the production of a regulated didactic discourse, parallel, though very different, to that of rhetoric, and, like rhetoric, belonging to the communicative use of language. Concepts like ‘quasi-synonymy’, ‘cultural stereotypes’, ‘prototypes’ or ‘morphosemantisms’ are only useful if they are integrated in a theory of usage. From this point of view, the opposition between scientific creativity and didactic reproduction, between transfer of knowledge and ideological investment, i.e. between didacticism and applications of rhetoric, are as important as logico-semantic content analyses. Since all syntheses on the subject of definition, contribute to the state of the epistemological thought of a particular historical period, any interpretation

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becomes questionable as soon as their logical, linguistic, cognitive and philosophical premises change. Nevertheless, when it reflects an internal coherence, each such synthesis merits close study. Moreover, if definition presupposes and deserves a theory, it still remains a technique. Because it is shaped by its social and logical context, this technique is as varied in its rules as in its objectives. Thus, the abstract and deductive definition of mathematics and logic stands out from all the others: we recall that for Kant it is the only ‘definition’ strictu sensu (an “arbitrary synthesis”). In the sciences based on experience and induction, definitions are built on different foundations; the definition of technical and scientific applications relies on sequences of procedures and production, resulting in an intricate descriptive statement which represents the technical object in question. The motivation of these definitions is based on an essential objective in the field of anthropology, that which defines homo faber. At a completely different level, we can elaborate a corpus of definitions which conceptualise a vision of the world; these definitions would be partly ‘scientific’, but also heuristic and, in a certain way, legal (in which case they are what Searle calls “speech acts”). These definitions convert writers’ and thinkers’ (Goethe’s, for example) general discourse about the world into a truly philosophical discourse (that of Kant, Hegel, or, to a lesser degree, that of Nietzsche — to stay within the conceptual universe circumscribed by the German language). The presence of terminological definitions can therefore be the hallmark of the technically “philosophical” status of a general and abstract discourse.1 These definitions also permit us to tackle prescriptive conceptual units, entirely consisting of obligatory assertions: i.e. the definitions of codes, laws, constitutions, the contents of jurisprudence. Law — but also religious discourse: books of revelation, or rituals — would not exist without defining statements; they are an integral part of its production. Theoreticians like Heinrich Rickert have fully understood the specific nature of legal definitions, but the conceptual framework offered by semiotics permits us to refine and possibly deepen these distinctions. Ranging from mathematical abstractions to the regulatory discourse on morals, beliefs (religions) and social activities (law, economics), each group of definitions corresponds to so many terminologies, which are seldom extralinguistic in nature (except in science and formalised knowledge), but are almost invariably embedded in the semantics of a language, which permits us to

1. See: Alain Rey ‘Lexico-logiques’, in: Encyclopédie philosophique universelle, vol 1. Paris: PUF, 1988.

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rediscover the linguistic dimension of the definition side by side its terminological, cognitive dimension which is meant to be interlinguistic. The epistemological continuity from Plato to John Stuart Mill and Heinrich Rickert hides vast differences of cultural and intellectual backgrounds. With the passage of centuries, this background grows and is enriched; definition moves from its role of unifying logical-philosophical foundation to that of a plurality of procedures required to guarantee the validity of the diverse types of discourse which organise the knowledge and the institutions of a particular civilisation. The difficulties encountered in “defining definition” exemplify the difficulty and the necessity of defining, which has become an essential prerequisite both for the production and the understanding of the most important manifestations of social discourse, — excepting, of course, the discourse of the indefinable, the untranslatable, indeed the unnamable (Samuel Beckett) which constitutes literary and poetical creation. For no effort of rational thought can exhaust the global powers of human language.

u

Plato

Theaetetus (Extracts) [202] Socrates: Methought that I too had a dream, and I heard in my dream that the primeval letters or elements out of which you and I and all other things are compounded, have no reason or explanation; you can only name them, but no predicate can be either affirmed or denied of them, for in the one case existence, in the other non-existence is already implied, neither of which must be added, if you mean to speak of this or that thing by itself alone. It should not be called itself, or that, or each, or alone, or this, or the like; for these go about everywhere and are applied to all things, but are distinct from them; whereas, if the first elements could be described, and had a definition of their own, they would be spoken of apart from all else. But none of these primeval elements can be defined; they can only be named, for they have nothing but a name, and the things which are compounded of them, as they are complex, are expressed by a combination of names, for the combination of names is the essence of a definition. Thus, then, the elements or letters are only objects of perception, and cannot be defined or known; but the syllables or combinations of them are known and expressed, and are apprehended by true opinion. When, therefore, any one forms the true opinion of anything without rational explanation, you may say that his mind is truly exercised, but has no knowledge; for he who cannot give and receive a reason for a thing, has no knowledge of that thing; but when he adds rational explanation, then he is perfected in knowledge and may be all that I have been denying of him. Was that the form in which the dream appeared to you?

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Theaetetus: Precisely. Soc. And you allow and maintain that true opinion, combined with definition or rational explanation, is knowledge? Theaet. Exactly. Soc. Then may we assume, Theaetetus, that today, and in this casual manner, we have found a truth which in former times many wise men have grown old and have not found? Theaet. At any rate, Socrates, I am satisfied with the present statement. Soc. Which is probably correct — for how can there be knowledge apart from definition and true opinion? And yet there is one point in what has been said which does not quite satisfy me. Theaet. What was it? Soc. What might seem to be the most ingenious notion of all: — that the elements or letters are unknown, but the combination or syllables known. Theaet. And was that wrong? Soc. We shall soon know; for we have as hostages the instances which the author of the argument himself used. Theaet. What hostages? Soc. The letters, which are the elements; and the syllables, which are the combinations; — he reasoned, did he not, from the letters of the alphabet? [203] Theaet. Yes; he did. Soc. Let us take them and put them to the test, or rather, test ourselves: — what was the way in which we learned letters? and, first of all, are we right in saying that syllables have a definition, but that letters have no definition? Theaet. I think so. Soc. I think so too; for, suppose that someone asks you to spell the first syllable of my name: — Theaetetus, he says, what is SO? Theaet. I should reply S and O. Soc. That is the definition which you would give of the syllable? Theaet. I should. Soc. I wish that you would give me a similar definition of the S. Theaet. But how can anyone, Socrates, tell the elements of an element? I can only reply that S is a consonant, a mere noise, as of the tongue hissing; B, and most other letters, again, are neither vowel sounds nor noises. Thus letters may be most truly said to be undefined; for even the most distinct of them, which are the seven vowels, have a sound only, but no definition at all.

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Soc. Then, I suppose, my friend, that we have been so far right in our idea about knowledge? Theaet. Yes; I think that we have. Soc. Well, but have we been right in maintaining that the syllable can be known, but not the letters? Theaet. I think so. Soc. And do we mean by a syllable two letters, or if there are more, all of them, or a single idea which arises out of the combination of them? Theaet. I should say that we mean all the letters. Soc. Take the case of the two letters S and O, which form the first syllable of my own name; must not he who knows the syllable, know both of them? Theaet. Certainly. Soc. He knows, that is, the S and the O? Theaet. Yes. Soc. But can he be ignorant of either singly and yet know both together? Theaet. Such a supposition, Socrates, is monstrous and unmeaning. Soc. But if he cannot know both without knowing each, then, if he is ever to know the syllable, he must know the letters first; and thus the fine theory has again taken wings and departed. Theaet. Yes, with wonderful celerity. Soc. Yes, we did not keep watch properly. Perhaps we ought to have maintained that a syllable is not the letters, but rather one single idea framed out of them, having a separate form distinct from them. … [205] Soc. … must not the alternative be that either the syllable is not the letters, and then the letters are not parts of the syllable, or that the syllable will be the same with the letters, and will therefore be equally known with them? Theaet. You are right. Soc. And, in order to avoid this, we suppose it to be different from them? Theaet. Yes. Soc. But if letters are not parts of syllables, can you tell me of any other parts of syllables, which are not letters? Theaet. No, indeed, Socrates; for if I admit the existence of parts in a syllable, it would be ridiculous in me to give up letters and seek for other parts. Soc. Quite true, Theaetetus, and therefore, according to our present view, a syllable must surely be an indivisible form?

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Theaet. True. Soc. But do you remember, my friend, that only a little while ago we admitted and approved the statement, that of the first elements out of which all other things are compounded there could be no definition, because each of them when taken by itself is uncompounded; nor can one rightly attribute to then the words ‘being’ or ‘this’, because they are alien and inappropriate words, and for this reason the letters or elements were indefinable and unknown? Theaet. I remember. Soc. And is not this also the reason why they are simple and indivisible? I can see no other. Theaet. No other reason can be given. Soc. Then is not the syllable in the same case as the elements or letters, if it has no parts and is one form? Theaet. To be sure. Soc. If, then, a syllable is a whole, and has many parts or letters, the letters as well as the syllable must be intelligible and expressible, since all the parts are acknowledged to be the same as the whole? Theaet. True. Soc. But if it be one and indivisible, then the syllables and the letters are alike undefined and unknown, and for the same reason? Theaet. I cannot deny that. Soc. We cannot, therefore, agree in the opinion of him who says that the syllable can be known and expressed, but not the letters. [206] Theaet. Certainly not; if we may trust the argument. Soc. Well, but will you not be equally inclined to disagree with him, when you remember your own experience in learning to read? … Then, if we argue from the letters and syllables which we know to other simples and compounds, we shall say that the letters or simple elements as a class are much more certainly known than the syllables, and much more indispensable to a perfect knowledge of any subject; and if someone says that the syllable is known and the letter unknown, we shall consider that either intentionally or unintentionally he is talking nonsense? Theaet. Exactly. Soc. And there might be given other proofs of this belief, if I am not mistaken. But do not let us in looking for them lose sight of the question before us, which is the meaning of the statement, that right opinion with rational definition or explanation is the most perfect form of knowledge.

THEAETETUS

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Theaet. Exactly. [207] … Soc. … but I want to know first, whether you admit the resolution of all things into their elements to be a rational explanation of them, and the consideration of them in syllables or larger combinations of them to be irrational — is this your view? Theaet. Precisely. Soc. Well, and do you conceive that a man has knowledge of any element who at one time affirms and at another time denies that element of something, or thinks that the same thing is composed of different elements at different times? Theaet. Assuredly not. Soc. And do you not remember that in your case and in that of others this often occurred in the process of learning to read? Theaet. You mean that I mistook the letters and misspelt the syllables? Soc. Yes. Theaet. To be sure; I perfectly remember, and I am very far from supposing that they who are in this condition have knowledge. [208] Soc. And yet he will have explanations, as well as right opinion, for he knew the order of the letters when he wrote; and this we admit to be explanation. Theaet. True. Soc. Then my friend, there is such a thing as right opinion united with definition or explanation, which does not yet attain to the exactness of knowledge. Theaet. It would seem so. Soc. And what we fancied to be a perfect definition of knowledge is a dream only. But we had better not say so as yet, for were there not three explanations of knowledge, one of which must, as we said, be adopted by him who maintains knowledge to be true opinion combined with rational explanation? And very likely there will be found someone who will not prefer this but the third. Theaet. You are quite right; there is still one remaining. the first was the image or expression of the mind in speech; the second, which has just been mentioned, is a way of reaching the whole by an enumeration of the elements. But what is the third definition? Soc. There is, further, the popular notion of telling the mark or sign of difference which distinguishes the thing in question from all others.

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Theaet. Can you give me any example of such a definition? Soc. As, for example, in the case of the sun, I think that you would be contented with the statement that the sun is the brightest of the heavenly bodies which revolve around the earth. Theaet. Certainly. Soc. Understand why: — the reason is, as I was just now saying, that if you get at the difference and distinguishing characteristic of each thing, then, as many persons affirm, you will get at the definition or explanation of it; but while you lay hold only of the common and not of the characteristic notion, you will only have the definition of those things to which this common quality belongs. Theaet. I understand you, and your account of definition is in my judgment correct. Soc. But he, who having right opinion about anything, can find out the difference which distinguishes it from other things will know that of which before he had only an opinion. Theaet. Yes; that is what we are maintaining. Soc. Nevertheless, Theaetetus, on a nearer view, I find myself quite disappointed; the picture, which at a distance was not so bad, has now become altogether unintelligible. Theaet. What do you mean? [209] Soc. I will endeavour to explain: I will suppose myself to have true opinion of you, and if to this I add your definition, then I have knowledge, but if not, opinion only. Theaet. Yes. Soc. The definition was assumed to be the interpretation of your difference. Theaet. True. Soc. But when I had only opinion, I had no conception of your distinguishing characteristics. Theaet. I suppose not. Soc. Then I must have conceived of some general or common nature which no more belonged to you than to another. Theaet. True. Soc. Tell me, now — How in that case could I have formed a judgment of you any more than of anyone else? Suppose that I imagine Theaetetus to be a man who has a nose, eyes, and mouth, and every other member complete; how would that enable me to distinguish Theaetetus from Theodorus, or from some outer barbarian? Theaet. How could it?

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Soc. Or if I had further conceived of you, not only as having nose and eyes, but as having a snub nose and prominent eyes, should I have any more notion of you than of myself and others who resemble me? Theaet. Certainly not. Soc. Surely I can have no conception of Theaetetus until your snub-nosedness has left an impression on my mind different from the snub-nosedness of all others whom I have seen, and until your other peculiarities have a like distinctness; and so when I meet you tomorrow the right opinion will be recalled. Theaet. Most true. Soc. Then right opinion implies the perception of differences? Theaet. Clearly. Soc. What, then, shall we say of adding reason or explanation to right opinion? If the meaning is, that we should form an opinion of the way in which something differs from another thing, the proposal is ridiculous. Theaet. How so? Soc. We are supposed to acquire a right opinion of the differences which distinguish one thing from another when we have already a right opinion of them, and so we go round and round: — the revolution of the scytal, or pestle, or any other rotary machine, in the same circles, is as nothing compared with such a requirement; and we may be truly described as the blind directing the blind; for to add those things which we already have, in order that we may learn what we already think, is like a soul utterly benighted. Theaet. Tell me; what were you going to say just now, when you asked the question? Soc. If, my boy, the argument, in speaking of adding the definition, had used the word ‘know’, and not merely ‘have an opinion’ of the difference, this which is the most promising of all the definitions of knowledge would have come to a pretty end, for to know is surely to acquire knowledge. [210] Theaet. True. Soc. And so, when the question is asked, What is knowledge? this fair argument will answer ‘Right opinion with knowledge,’ — knowledge, that is, of difference, for this, as the said argument maintains, is adding the definition. Theaet. That seems to be true. Soc. But how utterly foolish, when we are asking what is knowledge, that the reply should only be, right opinion with knowledge of difference or of anything! And so, Theaetetus, knowledge is neither sensation nor true opinion, nor yet definition and explanation accompanying and added to true opinion?

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Theaet. I suppose not. Soc. And are you still in labour and travail, my dear friend, or have you brought all that you have to say about knowledge to the birth? Theaet. I am sure, Socrates, that you have elicited from me a good deal more than ever was in me. Soc. And does not my art show that you have brought forth wind, and that the offspring of your brain are not worth bringing up? Theaet. Very true. Soc. But if, Theaetetus, you should ever conceive afresh, you will be able all the better for the present investigation, and if not, you will be soberer and humbler and gentler to other men, and will be too modest to fancy that you know what you do not know. These are the limits of my art; …

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LAWS

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Laws (Extract) Book X [895] Athenian Stranger: … you would admit that we have a threefold knowledge of things? Cleinias: What do you mean? Ath. I mean that we know the essence, and that we know the definition of the essence, and the name — these are the three; and there are two questions which may be raised about anything. Cle. How two? Ath. Sometimes a person may give the name and ask the definition; or he may give the definition and ask the name. I may illustrate what I mean in this way. Cle. How? Ath. Number like some other things is capable of being divided into equal parts; when thus divided, number is named ‘even’, and the definition of the name ‘even’ is ‘number divisible into two equal parts’? Cle. True. Ath. I mean, that when we are asked about the definition and give the name, or when we are asked about the name and give the definition — in either case, whether we give name or definition, we speak of the same thing, calling ‘even’ the number which is divided into two equal parts. Cle. Quite true. Ath. And what is the definition of that which is named ‘soul’? [896] Can we conceive of any other than that which has already been given — the motion which can move itself? Cle. You mean to say that the essence which is defined as the selfmoved is the same with that which has the name soul? Ath. Yes; and if this is true, do we still maintain that there is anything wanting in the proof that the soul is the first origin and moving power of all that is, or has become, or will be, and their contraries, when she has clearly been shown to be the source of change and motion in all things? Cle. Certainly not; the soul as being the source of motion, has been most satisfactorily shown to be the oldest of all things.

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Aristotle

Analytica Posteriora (Extracts)

BOOK I (75B, 30) 8 The same is true of definitions, since a definition is either a primary premiss or a conclusion of a demonstration, or else only differs from a demonstration in the order of its terms.

BOOK I (76A–76B) 10 I call the basic truths of every genus those elements in it the existence of which cannot be proved. As regards both these primary truths and the attributes dependent on them the meaning of the name is assumed. The fact of their existence as regards the primary truths must be assumed; but it has to be proved of the remainder, the attributes. Thus we assume the meaning alike of unity, straight, and triangular; but while as regards unity and magnitude we assume also the fact of their existence, in the case of the remainder proof is required. Of the basic truths used in the demonstrative sciences some are peculiar to each science, and some are common, but common only in the sense of analogous, being of use only in so far as they fall within the genus constituting the province of the science in question.

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Peculiar truths are, e.g., the definitions of line and straight; common truths are such as ‘take equals from equals and equals remain’. Only so much of these common truths is required as falls within the genus in question: for a truth of this kind will have the same force even if not used generally but applied by the geometer only to magnitudes, or by the arithmetician only to numbers. Also peculiar to a science are the subjects the existence as well as the meaning of which it assumes, and the essential attributes of which it investigates, e.g., in arithmetic units, in geometry points and lines. Both the existence and the meaning of the subjects are assumed by these sciences; but of their essential attributes only the meaning is assumed. For example arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, or of deflection or verging of lines, whereas the existence of these attributes is demonstrated by means of the axioms and from previous conclusions as premisses. Astronomy too proceeds in the same way. For indeed every demonstrative science has three elements: (1) that which it posits, the subject genus whose essential attributes it examines; (2) the so-called axioms, which are primary premisses of its demonstration; (3) the attributes, the meaning of which it assumes. Yet some sciences may very well pass over some of these elements; e.g. we might not expressly posit the existence of the genus if its existence were obvious (for instance, the existence of hot and cold is more evident than that of number); or we might omit to assume expressly the meaning of the attributes if it were well understood. In the same way the meaning of axioms, such as ‘Take equals from equals and equals remain’, is well known and so not expressly assumed. Nevertheless in the nature of the case the essential elements of demonstration are three: the subject, the attributes, and the basic premisses. That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate — I say ‘must believe’, because all syllogism, and therefore a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object. That which is capable of proof but assumed by the teacher without proof is, if the pupil believes and accepts it, hypothesis, though only in a limited sense hypothesis — that is, relatively to the pupil; if the pupil has no opinion or a contrary opinion on the matter, the same assumption is an illegitimate postulate. Therein lies the distinction between hypothesis and illegitimate postulate: the latter is the contrary of the pupil’s opinion, demonstrable, but assumed and used without demonstration. The definitions — viz, those which are not expressed as statements that anything is or is not — are not hypotheses: but it is in the premisses of a science

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that its hypotheses are contained. Definitions require only to be understood, and this is not hypothesis — unless it be contended that the pupil’s hearing is also an hypothesis required by the teacher.

u

BOOK II, (90a–98a) 3 It is clear, then, that all questions are a search for a ‘middle’. Let us now state how essential nature is revealed, and in what way it can be reduced to demonstration; what definition is, and what things are definable. And let us first discuss certain difficulties which these questions raise, beginning what we have to say with a point most intimately connected with our immediately preceding remarks, namely the doubt that might be felt as to whether or not it is possible to know the same thing in the same relation, both by definition and by demonstration. It might, I mean, be urged that definition is held to concern essential nature and is in every case universal and affirmative; whereas, on the other hand, some conclusions are negative and some are not universal; e.g., all in the second figure are negative, none in the third are universal. And again, not even all affirmative conclusions in the first figure are definable, e.g., ‘every triangle has its angles equal to two right angles’. An argument proving this difference between demonstration and definition is that to have scientific knowledge of the demonstrable is identical with possessing a demonstration of it: hence if demonstration of such conclusions as these is possible, there clearly cannot also be definition of them. If there could, one might know such a conclusion also in virtue of its definition without possessing the demonstration of it; for there is nothing to stop our having the one without the other. Induction too will sufficiently convince us of this difference; for never yet by defining anything — essential attribute or accident — did we get knowledge of it. Again, if to define is to acquire knowledge of a substance, at any rate such attributes are not substances. It is evident, then, that not everything demonstrable can be defined. What then? Can everything definable be demonstrated, or not? There is one of our previous arguments which covers this too. Of a single thing qua single there is a single scientific knowledge. Hence, since to know the demonstrable scientifically is to possess the demonstration of it, an impossible consequence will

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follow: — possession of its definition without its demonstration will give knowledge of the demonstrable. Moreover, the basic premisses of demonstrations are definitions, and it has already been shown that these will be found indemonstrable; either the basic premisses will be demonstrable and will depend on prior premisses, and the regress will be endless; or the primary truths will be indemonstrable definitions. But if the definable and the demonstrable are not wholly the same, may they yet be partially the same? Or is that impossible, because there can be no demonstration of the definable? There can be none, because definition is of the essential nature or being of something, and all demonstrations evidently posit and assume the essential nature — mathematical demonstrations, for example, the nature of unity and the odd, and all the other sciences likewise. Moreover, every demonstration proves a predicate of a subject as attaching or as not attaching to it, but in definition one thing is not predicated of another; we do not, e.g., predicate animal of biped nor biped of animal, nor yet figure of plane — plane not being figure nor figure plane. Again, to prove essential nature is not the same as to prove the fact of a connexion. Now definition reveals essential nature, demonstration reveals that a given attribute attaches or does not attach to a given subject; but different things require different demonstrations — unless the one demonstration is related to the other as part to whole: I add this because if all triangles have been proved to possess angles equal to two right angles, then this attribute has been proved to attach to isosceles; for isosceles is a part of which all triangles constitute the whole. But in the case before us the fact and the essential nature are not so related to one another, since the one is not a part of the other. So it emerges that not all the definable is demonstrable nor all the demonstrable definable; and we may draw the general conclusion that there is no identical object of which it is possible to possess both a definition and a demonstration. It follows obviously that definition and demonstration are neither identical nor contained either within the other: if they were, their objects would be related either as identical or as whole and part. 4 So much, then, for the first stage of our problem. The next step is to raise the question whether syllogism — i.e. demonstration — of the definable nature is possible or, as our recent argument assumed, impossible. We might argue it impossible on the following grounds: — (a) syllogism proves an attribute of a subject through the middle term; on the other hand (b) its definable nature is both ‘peculiar’ to a subject and predicated of it as belonging

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to its essence. But in that case (1) the subject, its definition, and the middle term connecting them must be reciprocally predicable of one another; for if A is ‘peculiar’ to C, obviously A is ‘peculiar’ to B and B to C — in fact all three terms are ‘peculiar’ to one another: and further (2) if A inheres in the essence of all B and B is predicated universally of all C as belonging to C’s essence, A also must be predicated of C as belonging to its essence. If one does not take this relation as thus duplicated — if, that is, A is predicated as being of the essence of B, but B is not of the essence of the subjects of which it is predicated — A will not necessarily be predicated of C as belonging to its essence. So both premisses will predicate essence, and consequently B also will be predicated of C as its essence. Since, therefore, both premisses do predicate essence — i.e. definable form — C’s definable form will appear in the middle term before the conclusion is drawn. We may generalize by supposing that it is possible to prove the essential nature of man. Let C be man, A man’s essential nature — two-footed animal, or aught else it may be. Then, if we are to syllogize, A must be predicated of all B. But this premiss will be mediated by a fresh definition, which consequently will also be the essential nature of man. Therefore the argument assumes what it has to prove, since B too is the essential nature of man. It is, however, the case in which there are only the two premisses — i.e. in which the premisses are primary and immediate — which we ought to investigate, because it best illustrates the point under discussion. Thus they who prove the essential nature of soul or man or anything else through reciprocating terms beg the question. It would be begging the question, for example, to contend that the soul is that which causes its own life, and that what causes its own life is a self-moving number; for one would have to postulate that the soul is a selfmoving number in the sense of being identical with it. For if A is predicable as a mere consequent of B and B of C, A will not on that account be the definable form of C: A will merely be what it was true to say of C. Even if A is predicated of all B inasmuch as B is identical with a species of A, still it will not follow: being an animal is predicated of being a man — since it is true that in all instances to be human is to be animal, just as it is also true that every man is an animal — but not as identical with being man. We conclude, then, that unless one takes both the premisses as predicating essence, one cannot infer that A is the definable form and essence of C: but if one does so take them, in assuming B one will have assumed, before drawing the conclusion, what the definable form of C is; so that there has been no inference, for one has begged the question.

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5 Nor, as was said in my formal logic, is the method of division a process of inference at all, since at no point does the characterization of the subject follow necessarily from the premising of certain other facts: division demonstrates as little as does induction. For in a genuine demonstration the conclusion must not be put as a question nor depend on a concession, but must follow necessarily from its premisses, even if the respondent deny it. The definer asks ‘Is man animal or inanimate?’ and then assumes — he has not inferred — that man is animal. Next, when presented with an exhaustive division of animal into terrestrial and aquatic, he assumes that man is terrestrial. Moreover, that man is the complete formula, terrestrial-animal, does not follow necessarily from the premisses: this too is an assumption, and equally an assumption whether the division comprises many differentiae or few. (Indeed as this method of division is used by those who proceed by it, even truths that can be inferred actually fail to appear as such.) For why should not the whole of this formula be true of man, and yet not exhibit his essential nature or definable form? Again, what guarantee is there against an unessential addition, or against the omission of the final or of an intermediate determinant of the substantial being? The champion of division might here urge that though these lapses do occur, yet we can solve that difficulty if all the attributes we assume are constituents of the definable form, and if, postulating the genus, we produce by division the requisite uninterrupted sequence of terms, and omit nothing; and that indeed we cannot fail to fulfil these conditions if what is to be divided falls whole into the division at each stage, and none of it is omitted; and that this — the dividendum — must without further question be (ultimately) incapable of fresh specific division. Nevertheless, we reply, division does not involve inference; if it gives knowledge, it gives it in another way. Nor is there any absurdity in this: induction, perhaps, is not demonstration any more than is division, yet it does make evident some truth. Yet to state a definition reached by division is not to state a conclusion: as, when conclusions are drawn without their appropriate middles, the alleged necessity by which the inference follows from the premisses is open to a question as to the reason for it, so definitions reached by division invite the same question. Thus to the question ‘What is the essential nature of man?’ the divider replies ‘Animal, mortal, footed, biped, wingless’; and when at each step he is asked ‘Why?’, he will say, and, as he thinks, prove by division, that all animal is mortal or immortal: but such a formula taken in its entirety is not definition; so that even if division does demonstrate its formula, definition at any rate does not turn out to be a conclusion of inference.

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6 Can we nevertheless actually demonstrate what a thing essentially and substantially is, but hypothetically, i.e. by premising (1) that its definable form is constituted by the ‘peculiar’ attributes of its essential nature; (2) that such and such are the only attributes of its essential nature, and that the complete synthesis of them is peculiar to the thing; and thus — since in this synthesis consists the being of the thing — obtaining our conclusion? Or is the truth that, since proof must be through the middle term the definable form is once more assumed in this minor premiss too? Further, just as in syllogizing we do not premise what syllogistic inference is (since the premisses from which we conclude must be related as whole and part) so the definable form must not fall within the syllogism but remain outside the premisses posited. It is only against a doubt as to its having been a syllogistic inference at all that we have to defend our argument as conforming to the definition of syllogism. It is only when some one doubts whether the conclusion proved is the definable form that we have to defend it as conforming to the definition of definable form which we assumed. Hence syllogistic inference must be possible even without the express statement of what syllogism is or what definable form is. The following type of hypothetical proof also begs the question. If evil is definable as the divisible, and the definition of a thing’s contrary — if it has one — is the contrary of the thing’s definition; then, if good is the contrary of evil and the indivisible of the divisible, we conclude that to be good is essentially to be indivisible. The question is begged because definable form is assumed as a premiss, and as a premiss which is to prove definable form. ‘But not the same definable form’ you may object. That I admit, for in demonstrations also we premise that ‘this’ is predicable of ‘that’; but in this premiss the term we assert of the minor is neither the major itself nor a term identical in definition, or convertible, with the major. Again, both proof by division and the syllogism just described are open to the question why man should be animal-biped-terrestrial and not merely animal and terrestrial, since what they premise does not ensure that the predicates shall constitute a genuine unity and not merely belong to a single subject as do musical and grammatical when predicated of the same man. 7 How then by definition shall we prove substance or essential nature? We cannot show it as a fresh fact necessarily following from the assumption of premisses admitted to be facts — the method of demonstration: we may not proceed as by

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induction to establish a universal on the evidence of groups of particulars which offer no exception, because induction proves not what the essential nature of a thing is but that it has or has not some attribute. Therefore, since presumably one cannot prove essential nature by an appeal to sense perception or by pointing with the finger, what other method remains? To put it another way: how shall we by definition prove essential nature? He who knows what human — or any other — nature is, must know also that man exists; for no one knows the nature of what does not exist — one can know the meaning of the phrase or name ‘goat-stag’ but not what the essential nature of a goat-stag is. But further, if definition can prove what is the essential nature of a thing, can it also prove that it exists? And how will it prove them both by the same process, since definition exhibits one single thing and demonstration another single thing, and what human nature is and the fact that man exists are not the same thing? Then too we hold that it is by demonstration that the being of everything must be proved — unless indeed to be were its essence; and, since being is not a genus, it is not the essence of anything. Hence the being of anything as fact is matter for demonstration; and this is the actual procedure of the sciences, for the geometer assumes the meaning of the word triangle, but that it is possessed of some attribute he proves. What is it, then, that we shall prove in defining essential nature? Triangle? In that case a man will know by definition what a thing’s nature is without knowing whether it exists. But that is impossible. Moreover it is clear, if we consider the methods of defining actually in use, that definition does not prove that the thing defined exists: since even if there does actually exist something which is equidistant from a centre, yet why should the thing named in the definition exist? Why, in other words, should this be the formula defining ‘circle’? One might equally well call it the definition of mountain copper. For definitions do not carry a further guarantee that the thing defined can exist or that it is what they claim to define: one can always ask why. Since, therefore, to define is to prove either a thing’s essential nature or the meaning of its name, we may conclude that definition, if it in no sense proves essential nature, is a set of words signifying precisely what a name signifies. But that were a strange consequence; for (1) both what is not substance and what does not exist at all would be definable, since even non-existents can be signified by a name: (2) all sets of words or sentences would be definitions, since any kind of sentence could be given a name; so that we should all be talking in definitions, and even the Iliad would be a definition: (3) no demonstration can prove that any particular name means any particular thing: neither, therefore, do definitions, in addition to revealing the meaning of a name, also reveal that the name has this meaning. It appears then from these considerations that neither

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definition and syllogism nor their objects are identical and further that definition neither demonstrates nor proves anything, and that knowledge of essential nature is not to be obtained either by definition or by demonstration. 8 We must now start afresh and consider which of these conclusions are sound and which are not, and what is the nature of definition, and whether essential nature is in any sense demonstrable and definable or in none. Now to know its essential nature is, as we said, the same as to know the cause of a thing’s existence, and the proof of this depends on the fact that a thing must have a cause. Moreover, this cause is either identical with the essential nature of the thing or distinct from it; and if its cause is distinct from it, the essential nature of the thing is either demonstrable or indemonstrable. Consequently, if the cause is distinct from the thing’s essential nature and demonstration is possible, the cause must be the middle term, and, the conclusion proved being universal and affirmative, the proof is in the first figure. So the method just examined of proving it through another essential nature would be one way of proving essential nature, because a conclusion containing essential nature must be inferred through a middle which is an essential nature just as a ‘peculiar’ property must be inferred through a middle which is a ‘peculiar’ property; so that of the two definable natures of a single thing this method will prove one and not the other. Now it was said before that this method could not amount to demonstration of essential nature but is actually a dialectical proof of it — so let us begin again and explain by what method it can be demonstrated. When we are aware of a fact we seek its reason, and though sometimes the fact and the reason dawn on us simultaneously, yet we cannot apprehend the reason a moment sooner than the fact; and clearly in just the same way we cannot apprehend a thing’s definable form without apprehending that it exists, since while we are ignorant whether it exists we cannot know its essential nature. Moreover we are aware whether a thing exists or not sometimes through apprehending an element in its character, and sometimes accidentally, as, for example, when we are aware of thunder as a noise in the clouds, of eclipse as a privation of light, or of man as some species of animal, or of the soul as a self-moving thing. As often as we have accidental knowledge that the thing exists, we must be in a wholly negative state as regards awareness of its essential nature; for we have not got genuine knowledge even of its existence, and to search for a thing’s essential nature when we are unaware that it exists is to search for nothing. On the other hand, whenever we apprehend an element in the thing’s character there is less difficulty.

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Thus it follows that the degree of our knowledge of a thing’s essential nature is determined by the sense in which we are aware that it exists. Let us then take the following as our first instance of being aware of an element in the essential nature. Let A be eclipse, C the moon, B the earth’s acting as a screen. Now to ask whether the moon is eclipsed or not is to ask whether or not B has occurred. But that is precisely the same as asking whether A has a defining condition; and if this condition actually exists, we assert that A also actually exists. Or again we may ask which side of a contradiction the defining condition necessitates: does it make the angles of a triangle equal or not equal to two right angles? When we have found the answer, if the premisses are immediate, we know fact and reason together; if they are not immediate, we know the fact without the reason, as in the following example: let C be the moon, A eclipse, B the fact that the moon fails to produce shadows though she is full and though no visible body intervenes between us and her. Then if B, failure to produce shadows in spite of the absence of an intervening body, is attributable to C, and A, eclipse, is attributable to B, it is clear that the moon is eclipsed, but the reason why is not yet clear, and we know that eclipse exists, but we do not know what its essential nature is. But when it is clear that A is attributable to C and we proceed to ask the reason of this fact, we are inquiring what is the nature of B: is it the earth’s acting as a screen, or the moon’s rotation or her extinction? But B is the definition of the other term, viz., in these examples, of the major term A; for eclipse is constituted by the earth acting as a screen. Thus, (1) ‘What is thunder?’ ‘The quenching of fire in cloud’ and (2) ‘Why does it thunder?’ ‘Because fire is quenched in the cloud’ are equivalent. Let C be cloud, A thunder, B the quenching of fire. Then B is attributable to C, cloud, since fire is quenched in it; and A, noise, is attributable to B; and B is assuredly the definition of the major term A. If there be a further mediating cause of B, it will be one of the remaining partial definitions of A. We have stated then how essential nature is discovered and becomes known, and we see that, while there is no syllogism — i.e. no demonstrative syllogism — of essential nature, yet it is through syllogism, viz. demonstrative syllogism, that essential nature is exhibited. So we conclude that neither can the essential nature of anything which has a cause distinct from itself be known without demonstration, nor can it be demonstrated; and this is what we contended in our preliminary discussions. 9 Now while some things have a cause distinct from themselves, others have not. Hence it is evident that there are essential natures which are immediate, that is

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are basic premisses; and of these not only that they are but also what they are must be assumed or revealed in some other way. This too is the actual procedure of the arithmetician, who assumes both the nature and the existence of unit. On the other hand, it is possible (in the manner explained) to exhibit through demonstration the essential nature of things which have a ‘middle’, i.e. a cause of their substantial being other than that being itself; but we do not thereby demonstrate it. 10 Since definition is said to be the statement of a thing’s nature, obviously one kind of definition will be a statement of the meaning of the name, or of an equivalent nominal formula. A definition in this sense tells you, e.g., the meaning of the phrase ‘triangular character’. When we are aware that triangle exists, we inquire the reason why it exists. But it is difficult thus to learn the definition of things the existence of which we do not genuinely know — the cause of this difficulty being, as we said before, that we only know accidentally whether or not the thing exists. Moreover, a statement may be a unity in either of two ways, by conjunction, like the Iliad, or because it exhibits a single predicate as inhering not accidentally in a single subject. That then is one way of defining definition. Another kind of definition is a formula exhibiting the cause of a thing’s existence. Thus the former signifies without proving, but the latter will clearly be a quasi-demonstration of essential nature, differing from demonstration in the arrangement of its terms. For there is a difference between stating why it thunders, and stating what is the essential nature of thunder; since the first statement will be ‘Because fire is quenched in the clouds’, while the statement of what the nature of thunder is will be ‘The noise of fire being quenched in the clouds’. Thus the same statement takes a different form: in one form it is continuous demonstration, in the other definition. Again, thunder can be defined as noise in the clouds, which is the conclusion of the demonstration embodying essential nature. On the other hand the definition of immediates is an indemonstrable positing of essential nature. We conclude then that definition is (a) an indemonstrable statement of essential nature, or (b) a syllogism of essential nature differing from demonstration in grammatical form, or (c) the conclusion of a demonstration giving essential nature. Our discussion has therefore made plain (1) in what sense and of what things the essential nature is demonstrable, and in what sense and of what things it is not; (2) what are the various meanings of the term definition, and in what sense and of what things it proves the essential nature, and in what sense and of

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what things it does not; (3) what is the relation of definition to demonstration, and how far the same thing is both definable and demonstrable and how far it is not. 11 We think we have scientific knowledge when we know the cause, and there are four causes: (1) the definable form, (2) an antecedent which necessitates a consequent, (3) the efficient cause, (4) the final cause. Hence each of these can be the middle term of a proof, for (a) though the inference from antecedent to necessary consequent does not hold if only one premiss is assumed — two is the minimum — still when there are two it holds on condition that they have a single common middle term. So it is from the assumption of this single middle term that the conclusion follows necessarily. The following example will also show this. Why is the angle in a semicircle a right angle? — or from what assumption does it follow that it is a right angle? Thus, let A be right angle, B the half of two right angles, C the angle in a semicircle. Then B is the cause in virtue of which A, right angle, is attributable to C, the angle in a semicircle, since B=A and the other, viz. C, =B, for C is half of two right angles. Therefore it is the assumption of B, the half of two right angles, from which it follows that A is attributable to C, i.e. that the angle in a semicircle is a right angle. Moreover, B is identical with (b) the defining form of A, since it is what A’s definition signifies. Moreover, the formal cause has already been shown to be the middle. (c) ‘Why did the Athenians become involved in the Persian war?’ means ‘What cause originated the waging of war against the Athenians?’ and the answer is, ‘Because they raided Sardis with the Eretrians’ since this originated the war. Let A be war, B unprovoked raiding, C the Athenians. Then B, unprovoked raiding, is true of C, the Athenians, and A is true of B, since men make war on the unjust aggressor. So A, having war waged upon them, is true of B, the initial aggressors, and B is true of C, the Athenians, who were the aggressors. Hence here too the cause — in this case the efficient cause — is the middle term. (d) This is no less true where the cause is the final cause. E.g., why does one take a walk after supper? For the sake of one’s health. Why does a house exist? For the preservation of one’s goods. The end in view is in the one case health, in the other preservation. To ask the reason why one must walk after supper is precisely to ask to what end one must do it. Let C be walking after supper, B the non-regurgitation of food, A health. Then let walking after supper possess the property of preventing food from rising to the orifice of the stomach, and let this condition be healthy; since it seems that B, the non-regurgitation of food, is attributable to C, taking a walk, and that A, health, is attributable to B. What, then, is the cause through which A, the final cause, inheres in C? It is B,

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the non-regurgitation of food; but B is a kind of definition of A, for A will be explained by it. Why is B the cause of A’s belonging to C? Because to be in a condition such as B is to be in health. The definitions must be transposed, and then the detail will become clearer. Incidentally, here the order of coming to be is the reverse of what it is in proof through the efficient cause: in the efficient order the middle term must come to be first, whereas in the teleological order the minor, C, must first take place, and the end in view comes last in time. The same thing may exist for an end and be necessitated as well. For example, light shines through a lantern (1) because that which consists of relatively small particles necessarily passes through pores larger than those particles — assuming that light does issue by penetration — and (2) for an end, namely to save us from stumbling. If, then, a thing can exist through two causes, can it come to be through two causes — as for instance if thunder be a hiss and a roar necessarily produced by the quenching of fire, and also designed, as the Pythagoreans say, for a threat to terrify those that lie in Tartarus? Indeed, there are very many such cases, mostly among the processes and products of the natural world; for nature, in different senses of the term ‘nature’, produces now for an end, now by necessity. Necessity too is of two kinds. It may work in accordance a with a thing’s natural tendency, or by constraint and in opposition to it; as, for instance, by necessity a stone is borne both upwards and downwards, but not by the same necessity. Of the products of man’s intelligence some are never due to chance or necessity but always to an end, as for example a house or a statue; others, such as health or safety, may result from chance as well. It is mostly in cases where the issue is indeterminate (though only where the production does not originate in chance, and the end is consequently good), that a result is due to an end, and this is true alike in nature or in art. By chance, on the other hand, nothing comes to be for an end. 12 The effect may be still coming to be, or its occurrence may be past or future, yet the cause will be the same as when it is actually existent — for it is the middle which is the cause — except that if the effect actually exists the cause is actually existent, if it is coming to be, so is the cause, if its occurrence is past the cause is past, if future, the cause is future. For example, the moon was eclipsed because the earth intervened, is becoming eclipsed because the earth is in process of intervening, will be eclipsed because the earth will intervene, is eclipsed because the earth intervenes. To take a second example: assuming that the definition of ice is solidified

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water, let C be water, A solidified, B the middle, which is the cause, namely total failure of heat. Then B is attributed to C, and A, solidification, to B: ice forms when B is occurring, has formed when B has occurred, and will form when B shall occur. This sort of cause, then, and its effect come to be simultaneous when they are in process of becoming, and exist simultaneously when they actually exist; and the same holds good when they are past and when they are future. But what of cases where they are not simultaneous? Can causes and effects different from one another form, as they seem to us to form, a continuous succession, a past effect resulting from a past cause different from itself, a future effect from a future cause different from it, and an effect which is coming-to-be from a cause different from and prior to it? Now on this theory it is from the posterior event that we reason (and this though these later events actually have their source of origin in previous events — a fact which shows that also when the effect is coming-to-be we still reason from the posterior event), and from the prior event we cannot reason (we cannot argue that because an event A has occurred, therefore an event B has occurred subsequently to A but still in the past — and the same holds good if the occurrence is future) — cannot reason because, be the time interval definite or indefinite, it will never be possible to infer that because it is true to say that A occurred, therefore it is true to say that B, the subsequent event, occurred; for in the interval between the events, though A has already occurred, the latter statement will be false. And the same argument applies also to future events; i.e. one cannot infer from an event which occurred in the past that a future event will occur. The reason of this is that the middle must be homogeneous, past when the extremes are past, future when they are future, coming to be when they are coming-to-be, actually existent when they are actually existent; and there cannot be a middle term homogeneous with extremes respectively past and future. And it is a further difficulty in this theory that the time interval can be neither indefinite nor definite, since during it the inference will be false. We have also to inquire what it is that holds events together so that the coming-to-be now occurring in actual things follows upon a past event. It is evident, we may suggest, that a past event and a present process cannot be ‘contiguous, for not even two past events can be ‘contiguous’. For past events are limits and atomic; so just as points are not ‘contiguous’ neither are past events, since both are indivisible. For the same reason a past event and a present process cannot be ‘contiguous’, for the process is divisible, the event indivisible. Thus the relation of present process to past event is analogous to that of line to point, since a process contains an infinity of past events. These questions, however, must receive a more explicit treatment in our general theory of change.

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The following must suffice as an account of the manner in which the middle would be identical with the cause on the supposition that coming-to-be is a series of consecutive events: for in the terms of such a series too the middle and major terms must form an immediate premiss; e.g. we argue that, since C has occurred, therefore A occurred: and C’s occurrence was posterior, A’s prior; but C is the source of the inference because it is nearer to the present moment, and the starting-point of time is the present. We next argue that, since D has occurred, therefore C occurred. Then we conclude that, since D has occurred, therefore A must have occurred; and the cause is C, for since D has occurred C must have occurred, and since C has occurred A must previously have occurred. If we get our middle term in this way, will the series terminate in an immediate premiss, or since, as we said, no two events are ‘contiguous’, will a fresh middle term always intervene because there is an infinity of middles? No: though no two events are ‘contiguous’, yet we must start from a premiss consisting of a middle and the present event as major. The like is true of future events too, since if it is true to say that D will exist, it must be a prior truth to say that A will exist, and the cause of this conclusion is C; for if D will exist, C will exist prior to D, and if C will exist, A will exist prior to it. And here too the same infinite divisibility might be urged, since future events are not ‘contiguous’. But here too an immediate basic premiss must be assumed. And in the world of fact this is so: if a house has been built, then blocks must have been quarried and shaped. The reason is that a house having been built necessitates a foundation having been laid, and if a foundation has been laid blocks must have been shaped beforehand. Again, if a house will be built, blocks will similarly be shaped beforehand; and proof is through the middle in the same way, for the foundation will exist before the house. Now we observe in Nature a certain kind of circular process of coming-tobe; and this is possible only if the middle and extreme terms are reciprocal, since conversion is conditioned by reciprocity in the terms of the proof. This — the convertibility of conclusions and premisses — has been proved in our early chapters, and the circular process is an instance of this. In actual fact it is exemplified thus: when the earth had been moistened an exhalation was bound to rise, and when an exhalation had risen cloud was bound to form, and from the formation of cloud rain necessarily resulted, and by the fall of rain the earth was necessarily moistened: but this was the starting-point, so that a circle is completed; for posit any one of the terms and another follows from it, and from that another, and from that again the first. Some occurrences are universal (for they are, or come-to-be what they are, always and in every case); others again are not always what they are but only as

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a general rule: for instance, not every man can grow a beard, but it is the general rule. In the case of such connexions the middle term too must be a general rule. For if A is predicated universally of B and B of C, A too must be predicated always and in every instance of C, since to hold in every instance and always is of the nature of the universal. But we have assumed a connexion which is a general rule; consequently the middle term B must also be a general rule. So connexions which embody a general rule — i.e. which exist or come to be as a general rule — will also derive from immediate basic premisses. 13 We have already explained how essential nature is set out in the terms of a demonstration, and the sense in which it is or is not demonstrable or definable; so let us now discuss the method to be adopted in tracing the elements predicated as constituting the definable form. Now of the attributes which inhere always in each several thing there are some which are wider in extent than it but not wider than its genus (by attributes of wider extent I mean all such as are universal attributes of each several subject, but in their application are not confined to that subject). I.e. while an attribute may inhere in every triad, yet also in a subject not a triad — as being inheres in triad but also in subjects not numbers at all — odd on the other hand is an attribute inhering in every triad and of wider application (inhering as it does also in pentad), but which does not extend beyond the genus of triad; for pentad is a number, but nothing outside number is odd. It is such attributes which we have to select up to the exact point at which they are severally of wider extent than the subject but collectively coextensive with it; for this synthesis must be the substance of the thing. For example every triad possesses the attributes number, odd, and prime in both senses, i.e. not only as possessing no divisors, but also as not being a sum of numbers. This, then, is precisely what triad is, viz, a number, odd, and prime in the former and also the latter sense of the term: for these attributes taken severally apply, the first two to all odd numbers, the last to the dyad also as well as to the triad, but, taken collectively, to no other subject. Now since we have shown above that attributes predicated as belonging to the essential nature are necessary and that universals are necessary, and since the attributes which we select as inhering in triad, or in any other subject whose attributes we select in this way, are predicated as belonging to its essential nature, triad will thus possess these attributes necessarily. Further, that the synthesis of them constitutes the substance of triad is shown by the following argument. If it is not identical with the being of triad, it must be related to triad as a genus named or nameless. It will then be of wider extent than triad —

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assuming that wider potential extent is the character of a genus. If on the other hand this synthesis is applicable to no subject other than the individual triads, it will be identical with the being of triad, because we make the further assumption that the substance of each subject is the predication of elements in its essential nature down to the last differentia characterizing the individuals. It follows that any other synthesis thus exhibited will likewise be identical with the being of the subject. The author of a hand-book on a subject that is a generic whole should divide the genus into its first infimae species — number e.g. into triad and dyad — and then endeavour to seize their definitions by the method we have described — the definition, for example, of straight line or circle or right angle. After that, having established what the category is to which the subaltern genus belongs — quantity or quality, for instance — he should examine the properties ‘peculiar’ to the species, working through the proximate common differentiae. He should proceed thus because the attributes of the genera compounded of the infimae species will be clearly given by the definitions of the species; since the basic element of them is the definition, i.e. the simple infimae species, and the attributes inhere essentially in the simple infimae species, in the genera only in virtue of these. Divisions according to differentiae are a useful accessory to this method. What force they have as proofs we did, indeed, explain above, but that merely towards collecting the essential nature they may be of use we will proceed to show. They might, indeed, seem to be of no use at all, but rather to assume everything at the start and to be no better than an initial assumption made without division. But, in fact, the order in which the attributes are predicated does make a difference — it matters whether we say animal — tame — biped, or biped — animal — tame. For if every definable thing consists of two elements and ‘animal-tame’ forms a unity, and again out of this and the further differentia man (or whatever else is the unity under construction) is constituted, then the elements we assume have necessarily been reached by division. Again, division is the only possible method of avoiding the omission of any element of the essential nature. Thus, if the primary genus is assumed and we then take one of the lower divisions, the dividendum will not fall whole into this division: e.g. it is not all animal which is either whole-winged or split-winged but all winged animal, for it is winged animal to which this differentiation belongs. The primary differentiation of animal is that within which all animal falls. The like is true of every other genus, whether outside animal or a subaltern genus of animal; e.g. the primary differentiation of bird is that within which falls every bird, of fish that within which falls every fish. So, if we proceed in this way, we

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can be sure that nothing has been omitted: by any other method one is bound to omit something without knowing it. To define and divide one need not know the whole of existence. Yet some hold it impossible to know the differentiae distinguishing each thing from every single other thing without knowing every single other thing; and one cannot, they say, know each thing without knowing its differentiae, since everything is identical with that from which it does not differ, and other than that from which it differs. Now first of all this is a fallacy: not every differentia precludes identity, since many differentiae inhere in things specifically identical, though not in the substance of these nor essentially. Secondly, when one has taken one’s differing pair of opposites and assumed that the two sides exhaust the genus, and that the subject one seeks to define is present in one or other of them, and one has further verified its presence in one of them; then it does not matter whether or not one knows all the other subjects of which the differentiae are also predicated. For it is obvious that when by this process one reaches subjects incapable of further differentiation one will possess the formula defining the substance. Moreover, to postulate that the division exhausts the genus is not illegitimate if the opposites exclude a middle; since if it is the differentia of that genus, anything contained in the genus must lie on one of the two sides. In establishing a definition by division one should keep three objects in view: (1) the admission only of elements in the definable form, (2) the arrangement of these in the right order, (3) the omission of no such elements. The first is feasible because one can establish genus and differentia through the topic of the genus, just as one can conclude the inherence of an accident through the topic of the accident. The right order will be achieved if the right term is assumed as primary, and this will be ensured if the term selected is predicable of all the others but not all they of it; since there must be one such term. Having assumed this we at once proceed in the same way with the lower terms; for our second term will be the first of the remainder, our third the first of those which follow the second in a ‘contiguous’ series, since when the higher term is excluded, that term of the remainder which is ‘contiguous’ to it will be primary, and so on. Our procedure makes it clear that no elements in the definable form have been omitted: we have taken the differentia that comes first in the order of division, pointing out that animal, e.g., is divisible exhaustively into A and B, and that the subject accepts one of the two as its predicate. Next we have taken the differentia of the whole thus reached, and shown that the whole we finally reach is not further divisible — i.e. that as soon as we have taken the last differentia to form the concrete totality, this totality admits of no division into species. For it is clear that there is no superfluous addition, since all these terms

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we have selected are elements in the definable form; and nothing lacking, since any omission would have to be a genus or a differentia. Now the primary term is a genus, and this term taken in conjunction with its differentiae is a genus: moreover the differentiae are all included, because there is now no further differentia; if there were, the final concrete would admit of division into species, which, we said, is not the case. To resume our account of the right method of investigation: We must start by observing a set of similar — i.e. specifically identical — individuals, and consider what element they have in common. We must then apply the same process to another set of individuals which belong to one species and are generically but not specifically identical with the former set. When we have established what the common element is in all members of this second species, and likewise in members of further species, we should again consider whether the results established possess any identity, and persevere until we reach a single formula, since this will be the definition of the thing. But if we reach not one formula but two or more, evidently the definiendum cannot be one thing but must be more than one. I may illustrate my meaning as follows. If we were inquiring what the essential nature of pride is, we should examine instances of proud men we know of to see what, as such, they have in common; e.g., if Alcibiades was proud, or Achilles and Ajax were proud, we should find, on inquiring what they all had in common that it was intolerance of insult; it was this which drove Alcibiades to war, Achilles to wrath, and Ajax to suicide. We should next examine other cases, Lysander, for example, or Socrates, and then if these have in common indifference alike to good and ill fortune, I take these two results and inquire what common element have equanimity amid the vicissitudes of life and impatience of dishonour. If they have none, there will be two genera of pride. Besides, every definition is always universal and commensurate: the physician does not prescribe what is healthy for a single eye, but for all eyes or for a determinate species of eye. It is also easier by this method to define the single species than the universal, and that is why our procedure should be from the several species to the universal genera — this for the further reason too that equivocation is less readily detected in genera than in infimae species. Indeed, perspicuity is essential in definitions, just as inferential movement is the minimum required in demonstrations; and we shall attain perspicuity if we can collect separately the definition of each species through the group of singulars which we have established — e.g., the definition of similarity not unqualified but restricted to colours and to figures; the definition of acuteness, but only of sound, and so proceed to the common universal with a careful avoidance of equivocation. We may add that if dialectical disputation must not employ metaphors, clearly

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metaphors and metaphorical expressions are precluded in definition: otherwise dialectic would involve metaphors. 14 In order to formulate the connexions we wish to prove we have to select our analyses and divisions. The method of selection consists in laying down the common genus of all our subjects of investigation — if e.g., they are animals, we lay down what the properties are which inhere in every animal. These established, we next lay down the properties essentially connected with the first of the remaining classes — e.g., if this first subgenus is bird, the essential properties of every bird — and so on, always characterizing the proximate subgenus. This will clearly at once enable us to say in virtue of what character the subgenera — man, e.g., or horse — possess their properties. Let A be animal, B the properties of every animal, C D E various species of animal. Then it is clear in virtue of what character B inheres in D — namely A — and that it inheres in C and E for the same reason: and throughout the remaining subgenera always the same rule applies. We are now taking our examples from the traditional class-names, but we must not confine ourselves to considering these. We must collect any other common character which we observe, and then consider with what species it is connected and what properties belong to it. For example, as the common properties of horned animals we collect the possession of a third stomach and only one row of teeth. Then since it is clear in virtue of what character they possess these attributes — namely their horned character — the next question is, to what species does the possession of horns attach? Yet a further method of selection is by analogy: for we cannot find a single identical name to give to a squid’s pounce, a fish’s spine, and an animal’s bone, although these too possess common properties as if there were a single osseous nature.

Topics Extracts BOOK I, (101b–103b) 4 Since, however, of what is peculiar to anything partly signifies its essence and partly not, let us divide the ‘peculiar’ into both the aforesaid parts, and call that part which indicates the essence a ‘definition’, while of the remainder let us

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adopt the terminology which is generally current about these things and speak of it as a ‘property’. What we have said, then, makes it clear that according to our present division, the elements turn out to be four, all told, namely either property or definition or genus or accident. Do not let anyone suppose us to mean that each of these enunciated by itself constitutes a proposition or problem, but only that it is from these that both problems and propositions are formed. The difference between a problem and a proposition is a difference in the turn of the phrase. For if it be put in this way, “‘An animal that walks on two feet’ is the definition of a man, is it not?” or “‘Animal’ is the genus of man, is it not?” the result is a proposition: but if thus “Is ‘An animal that walks on two feet’ definition of man or no?” [or “Is ‘animal’ his genus or no?”] the result is a problem. Similarly too in other cases. Naturally, then, problems and propositions are equal in number; for out of every proposition you will make a problem if you change the turn of the phrase. 5 We must now say what are ‘definition’, ‘property’,‘genus’ and ‘accident’. A ‘definition’ is a phrase signifying a thing’s essence. It is rendered in the form either of a phrase in lieu of a term, or of a phrase in lieu of another phrase; for it is sometimes possible to define the meaning of a phrase as well. People whose rendering consists of a term only, try it as they may, clearly do not render the definition of the thing in question, because a definition is always a phrase of a certain kind. One may, however, use the word ‘definitory’ also of such a remark as “The ‘becoming’ is ‘beautiful’”, and likewise also of the question, ‘Are sensation and knowledge the same or different?’, for argument about definitions is mostly concerned with questions of sameness and difference. In a word we may call ‘definitory’ everything that falls under the same branch of inquiry as definitions; and that all the above-mentioned examples are of this character is clear on the face of them. For if we are able to argue that two things are the same or are different, we shall be well supplied by the same turn of argument with lines of attack upon their definitions as well: for when we have shown that they are not the same, we shall have demolished the definition. Observe, please, that the converse of this last statement does not hold: for to show that they are the same is not enough to establish a definition. To show, however, that they are not the same is enough of itself to overthrow it. A ‘property’ is a predicate which does not indicate the essence of a thing, but yet belongs to that thing alone, and is predicated convertibly of it. That it is a property of man to be capable of learning grammar: for if A be a man, then he is capable of learning grammar, and if he is capable of learning grammar, he is

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a man. For no one calls anything a ‘property’ which may possibly belong to something else, e.g. ‘sleep’ in the case of man, even though at a certain time it may happen to belong to him alone. That is to say, if any such thing were actually to be called a property, it will be called not a ‘property’ absolutely, but a ‘temporary’ or a ‘relative’ property: for ‘being on the right hand side’ is a temporary property, while ‘two-footed’ is in point of fact ascribed as a property in certain relations; e.g., it is a property of man relatively to a horse and a dog. That nothing which may belong to anything else that A is a convertible predicate of A is clear: for it does not necessarily follow that if something is asleep it is a man. A ‘genus’ is what is predicated in the category of essence of a number of things exhibiting differences in kind. We should treat as predicates in the category of essence all such things as it would be appropriate to mention in reply to the question, ‘What is the object before you?’; as, for example, in the case of man, if asked that question, it is appropriate to say ‘He is an animal’. The question, ‘Is one thing in the same genus as another or in a different one?’ is also a ‘generic’ question; for a question of that kind as well falls under the same branch of enquiry as the genus: for having argued that ‘animal’ is the genus of man, and likewise also of ox, we shall have argued that they are in the same genus; whereas if we show that it is the genus of the one but not of the other, we shall have argued that these things are not in the same genus. An ‘accident’ is (1) something which, though it is none of the foregoing — i.e. neither a definition nor a property nor a genus — yet belongs to the thing: (2) something which may possibly either belong or not belong to any one and the self-same thing, as, e.g., the ‘sitting posture’ may belong or not belong to some self-same thing. Likewise also ‘whiteness’, for there is nothing to prevent the same thing to be at one time white, and at another not white. Of the definitions of accident the second is the better: for if he adopts the first, anyone is bound, if he is to understand it, to know already what ‘definition’ and ‘genus’ and ‘property’ are, whereas the second is sufficient of itself to tell us the essential meaning of the term in question. To Accident are to be attached also all comparisons of things together, when expressed in language that is drawn in any kind of way from what happens (accidit) to be true of them; such as, for example, the question: ‘Is the honourable or the expedient preferable?’ and ‘Is the life of virtue or the life of self-indulgence the pleasanter?’, and any other problem which may be phrased in terms like these. For in all such cases the question is ‘to which of the two does the predicate in question happen (accidit) to belong more closely?’ It is clear on the face of it that there is nothing to prevent an accident from becoming a temporary or a relative property. Thus the sitting

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sitting posture is an accident, but will be a temporary property, whenever a man is the only person sitting, while if he be not the only one sitting, it is still a property relatively to those who are not sitting. So then, there is nothing to prevent an accident from becoming both a relative and a temporary property; but a property absolutely it will never be. 6 We must not fail to observe that all remarks made in criticism of a ‘property’ and ‘genus’ and ‘accident’ will be applicable to ‘definitions’ as well. For when we have shown that the attribute in question fails to belong only to the term defined, as we do also in the case of a property, or that the genus rendered in the definition is not the true genus, or that any of the things mentioned in the phrase used does not belong, as would be remarked also in the case of an accident, we shall have demolished the definition; so that, to use the phrase previously employed, all the points we have enumerated might in a certain sense be called ‘definitory’. But we must not expect on this account to find a single line of inquiry which will apply universally to them all: for this is not an easy thing to find, and, even were one found, it would be very obscure indeed, and of little service to the treatise before us. Rather, a special plan of inquiry must be laid down for each of the classes we have distinguished, and then, starting from the rules that are appropriate in each case, it will probably be easier to make our way right through the task before us. So then, as was said before, we must outline a division of our subject, and other questions we must relegate each to the particular branch to which it most naturally belongs, speaking of them as ‘definitory’ and ‘generic’ questions. The questions I mean have practically been already assigned to their several branches. 7 First of all we must define the number of senses borne by the term ‘Sameness’. Sameness would be generally regarded as falling, roughly speaking, into three divisions. We generally apply the term numerically or specifically or generically — numerically in cases where there is more than one name but only one thing, e.g. ‘doublet’ and ‘cloak’; specifically, where there is more than one thing, but they present no differences in respect of their species, as one man and another, or one horse and another: for things like this that fall under the same species are said to be ‘specifically the same’. Similarly, too, those things are called generically the same which fall under the same genus, such as a horse and a man. It might appear that the sense in which water from the same spring is called ‘the same water’ is somehow different and unlike the senses mentioned above: but

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really such a case as this ought not be ranked in the same class with the things that in one way or another are called ‘the same’ in view of unity of species. For all such things seem to be of one family and to resemble one another. For the reason why all water is said to be specifically the same as all other water is because of a certain likeness it bears to it, and the only difference in the case of water drawn from the same spring is this, that the likeness is more emphatic: that is why we do not distinguish it from the things that one way or another are called ‘the same’ in view of unity of species. It is generally supposed that the term ‘the same’ is most used in a sense agreed on by everyone to be allied to what is numerically one. But even so, it is apt to be rendered in more than one sense; its most literal and primary use is found whenever the sameness is rendered in reference to an alternative name or definition, as when cloak is said to be the same as a doublet, or an animal that walks on two feet is said to be the same as a man: a second sense is when it is rendered with reference to a property, as when what can acquire knowledge is called the same as a man, and what naturally travels upward the same as fire: while a third use is found when it is rendered in reference to a term drawn from Accident, as when the creature who is sitting, or who is musical, is called the same as Socrates. For all these uses mean to signify numerical unity. That what I have just said is true may be best seen where one form of appellation is substituted for another. For often when we give the order to call one of the people who are sitting down, indicating him by name, we change our description, whenever the person to whom we give the order happens not to understand us; he will, we think, understand better from some accidental feature; so we bid him call to us ‘the man who is sitting’ or ‘who is conversing over there’ — clearly supposing ourselves to be indicating the same object by its name and by its accident. 8 Of ‘sameness’ then, has been said, three senses are to be distinguished. Now one way to confirm that the elements mentioned above are those out of which and through which and to which arguments proceed, is by induction: for if any one were to survey propositions and problems one by one, it would be seen that each of them was formed either from the definition of something or from its property or from its genus of from its accident. Another way to confirm it is through reasoning. For every predicate must of necessity be either convertible with its subject or not: and if it is convertible, it would be its definition or property, for if it signifies the essence, it is the definition; if not, it is a property: for this was what a property is, viz. what is predicated convertibly, but does not signify the essence. If, on the other hand, it is not predicated convertibly of the thing, it

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either is or is not one of the terms contained in the definition of the subject: and if it be one of those terms, then it will be a genus or the differentia, inasmuch as the definition consists of genus and differentiae; whereas, if it be not one of those terms, clearly it would be an accident, for accident was said to be what belongs as an attribute to a subject without being either its definition or its genus or a property.

u

BOOK VI (139a–151b) 1 The discussion of definitions falls into five parts. For you have to show either (1) that it is not true at all to apply the expression as well to that to which the term is applied (for the definition of man ought to apply to every man); or (2) that though the object has a genus, he has failed to put the object defined into the genus, or to put it into the appropriate genus (for the framer of a definition should first place the object in its genus, and then append its differences; for all the elements of the definition the genus is usually supposed to be the principal mark of the essence of what is defined); or (3) that the expression is not peculiar to the object (for, as we have said above as well, a definition ought to be peculiar); or else (4) see if, though he has observed all the aforesaid cautions, he has yet failed to define the object, that is, to express its essence. (5) It remains, apart from the foregoing, to see if he has defined it, but defined it incorrectly. Whether, then, the expression be not also true of that of which the term is true you should proceed to examine according to the commonplace rules that relate to Accident. For there too the question is always ‘Is so and so true or untrue?’: for whenever we argue that an accident belongs, we declare it to be true, while whenever we argue that it does not belong, we declare it to be untrue. If, again, he has failed to place the object in the appropriate genus, or if the expression be not peculiar to the object, we must go on to examine the case according to the commonplace rules that relate genus to property. It remains, then to prescribe how to investigate whether the object has been either not defined at all, or else defined incorrectly. First, then, we must proceed to examine if it has been defined incorrectly: for with anything it is easier to do it than to do it correctly. Clearly, then, more mistakes are made in the latter task

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on account of its great difficulty. Accordingly the attack becomes easier in the latter case than in the former. Incorrectness falls into two branches: (1) the use of obscure language (for the language of a definition ought to be the very clearest possible, seeing that the whole purpose of rendering it is to make something known) (2) if the expression used be longer than is necessary: for all additional matter in a definition is superfluous. Again, each of the aforesaid branches is divided into a number of others. 2 One commonplace rule, then, in regard to obscurity is: see if the meaning intended by the definition involves an ambiguity with any other, e.g. ‘Becoming is a passage into being’, or ‘Health is the balance of hot and cold elements’. Here ‘passage’ and ‘balance’ are ambiguous terms: it is accordingly not clear which of the several possible senses of the term he intends to convey. Likewise also, if the term defined be used in different senses and he has spoken without distinguishing between them: for it is not clear to which of them the definition rendered applies, and one can then bring a captious objection on the ground that the definition does not apply to all things whose definition he has rendered: and this kind of thing is particularly easy in the case where the definer does not see the ambiguity of his terms. Or, again, the questioner may himself distinguish the various senses of the term rendered in the definition, and then institute his argument against each: for if the expression used is not adequate to the subject in any of its senses, it is clear that he cannot have defined it in any sense aright. Another rule is: see if he has used a metaphorical expression, as, for instance, if he has defined knowledge as ‘unsupplantable’ or the earth as a ‘nurse’, or temperance as a ‘harmony’. For a metaphorical expression is always obscure. It is possible also to argue sophistically against the user of a metaphorical expression as though he had used it in its literal sense: for the definition stated will not apply to the term defined, e.g. in the case of temperance: for harmony is always found between notes. Moreover, if harmony be the genus of temperance, then the same object will occur in two genera of which neither contains the other: for harmony does not contain virtue, nor virtue harmony. Again, see if he uses terms that are unfamiliar, as when Plato describes the eye as ‘brow-shaded’, or a certain spider as ‘poison-fanged’, or the marrow as ‘boneformed’. For an unusual phrase is always obscure. Sometimes a phrase is used neither ambiguously nor yet metaphorically, nor yet literally, as when the law is said to be the ‘measure’ or ‘image’ of the things

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that are by nature just. Such phrases are worse than metaphor; for the latter does make its meaning to some extent clear because of the likeness involved; for there is no likeness to justify the description ‘measure’ or ‘image’, as applied to the law, nor is the law ordinarily so called in a literal sense. So then, if a man says that the law is literally a ‘measure’ or an ‘image’, he speaks falsely: for an image is something produced by imitation, and this is not found in the case of the law. If, on the other hand, he does not mean the term literally, it is clear that he has used an unclear expression, and one that is worse than any sort of metaphorical expression. Moreover, see if from the expression used the definition of the contrary be not clear; for definitions that have been correctly rendered also indicate their contraries as well. Or, again, see if, when it is merely stated by itself, it is not evident what it defines: just as in the works of the old painters, unless there were an inscription, the figures used to be unrecognizable. 3 If, then, the definition be not clear, you should proceed to examine on lines such as these. If, on the other hand, he has phrased the definition redundantly, first of all look and see whether he has used any attribute that belongs universally, either to real objects in general, or to all that fall under the same genus as the object defined: for the mention of this is sure to be redundant. For the genus ought to divide the object from things in general, and the differentia from any of the things contained in the same genus. Now any term that belongs to everything separates off the given object from absolutely nothing, while any that belongs to all the things that fall under the same genus does not separate it off from the things contained in the same genus. Any addition, then, of that kind will be pointless. Or see if, though the additional matter may be peculiar to the given term, yet even when it is struck out the rest of the expression too is peculiar and makes clear the essence of the term. Thus, in the definition of man, the addition ‘capable of receiving knowledge’ is superfluous; for strike it out, and still the expression is peculiar and makes clear his essence. Speaking generally, everything is superfluous upon whose removal the remainder still makes the term that is being defined clear. Such, for instance, would also be the definition of the soul, assuming it to be stated as a ‘self-moving number’; for the soul is just ‘the self-moving’, as Plato defined it. Or perhaps the expression used, though appropriate, yet does not declare the essence, if the word ‘number’ be eliminated. Which of the two is the real state of the case it is difficult to determine clearly: the right way to treat the matter in all cases is to be guided by convenience.

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Thus, e.g., it is said that the definition of phlegm is the ‘undigested moisture that comes first off food’. Here the addition of the word ‘undigested’ is superfluous, seeing that the ‘first’ is one and not many, so that even when ‘undigested’ is left out the definition will still be peculiar to the subject: for it is impossible that both phlegm and also something else should be both first to arise from the food. Or perhaps the phlegm is absolutely the first thing to come off food, but only the first of the undigested matter, so that the addition ‘undigested’ is required; for stated the other way the definition would not be true unless the phlegm comes first of all. Moreover, see if anything contained in the definition fails to apply to everything that falls under the same species: for this sort of definition is worse than those which include an attribute belonging to all things universally. For in that case, if the remainder of the expression be peculiar, the whole too will be peculiar, for absolutely always, if to something peculiar anything whatever that is true be added, the whole too becomes peculiar. Whereas if any part of the expression do not apply to everything that falls under the same species, it is impossible that the expression as a whole should be peculiar: for it will not be predicted convertibly with the object; e.g. ‘a walking biped animal six feet high’: for an expression of that kind is not predicated convertibly with the term, because the attribute ‘six feet high’ does not belong to everything that falls under the same species. Again, see if he has said the same thing more than once, saying e.g., ‘desire’ is a ‘conation for the pleasant’. For ‘desire’ is always ‘for the pleasant’, so that what is the same as desire will also be ‘for the pleasant’. Accordingly our definition of desire becomes ‘conation-for-the-pleasant for the pleasant’: for the word ‘desire’ is the exact equivalent of the words ‘conation-for-the-pleasant’, so that both alike will be ‘for the pleasant’. Or perhaps there is no absurdity in this; for consider this instance: — ‘Man is a biped’: therefore, what is the same as ‘man is a biped’: but ‘a walking biped animal’ is the same as man, and therefore ‘a walking biped animal is a biped’. But this involves no real absurdity. For ‘biped’ is not a predicate of ‘walking animal’: if it were, then we should certainly have ‘biped’ predicated twice of the same thing; but as a matter of fact the subject said to be a biped is ‘a walking biped animal’, so that the word ‘biped’ is only used as a predicate once. Likewise also in the case of ‘desire’ as well: for it is not ‘conation’ that is said to be ‘for the pleasant’, but rather the whole idea, so that there too the predication is only made once. Absurdity results, not when the same word is uttered twice, but when the same thing is more than once predicated of a subject; e.g., if he says, like Xenocrates, that wisdom defines and contemplates reality: for definition is a certain type of

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contemplation, so that by adding the words ‘and contemplates’ over again he says the same thing twice over. Likewise, too, those fail who say that ‘cooling’ is ‘the privation of natural heat’. For all privation is a privation of some natural attribute, so that the addition of the word ‘natural’ is superfluous; it would have been enough to say ‘privation of heat’, for the word ‘privation’ shows of itself that the heat meant is natural heat. Again see if a universal have been mentioned and then a particular case of it be added as well, e.g., ‘Equity is a remission of what is expedient and just’; for what is just is a branch of what is expedient and is therefore included in the latter term: its mention is therefore redundant, an addition of the particular after the universal has been already stated. So also, if he defines ‘medicine’ as ‘knowledge of what makes for health in animals and men’, or ‘the law’ as ‘the image of what is by nature noble and just’; for what is just is a branch of what is noble, so that he says the same thing more than once. 4 Whether, then, a man defines a thing correctly or incorrectly you should proceed to examine on these and similar lines. But whether he has mentioned and defined its essence or no, should be examined as follows: — First of all, see if he has failed to make the definition through terms that are prior and more intelligible. For the reason why the definition is rendered is to make known the term stated, and we make things known by taking not any random terms, but such as are prior and more intelligible, as is done in demonstrations (for so it is with all teaching and learning); accordingly, it is clear that a man who does not define through terms of this kind has not defined at all. Otherwise, there will be more than one definition of the same thing: for clearly he who defines through terms that are prior and more intelligible has also framed a definition, and a better one, so that both would then be definitions of the same object. This sort of view, however, does not generally find acceptance: for of each real object the essence is single: if, then, there are to be a number of definitions of the same thing, the essence of the object will be the same, inasmuch as the definitions are different. Clearly, then, any one who has not defined a thing through terms that are prior and more intelligible has not defined it at all. The statement that a definition has not been made through more intelligible terms may be understood in two senses, either supposing that its terms are absolutely less intelligible, or supposing that they are less intelligible to us: for either sense is possible. Thus absolutely the prior is more intelligible than the posterior, a point, for instance, than a line, a line than a plane, and a plane than

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a solid; just as also a unit is more intelligible than a number; for it is the prius and starting point of all number. Likewise, also, a letter is more intelligible than a syllable. Whereas to us it sometimes happens that the converse is the case: for the solid falls under perception most of all — more than a plane — and a plane more than a line, and a line more than a point; for most people learn things like the former earlier than the latter; for any ordinary intelligence can grasp them, whereas the others require an exact and exceptional understanding. Absolutely, then, it is better to try to make what is posterior known through what is prior, inasmuch as such a way of procedure is more scientific. Of course, in dealing with persons who cannot recognise things through terms of that kind, it may perhaps be necessary to frame the expression through terms that are intelligible to them. Among definitions of this kind are those of a point, a line, and a plane all of which explain the prior by the posterior; for they say that a point is the limit of a line, a line of a plane, a plane of a solid. One must, however, not fail to observe that those who define in this way cannot show the essential nature of the term they define, unless it so happens that the same thing is more intelligible both to use and also absolutely, since a correct definition must define a thing through its genus and differentiae, and these belong to the order of things which are absolutely more intelligible than, and prior to, the species. For annul the genus and differentia, and the species too is annulled, so that these are prior to the species. For if the species be known, the genus and differentia must of necessity be known as well (for any one who knows what a man is, knows also what ‘animal’ and ‘walking’ are), whereas if the genus or the differentia be known it does not follow of necessity that the species is known as well: thus the species is less intelligible. Moreover, those who say that such definitions, viz. those which proceed from what is intelligible to this, that, or the other man, are really and truly definitions, will have to say that there are several definitions of one and the same thing. For, as it happens, different things are more intelligible to different people, not the same things to all; and so a different definition would have to be rendered to each several person, if the definition is to be constructed from what is more intelligible to particular individuals. Moreover, to the same people different things are more intelligible at different times; first of all the objects of sense; then, as they become more sharp-witted, the converse; so that those who hold that a definition ought to be rendered through what is more intelligible to particular individuals would not have to render the same definition at all times even to the same person. It is clear, then, that the right way to define is not through terms of that kind, but through what is absolutely more intelligible: for only in this could the definition come always to be one and the same. Perhaps, also, what is absolutely intelligible is

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what is intelligible, not to all, but to those who are in a sound state of understanding, just as what is absolutely healthy is healthy to those in a sound state of body. All such points as this ought to be made very precise, and made use of in the course of discussion as occasion requires. The demolition of the definition will most surely win a general approval if the definer happens to have framed his expression neither from what is absolutely more intelligible nor yet from what is so to us. One form, then, of the failure to work through more intelligible terms is the exhibition of the prior through the posterior, as we remarked before. Another form occurs if we find that the definition has been rendered of what is at rest and definite through what is indefinite and in motion: for what is still and definite is prior to what is indefinite and in motion. Of the failure to use terms that are prior there are three forms: (1) The first is when an opposite has been defined through its opposite, e.g. good through evil: for opposites are always simultaneous by nature. Some people think, also, that both are objects of the same science, so that the one is not even more intelligible than the other. One must, however, observe that it is perhaps not possible to define some things in any other way, e.g. the double without the half, and all the terms that are essentially relative: for in all such cases the essential being is the same as a certain relation to something, so that it is impossible to understand the one term without the other, and accordingly in the definition of the one the other too must be embraced. One ought to learn up all such points as these, and use them as occasion may seem to require. (2) Another is — if he has used the term defined itself. This passes unobserved when the actual name of the object is not used, e.g., supposing any one had defined the sun as ‘a star that appears by day’. For in bringing in ‘day’ he brings in the sun. To detect errors of this sort, exchange the word for its definition, e.g., the definition of ‘day’ as ‘the passage of the sun over the earth’. Clearly, whoever has said ‘the passage of the sun over the earth’ has said ‘the sun’ so that in bringing in the ‘day’ he has brought in the sun. (3) Again, see if he has defined one co-ordinate member of a division by another, e.g. ‘an odd number’ as ‘that which is greater by one than an even number’, for both are differentia of ‘number’. Likewise also, see if he has defined a superior through a subordinate term, e.g. “An ‘even number’ is ‘a number divisible into halves’”, or “‘the good’ is a ‘state of virtue’”. For half is derived from ‘two’, and ‘two’ is an even number: virtue is also a kind of good, so that the latter terms are subordinated to the former. Moreover, in using the subordinate term one is bound to use the other as well: for whoever employs the term ‘virtue’ employs the term ‘good’ seeing that

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virtue is a certain kind of good: likewise, also, whoever employs the term ‘half’ employs the term ‘even’, for to be ‘divided in half’ means to be divided into two, and two is even. 5 Generally speaking, then, one commonplace rule relates to the failure to frame the expression by means of terms that are prior and more intelligible: and of this the subdivisions are those specified above. A second is, see whether, though the object is in a genus, it has not been placed in a genus. This sort of error is always found where the essence of the object does not stand first in the expression, e.g., the definition of ‘body’ as ‘that which has three dimensions’, or the definition of ‘man’, supposing any one to give it, as ‘that which knows how to count’: for it is not stated what it is that has three dimensions, or what it is that knows how to count; whereas the genus is meant to indicate just this, and is submitted first to the terms in the definition. Moreover, see if, while the term to be defined is used in relation to many things, he has failed to render it in relation to all of them; as e.g., if he defines ‘grammar’ as ‘the knowledge how to write from dictation’: for he ought also to say that it is a knowledge how to read as well. For in rendering it as ‘knowledge of writing’ he has no more defined it than by rendering it as ‘knowledge of reading’: neither in fact has succeeded, but only he who mentions both these things, since it is impossible that there should be more than one definition of the same thing. It is only, however, in some cases that what has been said corresponds to the actual state of things: in some it does not, e.g., all those terms which are not used essentially in relation to both things: as medicine is said to deal with the production of disease and health; for it is said essentially to do the latter, but the former only by accident: for it is absolutely alien to medicine to produce disease. Here, then, the man who renders medicine as relative to both of these things has not defined it any better than he who mentions the one only. In fact, he has done it perhaps worse, for any one besides the doctor is capable of producing disease. Moreover, in a case where the term to be defined is used in relation to several things, see if he has rendered it as relative to the worse rather than to the better; for every form of knowledge and potentiality is generally thought to be relative to the best. Again, if the thing in question be not placed in its own proper genus, one must examine it according to the elementary rules in regard to genera, as has been said before. Moreover, see if he uses language which transgresses the genera of the

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things he defines, defining, e.g., justice as ‘a state that produces equality’ or ‘distributes what is equal’: for by defining so he passes outside of the sphere of virtue, and so by leaving out the genus of justice he fails to express its essence: for the essence of a thing must in each case bring in its genus. It is the same thing if the object be not put into its nearest genus; for the man who puts it into the nearest one has stated all the higher genera, seeing that all the higher genera are predicated of the lower. Either, then, it ought to be put into its nearest genus, or else to the higher genus all the differentiae ought to be appended whereby the nearest genus is defined. For then he would not have left out anything: but would merely have mentioned the subordinate genus by an expression instead of by name. On the other hand, he who mentions merely the higher genus by itself, does not state the subordinate genus as well: in saying ‘plant’ a man does not specify ‘a tree’. 6 Again in regard to the differentiae, we must examine in like manner whether the differentiae, too, be those of the genus. For if a man has not defined the object by the differentiae peculiar to it, or has mentioned something such as is utterly incapable of being a differentia of anything, e.g., ‘animal’ or ‘substance’, clearly he has not defined it at all: for the aforesaid terms do not differentiate anything at all. Further, we must see whether the differentia stated possesses anything that is co-ordinate with it in a division; for, if not, clearly the one stated could not be a differentia of the genus. For a genus is always divided by differentiae that are co-ordinate members of a division, as, for instance, ‘animal’ by the terms ‘walking’, ‘flying’, ‘aquatic’ and ‘biped’. Or see if, though the contrasted differentia exists, it yet is not true of the genus; for then, clearly, neither of them could be a differentia of the genus; for differentiae that are co-ordinates in a division with the differentia of a thing are all true of the genus to which the thing belongs. Likewise, also, see if, though it be true, yet the addition of it to the genus fails to make a species. For then, clearly, this could not be a specific differentia of the genus: for a specific differentia, if added to the genus, always makes a species. If, however, this be no true differentia, no more is the one adduced, seeing that it is a co-ordinate member of a division with this. Moreover, see if he divides the genus by a negation, as those do who define a line as ‘length without breadth’: for this simply means that it has not any breadth. The genus will then be found to partake of its own species: for since of everything either an affirmation or its negation is true, length must always either lack breadth or possess it, so that ‘length’ as well, i.e. the genus of ‘line’, will be either with or without breadth. But ‘length without breadth’ is the definition of

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a species, as also is ‘length with breadth’: for ‘without breadth’ and ‘with breadth’ are differentiae, and the genus and differentia constitute the definition of the species. Hence the genus would admit of the definition of its species. Likewise also, it will admit of the definition of the differentia, seeing that one or the other of the aforesaid differentiae is of necessity predicated of the genus. The usefulness of this principle is found in meeting those who assert the existence of ‘Ideas’: for if absolute length exist, how will it be predicable of the genus that it has breadth or that it lacks it? For one assertion or the other will have to be true of ‘length’ universally, if it is to be true of the genus at all: and this is contrary to the fact: for there exist both lengths which have, and lengths which have not breadth. Hence the only people against whom the rule can be employed are those who assert that genus is always numerically one; and this is what is done by those who assert the real existence of the ‘Ideas’; for they allege that absolute length and absolute animal are the genus. It may be in some cases the definer is obliged to employ a negation as well, e.g. in defining privations. For ‘blind’ means a thing which cannot see when its nature is to see. There is no difference between dividing the genus by a negation, and dividing it by such an affirmation as is bound to have a negation as its co-ordinate in a division, e.g. supposing he had defined something as ‘length possessed of breadth’; for co-ordinate in the division with that which is possessed of breadth is that which possesses no breadth and that only, so that again the genus is divided by a negation. Again, see if he rendered the species as a differentia, as do those who define ‘contumely’ as ‘insolence accompanied by jeering’; for jeering is a kind of insolence, i.e. it is a species and not a differentia. Moreover, see if he has stated the genus as the differentia, e.g. ‘Virtue is a good and noble state’: for ‘good’ is the genus of ‘virtue’. Or possibly ‘good’ here is not the genus but the differentia, on the principle that the same thing cannot be in two genera of which neither contains the other: for ‘good’ does not include ‘state’ nor vice versa: for not every state is good nor every good a ‘state’. Both, then, could be genera, and consequently, if ‘state’ is the genus of virtue, clearly ‘good’ cannot be its genus: it must rather be the differentia. Moreover, ‘a state’ indicates the essence of virtue, whereas ‘good’ indicates not the essence but a quality: and to indicate a quality is generally held to be a function of the differentia. See, further, whether the differentia rendered indicates an individual rather than a quality: for the general view is that the differentia always expresses a quality. Look and see, further, whether the differentia belongs only by accident to

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the object defined, for the differentia is never an accidental attribute, any more than the genus is: for the differentia of a thing cannot both belong and not belong to it. Moreover, if either the differentia or the species, or any of the things which are under the species, is predicable of the genus, then he could not have defined the term. For none of the aforesaid can possibly be predicated of the genus, seeing that the genus is the term with the widest range of all. Again see, if the genus be predicated of the differentia; for the general view is that the genus is predicated, not of the differentia, but of the objects of which the differentia is predicated. Animal, e.g., is predicated of ‘man’ and ‘ox’ or other walking animals, not of the actual differentia itself which we predicate of the species. For, if ‘animal’ is to be predicated of each of its differentiae, then ‘animal’ would be predicated of the species several times over; for the differentiae are predicates of the species. Moreover, the differentiae will all be either species or individuals, if they are animals; for every animal is either a species or an individual. Likewise you must inquire also if the species or any of the objects that come under it is predicated of the differentia: for this is impossible seeing that the differentia is not a term with a wider range than the various species. Moreover, if any of the species be predicated of it, the result will be that the differentia is a species: if, for instance, ‘man’ be predicated, the differentia is clearly the human race. Again, see if the differentia fails to be prior to the species: for the differentia ought to be posterior to the genus, but prior to the species. Look and see also if the differentia mentioned belongs to a different genus, neither contained in nor containing the genus in question. For the general view is that the same differentia cannot be used of two non-subaltern genera. Else the result will be that the same species as well will be in two non-subaltern genera; for each of the differentiae imports its own genus, e.g., ‘walking’ and ‘biped’ import with them the genus ‘animal’. If, then, each of the genera as well is true of that of which the differentia is true, it clearly follows that the species must be in two non-subaltern genera. Or perhaps it is not impossible for the same differentia to be used of two non-subaltern genera, and we ought to add the words ‘except they both be subordinate members of the same genus’. Thus ‘walking animal’ and ‘flying animal’ are non-subaltern genera, and ‘biped’ is the differentia of both. The words ‘except they both be subordinate members of the same genus’ ought therefore to be added; for both these are subordinate to ‘animal’. From this possibility, that the same differentia may be used of two non-subaltern genera, it is clear also that there is no necessity for the differentia to carry with it the whole of the genus to which it belongs, but only the one or

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the other of its limbs together with the genera that are higher than this, as ‘biped’ carries with it either ‘flying’ or ‘walking animal’. See too if he has rendered ‘existence in’ something as the differentia of a thing’s essence: for the general view is that locality cannot differentiate between one essence and another. Hence, too, people condemn those who divide animals by means of the terms ‘walking; and ‘aquatic’, on the ground that ‘walking’ and ‘aquatic’ indicate mere locality. Or possibly in this case the censure is undeserved; for ‘aquatic’ does not mean ‘in’ anything; nor does it denote a locality, but a certain quality: for even if the thing be on dry land, still it is ‘aquatic’: and likewise a land animal, even though it be in the water, will still be a land- and not an aquatic-animal. But all the same, if ever the differentia does denote existence in something, he will have made a bad mistake. Again, see if he has rendered an affection as the differentia: for every affection, if intensified, subverts the essence of the thing, while the differentia is not of that kind: for the differentia is generally considered rather to preserve that which it differentiates; and it is absolutely impossible for a thing to exist without its own special differentia: for if there be no ‘walking’, there will be no ‘man’. In fact, we may lay down absolutely that a thing cannot have as its differentia anything in respect of which it is subject to alteration: for all things of that kind, if intensified, destroy its essence. If, then, a man has rendered any differentia of this kind, he has made a mistake: for we undergo absolutely no alteration in respect of our differentiae. Again, see if he has failed to render the differentia of a relative term relatively to something else; for the differentiae of relative terms are themselves relative, as in the case also of knowledge. This is classed as speculative, practical and productive; and each of these denotes a relation: for it speculates upon something, and produces something and does something. Look and see also if the definer renders each relative term relatively to its natural purpose: for while in some cases the particular relative term can be used in relation to its natural purpose only and to nothing else, some can be used in relation to something else as well. Thus sight can only be used for seeing, but a strigil can also be used to dip up water. Still, if any one were to define a strigil as an instrument for dipping water, he has made a mistake: for that is not its natural function. The definition of a thing’s natural function is ‘that for which it would be used by the prudent man, acting as such, and by the science that deals specially with that thing’. Or see if, whenever a term happens to be used in a number of relations, he has failed to introduce it in its primary relation: e.g. be defining ‘wisdom’ as the virtue of ‘man’ or of the ‘soul’, rather than of the ‘reasoning faculty’: for

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‘wisdom’ is the virtue primarily of the reasoning faculty: for it is in virtue of this that both man and his soul are said to be wise. Moreover, if the thing of which the term defined has been stated to be an affection or disposition, or whatever it may be, be unable to admit it, then the definer has made a mistake. For every disposition and every affection is formed naturally in that of which it is an affection or disposition, as knowledge, too, is formed in the soul, being a disposition of soul. Sometimes, however, people make bad mistakes in matters of this sort, e.g. all those who say that ‘sleep’ is a ‘failure of sensation’, or that ‘perplexity’ is a state of ‘equality between contrary reasonings’, or that ‘pain’ is a ‘violent disruption of parts that are naturally conjoined’. For sleep is not an attribute of sensation, whereas it ought to be, if it is a failure of sensation. Likewise, perplexity is not an attribute of opposite reasonings, nor pain of parts naturally conjoined: for then inanimate things will be in pain, since pain will be present in them. Similar in character, too, is the definition of ‘health’, say, as a ‘balance of hot and cold elements’: for then health will be necessarily be exhibited by the hot and cold elements: for a balance of anything is an attribute inherent in those things of which it is the balance, so that health would be an attribute of them. Moreover, people who define in this way put effect for cause, or cause for effect. For the disruption of parts naturally conjoined is not pain, but only a cause of pain: nor again is failure of sensation sleep, but the one is the cause of the other: for either we go to sleep because sensation fails, or sensation fails because we go to sleep. Likewise also an equality between contrary reasonings would be generally considered to be a cause of perplexity: for it is when we reflect on both sides of a question and find everything alike to be in keeping with either course that we are perplexed which of the two we are to do. Moreover, with regard to all periods of time look and see whether there be any discrepancy between the differentia and the thing defined; e.g., supposing the ‘immortal’ to be defined as a ‘living thing immune at present from destruction’. For a living thing that is immune ‘at present’ will be immortal ‘at present’. Possibly, indeed, in this case this result does not follow, owing to the ambiguity of the words ‘immune at present from destruction’: for it may mean either that the thing has not been destroyed at present, or that it cannot be destroyed at present, or that at present it is such that it can never be destroyed. Whenever, then, we say a living thing is at present immune from destruction, we mean that it is at present a living thing of such kind never to be destroyed: and this is equivalent to saying that it is immortal, so that it is not meant that it is immortal only at present. Still, if ever it does happen that what has been rendered according to the definition belongs in the present only or past, whereas what is meant

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by the word does not so belong, then the two could not be the same. So, then, this commonplace rule ought to be followed, as we have said. 7 You should look and see also whether the term being defined is applied in consideration of something other than the definition rendered. Suppose, e.g., a definition of ‘justice’ as the ‘ability to distribute what is equal’. This would not be right, for ‘just’ describes rather the man who chooses, than the man who is able, to distribute what is equal: so that justice could not be an ability to distribute what is equal: for then also the most just man would be the man with the most ability to distribute what is equal. Moreover, see if the thing admits of degrees, whereas what is rendered according to the definition does not, or, vice versa, what is rendered according to the definition admits of degrees while the thing does not. For either both must admit them or else neither, if indeed what is rendered according to the definition is the same as the thing. Moreover, see if, while both of them admit of degrees, they yet do not both become greater together: e.g., suppose sexual love to be the desire for intercourse: for he who is more intensely in love has not a more intense desire for intercourse, so that both do not become intensified at once: they certainly should, however, had they been the same thing. Moreover, suppose two things to be before you, see if the term to be defined applies more particularly to the one to which the content of the definition is less applicable. Take, for instance, the definition of ‘fire’ as ‘the body that consists of the most rarefied particles’. For ‘fire’ denotes flame rather than light, but flame is less the body that consists of the most rarefied particles than is light: whereas both ought to be more applicable to the same thing, if they had been the same. Again, see if the one expression applies alike to both the objects before you, while the other does not apply to both alike, but more particularly to one of them. Moreover, see if he renders the definition relative to two things taken separately: thus, ‘the beautiful’ is ‘what is pleasant to the eyes or the ears’: or ‘the real’ is ‘what is being capable of being acted upon or of acting’. For then the same thing will be beautiful and not beautiful, and likewise will be both real and not real. For ‘pleasant to the ears’ will be the same as ‘beautiful’, so that ‘not pleasant to the ears’ will be the same as ‘not beautiful’: for of identical things the opposites, too, are identical, and the opposite of ‘beautiful’ is ‘not beautiful’, while of pleasant to the ears’ the opposite is not ‘not pleasant to the ears’: clearly, then, ‘not pleasant to the ears’ is the same thing as ‘not beautiful’. If, therefore, something be pleasant to the eyes but not to the ears, it will be both

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beautiful and not beautiful. In like manner we shall show also that the same thing is both real and unreal. Moreover, of both genera and differentiae and all the other terms rendered in definitions you should frame definitions in lieu of the terms, and then see if there be any discrepancy between them. 8 If the term defined be relative, either in itself or in respect of its genus, see whether the definition fails to mention that to which the term, either in itself or in respect of its genus, is relative, e.g., if he has defined ‘knowledge’ as an ‘incontrovertible conception’ or ‘wishing’ as ‘painless conation’. For of everything relative the essence is relative to something else, seeing that the being of every relative term is identical with being in a certain relation with something. He ought, therefore, to have said that knowledge is ‘conception of a knowable’ and that wishing is ‘conation for a good’. Likewise also, if he has defined ‘grammar’ as ‘knowledge of letters’: whereas in the definition there ought to be rendered either the thing to which the term itself is relative, or that, whatever it is, to which its genus is relative. Or see if a relative term has been described not in relation to its end, the end in anything being whatever is best in it or gives its purpose to the rest. Certainly it is what is best or final that should be stated, e.g. that desire is not for the pleasant but for pleasure: for this is our purpose for choosing what is pleasant as well. Look and see also if that in relation to which he has rendered the term be a process or an activity: for nothing of that kind is an end, for the completion of the activity or process is the end rather than the process or activity itself. Or perhaps this rule is not true in all cases, for almost everybody prefers the present experience of pleasure to its cessation, so that they would count the activity as the end rather than its completion. Again see in some cases if he has failed to distinguish the quantity or quality or place or other differentiae of an object; e.g., the quality and quantity of the honour the striving for which makes a man ambitious: for all men strive for honour, so that it is not enough to define the ambitious man as him who strives for honour, but the aforesaid differentiae must be added. Likewise, also, in defining the covetous man the quantity of money he aims at, or in the case of the incontinent man the quality of the pleasures, should be stated. For it is not the man who gives way to any sort of pleasure whatever who is called incontinent, but only he who gives way to certain kind of pleasure. Or again, people sometimes define night as a ‘shadow on the earth’, or an earthquake as ‘movement of the earth’, or a cloud as ‘condensation of the air’, or a wind as‘movement of

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the air’; whereas they ought to specify as well quantity, quality, place, and cause. Likewise, also, in other cases of the kind: for by omitting any differentiae whatever he fails to state the essence of the term. One should always attack deficiency. For a movement of he earth does not constitute an earthquake, nor a movement of the air a wind, irrespective of its manner and the amount involved. Moreover, in the case of conations, see if the word ‘apparent’ is left out, e.g. ‘wishing is a conation after the good’, or ‘desire is a conation after the pleasant, — instead of saying ‘the apparently good’, or ‘pleasant’. For often those who exhibit the conation do not perceive what is good or pleasant, so that their aim need not be really good or pleasant, but only apparently so. They ought, therefore, to have rendered the definition also accordingly. On the other hand, anyone who maintains the existence of ideas ought to be brought face to face with his ideas, even though he does render the word in question: for there can be no Idea of anything merely apparent: the general view is that an Idea is always spoken of in relation to an Idea: thus absolute desire is for the absolutely pleasant, and absolute wishing is the absolutely good; they therefore cannot be for an apparently good or an apparently pleasant: for the existence of an absolutely-apparently-good or pleasant would be an absurdity. 9 Moreover, if the definition be of the state of anything, look at what is in the state, while it be of what is in the state, look at the state: and likewise also in other cases of the kind. Thus if the pleasant be identical with the beneficial, then, too, the man who is pleased is benefited. Speaking generally, in definitions of this sort it happens that what the definer defines is in a sense more than one thing: for in defining knowledge, a man, in a sense, defines ignorance as well, and likewise also what has knowledge and what lacks it, and what it is to know and to be ignorant. For if the first be made clear, the others become in a certain sense clear as well. We have, then, to be on our guard in all such cases against discrepancy, using the elementary principles drawn from consideration of contraries and of co-ordinates. Moreover, in the case of relative terms, see if the species is rendered as relative to a species of that which the genus is rendered as relative, e.g., supposing belief to be relative to some object of belief, see whether a particular belief is made relative to some particular object of belief: and, if a multiple be relative to a fraction, see whether a particular multiple be made relative to a particular fraction. For if it be not so rendered, clearly a mistake has been made. See, also, if the opposite of the term has the opposite definition, whether, e.g., the definition of ‘half’ is the opposite of that of ‘double’: for if ‘double’ is

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‘that which exceeds another by an equal amount to that other’, ‘half’ is ‘that which is exceeded by an amount equal to itself’. In the same way, too, with contraries. For to the contrary term will apply the definition that is contrary in some one of the ways in which contraries are conjoined. Thus, e.g., if ‘useful’ = ‘productive of good’, ‘injurious’ = ‘productive of evil’ or ‘destructive of good’, for one or the other of these is bound to be contrary to the term originally used. Suppose, then, neither of these things to be the contrary of the term originally used, then clearly neither of the definitions rendered later could be the definition of the contrary of the term originally defined: and therefore the definition originally rendered of the original term has not been rightly rendered either. Seeing, moreover, that of contraries, the one is sometimes a word formed to denote the privation of the other, as, e.g., inequality is generally held to be the privation of equality (for ‘unequal’ merely describes things that are not ‘equal’), it is therefore clear that the contrary whose form denotes the privation must of necessity be defined through the other; whereas the other cannot then be defined through the one whose form denotes the privation; for else we should find that each is being interpreted by the other. We must in the case of contrary terms keep an eye on this mistake, e.g., supposing any one were to define equality as the contrary of inequality: for then he is defining it through the term which denoted privation of it. Moreover, a man who so defines is bound to use in his definition the very term he is defining; and this becomes clear, if for the word we substitute its definition. For to say ‘inequality’ is the same as to say ‘privation of equality’. Therefore equality so defined will be ‘the contrary of the privation of equality’, so that he would have used the very word to be defined. Suppose, however, that neither of the contraries be so formed as to denote privation, but yet the definition of it be rendered in a manner like the above, e.g., suppose ‘good’ to be defined as ‘the contrary of evil’, then, since it is clear that ‘evil’ too will be ‘the contrary of good’ (for the definition of things that are contrary in this way must be rendered in a like manner), the result again is that he uses the very term being defined: for ‘good’ is inherent in the definition of ‘evil’. If, then, ‘good’ be the contrary of evil and evil be nothing other than the ‘contrary of good’, then ‘good’ will be the ‘contrary of the contrary of good’. Clearly, then, he has used the very word to be defined. Moreover, see if in rendering a term formed to denote privation, he has failed to render the term of which it is the privation, e.g., the state, or contrary, or whatever it may be whose privation it is: also if he has omitted to add either any term at all in which the privation is naturally formed, or else that in which it is naturally formed primarily, e.g. whether in defining ‘ignorance’ as a privation, he has failed to say that it is the privation of ‘knowledge’; or, has

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failed to add in what it is naturally formed, or, though he has added this, has failed to render the thing in which it is primarily formed, placing it, e.g., in ‘man’ or in ‘soul’, and not in the ‘reasoning faculty’: for if in any of these respects he fails, he has made a mistake. Likewise, also, if he has failed to say that ‘blindness’ is the ‘privation of sight in an eye’: for a proper rendering of its essence must state both of what it is the privation and what it is that is deprived. Examine further whether he has defined by the expression ‘a privation’ a term that is not used to denote a privation: thus a mistake of this sort also would be generally thought to be incurred in the case of ‘error’ by any one who is not using it as a merely negative term. For what is generally thought to be in error is not that which has no knowledge, but rather that which has been deceived, and for this reason we do not talk of inanimate things or children as ‘erring’. ‘Error’, then, is not used to denote a mere privation of knowledge. 10 Moreover, see whether the like inflexions in the definition apply to the like inflexions of the term; e.g., if ‘beneficial’ means ‘productive of health’, does ‘beneficially’ mean ‘productively of health’ and a ‘benefactor’ a ‘producer of health’? Look too, and see whether the definition given will apply to the Idea as well. For in some cases it will not do so; e.g., in the Platonic definition where he adds the word ‘mortal’ in his definition of living creatures: for the Idea (e.g. the absolute Man) is not mortal, so that the definition will not fit the Idea. So wherever the words ‘capable of acting on’ or ‘capable of being acted upon’ are added, the definition and the Idea are absolutely bound to be discrepant: for those who assert the existence of Ideas hold that they are incapable of being acted upon, or of motion. In dealing with these people even arguments of this kind are useful. Further, see if he has rendered a single common definition of terms that are used ambiguously. For terms whose definition corresponding to their common name is one and the same, are synonymous; if, then, the definition applies in a like manner to the whole range of the ambiguous term, it is not true of any one of the objects described by the term. This is, moreover, what happens to Dionysius’ definition of ‘life’ when stated as ‘a movement of a creature sustained by nutriment, congenitally present with it’: for this is found in plants as much as in animals, whereas ‘life’ is generally understood to mean not one kind of thing only, but to be one thing in animals and another in plants. It is possible to hold the view that life is a synonymous term and is always used to describe one thing only, and therefore to render the definition in this way on purpose: or it may quite well happen that a man may see the ambiguous character of the

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word, and wish to render the definition of the one sense only, and yet fail to see that he has rendered the definition common to both senses instead of one peculiar to the sense he intends. In either case, whichever course he pursues, he is equally at fault. Since ambiguous terms sometimes pass unobserved, it is best in questioning to treat such terms as though they were synonymous (for the definition of the one sense will not apply to the other, so that the answerer will be generally thought not to have defined it correctly, for to a synonymous term the definition should apply in its full range), whereas in answering you should yourself distinguish between the senses. Further, as some answerers call ‘ambiguous’ what is really synonymous, whenever the definition rendered fails to apply universally, and, vice versa, call synonymous what is really ambiguous supposing their definition applies to both senses of the term, one should secure a preliminary admission on such points, or else prove beforehand that so-and-so is ambiguous or synonymous, as the case may be: for people are more ready to agree when they do not foresee what the consequence will be. If, however, no admission has been made, and the man asserts that what is really synonymous is ambiguous because the definition he has rendered will not apply to the second sense as well, see if the definition of this second meaning applies also to the other meanings: for if so, this meaning must clearly be synonymous with those others. Otherwise, there will be more than one definition of those other meanings, for there are applicable to them two distinct definitions in explanation of the term, viz, the one previously rendered and also the later one. Again, if any one were to define a term used in several senses, and, finding that his definition does not apply to them all, were to contend not that the term is ambiguous, but that even the term does not properly apply to all those senses, just because his definition will not do so either, then one may retort to such a man that though in some things one must not use the language of the people, yet in a question of terminology one is bound to employ the received and traditional usage and not to upset matters of that sort. 11 Suppose now that a definition has been rendered of some complex term, take away the definition of one of the elements in the complex, and see if also the rest of the definition defines the rest of it: if not, it is clear that neither does the whole definition define the whole complex. Suppose, e.g., that some one has defined a ‘finite straight line’ as ‘the limit of a finite plane, such that its centre is in a line with its extremes’; if now the definition of a ‘finite line’ be the ‘limit of a finite plane’, the rest (viz. ‘such that its centre is in a line with its extremes’) ought to be a definition of‘straight’. But an infinite straight line has

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neither centre nor extremes and yet is straight, so that this remainder does not define the remainder of the term. Moreover, if the term defined be a compound notion, see if the definition rendered be equimembral with the term defined. A definition is said to be equimembral with the term defined when the number of the elements compounded in the latter is the same as the number of nouns and verbs in the definition. For the exchange in such cases is bound to be merely one of term for term, in the case of some, if not of all, seeing that there are no more terms used now than formerly; whereas in a definition terms ought to be rendered by phrases, if possible in every case, or if not, in the majority. For at that rate, simple objects too could be defined by merely calling them by a different name, e.g. ‘cloak’ instead of ‘doublet’. The mistake is even worse, if actually a less well known term be substituted, e.g. ‘pellucid mortal’ for ‘white man’: for it is no definition, and moreover is less intelligible when put in that form. Look and see also whether, in the exchange of words, the sense fails still to be the same. Take, for instance, the explanation of ‘speculative knowledge’ as ‘speculative conception’: for conception is not the same as knowledge — as it certainly ought to be if the whole is to be the same too: for though the word ‘speculative’ is common to both expressions, yet the remainder is different. Moreover, see if in replacing one of the terms by something else he has exchanged the genus and not the differentia, as in the example just given: for ‘speculative’ is a less familiar term than knowledge; for the one is the genus and the other the differentia, and the genus is always the most familiar term of all; so that it is not this, but the differentia, that ought to have been changed, seeing that it is the less familiar. It might be held that this criticism is ridiculous: because there is no reason why the most familiar term should not describe the differentia, and not the genus; in which case, clearly, the term to be altered would also be that denoting the genus and not the differentia. If, however, a man is substituting for a term not merely another term but a phrase, clearly it is of the differentia rather than of the genus that a definition should be rendered, seeing that the object of rendering the definition is to make the subject familiar; for the differentia is less familiar than the genus. If he has rendered the definition of the differentia, see whether the definition rendered is common to it and something else as well: e.g., whenever he says that an odd number is a ‘number with a middle’, further definition is required of how it has a middle: for the word ‘number’ is common to both expressions, and it is the word ‘odd’ for which the phrase has been substituted. Now both a line and a body have a middle, yet they are not ‘odd’; so that this could not be a

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definition of ‘odd’. If, on the other hand, the phrase ‘with a middle’ be used in several senses, the sense here intended requires to be defined. So that this will either discredit the definition or prove that it is no definition at all. 12 Again, see if the term of which he renders the definition is a reality, whereas what is contained in the definition is not. E.g., suppose ‘white’ to be defined as ‘colour mingled with fire’: for what is bodiless cannot be mingled with body, so that ‘colour’ ‘mingled with fire’ could not exist, whereas ‘white’ does exist. Moreover, those who in the case of relative terms do not distinguish to what the object is related, but have described it only so as to include it among too large a number of things, are wrong either wholly or in part; e.g., suppose some one to have defined ‘medicine’ as a ‘science of Reality’. For if medicine be not a science of anything that is real, the definition is clearly altogether false; while if it be a science of some real thing, but not of another, it is partly false; for it ought to hold of all reality, if it is said to be of Reality essentially and not accidentally; as is the case with other relative terms: for every object of knowledge is a term relative to knowledge: likewise, also, with other relative terms, inasmuch as all such are convertible. Moreover, if the right way to render account of a thing be to render it as it is not in itself but accidentally, then each and every relative term would be used in relation not to one thing but to a number of things. For there is no reason why the same thing should not be both real and white and good, so that it would be a correct rendering to render the object in relation to any one whatsoever of these, if to render what it is accidentally be a correct way to render it. It is, moreover, impossible that a definition of this sort should be peculiar to the term rendered: for not only medicine, but the majority of the other sciences too, have for their object some real thing, so that each will be a science of reality. Clearly, then, such a definition does not define any science at all; for a definition ought to be peculiar to its own term, not general. Sometimes, again, people define not the thing but only the thing in a good or perfect condition. Such is the definition of a rhetorician as ‘one who can always see what will persuade in the given circumstances, and omit nothing’; or of a thief as ‘one who pilfers in secret’: for clearly, if they each do this, then the one will be a good rhetorician, and the other a good thief: whereas it is not the actual pilfering in secret, but the wish to do it, that constitutes the thief. Again, see if he has rendered what is desirable for its own sake as desirable for what it produces or does, or as in any way desirable because of something else, e.g. by saying that justice is ‘what preserves the laws’ or that wisdom is ‘what produces happiness’; for what produces or preserves something else is one

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of the things desirable for something else. It might be said that it is possible for what is desirable in itself to be desirable for something else as well: but still to define what is desirable in itself in such a way is none the less wrong: for the essence contains par excellence what is best in anything, and it is better for a thing to be desirable in itself than to be desirable for something else, so that this is rather what the definition too ought to have indicated. 13 See also whether in defining anything a man has defined it as an ‘A and B’, or as a ‘product of A and B’ or as an ‘A + B’. If he defines it as ‘A and B’, the definition will be true of both and yet of neither of them; suppose, e.g., justice to be defined as ‘temperance and courage’. For if of two persons each has one of the two only, both and yet neither will be just: for both together have justice, and yet each singly fails to have it. Even if the situation here described does not so far appear very absurd because of the occurrence of this kind of thing in other cases also (for it is quite possible for two men to have a mina between them, though neither of them has it by himself), yet at least that they should have contrary attributes surely seems quite absurd; and yet this will follow if the one be temperate and yet a coward, and the other, though brave, be a profligate; for then both will exhibit both justice and injustice: for if justice be temperance and bravery, then injustice will be cowardice and profligacy. In general, too, all the ways of showing that the whole is not the same as the sum of its parts are useful in meeting the type just described; for a man who defines in this way seems to assert that the parts are the same as the whole. The arguments are particularly appropriate in cases where the process of putting the parts together is obvious, as in a house and other things of that sort: for there, clearly, you may have the parts and yet not have the whole, so that parts and whole cannot be the same. If, however, he has said that the term being defined is not ‘A and B’ but the ‘product of A and B’, look and see in the first place if A and B cannot in the nature of things have a single product: for some things are so related to one another that nothing can come of them, e.g. a line and a number. Moreover, see if the term that has been defined is in the nature of things found primarily in some single subject, whereas the things which he has said produce it are not found primarily in any single subject, but each in a separate one. If so, clearly that term could not be the product of these things: for the whole is bound to be in the same things wherein its parts are, so that the whole will then be found primarily not in one subject only, but in a number of them. If, on the other hand, both parts and whole are found primarily in some single subject, see if that medium is not the same, but one thing in the case of the whole and another in

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that of the parts. Again, see whether the parts perish together with the whole: for it ought to happen, vice versa, that the whole perishes when the parts perish; when the whole perishes, there is no necessity that the parts should perish too. Or again, see if the whole be good or evil, and the parts neither, or, vice versa, if the parts be good or evil and the whole neither. For it is impossible either for a neutral thing to produce something good or bad, or for things good or bad to produce a neutral thing. Or again, see if the one thing is more distinctly good than the other is evil, and yet the product be no more good than evil, e.g. suppose shamelessness be defined as ‘the product of courage and false opinion’: here the goodness of courage exceeds the evil of false-opinion; accordingly the product of these ought to have corresponded to this excess, and to be either good without qualification, or at least more good than evil. Or it may be that this does not necessarily follow, unless each be in itself good or bad; for many things that are productive are not good in themselves, but only in combination; or, per contra, they are good taken singly, and bad or neutral in combination. What has just been said is most clearly illustrated in the case of things that make for health or sickness; for some drugs are such that each taken alone is good, but if they are both administered in a mixture, bad. Again, see whether the whole, as produced from a better and worse, fails to be worse than the better and better than the worse element. This again, however, need not necessarily be the case, unless the elements compounded be in themselves good; if they are not, the whole may very well not be good, as in the cases just instanced. Moreover, see if the whole be synonymous with one of the elements: for it ought not to be, any more than in the case of syllables: for the syllable is not synonymous with any of the letters of which it is made up. Moreover, see if he has failed to state the manner of their composition: for the mere mention of its elements is not enough to make the thing intelligible. For the essence of any compound thing is not merely that it is a product of so-andso, but that it is a product of them compounded in such and such a way, just as in the case of a house: for here the materials do not make a house irrespective of the way they are put together. If a man has defined an object as ‘A + B’, the first thing to be said is that ‘A + B’ means the same either as ‘A and B’, or as the ‘product of A and B’. For ‘honey + water’ means either the honey and the water, or the ‘drink made of honey and water’. If, then, he admits that ‘A + B’ is the same as either of these two things, the same criticisms will apply as have already been given for meeting each of them. Moreover, distinguish between the different senses in which one thing may be said to be ‘+’ another, and see if there is none of them in which A could be said to exist ‘+ B’. Thus, e.g., supposing the expression to mean that

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they exist either in some identical thing capable of containing them (as e.g., justice and courage are found in the soul), or else in the same place or in the same time, and if this be in no way true of the A and B in question, clearly the definition rendered could not hold of anything, as there is no possible way in which A can exist ‘+B’. If, however, among the various senses above distinguished, it be true that A and B are each found in the same time as the other, look and see if possibly the two are not used in the same relation. Thus e.g., suppose courage to have been defined as ‘daring with right reasoning’: here it is possible that the person exhibits daring in robbery, and right reasoning in regard to the means of health: but he may have ‘the former quality + the latter’ at the same time, and not as yet be courageous! Moreover, even though both be used in the same relation as well, e.g. in relation to medical treatment (for a man may exhibit both daring and right reasoning in respect of medical treatment), still, none the less, not even this combination of ‘the one + the other’ makes him ‘courageous’. For the two must not relate to any casual object that is the same, any more than each to a different object; rather, they must relate to the function of courage, e.g., meeting the perils of war, or whatever is more properly speaking its function than this. Some definitions rendered in this form fail to come under the aforesaid division at all, e.g., a definition of anger as ‘pain with a consciousness of being slighted’. For what this means to say is that it is because of a consciousness of this sort that the pain occurs; but to occur ‘because of’ a thing is not the same as to occur ‘+ a thing’ in any of its aforesaid senses. 14 Again, if he have described the whole compounded as the ‘composition’ of these things (e.g. ‘a living creature’ as a ‘composition of soul and body’), first of all see whether he has omitted to state the kind of composition, as, e.g., in a definition of ‘flesh’ or ‘bone’ as the ‘composition of fire, earth, and air’. For it is not enough to say it is a composition, but you should also go on to define the kind of composition: for these things do not form flesh irrespective of the manner of their composition, but when compounded in one way they form flesh, when in another, bone. It appears, moreover, that neither of the aforesaid substances is the same as a ‘composition’ at all: for a composition always has a decomposition as its contrary, whereas neither of the aforesaid has any contrary. Moreover, if it is equally probable that every compound is a composition or else that none is, and every kind of living creature, though a compound, is never a composition, then no other compound could be a composition either. Again, if in the nature of a thing two contraries are equally liable to occur,

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and the thing has been defined through the one, clearly it has not been defined; else there will be more than one definition of the same thing; for how is it any more a definition to define it through this one than through the other, seeing that both alike are naturally liable to occur in it? Such is the definition of the soul, if defined as a substance capable of receiving knowledge: for it has a like capacity for receiving ignorance. Also, even when one cannot attack the definition as a whole for lack of acquaintance with the whole, one should attack some part of it, if one knows that part and sees it to be incorrectly rendered: for if the part be demolished, so too is the whole definition. Where, again, a definition is obscure, one should first of all correct and reshape it in order to make some part of it clear and get a handle for attack, and then proceed to examine it. For the answerer is bound either to accept the sense as taken by the questioner, or else himself to explain clearly whatever it is that his definition means. Moreover, just as in the assemblies the ordinary practice is to move an emendation of the existing law and, if the emendation is better, they repeal the existing law, so one ought to do in the case of definitions as well: one ought oneself to propose a second definition: for if it is seen to be better, and more indicative of the object defined, clearly the definition already laid down will have been demolished, on the principle that there cannot be more than one definition of the same thing. In combating definitions it is always one of the chief elementary principles to take by oneself a happy shot at a definition of the object before one, or to adopt some correctly expressed definition. For one is bound, with the model (as it were) before one’s eyes, to discern both any shortcoming in any features that the definition ought to have, and also any superfluous addition, so that one is better supplied with lines of attack. As to definitions, then, let so much suffice.

BOOK VII (151b, 25–155a, 40) 1 Whether two things are ‘the same’ or ‘different’, in the most literal of the meanings ascribed to ‘sameness’ (and we said that ‘the same’ applies in the most literal sense to what is numerically one), may be examined in the light of their inflexions and coordinates and opposites. For if justice be the same as courage, then too the just man is the same as the brave man, and ‘justly’ is the same as ‘bravely’. Likewise, too, in the case of their opposites: for if two things be the same, their opposites also will be the same, in any of the recognized forms of

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opposition. For it is the same thing to take the opposite of the one or that of the other, seeing that they are the same. Again it may be examined in the light of those things which tend to produce or to destroy the things in question, of their formation and destruction, and in general of any thing that is related in like manner to each. For where things are absolutely the same, their formations and destructions also are the same, and so are the things that tend to produce or to destroy them. Look and see also, in a case where one of two things is said to be something or other in a superlative degree, if the other of these alleged identical things can also be described by a superlative in the same respect. Thus Xenocrates argues that the happy life and the good life are the same, seeing that of all forms of life the good life is the most desirable and so also is the happy life: for ‘the most desirable’ and ‘the greatest’ apply but to one thing. Likewise also in other cases of the kind. Each, however, of the two things termed ‘greatest’ or ‘most desirable’ must be numerically one: otherwise no proof will have been given that they are the same; for it does not follow because Peloponnesians and Spartans are the bravest of the Greeks, that Peloponnesians are the same as Spartans, seeing that ‘Peloponnesian’ is not any one person nor yet ‘Spartan’, it only follows that the one must be included under the other as ‘Spartans’ are under ‘Peloponnesians’: for otherwise, if the one class be not included under the other, each will be better than the other. For then the Peloponnesians are bound to be better than the Spartans, seeing that the one class is not included under the other; for they are better than anybody else. Likewise also the Spartans must perforce be better than the Peloponnesians; for they too are better than anybody else; each then is better than the other! Clearly therefore what is styled ‘best’ and ‘greatest’ must be a single thing, if it is to be proved to be ‘the same’ as another. This also is why Xenocrates fails to prove his case: for the happy life is not numerically single, nor yet the good life, so that it does not follow that, because they are both the most desirable, they are therefore the same, but only that the one falls under the other. Again, look and see if, supposing the one to be the same as something, the other also is the same as it: for if they be not both the same as the same thing, clearly neither are they the same as one another. Moreover, examine them in the light of their accidents or of the things of which they are accidents: for any accident belonging to the one must belong also to the other, and if the one belong to anything as an accident, so must the other also. If in any of these respects there is a discrepancy, clearly they are not the same. See further whether, instead of both being found in one class of predicates, the one signifies a quality and the other a quantity or relation. Again, see if the

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genus of each be not the same, the one being ‘good’ and the other ‘evil’, or the one being ‘virtue’ and the other ‘knowledge’: or see if, though the genus is the same, the differentiae predicted of either be not the same, the one, e.g., being distinguished as a ‘speculative ‘science, the other as a ‘practical’ science. Likewise also in other cases. Moreover, from the point of view of ‘degrees’, see if the one admits an increase of degree but not the other, or if though both admit it, they do not admit it at the same time; just as it is not the case that a man desires intercourse more intensely, the more intensely he is in love, so that love and the desire for intercourse are not the same. Moreover, examine them by means of an addition, and see whether the addition of each to the same thing fails to make the same whole; or if the subtraction of the same thing from each leaves a different remainder. Suppose, e.g., that he has declared ‘double a half’ to be the same as ‘a multiple of a half’: then, subtracting the words ‘a half’ from each, the remainders ought to have signified the same thing: but they do not; for ‘double’ and ‘a multiple of’ do not signify the same thing. Inquire also not only if some impossible consequence results directly from the statement made, that A and B are the same, but also whether it is possible for a supposition to bring it about; as happens to those who assert that ‘empty’ is the same as ‘full of air’: for clearly if the air be exhausted, the vessel will not be less but more empty, though it will no longer be full of air. So that by a supposition, which may be true or may be false (it makes no difference which), the one character is annulled and not the other, showing that they are not the same. Speaking generally, one ought to be on the look-out for any discrepancy anywhere in any sort of predicate of each term, and in the things of which they are predicated. For all that is predicated of the one should be predicated also of the other, and of whatever the one is a predicate, the other should be a predicate of it as well. Moreover, as ‘sameness’ is a term used in many senses, see whether things that are the same in one way are the same also in a different way. For there is either no necessity or even no possibility that things that are the same specifically or generically should be numerically the same, and it is with the question whether they are or are not the same in that sense that we are concerned. Moreover, see whether the one can exist without the other; for, if so, they could not be the same. 2 Such is the number of the commonplace rules that relate to ‘sameness’. It is

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clear from what has been said that all the destructive commonplaces relating to sameness are useful also in questions of definition, as was said before: for if what is signified by the term and by the expression be not the same, clearly the expression rendered could not be a definition. None of the constructive commonplaces, on the other hand, helps in the matter of definition; for it is not enough to show the sameness of content between the expression and the term, in order to establish that the former is a definition, but a definition must have also all the other characters already announced. 3 This then is the way, and these the arguments, whereby the attempt to demolish a definition should always be made. If, on the other hand, we desire to establish one, the first thing to observe is that few if any who engage in discussion arrive at a definition by reasoning: they always assume something of the kind as their starting point — both in geometry and in arithmetic and the other studies of that kind. In the second place, to say accurately what a definition is, and how it should be given, belongs to another inquiry. At present it concerns us only so far as is required for our present purpose, and accordingly we need only make the bare statement that to reason to a thing’s definition and essence is quite possible. For if a definition is an expression signifying the essence of the thing and the predicates contained therein ought also to be the only ones which are predicated of the thing in the category of essence; and genera and differentiae are so predicated in that category: it is obvious that if one were to get an admission that so and so are the only attributes predicated in that category, the expression containing so and so would of necessity be a definition; for it is impossible that anything else should be a definition, seeing that there is not anything else predicated of the thing in the category of essence. That a definition may thus be reached by a process of reasoning is obvious. The means whereby it should be established have been more precisely defined elsewhere, but for the purposes of the inquiry now before us the same commonplace rules serve. For we have to examine into the contraries and other opposites of the thing, surveying the expressions used both as wholes and in detail: for if the opposite definition defines that opposite term, the definition given must of necessity be that of the term before us. Seeing, however, that contraries may be conjoined in more than one way, we have to select from those contraries the one whose contrary definition seems most obvious. The expressions, then, have to be examined each as a whole in the way we have said, and also in detail as follows. First of all, see that the genus rendered is correctly rendered; for if the contrary thing be found in the contrary genus to that stated in the definition, and the thing

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before you is not in that same genus, then it would clearly be in the contrary genus: for contraries must of necessity be either in the same genus or in contrary genera. The differentiae, too, that are predicated of contraries we expect to be contrary, e.g., those of white and black, for the one tends to pierce the vision, while the other tends to compress it. So that if contrary differentiae to those in the definition are predicated of the contrary term, then those rendered in the definition would be predicated of the term before us. Seeing, then, that both the genus and the differentiae have been rightly rendered, clearly the expression given must be the right definition. It might be replied that there is no necessity why contrary differentiae should be predicated of contraries, unless the contraries be found within the same genus: of things whose genera are themselves contraries it may very well be that the same differentia is used of both, e.g., of justice and injustice; for the one is a virtue and the other a vice of the soul: ‘of the soul’, therefore, is the differentia in both cases, seeing that the body as well has its virtue and vice. But this much at least is true, that the differentiae of contraries are either contrary or else the same. If, then, the contrary differentia to that given be predicated of the contrary term and not of the one in hand, clearly the differentia stated must be predicated of the latter. Speaking generally, seeing that the definition consists of genus and differentiae, if the definition of the contrary term be apparent, the definition of the term before you will be apparent also: for since its contrary is found either in the same genus or in the contrary genus, and likewise also the differentiae predicated of opposites are either contrary to, or the same as, each other, clearly of the term before you there will be predicated either the same genus as of its contrary, while, of its differentiae, either all are contrary to those of its contrary, or at least some of them are so while the rest remain the same; or, vice versa, the differentiae will be the same and the genera contrary; or both genera and differentiae will be contrary. And that is all; for that both should be the same is not possible; else contraries will have the same definition. Moreover, look at it from the point of view of its inflexions and coordinates. For genera and definitions are bound to correspond in either case. Thus if forgetfulness be the loss of knowledge, to forget is to lose knowledge, and to have forgotten is to have lost knowledge. If, then, any one whatever of these is agreed to, the others must of necessity be agreed to as well. Likewise, also, if destruction is the decomposition of the thing‘s essence, then to be destroyed is to have its essence decomposed, and ‘destructively’ means ‘in such a way as to decompose its essence’; if again ‘destructive’ means ‘apt to decompose something’s essence’, then also ‘destruction’ means ‘the decomposition of

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its essence’. Likewise also with the rest: get an admission of any one of them whatever, and all the rest are admitted too. Moreover, look at it from the point of view of things that stand in relations that are like each other. For if ‘healthy’ means ‘productive of health’, ‘vigorous’ too will mean ‘productive of vigour’, and ‘useful’ will mean ‘productive of good’. For each of these things is related in like manner to its own peculiar end, so that if one of them is defined as ‘productive of’ that end, this will also be the definition of each of the rest as well. Moreover, look at it from the point of view of greater and like degrees, in all the ways in which it is possible to establish a result by comparing two and two together. Thus if A defines a better than B defines b, and B is a definition of b, so too is A of a. Further, if A’s claim to define a is like B’s to define b and B defines b, then A too defines a. This examination from the point of view of greater degrees is of no use when a single definition is compared with two things, or two definitions with one thing; for there cannot possibly be one definition of two things or two of the same thing. 4 The most handy of all the commonplace arguments are those just mentioned and those from coordinates and inflexions, and these therefore are those which it is most important to master and to have ready to hand: for they are the most useful on the greatest number of occasions. Of the rest, too, the most important are those of most general application: for these are the most effective, e.g., that you should examine the individual cases, and then look to see in the case of their various species whether the definition applies. For the species is synonymous with its individuals. This sort of inquiry is of service against those who assume the existence of Ideas, as has been said before. Moreover see if a man has used a term metaphorically, or predicated it of itself as though it were something different. So too if any other of the commonplace rules is of general application and effective, it should be employed. 5 That it is more difficult to establish than to overthrow a definition, is obvious from considerations presently to be urged. For to see for oneself, and to secure from those whom one is questioning, an admission of premisses of this sort is no simple matter, e.g. that of the elements of the definition rendered the one is genus and the other differentia, and that only the genus and differentiae are predicated in the category of essence. Yet without these premisses it is impossible to reason to a definition; for if any other things as well are predicated of the

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thing in the category of essence, there is no telling whether the formula stated or some other one is its definition, for a definition is an expression indicating the essence of a thing. The point is clear also from the following: It is easier to draw one conclusion than many. Now in demolishing a definition it is sufficient to argue against one point only (for if we have overthrown any single point whatsoever, we shall have demolished the definition); whereas in establishing a definition, one is bound to bring people to the view that everything contained in the definition is attributable. Moreover, in establishing a case, the reasoning brought forward must be universal: for the definition put forward must be predicated of everything of which the term is predicated, and must moreover be convertible, if the definition rendered is to be peculiar to the subject. In overthrowing a view, on the other hand, there is no longer any necessity to show one’s point universally: for it is enough to show that the formula is untrue of any one of the things embraced under the term. Further, even supposing it should be necessary to overthrow something by a universal proposition, not even so is there any need to prove the converse of the proposition in the process of overthrowing the definition. For merely to show that the definition fails to be predicated of every one of the things of which the term is predicated, is enough to overthrow it universally: and there is no need to prove the converse of this in order to show that the term is predicated of things of which the expression is not predicated. Moreover, even if it applies to everything embraced under the term, but not to it alone, the definition is thereby demolished. The case stands likewise in regard to the property and genus of a term also. For in both cases it is easier to overthrow than to establish. As regards the property this is clear from what has been said: for as a rule the property is rendered in a complex phrase, so that to overthrow it, it is only necessary to demolish one of the terms used, whereas to establish it is necessary to reason to them all. Then, too, nearly all the other rules that apply to the definition will apply also to the property of a thing. For in establishing a property one has to show that it is true of everything included under the term in question, whereas to overthrow one it is enough to show in a single case only that it fails to belong: further, even if it belongs to everything falling under the term, but not to that only, it is overthrown in this case as well, as was explained in the case of the definition. In regard to the genus, it is clear that you are bound to establish it in one way only, viz, by showing that it belongs in every case, while of overthrowing it there are two ways: for if it has been shown that it belongs either never or not in a certain case, the original statement has been demolished. Moreover, in establishing a genus it is not enough to show that it belongs, but also that it belongs as genus has to be shown; whereas in overthrowing it, it is enough to

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show its failure to belong either in some particular case or in every case. It appears, in fact, as though, just as in other things to destroy is easier than to create, so in these matters too to overthrow is easier than to establish. In the case of an accidental attribute the universal proposition is easier to overthrow than to establish; for to establish it, one has to show that it belongs in every case, whereas to overthrow it, it is enough to show that it does not belong in one single case. The particular proposition is, on the contrary, easier to establish than to overthrow: for to establish it, it is enough to show that it belongs in a particular instance, whereas to overthrow it, it has to be shown that it never belongs at all. It is clear also that the easiest thing of all is to overthrow a definition. For on account of the number of statements involved we are presented in the definition with the greatest number of points for attack, and the more plentiful the material, the quicker an argument comes: for there is more likelihood of a mistake occurring in a large than in a small number of things. Moreover, the other rules too may be used as means for attacking a definition: for if either the formula be not peculiar, or the genus rendered be the wrong one, or something included in the formula fail to belong, the definition is thereby demolished. On the other hand, against the others we cannot bring all of the arguments drawn from definitions, nor yet of the rest: for only those relating to accidental attributes apply generally to all the aforesaid kinds of attribute. For while each of the aforesaid kinds of attribute must belong to the thing in question, yet the genus may very well not belong as a property without as yet being thereby demolished. Likewise also the property need not belong as a genus, nor the accident as a genus or property, so long as they do belong. So that it is impossible to use one set as a basis of attack upon the other except in the case of definition. Clearly, then, it is the easiest of all things to demolish a definition, while to establish one is the hardest. For there one both has to establish all those other points by reasoning (i.e. that the attributes stated belong, and that the genus rendered is the true genus, and that the formula is peculiar to the term), and moreover, besides this, that the formula indicates the essence of the thing; and this has to be done correctly. Of the rest, the property is most nearly of this kind: for it is easier to demolish, because as a rule it contains several terms; while it is the hardest to establish, both because of the number of things that people must be brought to accept, and, besides this, because it belongs to its subject alone and is predicated convertibly with its subject. The easiest thing of all to establish is an accidental predicate: for in other cases one has to show not only that the predicate belongs, but also that it belongs

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in such and such a particular way: whereas in the case of the accident it is enough to show merely that it belongs. On the other hand, an accidental predicate is the hardest thing to overthrow, because it affords the least material: for in stating an accident a man does not add how the predicate belongs; and accordingly, while in other cases it is possible to demolish what is said in two ways, by showing either that the predicate does not belong, or that it does not belong in the particular way stated, in the case of an accidental predicate the only way to demolish it is to show that it does not belong at all. The commonplace arguments through which we shall be well supplied with lines of argument with regard to our several problems have now been enumerated at about sufficient length.

Physics (Extract) BOOK I (184b) Thus we must advance from generalities to particulars; for it is a whole that is best known to sense-perception, and a generality is a kind of whole, comprehending many things within it, like parts. Much the same thing happens in the relation of the name to the formula. A name, e.g. ‘round’, means vaguely a sort of whole: its definition analyses this into its particular senses. Similarly a child begins by calling all men ‘father’, and all women ‘mother’, but later on distinguishes each of them.

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Metaphysics (Extracts) CHAPTER Z (1029b-1031a) 4 Therefore there is an essence only of those things whose formula is a definition. But we have no definition where we have a word and a formula identical in meaning (for in that case all formulae or sets of words would be definitions; for

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there will be some name for any set of words whatever, so that even the Iliad will be a definition), but where there is a formula of something primary; and primary things are those which do not imply the predication of one element in them of another element. Nothing, then, which is not a species of a genus will have an essence — only species will have it, for these are thought to imply not merely that the subject participates in the attribute and has it as an affection, or has it by accident; but for everything else as well, if it has a name, there will be a formula of its meaning — viz. that this attribute belongs to this subject; or instead of a simple formula we shall be able to give a more accurate one; but there will be no definition nor essence. Or has ‘definition’, like ‘what a thing is’ several meanings? ‘What a thing is’ in one sense means substance and the ‘this’, in another one or other of the predicates, quality, quantity, and the like. For as ‘is’ belongs to all things, not however in the same sense, but to one sort of thing primarily and to others in a secondary way, so too ‘what a thing is’ belongs in the simple sense to substance, but in a limited sense to other categories. For even of a quality we might ask what it is, so that quality also is a ‘what a thing is’, — not in the simple sense, however, but just as, in the case of that which is not, some say, emphasizing the linguistic form, that that which is not is — not is simply, but is non-existent; so too with quantity. We must no doubt inquire how we should express ourselves on each point, but certainly not more than how the facts actually stand. And so now also, since it is evident what language we use, essence will belong, just as ‘What a thing is’ does, primarily and in the simple sense of substance, and in a secondary way to the other categories also, — not essence in the simple sense, but the essence of a quality or of a quantity. For it must be either by an equivocation that we say these are, or by adding to and taking from the meaning of ‘are’ (in the way in which that which is not known may be said to be known), — the truth being that we use the word neither ambiguously nor in the same sense, but just as we apply the word ‘medical’ by virtue of a reference to one and the same thing, not meaning one and the same thing, nor yet speaking ambiguously; for a patient and an operation and an instrument are called medical neither by an ambiguity nor with a single meaning, but with reference to a common end. But it does not matter at all in which of the two ways one likes to describe the facts; this is evident, that definition and essence in the primary and simple sense belong to substances. Still they belong to other things as well, only not in the primary sense. For if we suppose this it does not follows that there is a definition of every word which means the same as any formula; it must mean the same as a particular kind of formula; and this condition is satisfied if it is a formula of

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something which is one, not by continuity like the Iliad or the things that are one by being bound together, but in one of the main senses of ‘one’, which answers to the sense of ‘is’; now ‘that which is’ in one sense denotes a ‘this’, in another a quantity, in another a quality. And so there can be a formula or definition even of a white man, but not in the sense in which there is a definition either of white or of a substance. 5 It is a difficult questions, if one denies that a formula with an added determinant is a definition, whether any of the terms that are not simple but coupled will be definable. For we must explain them by adding a determinant. E.g., there is the nose, and concavity, and snubness, which is compounded out of the two by the presence of the one in the other, and it is not by accident that the nose has the attribute either of concavity or of snubness, but in virtue of its nature; nor do they attach to it as whiteness does to Callias, or to a man (because Callias, who happens to be a man, is white) but as ‘male’ attaches to animal and ‘equal’ to quantity, and as all so-called ‘attributes propter se’ attach to their subjects. And such attributes are those in which is involved either the formula or the name of the subject of the particular attribute, and which cannot be explained without this; e.g white can be explained apart from man, but not female apart from animal. Therefore there is either no essence and definition of any of these things, or if there is, it is in another sense, as we have said. But there is also a second difficulty about them. For if snub nose and concave nose are the same thing, snub and concave will be the same thing; but if snub and concave are not the same (because it is impossible to speak of snubness apart from the thing of which it is an attribute propter se, for snubness is concavity-in-a-nose), either it is impossible to say ‘snub nose’ or the same thing will have been said twice, concave-nose nose; for snub nose will be concave-nose nose. And so it is absurd that such things should have an essence; if they have, there will be an infinite regress; for in snub-nose nose yet another ‘nose’ will be involved. Clearly, then, only substance is definable. For if the other categories are also definable, it must be by addition of a determinant, e.g., the qualitative is defined thus, and so is the odd, for it cannot be defined apart from number; nor can female be defined apart from animal. (When I say ‘by addition’ I mean the expressions by which it turns out that we are saying the same thing twice, as in these instances.) And if this is true, coupled terms also, like ‘odd number’, will not be definable (but this escapes our notice because our formulae are not accurate). But if these are also definable, either it is in some other way, or, as we

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said, definition and essence must be said to have more than one sense. Therefore in one sense nothing will have a definition and nothing will have an essence, except substances, but in another sense other things will have them. Clearly, then, definition is the formula of the essence, and essence belongs to substances either alone or chiefly and primarily and in the unqualified sense.

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(1034b) 10 Since a definition is a formula, and every formula has parts, and as the formula is to the thing, so is the part of the formula to the part of the thing, the question is already being asked whether the formula of the parts must be present in the formula of the whole or not. For in some cases the formulae of the parts are seen to be present, and in some not. The formula of the circle does not include that of the segments, but that of the syllable includes that of the letters; yet the circle is divided into segments as the syllable is into letters. — And further if the parts are prior to the whole, and the acute angle is a part of the right angle and the finger a part of the animal, the acute angle will be prior to the right angle and the finger to the man. But the latter are thought to be prior; for in formula the parts are explained by reference to them, and in respect also of the power of existing apart from each other the wholes are prior to the parts.

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(1037b-1038a) 12 Now let us treat first of definition, in so far as we have not treated of it in the Analytics (An. Post.ii,3–10,13.); for the problem stated in them is useful for our inquiries concerning substance. I mean this problem: — wherein can consist the unity of that, the formula of which we call a definition, as for instance, in the case of man, ‘two-footed animal’; for let this be the formula of man. Why, then, is this one, and not many, viz. ‘animal’ and ‘two-footed’? For in the case of

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‘man’ and ‘pale’ there is a plurality when one term does not belong to the other, but a unity when it does belong and the subject, man, has a certain attribute; for then a unity is produced and we have ‘the pale man’. In the present case, on the other hand, one does not share in the other; the genus is not thought to share in its differentiae (for then the same thing would share in contraries; for the differentiae by which the genus is divided are contrary). And even if the genus does share in them, the same argument applies, since the differentiae present in man are many, e.g. endowed with feet, two-footed, featherless. Why are these one and not many? Not because they are present in one thing; for on this principle a unity can be made out of all the attributes of a thing. But surely all the attributes in the definition must be one; for the definition is a single formula and a formula of substance, so that it must be a formula of some one thing; for substance means a ‘one’ and a ‘this’, as we maintain. We must first inquire about definitions reached by the method of division. There is nothing in the definition except the first-named genus and the differentiae. The other genera are the first genus and along with this the differentiae that are taken with it, e.g. the first may be ‘animal’ the next ‘animal which is two-footed’, and again ‘animal which is two-footed and featherless’, and similarly if the definition includes more terms. And in general it makes no difference whether it includes many or few terms — nor, therefore, whether it includes few or simply two; and of the two one is the differentiae and the other genus; e.g. in ‘two-footed animal’ ‘animal, is genus, and the other is differentiae. If then the genus absolutely does not exist apart from the species-of-a genus, or if it exists but exists as matter (for the voice is genus and matter, but its differentiae make the species, i.e. the letters, out of it), clearly the definition is the formula which comprises the differentiae. But it is also necessary that the division be by the differentiae of the differentiae; e.g., ‘endowed with feet’ is a differentia of ‘animal’; again the differentia of ‘animal endowed with feet’ must be of it qua endowed with feet. Therefore we must not say, if we are to speak rightly, that of that which is endowed with feet one part has feathers and one is featherless (if we do this we do it through incapacity); we must divide it only into cloven-footed and notcloven; for these are differentiae in the foot; clove-footedness is a form of footedness. And the process wants always to go on so till it reaches the species that contain no differences. And then there will be as many kinds of foot as there are differentiae, and the kinds of animals endowed with feet will be equal in number to the differentiae. If then this is so, clearly the last differentia will be the substance of the thing and its definition, since it is not right to state the same things more than once in our definitions; for it is superfluous. And this

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does happen; for when we say ‘animal endowed with feet and two-footed’ we have said nothing other than ‘animal having feet, having two feet’; and if we divide this by the proper division, we shall be saying the same thing more than once — as many times as there are differentiae. If then a differentia of a differentia be taken at each step, one differentia — the last — will be the form and the substance; but if we divide according to accidental qualities, e.g. if we were to divide that which is endowed with feet into the white and the black, there will be as many differentiae as there are cuts. Therefore it is plain that the definition is the formula which contains the differentiae, or, according to the right method, the last of these. This would be evident, if we were to change the order of such definitions, e.g., of that of man, saying ‘animal which is two-footed and endowed with feet’; for ‘endowed with feet’ is superfluous when ‘two-footed’ has been said. But there is no order in the substance; for how are we to think the one element posterior and the other prior? Regarding the definitions, then, which are reached by the method of divisions, let this suffice as our first attempt at stating their nature.

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(1039b-1040b) 15 Since substance is of two kinds, the concrete thing and the formula (I mean that one kind of substance is the formula taken with the matter, while another kind is the formula in its generality), substances in the former sense are capable of destruction (for they are capable also of generation), but there is no destruction of the formula in the sense that it is ever in course of being destroyed (for there is no generation of it either; the being of house is not generated. but only the being of this house), but without generation and destruction formulae are and are not; for it has been shown that no one begets or makes these. For this reason, also, there is neither definition of nor demonstration about sensible individual substances, because they have matter whose nature is such that they are capable both of being and of not being; for which reason all the individual instances of them are destructible. If then demonstration is of necessary truths and definition is a scientific process, and if, just as knowledge cannot be sometimes knowledge and sometimes ignorance, but the state which varies thus is opinion, so too demonstration and definition cannot vary thus, but it is opinion that deals with

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that which can be otherwise than as it is, clearly there can neither be definition of nor demonstration about sensible individuals. For perishing things are obscure to those who have the relevant knowledge, when they have passed from our perception; and though the formulae remain in the soul unchanged, there will no longer be either definition or demonstration. And so when one of the definitionmongers defines any individual, he must recognise that his definition may always be overthrown; for it is not possible to define such things. Nor it is possible to define any Idea. For the Ideas is, as its supporters say, an individual, and can exist apart; and the formula must consist of words; and he who defines must not invent a word (for it would be unknown), but the established words are common to all the members of a class; these then must apply to something besides the thing defined; e.g., if one were defining you, he would say ‘an animal which is lean’ or ‘pale’, or something else which will apply also to some one other than you. If any one were to say that perhaps all the attributes taken apart may belong to many subjects, but together they belong only to this one, we must reply first that they belong also to both the elements; e.g., ‘twofooted animal’ belongs to animal and to the two-footed. (And in the case of eternal entities this is even necessary, since the elements are prior to and parts of the compound; nay more, they can also exist apart, if ‘man’ can exist apart. For either neither or both can. If neither can, the genus will not exist apart from the various species; but if it does, the differentia will also.) Secondly, we must reply that ‘animal’ and ‘two-footed’ are prior in being to ‘two-footed animal’; and things which are prior to others are not destroyed when the others are. Again, if the Ideas consist of Ideas (as they must, since elements are simpler than the compound), it will be further necessary that the elements also of which the Idea consists, e.g. ‘animal’ and ‘two-footed’, should be predicated of many subjects. If not, how will they come to be known? For there will then be an Idea which cannot be predicated of more subjects than one. But this is not thought possible — every Idea is thought to be capable of being shared. As has been said, then, the impossibility of defining individuals escapes notice in the case of eternal things, especially those which are unique, like the sun or the moon. For people err not only by adding attributes whose removal the sun would survive, e.g., ‘going round the earth’ or ‘night-hidden’ (for from their view it follows that if it stands still or is visible, it will no longer be the sun; but it is strange, if this is so; for ‘the sun’ means a certain substance); but also by the mention of attributes which can belong to another subject; e.g., if another thing with the stated attributes comes into existence, clearly it will be a sun; the formula therefore is general. But the sun was supposed to be an individual, like Cleon or Socrates. After all, why does not one of the supporters of the Ideas

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produce a definition of an Idea? It would become clear, if they tried, that what has now been said is true.

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CHAPTER H (1045a-1045b) 6 To return to the difficulty which has been stated with respect both to definitions and to numbers, what is the cause of their unity? In the case of all things which have several parts and in which the totality is not, as it were, a mere heap, but the whole is something besides the parts, there is a cause; for even in bodies contact is the cause of unity in some cases, and in others viscosity or some other such quality. And a definition is a set of words which is one not by being connected together, like the Iliad, but by dealing with one object. — What, then, is it that makes man one; why is he one and not many, e.g. animal + biped, especially if there are, as some say, an animal-itself and a biped-itself. Why are not those Forms themselves the man, so that men would exist by participation not in man, nor in one Form, but in two, animal and biped, and in general man would not be one but more than one thing, animal and biped? Clearly, then, if people proceed thus in their usual manner of definition and speech, they cannot explain and solve the difficulty. But if, as we say, one element is matter and another is form, and one is potentially and the other is actually, the question will no longer be thought a difficulty. For this difficulty is the same as would arise if ‘round bronze’ were the definition of ‘cloak’; for this word would be a sign of the definitory formula, so that the question is, what is the cause of the unity of ‘round’ and ‘bronze’? The difficulty disappears, because the one is matter, the other form. What, then, causes this — that which was potentially to be actually — except, in the case of things which are generated, the agent? For there is no other cause of the potential sphere’s becoming actually a sphere, but this was the essence of either. Of matter some is intelligible, some perceptible, and in a formula there is always an element of matter as well as one of actuality; e.g., the circle is a ‘plane figure’. But of the things which have no matter, either intelligible or perceptible, each is by its nature essentially a kind of unity, as it is essentially a kind of being — individual substance, quality, or quantity (and so neither ‘existent’ nor ‘one’ is present in their definitions), and the essence of each of them is by its very nature a kind of

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unity as it is a kind of being — and so none of these has any reason outside itself for being one, nor for being a kind of being; for each is by its nature a kind of being and a kind of unity, not as being in the genus ‘being’ or ‘one’ nor in the sense that being and unity can exist apart from particulars. Owing to the difficulty about unity some speak of ‘participation’, and raise the question, what is the cause of participation and what is it to participate; and others speak of ‘communion’ as Lycophron says knowledge is a communion of knowing with the soul; and others say life is a ‘composition’ or ‘connexion’ of soul with body. Yet the same account applies to all cases; for being healthy, too, will on this showing be either a ‘communion’ or a ‘connexion’ or a ‘composition’ of soul and health, and the fact that the bronze is a triangle will be a ‘composition’ of bronze and triangle, and the fact that a thing is white will be a ‘composition’ of surface and whiteness. The reason is that people look for a unifying formula, and a difference, between potency and complete reality. But, as has been said, the proximate matter and the form are one and the same thing, the one potentially, and the other actually. Therefore it is like asking what in general is the cause of unity and of a thing’s being one; for each thing is a unity, and the potential and the actual are somehow one. Therefore there is no other cause here unless there is something which caused the movement from potency into actuality. And all things which have no matter are without qualification essentially unities.

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Isidore of Seville

Etymologiae1 (Extract)

BOOK II: OF RHETORIC AND DIALECTICS Chapter XXIX: On the division of the definitions, extracted from the book by Marius Victorinus2 1. The definition of the philosopher explains what a thing is, how it is and of how many parts it should be composed. It has the form of a short sentence which includes the specific meaning of the nature of a thing, separating it from what is characteristic of another. There are fifteen kinds of definitions. 2. The first kind of definition is that which in Greek is called usiodes, and in Latin substantialis, and it is what is truly and properly called a definition. As, for example, ‘Man is a rational, mortal animal capable of feeling and discipline’. Because this definition proceeds by species and differentia, it presents what is the essence and determines fully what a man is.

1. Isidore was Bishop of Seville and advisor to a number of the Spanish Gothic kings in the seventh century AD. His Etymologies is best characterised as an encyclopedia of classical and contemporary knowledge, divided into 20 books. 2. For this chapter, Isidore is said to have used two sources: Cassiodorus Cassiodori Senatoris Institutiones and a text by Marius Victorinus himself, namely De definitionibus liber.

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3. The second kind of definition is that which in Greek is called ennoematiké, and in Latin means notio; which we can call ‘notion’, with a general name and not a specific one. It is always formulated in this way: ‘A man is that which by reasoned understanding and its application is above all animals’. It does not say what thing man is; by stating what he does, it guides us by precise indications to the idea of what he is. In this kind and in the following definitions we are given a certain idea of the thing, not a substantial explanation; and because the first one is substantive it occupies the first place among all definitions. 4. The third kind of definition is that which in Greek is called poiotes and in Latin qualitativa, its name comes from ‘quality’ because it clearly indicates what and how the defined thing is, as, for example: ‘Man is one who stands out by his intellect, is distinguished in the arts, who by knowledge of things chooses how he should act, and condemns what his perception indicates to be futile. In this way it defines and expresses man by his qualities. 5. The fourth kind of definition is that which in Greek is called hypografiké, and which Tulio (Cicero) called descriptio; it defines the thing by a description, using enumeration of statements and facts. If the question is, for example, what a dissolute, what a cruel and what a gluttonous person is, the nature of dissolute, miserly and cruel persons is described; and accordingly, if we wanted to define a dissolute person, we would say: ‘A dissolute person is one who likes unnecessary food, but favours rich and expensive food, indulges in pleasures and is always ready for immoderate passion’. Here, a dissolute person is defined, but by means of description; this kind of definition is more suitable for orators than for dialecticians because it is expansive and applicable both to good and bad things. 6. The fifth kind of definition is that which in Greek is called kata antilexin and in Latin ad verbum. In this kind the meaning of a word is expressed by means of another, unique and different word, and in this way it states the meaning of one word by the use of another, as when one says: ‘conticescere’ is ‘tacere’ (to be silent); or when we say ‘terminus’ which is ‘limit’; or ‘devastated’ which is ‘destroyed’. 7. The sixth kind of definition is that which in Greek is called kata diaphorán and in Latin per differentiam. Professional writers call this definition of eodem et de altero (from one and the other) as when one asks what difference there is between a king and a tyrant, and by means of the difference one is defined as well as the other, saying: ‘The king in moderate and good-tempered the tyrant is godless and cruel.

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8. The seventh kind of definition is that which in Greek is called kata metaphorán and in Latin per translationem, as in Cicero’s Topics (Top.,32): ‘The beach is the place where the waves run out’. It can be used for various ends; one is for warning, like: ‘Nobility is the virtue of the elders and a burden for those who follow’; at other times it is for designating, like: ‘the head is the castle of the body’; at others for praising: ‘Adolescence is the flower of the ages’; at others for condemning, like: ‘Riches are the generous allowance for a short life’. 9. The eighth kind of definition is that which the Greeks call kata aphairesin tu enantiu, and the Romans por privantiam contrarii eius quod definitur (the denial of the opposite of what is being defined), as, for example: ‘Good is what is not bad’. ‘Just is what is not injust’ and the like. This type of definition should be used when the opposite is already known, as in: ‘If good is what is advantageous and honourably is good, what is not so is bad’. 10. The ninth kind of definition is that which the Greeks call kata hypotyposin and in Latin it is called per quandam imaginationem (by imagination), as when one says:‘Aeneas is the son of Anchises and Venus’. This definition always refers to individuals which the Greeks call átoma. 11. The tenth kind of definition is that which is called in Greek kata analogian and in Latin per analogiam or iuxta rationem (according to reasoning), as when one asks: ‘what is an animal’ and the answer is ‘like a man’. The example given identifies the thing looked for. It is proper for definitions to clarify the thing that has been queried. 12. The eleventh kind of definition is that which the Greeks call kata elleipés oleokleru homoiu genus, which in Latin is per indigentiam pleni ex eodem genere (incomplete within the same genus); as, if when someone asks ‘what is a third part’ and one were to answer:‘that which is lacking two thirds to make it a whole’. 13. The twelfth kind of definition is that which the Greeks call kata epainon, which means ‘per laudem’ (by praising), as in Tulio Cicero’s pro Cluentio (146): ‘The law is the mind, the counsel, the spirit and the judgment of the state’, or that other one (Cicero, Phil., 2, 113): ‘Peace is quiet liberty’. Definitions can also be made ‘by vituperation’, which the Greeks call psogon, as in (ibid.) ‘Slavery is the worst of all evils, which has to be resisted not only with war, but even with death’. 14. The thirteenth kind of definition is that which the Greeks call kata to pros ti, and the latins definitio segundum quid (by relation) such as: ‘A father is one who has a son’; ‘A master is he who has a slave’.

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15. The fourteenth kind of definition is that which the Greeks call kata ton oron, and in Latin per totum, as it is employed by Cicero in In Rhetoricis (Inv.,1, 42): ‘Genus is what covers many parts’. And this one: ‘The part is that which is included in the genus’. 16. The fifteenth kind of definition is that which the Greeks call kata aitiologian and in Latin it is secundum rei rationem (according to the reason of the thing) as in: ‘When it is day the sun is above the earth; At night the sun is below the earth’. We have to keep in mind that these kinds definition are rightly associated with the Topics, because they are among some classes of arguments and these are discussed in the Topics. Let us now speak of the topics which are the bases of arguments, the sources of opinions and the origin of sentences.

u

Translator’s note The translation is based on the Latin and Spanish versions cited in the bibliography. The punctuation is adapted to modern usage. Since it seems generally assumed that Isidoro did not write Greek, a composite Roman transliteration based on Luis Cortés y Góngora has been adopted for the Greek citations.

Blaise Pascal

The Spirit of Geometry THOUGHTS ABOUT GEOMETRY IN GENERAL First part, containing the spirit of geometry or the true method In the study of truth, we may pursue three main objectives: first, to discover it when we search for it; secondly, to prove it when we possess it; thirdly, to discern it from falsehood when we examine it. I am not speaking of the first, I shall discuss the second in detail and it encompasses the third. For, if we know the method for proving truth, we shall at the same time have the method for discerning it, because, while checking whether the proof we give of it conforms to the known rules, we shall discover whether it has been precisely proved. Geometry, which excels in these three objectives, explains the art of discovering unknown truths — a process it calls analysis — which need not be discussed here since there are so many excellent works on the subject. The only arts I shall present here is that of proving already discovered truths, and the art of clarifying them in such a way that their proof be irrefutable. To this end I only have to explain the method which geometry uses for this purpose: because it teaches it perfectly by its examples, even though it does not express it in words. And, because this art consists of two main things, one, to prove each proposition in particular, the other, to present all propositions in the best possible sequence, I shall divide my book into two sections, the first of which will contain the rules for the conduct of geometrical, i.e. methodical and perfect proofs, and the second will contain the rules for their geometrical, i.e.

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methodical and accomplished sequence; so that, both parts encompass everything necessary for carrying out the thought processes for proving and discerning the truths, which I intend to give completely.

u

THE SPIRIT OF GEOMETRY First Section: Of the Methods for Geometrical, i.e. methodical and perfect proofs Reflections on Geometry in General I can provide no better understanding of the procedure to be followed in the presentation of convincing proofs than to explain those of geometry. But first I must present the idea of an even more eminent and accomplished method which man will never be capable of: because what surpasses geometry, surpasses us; nevertheless, it is necessary to say a few words about it, even though it be impossible to apply it. I have chosen geometry only in order to achieve this objective, because it is the only science which knows the true rules of reasoning, and — without dwelling on the rules of syllogisms, which are so natural that they cannot be ignored — it examines and establishes the true method of reasoning in all things, which most people ignore, and which is so beneficial that we observe from experience that among equal minds and equal conditions, he who knows geometry is superior and derives from it a totally new strength. I therefore want to explain what a proof is on the example of geometry, for geometry is almost the only human science which produces infallible proofs because it alone follows the true method, in contrast to all the other sciences which, because of natural necessities, find themselves in some sort of confusion which only geometers know how to dispel in every detail. This true method, which would establish the most perfect proofs, if it were possible to attain it, would consist of two main elements: one, never to use a term whose meaning had not been previously clearly determined; the other, never to advance a proposition that had not previously been demonstrated as true by means of existing known truths; which means, in a word, to define all terms and

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to prove all propositions. But, in order to follow the order which I preconise, I must explain what I mean by ‘definition’. In geometry, only those definitions are recognised which logicians call nominal definitions, i.e. the simple allocation of names to things which have been clearly determined in perfectly known terms. I am referring only to these definitions. Their usefulness and application consists in clarifying and shortening discourse, by expressing, by the one given name, that which could only be expressed by several terms; in a manner, however, that the allocated name remain free of all other meanings, if it were to have any, in order to retain only the one it has been given. Here is an example: When it is necessary to differentiate among the numbers those which are divisible by two from those which are not, we give them the following names in order to avoid frequent repetition of these conditions: I name all number which are divisible by two without a remainder, ‘even’. This is a geometrical definition, because, after clearly having determined a thing, i.e. every number divisible by two without a remainder, it is given a name which one denudes of all other meanings, if it has any, in order to give it the name of the thing determined. From these observations it appears that definitions are quite arbitrary and that they can never incur contradiction, because there is no greater freedom than to give any name we choose to a thing we have clearly determined. We only have to be careful not to abuse this freedom of naming by giving the same name to two different things. This is, however, not prohibited, provided we do not confuse the resulting consequences nor extend from the name to the definition. But if one succumbs to this weakness, it is possible to apply a sure and infallible remedy. This consists in mentally substituting the definition for the definiendum, and having the definition clearly in mind so that every time we speak of, say, even numbers, we understand clearly that we mean a number divisible by two without a remainder, and that these two things are so united and inseparable in our thoughts that every time one is mentioned our mind immediately associates it with the other. For geometers and all those who proceed methodically only allocate names to things in order to shorten their discourse and not to diminish or change the ideas of the things they are discussing. For they demand that the mind should at all times substitute the full definition for the short terms they use in order to avoid the confusion that the use of a multitude of words can bring about. Nothing can more promptly and more forcefully dispel the captious surprises of the sophists than this method which one must have present at all times and which alone suffices to banish all manner of difficulties and equivocations.

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Once all this has been well understood, I return to explaining the true order which consists, as I have already stated, in defining everything and proving everything. While this method would be admirable, it is absolutely impossible: because it is evident that the first terms we would want to define, would presuppose precedents which would serve for their explanation and that in their turn the first propositions we would want to prove would presuppose others which had preceded them; in this way it is obvious that we would never arrive at the first ones. Equally, by deeper and deeper research we would necessarily arrive at primitive words which can no longer be defined, and principles that are so self-evident that we would find no others to prove them. Hence, it appears that man is faced with the natural and unalterable impossibility of dealing with any science in an absolutely accomplished order. But it does not follow that we should therefore abandon all order. For there is one, that of geometry, which is, indeed, inferior inasmuch as it is less convincing but not inasmuch as it is less certain. Geometry does not define everything nor prove everything and this is its limitation. But it only presupposes things which are clear and constant to natural insight and for this reason it is completely true, because it is supported by nature instead of by discourse. This order, the most perfect known to man, does not consist in defining everything nor or proving everything, nor in defining nothing and proving nothing, but in maintaining the middle position of not defining what is clear and understood by all people, and defining everything else; and not proving the things known by all people, but proving all others. Against this order transgress both all those who undertake to define everything and to prove everything and those who neglect to do this with the things which are not self-evident. This is what geometry teaches to perfection. It does not define ‘space’, ‘time’, ‘movement’, ‘number’, ‘equality’ nor similar things of which there are many, because these terms designate the things they represent so perfectly for us that any explanation we might want to give would add more obscurity than instruction. For there is nothing weaker than the discourse of those who want to define these basic words. What need is there to explain what is meant by the word ‘man’? Does one not know sufficiently well what one wants to designate with this term; and what benefit did Plato intend to provide by saying that man was a ‘featherless biped’?1 As if the idea of ‘man’ which I naturally have and which I do not know how to express were not clearer and more precise than the one he gives by his useless and even ridiculous explanation. For a man does not lose his

1. This definition is frequently attributed to Plato, but there seems to be no textual evidence for it.

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human nature if he loses his two legs and a capon does not become a man if it loses its feathers. There are those who go to the extreme absurdity of explaining a word by the self-same word. I know of those who have defined light thus: la lumière est un mouvement luminaire des corps lumineux (light is a luminary movement of luminous bodies), as if were possible to understand the words luminaire and lumineux without the word lumière. We cannot undertake to define ‘being’ without falling into this absurdity. Because we cannot define a word without the word ‘is’ whether it be expressed or understood. Consequently, for defining ‘being’ one has to say ‘is’ and thus employ the word to be defined in the definition. From this we see clearly that there are words that cannot be defined. And if nature had not provided a remedy for this shortcoming by the same idea which it has given to all men, all our expressions would be confused; instead of which we can use them with the same assurance and the same certainty as if they had been explained in a manner free of all misunderstanding. For without using words, nature has given us a clearer understanding than art can supply through our explanations. It is not because all men have the same idea of the essence of things that I affirm that it is impossible and pointless to define. ‘Time’, for example, is of this type. Who can define it? And why do it, since all men understand what we mean when we speak about time without previously having specified it? Nevertheless there are many different opinions about the essence of time. Some say it is the movement of a created thing; others say it is the measure of movement, etc. I do not maintain that the nature of these indefinable things is common to all people: it is only the relation between the name and the thing, in such a manner that when we hear the expression ‘time’ we turn our thoughts to the same thing; which suffices to cause that this term has no need for being defined, even though afterwards, when we establish what time is, opinions may differ after having given the matter some thought. Because definitions are made only for denoting the things we have named and not for explaining their nature. It is certainly not prohibited to give the name of ‘time’ to the movement of a created thing, for, as I have just said, nothing is freer than definitions. But, according to this definition, there would be two things that would be called by the name of ‘time’: one, what everybody understands naturally by this word and which everybody who speaks our language denotes by the term; the other, the movement of a created thing, because it would also have this name according to the new definition. It is therefore necessary to avoid ambiguity and not to

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confuse consequences. For it does not follow that the thing we normally associate quite naturally with the word ‘time’ is the movement of a created thing. We were free to give these two things the same name, but we are not free to make them agree in their meanings as well. When we produce the sentence ‘time is the movement of a created thing’, we have, therefore, to ask what the word ‘time’ means, i.e. whether we leave it with its common and generally agreed meaning or whether we strip it of its meaning in order to give it, on this occasion, the meaning of ‘the movement of a created thing’. If we deprive it of every other meaning, there is no contradiction. There will then be a free definition resulting from which, as I have stated above, there will be two things with the same name. But if its normal meaning is preserved, and we nevertheless propose that what we understand by this word is ‘the movement of a created thing’, there is a contradiction. It is then no longer a free definition, but a proposition that has to be proved, unless it is self-evident. In this case it will be a principle and an axiom, but never a definition because with this utterance we do not want to state that the word ‘time’ means the same as ‘the movement of a created thing’, but that what one understands by the term ‘time’ is the aforementioned movement. If I did not know how necessary it is to understand this fully, and how often situations like the ones I have exemplified occur in every day speech and in scientific discourse, I would not have paused here. But my experience of confused discussions tells me that we can never persevere enough with this precision to which I devote this treatise rather than the subject matter of its content. For how many people are there who believe having defined ‘time’ by saying that it is the measure of movement, while still leaving it its common meaning! Nevertheless, they have only formulated a proposition and not a definition. Equally, how many people are there who believe having defined ‘movement’ by saying: motus nec simpliciter actus nec mera potentia est, sed actus entis in potentia2 (Movement is not simply an act nor a mere potentiality, but the realisation of a potentiality). Still, if they leave the word ‘movement’ with its common meaning, as they do, this is not a definition but a proposition. By confusing in this way the definitions which they call nominal, which are the genuine, free, admissible and geometrical definitions, with those which they call definitions of things which really are propositions which are not free but subject to contradiction, they take the liberty of also formulating these as well as others; and by the same things each in his own way — with a freedom as little admissible in this type of definition as it is permitted in the former — they muddle up

2. Aristotle, Phys. III 2, 201.

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everything. And, losing all order and insight, they lose themselves and get caught up in inexplicable difficulties. One will never fall into this type of error if one follows the order of geometry. This judicious science stays clear of defining these basic words ‘space’, ‘time’, ‘movement’, ‘equality’, ‘majority’, ‘diminution’, ‘all’ and others which people understand by themselves. But, beyond these, the other terms used in geometry are so clearly explained and defined that one does not need a dictionary for understanding any of them. In a word, all these terms are fully intelligible either by natural intuition or by their given definitions. This is the manner by which geometry avoids all the pitfalls one may encounter in connection with the first point which consists in defining only those thing which need it. Geometry proceeds in the same way regarding the second point which consists in proving the propositions which are not self-evident. For, when it has arrived at the first established truths, it stops and demands that they be acknowledged because there is nothing clearer to prove them. In this way, everything which geometry presents is explained either through natural intuition or through proofs. From this follows that, if this science does not define and prove everything, it is for the sole reason that this is impossible for us. (But since nature provides everything which this science does not, its order is not superhumanly perfect but as perfect as the human mind can make it. It seems appropriate to state this right from the beginning of this treatise …) It may perhaps be considered strange that geometry should not be able to define any of the things which are its principal objects. For it can neither define ‘movement’ nor ‘number’ nor ‘space, though it studies these three things in particular. And, according to their examination, it adopts the three different names of its subdivision mechanics, arithmetic and geometry. This last name is applicable both to the genus and the species. This should not be a cause of surprise, when it is remarked that this admirable science is only concerned with the simplest of things and that the very property which makes them worthy of study also renders them incapable of being defined. So, the absence of definitions is a virtue rather than a fault because it does not arise from their obscurity but on the contrary from their extreme evidence which is of the sort that — though it does not have the force of proofs — it has their absolute certainty. It presupposes, therefore, that we know what is meant by the words ‘movement’, ‘number’, ‘space’, and, without wasting time in definitions, it penetrates their nature and discovers their marvellous properties. These three things, which, according to the phrase Deus fecit omnia in

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pondere, in numero et mensura (God made everything according to weight, number and measure),3 comprise the entire universe, have a necessary and reciprocal relationship. For we cannot imagine movement without something that moves; and this something being ‘one’, this unit is the origin of all numbers. And, finally, since movement cannot occur without space, it is clear that these three things are comprised in the first. Even time is included, because ‘movement’ and ‘time’ are relative to each other, speed or slowness which are the differences of movement having a necessary relation to time. Thus there are properties which are common to all things, and their understanding opens our minds to the greatest miracles of nature. The most important of these are the two infinities which are found in all of them: largeness and smallness. For however fast a movement may be, we can always conceive of one which is faster, and accelerate the latter and so on to infinity without ever getting to one to which it would not be possible to add. And vice versa, however slow a movement may be we can slow it down further and also this one, and so to infinity without ever getting to a degree of slowness to which one could not add an infinity of others without coming to rest. Equally, however large a number, one can always imagine a larger one, and yet another and one which exceeds the latter and so on to infinity without ever getting to one which we could not increase. And vice verse, however small a number, like the hundredth or the ten thousandth part, one can imagine yet a smaller one and so on to infinity without getting to zero or nothing. Equally, however large a space may be, one can imagine a larger one and yet another larger one, and so on to infinity without ever getting to one which could not be enlarged. And vice versa, however small a space may be, one can always imagine a smaller one, and so forth to infinity, without ever arriving at an indivisible one which no longer has any extension. It is the same with ‘time’. One can always think of a larger timespan without an ultimate one, or a smaller timespan without arriving at a moment and to a pure nothing of duration. This means in a word, whatever movement, whatever number, whatever space, whatever time, there will always be a larger and a smaller one: such that they are held between infinity and nothing, being always infinitely distant from these extremes. All these truths cannot be proved, yet they are the foundation and the

3. Old Testament: Sap. XI, 2.

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principles of geometry. But since the reason why they are incapable of proof, does not lie in their obscurity, but, on the contrary, in their extreme evidence, this lack of proof is not a fault, but rather a perfection. Thus, one will note that geometry can neither define its objects nor prove its principles; but for this singular and advantageous reason, that both are of an extreme natural clarity, which convinces reason more forcefully than discourse. For what is there more evident than this truth, that a number, whatever it may be, can be enlarged? Can it not be doubled? And that the speed of a movement can be doubled, and that a space can equally be doubled? And who can doubt that a number, whatever it may be, cannot be halved, and this half halved yet again? For should this half be a nothing? And how should these two halves, which would be two zeros, result in a number? Equally, a movement, however slow it may be, can it not be slowed down by half, such that it would cover the same space in twice the time, and yet again? For would this be absolute rest? And how would it be possible that these two halves of speed, which would be two rests, result in the original speed? Likewise, a space, however small it may be, can it not be divided in two, and these halves yet again? And how would it be possible that these halves should be indivisible without any extension, which combined had yielded the original extension? There is no natural human knowledge which would precede this knowledge and surpass it in clarity. Nevertheless, so that there may be an example for everything, there are people, distinguished in every other respect, who are disturbed by these infinities, and who find no way of agreeing to them. I have never encountered anyone who thought that a space could not be enlarged. But I have met some, very able otherwise, who maintain that a space could be divided into two indivisible parts, whatever absurdity may result. I have tried to investigate what might be the origin of their confusion, and I have found one main reason, namely that they cannot conceive of something infinitely divisible: from which they conclude that it cannot be thus divisible. It is a natural human weakness for people to think that they possess truth directly. And for this reason they are always ready to deny everything they cannot understand, whereas in fact they naturally know only untruths, but they should only accept as truth those things the opposite of which seems false. And for this reason, we must suspend judgment every time a proposition seems incomprehensible, and not deny it for this reason; but we should examine its opposite. And if we find it manifestly false, we can boldly affirm the original proposition, however incomprehensible it might appear to be. Let us apply this rule to our topic.

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There is no geometrician who does not accept that space is infinitely divisible. One can no more be a geometrician without accepting this principle, than a human being can exist without a soul. And nevertheless, there is nobody who can understand infinite divisibility; we can only be assured of this inconceivable truth by the following single reasoning, which is, however, sufficient: we understand clearly that it is false that by dividing a space one might arrive at an indivisible, i.e. at a space without any extension. For what is there more absurd than to maintain that by constantly dividing a space we arrive finally at a division, like the division in two, in which each half would be indivisible without any extension, so that, in this way, these two nothings in extension together would make up an extension? For I should like to ask those who have this idea whether they can clearly conceive of two indivisibles touching each other: if this happens everywhere they are only one thing, and consequently both together are indivisible; if this does not happen everywhere, it only happens in one part: consequently they have parts, hence they are not indivisible. Let them admit, as under pressure they do, that their proposition is as untenable as the other, let them accept that we must not judge the truth of things according to our capacity for conceiving them: for nevertheless, as both these opposites are inconceivable, it is necessarily certain that one of them is true. But let them oppose these chimeric difficulties, which are only proportional to our weakness, with these natural insights and these solid truths: if it were true that space be composed of a specific finite number of indivisibles, it would follow that in two spaces, each of which would be square, i.e. equilateral, one being twice the size of the other, one would contain twice the number of indivisibles compared to the number of indivisibles in the other. Let them remember well the following conclusion and afterwards practice ordering points into squares until they find two squares of which one contains twice as many points as the other, then I shall make all geometricians in the world defer to them. But if this enterprise proves naturally impossible, i.e. when there is an unsurmountable obstacle to order to make squares out of points, of which one has twice as many as the other, as I might prove here myself, if the matter merited the time it would take, then let them draw the conclusion. And in order to console them for the trouble which certain cases would cause them, such as, conceiving that a space should contain an infinity of divisibles, since one traverses them in such a short space of time, during which one would have crossed this infinity of divisibles, they should be told that they must not compare such unequal things as the infinity of divisibles of space with the brevity of time to traverse them; let them rather compare the whole space with the whole of time, and the infinity of divisibles of space with the infinity of

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moments of time. In this way they would discover that one crosses an infinity of divisibles of space in an infinity of moments of time and a little space in a little time; then they would no longer find the disproportion which had astonished them. In short, if they should find it strange that a small space should have as many parts as a large one, let them also understand that these are correspondingly smaller ones. And let them look at the firmament through a small glass in order to become familiar with this cognition, by seeing each part of the sky in each part of the glass. But if they cannot grasp that parts so little that they are imperceptible to us, can still be divided like the sky, there is no better cure than to let them look at the firmament through glasses which magnify this little point to a large mass; then they will easily understand that, with the aid of a still more finely cut glass, it is possible to enlarge these points to such an extent that they are equal to the firmament the extension of which they admire. And when now these objects seem to be easily divisible, let them remember that nature is infinitely more powerful than art. For who assures them that these glasses have changed the natural size of these objects? Or, that they have not, on the contrary, reestablished their natural size which the shape of our eyes had changed and reduced, like spectacles which diminish? It is annoying to waste time over such trivialities, but there is a time for foolish talk. It suffices to tell clear minds in these matters that two nothings of extension cannot result in an extension. But since there are those who pretend to escape from this insight by the fantastic reply that two nothings of extension can produce an extension just as two units neither of which is a number, can be combined into a number, they have to be told in reply that they might just as well say that twenty thousand men make an army, though none of them individually is an army, or that a thousand houses form a town, even though none of them is a town, or that the parts make a whole, even though no part is the whole, or, to stay with the comparison of numbers, that ten tens make a hundred, and that twelve ones make a dozen, even though none of them is. But no clear mind will confuse the immutable nature of things by such uneven comparisons with their free and arbitrary names which are dependent on the whim of the men who have formed these names. For it is obvious that in order to facilitate discourse, the name of army has been given to twenty thousand men, that of town to a group of houses, that of a dozen to twelve units. And that this freedom is the cause of the names of the units, ‘dozen’, ‘ten’, ‘hundred’, all different according to our imagination, even though, because of their invariable nature these things are in fact of the same kind, and are all proportional to each other and only vary by more or less, and even though because of these names ‘ten’ is not ‘a hundred’,

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nor a ‘house’ a ‘town’, as little as a ‘town’ is a ‘house’. But then also, even though a house is not a town, it is nevertheless not a nothing of town. There is a difference between not being something and being nothing. For, in order to understand this matter properly, it is necessary to know that the only reason why the unit ‘one’ is not in the list of numbers is that Euclid and the first authors writing about arithmetic excluded ‘one’ from the list of numbers when they had to assign several properties which belong to all numbers except the ‘one’; they did this in order to avoid having to say all the time that this or that property belonged to all numbers except the ‘one’. This was possible, as we have already stated, because of the freedom we have to form definitions according to our free will. If they had wished to do so, they could also have excluded the ‘two’ or the ‘four’ or anything else they fancied, because this was their freedom provided they had this point clear; just as, on the contrary, we could now include the ‘one’ among the list of numbers, and even the fractions, if we so wished. And, we are, in fact, obliged to do this with general propositions in order to avoid having to say every time: “Every number and ‘one’ and the fractions have this or that property”; and I have used ‘number’ in this indefinite sense in everything I have written about the subject. But even Euclid, who excluded ‘one’ from the numbers, has used his freedom to define by clarifying that ‘one’ is not nothing, but on the contrary, is of the same nature; so he defined homogenous magnitudes by stating: the magnitudes are of the same kind when the one, repeatedly multiplied finally can exceed the other. And, since ‘one’ repeatedly multiplied can exceed any number, it is of the same kind as the numbers precisely because of its essence and its immutable nature, in the sense of Euclid himself, who wanted that it should not be called a number. The same applies to an indivisible with regard to an extension. For it not only differs in name, which is arbitrary, but it differs in kind because, no matter how many times one multiplies an indivisible unit, it is still far away from surpassing an extension because it can never form more than one and only one indivisible unit; this is natural and necessary, as we have already demonstrated. And since this last proof is founded on the definition of these two things ‘indivisible’ and ‘extension’, we shall complete and finish the proof. An indivisible is something which has no parts, and an extension is something which has several separate parts. On the basis of these definitions I state that two indivisibles combined do not make an extension. For, if they are combined each one touches the other in one part; consequently, the parts in which they touch are not separated, for otherwise they

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would not touch. But, according to the definition, they have no other parts, hence they do not have separate parts, hence they are not an extension according to the definition of extension which includes the separation of parts. For the same reason it is possible to prove the same for all indivisibles that are joined on. And consequently an indivisible, no matter how many times one multiplies it, will never become an extension. Hence, it is not of the same kind as an extension, by the definition of things of the same kind. This is the way to prove that indivisibles are not of the same kind as numbers. This is also the reason why two ones can make a number, because they are of the same kind, and why two indivisibles do not make an extension because they are not of the same kind. This shows how little cause there is for comparing the relation between ‘one’ and the numbers with that of indivisibles and extension. But if we want to take a comparison from numbers which properly represents what we consider in extension, we have to resort to the relation of zero to the numbers. For the zero is not of the same kind as the numbers, because multiplied it cannot exceed them: thus it is a true indivisible of numbers, just like the indivisible is a true zero of extension. And one will find the same relation between rest and movement, and between the moment and time. For all these things are heterogeneous in their magnitude, because infinitely multiplied, they can only yield indivisibles, and for the same reason. And then one will find a perfect correspondence between these things. For all these magnitudes are infinitely divisible without becoming indivisibles, so that they all stand in the middle between the infinite and nothing. This is the admirable relation which nature has established among things and these are the two marvellous infinities it has presented to mankind, not for their understanding but for their admiration. And, in order to conclude this reflection with a last observation, I shall add that these two infinities, though infinitely different, are yet related to each other so that knowledge of the one inevitable leads to knowledge of the other. For in the case of numbers, from the fact that they can always be increased, it follows absolutely that they can always be reduced. This is quite clear, for if one can multiply a number by a hundred thousand, for example, one can also take a hundred thousandth part by dividing it by the same number that one multiplies it with; so every expression of increase becomes an expression of division, when the whole number is changed into a fraction. Hence, infinite multiplication necessarily implies infinite division. And in space the same relation is visible between these two opposite infinities; in the sense that from the fact that a space can be infinitely extended

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it follows that it can be infinitely reduced, as in the following example: if one looks at a ship through a glass which moves away in the same steady direction, it is obvious that the place of the lens through which one sees any particular point of the ship will constantly rise in the measure as the ship moves further away. Hence, if the trajectory of the ship is constantly extended to infinity, this point will rise continuously; yet it will never reach the point where the horizontal ray falls which leads from the eye to the lens. So it will constantly approximate it without ever reaching it, constantly dividing the space which remains below this horizontal point. This leads to the necessary conclusion which follows from the infinity of the extension of the ship’s course to the infinite and infinitely small division of the little space remaining below this horizontal point. Those who are not convinced by these reasons and who hold on to the belief that space is not infinitely divisible, cannot lay claim to geometrical proofs. And though they may be very enlightened in other matters, they are no so in this respect: for it is possible to be very able while being ignorant of geometry. But those who clearly recognise these truths, can admire the grandeur and power of nature in this double infinity which surrounds us everywhere, and by reflection on these marvels they may get to know themselves, by seeing themselves located between an infinity and a nothing of extension, between an infinity and a nothing of number, between an infinity and a nothing of movement, between an infinity and a nothing of time. From which one can learn to estimate one’s true value, and to reflect on things which are worth more than all the rest of geometry. I have felt obliged to undertake this long reflection for the benefit of those who do not immediately grasp this double infinity, but who are capable of becoming convinced of it. And, though some people have enough insight not to need it, it may nevertheless happen that this treatise which is necessary for some, may not be totally useless for others.

The Art of Persuasion The art of persuasion is necessarily related to the way in which we agree to what is presented to us and to the nature of the things which we are expected to believe. Everybody knows that there are two entrances through which opinions are accepted by the soul — which are its two main forces — reason and will. The most natural one is reason because one should only ever agree to proven truths; but the most common one, even though contrary to nature, is will. For all people are almost always inclined to believe not by proof but by preference. This is a

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base, unworthy and alien way. Also everybody repudiates it. Everyone professes to believe and even to love only what he knows to be meritorious. I am not speaking here of divine truths which I would take great care not to be subsumed under the art of persuasion because they stand infinitely above nature. Only God can place them in our souls and in the manner He pleases. I know that He wanted them to reach the mind through the heart, and not the heart through the mind, in order to humiliate this haughty power of reason which claims to be the judge of things chosen by the will; and to heal the weak will which is completely corrupted by its vile connections. This is the origin of the saying which has become a proverb, that, speaking of human concerns, ‘it is necessary to know things before loving them’; the saints, by contrast, when speaking of divine concerns, exhort us to love them in order to know them and that we can attain truth only through charity. This has become one of their most useful sayings. This explains why God has established this supernatural order quite contrary to the order which was to be natural for people in connection with natural things; however, people have corrupted it by doing with profane things what they should be doing with holy things, because, in fact, we believe almost nothing except that which pleases us. And this is the cause for our reluctance to accept the truths of the Christian religion which are entirely contrary to our pleasures. Tell us pleasant things and we shall listen to you, the Jews said to Moses, as if what is pleasant should direct our beliefs. And God, to punish this disorder by imposing an order of His own making, pours his light into our minds only after he has tamed the rebellious will by a very celestial sweetness which charms and conquers. I am therefore only speaking of truths within our reach; and of these I say that the mind and the heart are like two doors by which they enter into the soul, but very few enter through the mind, whereas they are introduced in large numbers by the will’s daring whims without the counsel of reason. Each of these forces have their principles and the initial stimuli for their actions. Those of the mind are natural and generally known truths such as ‘the whole is bigger than a part’, beside several specific axioms which some accept and others do not, but which, once they are accepted, are as powerful, though false, to engage beliefs more strongly than the most certain axioms. The forces of the will are certain common and generally known desires, such as the desire to be happy, which nobody can be without, beside several specific aims which everyone pursues in order to attain them and which, having the power to please us, though harmful in their effect, are as strong in driving the will to action as if they represented its true happiness. So much about the forces which lead us to agreement.

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As for the properties of the things of which we must persuade others, they are very diverse. Some necessarily follow from common principles and accepted truths. These may easily persuade, for, revealing their relation to agreed principles they inevitably lead to persuasion. It is impossible that they should not be accepted by the soul as soon as they are enlisted in the truths it has already accepted. There are some which are closely related to the objects of our desire, and these, too, are firmly accepted, for as soon as one shows the soul that something can lead it to what it loves above all else, it will inevitably yield to it with joy. Those properties, however, which combine these relationships, both with the acknowledged truths and the wishes of the heart, are so sure of their effect that there is nothing in nature to equal them. Just as, to the contrary, that which has no connection with our beliefs and our pleasures is disagreeable to us and completely strange. In any of these encounters there is no doubt; but doubts exist where the things we want others to believe are well-founded on known truths but, at the same time, contrary to our most cherished pleasures: in these cases, by an experience which is only too common, as I have already stated, there is great danger of showing that this imperious soul, which claimed only to be guided by reason, follows the wishes of a corrupt will by a shameful and foolhardy choice, regardless of the resistance which an enlightened mind might offer. On such occasions a dubious toing and froing emerges between truth and lust and the knowledge of the one and the feeling of the other wage a war with a very uncertain outcome, for in order to judge it, we would have to know the innermost secrets of man which he himself hardly ever knows. From the foregoing we learn that whatever we want to convince others of, we must consider the persons we want to persuade and know their minds and hearts, what principles they accept, what things they love; and observe what relation there might be between the attractiveness attributed to the matter in hand and the accepted principles or preferred objects. Consequently the art of persuading consists as much in pleasing as in convincing because men are guided more by whims than by reason. Now, of these two methods, the first to convince and the second to please, I shall here give only the rules for the first, and only on the basis that these principles have been acknowledged and that they are faithfully being adhered to. Otherwise I would doubt whether there is an art consisting of adapting proofs to the shiftiness of our whims. But the art of pleasing is incomparably more difficult, more subtle, more useful and admirable. Also, I am not dealing with this subject because I am

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incapable of doing so. I feel so inadequate to this task that it strikes me as the most impossible thing to do. Not that I believe that the rules for pleasing are any less certain than those for proving and that people who master and practice them will not as fully succeed in being appreciated by kings and all sorts of other men, as by demonstrating the elements of geometry to those who have sufficient imagination to understand the hypotheses. But I think — and it may be my inadequacy which makes me believe this — that it is impossible to succeed. I know at least that if anyone is capable of doing so, it will be someone of my acquaintance and that no one else has so clear and rich insights. The cause for this extreme difficulty lies in the fact that the principles of pleasure are neither firm nor stable. They are different in all people and in each one so widely variable that at various times no man is more different from another than from himself. Men differ in their pleasures from women; so do rich and poor; a prince, a warrior, a merchant, a gentlemen, a peasant, the old, the young, the healthy, the infirm, all vary; the least occurrence changes them. There is, however, an art, the one I shall describe here, for showing the connection of truths with their principles, be they of certainty or of pleasure, provided that the principles which one has accepted at one time remain firm and will never be denied. But since there are few principle of this kind, and since beside geometry, which only considers very simple lines, there is hardly any truth about which we are always agreed, and even fewer objects of pleasure which we do not change hourly, I do not know whether it is possible to present definite rules for adapting discourse to the inconstancy of our whims. This art, which I call the art of persuasion, and which in reality is only the conduct of methodologically perfect proofs, consists of three essential parts: to define the terms one uses by clear definitions; to propose evident principles or axioms for proving the matter in hand; and, in the demonstrations, always to substitute the definitions for the defined concept. The reason for this method is evident, since it would be useless to present something that one wants to prove and then to proceed to the proof, unless one has previously clearly defined all the unintelligible terms; and equally, the proof must also be preceded by a demand for self-evident principles which are necessary for the proof, for if one does not shore up the foundations one cannot hold up the building; and finally, during the proof one must mentally substitute the definitions for the definienda, since otherwise one might misuse the different meanings encountered in the terms. It is clear that by following this method one is sure of convincing, since the invincible force of the conclusions cannot fail to

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have its effect if the terms are fully understood and through their definitions perfectly free from misunderstanding, if the principles are agreed and if in the proofs one always mentally substitutes the definition for the thing defined. For this reason, a proof which respects these conditions has never ever been questioned and proofs which do not respect them can never convince. It matters therefore that one should understand and follow these rules, and for this reason, to make matters simpler and more straightforward, I shall give them all in the following few rules which encompass everything necessary for the completeness of the definitions, the axioms and proofs, and consequently the whole method of geometrical proofs of the art of persuasion. Rules for definitions 1. Do not attempt to define the things which are so well-known by themselves that we do not have clearer terms for explaining them. 2. Do not admit any terms which are somewhat obscure or ambiguous without defining them. 3. In definitions of terms, use only well-known words or already explained words. Rules for axioms 1. Do not accept any necessary principle without first having enquired whether it is generally acknowledged, however clear and evident it might appear to be. 2. Do not admit in axioms things which are not perfectly self-evident. Rules for proofs 1. Do not try to prove things which are so self-evident that we do not have anything clearer for proving them. 2. Prove all somewhat obscure propositions and use only very evident axioms or already proved propositions. 3. Always mentally substitute the definition for the definiendum, in order not be deceived by the ambiguity of terms limited by definitions. These eight rules enclose all the precepts for solid and immutable proofs. Three of these are not absolutely necessary and may be neglected without incurring in error; it is even difficult to follow them precisely, though it is better if one does adhere to them as much as possible. They are the three first of each group. For definitions Do not define perfectly well-known terms. For axioms Permit that only simple and self-evident axioms are demanded. For proofs Do not prove anything which is already well-known.

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Undoubtedly, no great harm can be done by defining and clearly explaining things which are already clear, nor in demanding at the outset axioms which cannot be rejected where they are needed, nor, finally, proving propositions which one would accept without proof. But the other five rules are absolutely necessary and one cannot dispense with them without incurring basic mistakes and often serious error. For this reason I repeat them here: Necessary rules for definitions 2. Do not admit any terms which are somewhat obscure or ambiguous without defining them. 3. In definitions of terms, use only well-known words or already explained words. Necessary rules for axioms 2. Do not admit in axioms things which are not perfectly self-evident. Necessary rules for proofs 2. Prove all somewhat obscure propositions and use only very evident axioms or already proved propositions. 3. Always mentally substitute the definition for the definiendum, in order not be deceived by the ambiguity of terms limited by definitions. These five rules constitute everything necessary for making proofs convincing, immutable and, which is all inclusive, geometrical. The eight rules altogether make them even more perfect. I am now proceeding to the order in which propositions have to be presented in order to arrange them in a perfect and geometrical sequence. The art of persuasion is contained in these two rules: Define all names which are used; prove everything by mentally substituting the definition for the definiendum. Here it seems indicated to present three possible objections. One, that this method is nothing new; the second, that it is easy to acquire them without having first to learn the elements of geometry since they consist of the two sentences given above which one knows after a first reading; and finally that it is quite useless because its application is almost entirely limited to geometry. In reply one must argue that there is nothing so little known, nothing more difficult to apply, nor more useful and universal. Concerning the first objection, which says, that these rules of defining and proving everything are quite common and that the logicians themselves have included them in the principles of their art, I wish that this were true, that they

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were indeed so widely known that I would not have to go to the trouble of searching so carefully for the causes of the many faults of reasoning which are so very common. But this is so little the case that, with the exception of the geometricians of which there are so few that they are rare in a whole population and in time, there is nobody else who also knows them. It will be easy to transmit them to those who have perfectly understood what little I have so far said about them. If they have not fully understood me, I admit that they will have nothing to learn from them. But if they have entered into the spirit of these rules and these rules have impressed themselves upon them to such an extent that they have taken root and become firmly established, they will notice what a big difference there is between what has been said here and that which, in a similar vein, some logician have perhaps casually written about it in one of their works. Those who have the power of discernment know what difference there can be between two similar words, according to the place where they are used and the circumstances which surround them. Does one really believe that two people who have learnt the same book by heart know it equally well, whether the one does not understand to the extent of knowing all the principles, the force of conclusions, the answers to possible objections and the whole structure of the work, whereas another may understand only dead words and seeds which, though equal to those which have produced such fertile trees, have remained dry and fruitless in the sterile mind which has received them in vain? Those who say the same things do not possess them in equal manner. For this reason the incomparable author of The Art of Conversation4 devotes so much care to making us accept that one should not judge the capabilities of a man by the excellence of a felicitous phrase he has uttered: but instead of extending one’s admiration of a good speech to the speaker, he says, one should discover the mind it originates from; one should try to ascertain if it comes from his memory or a happy coincidence; one should receive him coldly and with disdain in order to test whether he resents that one does not give him the credit he believes to deserve. In most cases one will see that he quickly denies it and that one can distract him quite far from this idea which is better than he believes, and engage him in a quite low and ridiculous idea. We must, therefore discover how this thought is lodged in the author; how, whence and to what extent it is his own. Otherwise a hasty judgment will be considered foolhardy. I should like to ask judicious people whether the two following principle are indeed the same in both Descartes’s and St. Augustine’s mind, who some 1200

4. Michel de Montaigne, ‘De l’art de conférer’ in: Essais III, 8.

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years earlier said the same thing: Matter is by its nature incapable of thought and I think, therefore I am. Truly, I am far from maintaining that Descartes is not the real author, even though he might have acquired this thought during his reading of the works of the great saint. For I know the difference between writing a phrase by chance, without a longer and extended reflection, and discovering in this expression an admirable sequence of conclusions which prove the difference between matter and mind and to transform them into a firm principle backed up by a wholly new physics, as Descartes claimed. Without examining whether he has fully succeeded in his claim, I assume he has; and on the basis of this assumption I say that this proposition found in his writings differs from the same expression written incidentally by others, as much as a dead body differs from that of a man full of life and vigour. A person may say something impromptu, without being aware of its excellence, in which someone else perceives a wonderful sequence of conclusions, so that we boldly say that it is no longer the same expression and that he owes it as little to the person from whom he has learnt it than a beautiful tree belongs to the man who, unwittingly, dropped the seed on rich soil which knew how to benefit from its fertility. Also, the same thoughts sometimes grow in someone else quite differently than they do in their originator: infertile in their natural soil, abundant after having been transplanted. But, quite frequently it also happens that an enlightened mind reaps from his own thoughts all the fruit they are capable of, and that, subsequently, others, who have heard them being praised, borrow them and adorn themselves with them without knowing of their excellence: it is on these occasions that the differences of an expression in other mouths become most obvious. In this manner logic may have borrowed the rules of geometry without understanding their power. And, having let them loose among its own rules, it does not follow that they have penetrated into the spirit of geometry. Unless they produce other evidence than having said them in passing, I shall be far from putting them on the same level as the science which teaches the true method of reasoning. But, on the contrary, I shall be strongly inclined to exclude them from it and almost irrevocably. For, having said something in passing, without attention to anything it contains, and instead of following these insights, to lose oneself in futile searches, running after what they neither offer nor give, this means truly to show that one is not clairvoyant, more so for not being aware of them than for having missed to follow them.

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Everybody is in search of the method for avoiding error. The logicians assure us that they know the way. Only the geometricians get there, and outside their science and those sciences which imitate it, there are no veritable proofs, and all the art is contained in the precepts we have enunciated. They alone suffice. They alone prove. All other rules are useless or harmful. I know this from long experience of all sorts of books and people. And for this reason I pronounce the same judgment both on those who say that the geometricians offer them nothing new with these rules because they had them already, though mixed up among a multitude of other useless of false ones among which they could not discern them, and on those who search for a priceless diamond among a pile of false ones which they cannot tell apart and who, by having them altogether, pride themselves on having the good one; and on that other who, without tarrying over the worthless pile, picks out the good stone which was looked for and because of which the rest of them had not been thrown away. The affliction of false reasoning is a disease which can be cured by these two remedies. Another remedy has been cooked up from a mass of useless herbs among which the good ones are hidden and where they are ineffectual because of the poor quality of the whole mixture. In order to discover all sophisms and all the errors of captious reasoning, they have invented barbaric names which astound those who hear them. And whereas the only way of unravelling all the twists of this complicated knot is to pull from the end indicated by the geometricians, they have marked an odd number of others, among which the former is included, but they do not know which is the right one. And so, when we are shown a number of different ways which they assure us will lead to where we want to go, even though there are only two which take us there, one has to know how to mark the right ones. It may be objected that geometry, which indicates them precisely, only offers what was already known from others because they did the same thing and more, without paying attention to the fact that this gift lost its value because of its abundance and that, while adding, they took away. Nothing is more common than good things: it is only a matter of discerning them. And it is true that they are all natural and within our reach and even generally known. But one does not know how to differentiate them. That is universal. Excellence of whatever kind is not found in exceptional and peculiar things. One rises to get closer but instead gets farther away. In most case one must humble oneself. The best books are those which readers think they could have written themselves. Nature, which alone is good, is completely familiar and common.

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I therefore leave no doubt that these rules by being the true ones, must be as simple, naive and natural as they are. Neither Barbara nor Baralipton5 shape judgments. One must not put a straitjacket on the spirit. Tortuous and painful manners fill it with foolish pretensions by an idle presumption and ridiculous inflation instead of healthy and nourishing food. One of the principal reasons which lead those astray who acquire the knowledge of the true path to follow is that they have qualified the good things as ‘great’, ‘high’, ‘elevated’ and ‘sublime’ and hence have convinced themselves that they are beyond their reach. This spoils everything. I would call them low, common, familiar. These names fit better. I hate inflated words.

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Translator’s note This translation is based on the fragmentary text of the original unique manuscript copy of the essays, as presented in the critical edition prepared by Jean-Pierre Schobinger. Three insignificant sentence fragments have been omitted because they do not add to the text. Some additional punctuation has been introduced for greater ease of reading.

5. Barbara Baralipton are mnemotechnical expressions for the first and the fifth modus of the first syllogistic figure. The use of these expressions indicates Pascal’s rejection of syllogisms.

Benedict de Spinoza

Of the Improvement of the Understanding (Extract)

Conditions of definitions … It is necessary (as we have said) for our purpose that everything should be conceived, either solely through its essence, or through its proximate cause. If the thing be self-existent, or, as is commonly said, the cause of itself, it must be understood through its essence only; if it be not self-existent, but requires a cause for its existence, it must be understood through its proximate cause. For in reality the knowledge of an effect is nothing else than the acquisition of more perfect knowledge of its cause. Therefore, we may never, while we are concerned with inquiries into actual things, draw any conclusions from abstractions; we shall be extremely careful not to confound that which is only in the understanding with that which is in the thing itself. The best basis for drawing a conclusion will be either some particular affirmative essence, or a true and legitimate definition. For the understanding cannot descend from universal axioms by themselves to particular things, since axioms are of infinite extent, and do not determine the understanding to contemplate one particular thing more than another. Thus the true method of discovery is to form thoughts from some given definition. This process will be more fruitful and easy in proportion as the thing given be better defined. Wherefore, the cardinal point of all this second part of method consists in the knowledge and the conditions of good definition, and the means of finding them. I will first treat of the conditions of definition. A definition, if it is to be called perfect, must explain the inmost essence of

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a thing, and must take care not to substitute for this any of its properties. In order to illustrate my meaning, without taking an example which would seem to show a desire to expose other people’s errors, I will choose the case of something abstract, the definition of which is of little moment. Such is a circle. If a circle be defined as a figure, such that all straight lines drawn from the centre to the circumference are equal, everyone can see that such a definition does not in the least explain the essence of a circle, but solely one of its properties. Though, as I have said, this is of no importance in the case of figures and other abstractions, it is of great importance in the case of physical beings and realities: for the properties of things are not understood so long as their essences are unknown. If the latter be passed over, there is necessarily a perversion of the succession of ideas which should reflect the succession of nature, and we go far astray from our object. In order to be free from this fault, the following rules should be observed in definition: I.

II.

If the thing in question be created, the definition must (as we have said) comprehend the proximate cause. For instance, a circle should, according to this rule, be defined as follows: the figure described by any line whereof one end is fixed and the other free. This definition clearly comprehends the proximate cause. A conception of definition of a thing should be such that all the properties of that thing, in so far as it is considered by itself, and not in conjunction with other things, can be deduced from it, as may be seen in the definition given of a circle: for from that it clearly follows that all straight lines drawn from the centre to the circumference are equal. That this is a necessary characteristic of a definition is so clear to anyone, who reflects on the matter, that there is no need to spend time in proving it, or in showing that, owing to this second condition, every definition should be affirmative. I speak of intellectual affirmation, giving little thought to verbal affirmations which, owing to the poverty of language, must sometimes, perhaps, be expressed negatively, though the idea contained is affirmative.

The rules for the definition of an uncreated thing are as follows: I. II.

The exclusion of all ideas of cause — that is, the thing must not need explanation by anything outside itself. When the definition of the thing has been given, there must be no room for doubt as to whether the thing exists or not.

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III. It must contain, as far as the mind is concerned, no substantives which could be put in an adjectival form; in other words, the object defined must not be explained through abstractions. IV. Lastly, though this is not absolutely necessary, it should be possible to deduce from the definition all the properties of the thing defined. All these rules become obvious to anyone giving strict attention to the matter. I have also stated that the best basis for drawing a conclusion is a particular affirmative essence. The more specialised the idea is, the more is it distinct, and therefore clear. Wherefore a knowledge of particular things should be sought for as diligently as possible.

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Correspondence between Spinoza and Simon de Vries Extracts

Simon de Vries1 to Spinoza

24–2–1663

Most Honourable Friend, … Clavius,2 however, whose opinion he (Jacquet)3 quotes, thinks as follows: ‘Definitions’, he says, ‘are artificial phrases, nor is there any need in reasoning that a thing should be defined in a particular way; but it is sufficient that a thing defined should never be said to agree with another thing, until it has been shown that its definition also agrees therewith.’ Thus, according to Borel,4 the definition of a given thing should consist as regards its construction or passive quality in something thoroughly known to us and true. Clavius, on the other hand, holds that it is a matter of indifference, whether the construction or passive quality be well known and true, or the

1. Simon de Vries, a pupil who repeatedly, but unsuccessfully, tried to assist Spinoza financially. 2. Christopher Clavius, 1537–1612, Professor of Mathematics in Rome. 3. Andrew Jacquet, 1611–1660, Professor of Mathematics in Antwerp. 4. Peter Borel, 1620–1689, physician to the king of France.

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reverse; so long as we do not assert, that our definition agrees with anything, before it has been proved. I should prefer Borel’s opinion to that of Clavius. I know not which you would assent to, if to either. As these difficulties have occurred to me with regard to the nature of definition, which is reckoned among the cardinal points of demonstration, and as I cannot free my mind from them, I greatly desire, and earnestly beg you, when you have leisure and the opportunity, to be kind enough to send me your opinion on the matter, and at the same time to tell me the distinction between axioms and definitions. Borel says that the difference is purely nominal, But I believe you decide otherwise.

Spinoza to Simon de Vries

n.d.

Respected Friend, … As for the questions propounded in your club, which is wisely enough ordered, I see that your difficulties arise from not distinguishing between kinds of definition: that is between a definition serving to explain a thing, of which the essence only is sought and in question, and a definition which is put forward only for purposes of inquiry. The former, having a definite object ought to be true, the latter need not. For instance, if someone asks me for a description of Solomon’s temple, I am bound to give him a true description, unless I want to talk nonsense to him. But if I have constructed, in my mind, a temple which I desire to build, and infer from the description of it that I must buy such and such a site and so many thousand stones and other materials, will any sane person tell me that I have drawn a wrong conclusion because my definition is possibly untrue? or will anyone ask me to prove my definition? Such a person would simply be telling me, that I had not conceived that which I had conceived, or be requiring me to prove, that I had conceived that which I had conceived; in fact, evidently trifling. Hence, a definition either explains a thing, in so far as it is external to the intellect, in which case it ought to be true and only to differ from a proposition or an axiom in being concerned merely with the essences of things, or the modification of things, whereas the latter has a wider scope and extends also to external truths. Or else it explains a thing, as it is conceived or can be conceived by us; and then it differs from an axiom or proposition, inasmuch as it only requires to be conceived absolutely, and not like an axiom as true. Hence a bad definition is one which is not conceived. To explain my meaning, I will take Borel’s example — a man saying that two straight lines enclosing a space

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shall be called ‘figurals’. If the man means by a straight line the space as the rest of the world means by a curved line, his definition is good (for by the definition would be meant such a figure as (), or the like); so long as he does not afterwards mean a square or other kind of figure. But if he attaches the ordinary meaning to the word ‘a straight line’, the thing is evidently inconceivable, and therefore there is no definition. These considerations are plainly confused by Borel, to whose opinion you incline. I give another example, the one you cite at the end of your letter. If I say that each substance has only one attribute, this is an unsupported statement and needs proof. But, if I say that I mean by substance that which consists only in one attribute, the definition will be good, so long as entities consisting of several attributes are afterwards styled by some name other than substance.

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John Locke

An Essay concerning Human Understanding (Extracts)

BOOK III: OF WORDS Chapter III: Of General Terms § 9 … To conclude, this whole mystery of Genera and Species, which make such a noise in the Schools, and are, with Justice, so little regarded out of them, is nothing else but abstract Ideas, more or less comprehensive, with names annexed to them. In all which, this is constant and unvariable, That every more general term stands for such an Idea, as is but a part of any of those contained under it. § 10 This may shew us the reason why, in the defining of Words, which is nothing but declaring their signification, we make use of the Genus, or next general Word that comprehends it. Which is not out of necessity, but only to save the labour of enumerating the several simple Ideas, which the next general Word or genus stands for, or, perhaps, sometimes the shame of not being able to do it. But though defining by Genus and Differentia (I crave leave to use these terms of Art, though originally Latin, since they most properly suit those Notions they are applied to; I say, though defining by the Genus be the shortest way; yet, I think, it may be doubted, whether it be the best. This I am sure, it is not the only, and so not absolutely necessary. For, Definition being nothing but making another understand by Words what Idea the term defined stands for, a Definition is best made by enumerating those simple Ideas that are combined in the signification

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of the term Defined: and if, instead of such an enumeration, Men have accustomed themselves to use the next general term, it has not been out of necessity, or for greater clearness; but for quickness and dispatch sake. For, I think, that to one who desired to know what Idea the Word Man stood for; if it should be said, that Man was a solid extended Substance, having Life, Sense, spontaneous Motion, and the Faculty of Reasoning, I doubt not but the meaning of the term Man would be as well understood, and the Idea it stands for be at least as clearly made known, as when it is defined to be a rational Animal; which by the several Definitions of Animal, Vivens, and Corpus, resolves it self into those enumerated Ideas. I have, in explaining the term Man, followed here the ordinary Definition of the Schools: which though, perhaps, not the most exact, yet serves well enough to my present purpose. And one may, in this instance, see what gave occasion to the Rule that a Definition must consist of Genus and Differentia: and it suffices to shew us the little necessity there is of such a Rule, or advantage in the strict observing of it. For Definitions, as has been said, being only the explaining of one Word, by several others, so that the meaning, or Idea it stands for, may be certainly known, Languages are not always so made, according to the Rules of Logick, that every term can have its signification, exactly and clearly expressed by two others. Experience sufficiently satisfies us to the contrary; or else those who have made this Rule have done ill, that they have given us so few Definitions conformable to it. But of Definitions, more in the next Chapter. § 11 To return to general Words, it is plain, by what has been said, That General and Universal, belong not to the real existence of Things; but are the Inventions and Creatures of the Understanding, made by it for its own use, and concern only Signs, whether Words or Ideas. Words are general, as has been said, when used for Signs of general Ideas, and so are applicable indifferently to many particular Things; And Ideas are general when they are set up as the Representatives of many particular Things: but universality belongs not to Things themselves, which are all of them particular in their Existence, even those Words, and Ideas which in their signification, are general. When therefore we quit Particulars, the Generals that rest, are only Creatures of our own making, their general Nature being nothing but the Capacity they are put into, by the Understanding, of signifying or representing many particulars. For the signification they have, is nothing but a relation, that by the mind of Man is added to them. § 12 The next Thing therefore to be considered is, What kind of signification it is that general Words have. For as it is evident, that they do not signify barely one

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particular Thing, for then they would not be general Terms but proper Names, so on the other side ‘tis as evident, they do not signify a plurality; for Man and Men would then signify the same, and the distinction of numbers (as Grammarians call them) would be superfluous and useless. That then which general Words signify is a sort of Things; and each of them does that by being a sign of an abstract Idea in the mind, to which Idea, as Things existing are found to agree, so they come to be ranked under that name; or, which is all one, be of that sort. Whereby it is evident that the Essences of the sorts, or (if the Latin Word pleases better) Species of Things, are nothing else but these abstract Ideas. … § 20 To conclude, this is that, which in short I would say, (viz.) that all the great Business of Genera and Species, and their Essences, amounts to no more but this, That Men making abstract Ideas, and settling them in their Minds, with Names annexed to them, do thereby enable themselves to consider Things, and Discourse of them, as it were in bundles, for the easier and readier improvement, and communication of their Knowledge, which would advance but slowly, were their Words and Thoughts confined only to particulars.

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CHAPTER IV: OF THE NAMES OF SIMPLE IDEAS § 1 Though all Words, as I have shewn, signify nothing immediately, but the Ideas in the Mind of the Speaker; yet upon a nearer survey, we shall find that the Names of simple Ideas, mixed Modes (under which I comprise Relations too,) and natural Substances, have each of them something peculiar, and different from the other. For example: § 2 First, The Names of simple Ideas and Substances, with the abstract Ideas in the Mind which they immediately signify, intimate also some real Existence, from which was derived their original pattern. But the Names of mixed Modes, terminate in the Idea that is in the Mind and lead not the Thoughts any farther, as we shall see more at large in the following Chapter. § 3 Secondly, The Names of simple Ideas and Modes signify always the real, as well as nominal Essence of their Species. But the Names of natural Substances signify

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rarely, if ever, anything but barely the nominal Essences of those Species, as we shall shew in the Chapter, that treats of the Names of Substances in particular. § 4 Thirdly, The Names of simple Ideas are not capable of any definitions; the Names of all complex Ideas are. It has not, that I know, hitherto been taken notice of by any Body, what Words are, and what are not capable of being defined: the want whereof is (as I am apt to think) not seldom the occasion of great wrangling and obscurity in Men’s Discourses, whilst some demand definitions of Terms that cannot be defined; and others think, they ought to rest satisfied, in an Explication made by a more general Word, and its Restriction, (or to speak in Terms of Art by a Genus and Difference,) when even after such Definition made according to rule, those who hear it, have often no more a clear Conception, of the meaning of the Word, than they had before. This at least, I think, that the shewing what Words are, and what are not capable of Definitions and wherein consists a good Definition is not wholly besides our present purpose; and perhaps, will afford so much Light to the Nature of these Signs and our Ideas, as to deserve a more particular Consideration. § 5 I will not here trouble my self, to prove that all Terms are not definable from that Progress, in infinitum, which it will visibly lead us into, if we should allow, that all Names could be defined. For if the Terms of one Definition, were still to be defined by another, Where at last should we stop? But I shall from the Nature of our Ideas and the Signification of our Words shew, why some Names can, and others cannot be defined, and which they are. § 6 I think it is agreed that a Definition is nothing else but the shewing the meaning of one Word by several other not synonymous Terms. The meaning of Words, being only the Ideas they are made to stand for by him that uses them; the meaning of any Term is then shewed, or the Word is defined when by other Words, the Idea it is made the Sign of, and annexed to in the Mind of the Speaker, is as it were represented, or set before the view of another; and thus its Signification ascertained: This is the only use and end of Definitions; and therefore the only measure of what is, or is not a good Definition. § 7 This being premised, I say, that The Names of simple Ideas, and those only, are incapable of being defined. The reason whereof is this, That the several Terms of a Definition, signifying several Ideas, they can altogether by no means represent an Idea which has no Composition at all: And therefore a Definition, which is properly nothing but the shewing the meaning of one Word by several

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others, not signifying each the same Thing, can in the Names of simple Ideas have no Place. § 8 The not observing this difference in our Ideas, and their Names, has produced that eminent trifling in the Schools, which is so easy to be observed, in the definitions they give us of some few of these simple Ideas. For as to the greatest part of them, even those Masters of Definitions, were fain to leave them untouch’d, meerly by the impossibility they found in it. What more exquisite Jargon could the Wit of Man invent, than this Definition, The Act of a being in Power, as far forth as in Power, which would puzzle any rational Man, to whom it was not already known by its famous absurdity, to guess what Word it could ever be supposed to be the Explication of. If Tully asking a Dutchman what ‘Beweeginge’ was, should have received this Explication in his own Language, that it was Actus entis in potentia quatenus in potentia; I ask whether any one can imagine he could thereby have understood what the Word ‘Beweeginge’ signified, or have guessed what Idea a Dutchman ordinarily had in his Mind and would signify to another, when he used that sound. § 9 Nor have the Modern Philosophers, who have endeavoured to throw off the Jargon of the Schools and speak intelligibly, much better succeeded in defining simple Ideas, whether by explaining their Causes or any otherwise. The Atomists, who define Motion to be a passage from one place to another, What do they more than put one synonymous Word for another? For what is Passage other than Motion? And if they were asked what Passage was, How would they better define it than by Motion? For is it not at least as proper and significant to say Passage is a Motion from one place to another as to say Motion is a passage, etc. This is to translate, and not to define, when we change two Words of the same Signification one for another; which, when one is better understood than the other, may serve to discover what Idea the unknown stands for; but is very far from a Definition, unless we will say, every English Word in the Dictionary is the definition of the Latin Word it answers, and that Motion is a definition of Motus. Nor will the successive Application of the parts of the Superficies of one Body, to those of another, which the Cartesians give us, prove a much better Definition of Motion when well examined. § 10 The act of perspicuous, as far forth as perspicuous is another Peripatetick Definition of a simple Idea; which though not more absurd than the former of Motion, yet betrays its Uselessness and Insignificancy more plainly, because Experience will easily convince any one, that it cannot make the meaning of the

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Word Light (which it pretends to define) at all understood by a blind Man; but the definition of Motion appears not at first sight so useless, because it escapes this way of Trial. For this simple Idea, entering by the Touch as well as Sight; ‘tis impossible to shew an Example of any one, who has no other way to get the Idea of Motion, but barely by the definition of that Name. Those who tell us, that Light is a great number of little Globules, striking briskly on the bottom of the Eye, speak more intelligibly than the Schools; but yet these Words never so well understood, would make the Idea, the Word Light stands for, no more known to a Man that understands it not before, than if one should tell him that Light was nothing but a Company of little Tennis-balls, which Fairies all day long struck with Rackets against some Men’s Fore-heads, whilst they passed by others. For granting this explication of the Thing to be true; yet the Idea of the cause of Light, if we had it never so exact, would no more give us the Idea of Light it self, as it is such a particular perception in us, than the Idea of the Figure and Motion of a sharp piece of Steel, would give us the Idea of that Pain, which it is able to cause in us. For the cause of any Sensation and the Sensation it self, in all the simple Ideas of one Sense, are two Ideas, and two Ideas so different, and distant one from another, that no two can be more so. And therefore should Des Carte’s Globules strike never so long on the retina of a Man, who was blind by a Gutta Serena, he would thereby never have any Idea of Light or anything approaching to it, though he understood what little Globules were, and what striking on another Body was, never so well. And therefore the Cartesians very well distinguish between that Light which is the Cause of that Sensation in us, and the Idea which is produced in us by it, and is that which is properly Light. § 11 Simple Ideas, as has been shewn, are only to be got by those impressions Objects themselves make on their Minds, by the proper Inlets appointed to each sort. If they are not received in this way, all the Words in the World, made use of to explain, or define any of their Names, will never be able to produce in us the Idea it stands for. For Words, being Sounds, can produce in us no other simple Ideas, than of those very Sounds; nor excite any in us, but by that voluntary connexion, which is known to be between them, and those simple Ideas, which common Use has made them Signs of. He that thinks otherwise, let him try if any Words can give him the taste of a Pine Apple, and make him have the true Idea of the Relish of that celebrated delicious Fruit. So far as he is told it has a resemblance with any Tastes, whereof he has the Ideas already in his Memory, imprinted there by sensible Objects not Strangers to his Palate, so far may he approach that resemblance in his Mind. But this is not giving us that Idea by a

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Definition, but exciting in us other simple Ideas, by their known Names; which will be still very different from the true taste of that fruit it self. In Light and Colours, and all other simple Ideas, it is the same thing: for the signification of Sounds, is not natural, but only imposed and arbitrary. And no Definition of Light or Redness, is more fitted, or able to produce either of those Ideas in us, than the sound Light, or Red, by it self. For to hope to produce an Idea of Light, or Colour, by a Sound, however formed, is to expect that Sounds should be visible, or Colours audible; and to make the Ears do the Office of all the other senses. Which is all one as to say, that we might Taste, Smell. and See by the Ears: a sort of Philosophy worthy only of Sanco Panca, who had the Faculty to see Dulcinea by Hearsay. And therefore he that has not before received into his Mind, by the proper Inlet, the simple Idea which any Word stands for, can never come to know the signification of that Word, by any other words, or Sounds, whatsoever put together, according to any Rules of Definition. The only way is, by applying to his Senses the proper Object; and so producing that Idea in him, for which he has learn’d the name already. A studious blind Man, who had mightily beat his Head about visible Objects, and made use of the explication of his Books and Friends, to understand those names of Light, and Colours, which often came in his way; bragg’d one day, That he now understood what Scarlet signified. Upon which his Friend demanding, what Scarlet was? the blind Man answered, It was like the Sound of a Trumpet. Just such an Understanding of the name of any other simple Idea will he have, who hopes to get it only from a Definition, or other Words made use of to explain it. § 12 The case is quite otherwise in complex Ideas; which consisting of several simple ones, it is in the power of Words, standing for the several Ideas, that make the Composition, to imprint complex Ideas in the Mind, which were never there before, and so make their Names be understood. In such Collections of Ideas, passing under one name, Definitions, or the teaching of the signification of one word, by several others, has place, and may make us understand the Names of Things, which never come within the reach of our Senses; and frame Ideas suitable to those in other Men’s Minds, when they use those Names: provided that none of the terms of the Definition stand for any such simple Ideas, which he to whom the Explication is made, has never yet had in his Thoughts. Thus the word Statue may be explained to a blind Man by other words, when Picture cannot, his Senses have given him the Idea of Figure, but not of Colours, which therefore Words cannot excite in him. This gain’d the Prize to the Painter, against the Statuary; each of which contending for the excellency of his Art, and the Statuary bragging, that his was to be preferred, because it reached farther,

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and even those who had lost their Eyes, could yet perceive the excellency of it. The Painter agreed to refer himself to the Judgment of a blind Man; who being brought where there was a Statue being made by the one, and a Picture drawn by the other; he was first led to the Statue, in which he traced with his Hands, all the Lineaments of the Face and the Body; and with great admiration, applauded the Skill of the Work-man. But being led to the Picture, and having his hands laid upon it, was told, That now he touched the head, and the Forehead, Eyes, Nose, etc. as his hand moved over the parts of the Picture on the Cloth, without finding any the least distinction: Whereupon he cried out, that certainly that must needs be a very admirable and divine piece of Workmanship, which could represent to them all those Parts, where he could neither feel nor perceive any thing. § 13 He that should use the word Rainbow, to one who knew all those Colours, but yet had never seen that Phaenomenon, would, by enumerating the Figure, Largeness, Position, and Order of the Colours, so well define that word, that it might be perfectly understood. But yet that Definition, how exact and perfect soever, would never make a blind Man understand it; because several of the simple Ideas that make that complex one, being such as he never received by Sensation and Experience, no words are able to excite them in his Mind. § 14 Simple Ideas, as has been shewed, can only be got by Experience, from those Objects, which are proper to produce in us those Perceptions. When by this means we have our Minds stored with them, and know the Names for them, then we are in a condition to define, and by Definition to understand the Names of complex Ideas, that are made up of them. But when any term stands for a simple Idea, that a Man has never yet had in his Mind, it is impossible, by any Words, to make known its meaning to him. When any term stands for an Idea which he has been accustomed to, may make him understand its meaning. But in no case whatsoever, is any name, of any simple Idea, capable of a Definition.

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CHAPTER XI: OF THE REMEDIES OF THE FOREGOING IMPERFECTIONS AND ABUSES § 17 This I have here mentioned by the by, to shew of what Consequence it is for Men in their names of mixed Modes and, consequently, in all their moral

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Discourses, to define their Words when there is Occasion: Since thereby moral Knowledge may be brought, to so great Clearness and Certainty. And it must be great want of Ingenuity (to say no worse of it) to refuse to do it: Since a Definition is the only way, whereby the precise Meaning of moral Words can be known; and yet a way, whereby their Meaning may be known certainly, and without leaving any room for any contest about it. And therefore the Negligence or Perverseness of Mankind cannot be excused, if their Discourses in Morality be not much more clear than those in natural Philosophy: since they are about Ideas in the Mind, which are none of them false or disproportionate; they having no external Beings for Archetypes which they are referr’d to and must correspond with. It is far easier for Men to frame in their minds an Idea which shall be the Standard to which they will give the Name Justice, with which Pattern so made, all Actions that agree shall pass under that denomination, than, having seen Aristides, to frame an Idea that shall in all things be exactly like him who is as he is, let Men make what Idea they please of him. For the one, they need but know the combination of Ideas, that are put together within in their own Minds; for the other, they must inquire into the whole Nature and abstruse hidden Constitution and various Qualities of a Thing existing without them. § 18 Another Reason that makes the defining of mixed Modes so necessary, especially of moral Words, is what I mentioned a little before, viz. That it is the only way whereby the signification of the most of them can be known with certainty. For the Ideas they stand for, being for the most part such, whose component Parts no where exist together, but scattered and mingled with others, it is the Mind alone that collects them and gives them the Union of one Idea: and it is only by Words, enumerating the several simple Ideas which the Mind has united, that we can make known to others,what their Names stand for; the assistance of the senses in this case not helping us, by the proposal of sensible Objects, to shew the Ideas which our names of this kind stand for, as it does often in the names of sensible simple Ideas, and also to some degree in those of Substances. § 19 Thirdly, For the explaining the signification of the Names of Substances, as they stand for the Ideas we have their distinct Species, both the fore-mentioned ways, viz. of shewing and defining, are requisite in many cases, to be made use of. For there being ordinarily in each Sort some leading Qualities, to which we suppose the other Ideas, which make up pure complex Idea of that Species, annexed, we forwardly give the specifick Name to that Thing, wherein that characteristical Mark is found, which we take to be the most distinguishing Idea of that Species. These leading or characteristical (as I may call them) Ideas, in

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the sorts of Animals and Vegetables is (as has been before remarked, (ch vi, § 29 & ch.ix, § 15) mostly Figure, and in inanimate Bodies Colour, and in some both together. Now, § 20 These leading sensible qualities are those, which make the chief Ingredients of our Specifick Ideas, and consequently the most observable and invariable part in the Definitions of our specifick Names, as attributed to Sorts of Substances coming under our Knowledge. For though the Sound Man, in its own Nature, be as apt to signify a complex Idea made up of Animality and Rationality, united in the same Subject, as to signify any other combination; yet used as a mark to stand for a sort of Creatures we count of our own kind, perhaps the outward shape is as necessary to be taken into our complex Idea, signified by the word Man, as any other we find in it. And therefore why Plato’s animal in plume bipes latis unguibus, should not be as good a Definition of the Name Man, standing for that sort of Creatures, will not be easy to shew: for ‘tis the Shape, as the leading Quality, that seems more to determine that Species than a Faculty of Reasoning, which appears not at first and in some never. And if this be not allow’d to be so, I do not know how they can be excused from murther, who kill monstrous Births (as we call them,) because of an unordinary Shape, without knowing whether they have a Rational Soul or no, which can be no more discerned in a wellformed than ill-shaped Infant as soon as born. And who is it that has informed us, that a Rational Soul can inhabit no Tenement, unless it has just such a sort of Frontispiece or can join it self to, and inform no sort of Body, but one that is just of such an outward Structure? § 21 Now these leading Qualities are best made known by shewing, and can hardly be made known otherwise. For the shape of an Horse, or Cassuary, will be but rudely and imperfectly imprinted on the Mind by Words, the sight of the Animals doth it a thousand times better: And the idea of the particular Colour of Gold, is not to be got by any description of it, but only the frequent exercise of the Eyes about it; as is evident in those who are used to this Metal, who will frequently distinguish true from counterfeit, pure from adulterate, by the sight, where others (who have as good Eyes but yet by use have not got the precise nice Idea of that particular Yellow) shall not perceive any difference. The like may be said of the other simple Ideas, peculiar in their kind to any Substance; for which precise ideas there are no peculiar Names. The particular ringing sound there is in Gold, distinct from the sound of other Bodies, has no particular Name annexed to it, no more than the particular Yellow that belongs to that metal.

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§ 22 But because many of the simple Ideas that make up our specifick Ideas of Substances are Powers, which lie not obvious to our Senses in the Things as they ordinarily appear; therefore, in the signification of our Names of Substances, some part of the signification will be better made known by enumerating those simple Ideas than in shewing the Substance it self. For he that, to the yellow shining Colour of Gold got by sight, shall, from my enumerating them, have the Ideas of greater Ductility, Fusibility, Fixedness, and Solubility in Aqua Regia, will have a perfecter Idea of Gold, than he can have by seeing a piece of Gold, and thereby imprinting in his Mind only its obvious Qualities. But if the formal Constitution of this shining, heavy, ductile Thing (from whence all its Properties flow) lay open to our Senses, as the formal Constitution or Essence of a triangle does, the signification of the word Gold might as easily be ascertained, as that of Triangle. § 23 Hence we may take notice, how much the Foundation of all our Knowledge of corporeal Things lies in our Senses. For how Spirits, separate from Bodies, (whose Knowledge and Ideas of these Things is certainly much more perfect than ours) know them we have no Notion, no Idea at all. The whole extent of our Knowledge, or Imagination reaches not beyond our own Ideas, limited to our ways of Perception. Though yet it be not to be doubted, that Spirits of a higher rank than those immersed in Flesh, may have as clear Ideas of the radical Constitution of Substances as we have of a Triangle, and so perceive how all their Properties and Operations flow from thence: but the Manner how they come by that Knowledge exceeds our Conceptions. § 24 But though Definitions will serve to explain the Names of Substances as they stand for our Ideas, yet they leave them not without great imperfection as they stand for Things. For our Names of Substances being not put barely for our Ideas, but being made use of ultimately to represent Things, and so are put in their place, their signification must agree with the Truth of Things as well as with Men’s Ideas. And therefore in Substances, we are not always to rest in the ordinary complex Idea commonly received as the signification of that Word, but must go a little further, and inquire into the Nature and Properties of the Things themselves, and thereby perfect, as much as we can, our Ideas of their distinct Species, or else learn them from such as are used to that sort of Things and are experienced in them. For since ‘tis intended their Names should stand for such Collections of simple Ideas as do really exist in Things themselves, as well as for the complex Idea in other Men’s Minds which in their ordinary acceptation they stand for: therefore to define their Names right, natural History is to be inquired into; and their Properties are, with care and examination, to be found

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out. For it is not enough, for the avoiding Inconveniences in Discourses and Arguings about natural Bodies and substantial Things, to have learned, from the Propriety of the Language, the common but confused or very imperfect Idea to which each Word is applied, and to keep them to that idea in our use of them; but we must, by acquainting ourselves with the History of that sort of Things, rectify and settle our complex Idea belonging to each specific Name; and in Discourse with others (if we find them mistake us) we ought to tell what the complex Idea is that we make such a Name stand for. This is the more necessary to be done by all those who search after Knowledge and philosophical Verity in that Children, being taught Words whilst they have but imperfect Notions of Things, apply them at random and without much thinking and seldom frame determined Ideas to be signified by them. Which Custom (it being easy and serving well enough for the ordinary Affairs of Life and Conversation) they are apt to continue when they are Men, and so begin at the wrong end, learning Words first and perfectly, but make the Notions, to which they apply those Words afterwards, very overtly. By this means it comes to pass that Men speaking the proper Language of their Country, i.e. according to Grammar-Rules of that Language, do yet speak very improperly of Things themselves and, by their arguing one with another, make but small progress in the discoveries of useful Truths and the Knowledge of Things as they are to be found in themselves, and not in our Imaginations; and it matters not much, for the improvement of our Knowledge, how they are call’d. § 25 It were therefore to be wished, That Men, versed in physical Enquiries and acquainted with the several sorts of natural Bodies, would set down those simple Ideas wherein they observe the Individuals of each sort constantly to agree. This would remedy a great deal of that confusion which comes from several Persons applying the same Name to a Collection of a smaller or greater number of sensible Qualities, proportionably as they have been more or less acquainted with or accurate in examining the Qualities of any sort of Things which come under one denomination. But a Dictionary of this sort, containing, as it were, a Natural History, requires too many hands as well as too much time, cost, pains, and sagacity ever to be hoped for; and till that be done, we must content ourselves with such Definitions of the Names of Substances as explain the sense Men use them in. And ‘twould be well, where there is occasion, if they would afford us so much. This yet is not usually done; but Men talk to one another and dispute in Words whose meaning is not agreed between them, out of a mistake that the signification of common Words are certainly established and the precise Ideas they stand for perfectly known, and that it is a shame to be ignorant of them.

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Both which Suppositions are false: no Names of complex Ideas having so settled determined Significations that they are constantly used for the same precise Ideas. Nor is it a shame for a Man not to have a certain Knowledge of anything but by the necessary ways of attaining it; and so it is no discredit not to know what precise Idea any Sound stands for in another Man’s Mind, without he declare it to me by some other way than barely using that Sound, there being no other way without such a Declaration certainly to know it. Indeed, the necessity of Communication by Language brings Men to an agreement in the signification of common Words, within some tolerable latitude, that may serve for ordinary Conversation; and so a Man cannot be supposed wholly ignorant of the Ideas which are annexed to Words by common use, in a Language familiar to him. But common use, being but a very uncertain Rule, which reduces itself at last to the Ideas of particular Men, proves often but a very variable Standard. But though such a Dictionary as I have above mentioned will require too much time, cost, and pains to be hoped for in this Age, yet methinks it is not unreasonable to propose that Words standing for Things which are known and distinguished by their outward shapes should be expressed by little Draughts and Prints made of them. A Vocabulary made after this fashion would perhaps, with more ease and in less time, teach the true signification of many Terms, especially in Languages of remote Countries or Ages, and settle truer Ideas in Men’s minds of several Things, whereof we read the Names in ancient Authors, than all the large and laborious Comments of learned Criticks. Naturalists that treat of Plants and Animals have found the benefit of this way: And he that has had occasion to consult them will have reason to confess that he has a clearer Idea of Apium or Ibex, from a little Print of that Herb or Beast, than he could have from a long Definition of the Names of either of them. And so, no doubt, he would have of Strigil and Sistrum if, instead of a Curry-comb, and Cymbal, which are the English Names Dictionaries render them by, he could see stamp’d in the Margin small Pictures of these Instruments as they were in use amongst the Ancients. Toga, Tunica, Pallium are Words easily translated by Gown, Coat, and Cloak; but we have thereby no more true Ideas of the Fashion of those Habits amongst the Romans than we have of the Faces of the Taylors who made them. Such Things as these, which the Eye distinguishes by their shapes, would be best let into the Mind by Draughts made of them, and more determine the signification of such Words than any other Words set for them, or made use of to define them. But this only by the by. § 26 Fifthly, If Men will not be at pains to declare the meanings of their Words, their Terms are not to be understood; yet this is the least that can be expected,

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that in all Discourses, wherein one Man pretends to instruct or convince another, he should use the same Word constantly in the same sense: If this were done, (which no body can refuse, without great disingenuity) many of the Books extant might be spared; many of the Controversies in Dispute would be at an end; several of those great volumes, swollen with ambiguous Words, now used in one sense, and by and by in another, would shrink into very narrow compass; and many of the Philosophers (to mention no other,) as well as Poet’s Works, might be contained in a Nut-shell. § 27 But, after all, the provision of Words is so scanty in respect of that infinite variety of Thoughts, than Men, wanting Terms to suit their precise Notions, will, notwithstanding their utmost caution, be forced often to use the same Word, in somewhat different Senses. And though in the continuation of a Discourse, or the pursuit of an Argument, there be hardly room to digress into a particular definition, as often as a Man varies the signification of any Term; Yet the import of the Discourse will, for the most part, if there be no designed fallacy, sufficiently lead candid and intelligent Readers, into the true meaning of it: but where that is not sufficient to guide the Reader, there it concerns the Writer to explain his meaning, and shew in what sense he there uses that Term.

BOOK IV: OF KNOWLEDGE AND OPINION Chapter VIII: Of Trifling Propositions § 3 But if Men will call Propositions Identical, wherein the same Term is not affirmed of itself, whether they speak more properly than I, others must judge; This is certain: all that they say of Propositions that are not Identical, in my sense, concerns not me, nor what I have said; all that I have said relating to those Propositions, wherein the same Term is affirmed of it self. And I would fain see an Instance wherein any such can be made use of, to the advantage and improvement of anyone’s Knowledge. Instances of other kinds, whatever use may be made of them, concern not me, as not being such as I call Identical. § 4 Secondly, Another sort of Trifling Propositions is, when part of the complex Idea is predicated of the Name of the whole: a part of the Definition of the Word defined. Such are all Propositions wherein the Genus is predicated of the Species, or more comprehensive of less comprehensive Terms: For what Information, what Knowledge carries this Proposition in it, viz. Lead is a Metal, to a Man who

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knows the complex Idea the Name Lead stands for. All the simple Ideas that go to the complex one signified by the Term Metal, being nothing but what he before comprehended and signified by the Name Lead. Indeed, to a Man that knows the Signification of the Word Metal, and not of the Word Lead, it is a shorter way to explain the Signification of the Word Lead, by saying it is a Metal, which at once expresses several of its simple Ideas, than to enumerate them one by one, telling him it is a Body very heavy, fusible, and malleable. § 5 Alike trifling it is to predicate any other part of the Definition of the Term defined, or to affirm any one of the simple Ideas of a complex one, of the Name of the whole complex Idea; as All Gold is fusible. For Fusibility being one of the simple Ideas that goes to the making up the complex one the sound Gold stands for, what can it be but playing with Sounds, to affirm that of the Name Gold which is comprehended in its received Signification? ‘twould be thought little better than ridiculous, to affirm gravely, as a Truth of moment, that Gold is yellow; and I see not how it is any jot more material to say It is fusible, unless that Quality be left out of the complex Idea of which the Sound Gold is the mark in ordinary Speech. What Instruction can it carry with it, to tell one that which he hath been told already or he is supposed to know before? For I am supposed to know the Signification of the Word another uses to me, or else he is to tell me. And if I know that the Name Gold stands for this complex Idea of body, yellow, heavy, fusible, malleable, ‘twill not much instruct me to put it solemnly afterwards in a Proposition and gravely say All Gold is fusible. Such Propositions can only serve to shew the Disingenuity of one who will go from the Definition of his own Terms, by reminding him sometimes of it; but carry no Knowledge with them but of the Signification of Words, however certain they be. § 6 Every Man is an animal or living Body, is as certain a Proposition as can be; but no more conducing to the Knowledge of Things than to say A palfry is an ambling horse, or a neighing, ambling animal: both being only about the Signification of Words, and make me know but this; that Body, Sense, and Motion or power of Sensation and moving, are three of those Ideas that I always comprehend and signify by the Word Man; and where they are not to be found together, the Name Man belongs not to that Thing: And so of the other, that Body, Sense, and a certain way of going, with a certain kind of Voice, are some of those ideas which I always comprehend and signify by the Word Palfry; and when they are not to be found together, the name Palfry belongs not to that Thing. ‘Tis just the same, and to the same purpose, when any Term, standing for any one or more of the simple Ideas that altogether make up that complex Idea

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which is called a Man, is affirmed of the term Man: v.g. suppose a Roman signified by the Word Homo all these distinct Ideas united in one subject, Corporietas, Sensibilitas, Potentia se movendi, Rationalitas, Risibilitas, he might no doubt with great certainty universally affirm one, more, or all of these together of the Word Homo, but did no more than say that the Word Homo in his Country, comprehended in its signification, all these Ideas. Much like a Romance knight, who by the word Palfry signified these Ideas; body of a certain figure, four-legged, with Sense, motion, ambling, neighing, white, used to have a woman on his back, might with the same certainty, universally affirm also any or all of these of the word Palfry: but did thereby teach no more but that the word Palfry, in his, or Romance Language, stood for all these, and was not to be applied to anything where any of these was wanting. But he that shall tell me that, in whatever Thing Sense, Motion, Reason, and Laughter were united, that Thing had actually a Notion of GOD or would be cast into a sleep by Opium, made indeed an instructive Proposition: because, neither having the Notion of GOD nor being cast into sleep by Opium, being contained in the Idea signified by the Word Man, we are by such Propositions taught something more than barely what the Word Man stands for; and therefore the Knowledge contained in it is more than verbal. § 7 Before a Man makes any Proposition, he is supposed to understand the terms he uses in it, or else he talks like a Parrot, only making a noise by imitation, and framing certain Sounds which he has learnt of others, but not, as a rational Creature, using them for Signs of Ideas which he has in his mind. The Hearer also is supposed to understand the Terms as the Speaker uses them, or else he talks jargon and makes an unintelligible noise. And therefore he trifles with Words who makes such a Proposition which, when it is made, contains no more than one of the Terms does, and which a Man was supposed to know before: v.g. A triangle hath three sides, or Saffron is yellow. And this is no further tolerable than where a Man goes to explain his Terms, to one who is supposed or declares himself not to understand him: and then it teaches only the Signification of that Word and the use of that Sign. § 8 We can know then the Truth of two sorts of Propositions with perfect certainty; the one is, of those trifling Propositions which have a certainty in them, but ‘tis but a verbal Certainty, but not instructive. And, secondly, we can know the Truth, and so may be certain in Propositions which affirm something of another, which is a necessary consequence of its precise complex Idea, but not contained in it. As that the external Angle of all Triangles is bigger than either of

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the opposite internal Angles; which relation of the outward angle to either of the opposite internal angles making no part of the complex Idea signified by the name Triangle, this is a real Truth and conveys with it instructive real Knowledge. § 9 We having little or no Knowledge of what Combinations there be of simple Ideas existing together in Substances but by our Senses, we cannot make any universal certain Propositions concerning them, any farther than our nominal Essences lead us: which being to a very few and inconsiderable Truths, in respect of those which depend on their real Constitutions, the general Propositions that are made about Substances, if they are certain, are for the most part but trifling; and if they are instructive, are uncertain, and such as we can have no Knowledge of their real Truth, how much soever constant Observation and Analogy may assist our Judgments in guessing. Hence it comes to pass, that one may often meet with very clear and coherent Discourses that amount yet to nothing. For it is plain that Names of substantial beings, as well as others as far as they have relative Significations affixed to them, may, with great Truth, be joined negatively and affirmatively in Propositions, as their relative Definitions make them fit to be so joined; and Propositions consisting of such Terms may, with the same clearness, be deduced one from another as those that convey the most real Truths, and all this without any Knowledge of the Nature or reality of Things existing without us. By this method one may make Demonstrations and undoubted Propositions in Words, and yet thereby advance not one jot in the Knowledge of the Truth of Things: v.g. he that, having learnt these following Words with their ordinary mutually relative acceptations annexed to them; v.g. Substance, Men, animal, form, soul, vegetative, sensitive, rational, may make several undoubted Propositions about the Soul without knowing at all what the Soul really is; and of this sort a Man may find an infinite number of Propositions, Reasonings, and Conclusions, in Books of Metaphysicks, School-divinity, and some sort of natural Philosophy, and, after all, know as little of GOD, Spirits, or Bodies as he did before he set out. § 10 He that hath liberty to define, i.e. determine, the signification of his Names of Substances (as certainly every one does in effect, who makes them stand for his own Ideas) and makes their Significations at a venture, taking them from his own or other Men’s Fancies and not from an Examination or Enquiry into the Nature of Things themselves, may with little Trouble demonstrate them one of another, according to those several respects and mutual Relations he has given them one to another wherein, however Things agree or disagree in their own Nature, he needs mind nothing but his own Notions, with the Names he hath

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bestowed upon them; but thereby no more increases his own Knowledge than he does his Riches who, taking a Bag of Counters, calls one in a certain place a Pound, another in another place a Shilling, and a third in a third place a Penny, and so proceeding, may undoubtedly reckon right and cast up a great sum, according to his Counters so placed and standing for more or less as he pleases, without being one jot the richer or without even knowing how much a Pound, Shilling, or Penny is, but only that one is contained in the other twenty times, and contains the other twelve; which a Man may also do in the signification of Words, by making them, in respect of one another, more, or less, or equally comprehensive. § 11 Though yet, concerning most Words used in Discourses, especially argumentative and controversial, there is this more to be complained of, which is the worst sort of Trifling, and which sets us yet further from the certainty of Knowledge we hope to attain by them or find in them, viz. that most Writers are so far from instructing us in the Nature and Knowledge of Things that they use their Words loosely and uncertainly, and do not, by using them constantly and steadily in the same significations, make plain and clear deductions of Words one from another and make their Discourses coherent and clear (how little soever it were instructive,) which were not difficult to do, did they not find it convenient to shelter their Ignorance or Obstinacy under the Obscurity and perplexedness of their Terms; to which, perhaps, Inadvertency and ill Custom does in many men much contribute. § 12 To conclude, barely verbal Propositions may be known by these following Marks: First, All Propositions, wherein two abstract Terms are affirmed one of another are barely about the signification of Sounds. For since no abstract Idea can be the same with any other but it self, when its abstract Name is affirmed of any other Term, it can signify no more but this: that it may, or ought to be called by that Name, or that these two Names signify the same Idea. Thus, should anyone say that Parsimony is Frugality, that Gratitude is justice, that this or that action is or is not temperance: however specious these and the like Propositions may at first sight seem, yet when we come to press them and examine nicely what they contain, we shall find that it all amounts to nothing but the signification of those Terms. § 13 Secondly, All Propositions, wherein a part of the complex Idea which any Term stands for is predicated of that Term, are only verbal, v.g. to say that Gold

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is a Metal, or heavy. And thus all Propositions wherein more comprehensive Words, called Genera, are affirmed of subordinate or less comprehensive, called Species or Individuals, are barely verbal. When by these two Rules we have examined the Propositions that make up the Discourses we ordinarily meet with, both in and out of books, we shall perhaps find that a greater part of them than is usually suspected are purely about the signification of Words, and contain nothing in them but the Use and Application of these Signs. This I think I may lay down for an infallible Rule: that wherever the distinct Idea any Word stands for is not known and considered, and something not contained in the Idea is not affirmed or denied of it, there our Thoughts stick wholly in Sounds and are able to attain no real Truth or Falsehood. This, perhaps, if well heeded, might save us a great deal of useless Amusement and Dispute; and very much shorten our Trouble and wandring in the search of real and true Knowledge.

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Gottfried Wilhelm Leibniz

Correspondence (Extracts)

Leibniz to Hermann Conring1

19-3-1678

… Demonstration is reasoning by which some proposition is made certain. This is achieved whenever it is shown that the proposition necessarily follows from certain suppositions (which are assumed to be certain). By necessarily I mean in such a way that its contrary implies a contradiction; this is the true and unique mark of impossibility. Just as necessity corresponds to impossibility, furthermore, so an identity corresponds to a proposition which implies a contradiction. For the primary impossibility in propositions is this: A is not A; just so the primary necessity in propositions is this: A is A. Hence only identities are indemonstrable, but all axioms are demonstrable, even though they are mostly so clear and easy that they do not need demonstration; nevertheless, they are demonstrable in the sense that if the terms are understood (i.e., by substituting the definitions for the terms defined), it becomes clear that they are necessary or that their contrary implies a contradiction in terms. This is also the opinion of the Scholastics. But we know that identical propositions are necessary propositions without any understanding or analysis of their terms, for I know that A is A, whatever may be understood by A. All propositions, however, whose truth must be shown by further analysing and understanding their terms are demonstrable by such analysis, that is, by definitions. So it is clear that demonstration is a chain of

1. Hermann Conring (1606–1681), professor at Helmstädt.

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definitions. For in the demonstration of any proposition, nothing is used but definitions, axioms (with which I here include postulates), theorems which have been demonstrated previously, and observations. Since the theorems again must themselves be demonstrated, and axioms, except for identities, can also all be demonstrated, it follows that all truths can be resolved into definitions, identical propositions, and observations — though purely intelligible truths do not need observations. After the analysis has been completed, it will become manifest that the chain of demonstration begins with identical propositions or observations and ends in a conclusion but that the beginning is connected with the conclusion through intervening definitions. In this sense I said that a demonstration is a chain of definitions. The definition of a compound idea, moreover, is an analysis of it into its parts, just as a demonstration is nothing but the analysis of a truth into other truths which are already known. And the solution of any problem which is to be worked out is the reduction of the problem to others which are easier or already known to be within one’s power. This is my analysis, which has been tested in mathematics and in other sciences and will succeed. If anyone has another, I shall be surprised if it does not reduce finally to this one, or prove to be a part or corollary of it. …

u Leibniz to Walter von Tschirnhaus2

-3-1678

… You say that it is difficult to set up definitions of things; perhaps you mean in the most simple and the primitive concepts, so to speak. These, I admit, it is difficult to give. We must realize, indeed, that there are several definitions of the same thing, that is, reciprocal properties which distinguish one thing from all other things and that from each one we can derive all the other properties of the thing defined. You are not unaware of this, but some of these definitions are more perfect than others, that is, they come nearer to the primary and adequate notions. Indeed, I hold this to be a certain criterion of a perfect and adequate definition: that when the definition is once grasped, we cannot further doubt whether the thing defined in it is possible or not. Besides, anyone who wishes to construct a characteristic or universal

2. Walter von Tschirnhaus, a mathematician who worked with Leibniz in Paris at the time of the discoveries of the calculus.

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analytic can use any definitions whatever in the beginning, since all will eventually lead to the same result when the analysis is continued. …

u Leibniz to Antoine Arnauld

14-7-1686

… I believe that the sign of a true idea is that its possibility can be proved, either a priori, by stating its cause or reason, or a posteriori, when we know of its existence from experience. Therefore definitions are real when we know the possibility of the thing. If this is not the case definitions are nominal and unreliable. If the definition contains a contradiction two contradictory propositions can be derived from it. For this reason, you rightly informed P. Malebranche and others that a distinction must be made between true and false ideas …

u Leibniz to De Volder3

1699

… You say there are two kinds of concepts, some representing a single unified something from which nothing can be separated without destroying the whole; in your opinion this is the concept of substance, and you say that extension is such a concept. The other kind of concept may represent two or more things. This is a little obscure to me. Surely every concept or definition is such that you cannot remove anything without destroying the whole definition; yet in that case another concept may come into being in the definition. Thus if you remove the concept of equal sides from the definition of a square, the square is destroyed but a rectangle remains. A concept from which nothing can be removed must be simple and primitive, but I do not think that the concept of substance should be established in this way or that the concept of extension is of this kind. …

3. Burcher de Volder (1643–1709) professor of philosophy, physics and mathematics at the University of Leyden.

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Preface to an Edition of Nizolius4 (Extract) … For a definition is nothing but the expressed meaning of a word or more briefly, the meaning signified. In a definition care must be taken not only that the definition be reciprocally true but also that it be clear. Technical terms are therefore to be shunned as worse than dog or snake, and one must abstain particularly from those words for categories which are far removed from Latin usage. Once set up, the definition is to be adhered to consistently, so that wherever you substitute the definition for the term defined, there results no absurd statement. Even if no definition is given beforehand, the use of a word should be uniform so that the same definition could be substituted anywhere. Thus, for any given word, we should see what meaning is to be attached to it and conversely, what word should be attached to a given meaning. In this, both brevity and clarity must be respected. The greatest clarity is found in commonplace terms with their popular usage retained. There is always a certain obscurity in technical terms. … An analogy should be both generally accepted and fitting, so that the definition of the new word which we intend can be moulded from the meaning of the root and the analogy. … Technical terms are to be avoided, as I have said; indeed, they are to be used with care whenever possible. But this is not always possible because of the prolixity which would result if popular terms were always used. For example, a square is a quadrilateral, equilateral, and rectangular, but the words ‘quadrilateral’, ‘equilateral’ and ‘rectangular’ (not to mention ‘plane’) are technical in their turn. Hence they can be further resolved. That is quadrilateral which has only four sides. A side is a bounding line. That is rectangular all of whose angles are right. An angle is the intersection of lines; right is that which is equal on both sides. Thus if we are to avoid technical terms, we shall have to put all these words in place of the word square: that figure, all of whose bounding lines are equal and whose bounding lines are only four, and in which all intersections of terminating lines are equal on both sides. If even greater rigor is demanded, the words line, bounding, intersection, and equality must be further resolved, for their popular usage does not exactly fit the concepts of geometry. … I believe that even the blind can see how annoying it would be, and how awkward, to

4. Marius Nizolius (1498–1576) Italian Humanist, published works attacking scholasticism which were later published by Leibniz in 1670 under the title Antibarbarus philosophicus.

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have always to use all these words in place of the word ‘square’ in our demonstrations. To this can be added what I have already said in many passages of the Art of Combinations. Our judgments are thus rendered more reliable by this process of analysing technical terms into merely popular ones; hence a perfect demonstration merely carries out such analysis to the ultimate and best-known elements. But if this entire analysis were done in one place — the subject and predicate of each judgement into their definitions, and the ingredient terms of the definitions into further definitions — or if we had consistently to return to other definitions or demonstrations we had already given or to works of some author who had done this, our memory would be overtaxed.

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On Universal Synthesis and Analysis, or the Art of Discovery and Judgment (Extract) … The primary concepts from whose combination the rest are made are either distinct or confused. Those are distinct which are understood through themselves, such as ‘being’. Those are confused though clear, which are perceived through themselves, such as colour, because we can only explain them to someone else by showing them to him. For though the nature of colour is analyzable since it has a cause, we cannot sufficiently describe or recognise it by any concepts that are separately explained; it is known only confusedly and hence cannot be given a nominal definition. A nominal definition consists in the enumeration of signs or elements sufficient to distinguish the thing defined from everything else. If we proceed to seek the elements of the elements, we shall at last come to primitive concepts which have no elements at all, or none which we can explain to a sufficient degree. This is the art of dealing with distinct concepts. The art of dealing with confused concepts, however, must discover the distinct concepts which accompany the confused ones, whether these distinct concepts can be understood through themselves or can at least be resolved into such as are understood, for with their help we can sometimes arrive at some cause or resolution of the confused notion. All derivative concepts, moreover, arise from a combination of primitive ones, and the more composite concepts from the combination of less composite

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ones. But one must take care that the combinations do not become useless through the joining together of incompatible concepts. This can be avoided only by experience or by resolving them into distinct single concepts. One must be especially careful in setting up real definitions, to establish their possibility, that is, to show that the concepts from which they are formed are compatible with each other. So while every reciprocal property of a thing can serve as its nominal definition, since all the other attributes of the thing can be demonstrated from it, not every such property suffices for a real definition. For as I have pointed out, there are certain properties which I call paradoxical, whose possibility can be doubted. For example, it can be doubted whether there is a curve for which it is true that given any segment and any point on the curve, the lines connecting this point with the ends of the segment will always form the same angle. For assuming that we have so adjusted the points of the curve to one segment, we still cannot foresee that what may seem to have succeeded by chance in one case will succeed in others, namely that the same points on the curve will satisfy this condition with respect to another segment as well, since all of the points are now determined and no further ones can be assumed. Yet we know that this is the nature of a circle. So, although someone might give a name to the curve having this property, it would not yet be certain that such a curve is possible, and hence that its definition is real. But the concept of the circle set up by Euclid, that of a figure described by the motion of a straight line in a plane about a fixed end, affords a real definition, for such a figure is evidently possible. Hence it is useful to have definitions involving the generation of a thing, or if this is impossible, at least its constitution, that is, a method by which the thing appears to be producible or at least possible. … Obviously we cannot build a secure demonstration on any concept unless we know that this concept is possible, for from impossibles or concepts involving contradictions contradictory propositions can be demonstrated. This is an a priori reason why possibility is a requisite in a real definition. A difficulty raised by Hobbes can also be answered on this basis. For Hobbes saw that all truths can be demonstrated from definitions but held that all definitions are arbitrary and nominal, since we impose arbitrary names upon things. He therefore concluded that truths also consist merely in names and are arbitrary. But we must recognize that if we are to have a real definition, we cannot combines notions arbitrarily, but the concept we form out of them must be possible. Hence every real definition must contain at least the affirmation of some possibility. Furthermore, although names are arbitrary, once they are adopted, their consequences are necessary, and certain truths arise which are real even though they depend on the characters which have been imposed. For example, the rule of nine depends on

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characters imposed by the decimal system, yet it contains real truth. Moreover, to set up a hypothesis to explain the method of production is merely to demonstrate the possibility of a thing, and this is useful even though the thing in question often has not been generated in that way. Thus the same ellipse can be thought of either as described in a plane with the aid of two foci and the motion of a thread about them or as a conic or cylindrical section. Once a hypothesis or a manner of generation is found, one has a real definition from which others can also be derived, and from them those can be selected which best satisfy the other conditions, when a method of actually producing the thing is sought. Those real definitions are most perfect, furthermore, which are common to all the hypotheses or methods of generation and which involve the proximate cause of a thing, and from which the possibility of the thing is immediately apparent without presupposing any experiment or the demonstration of any further possibilities. In other words, those real definitions are most perfect which resolve the thing into simple primitive notions understood in themselves. Such knowledge I usually call adequate or intuitive, for, if there were any inconsistency, it would appear here at once, since no further resolution can take place.

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Discourse on Metaphysics (Extract) … It is well also to distinguish nominal and real definitions. I call a definition nominal when it can still be doubted that the defined concept is possible. So, for example, if I say that an endless screw is a line in three dimensions whose parts are congruent or can be brought to coincide with each other, anyone who does not know from another source what an endless screw is could doubt whether such a line is possible, even though this is in fact one of the reciprocal properties of the endless screw, for the other lines whose parts are congruent to each other (there are only two, the circumference of a circle and a straight line) are plane figures, that is to say, they can be drawn in a plane. This shows us that every reciprocal property can serve as a nominal definition but that when the property makes us understand the possibility of a thing, it establishes a real definition. As long as we have only a nominal definition, we cannot be sure of the consequences drawn from it, for if it concealed some contradiction or impossibility, we could draw conflicting conclusions. This is why truths do not depend on

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names and are not arbitrary as some modern philosophers have thought. Nevertheless, there is still a great difference between the kinds of real definitions, for when possibility is proved only through experience, the definition is only real and nothing more; as in the definition of quicksilver, the possibility of which we recognize because we know that such a body, extremely heavy and yet rather volatile, is actually found. But when the proof of possibility is presented a priori, the definition is both real and causal, as when it contains the possible production of the thing. And when the definition pushes its analysis back to the primitive concepts without assuming anything which needs an a priori proof of its possibility, it is perfect or essential.

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New Essays on Human Understanding1 (Extracts) BOOK III: OF WORDS Chapter III: Of General Terms § 10 Philalethes: This may shew us the reason why, in the defining of Words, which is nothing but declaring their signification, we make use of the Genus, or next general Word that comprehends it. Which is not out of necessity, but only to save the labour of enumerating the several simple Ideas, which the next general Word or genus stands for, or, perhaps, sometimes the shame of not being able to do it. But though defining by Genus and Differentia yet, I think, it may be doubted, whether it be the best. This I am sure, it is not the only one, and so not absolutely necessary. The meaning of the term Man would be as well understood, when it is defined to be a rational Animal; which by the several Definitions of

1. This work was written in French in the form of a dialogue between Philalethes (Locke) and Theophilus (Leibniz) and is a detailed comment on specific passages of Locke’s Essay concerning Human Understanding, in the French translation available to Leibniz. Philalethes cites specific passages of Locke’s work to which Theophilus replies. This translation reproduces the cited sections of Locke’s original text with appropriate references to Books, Chapters and paragraphs (§), and gives an English rendering of Philalethes’ introductory sentences (in round brackets) and Leibniz’ replies of selected passages dealing with questions concerning definitions.

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Animal, Vivens, and Corpus, resolves it self into those enumerated Ideas. It suffices to shew us the little necessity there is of such a Rule, or advantage in the strict observing of it. Languages are not always so made, according to the Rules of Logick, that every term can have its signification, exactly and clearly expressed by two others. Experience sufficiently satisfies us to the contrary; or else those who have made this Rule have done ill, that they have given us so few Definitions conformable to it. Theophilus: I agree, yet for many reasons it is convenient that definitions consist of two parts: this would surely represent a considerable shortening, and all divisions could be reduced to dichotomies which are most suitable for this purpose and are very useful for invention, judgment and memory. Nevertheless, I do not believe that the Logicians always require that the Genus and Differentia be expressed in a single word; for example, the term regular Polygon can be referred to the Genus Square, for a Circle the Genus could be given as plane curved figure, and the Differentia would be that the points of the circumference be equal from a central point. In addition, it is useful to observe that often the Genus can be changed for the Differentia and the Differentia for the Genus; for example, a Square is a regular quadrilateral or a quadrilateral regular; so that genus and differentia only differ to the extent that adjective and noun differ. As if, instead of saying Man is a rational animal, language would permit us to say an animal rational, i.e. a rational substance endowed with animal nature; in contrast to the spirits which are rational substances but without animal nature. And this change of genus and Differentia depends on the variations of the order of the subdivisions. § 11 Philalethes. To return to general Words, it is plain, by what has been said, That General and Universal, belong not to the real existence of Things; but are the Inventions and Creatures of the Understanding. § 12 Whereby it is evident that the Essences of the sorts, or (if the Latin Word pleases better) Species of Things, are nothing else but these abstract Ideas. Theophilus. I cannot quite see the conclusion you draw. For generality consists in the resemblance of individual objects and this resemblance is a reality. § 15 Philalethes. ‘Tis true, there is ordinarily a real Constitution of the sorts of Things; and ‘tis past doubt, there must be some real Constitution, on which any Collection of simple Ideas co-existing, must depend. But it being evident, that Things are ranked under Names into sorts or Species, only as they agree to

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certain abstract Ideas, to which we have annexed those Names, the Essence of each Genus, or Sort, comes to be nothing but that abstract Idea, which the General, or Sortal (if I may have leave so to call it from Sort, as I do General from Genus,) Name stands for. And this we shall find to be that, which the word Essence imports, in its most familiar use. These two sorts of Essences, I suppose, may not unfitly be termed, the one Real, the other the Nominal Essence. Theophilus. It seems that your language innovates greatly in its manner of expression. It is true that hitherto we have spoken of real and nominal definitions, but not, as far as I am aware, of anything other than real essences: unless by nominal Essences one has meant false and impossible Essences which appear to be Essences, but which are not; As, for instance, would be the case of the regular decahedron, i.e. a regular body consisting of ten planes or hedrons. Essence is basically the possibility of what one proposes. What we suppose to be possible is expressed by the definition, but this definition is only nominal if it does not at the same time express the possibility, for otherwise one may doubt whether the definition expresses something real, i.e. possible; until experience comes to our aid to make this reality known to us a posteriori when the thing is really in this world, which suffices instead of reason which makes the reality known a priori by showing the cause or the possible origin of the defined thing. It is therefore not for us to link ideas as we please, unless this combination be justified either by reason which shows it to be possible, or by experience which shows it to be real and consequently also possible. In order better to differentiate Essence and definition, we have to consider that a thing has only one essence but several definitions which express the same essence, just as the same structure or the same city can be represented by different scenographies according to the different angles of viewing.

Chapter IV: Of the Names of simple Ideas § 2 Philalethes. (I must admit that I have always believed that the formation of the modes was arbitrary but that) The Names of simple Ideas and Substances, with the abstract Ideas in the Mind which they immediately signify, intimate also some real Existence. Theophilus. I cannot see any need for this. God has the ideas before creating the objects of these ideas, and nothing prevents him from still communicating these ideas to intelligent beings. It has not even been proved that the objects of our

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mind and simple ideas presented by our senses are outside us. What happens, above all, with those who, like the Cartesians and your celebrated author, believe that our simple ideas of intuitive properties bear no resemblance to what is outside us in the objects: there would, therefore, be nothing which would require these ideas to be founded in some sort of real existence. § 4 Philalethes. (You will surely concede the following difference between simple and complex ideas, i.e. that) the Names of simple Ideas are not capable of any definitions; the Names of all complex Ideas are. § 5 I will not here trouble myself to prove that all terms are not definable from this progress in infinitum which it will visibly lead us to if we should allow that all names could be defined. For if the Terms of one Definition, were still to be defined by another, Where at last should we stop? (something, which, I believe, nobody has observed until now?) § 6 I think it is agreed that a Definition is nothing else but the shewing of the meaning of one Word by several other not synonymous Terms. § 7 This being premised, I say, that The Names of simple Ideas, and those only, are incapable of being defined. Theophilus. In my little essay on ideas, published in the Actes de Leipzic approximately twenty years ago, I have also stated that simple terms cannot have nominal definitions: But I added at the same time that these terms, when they are simple only in our eyes (because we do not have the means of analysing them into the elementary perceptions of which they are composed), like hot, cold, yellow, green can be given a real definition which explains their cause: thus the real definition of green is to be composed of a mixture of blue and yellow, even though green is no more capable of being defined nominally than blue or yellow. But terms which are absolutely simple, i.e. whose conception is clear and distinct, cannot have either nominal or real definitions. You will find in my little essay in the Actes de Leipzic the basis for a substantial part of the doctrine about the understanding presented here in abbreviated form. Philalethes. (It is useful to explain this and to point out what can and what cannot be defined. And I am inclined to believe that this frequently leads to serious disputations and to much nonsense in the discourse of Men who are not mindful of this.) § 8 The not observing this difference in our Ideas, and their Names, has produced that eminent trifling in the Schools, which is so easy to be observed, in the definitions they give us of some few of these simple Ideas. For as to the greatest part of them, even those Masters of Definitions, were fain to leave them untouch’d, merely by the impossibility they found in it. What more

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exquisite Jargon could the Wit of Man invent, than this Definition, The Act of a being in Power, as far forth as in Power. § 9 Nor have the Modern Philosophers, who define Motion to be a passage from one place to another, What do they more than put one synonymous Word for another? Theophilus. In one of our previous exchanges I have already remarked that in your country some ideas are considered to be simple when in fact they are not: movement is of this type and I believe it to be definable, and the definition which says that it is a change of place is not to be despised. Aristotle’s definition is not as absurd as some people think, when it is understood that the Greek ~iÈnhÈsi| did not in his time mean what we understand by movement, but rather what we express by the word change, which explains why he gives it such an abstract and metaphysical definition, so that what we call movement was for Aristotle jo}aÌ, latio, and is found as a species of change. § 10 Philalethes. (But surely you will not excuse) the same author’s definition of light as the act of perspicuous, as far forth as perspicuous. Theophilus. Like you I find it quite useless; he uses ‘act’ too often which does not tell us very much. For him, diaphane is a medium through which one might see, and according to him, light is what consists of the actual course. § 11 Philalethes. ( We agree therefore, that simple Ideas cannot have nominal definitions, just as we cannot know the taste of pine-apple by the account of travellers, unless we have the skill of Sancho Panza who could see Dulcinea by hearsay, or like the blind man who has often heard of the clash of scarlet, thought that it should be like the sound of a trumpet.) Theophilus. You are right and all the travellers’ accounts of the world could not have given us what we owe a gentleman of this country who has successfully grown pineapples at three leagues from Hanover almost on the banks of the Weser, and who has discovered means of propagating them so that we may possibly have them in such abundance as oranges from Portugal, even though there seems to be some loss of taste.

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Chapter XI: Of the Remedies of the foregoing Imperfections and Abuses § 17 Philalethes. Since a Definition is the only way, whereby the precise Meaning of moral Words can be known; It is far easier for Men to frame in their minds an Idea which shall be the Standard to which they will give the Name Justice, than, having seen Aristides, to frame an Idea that shall in all things be exactly like him who is as he is. § 18 For the Ideas they stand for, being for the most part such, whose component Parts no where exist together, (they can only be fixed by definition from what is) scattered and mingled with others. § 19 Thirdly, For the explaining the signification of the Names of Substances, as they stand for the Ideas we have their distinct Species, both the fore-mentioned ways, viz. of shewing and defining, are requisite in many cases, to be made use of. For there being ordinarily in each Sort some leading Qualities, to which we suppose the other Ideas, which make up pure complex Idea of that Species, annexed, we forwardly give the specifick Name to that Thing, wherein that characteristical Mark is found, which we take to be the most distinguishing Idea of that Species. These leading or characteristical (as I may call them) Ideas, in the sorts of Animals and Vegetables is mostly Figure, and in inanimate Bodies Colour, and in some both together. § 20 And therefore why Plato’s animal in plume bipes latis unguibus, should not be as good a Definition of the Name Man, standing for that sort of Creatures, will not be easy to shew: And if this be not allow’d to be so, I do not know how they can be excused from murther, who kill monstrous Births (as we call them,) because of an unordinary Shape, without knowing whether they have a Rational Soul or no, which can be no more discerned in a well-formed than ill-shaped Infant as soon as born. § 21 Now these leading Qualities are best made known by shewing, as is evident in those who are used to Gold, who will frequently distinguish true from counterfeit, pure from adulterate, by the sight. Theophilus. It all depends on the definitions which can proceed up to simple ideas. The same thing can have several definitions but in order to know which are suitable, we have to learn by reason, demonstrating one definition by another… I am, however, surprised that you are returning to the definition of man attributed to Plato, since you have stated yourself that as to morals, man has to be taken as a corporal and rational being without consideration of his external shape.

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BOOK IV: OF KNOWLEDGE AND OPINION Chapter VIII: Of trifling propositions § 12 Philalethes. All Propositions, wherein two abstract Terms are affirmed one of another are barely about the signification of Sounds. For since no abstract Idea can be the same with any other but it self, when its abstract Name is affirmed of any other Term, it can signify no more but this: that it may, or ought to be called by that Name, or that these two Names signify the same Idea. Thus, should anyone say that Parsimony is Frugality, that Gratitude is Justice: however specious these and the like Propositions may at first sight seem, yet when we come to press them and examine nicely what they contain, we shall find that it all amounts to nothing but the signification of those Terms. Theophilus. But the meanings of terms, i.e. the definitions together with the identical axioms express the principles of all demonstrations: and as these definitions can make known both the ideas and their possibility, it is obvious that what depends on it is not always purely verbal. Regarding your example, that Gratitude is Justice, or rather a part of Justice, it is not to be despised, because it shows that what is called actio ingrati or the complaint one can make of ingrates, should be less neglected by the courts of justice. But, I have already said elsewhere that abstract ideas can be attributed both to the genus and the species, as in: duration is a continuity, virtue is a habit: but universal justice is not only a virtue, but the whole of moral virtue.

George Berkeley

A Treatise concerning the Principles of Human Knowledge (Extract) Introduction 18. I come now to consider the source of this prevailing notion, and that seems to me to be language. And surely nothing of less extent than reason itself could have been the source of an opinion so universally received. The truth of this appears as from other reasons so also from the plain confession of the ablest patrons of abstract ideas, who acknowledge that they are made in order to permit naming; from which it is clear consequence that if there had been no such thing as speech or universal signs, there never had been any thought of abstraction. See B.iii. ch.6 §39, and elsewhere of the Essay on Human Understanding.1 Let us examine the manner wherein Words have contributed to the origins of that mistake. First then, it is thought that every name has, or ought to have, one only precise and settled signification; which inclines men to think there are certain abstract determinate ideas that constitute the true and only immediate signification of each general name; and that it is by the mediation of these abstract ideas that a general name comes to signify any particular thing. Whereas, in truth, there is no such thing as one precise and definite signification annexed to any general name, they all signifying indifferently a great number of particular ideas. All which does evidently follow from what has been already

1. In difference to Locke, whose Essay on Human Understanding he refers to, Berkeley considers all ideas to be particular. He believes that we rise above particular ideas by an intellectual apprehension of their relations and not by forming ‘abstract pictures’, which he considered to be contradictory absurdities.

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said, and will clearly appear to any one by a little reflexion. To this it will be objected that every name that has a definition is thereby restrained to one certain signification. For example, a triangle is defined to be ‘a plain surface comprehended by three right lines’; by which that name is limited to denote one certain idea and no other. To which I answer, that in the definition it is not said whether the surface be great or small, black or white, nor whether the sides are long or short, equal or unequal, nor with what angles they are inclined to each other; in all which there may be great variety, and consequently there is not one settled idea which limits the signification of the word triangle. It is one thing for to keep a name constantly to the same definition, and another to make it stand everywhere for the same idea: The one is necessary, the other useless and impracticable. 19. But, to give a farther account how words came to produce the doctrine of abstract ideas, it must be observed that it is a received opinion that language has no other end but the communicating of ideas, and that every significant name stands for an idea. This being so, and it being withal certain that names which yet are not thought altogether insignificant do not always mark out particular conceivable ideas, it is straightaway concluded that they stand for abstract notions. That there are many names in use amongst speculative men which do not always suggest to others determinate, particular ideas, or in truth anything at all, is what nobody will deny. And a little attention will discover that it is not necessary (even in the strictest reasonings) that significant names which stand for ideas should, every time they are used, excite in the understanding the ideas they are made to stand for: in reading and discoursing, names being for the most part used as letters are in Algebra, in which, though a particular quantity be marked by each letter, yet to proceed right it is not requisite that in every step each letter suggests to your thoughts that particular quantity it was appointed to stand for. 20. Besides, the communicating of ideas marked by words is not the chief and only end of language, as is commonly supposed. There are other ends, as the raising of some passion, the exciting to or deterring from an action, the putting the mind in some particular disposition; to which the former is in many cases barely subservient, and sometimes entirely omitted, when these can be obtained without it, as I think doth not unfrequently happen in the familiar use of language. I entreat the reader to reflect with himself, and see if it doth not often happen, either in hearing or reading a discourse, that the passion of fear, love, hatred, admiration and disdain, and the like, arise immediately in his mind upon the perception of certain words, without any ideas coming between them. At first, indeed, the words might have occasioned ideas that were fitting to produce

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those emotions; but, if I mistake not, it will be found that, when language is once grown familiar, the hearing of the sounds or sight of the characters is oft immediately attended with those passions which at first were wont to be produced by the intervention of ideas that are now quite omitted. May we not, for example, be affected with the promise of a good thing, though we have not an idea of what it is? Or is not the being threatened with danger sufficient to excite a dread, though we think not of any particular evil likely to befall us, not yet frame to ourselves an idea of danger in abstract? If any one shall join ever so little reflection of his own to what has been said, I believe that it will evidently appear to him that general names are often used in the propriety of language without the speakers designing them for marks of ideas in his own, which he would have them raise in the mind of the hearer. Even proper names themselves do not seem always spoken with a design to bring into our view the ideas of those individuals that are supposed to be marked by them. For example, when a schoolman tells me ‘Aristotle hath said it’, all I conceive he means by it is to dispose me to embrace his opinion with the deference and submission which custom has annexed to that name. And this effect may be so instantly produced in the minds of those who are accustomed to resign their judgment to authority of that philosopher, as it is impossible any idea either of his person, writings, or reputation should go before. So close and immediate a connexion may custom establish betwixt the very word Aristotle and the motions of assent and reverence in the minds of some men. Innumerable examples of this kind may be given, but why should I insist on those things which every one’s experience will, I doubt not, plentifully suggest unto him? 21. We have I think, shewn the impossibility of Abstract Ideas. We have considered what has been said for them by their ablest patrons; and endeavoured to shew they are of no use for those ends to which they are thought necessary. And lastly, we have traced them to the source from whence they flow, which appears evidently to be Language. It cannot be denied that words are of excellent use, in that by their means all that stock of knowledge which has been purchased by the joint labours of inquisitive men in all ages and nations may be drawn into the view and made the possession of one single person. But at the same time it must be owned that most parts of knowledge have been so strangely perplexed and darkened by the abuse of words, and general ways of speech wherein they are delivered, that it may almost be made a question whether language has contributed more to the hindrance or advancement of the sciences…

Immanuel Kant

The only possible argument in support of a demonstration of the existence of God (Extract)

[2:70] SECTION I: IN WHICH IS FURNISHED THE ARGUMENT IN SUPPORT OF A DEMONSTRATION OF THE EXISTENCE OF GOD First reflection: Of existence in general Even in the profoundest of treatises, the rule of thoroughness does not always demand that every concept employed should be developed or defined. No such requirement exists, namely, if one is assured that the clear and ordinary concept by itself can occasion no misunderstanding in the context in which it is employed. Such is the case with the geometer who with the greatest certainty uncovers the most secret properties and relations of that which is extended, even though in doing so he merely makes use of the ordinary concept of space. And such is also the case in the deepest science of all, where the word ‘representation’ in understood with sufficient precision and employed with confidence, even though its meaning can never be analysed by means of definition. Hence in these reflections I should not aspire to analyse the very simple and well-understood concept of existence, were it not for the fact that the present case is one in which such omission could occasion confusion and lead to serious errors. It is certain that anywhere else in philosophy the concept could confidently be employed in the undeveloped form in which it occurs in ordinary usage. The one exception is [2:71] the question concerning absolutely necessary

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existence and contingent existence. In this one case, an investigation of a subtler sort has drawn erroneous conclusions from an unhappily contrived but otherwise very pure concept. These erroneous conclusions have extended themselves over one of the most sublime parts of philosophy. It is not to be expected that I shall begin by offering a formal definition of existence. Such a procedure is always undesirable when the correctness of the suggested definition is so uncertain. This situation arises more frequently than one perhaps realises. My procedure will be like that of someone who is searching for a definition and who first of all assures himself of what can be said with certainty, either affirmatively or negatively about the object of the definition, even though he has not yet established the concept of the object in detail. Long before one ventures a definition of one’s object, and even when one lacks the courage to offer a definition at all, there is still a great deal which can be asserted with the highest degree of certainty about the object in question. I doubt whether anyone has even correctly defined what space is. But, without getting involved in such a definition, I am certain that where space exists external relations must also exist, that it cannot have more than three dimensions, and so on. Whatever a desire may be, it is based upon some representation or other, it presupposes pleasure in the object of the desire, and so on. From that which is known with certainty and prior to the definition of a thing, it is frequently possible to infer with complete certainty that which is relevant to the purpose of our investigation. To aspire to a definition is to venture upon unnecessary difficulties. The mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned against them, or that they will learn to be more circumspect as a result. By this method alone I hope to arrive at the enlightenment which I have vainly sought in others. As for the flattering idea that one’s greater perspicacity will secure one the success which has been denied to others: it is well to remember that this has always been the style of those whose wish it has been to lead us from the errors made by others to errors of their own devising.

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Inquiry concerning the distinctness of the principles of natural theology and morality (Extracts)

[2:273] First reflection: General comparison of the manner in which certainty is attained in mathematical cognition with the manner in which certainty is attained in philosophical cognition [2:279] § 3. In mathematics, unanalysable concepts and indemonstrable propositions are few in number, whereas in philosophy they are innumerable [2:281] … If a comparison were to be made between this (mathematics) and philosophy, and, in particular between this and metaphysics, I should like to see drawn up a table of the indemonstrable propositions which lie at the foundation of these sciences throughout their whole extent. Such a table would constitute a scheme of immeasurable scope. But the most important business of higher philosophy consists in seeking out these indemonstrable fundamental truths; and the discovery of such truths will never cease as long as cognition of such a kind as this continues to grow. For, no matter what the object may be, those characteristic marks, which the understanding initially and immediately perceives in the object, constitute the data for exactly the same number of indemonstrable propositions, which then form the foundation on the basis of which definitions can then be drawn up. Before I set about the task of defining what space is, I clearly see that since this concept is given to me, I must first of all, by analysing it, seek out those characteristic marks which are initially and immediately thought in that concept. Adopting this approach, I notice that there is a manifold in space of which the parts are external to each other; I notice that this manifold is not constructed by substances, for the cognition I wish to acquire relates not to things in space but to space itself; and I notice that space can only have three dimensions etc. Propositions such as these can well be explained if they are examined in concreto so that they become cognised intuitively; but they can never be proved. For on what basis could such a proof be constructed, granted that these propositions constitute the first and simplest thoughts I can have of my object, when I first call it to mind? In mathematics, the definitions are the first thought which I can entertain of the thing defined, for my concept of the object only comes into existence as a result of the definition. It is, therefore, absolutely absurd to regard the definitions as capable of proof. In philosophy, where the

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concept of the thing to be [2:282] defined is given to me, that which is initially and immediately perceived in it must serve as an indemonstrable fundamental judgment. For since I do not yet possess a complete and distinct concept of the thing, but am only now beginning to look for such a concept, it follows that the fundamental judgment cannot be proved by reference to this concept. On the contrary, such a judgment serves to generate this distinct cognition and to produce the definition of the things under examination. And here the only error which can occur beforehand is that of mistaking a derivative characteristic mark for one which is primary and fundamental. The following reflection will contain some considerations which will put this claim beyond doubt. …

[2:283] Second reflection: The only method for attaining the highest possible degree of certainty in metaphysics … In mathematics I begin with the definition of my object, for example of a triangle, or a circle, or whatever. In metaphysics I may never begin with a definition. Far from being the first thing I know about the object, the definition is nearly always the last thing I come to know. In mathematics, namely, I have no concept of my object at all until it is furnished by the definition. In metaphysics, I have a concept which is already given to me, although it is a confused one. My task is to search for the complete, distinct and determinate concept. How then am I to begin? Augustine said: ‘I know perfectly well what time is, but if somebody asks me what is it I do not [2:284] know. In such a case as this, many operations have to be performed in unfolding obscure ideas, in comparing them with each other, in subordinating them to each other and in limiting them by each other. And I would go as far as to say that, although much that is true and much that is penetrating has been said about time, nonetheless no real definition has ever been given of time. For, as far as the nominal definition is concerned, it is of little or no use to us, for even without the nominal definition the word is understood well enough not to be misused. If we had as many correct definitions of time as there are definitions to be found in the books devoted to the subject, with what certainty could inferences be made and conclusions drawn. But experience teaches us the opposite. In philosophy and in particular in metaphysics, one can often come to know a great deal about an object with distinctness and certainty, and even establish reliable conclusions on that basis prior to having a definition of that object, and

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even, indeed, when one had no intention of furnishing one. In the case of any particular thing, I can be immediately certain about a number of predicates, even though I am not acquainted with a sufficiently large number of them to be able to furnish a completely determinate concept of the thing, in other words, a definition. Even if I had never defined what an appetite was, I should still be able to say with certainty that every appetite presupposed the representation of the object of the appetite; that this representation was an anticipation of what was to come in the future; that the feeling of pleasure was connected with it; and so forth. Everyone is constantly aware of all this in the immediate consciousness of appetite. One might perhaps eventually be able to arrive at a definition of appetite on the basis of such remarks as these, once they had been compared with each other. But as long as it is possible to establish what one is seeking by inference from a few immediately certain characteristic marks of the thing in question, and to do so without a definition, there is no need to venture on an undertaking which is so precarious. In mathematics, as is known, the situation is completely different. In mathematics the significance of the signs employed is certain, for it is not difficult to know what the significance was which one wished to attribute to those signs. In philosophy generally and in metaphysics in particular, words acquire their meaning as a result of linguistic usage, unless, that is, the meaning has been more precisely determined by means of a logical limitation. But it frequently happens that the same words are employed for concepts which, while very similar, nonetheless conceal within themselves considerable differences. For this reason, whenever such a concept is applied, even though one’s terminology may seem to be fully sanctioned by linguistic usage, one must still pay careful attention to whether it is really the same concept which is connected here with the same sign. We say that a person distinguishes gold from brass if, for example, he recognises that the density to be found in the one metal is not to be found in the other. We also say that an animal distinguishes one kind of provender from another it if eats the one and leaves the other untouched. Here, the word ‘distinguishes’ is being used in both cases even though, in the first case, it means ‘recognise the difference’, which is something which can never occur without judging, whereas in the second case it merely signifies that different actions are performed when different representations are present, and in this case it is not necessary that a judgment should occur. All that we perceive in the case of the animal is that it is impelled to perform direct actions by different sensations; and that is something which is perfectly possible without in the least needing to make a judgment about similarity or difference. From all this there follow quite naturally the rules which govern the method

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by which alone the highest possible degree of metaphysical certainty can be attained. These rules are quite different from those which have hitherto been followed. They promise, if they are adopted, to produce a happier outcome that could ever have been expected on a different path. The first and most important rule is this: one ought not to start with definitions, unless that is, one is merely seeking a nominal definition, such as, for example, the definition: that of which the opposite is impossible is necessary. But even then there are only a few cases where one can confidently establish a distinctly determinate concept right at the very beginning. One ought, rather, begin by carefully searching out what is immediately certain in one’s object, even before one has its definition. Having established what is immediately certain in the object of one’s enquiry, one then proceeds to draw conclusions from it. One’s chief concern will be to arrive at judgments about the object which are true and completely certain. And in doing this, one will not make an elaborate parade of one’s hope of arriving at a definition. Indeed, one will never venture to offer such a definition, until one has to concede the definition, once it has presented itself on the basis of the most certain judgments. …

2:290 Third reflection: On the nature of metaphysical certainty § 1. Philosophical certainty is altogether different in nature from mathematical certainty One is certain if one knows that it is impossible that a cognition should be false. The degree of this certainty, [2:291] taken objectively, depends upon the sufficiency in the characteristic marks of the necessity of a truth. But taken subjectively, the degree of certainty increases with the degree of intuition to be found in the cognition of this necessity. In both respects, mathematical certainty is of as different kind to philosophical certainty. I shall demonstrate this with the greatest possible clarity. The human understanding, like any other force of nature, is governed by certain rules. Mistakes are made, not because the understanding combines concepts without rule, but because the characteristic mark which is not perceived in a thing is actually denied of it. One judges that that of which one is not conscious in a thing does not exist. Now, firstly, mathematics arrives at its concepts synthetically; it can say with certainty that what it did not intend to represent in the object by means of the definition is not contained in that object. For the concept of what has been defined only comes into existence by means of

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the definition; the concept has no other significance at all apart from that which is given to it by the definition. Compared with this, philosophy and particularly metaphysics are a great deal more uncertain in their definitions, should they venture to offer any. For the concept of that which is to be defined is given. Now, if one should fail to notice some characteristic mark or other, which nonetheless belongs to the adequate distinguishing of the concept in question, and if one judges that no such characteristic mark belongs to the complete concept, then the definition will be wrong and misleading. Numberless examples of such errors could be adduced, and for that very reason I refer only to the above example of touching. Secondly, mathematics, in its inferences and proofs, regards its universal knowledge under signs in concreto, whereas philosophy always regards its universal knowledge in abstracto, as existing alongside signs. And this constitutes a substantial difference in the way in which the two enquiries attain to certainty. For since signs in mathematics are sensible means to cognition, it follows that one can know that no concept has been overlooked, and that each particular comparison has been drawn in accordance with easily observed rules etc. And these things can be known with the degree of assurance characteristic of seeing something with one’s own eyes. And in this, the attention is considerably facilitated by the fact that it does not have to think things in their universal representation; it has rather to think the signs as they occur in their particular cognition which, in this case, is sensible in character. By contrast, the only help which words, construed as the signs of philosophical cognition, afford is that of reminding us of the universal concepts which they signify. It is at all times necessary to be immediately aware of their significance. The pure understanding must be maintained in a state of constant attention; how easy it is for the characteristic mark of an abstracted concept to escape our attention without our noticing, for there is nothing sensible which can reveal to us the fact that the characteristic mark has been overlooked. And when that happens, different things are taken to be the same thing, and the result is error. … [2:292] § 2. Metaphysics is capable of a certainty which is sufficient to produce conviction … Errors do not arise simply because we do not know certain things. We make mistakes because we venture to make judgments, even though we do not know everything which is necessary for doing so. A large number of errors, indeed almost all of them, are due to this latter kind of overhastiness. You have certain [2:293] knowledge of some of the predicates of a thing. Very well! Base your

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conclusions on this certain knowledge and you will not go wrong. But you insist on having a definition at all costs. And yet you are not sure that you know everything that is necessary for drawing up such a definition; nonetheless, you venture on such an undertaking and thus you fall into error. It is therefore possible to avoid errors, provided that one seeks out cognitions which are certain and distinct, and provided that one does not so lightly lay claim to be able to furnish definitions. Furthermore, you could also establish a substantial part of an indubitable conclusion, and do so with certainty; but do not, on any account, permit yourself to draw the whole conclusion, no matter how slight the difference may appear to be. … § 3. The certainty of the first fundamental truths of metaphysics is not of a kind different from that of any other cognition, apart from mathematics … [2:294] The proposition, a body is divisible, is demonstrable, for the identity of the predicate and the subject can be shown by analysis and therefore indirectly: a body is compound, but what is compound is divisible, so a body is divisible. The intermediate characteristic mark here is being compound. Now, in philosophy there are, as we have [2:295] said above, many indemonstrable propositions. All these demonstrable propositions are subsumed under the formal first principles, albeit immediately. However, insofar as they also contain the grounds of other cognitions, they are also the first material principles of human reason. For example: a body is compound is an indemonstrable proposition, for the predicate can only be thought as an immediate and primary characteristic mark in the concept of a body. Such material principles constitute, as Crusius1 rightly says, the foundation of human reason and the guarantor of its stability. For, as we have mentioned above, they provide the stuff for definitions and, even when one is not in possession of a definition, the data from which conclusions can be reliably drawn. … Accordingly, metaphysics has no formal grounds of certainty which are different in kind from those of geometry. In both metaphysics and geometry, the formal element of the judgments exists in virtue of the laws of agreement and [2:296] contradiction. In both sciences, indemonstrable propositions constitute the foundation on the basis of which conclusions are drawn. But whereas in mathematics

1. Christian August Crusius (1715–1775), Professor of Theology in Leipzig, a philosopher who had a profound influence on Kant.

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the definitions are the first indemonstrable concepts of the thing defined, in metaphysics, the place of these definitions is taken by a number of indemonstrable propositions which provide the primary data. Their certainty may be just as great as that of the definitions of geometry. They are responsible for furnishing either the stuff, from which the definitions are formed, or the foundation, on the basis of which reliable conclusions are drawn. Metaphysics is as much capable of the certainty which is necessary to produce convictions as mathematics. The only difference is that mathematics is easier and more intuitive in character.

Editor’s note The numbers in square brackets refer to the pagination of the standard edition of these works, i.e. Vols. I and II of Immanuel Kants Gesammelte Schriften, Berlin: Akademie der Wissenschaften.

John Stuart Mill

A System of Logic, Ratiocinative and Inductive (Extract) BOOK I: OF NAMES AND PROPOSITIONS Chapter viii. Of definition § 1. [A definition, what] One necessary part of the theory of Names and of Propositions remains to be treated of in this place: the theory of Definitions. As being the most important of the class of propositions which we have characterized as purely verbal, they have already received some notice in the chapter preceding the last. But their fuller treatment was at that point postponed, because definition is so closely connected with classification, that, until the nature of the latter process is in some measure understood, the former cannot be discussed to much purpose. The simplest and most correct notion of a Definition is, a proposition declaratory of the meaning of a word; namely, either the meaning which it bears in common acceptation, or that which the speaker or writer, for the particular purposes of his discourse, intends to annex to it. The definition of a word being the proposition which enunciates its meaning, words which have no meaning are unsusceptible of definition. Proper names, therefore, cannot be defined. A proper name being a mere mark put upon an individual, and of which it is the characteristic property to be destitute of meaning, its meaning cannot of course be declared; though we may indicate by language, as we might indicate still more conveniently by pointing with the finger, upon what individual that particular mark has been, or is intended to be, put. It is no definition of “John Thomson” to say he is “the son of General

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Thomson”; for the name John Thomson does not express this. Neither it is any definition of “John Thomson” to say he is “the man now crossing the street”. These propositions may serve to make known who is the particular man to whom the name belongs, but that may be done still more unambiguously by pointing to him, which, however, has not been esteemed one of the modes of definition. In the case of connotative names, the meaning, as has been so often observed, is the connotation; and the definition of a connotative name, is the proposition which declares its connotation. This might be done either directly or indirectly. The direct mode would be by a proposition in this form: “Man” (or whatsoever the word may be) “is a name connoting such and such attributes”, or “is a name which, when predicated of anything, signifies the possession of such and such attributes by that thing”. Or thus: Man is everything which possesses such and such attributes: Man is everything which possesses corporeity, organization, life, rationality, and certain peculiarities of external form. This form of definition is the most precise and least equivocal of any; but it is not brief enough, and is besides too technical for common discourse. The more usual mode of declaring the connotation of a name, is to predicate of it another name or names of known signification, which connote the same aggregation of attributes. This may be done either by predicating of the name intended to be defined, another connotative name exactly synonymous, as, “Man is a human being,” which is not commonly accounted a definition at all; or by predicating two or more connotative names, which make up among them the whole connotation of the name to be defined. In this last case, again, we may either compose our definition of as many connotative names as there are attributes, each attribute being connoted by one, as, Man is a corporeal, organized, animated, rational being, shaped so and so; or we may employ names which connote several of the attributes at once, as, Man is a rational animal, shaped so and so. The definition of a name, according to this view of it, is the sum total of the essential propositions which can be framed with that name for their subject. All propositions the truth of which is implied in the name, all those which we are made aware of by merely hearing the name, are included in the definition, if complete, and may be evolved from it without the aid of any other premises; whether the definition expresses them in two or three words, or in a larger number. It is, therefore, not without reason that Condillac and other writers have affirmed a definition to be an analysis. To resolve any complex whole into the elements of which it is compounded, is the meaning of analysis: and this we do when we replace one word which connotes a set of attributes collectively, by two or more which connote the same attributes singly, or in smaller groups.

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§ 2. [Every name can be defined, whose meaning is susceptible of analysis] From this, however, the question naturally arises, in what manner are we to define a name which connotes only a single attribute: for instance, “white”, which connotes nothing but whiteness; “rational”, which connotes nothing but the possession of reason. It might seem that the meaning of such names could only be declared in two ways; by a synonymous term, if any such can be found; or in the direct way already alluded to: “White is a name connoting the attribute whiteness.” Let us see, however, whether the analysis of the meaning of the name, that is, the breaking down of that meaning into several parts, admits of being carried farther. Without at present deciding this question as to the word white, it is obvious that in the case of rational some further explanation may be given of its meaning than is contained in the proposition, “Rational is that which possesses the attribute of reason”; since the attribute reason itself admits of being defined. And here we must turn our attention to the definitions of the attributes, or rather of the names of attributes, that is, of abstract names. In regard to such names of attributes as are connotative, and express attributes of those attributes, there is no difficulty: like other connotative names they are defined by declaring their connotation. Thus the word fault may be defined, “a quality productive of evil or inconvenience.” Sometimes, again, the attribute to be defined is not one attribute, but an union of several: we have only, therefore, to put together the names of all the attributes taken separately, and we obtain the definition of the name which belongs to them all taken together; a definition which will correspond exactly to that of the corresponding concrete name. For, as we define a concrete name by enumerating the attributes which it connotes, and as the attributes connoted by a concrete name form the entire signification of the corresponding abstract name, the same enumeration will serve for the definition of both. Thus, if the definition of a human being be this “a being, corporeal, animated, rational, shaped so and so,” the definition of humanity will be corporeity, and animal life, combined with rationality, and with such and such a shape. When, on the other hand, the abstract name does not express a complication of attributes, but a single attribute, we must remember that every attribute is grounded on some fact or phenomenon, from which, and which alone, it derives its meaning. To that fact or phenomenon, called in a former chapter the foundation of the attribute, we must, therefore, have resource for its definition. Now, the foundation of the attribute may be a phenomenon of any degree of complexity, consisting of many different parts, either coexistent or in succession. To obtain a definition of the attribute, we must analyze the phenomenon into these parts. Eloquence, for example, is the name of one attribute only; but this attribute

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is grounded on external effects of a complicated nature, flowing from the acts of the person to who we ascribe the attribute; and by resolving the phenomenon of causation into its two parts, the cause and the effect, we obtain a definition of eloquence, viz. the power of influencing feelings by speech or writing. A name, therefore, whether concrete or abstract, admits of definition, provided we are able to analyze, that is, to distinguish into parts, the attribute or set of attributes which constitute the meaning both of the concrete name and of the corresponding abstract: if a set of attributes, by enumerating them; if a single attribute, by dissecting the fact or phenomenon (whether of perception or of internal consciousness) which is the foundation of the attribute. But, further, even when the fact is one of our simple feelings or states of consciousness, and therefore unsusceptible of analysis, the names both of the object and the attribute still admit of definition: or rather, would do so if all our simple feelings had names. Whiteness may be defined, the property or power of exciting the sensation of white. A white object may be defined, an object which excites the sensation of white. The only names which are unsusceptible of definition, because their meaning is unsusceptible of analysis, are the names of the simple feelings themselves. These are in the same condition as proper names. They are not indeed, like proper names, unmeaning; for the words sensation of white signify that the sensation which I so denominate resembles other sensations which I remember to have had before, and to have called by that name. But as we have no words by which to recall those former sensations, except the very word which we seek to define, or some other which, being exactly synonymous with it, requires definition as much, words cannot unfold the signification of this class of names; and we are obliged to make a direct appeal to the personal experience of the individual whom we address. § 3. [Complete, how distinguished from incomplete definitions] Having stated what seems to be the true idea of a Definition, I proceed to examine some opinions of philosophers, and some popular conceptions of the subject, which conflict more or less with that idea. The only adequate definition of a name is, as already remarked, one which declares the facts, and the whole of the facts, which the name involves in its signification. But with most persons the object of the definition does embrace so much; they look for nothing more, in a definition, than a guide to the correct use of the term — a protection against applying it in a manner inconsistent with custom and convention. Anything, therefore, is to them a sufficient definition of a term, which will serve as a correct index to what the term denotes; though not embracing the whole, and sometimes, perhaps, not even any part, of what it

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connotes. This gives rise to two sorts of imperfect, or unscientific definitions; Essential but incomplete Definitions, and Accidental Definitions, or Descriptions. In the former, a connotative name is defined by a part only of its connotation; in the later, by something which forms no part of the connotation at all. An example of the first kind of imperfect definition is the following: Man is a rational animal. It is impossible to consider this as a complete definition of the word Man, since, if we adhered to it, we should be obliged to call the Houyhnhnms men; but as there happen to be no Houyhnhnms, this imperfect definition is sufficient to mark out and distinguish from all other things, the objects at present denoted by “man”; all the beings actually known to exist, of whom the name is predicable. Though the word is defined by some only among the attributes which it connotes, not by all, it happens that all known objects which possess the enumerated attributes, possess also those which are omitted; so that the field of predication which the word covers, and the employment of it which is conformable to usage, are as well indicated by the inadequate definition as by an adequate one. Such definitions, however, are always liable to be overthrown by the discovery of new objects in nature. Definitions of this kind are what logicians have had in view, when they laid down the rule, that the definition of a species should be per genus et differentiam. Differentia being seldom taken to mean the whole of the peculiarities constitutive of the species, but some one of those peculiarities only, a complete definition would be per genus et differentiae rather than differentiam. It would include, with the name of the superior genus, not merely some attribute which distinguishes the species intended to be defined from all other species of the same genus, but all the attributes implied in the name of the species, which the name of the superior genus has not already implied. The assertion, however, that a definition must of necessity consist of a genus and differentiae, is not tenable. It was early remarked by logicians, that the summum genus in any classification, having no genus superior to itself, could not be defined in this manner. Yet we have seen that all names, except those of our elementary feelings, are susceptible of definition in the strictest sense; by setting forth in words the constituent parts of the fact or phenomenon, of which the connotation of every word is ultimately composed. § 4. [And how complete definitions are distinguished from descriptions] Although the first kind of imperfect definition, (which defines a connotative term by a part only of what it connotes, but a part sufficient to mark out correctly the boundaries of its denotation) has been considered by the ancients, and by logicians in general, as a complete definition; it has always been deemed necessary that the attributes employed should really form part of the connotation; for the rule was

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that the definition must be drawn from the essence of the class; and this would not have been the case if it had been in any degree made up of attributes not connoted by the name. The second type of imperfect definition, therefore, in which the name of a class is defined by any of its accidents — that is, by attributes which are not included in its connotation — has been rejected from the rank of the genuine Definition by all logicians, and has been termed description. This kind of imperfect definition, however, takes its rise from the same cause as the other, namely, the willingness to accept as a definition anything which, whether it expounds the meaning of the name or not, enables us to discriminate the things denoted by the name from all other things, and consequently to employ the term in predication without deviating from established usage. This purpose is duly answered by stating any (no matter what) of the attributes which are common to the whole of the class, and peculiar to it; or any combination of attributes which happens to be peculiar to it, though separately each of these attributes may be common to it with some other things. It is only necessary that the definition (or description) thus formed, should be convertible with the name which it professes to define; that is, should be exactly co-extensive with it, being predicable of everything of which it is predicable, and of nothing of which it is not predicable; though the attributes specified may have no connection with those which mankind had in view when they formed or recognised the class, and gave it a name. The following are correct definitions of Man, according to this test: Man is a mammiferous animal, having (by nature) two hands (for the human species answers to this description, and no other animal does): Man is an animal who cooks his food: Man is a featherless biped. What would otherwise be a mere description, may be raised to the rank of a real definition by the peculiar purpose which the speaker of writer has in view. As was seen in the preceding chapter, it may, for the ends of a particular art or science, or for the more convenient statement of an author’s particular doctrines, be advisable to give some general name, without altering its denotation, a special connotation, different from its ordinary one. When this is done, a definition of the name by means of the attributes which make up the special connotation, though in general a mere accidental definition or description, becomes on the particular occasion and for the particular purpose a complete and genuine definition. This actually occurs with respect to one of the preceding examples, “Man is a mammiferous animal having two hands,” which is the scientific definition of man, considered as one of the species in Cuvier’s distribution of the animal kingdom. In cases of this sort, though the definition is still a declaration of the meaning which in the particular instance the name is appointed to convey, it

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cannot be said that to state the meaning of the word is the purpose of the definition. The purpose is not to expound a name, but a classification. The special meaning which Cuvier assigned to the word Man, (quite foreign to its ordinary meaning, though involving no change in the denotation of the word), was incidental to a plan of arranging animals into classes on a certain principle, that is, according to a certain set of distinctions. And since the definition of Man according to the ordinary connotation of the word, though it would have answered every other purpose of a definition, would not have pointed out the place which the species ought to occupy in that particular classification; he gave the word a special connotation, that he might be able to define it by the kind of attributes on which, for reasons of scientific convenience, he had resolved to found his division of animated nature. Scientific definitions, whether they are definitions of scientific terms, or of common terms used in a scientific sense, are almost always of the kind last spoken of: their main purpose is to serve as the landmarks of scientific classification. And since the classifications in any science are continually modified as scientific knowledge advances, the definitions in the sciences are also constantly varying. A striking instance is afforded by the words Acid and Alkali, especially the former. As experimental discovery advanced, the substances classed with acids have been constantly multiplying, and by a natural consequence the attributes connoted by the word have receded and become fewer. At first it connoted the attributes, of combining with an alkali to form a neutral substance (called a salt); being compounded of a base and oxygen; causticity to the taste and touch; fluidity, &c. The true analysis of muriatic acid, into chlorine and hydrogen, caused the second property, composition from a base and oxygen, to be excluded from the connotation. The same discovery fixed the attention of chemists upon hydrogen as an important element in acids; and more recent discoveries having led to the recognition of its presence in sulphuric, nitric, and many other acids, where its existence was not previously suspected, there is now a tendency to include the presence of this element in the connotation of the word. But carbonic acid, silica, sulphurous acid, have no hydrogen in their composition; that property cannot therefore be connoted by the term, unless those substances are no longer to be considered acids. Causticity and fluidity have long since been excluded from the characteristics of the class, by the inclusion of silica and many other substances in it; and the formation of neutral bodies by combination with alkalis, together with such electro-chemical peculiarities as this is supposed to imply, are now the only differentiae which form the fixed connotation of the word Acid, as a term of chemical science. What is true of the definition of any term of science, is of course true of the

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definition of a science itself; and accordingly, the definition of a science must necessarily be progressive and provisional. Any extension of knowledge or alteration in the current opinions respecting the subject matter, may lead to a change more or less extensive in the particulars included in the science; and its composition being thus altered, it may easily happen that a different set of characteristics will be found better adapted as differentiae for defining its name. In the same manner in which a special or technical definition has for its object to expound the artificial classification out of which it grows; the Aristotelian logicians seem to have imagined that it was also the business of ordinary definition to expound the ordinary, and what they deem the natural, classification of things, namely, the division of them into Kinds; and to show the place which each Kind occupies, as superior, collateral, or subordinate, among other Kinds. This notion would account for the rule that all definition must necessarily be per genus et differentiam, and would explain why a single differentia was deemed sufficient. But to expound, or express in words, a distinction of Kind, has already been shown to be an impossibility: the very meaning of a Kind is, that the properties which distinguish it do not grow out of one another, and cannot therefore be set forth in words, even by implication, otherwise than by enumerating them all: and all are not known, nor are they ever likely to be so. It is idle, therefore, to look at this as one of the purposes of a definition: while, if it be only required that a definition of a Kind should indicate what kinds include it or are included by it, and definitions which expound the connotation of the names will do this: for the name of each class must necessarily connote enough of its properties to fix the boundaries of the class. If the definition, therefore, be a full statement of the connotation, it is all that a definition can be required to be. § 5. [What are called definitions of Things, are definitions of Names with an implied assumption of the existence of Things corresponding to them] Of the two incomplete and popular modes of definition, and in what they differ from the complete or philosophical mode, enough has now been said. We shall next examine an ancient doctrine, once generally prevalent and still by no means exploded, which I regard as the source of a great part of the obscurity which hanging over some of the most important processes of the understanding in the pursuit of truth. According to this, the definitions of which we have now treated are only one of two sorts into which definitions may be divided, viz. definitions of names, and definitions of things. The former are intended to explain the meaning of a term; the latter, the nature of a thing; the last being incomparably the more important.

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This opinion was held by the ancient philosophers, and by their followers, with the exception of the Nominalists; but as the spirit of modern metaphysics, until a recent period, has been on the whole a Nominalist spirit, the notion of definition of things has been to a certain extent in abeyance, still continuing, however, to breed confusion in logic, by its consequences indeed rather than by itself. Yet the doctrine in its own proper form now and then breaks out, and has appeared (among other places) where it was scarcely to be expected, in a justly admired work, Archbishop Whately’s Logic. In a review of that work, published by me in the Westminster Review for January 1828 and containing some opinions which I no longer entertain, I find the following observations on the question now before us; observations with which my present view of that question is still sufficiently in accordance. [The text omitted here is that quoted by the author from his own review article of the book cited.]

There is a real distinction, then, between definitions of names, and what are erroneously called definitions of things; but it is, that the latter, along with the meaning of a name, covertly asserts a matter of fact. This covert assertion is not a definition, but a postulate. The definition is a mere identical proposition, which gives information only about the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact, which may lead to consequences of every degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition; and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truth. We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of Geometry is deduced from definitions. This, so long as a definition was considered to be a proposition “unfolding the nature of the thing”, did well enough. But Hobbes followed, and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name; yet he continued to affirm as broadly as any of his predecessors, that the aÎr~aiÌ, principia, or original premises of mathematics, and even of all science, are definitions; producing the singular paradox, that systems of scientific truth, nay, all truths whatever at

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which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words. To save the credit of the doctrine that definitions are the premises of scientific knowledge, the proviso is sometimes added, that they are so only under a certain condition, namely, that they be framed conformably to the phenomena of nature; that is, that they ascribe such meanings to terms as shall suit objects actually existing. But this is only an instance of the attempt so often made, to escape from the necessity of abandoning old language after the ideas which it expresses have been exchanged for contrary ones. From the meaning of a name (we are told) it is possible to infer physical facts, provided the name has corresponding to it an existing thing. But if this proviso be necessary, from which of the two is the inference really drawn? From the existence of a thing having the properties, or from the existence of a name meaning them? Take, for instance, any of the definitions laid down as premises in Euclid’s Elements; the definition, let us say, of a circle. This, being analyzed, consists of two propositions; the one an assumption with respect to a matter of fact, the other a genuine definition. “A figure may exist, having all the points in the line which bounds it equally distant from a single point within it:” “Any figure possessing this property is called a circle.” Let us look at one of the demonstrations which are said to depend on this definition, and observe to which of the two propositions contained in it the demonstration really appeals. “About the centre A, describe the circle B C D.” Here is an assumption that a figure, such as the definition expresses, may be described; which is no other than the postulate, or covert assumption, involved in the so-called definition. But whether that figure be called a circle or not is quite immaterial. The purpose would be as well answered, in all respects except brevity, were we to say, “Through the point B, draw a line returning into itself, of which every point shall be at an equal distance from the point A.” By this the definition of a circle would be got rid of, and rendered needless; but not the postulate implied in it; without that the demonstration could not stand. The circle being now described, let us proceed to the consequence. “Since B C D is a circle, the radius B A is equal to the radius C A.” B A is equal to C A, not because B C D is a circle, but because B C D is a figure with the radii equal. Our warrant for assuming that such a figure about the centre A, with the radius B A, may be made to exist, is the postulate. Whether the admissibility of these postulates rests on intuition. or on proof, may be a matter of dispute; but in either case they are the premises on which the theorems depend; and while these are retained it would make no difference in the certainty of geometrical truths, though every definition in Euclid, and every technical terms therein defined, were laid aside.

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It is perhaps, superfluous to dwell at so much length on what is so nearly self-evident; but when a distinction, obvious as it may appear, has been confounded, and by powerful intellects, it is better to say too much than too little for the purpose of rendering such mistakes impossible in future. I will, therefore, detain the reader while I point out one of the absurd consequences flowing from the supposition that definitions, as such, are the premises in any of our reasonings, except such as relate to words only. If this supposition were true, we might argue correctly from true premises, and arrive at a false conclusion. We should only have to assume as a premise the definition of a nonentity; or rather of a name which has no entity corresponding to it. Let this, for instance, be our definition: A dragon is a serpent breathing flame. This proposition, considered only as a definition, is indisputably correct. A dragon is a serpent breathing flame: the word means that. The tacit assumption, indeed, (if there were any such understood assertion), of the existence of an object with properties corresponding to the definition, would, in the present instance, be false. Out of this definition we may carve the premises of the following syllogism: A dragon is a thing which breathes flame: A dragon is a serpent: From which the conclusion is, Therefore some serpent or serpents breathe flame: an unexceptionable syllogism in the first mode of the third figure, in which both premises are true and yet the conclusion false; which every logician knows to be an absurdity. The conclusion being false and the syllogism correct, the premises cannot be true. But the premises considered as parts of a definition are true. Therefore, the premises considered as parts of a definition cannot be the real ones. The real premises must be: A dragon is a really existing thing which breathes flame: A dragon is a really existing serpent: which implied premises being false, the falsity of the conclusion presents no absurdity. If we would determine what conclusion follows from the same ostensible premises when the tacit assumption of real existence is left out, let us, according to the recommendation in a previous page, substitute means for is. We then have:

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Dragon is a word meaning a thing which breathes flame: Dragon is a word meaning a serpent: From which the conclusion is, Some word or words which mean a serpent, also mean a thing which breathes flame: where the conclusion (as well as the premises) is true, and is the only kind of conclusion which can ever follow from a definition, namely, a proposition relating to the meaning of words. There is still another shape into which we may transform this syllogism. We may suppose the middle term to be the designation neither of a thing nor of a name, but of an idea. We then have: The idea of a dragon is an idea of a thing which breathes flame: The idea of a dragon is an idea of a serpent: Therefore, there is an idea of a serpent, which is an idea of a thing breathing flame. Here the conclusion is true, and also the premises; but the premises are not definitions, they are propositions affirming that an idea existing in the mind, includes certain ideal elements. The truth of the conclusion follows from the existence of the psychological phenomenon called the idea of a dragon; and therefore still from the tacit assumption of a matter of fact. When, as in this last syllogism, the conclusion is a proposition respecting an idea, the assumption on which it depends may be merely that of the existence of an idea. But when the conclusion is a proposition concerning a Thing, the postulate involved in the definition which stands as the apparent premise, is the existence of a thing conformable to the definition, and not merely of an idea conformable to it. This assumption of real existence will always convey the impression that we intend to make, when we profess to define any name which is already known to be a name of really existing objects. On this account it is, that the assumption was not necessarily implied in the definition of a dragon, while there was no doubt of its being included in the definition of a circle. § 6. [What are called definitions of Things are definitions of Names even when such Things do not in reality exist] One of the circumstances which have contributed to keep up the notion, that demonstrative truths follow from definitions rather than from the postulates implied in those definitions, is that the postulates, even in those sciences which are considered to surpass all others in demonstrative certainty, are not always exactly true. It is not true that a circle exists, or can be

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described, which has all ita radii exactly equal.Such accuracy is ideal only; it is not found in nature, still less can it be realised by art. People had a difficulty, therefore, in conceiving that the most certain of all conclusions could rest on premises which, instead of being certainly true, are certainly not true to the full extent asserted. This apparent paradox will be examined when we come to treat of Demonstration; where we shall be able to show that as much of the postulate is true, as is required to support as much as is true of the conclusion. Philosophers, however, to whom this view had not occurred, or whom it did not satisfy, have thought it indispensable that there should be found in definitions something more certain, or at least more accurately true, than the implied postulate of the real existence of a corresponding object. And this something they had flattered themselves they had found, when they laid it down that a definition is a statement and analysis not of the mere meaning of a word, nor yet of the nature of a thing, but an idea. Thus, the proposition “A circle is a plain figure bounded by a line all the points of which are at an equal distance from a given point within it,” was considered by them, not as an assertion that any real circle has that property, (which would not be exactly true,) but that we conceive of a circle as having it; that our abstract idea of a circle is an idea of a figure with its radii exactly equal. Conformable to this it is said, that the subject-matter of mathematics, and of every other demonstrative science, is not things as they really exist, but abstractions of the mind. A geometrical line is a line without breadth; but no such line exists in nature. The definition (it is said) is a definition of this mental line, not of any actual line: and it is only of the mental line, not of any line existing in nature, that the theorems of geometry are accurately true. Allowing this doctrine respecting the nature of demonstrative truth to be correct (which, in a subsequent place, I shall endeavour to prove that it is not) even on the supposition, the conclusions which seem to follow from a definition, do not follow from the definition as such, but from an implied postulate. Even if it be true that there is no object in nature answering to the definition of a line, and that the geometrical properties of lines are not true on any lines in nature, but only of the idea of a line; the definition, at all events, postulates the real existence of such an idea; it assumes that the mind can frame, or rather has framed, the notion of length without breadth, and without any other sensible property whatever. To me, indeed, it appears that the mind cannot form any such notion; it cannot conceive length without breadth; it can only, in contemplating objects, attend to their length, exclusively of their other sensible qualities, and so determine what properties may be predicated of them in virtue of their length alone. If this be true, the postulate involved in the geometrical definition of a

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line, is the real existence, not of length without breadth, but merely of length, that is, of long objects. This is quite enough to support all the truths of geometry, since every property of a geometrical line is really a property of all physical objects in so far as possessing length. But even what I hold to be the false doctrine on the subject, leaves the conclusion that our reasonings are grounded on the matters of fact postulated in definitions, and not on the definitions themselves, entirely unaffected; and accordingly this conclusion is one which I have in common with Dr. Whewell, in his Philosophy of the Inductive Sciences: though on the nature of demonstrative truth, Dr. Whewell’s opinions are greatly at variance with mine. And here, as in many other instances, I gladly acknowledge that his writings are eminently serviceable in clearing from confusion the initial steps in the analysis of the mental processes, even where his views respecting the ultimate analysis are such as (though with unfeigned respect) I cannot but regard as fundamentally erroneous. § 7. [Definitions, though of names only, must be grounded on knowledge of the corresponding things] Although, according to the opinion here presented, Definitions are properly of names only, and not of things, it does not follow from this that definitions are arbitrary. How to define a name, may not only be an inquiry of considerable difficulty and intricacy, but may involve considerations going deep into the nature of the things which are denoted by the name. Such, for instance, are the inquiries which form the subjects of the most important of Plato’s dialogues; as, “What is Rhetoric?” the topic of the “Gorgias,” or “What is Justice?” that of the “Republic.” Such, also, is the question scornfully asked by Pilate, “What is truth?” and the fundamental question with speculative moralists in all ages, “What is virtue?” It would be a mistake to represent these difficult and noble inquiries as having nothing in view beyond ascertaining the conventional meaning of a name. They are inquiries no so much to determine what is, as what should be, the meaning of a name; which, like other practical questions of terminology, requires for its solution that we should enter, and sometimes enter very deeply, into the properties not merely of names but of the things named. Although the meaning of every concrete general name resides in the attributes which it connotes, the objects were named before the attributes; as appears from the fact that in all languages, abstract names are mostly compounds or other derivatives of the concrete names which correspond to them. Connotative names, therefore, were, after proper names, the first which were used: and in the simpler cases, no doubt, a distinct connotation was present to the minds of those who first used the name, and was distinctly intended by them to be

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conveyed by it. The first person who used the word white, as applied to snow or to any other object, knew, no doubt, very well what quality he intended to predicate, and had a perfectly distinct conception in his mind of the attribute signified by the name. But where the resemblances and differences on which our classifications are founded are not of this palpable and easily determined kind; especially where they consist not in any one quality but in a number of qualities, the effects of which, being blended together, are not very easily discriminated, and referred each to its true source; it often happens that names are applied to nameable objects, with no distinct connotation present to the minds of those who apply them. They are only influenced by a general resemblance between the new object and all or some of the old familiar objects which they have been accustomed to call by that name. This, as we have seen, is the law which even the mind of the philosopher must follow, in giving names to the simple elementary feelings of our nature: but, where the things to be named are complex wholes, a philosopher is not content with noticing a general resemblance; he examines what the resemblance consists in: and he only gives the same name to things which resemble one another in the same definite particulars. The philosopher, therefore, habitually employs his general names with a definite connotation. But language was not made, and can only in some small degree be mended, by philosophers. In the minds of the real arbiters of language, general names, especially where the classes they denote cannot be brought before the tribunal of the outward senses to be identified and discriminated, connote little more than a vague gross resemblance to the things which they were earliest, or have been most, accustomed to call by those names. When, for instance, ordinary persons predicate the word just or unjust of any action, noble or mean of any sentiment, expression or demeanour, statesman or charlatan of any personage figuring in politics, do they mean to affirm of those various subjects any determinate attributes, of whatever kind? No: they merely recognise, as they think, some likeness, more or less vague or loose, between these and some other things which they have been accustomed to denominate or to hear denominated by those appellations. Language, as Sir James Mackintosh used to say of governments, “is not made, but grows.” A name is not imposed at once and by previous purpose upon a class of objects, but is first applied to one thing, and then extended by a series of transitions to another and another. By this process (as has been remarked by several writers, and illustrated with great force and clearness by Dugald Stewart in his Philosophical Essays) a name not infrequently passes by successive links of resemblance from one object to another, until it becomes applied to things having nothing in common with the first things to which the name was given;

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which, however, do not, for that reason, drop the name; so that it at last denotes a confused huddle of objects, having nothing whatever in common; and connotes nothing, not even a vague and general resemblance. When a name has fallen into this state, in which by predicating it of any object we assert literally nothing about the object, it has become unfit for the purposes either of thought or of the communication of thought; and can only be made serviceable by stripping it of some part of its multifarious denotation, and confining it to objects possessed of some attributes in common, which it may be made to connote. Such are the inconveniences of a language which “is not made, but grows.” Like the governments which are in a similar case, it may be compared to a road which is not made but has made itself: it requires continual mending in order to be passable. From this it is already evident, why the question respecting the definition of an abstract name is often one of so much difficulty. The question, What is justice? is, in other words, What is the attribute which mankind mean to predicate when they call an action just? To which the first answer is, that having come to no precise agreement on the point, they do not mean to predicate distinctly any attribute at all. Nevertheless, all believe that there is some common attribute belonging to all the actions which they are in the habit of calling just. The question then must be, whether there is any such common attribute? and, in the first place, whether mankind agree sufficiently with one another as to the particular actions which they do or do not call just, to render the inquiry, what quality those actions have in common, a possible one: if so, whether the actions really have any quality in common; and if they have, what it is. Of these three, the first alone is an inquiry into usage and convention; the other two are inquiries into matter of fact. And if the second question (whether the actions form a class at all) has been answered negatively, there remains a fourth, often more arduous than all the rest, namely, how best to form a class artificially, which the name may denote. And here it is fitting to remark, that the study of the spontaneous growth of languages is of the utmost importance to those who would logically remodel them. The classifications rudely made by established language, when retouched, as they almost all require to be, by the hands of the logician, are often in themselves excellently suited to his purposes. As compared with the classifications of a philosopher, they are like the customary law of a country, which has grown up as it were spontaneously, compared with laws methodized and digested into a code: the former are a far less perfect instrument than the latter; but being the result of a long, though unscientific, course of experience, they contain a mass of materials which may be made very usefully available in the formation of the systematic body of written law. In like manner, the established grouping

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of objects under a common name, even when founded only on a gross and general resemblance, is evidence, in the first place, that the resemblance is obvious, and therefore considerable; and, in the next place, that it is a resemblance which has struck great numbers of persons during a series of years and ages. Even when a name, by successive extensions, has come to be applied to things among which there does not exist this gross resemblance common to them all, still at every step in its progress we shall find such a resemblance. And these transitions of the meaning of words are often an index to real connexions between the things denoted by them, which might otherwise escape the notice of thinkers; of those at least who, from using a different language, or from any difference in their habitual associations, have fixed their attention in preference on some other aspect of the things. The history of philosophy abounds in examples of such oversights, committed for want of perceiving the hidden link that connected together the seemingly disparate meanings of some ambiguous word. Whenever the inquiry into the definition of the name of any real object consists of anything else than a mere comparison of authorities, we tacitly assume that a meaning must be found for the name, compatible with its continuing to denote, if possible all, but at any rate the greater or the more important part, of the things of which it is commonly predicated. The inquiry, therefore, into the definition, is an inquiry into the resemblances and differences among those things: whether there be any resemblance running through them all; if not, through what portion of them such a general resemblance can be traced: and finally, what are the common attributes, the possession of which gives to them all, or to that portion of them, the character of resemblance which has led to their being classed together. When these common attributes have been ascertained and specified, the name which belongs in common to the resembling objects acquires a distinct instead of a vague connotation; and by possessing this distinct connotation, becomes susceptible to definition. In giving a distinct connotation to the general name, the philosopher will endeavour to fix upon such attributes as, while they are common to all the things usually denoted by the name, are also of greatest importance in themselves; either directly, or from the number, the conspicuousness, or the interesting character, of the consequences to which they lead. He will select, as far as possible, such differentiae as lead to the greatest number of interesting propria. For these, rather than the more obscure and recondite qualities on which they often depend, give that general character and aspect to a set of objects, which determine the group into which they naturally fall. But to penetrate to the more hidden agreement on which these obvious and superficial agreements depend, is often one of the most difficult of scientific problems. As it is among the most

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difficult, so it seldom fails to be among the most important. And since upon the result of inquiry respecting the causes of the properties of a class of things, there incidentally depends the question what shall be the meaning of a word; some of the most profound and most valuable investigations which philosophy presents to us, have been introduced by, and have offered themselves under the guise of, inquiries into the definition of a name.

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Heinrich Rickert

The Theory of Definitions Preface to the first edition I am indebted to Professor D. Windelband for suggesting that I undertake a special study of the concept of definition. The form of this study may cause offence because the most important element, i.e. the attempt to reshape the traditional theory of the concept, exceeds the scope of the topic and is therefore only touched upon superficially. In a systematic representation, the order and realisation of thoughts would have to be substantially altered. I have, nevertheless, preferred to keep the present form because it indicates the route by which, in the course of my research, I have been led to the conclusion that a mistaken view of the concept has caused great confusion in the theory. In order to prove the accuracy of these results, it seemed to me that a form which reveals the evolution of my convictions would be the most suitable. For specialists in the field, it is hardly necessary to point out that among all recent studies in logic the work of Sigwart has had the strongest influence on me. — For the method of my study I am indebted to my venerated teacher, Professor D. Windelband, whom I should like to thank warmly for his many suggestions. Straßburg, June 1888.

Extracts from the Preface to the second edition I hesitated a long time before deciding to re-issue my doctoral dissertation which

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was written more that twenty seven years ago. Only the repeated information of my esteemed publisher that there was a public demand for this little study and his friendly wish for a second edition, made me examine it with a view to its republication. There was, obviously, no question of my reshaping it the way I would write it today. This would have produced a totally new book, which is not within my current plan of activities. I have limited myself to some corrections, most of which are purely external, cut a few sentences for which I would not take responsibility today, and made some additions in which I attend to the detailed criticism which, twenty five years ago, Sigwart dedicated to this little book and which at that time gave me great pleasure. I have nothing much to add of substance. I no longer attach much importance to the question of the best use of the word ‘definition’. Like Sigwart, one may wish to denote with it only the sentence which equates the meanings of two expressions. In this case the theory of definition is unimportant for logic. If, however, one wishes to retain the original meaning of ‘horismos’, in other words, if definition is taken to be not only definition of words but also the specification of the concept, I would now not have to defend what I wrote then against Sigwart’s criticism. In my book on the limits of concept-formation in the natural sciences, and quite independently of any theory of definition, I developed and substantiated the theory of the concept, which I presented for the first time in this book. It was a continuation of what had been started here and for this reason this, my first publication still seems to retain some value. Besides, there is no generally acknowledged monograph on definition, and this too has caused me to permit reprinting of this essay. The questions raised in it still require clarification. Freiburg, 1st April 1915.

Extract from the Preface to the third edition What I said in the Preface to the second edition of this little book in 1915 also applies to this edition. When I examined the text in view of its new edition, I could not conceive of reshaping my first scientific study in the way I would have to present a monograph on the theory of definition today. For the most part I have limited myself to stylistic revisions; the content of the few added sentences follows the same orientation which was decisive for this work from the start. Still, I do not believe that this old study is completely out of date. I had no need, therefore, to feel apprehensive about permitting a third impression of the book with which I started my academic career more than four decades ago. Indeed, it

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almost seems to me as if it has, precisely today, once more become especially important to emphasise what I always considered to be the most important feature of my doctoral dissertation, and I want to state briefly what this is. If one relates the theory of definition to the logical problems which are much discussed these days, one can highlight the contentious question of the relation of “experience” and “thought” or “intuition” and “concept”. Every truth of which we are explicitly aware or which we have come to know takes the form of a judgment, and to its logical content there necessarily belongs both an “intuitive” and a “discursive” factor. One is as indispensable as the other, even though each must be present for different reasons. It would certainly be a mistake to suppose that we could come to grasp a theoretical truth by means of either empirical intuition alone or rational thought alone. This has been proved convincingly and repeatedly in the course of the history of philosophy, and one would think that since Kant this would be no longer in doubt. Nevertheless, from time to time there is a tendency to emphasise one of the moments of truth at the expense of the other, and currently there is a particular tendency to overestimate intuition “phenomenologically”, i.e. to believe that it is possible to grasp a theoretical truth merely by “seeing”. In post-Kantian philosophy, Fries in particular has, in his “anthropological critique of reason” emphasised “immediate knowing”; and in order to justify its importance, he presents the judgment as something logically secondary: it merely repeats, before consciousness, the other immediate knowledge. Such intuitionising tendencies seem to assert themselves wherever philosophy shrinks to mere “anthropology” or to the description of “experiences”. In such cases, judgments as the real bearers of theoretical knowledge are distinctly suspect, and devalued as “death of truth”. Given these circumstances, and fully acknowledging the intuitive factor in knowledge, special emphasis still needs to be given to the fact that intuition on its own would never be sufficient to grasp theoretical truth. At the present time, it would perhaps be advisable not to appeal to a “logician” like Kant who tended towards “constructions”, and was therefore from the start suspect to the friends of intuition; one would be better advised to cite Goethe, who was wholly “eye” and who more than most valued intuition in science. In his scientific work, and especially in his theory of colour, in which “sight” was clearly in the foreground, he said: “Merely looking at a thing does not advance us” and in this connection he saw quite clearly “that with each attentive look into the world, we theorise already”. This is indeed the case, and the theory of science, therefore, has the task of repeatedly re-examining how far we can go in the knowledge of truth with mere “seeing”, and where theorising, which can no longer be traced back to pure intuition, starts. The present study is also concerned with these problems. It treats

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definition as concept-formation and concept-analysis, and it seeks to show that the finished or “defined” concept is not, for example, logically prior to the judgment, but that, in respect of its logical content, it has to be understood as the product of a judgment. Judgments which are cast in scientific form thus bring into relation with each other structures which are themselves already the results of judgments. Once this is understood, it must become clear from one particular perspective how impossible it is to rely, in knowing, on intuition, and then to regard the judgment, which can never be mere intuition, as secondary. As long as one regards a combination of concepts in a judgment as mere “representations”, one may, of course, identify the content, essential to knowledge, with the representations which the judgment relates to each other, and one may then come to regard them as intuitive in character. Indeed, on this assumption, one may easily arrive at the conclusion that the relation of the representations, which must themselves be more than intuitive, is something inessential to the truth-content of the judgment, and that, ultimately, only the intuitive factors in the representations are relevant to the truth of the judgment. If, on the other hand, one has come to recognise that the structures, which as concepts of judgments are related to each other, only come into existence as defined concepts through judgments, and if one has come to recognise that these structures are therefore far from being merely representational or purely intuitive in character — if one has come to recognise all this, then it must become apparent that the non-intuitive, discursive factor is indispensable to any scientific knowledge. A more detailed discussion of these ideas would by far exceed the framework of a preface, and this is not the place to ask where, adopting the view that all knowledge is logically structured in this way, there would be a place in scientific judgments for the intuitive factor. I am limiting myself to pointing out the indispensability of the discursive moment, and in so doing, I merely wanted to indicate that my earliest work could be linked to questions which are much discussed today. From the beginning, my dissertation was engaged in a battle against intuitionism; and it fought, as I have always fought, precisely in order to support scientifically fruitful intuition. In other words, by pointing out the existence of methodological differences, it sought to draw attention to the diversity and richness of intuition — a diversity which can so easily be forgotten in the face of the one-sided constructions of intuitionism. Heidelberg, 16th September 1929

Heinrich Rickert Doktor der Kulturwissenschaften

THE THEORY OF DEFINITION

CONTENTS 1 Introduction 1.1 Task and method 1.2 Origin and original meaning of definition 2 General specification of definitions 2.1 Word-explanation and definition 2.2 The purpose of definition 3 Essential and inessential characteristics 3.1 The inadequacy of existing theories 3.2 Definitions in law 3.3 The definition of the natural sciences 3.4 The definition of mathematics 4 Definition and concept 4.1 Analytical and synthetic definition 4.2 Concept and judgment 4.3 The inadequacy of the existing theories of the concept 5 Genus proximum and differentia specifica 5.1 Genus and essence in the empirical sciences 5.2 Genus in mathematics 6 Nominal and real definitions 6.1 Name, thing and concept 6.2 Provisional and definitive definitions

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Chapter One: Introduction 1.1 Task and method Among the various forms of scientific thought there is hardly another about which opinions diverge as widely as about definitions. Though the words ‘definition’ and ‘define’ are thoroughly familiar to us, modern textbooks of logic encounter great difficulties in establishing what these words mean in logic. The wide diversity of opinion on this point is all the more surprising since the theories of definition are normally presented with such a brevity and certainty, as if there could no be question of controversy. Very few authors feel the need to consider and to discuss other opinions. In fact, wherever definitions are mentioned, we find a few undisputed and agreed formulae which appear to be a common property of all logical systems. On closer inspection, however, if one tries to develop a particular view about the concept of definition on the basis of these formulae, and especially about its position in the system of logic, it will be noticed that these formulae acquire a meaning only through their subsequent interpretations; and these interpretations exhibit such wide divergences that practically nothing remains of the apparent agreement. The reason for this divergence can easily be shown. The constantly recurring formulae originate from Aristotle and are so closely related to his metaphysics, that they are meaningless without it. Modern logic, which, with a few exceptions, is no longer based on Aristotle’s metaphysics, has nevertheless retained these logical formulae and given them a new meaning which, if it is not to be devoid of content, must depend on some other epistemological or metaphysical assumptions. It is probably undisputed that Aristotelian logic is never a “formal” logic in the sense of being independent of all metaphysical and other objective presuppositions. It is accordingly to be expected that a relation between logical forms and metaphysical or material opinions, similar to that found in the case of definitions, will have to be present in the case of all the logical forms which modern branches of knowledge have borrowed from Aristotle. The reason why this is not the case cannot be demonstrated here in detail. But let us, right at the outset, point out the unusual position occupied by definition in Aristotle’s system, and thereby indicate why it is easier to detach other logical forms from their metaphysical presuppositions than it is to detach definition from its presuppositions.

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Whereas the other logical forms are links in the process of scientific research and representation — the syllogism, for example, is a tool by the aid of which one progresses from one thought to the next — for Aristotle the task of the definition is to conclude the inquiry and definitively to fix the “essence” of the respective object of investigation. This difference is crucially important for the more than formal character of the definition. While it may be possible to separate all other forms of thinking from their content and to consider the truth they reveal as merely hypothetical, it is not possible in the case of definition without depriving it of the meaning it has for Aristotle. The concept of genus, for instance, plays an important role in the theory of definitions. This point alone serves to demonstrate how many diverse interpretations the old theory of definition has undergone in modern times so that the Aristotelian formulae could be preserved. Aristotle said ÏO}ismoÈ| eÑsti loÈgo| oÏ toÌ tiÈ hÕn eiÕnai shmaiÈnwn, (the definition is the expression which indicates the essence of a thing) (Topics VII, 5), which is usually translated as: the definition is the concept which indicates the essence. It is also specifically called “ouÎsiÈa| gnw}ismoÈ|”. This seems to be its task: to provide the knowledge of the “essence” of a thing, i.e. it should specify the general, timelessly valid concept, the specific manifestation of which is the individual thing of our world of perception. From this task we can derive the form in which it must appear if it is to serve its purpose; Aristotle specified this form as precisely as its content: “oÏ oÏ}ismoÌ| eÎk geÈnou| ~aiÌ diaforw Ín eÎstiÈn” (the definition consists of the generic concept and the differentiae) (Topics I, 8). The generic concept indicates the ‘essence’ to which we must subordinate the thing to be defined if we are to recognise its nature or essence. The addition of the differentia serves to indicate the specific type in which the essence manifests itself. In modern philosophy it is no longer customary to identify the essence and the genus, and for this reason modern logicians do not find it easy to assign to the definition the task of indicating the essence of a thing by means of its genus and differentia. While under certain material presuppositions of a metaphysical kind this is unambiguous and clear for the Aristotelian system, it is incomprehensible in a modern logic which does not already presuppose a metaphysics. Nevertheless, we read almost everywhere: definition consists of the specification of the genus proximus and the differentia specifica; we must therefore ask, what is the point of asking for a genus without metaphysical presuppositions. The answers differ greatly, as can easily be shown by a number of examples. Ueberweg (1882: para. 60, p. 165) seems closest to Aristotle. For him, definitions are “der Ausdruck des Wesens (der ‘essentia’) der Objekte des

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Begriffs” (the expression of the essence of the objects of the concept), inasmuch as it gives “alle wesentlichen Inhaltselemente oder alle wesentlichen Merkmale der Objekte des Begriffs” (all the essential elements of the content or all the essential characteristics of the objects of the concept). Ueberweg says “die wesentlichen Inhaltselemente sind teils solche, die der zu definierende Begriff mit den ihm nebengeordneten Begriffen teilt, und die demgemäß auch den Inhalt des übergeordneten Begriffs ausmachen, teils solche, wodurch er sich von den nebengeordneten und von den übergeordneten unterscheidet. Indem nun der Gegensatz von Gattung (genus) und Art (species) auch zur allgemeinen Bezeichnung des Gegensatzes irgend einer höheren Klasse zu einer niederen dient, sofern diese jener unmittelbar untergeordnet wird, so können hiernach die wesentlichen Inhaltselemente des zu definierenden Begriffs in generische und spezifische eingeteilt werden. Hierauf beruht die Forderung, daß die Definition den übergeordneten oder Gattungsbegriff und die spezifische Differenz oder den Artunterschied enthalte.” (the essential elements of the content are partly those which the definiendum shares with its coordinated concepts, and which therefore also comprise the content of the superordinate concept, and partly those by means of which it differs from the coordinate and superordinate ones. Whilst the contrast of genus and species also serves for the general designation of the contrast between higher and lower classes, as long as the latter are directly subordinated to the former, the essential elements of the content of the definiendum of a concept can be divided into generic and specific. This is the reason for the demand that a definition should contain the superordinate concept or the concept of the genus and the specific difference or the distinction of the species.) It is obvious that these sentences are meaningless until we know what these essential characteristics are. Ueberweg (1882 para. 55, p. 147) explains them by saying: “Wesentlich (essentialia) sind diejenigen Merkmale, welche a) den gemeinsamen und bleibenden Grund einer Mannigfaltigkeit anderer enthalten, und von welchen b) das Bestehen des Objektes und der Wert und die Bedeutung abhängt, die demselben teils als einem Mittel für anderes, teils und vornehmlich an sich oder als einem Selbstzweck in der Stufenreihe der Objekte zukommt”. (Essential characteristics (essentialia) are those which a) contain the common and permanent ground of a variety of other characteristics, and on which b) the existence of object depends, along with the value and the meaning which belongs to it, partly as a means to something else, and partly and especially in itself or as an end in itself in the hierarchy of objects.) These sentences, if they have any meaning at all, only make sense in the context of a metaphysics or an epistemology.

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In contrast, Lotze’s theories (1874: para.160, p.198) of definition are by and large free of epistemological presuppositions. He calls definition methodical description, by which he means that the demand for a generic concept restricts the “willkürliche und launenhafte Beschreibung” (arbitrary and capricious course of the description). “Ohne die Anwendung vieler Allgemeinbegriffe würde indessen auch sie nicht zum Ziele kommen; anstatt diese nun willkürlich zu wählen, verlangt die Definition, daß man von demjenigen Allgemeinen ausgehe, in welchem der größte Teil der zu leistenden Konstruktionsarbeit schon fertig und vollzogen vorliegt und welches durch einen eindeutigen Namen sprachlich bezeichnet, in jedem Bewußtsein als eine bekannte Anschauung vorausgesetzt werden kann, geeignet als Grundriß für die Einzeichnung der Einzelmerkmale zu dienen, durch welche das mitzuteilende Bild vollendet wird.” (Without the application of many general concepts definition would not succeed either; instead of arbitrarily selecting such general concepts, definition requires that one should start from the general concept in which most of the building work is already done and complete, and which has been linguistically designated by an unambiguous name, so that it can be presupposed in every consciousness as a familiar intuition, which is suitable to serve as a framework, within which the individual characteristics, completing the picture of the conveyed, can be described.) For Lotze the generic concept does not seem to be indispensable but is mainly required for the sake of brevity and precision of the definition. According to Sigwart (1873/1911: 385), finally, the definition is “ein Urteil, in welchem die Bedeutung eines, einen Begriff bezeichnenden Wortes angegeben wird” (a judgment in which the meaning of a word designating a concept is given), and the specification of the next highest genus and the specific difference have the task — independent of Sigwart’s ”definition”–of allocating the concept to its position in the ordered system of concepts. There is not much left here of Aristotle. As we shall demonstrate later in detail, for Sigwart definition is simply a means for communicating or fixing of thoughts by means of language. Although, as we have already seen, theories of definition differ greatly regarding genus and differentiae, there are even greater divergencies of opinion when it comes to nominal and real definition, and it might be said that there are no two modern logicians with the same theory about definitions. Furthermore, when Sigwart (1873: 387), for example, says: “Nennt man die Angabe aller Merkmale eines Begriffes oder des Genus proximum und der Differentia specifica Definition, so ist klar, daß es sich darin nicht um eine Begriffserklärung, sondern, sofern etwas erklärt wird, nur um eine Worterklärung handeln kann” (If the listing of all the characteristics of a concept or of the

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genus proximum and the differentia specifica is called definition, it is obvious that this cannot be a definition of the concept, but only, inasmuch as anything is being defined, a definition of a word). Lotze (1874:201), on the other hand, rightly states: “Namen lassen sich aussprechen und übersetzen, definieren können wir aber immer nur ihren Inhalt: unsere Vorstellung nämlich von dem, was sie bezeichnen sollen” (Names can be uttered and translated, but we can only define their content: our idea of what they are meant to denote). It would appear that these two logicians mean two quite different things by ‘definition’, which share not much more than the name, with the result that one of them means by definition something which cannot be identical with the originally named concept. This state of the theory of the definition is largely explained by the fact that at present relatively little attention is given to the form of thought, which in earlier times was the centre of interest. Not infrequently the theory of definition is considered a meaningless appendage of the theory of concepts: to relate it to and understanding it in terms of the fundamental questions of logic is regarded as unnecessary. Since it is not treated as an organic part of a whole, it can be neglected with relatively little disadvantage, for the resultant errors are not too serious for the formation of the system of logic as a whole. But the disadvantage is perhaps not as small as it might seem. The reason, why it is necessary to pay particular attention to definitions, and to form a generally acceptable view about them, is connected with an increasingly evident trend in modern treatises of logic, and which will hopefully never again disappear. Sigwart calls his important work an attempt “die Logik unter dem Gesichtspunkt der Methodenlehre zu gestalten und sie dadurch in lebendige Beziehung zu den wissenschaftlichen Aufgaben der Gegenwart zu setzen” (to treat logic from the point of view of methodology and thereby to place it in a dynamic relationship to the scientific endeavours of the present time). But one does not need to believe that logic is only concerned with methodological problems; one may even think that the methodological viewpoint has made Sigwart’s logic rather one-sided. Nevertheless, as part of logic, methodology must always be very important. Just how important for logic an insight into the essence of definitions actually is becomes immediately evident if one considers the methodological considerations which, in the individual sciences, are developed at the beginning or in the course of special investigations. Definition often plays an important role in these methodological considerations, and what is said about them by individual subject specialists, e.g. by Jhering, seems to be more important for methodology than most of the writings in textbooks by logicians. For this reason it seems useful to devote a particular study to definitions from a purely methodological point of view.

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The following investigation, which undertakes this task, will first have to establish which mental structure is designated as ‘definition’, because some logicians seem to have lost sight of the phenomenon. This enquiry will, of necessity, have to be historical. We shall therefore ask what the definition was for the Greeks in order to specify what mental structure, in a logic independent of Aristotelian metaphysics, occupies the kind of position that must be denoted by the term ‘definition’. Before we turn to this historical investigation, we must characterise and justify in a few words the method we shall be using in what follows. We have already indicated the main focus of our study. If, like Sigwart, one emphasises methodology, one can start with the objectives pursued by the sciences, and enquire about the means human thought must apply in order to reach these objectives. This is certainly one justified point of view among others. For methodology, definition is then a means to a scientific end; In the case of Aristotle we have without difficulty identified what distinguishes definition from the other forms of thought by considering the task of definition. In the same way, the view that a methodological form is to be understood in terms of its purpose may govern the whole methodological investigation. The intentional ‘thought’, guided by the will, which in the spiritual life of man is opposed to the ‘natural flow of representations’ conditioned by psychological laws, wishes to know and, to this end, to be logical. According to ability and inclination this attempt is directed at a smaller or larger part of what is called the world or reality, but the purpose is always to identify the correct or true view from among the many possible views of things to which thinking applies itself. There is a second purpose, equally obvious but not always sufficiently observed and precisely delineated. Human beings do not generally think for themselves alone, but try to communicate the results of their thinking to others, and this they can only do by means of language. The thoughts which we have found to be correct or true are only scientifically complete if they have found an intelligible expression in words. Anyone who has undertaken scientific research knows that we often fail in these two efforts, that of seeking the truth, as we may briefly say, and that of linguistically formulating it for the purpose of communication to others. Everyone has, on occasion, erred and then become aware of the mistake, everyone has, on occasion, been misunderstood and then become aware of having been misunderstood. Once this is acknowledged, one has to realise that one’s thinking had gone wrong in some way that prevented it from reaching its objective. This error

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which human thought sometimes incurs in its search for truth is not a psychological law; in fact, it cannot be a law of nature, because if it were possible to violate it, it would no longer be a law of nature. From a psychological point view, all thought is equally necessary. In the search for truth and its communication it is therefore something different, let us say a norm, a rule, a prescription which does not have to be but which ought to be observed, and which we recognize as binding as long as we are concerned with discovering the truth and communicating it to others. Consequently, once we have fully understood these rules, we believe that we are certain of avoiding errors. From this thought arose the attempt to establish a system of such rules of thought. Methodology, understood in the form of this logical task, will never want to establish psychologically necessary laws of thinking; it will only ever investigate the forms of thought in relation to the purpose one has in mind when they are being applied, and specify how they must be constituted if thinking is to attain its end by this means. Its necessity is therefore teleological (Windelband 1884: 109). This teleology is obviously quite different from that which declares the purpose to be the ‘explanatory’ principle of being. Here, we are not concerned with purpose in general, but with a specific purpose which we must identify in order to be able to use it scientifically. Our investigation will therefore proceed in the following way: we shall first indicate the purpose of definition. We shall specify its precise role in the thought-process which seeks truth and wants to communicate it, and only then shall we determine the rules which govern it. Consequently, we start with investigating the purpose which led to the introduction of definition in Greek philosophy. The thought-structure which serves the same purpose in modern science will then also have to be called definition.

1.2 Origin and original meaning of definition Our first question is: What preoccupation of human thought has led to definition. Turning to the development of Greek thought in the search for an answer, this study does not make a claim for historical completeness. All that matters is to highlight the basic motives which lead to logical consciousness in human thought and with it to definition; it is therefore sufficient if our description is merely schematic and leaves aside individual characteristics of Greek theorems, however important they may be in themselves. We are naturally only interested in Greek philosophy from the moment it begins to turn to the human thought-process. This occurs with the sophists. The metaphysics of Heraclitus and the Eleatics was still somehow externally oriented,

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but when in their study of the natural science their theories about the nature of the world showed distinct contradictions with the facts, it awakened doubts about the capacity of human thought to discover truth and forced our mind to investigate its own activity. This happened in a strange way. In some metaphysical systems of the natural philosophers, the principal driving force was the avoidance of contradiction. The Eleatics could not deal with the concept of ‘becoming’ because it was full of contradictions; for Heraclitus, on the other hand, there was no stasis but only movement. In this way the concept of contradiction was founded metaphysically, but no philosopher had explicitly faced up to the principle. Whether one supported one or the other theory, a single consequence seemed to result from both: the world of our senses, as it represents itself to us as a mixture of stasis and flux, stability and transformation, is an illusion. We do not recognise things the way they are, but the way they appear to us, in fact as they appear to the individual human being. For this reason objective knowledge is impossible, we only have subjective opinions. Protagoras argued against the possibility of objective knowledge on the basis of Heraclitan metaphysics, and Gorgias did the same on the basis of the metaphysics of the Eleatics. The philosophers who, on the basis of their involuntary metaphysical hypostatisation of the principle of contradiction, despaired about the possibility of any knowledge, then started to doubt the validity of this principle itself. If there is only opinion, then there is no difference between truth and error and contradictory opinions are equally valid. These philosophers negated the validity of the very principle by which they had proved their argument, and this was highly significant. If this had not at one time been questioned, it would have been impossible to discover the logical rules which, unknowingly, had been followed hitherto. In this way the sophists assisted our thinking by making us aware of logic. In a certain sense, the consciousness of the first metaphysicians had been a-logical, the consciousness of the sophists was anti-logical, and at this moment logic was awakened by the logical consciousness of Socrates. We had to remind ourselves briefly of this sequence of events in order to appreciate the full significance of the need for definitions in these circumstances. If one wanted to discuss Socrates’ theoretical philosophy, which must be separated from his views on ethics, one could roughly outline its development as follows: Socrates agreed with the sophists that knowledge did not, in fact, exist. But he was at the same time firmly convinced that all the different opinions contained something in which they agreed; the very fact of the existence of this conflict of opinions was indicative of this. The truth does not lie in a discovery by an individual, but truth is what these discoveries have in common, and the way to

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establish truth is to determine what is agreed among the many different opinions. Pursuing this idea, and stimulated by the strange methods of his opponents, Socrates felt the need which gave rise to the definition. As proof for the relativity of all opinions, the sophists had used the fact that a word can be used to denote several different concepts. Socrates accepted this and therefore demanded of every disputant the precise determination of the concepts which were to be connected with the words used. He recognised that it was only possible to reach insight if the concepts used in the investigation were determinate and agreed upon. For Socrates, definition was the means to create determinate and unambiguously designated concepts. This fact may be considered controversial, but this is only because with Socrates definition appeared in an unusual form, a form which was, in its turn, determined by the procedures of his opponents. The sophists proved by means of words and they were deceived about their errors through language, which has also been the cause of basic errors in a number of other philosophical systems. Socrates tried to penetrate the confusion of language and the multiple meanings of names in order to arrive at determinate concepts. To this end it was, however, necessary that in his investigations he should always start from the name and link the definition of concepts to it. Since definitions arose from dialogue, in which the linguistic formulation of thoughts is as important as the thought itself, this act of thought was so closely related to the word, as to suggest that the main function of definition was to indicate the meaning of a word. But for Socrates definition served as an explanation of names only to the extent that it was necessary to eliminate the errors which arose in discussion through the illogical elements of language. Its real purpose was to determine the concept.1 This is the first important point in our argument. Plato’s development of definition beyond Socrates is significant for two reasons. So far we have considered definition as a means for establishing determinate concepts by specifying the unusual, which was of value to knowledge because it was what was common to all particulars. A new theoretical evaluative standpoint is now established; it can best be understood with respect to the effort to overcome relativism, such as had developed with Protagoras in connection with the metaphysics of Heraclitus. Plato agreed with Heraclitus that the individual things of the sensible world are not but only coming to be, so that knowledge of the particular was not giving knowledge for him at all. But particular things also have something in common and this is what is permanent

1. See: Xenophanes, Memorabilia IV,6.

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in them. It not only comes to be, it also is, and it is on this that our cognition must be focused, if it is to be the knowledge of what truly is. Plato’s insight loses none of its significance by having been immediately given a peculiar re-interpretation, so that to the universal as ‘idea’ there is ascribed an existence independent of particular things; indeed, the universal is actually made into the ‘cause’ (aiÎtiÈa) of particulars. For the theory of definition, it is important to single out the purely logical element. Definition, which was for Socrates a means to true knowledge, in virtue of its forming the universal concept, now furnishes knowledge in virtue of its determining the universal idea, the manifestation of which is the particular which is to be known. Plato made a second contribution to logic, as a result of which definition really first became what it later was for Aristotle. He not only tried to identify each one of the various true ideas, but he made the first attempt to organise these ideas into a system. Just as the ideas contain the individual objects beneath them, they themselves can, in their turn, be united under a higher idea, and, if Plato had developed it further, he would have constructed the pyramid of ideas, the apex of which is the idea of the good as the real world principle.2 This is what gives the platonic definition its special form. To recognise an object means to assign it to its place in this pyramid. It is subordinated to an idea and assigns it to that which differentiates the object from the other objects subordinated to the same idea. This completes the acquisition of the knowledge of an object, for, according to Plato, it has been assigned its position relatively to the world principle. Herein lies the origin of what later became definition by genus proximus and differentia specifica. ‘ÏO}ismoÈ|’ is thus for Plato always knowledge of the essence of an object through the specification of its superordinate idea, of which it is the manifestation, in which it participates, or however else one may wish to express it. Thus we see the definition also exists in a particular form. Plato uses a scientific method which he handles with assurance. But he nowhere made this method itself the object of a special investigation; he did not develop a theory about the form of his knowledge, in other words, he did not give a definition of definition. This was done first by Aristotle. He did not invent definition; he only needed to analyze what Plato had already done. Plato had asked: What is the

2. Plato has not completely developed the thought that the good is the point of the pyramid of ideas, because the “aÎgaqoÌn” is still located ‘beyond the ousia’. Only Aristotle developed the thought to the extent that the deity, as “noÈhsi| nohÈsew|”, represents the point of the pyramid in every respect. See: Rickert, Die Erkenntnis der intelligibeln Welt und das Problem der Metaphysik, in: Logos, XVI, pp. 185–6.

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object of true knowledge? and he gave the answer: ‘the idea’. Aristotle, by contrast, asked: How do we know? and he replied: we determine the concept by specifying the genus and difference. From a methodological point of view, there is, in principle, no difference between Plato and Aristotle in respect of definition: for both ‘idea’ and ‘concept’ have the task of indicating the essence of a thing, and for this reason they must be used in giving a definition. The metaphysical difference between ‘idea’ and ‘concept’ does not concern us here. On the other hand, of course, it is also immediately apparent how closely Aristotle’s metaphysical view is intertwined with his theory of definition. The form is empty and arbitrary without the specific metaphysical presupposition. Nevertheless, and this is what matters above all here, for Socrates, Plato and Aristotle the word ‘o}ismoÈ|’ always designates the thought structure which has the task of determining the concept; and we may, therefore, and, indeed, we must use the word definition for the determination of a concept. To what extent we may also ascribe to definition that other task, which for us no longer has the same unambiguous meaning it had for the Greeks, that namely, of specifying the essence of a thing — that is something we cannot yet show here.

Chapter Two: General specification of definition 2.1 Word-explanation and definition We now turn to a systematic examination of the concept of definition. Most logicians start their explanation of definitions with a discussion of the wellknown fact that in the communication of thoughts, the words of a language do not always signify the same thing for the sender as for the recipient. Even those authors who hold that defining is not concerned with the definition of names, are inclined to emphasise the view that the task of definition is to remedy the misunderstandings produced by language.1 Their position can easily be understood from the history of logic: they are thereby strongly endorsing the peculiar form in which, as we have already seen, definition had first to make its appearance with Socrates. It was not by accident that we repeatedly stated, at the beginning of our investigation, the apparently self-evident thought that the search for truth and

1. See: Lotze 1880:192ff.

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the linguistic formulation for the purpose of communication are two quite different things. These two structures and their purposes cannot be distinguished from each other sharply enough; for, even though their difference, in the form we have just stated, is undoubtedly generally accepted, the matter looks quite different if we turn to a question closely related to this distinction, namely the question of the relationship between language and thought in general. This much discussed topic has at least to be touched upon here before we start an investigation of the concept of definition. It is undoubtedly true that we learn to think by means of language and that we constantly think with the assistance of language. We may even say that we cannot think logically without language, or at least not properly. It is therefore unacceptable in logic to ignore language. But the reason does not lie in the fact that thinking and speaking coincide. A simple reflection will explain this. On the one hand, there are the words and sentences we understand, and on the other, there are the words and sentences which are incomprehensible to us: hearing them does not and cannot make us think. The difference lies in the fact that for a listener some words and sentences have meaning or sense and others do not. However closely words and their meaning may be connected, they can be differentiated conceptually: they must indeed be different from each other since we can say that they are connected with each other. Speech alone has no meaning and is without thought. By its very nature, thinking does not operate in the realm of words and sentences but in that of meanings and sense structures. This does not mean, as we have already stressed, that we can think without an instrument such as language, but only that language is not conceptually inseparable from thinking. Whether we might, through exercise, learn to think without words, is irrelevant. Because thinking and speaking, words and meaning, sense and sentence are not identical, what matters is that language used for thinking, independently of the communication of thoughts to others, must play a quite different role than in the effort specifically directed towards expressing the results of thinking in a form others can understand. If in the first case, language is secondary and external to the process, even though in effect indispensable, in the second case, it is the very object on which our thought is focused. This needed to be emphasised in order to show that it is legitimate to discuss thinking in respect of its sense or meaning regardless of the linguistic formulation which has been adopted for the purpose of communication with others; for it becomes apparent from this that the definition which aims exclusively at the meaning of a word is something fundamentally different from the definition which aims at defining the content of a concept. In the former case, what is at issue is the word itself; the concept which is to be associated

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with the word is already presupposed as complete. In the latter case, by contrast, we are concerned with the meanings and sense-structures attaching to the words and which we mean or understand when speaking or listening; it is precisely the concept, which we are to think and connect with a word, which is in question. But let us first consider the linguistically formulated definition and try to determine the task which it has to perform. From the point of view of logic, this is quite simple. Anyone who utters a sentence feels the need to be understood, i.e. must wish his hearer or reader to associate the same meanings or concepts with the words employed as he does himself, or, as one commonly says, have the same ‘representations’ or ‘ideas’. In as far as it is a matter of simple meanings not requiring further analysis, he must either presuppose that the words and what they designate are familiar, or, if this is not the case, he must be able to point to the object he means by the words, or in any case somehow bring it about that his hearer directly experiences what he wishes to say; for apart from this, he has no means of conveying his thoughts. If a word, however, designates a compound concept, the matter is different. He can analyse such a concept into simple meanings or into concepts which are themselves compound, specifying that the name he has used in this case is intended to designate the concept which consists of such and such meanings or concepts, which are presupposed as familiar in respect of both their content and designation; or, if the concepts used to this end are in their turn not designated unambiguously, he can further analyse them until the entire concept has finally been analysed into the simple meanings of words of which he can then make his hearer or reader aware either by pointing to the objects meant or by uttering words. Once this process is complete, he can be sure that his listener thinks the same thing when he hears the words used by him, as he does himself. Obviously, this process of decomposing concepts into simple word-meanings is at times very cumbersome and will only be used when there is no other solution. But since most people can be assumed to know a large number of concepts together with their corresponding designations, it is normally sufficient to indicate the meaning of a word by reference to another word which will evoke in a listener a large number of the characteristics of the concept to be communicated, and then to add those words which evoke the remainder of the meanings intended by the speaker. The form of transmission will normally be thus: this or that name designates a concept, the elements of which have the meanings designated by these or those other names. In logic, this process is also called definition and Sigwart (1873:387) thinks that definitions can only be explanations of words and not of concepts. He says:

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“Das Wort allein, das dem Begriffe gegenüber äußerlich und zufällig ist, bedarf einer Erklärung, einer immer erneuten Erinnerung an seinen Gehalt” (The word alone, which is purely external and contingent in relation to the concept, needs an explanation, a constantly renewed reminder of its content). If we wanted to accept Sigwart’s statement, we would have to conclude that the theory of definition belongs only to that part of logic which deals with the linguistic formulation of thoughts, and that logic can only state the rules which tell us how best to express them linguistically. The concept is not to be defined, in Sigwart’s view, but must be already present if the nominal definition is to be possible. The designation ‘definition’ would then be justified for the explanation of words in as much as it would be concerned with limiting the scope of application of a word to specific meanings or concepts, i.e. in a certain sense to de-limit or de-fine a word. In saying this, however, it is to be noted that the theory of this definitions is methodologically completely exhausted by what has been just presented. According to Sigwart, logic, in its methodological function, only has to tell us what should be done if a certain purpose is intended. The purpose of this definition consists in eliciting the thought of certain meanings through the naming of a word. It necessarily follows that words associated with a number of or even no meanings, which make them ambiguous or nonsensical, should be replaced by words which listeners associate with only one meaning. Since, in addition, the attention will be directed towards reaching this goal as quickly and as simply as possible, logic can add the rule that words should be looked for which indicate as many of the intended meanings as possible at the same time, so that as few words as possible will be needed in order to exhaust the totality of meanings which speakers wish to evoke in listeners. All this is logically of secondary importance. But from this purpose of definitions, we can on no account, without further presuppositions, derive the rule that definitions should be made via the genus proximum and the differentia specifica. On the contrary, it is possible to imagine situations in which one can reach one’s goal much faster by referring to a coordinated or even subordinated concept, than by reference to the superordinate concept. And, in this context, the requirement that one should indicate the essential characteristics of an object does not yet make sense. All that matters in this type of definition is that, by naming a word, I evoke in the other person the meanings I already have and which I wish him to have as well. It is my will alone which determines what meanings the other person is to think; and I can only will that he should think the meanings, and, indeed, all the meanings, which constitute the elements of my concept, for otherwise he would not completely

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have my concept. I must, therefore, so designate by words what I have included in my concept, that it will be understood by the other; and everything in it will be equally essential, for if it were not, it would not have been included in my concept, and still less designated with a word in my definition. One may legitimately object that it is customary to define differently, that it is not simply a matter of using definitions to indicate the meaning of a word, but, on the contrary, that whoever defines starts with certain scientific presuppositions and must present his definition on the basis of the genus and differentia and essential features. Sigwart (1911: 388) himself says: “Bloß sprachliche Erklärungen, wie Logik heißt Denklehre, Demokratie heißt Volksherrschaft, oder Erklärungen sprachlicher Abkürzungen, wie eine Grade ist eine gerade Linie, nennt niemand Definition.” (nobody calls definition what are merely linguistic explanations such as: logic is theory of thinking, democracy is government by the people, or explanations of linguistic abbreviations, such as: the straight is a straight line). This is certainly true. But it is not clear, from Sigwart’s theory why we do not call these explanations of words definitions. After all, according to him, the concept is not defined but the word, and what else can a worddefinition be than a “purely linguistic explanation”? What is the difference between this and definitions according to Sigwart? There is none, and there cannot be one, if definition is simply an explanation of words and not the determination of a concept. Sigwart has here refuted himself, and it is precisely his remark that nobody considers simple translations definitions, which leads us to the problems this book is intended to clarify. Before someone can specify the meaning of a word designating a concept, a process of thought must already have taken place in the realm of the logical sense; This process of thought can only then find its linguistic expression; and calling this linguistic expression alone a definition is completely arbitrary. Aristotle’s word oÎ}ismoÈ| (horismos) does not designate the explanation of words alone, nor do we today use the word definition in this sense. Rather, it is used for both the logical thought-process and the linguistic expression. In current usage this thought-process is nothing other than the formation of the concept. The logical act of thinking as the real definition of a concept, must therefore already be complete, before it is linguistically formulated, for not until I have completely determined a concept can I utter a proposition in which I say that a specific name is to be used in language as the sign for this concept as defined by me. According to Lotze (1880: § 28, p. 47), each definition can be generally expressed by the formula S=f(a,b,c…). If it is linguistically formulated it can be dissolved into two judgments: 1. f(a,b,c…) is a concept; 2. this concept shall be

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named S. For logic, the essential part is the logically meaningful act of thought by which a concept is formed, and therefore it is not arbitrary if we call it the definition proper. What Sigwart calls definition is the linguistic formulation of the antecedent thought-process; it is tantamount to a “translation” in the widest sense of this word, i.e. it is tantamount to the naming of an intelligible name for what in unintelligible or is not understood. The reason why logical structures have not always been precisely distinguished from linguistic propositions, may be found in the fact that language also plays a distinctive role in the logical thought-process, which, as we have seen, was confused with the role of language in definitions intended as a means of transmitting thoughts. This relationship between language and thinking directed towards finding and representation of truth, will be examined in Section 4.4, below, where the importance of the distinction made here will become even clearer. We shall then recognize that the word is, in any case, in certain respects, indispensable for definition as the determination of concepts, even independently of the communication of thought.

2.2 The purpose of definition Let us first turn to the logical thought-act which has always been called definition, i.e. the determination of concepts, explicitly excluding all consideration of the efforts directed towards fixing the meaning of words for the purpose of unambiguous communication. We regard the act of defining as the mental process of concept-formation, unrelated to the exchange of ideas as tool and aid in scientific descriptions. We think that we have shown that this way of looking at definition is justified. For the other areas of methodology such a proof would hardly have been necessary, even though in practice all thought is connected with words and sentences. Definitions still carry a trace of their origin in disputations about truth, which brings them into very close contact with language. But their origin is not decisive for their logical nature. They are a means to an end which does not solely consist in specifying the meanings of words; and according to our approach, it is, therefore, first a matter of acquainting ourselves more closely with this purpose of definitions, which in general terms is that of determination of concepts. Only then will we be able to understand the logical nature of definitions. There is a great divergence of opinion about the ultimate goal of human knowledge, perhaps less in respect of what is desirable, as in respect of what is

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possible; and connected with this is the fact that many people no longer wish to have goals pursued which they consider impossible of achievement and which they attack as superfluous. Nevertheless, whether one, in Lotze’s words (1880: 608), limits oneself to “calculating the course of the world” or whether one goes further and wishes to understand it, on certain points there will be agreement among all those who have not remained pre-theoretical ‘pragmatists’, i.e. all those who strive in general for knowledge without taking account of practical purposes. “Niemand versucht es, eine Wissenschaft zustande zu bringen, ohne daß ihm eine Idee zugrunde liege, und unter der Regierung der Vernunft dürfen unsere Erkenntnisse überhaupt keine Rhapsodie, sondern sie müssen ein System ausmachen.”2 (Nobody attempts to create a branch of knowledge unless he has an underlying idea, and under the government of reason our knowledge, in general, may not be a rhapsody but must constitute a system). This has never been disputed by scientifically serious people. In this respect there will be agreement between the strictest Hegelian and the positivist for whom philosophy is nothing more than “Denken der Welt gemäß dem Prinzip des kleinsten Kraftmaßes” (Avenarius 1876) (thinking about the world in accordance with the principle of minimum force). Prescientific, atheoretical people content themselves with the aggregate of knowledge necessary for daily life. Scientific people aim at the emergence of a system out of the aggregate of items of knowledge which they possess; real progress in science is for them a step in that direction.3 Regarding its logical form, this goal is usually expressed as follows: our knowledge will be complete when we have fitted it into an all-embracing system of judgments, the subjects and predicates of which are completely determined concepts. It follows necessarily that definition, as the determination of concepts, must form concepts in such a way that it is possible to create such a system of judgments. Definition is thus a tool for shaping the components, from which the scientific system is built, and we must seek to understand this tool in respect of this its purpose.

2. See: I. Kant, Kritik der reinen Vernunft. S. W. Hartenstein, III p. 549. 3. When I wrote these sentences in my youth, I could not know, that one would attempt to deny that philosophy was the striving for a system and deprive it of its characteristic as a science or that this regression to a pre-scientific stage of development would be regarded as progress. I therefore did not consider the need for a justification. The reference to Kant seemed sufficient. Even now, I am only addressing readers who choose a scientific approach to philosophy. Where this will is lacking there is no point in a logical discussion.

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First, however, it will be necessary to make a distinction which we must strictly observe in what follows. Apart from the fact that mere word-explanations are called definitions, there is another ambiguity in the word ‘definition’, which we frequently encounter in language. Let us take the two sentences: ‘the building of this house is making rapid progress’ and: ‘this is a splendid building’. We see immediately that the word ‘building’ is used with two separate meanings. On the one hand, it designates the process by means of which a house comes to be, and on the other, it designates the house itself. Most words ending in ‘-ion’ or ‘-ing’ have this double meaning and this applies also to the word ‘definition’. However obvious this may appear to be, it is important to note that ‘definition’ can mean both the act of defining (definitio) and the product of the act of defining (definitum). This distinction, perhaps precisely because it is self-evident, has never been explicitly made or fixed by logic.4 We have made this point here in order to explain that in what immediately follows we speak of the act of defining unless otherwise stated. One builds a house, but, to start with, the builders have to deal, not with the house, but with the wood and bricks and the plans according to which they put them together. This raises two questions: What it the building matter of definition? and How is definition to form concepts from this matter? We shall first try to become acquainted with the matter. Logic usually subsumes concepts under representations. Ueberweg (1882:147) says: “Der Begriff (notio, conceptus) ist diejenige Vorstellung, in welcher die Gesamtheit der wesentlichen Merkmale oder das Wesen (essentia) der betreffenden Objekte dargestellt wird.” (The concept (notio, conceptus) is the representation which presents the totality of the essential characteristics or the essence (essentia) of the relevant objects). Since it is considered essential for the concept to be general, it will, on this assumption, be subsumed under the general representation. The task, therefore, of definition, would, first of all, be that of making concepts out of general representations. Pre-scientific thinking differs from scientific thinking in that the former operates with general representations and the latter with concepts. The material definition has to work on is therefore the so-called general representation. It has, however, been claimed that there can be no such thing as a general representation. Each representation is said to be individual. Nobody has a general

4. We only have to follow up its consequences in order to understand that we have to separate the mental process of thinking from the logical content of the thought.

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representation of a flower, but only ever of a particular flower, a rose, a carnation etc. and not either of a rose in general, but only of a particular rose with a quite particular form, size, etc. etc. This may be true. Psychologically speaking, it is possible that representations are only determined individually. But if this is so, we have to ask whether the meanings of the words we understand, without the concept being defined, can be called “representations” in the psychological sense. Without going into detail, we can point to the fact that even a scientifically naive person, with no defined concepts available, subsumes things which he has not yet seen under general word-meanings. This is evident from the fact that he designates such unfamiliar objects with the same name as the objects with which he is already familiar. This is not to say that he has made himself explicitly aware of which features are and which are not essential to the object in question, and which are thus constitutive of the general meaning of the word he uses. A word can have several different meanings, so that we think now of this and now of that meaning. What we are thinking at that moment is then highly indeterminate, so that we can say that an indeterminate semantic content is attached to the word we use for designating the objects, or that a word has an ‘indeterminate meaning’. Because of this indeterminacy, there will be wide divergences in the subsumption of objects between scientifically trained people and others. The prescientific classification of whales will always diverge from the scientific one, as the word “whale-fish” already suggests.5 Now, the concept differs from the word-meanings of so-called ‘general representations’ which are indeterminate in the sense specified and are subject to change. They differ because the constituent parts or elements constitutive of the concepts have been explicitly identified, and because only the one precisely specified semantic content may there be associated with the word in question. In this way it has been possible to consider the concept as differing only in degree from the so-called general representation or the word-meaning already existing independently of definition, and to consider it as the culmination of the mental labour which had already started in the prescientific consciousness in the formation of the general word-meaning. We shall leave aside for the moment how true this is. In one sense, however, the concept is certainly quite different in principle from the indeterminate word-meanings as explained here, and this difference is based on the logical value they have for knowledge of what is true. Whereas indeterminate word-meanings yield not only an uncertain knowledge,

5. Translator’s note: The author exemplifies this point by reference to the German word “Walfisch” (whale), which by its German name (Fisch = Fish) may be considered to be a fish rather than a mammal.

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but, from a scientific point of view, may even lead to errors, the concept is distinctive in respect of the fact that we can, with absolute certainty and necessity, see which objects are to be subsumed under it. The limiting value of definitions resides in the fact that they accurately formulate the area of validity of concepts, and the value of the definition as a delimitation depends precisely on the fact that it exactly specifies the applicability of the concept. Human thinking forms concepts by analysing the objects subsumed under general wordmeanings, and, guided by principles yet to be discussed, assembles into concepts a number of “characteristics” of these objects, in the awareness that these characteristics belong together as the elements of concepts. Once this has happened, then it is not the word or the name but the meaning, or the ‘general representation’ attaching to the word or name that has been defined, i.e. it is sharply delimited against other meanings or representations and can now be used scientifically as a ‘concept’. Sigwart, too, concedes that constancy is the distinctive characteristic which differentiates concepts from general representations. Besides, we can obviously form concepts without the necessity of elements required for their specification having to be present together in a so-called general representation. They can be brought together regardless of their origin. Such compositions, too, are a definition, for by this means, too, the concept is precisely determined, in that its content is specified and thereby sharply distinguished from other concepts, so that they can be used in scientific thinking.

Chapter Three: Essential and inessential characteristics 3.1 The inadequacy of existing theories Corresponding to the two forms of concept-formation already indicated, logic distinguishes between ‘analytical’ and ‘synthetic’ sciences. The former, the vast majority, are named in this way because the scientific work in these disciplines starts with an analysis. The material are the ‘general representations’, introduced in the previous chapter, which comprise a wide variety of objects and which already exist before we begin to think scientifically. If they admit of analysis, they consist of complexes of elementary word-meanings, i.e. they already contain some spontaneously generated syntheses of elements which are characteristics of

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the objects in their extension.1 The sciences submit these syntheses to a critical examination because scientists do not want to accept these complexes as given, but to account for the conjunction of their components. It therefore analyses them in order to recombine what appears essential, in the full awareness of the reason for their unity. The synthetic sciences proceed differently. They do not locate their material as described above, but produce it. Their work starts with a synthesis of elements and the basis of the next step of investigation are self-created concepts of objects in which the elements of these new concepts are found as characteristics of objects. The perfect example of such a synthetic science is mathematics. Let us first examine the so-called analytical sciences. Their task, as we have seen, consists in forming specific concepts from general word-meanings or representations, which are indeterminate in the sense described above. This process is also called abstraction because it abstracts or eliminates the purely individual characteristics of objects. The characteristics which are common to all objects, on the other hand, are combined into the concept being specified, as the elements of the meaning content which is to be established. The individual elements are also called accidental or inessential. The elements used for forming concepts and which have to be expressed in their linguistically formulated definitions are called the essential characteristics. The answer to the second question of how definitions have to process general word-meanings or “representations”, formulated in the previous section, would accordingly be provided by logic: definitions must specify the essential characteristics of objects and use them for the formation of concepts. This answer, however, makes a presupposition which must be examined. We had identified the essential characteristics with the common characteristics found in all relevant objects. Now the question arises, which things are to be subsumed under the same concept? What criteria can we apply in order to recognize that these and no other objects are subsumed under one concept? The only criterion we can give without scientific presuppositions is language. We form concepts for those objects which language designates with the same name. But this criterion is far from adequate. Up to a point it can serve as

1. The concepts ‘element’ and ‘characteristic’ are deliberately not rigorously separated here. It would be possible to speak of ‘elements’ of concepts as distinct from ‘characteristics’ of objects, but in order to be linguistically expressed, the characteristics of objects, just like the elements of concepts, must be word-meanings, and to that extent elements of concepts and characteristics of objects coincide. Besides, the whole theory of characteristics presented here is only provisional and will be discussed further in Chapter 4.2.

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an signpost; indeed, if a certain classification had not taken place in prescientific thinking with the aid of general word-meanings, the scientific work of conceptformation in the analytical sciences would not have a starting point. Primitive people have already noted certain characteristics of objects and for this reason the objects which exhibit these characteristics were grouped into a class and given the same name, i.e. they were subordinated to the same general wordmeaning. But scientific research must also examine the reasons why attention has been focused more on certain characteristics than on others, and then it occurs that science often finds it necessary to consider essential other characteristics of objects than those that were selected by prescientific people, i.e. it combines other objects under a common concept, than prescientific thought had previously combined under a common name, e.g. not to count the whale among fishes. What are these criteria? When is a characteristic essential and when is it not so? Traditional logic does not provide a satisfactory answer. It usually states only that those characteristics are called essential which an object shares with the concept which represents it. But the concept could only be formed if one already knew what were the essential characteristics. The answer is circular. If we want to know the proper function of definition and how it has to form the concept, we cannot accept the initially empty answer that they have to state the essential elements of objects. We must examine which characteristics a scientific concept must have, and how they are found to be essential, without letting our thinking be guided by a linguistic designation, or already presupposing the existence of the concept which has yet to be formed. The distinction between essential and inessential characteristics has frequently been considered invalid, for the reason that an intelligence which has understood the whole world will consider everything equally essential or inessential. This may be true. But not to acknowledge the validity of this distinction in logic and especially in methodology would only be justified if one intended to discover a universal method of scientific cognition with the help of which the human mind was to gain control of the world in its totality. The search for such a universal method has, at least for the time being, probably been abandoned. If some champions of the uniquely justified procedure of ‘natural science’ should still think that they are in possession of a method, by means of which one could come to know the ‘whole world’ in its unity, then that is to be taken more as a sign of a low degree of logical-philosophical culture than as a phenomenon to be taken seriously by methodology. Each science has its own method, which it creates for itself and which must be appropriate to its particular objectives. In order to understand what essential and inessential characteristics are, therefore, we must examine each science individually. For a universal method everything

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would undoubtedly be equally essential. For the methods of a particular science with its own limited objectives, only a part of the world is relevant for the formation of concepts and for that reason the distinction between what is essential and inessential is unavoidable. A criterion for this distinction can, however, only be gained from the goals a science has set itself.2

3.2 Definitions in law The purpose of a science cannot always, of course, be specified with the same degree of distinctness. There is, however, one science which has always been well-known for the logical clarity of its propositions and the precision of its concepts, namely jurisprudence. We shall try to show, by means of this example, what it means to say that a definition has to specify the essential characteristics. Jurisprudence can be counted among the analytical sciences. Its material is the sum of thoughts collected under the name of ‘law’; it consists of a set of legal principles containing more or less specific word-meanings which have been combined into judgments. The validity of these judgments depends on the ‘will of the legislator’; for even if historical research has established that, contrary to the earlier assumptions, law does not owe its existence merely to arbitrary determinations, but that the law is discovered rather than created by men, this is of no relevance to our inquiry (Jhering 1873:I, 26). The individual legal propositions must be accepted by man, before there can be a science of jurisprudence. It follows that its validity for the jurist consists in the purposeful, conscious intention, and this is fully the case when the law has been transformed into a particular legal disposition. Whatever its origin, its validity, and hence the necessity of the combination of its component parts, depends on the will of the legislator.

2. A detailed discussion of these propositions can be found in my book about the limitations of scientific concept-formation, (1896–1902, 5th ed. 1929). The main reason can be summarised as follows: The immediately given matter of all knowledge is endlessly diverse or ‘infinite’. By contrast the capacity for knowledge of finite human beings is finite. If, nevertheless, we want to approximate to universal knowledge, we must consider the world from several viewpoints. Only a diversity of methods can do justice to the intuitively endless richness of the world. It follows that all knowledge needs concept-formation, or expressed differently, there can be no purely intuitive knowledge in science. The necessary finitude of human knowledge involves the rejection of all intuitionism in epistemology. Intuitive knowledge would have to be infinite.

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There is no need to describe the full extension of the concept of jurisprudence as science. It suffices to state that one of the tasks jurists have to perform is to ensure that the will of the legislator is always expressed. Or, as Jhering (1873:II,2, 322) says: “Das Recht ist dazu da, daß es sich verwirkliche.” (The law is there to be enacted). We can therefore fully demonstrate the purpose of the scientific task of this part of jurisprudence, and we shall limit our investigation to this type of legal concept-formation. All legal propositions, even when they do not appear in this form, contain a ‘hypothetical judgment’ (Jhering 1873: I, 52; Rümelin: n.d.,9). If someone has done this or that, this or that must happen. It is thus always a matter of a premiss and a consequence, which latter the legislator wants to connect to the premiss. If the will of the legislator is to succeed, it is obviously necessary that the concepts used in the legal pronouncements be sharply and precisely defined. The legal pronouncements cannot be applied, before the appearance of reality subsumed under it, and the concepts used in legal pronouncements must therefore be composed of such elements or characteristics that each appearance, with which the legislator wishes to connect a particular consequence, can clearly be subsumed under the concept occurring in the relevant legal pronouncement. If the premiss in the hypothetical legal judgment is only a general, imprecise wordmeaning, there will always be room for disagreement as to whether an appearance of reality falls under it, and as to whether, therefore, the consequence desired by the legislator can be connected to it. If, on the contrary, their characteristics are precisely fixed in a concept, then one only needs to connect the appropriate consequence to each state of affairs which exhibits the same characteristics as the concept used in the legal proposition in order to be sure that the legislator’s intention is accomplished. The essential characteristics of a legal concept are, therefore, those which contribute to the accomplishment of the will of the legislator, or that “the law be enacted”.3 Legal propositions expressing the will of the legislator must therefore consist of concepts which can be unambiguously related to what happens in reality. In the form in which they are available to jurists, legal pronouncements may, however, be related to transient appearances, and it can happen that the concepts used in them, which at one time were unambiguous, can no longer be reliably used in the changed circumstances, indeed, cannot any longer be

3. Rümelin has a similar conception of what has to be included in a legal concept. But in his explanation, the thoughts referring only to the linguistic formulation are not always clearly separated from those occupied with the formation of the concept proper. This occurs especially in those situations where he adopts Sigwart’s diagnostic definition.

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understood. Jurists, therefore, have firstly to determine the original ‘sense’ of these pronouncements, i.e. to determine the will of the legislator, and secondly to examine the circumstances of reality which are subject to change, if they are to be able to form concepts from these two factors. Once these investigations are concluded, the jurist should no longer be in any doubt as to which characteristics will have to be incorporated as essential characteristics. Let us illustrate this process with a simple example. The legislator decides that forgery of ‘money’ should incur a specific penalty.4 At a time when money consisted exclusively of metal, the intention may have been expressed as: someone who forges coins will be punished in such and such a way. Since it was known that ‘coins’ meant pieces of metal with a particular image impressed on them, the application of this proposition offered no difficulties. The concept of money was defined in such a way that every punishable act was covered by it. Its essential characteristics were ‘minted metal’. The moment paper money came into circulation, this definition became useless and the jurist had to change it. The legislator’s intention was obviously not to punish the forgery of coins as such; but the punishment of the forgery of money was for him a means of securing the exchange of money and similar transactions, and since paper money did not yet exist, the legislator’s intention was unambiguous and could be expressed by defining money as ‘minted metal’. But this definition which was once correct, now contains inessential characteristics and no longer suffices to characterise all the punishable appearances. It had to be transformed so that it would comprise paper money. But there is more. Since it was recognised that it is not essential for the concept of money that it consist of metal or of any specific material, but that it is in its role of medium of exchange that it must not be forged, the specification of the material would no longer be included in the definition of money. Since, in addition, the characteristic of ‘minted’ disappeared, it became clear that the concept of minting was relevant in the earlier definition only to the extent that the falsification of the impression was punishable, as the legislator wanted to punish the falsification of any such minting impression which gave the metal its value as means of exchange; and since, finally, the concept of a certified piece of paper would also include bills of exchange, the falsification of which the legislator wanted to punish in a different way to that of money, it was necessary to add to the characteristic of certification that of publicness, which produced the definition ‘a publicly endorsed means of exchange’. Now we can be sure that everything which is a means of exchange

4. Cf. Jhering, 1873: I, 33, where the same example is used in a different context.

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and publicly endorsed, is covered by the concept of those objects for the forgery of which the legislator wants to impose certain penalties. Thus we have arrived at a new set of essential characteristics for the concept of money. It is self-evident that it is not always necessary to have a new invention in order to change or complete a legal concept. Even without an external cause it is possible to examine whether a definition contains a characteristic which could be omitted or be more general, without affecting the safety of the application of the concept. Here we merely wanted to demonstrate that omitting characteristics or their generalization must find its limit the moment the application of the concept becomes unsafe because in such cases the will of the legislator is no longer clearly expressed. The purpose of the law to be enacted is the decisive criterion whether a characteristic is essential or not for the formation of a legal concept. This example will also have shown that the specification of essential characteristics is in principle different from the theories of logic, which declare as essential those characteristics which concepts have in common with their superordinate class concept or those which are shared by the objects which in language share a common name. Both criteria also apply here. Language has the name ‘money’ for both ‘coins’ and ‘bills’; but the reason for the definition of money as a publicly endorsed means of exchange is not because this is the common feature of the objects which language designates as ‘money’; on the contrary, because they are publicly endorsed means of exchange, both coins and bills are given the name ‘money’, a fact which is still not widely understood. Equally, it is not essential for coins and bills to be publicly endorsed means of exchange because these are the characteristics of the superordinate concept, but the class concept ‘money’ has been given the specific characteristics because we can now safely subsume under it every action which the legislator wants to penalize in a particular way. In this way we have avoided the circular argument normally found in logic when it comes to theories of essential characteristics. We have shown, that in order to distinguish essential from inessential characteristics a specific purpose is required. In this way the claim that definition has to indicate the essential characteristics of objects does make sense with respect to jurisprudence. At the same time we have demonstrated that it is impossible to differentiate essential from inessential characteristic by purely logical reflections, without recourse to a material point of view. It follows necessarily that in the other socalled analytical sciences it is impossible to form concepts without such a point of view, and that in effect nobody can attempt to ‘establish a science without a fundamental idea’. Jhering (1873: II, 2, 311) may be right for this part of jurisprudence when he says: “mit derselben apodiktischen Gewißheit, mit der

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man behaupten kann, daß die Grundsätze der mathematischen Methode für alle Zeiten unwandelbar dieselben bleiben werden, läßt sich ein Gleiches auch für die juristische Methode behaupten” (with the same apodictic certainty with which one can claim that the principles of the mathematical method will remain unchanged for all times, we can claim that the same applies to the method of jurisprudence.) The reason is that this fundamental ‘idea’, to which the formation of concepts owes its existence must always be decisive for them, if they are to have any sense at all.

3.3 The definition of the natural sciences But what is the position of the other analytical sciences, especially the natural sciences? What is the idea which grounds them, and in accordance with which concepts are formed in them, and essential characteristics distinguished from inessential ones? Modern scientific methodology energetically refutes the concept of purpose as a principle of explanation, and it is most probably right to do so. Nor can there be any doubt but that the natural sciences employ their concepts in order to attain ‘knowledge’ in a way which is quite different to that in which jurisprudence employs the concepts we have just been examining. The validity of the concepts of the natural sciences does not depend on the positing of a purpose by the will. Nevertheless, not even the natural sciences can manage without a purpose. They not only need the general purpose, which they share with all sciences, namely that of attaining knowledge; they also, like jurisprudence, have, in addition, particular perspectives which guide them, specify their general purpose more precisely, and without which they could not attain their goal of knowledge. These perspectives, which differentiate the different disciplines from each other, and are subject to much change, cannot all be specified here. Generally, however, we can state the following. What for jurisprudence is the purpose of the law, is for the natural sciences, unless it uses language as the guiding principle for the formation of concepts, either a simple classification of their objects — something one usually tries to avoid because of its arbitrariness — or a theory consisting of general judgments in the form of a scientific ‘hypothesis’. When, for example, chemists define water as the substance the molecules of which consist of two atoms of hydrogen and one atom of oxygen, they have included hydrogen and oxygen as essential characteristics in the concept of water, because the general theory of chemical processes includes them among the so-called chemical ‘elements’, i.e. hypothetically indivisible units; in

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addition, the specification of chemical concepts presupposes molecular and atomic theory as a more general hypothesis. By contrast, those characteristics of water, which emerge when it is compared with steam or ice, are irrelevant for the chemist; but they become relevant for physicists when they examine water from the point of view of the general theories or hypotheses of the states of compounds. Emphasizing the importance of general assumptions or hypotheses for the natural sciences may raise some objections, because anything hypothetical has fallen into disrepute among natural scientists. They would like to deal only with “facts”, and this desire is easily understood from the contrast the natural sciences want to establish with the natural philosophy of the past. A reaction against this latter type of research was undoubtedly justified. But natural scientists are deceived if they believe that they can proceed entirely without hypotheses, i.e. without general assumptions which contain more than facts. The individual sensible appearances that have to be collected under concepts are endlessly diverse as individual intuitions and would thus mock any scientific treatment if they could not be structured and simplified, so that the essential is retained and the inessential left aside. Such a conceptual structuring, however, of necessity requires a general guiding principle. For this reason, the natural scientist is often in a position of having to adopt a perspective, which enables him to distinguish essential from inessential characteristics of objects, and thus to form concepts, e.g. the decision to consider the number of stamens in a flower as essential characteristics. Such an arbitrary and purely classificatory concept-formation was clearly recognised as unsatisfactory. The great success of Darwin’s thoughts are partly explained by the fact that he provided a perspective for the conceptual treatment of the organic world according to which the essential could be separated from the endless riches of the numerous appearances, each of which was itself an endless and manifold intuition. It is often said that Darwinism has made definitions in the disciplines of botany and zoology impossible. Quite the contrary is true. It was evolutionary theory which first rendered the formation of proper scientific concepts possible, for it replaced an arbitrary classification with a well-founded hypothesis, which provided a‘natural’ perspective for the specification of the essential characteristics of things and their synthesis into concepts, regardless of what one may think about Darwin’s hypothesis. Now, the different forms of the organic world are no longer accepted as simply given and grouped into classes, but attempts are being made to understand them as the different links in an evolutionary process and to establish relations between them, based on the recognition of a universal, general “causal” or law-like system. One no longer groups phenomena together under a

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single concept because they seem to belong together by virtue of certain external characteristics; one rather gains the perspective necessary for the conceptual articulation from the observation of the different stages in the evolutionary process, this or that organism presenting itself to us as an exemplification of such an evolutionary stage of development. It is hardly necessary to make explicit that Darwin’s hypothesis loses all its value when it is applied outside the field of biology. But even such transgressions can be logically interesting. Current attempts to separate the essential from the inessential in the fields of philosophy and history according to Darwinian principles, or the efforts to base ethics on biogenetic laws exhibit the same methodological irresponsibility as the speculations in natural philosophy of the early nineteenth century which were characterised by uncritical generalisation. Moreover, and perhaps more interesting for our purpose is the fact that these attempts also reveal the same striving for unity of the human mind which requires a point of view according to which it is possible to grasp what, from the whole realm of his consciousness, is essential and can be combined into concepts. A closer examination of the process of concept-formation in the separate branches of the natural sciences would not add to the clarity of the argument, with which we are here concerned. It would always be a matter of pointing out the guiding perspective of a particular field of investigation and of observing how the concepts of the respective science incorporate what, according to this perspective, is considered essential. Without a principle of selection the distinction between the essential and inessential would lose its sense, and without this distinction there would be no science. Let us confirm this statement with an example, examining, in particular, how the inessential is separated out by Cohnheim’s definition of disease. Pathology is the science of diseased life, and so the question arises, what is disease? A layman will simply define disease as the opposite of health, and because he has a rough idea of what a healthy person looks like, he will be satisfied with this definition. Everything which is not healthy, is diseased. Such a definition is insufficient for science. Initially it can hardly do better when it says that disease is an abnormality. But even if the concept of normalcy were to be defined precisely, this definition would not be sufficient. A person with a harelip represents a deviation from the normal type, but is not sick. Once we know the guiding principle of the science with respect to which disease is to be defined, and once we know that pathology is intended to achieve for diseased life what physiology produces for healthy life, it is obviously essential for the concept of disease that it be defined as an abnormal process, not just any abnormality. Now we can understand that a harelip does not fall under the

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concept of disease. Since it is essential for the concept of disease that something happens, Cohnheim (1877:3) defines disease as a “Abweichung von dem regelmäßigen, d.h. gesunden Lebensprozeß” (deviation from the regular, i.e. healthy process of living). With this definition abnormalities such as a harelip are excluded from the number of diseases. The discussion of concept-formation in the analytical sciences has shown that only the guiding perspective of a specific science can provide the definitive criterium for determining what part of the empirical material should be combined into a scientific concept. In this way we have arrived at an identification of essential characteristics which is free from any metaphysical presuppositions and which is only founded on the fact, or let us say hypothesis, that a universal method for understanding the world in its totality has not yet been discovered. The consequence of this conclusion for the natural sciences is a certain relativity of concept-formation not only because the addition of new empirical material can alter concepts — this obviously happens with all sciences — but also, because the guiding perspectives of the individual sciences can change; for example, the total restructuring of biology as a result of Darwin’s hypotheses. But these circumstances do not permit a challenge to the explanation presented here. If knowledge of reality is to advance, it must not allow its conceptual apparatus to become fossilized.

3.4 The definitions of mathematics Turning now to the synthetic sciences, especially to mathematics, it is obvious from the start that the formation of definitions must take place in a different manner to that in which it happens in the analytical sciences. At the start of its investigation, mathematics does not have a given material from which to select what is essential for its concepts. Mathematics creates its own material from which it follows that it will create nothing inessential in the sense already specified, which would have to be omitted later. There can be no question, therefore, of concept-formation by means of abstraction. Though it might appear that geometry, in contemplating its figures, would have to abstract from the material on which they are drawn and from the colours with which the drawings are made, and, indeed, also from the imperfectness of the lines and points drawn, which, after all, are always coloured surfaces. But this kind of abstraction does not coincide with the types we have examined earlier. Before I can draw a geometrical figure, I must already have formed the concept of that figure. If elements which are not essential to its nature get mixed in with its sensible

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presentation at a later stage, this does not affect the concept and there is no need to abstract explicitly from them such inessential representational elements. The position of the extreme empiricists is quite different. They describe mathematics as another analytical discipline and therefore consider the logical character of mathematics as a very uncomfortable negative counter-instance of their theories. Even if we were to agree with them, there is no need to give special consideration to mathematics because we are here unconcerned with the epistemological and transcendental philosophical aspects of these theories. In such a case, mathematical definitions would not differ from inductively obtained definitions in the natural sciences. But if we postulate that mathematics does not arrive at its concepts by means of abstraction, but by construction — for some concepts this cannot even be denied by the most ardent sensualists — it is clear that there can be no distinction between essential and inessential characteristics of objects in the meaning of the words already considered. Most writings on mathematical definitions belong to the field of epistemology and transcendental philosophy which are concerned with the problem of the truth or “objectivity” of the concepts. Leaving this question aside, a mathematical concept can be formed by any means: every element in it will be equally essential and it is therefore impossible for methodology to establish rules for the formation of mathematical concepts, similar to those for the concepts of the analytical sciences. This statement should not be misunderstood. It does not mean that there are no fixed logical boundaries for the formation of mathematical concepts, from which it would follow that mathematics was a game without scientific value. We only wish to state that mathematical concepts do not, like the concepts of the natural sciences, relate to real sensible objects, from the immense diversity of which certain characteristics have to be separated out as essential. It is rather the case that mathematical concepts deal with an ‘ideal’ being, in which everything is equally essential or in which the difference between essential and inessential disappears. For this reason a mathematical concept can never be formed ‘wrongly’ in the way in which a legal of a scientific concept may contain inessential characteristics of the objects falling under a given concept. The problem of the correctness of mathematical concepts lies completely outside our methodological investigation of definition. It has been mentioned here only because there is a relatively large body of literature on mathematical definitions dealing with the so-called nominal and real definitions and discussing the attempts to define the simplest mathematical concepts like a straight line. We shall have to return to mathematical definitions in another context. In this chapter on essential characteristics, it was only necessary to state that there is a type of concept-formation which can be called free construction by means of synthesis

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of elements of concepts which is not preceded by an analysis which would have to separate out the characteristics essential for the formation of a concept from a mass of partly inessential ones. The work which precedes the mathematical synthesis relating to elements of mathematical concepts, is not a methodological problem of the theory of definition, and therefore does not belong here.

Chapter Four: Definition and concept 4.1 Analytical and synthetic definition So far we have encountered two types of definitions, which, according to the type of science in which they are used, are commonly called analytic and synthetic. But these names are relatively uninformative. In Ueberweg (1882:172), we find the observation that the difference does not so much lie in the nature of the definition as in the mode of its origin in the subject. But even this is only partly true. Since definitions, inasmuch as they are formations of concepts, are always only assemblies of characteristics, the genesis of the definition cannot, strictly speaking, be called analytic. Rather, the distinction lies in the preliminary work for concept-formation: in the case of the analytical definition, the material of elements has been obtained from ‘general representations’ by analysis and the elimination of inessential characteristics; in the case of the synthetic definition, elements, which had not previously been brought together into a ‘representation’ and therefore did not have to be analyzed, are assembled into a concept. It seems inappropriate to call a definition analytic after a thought-process which must, necessarily, precede it but which must also be completed before the actual specification of a concept can be undertaken in the definition. It will be seen as even less suitable, when we consider that in the process of forming a concept, a definition always represents a synthesis. In its concept forming function, the analytical definition would accordingly be an analytical synthesis. But even this designation will have to be abandoned if we can demonstrate that these two types of definitions can be contrasted with another thought-process which we have not as yet taken into account. This act combines definition with analysis and should therefore more properly be called analytical definition. At the end of Section 2.1 we have shown that, before a definition can find its linguistic expression, it must be preceded by a logical thought-process by means of which the concept is formed. So far we have only been concerned with this process of concept-formation, and definition, as far as it has been dealt with, presented itself as the synthesis into concepts of the essential characteristics

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of objects. These concepts are supposed to serve as the subjects or predicates of judgments, a system of such judgments constituting what we would call a science. A science does not come into existence all of a sudden, but by a process of constant elaboration of experience or through the construction of concepts. Every concept constitutes one of the building blocks for the construction of the system. It crystallises, as it were, the result of scientific work. This was the point we had arrived at. But this is insufficient to conclude our study of definitions. If its significance for scientific knowledge is to be fully recognised, the concept must be analysed into its constituent parts. This is the place for definitions, in the widely accepted and almost exclusively considered sense of concept-analysis. In this form definitions occur as propositions which specify the meaning of a word. But even then a definition is not an explanation of a word but it ‘explains’ the concept evoked by the word. This is necessary because the concept as such, which, as we know, is a synthesis of the essential characteristics of objects, remains unproductive for scientific investigation if it is only considered in its function as summary. Definition, qua synthesis, has deposited into the concept the results of preceding scientific work, and it is meant to store these results until they are needed for further work. But if one is to be able to use them, one must extract them from the finished rigid concept, to bring it back to life, as it were. The act of thought which analyses the concept to this end, can but be designated analytic definition, to distinguish it from concept-formation, or synthetic definition, which has preceded it. From now on we shall use the expressions analytical and synthetic definition in quite a different sense to that in which logic uses them. What is at issue is not the differentiation of two types of sciences, of which one starts its investigations with the analysis of given objects, and the other with free constructions by synthesis of elements, from which processes they have derived the name analytical and synthetic; what is rather at issue are two acts of concept-formation and concept-analysis which occur uniformly in both types of science and which achieve in mathematics exactly what they achieve in any empirical science. These two acts of thought must be sharply distinguished from each other, and we need two separate terms to designate these processes, of which the one is a synthesis of the essential characteristics of objects into a concept, and the other an analysis of the concept into its characteristics. After what has been said, it is obvious that an analytical definition can only occur if it has been preceded by a synthetic definition, and that it therefore includes the synthetic definition. We can, therefore, use ‘definition’ without further qualification to designate the entire act of thought which includes both synthesis and analysis.

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4.2 Concept and judgment Before leaving synthetic definition as the preliminary stage to proper, analytical definition, and determining the rules for definition as concept-analysis, we have to examine the place occupied by these two acts of thought in the system of logical forms in general. This does not require much more than drawing a few conclusions from what has already been said. Up to now our discussion of definition has always referred to the act of defining. But as we know, language also uses the word definition to designate the product, the definitum, resulting from the act of defining. After what has been said, it is undoubtedly the case that definition in this second sense is completely identical with the concept. Obvious as it might appear, this insight was first explicitly expressed by Sigwart (1911: 387). He said “Eine Vorstellung ist nur dann ein Begriff wenn sie klar ist, d.h. wenn, was darin gedacht wird, vollkommen bewußt ist. Die Definition ist also der Begriff selbst, nicht etwas vom Begriff Verschiedenes.” (A representation is a concept only when it is clear, i.e. if what is thought in it is completely conscious. The definition is therefore the concept itself, not something different from the concept). This sentence of Sigwart is not consistent with his determination of definition as the mere clarification of a word, but it is certainly true. It is only surprising that Sigwart, who except for this instance, calls definition a judgment, did not draw the conclusion which was not only obvious but represents a development of logic in the direction he had already taken. A definition, as product or definitum, is, as we know, a concept. What then is the concept itself? When we examine the process of concept-analysis, it presents itself to us in the form of a judgment, and the analytical definition, usually simply called definition, is, in logic, also considered a judgment, where judgment means the whole thought-structure meant by or corresponding to a declarative proposition, as distinct from the meaning attached to a single word which represents only a part of the logical content. In most cases, strictly speaking, definition which lists several characteristics should be called a complex of judgments, for the specification of a characteristic is always a judgment. Definition is thus a complex of ‘analytical judgments’ which explicitly emphasise what was already thought in the concept. Accordingly, analytical definition converts the concept into a judgment or a series of judgments, the subject of which is always the concept to be analysed and the predicates of which are the characteristics which the synthetic definition has incorporated in it as essential. When we now see that the specification of the content of a concept, which

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logic calls the enumeration of its characteristics, consists of a series of judgments, we can conclude that synthetic definitions, which we have described as combinations of characteristics, must also consist of a series of judgments. We are not so conscious of this, because we never have reason to express this act of concept-formation linguistically in propositions, but it is obvious, that the synthesis of elements can only occur by means of judgments. Therefore, synthetic definition, which assembles the characteristics, appears to us as the act of thought which analytical definition only needs to reverse, in order to analyse the concept into its judgments. Consequently we can describe synthetic definition as the transition from judgment to concept, and conversely, analytical definition, which isolated the characteristics again, as the transition from concept to judgment. Now, we know that the logical ideal of our knowledge consists of a complete system of judgments, the subjects and predicates of which are constants, i.e. defined concepts. Let us imagine that this systematisation of our knowledge has been complete in every direction. In this case we can compare the content of our knowledge with a network of threads, in which the nodes are the concepts, while the threads which connect the nodes are the relations between concepts, i.e. the judgments. If one thinks of the threads as tending in the direction of their nodes, one has an analogy for synthetic definition; for here we have judgments combining into a concept. On the other hand, viewing the matter differently, one could construe the threads as in a way radiating from a node in various directions, and this would yield the likeness of an analytical definition, for here the concept is analysed into its judgments. Human thought, were we to imagine the scientific systematisation of its content as complete, would never “intuit” that content in its totality, or grasp it intuitively; it would only ever be able to traverse it either forming concepts from the elements related to each other, i.e. judgments, or by analysing these concepts into judgments again, in other words it would always proceed discursively. Human thought, strictly speaking, would move only in judgments, a fact which illuminates the theory of concepts. If judging seems to be the basic function of our thinking, with which we grasp the truth, then concepts, just like the nodes in our network which consist only of threads, are nothing but transit points of intersecting judgments. Living thought cannot, in truth, linger for a single moment over a single concept. It can only ever form a concept in judging, only to analyse it immediately afterwards in judging again; and, if knowledge had been completely systematised, it would only ever move in synthetic and analytical definitions. Therefore, from the point of view of its logical content, a concept does not differ from the judgments that constitute it. It represents at best the ideal point at which individual judgments intersect. If one leaves aside the judgments, we are left only with the thought

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that the judgments should be thought together into a unity. This demand, however, to think of the concept as a unity, is beyond the capacity of human thinking; we can, accordingly call the concept an Idea, in the Kantian sense, the idea, namely, of a task for human thinking, which, as soon as the circumstances have become clear, must be accompanied by the awareness of its insolubility. Whenever we speak of the concept as something unitary and constant, we create, strictly speaking, a fiction, albeit a fiction of great logical value. We behave as if we had executed a task which we can never perform, and therefore it is best to describe the concept as a complex of judgments thought of as at rest. This view, which equates the logical content of the concept with the content of the judgment, contradicts the traditional teachings of logic.1 Usually the concept is regarded as an earlier stage of thinking, and the judgment as a relation between two concepts. It may seem paradoxical to say that the defined concept is, in respect of its logical content, nothing but a judgment in an unusual form, a judgment which, so to speak, has been put aside for later use; and if, according to what has been said above, one also perhaps admits that the definition as product differs from the act of definition only in respect of the fact that one tries to construe the judgments in the definition as units, one will regard the concept itself as something differing from the judgment in an entirely different manner. We shall try to explain the reason why this erroneous conception is widespread and, in doing so, to clarify our own position from another angle, so as to arrive at a definitive view of the definition and its logical significance.

4.3 The inadequacy of the existing theories of the concept We know that logic considers the concept as a general representation which differs from other general representations by its constancy. It is customary to illustrate the relationship between a logical concept and a general representation

1. In this respect, this tradition has, in principle, already been abandoned by some authors. In my opinion, Sigwart (1880:456), for example, is necessarily forced into the view of concepts as here explained, which, however, judging from his comments against Wundt, he does not seem inclined to admit. Schuppes (1878:117) expresses the opinion that concepts consist of judgments. Riehl (1876:224) also says: “Concepts are potential judgments. Later, however, (1892:14) he states that “definitions are not assertions, even though they have the form of assertions”. Consequently, in the new edition of Philosophischer Kritizismus (1887: II, 259), the sentence has been altered to “concepts are potential definitions”. Windelband (1884:180) has initiated a modification of the traditional theory, but from a different point of view. For further references see also: Lask (1912: 49ff).

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by examples from the descriptive natural sciences, just as we have done. But one must be careful with these examples, for if one uses the concept of a plant or an animal to evaluate these relations, a specific sensible image may easily intrude on what we thereby think, and propositions about the concept may then give the impression that the concept is not a general representation in the sense of a word-meaning, but a general, though precisely specified intuitive image. If, for example, one speaks of the general representation of a tree, on the one hand, and of the concept of tree, on the other, and tries to grasp one or the other, concept or representation, one may perhaps succeed best if one imagines the intuitive image of a tree while thinking that this or the other property of the image is irrelevant. In the case of the representation the relevant properties are left undecided; today one may think of one property and tomorrow of another. In the case of a concept, on the other hand, the ‘characteristics’ are permanently and precisely specified as the essential constituents of the thing. As long as we are dealing with similar things, such as trees, this way of looking at the problem seems quite convincing. But is it possible to treat the problem exhaustively with such examples? Lotze (1880:49), whose theory of the concept has corrected many misconceptions, already explicitly distinguished between general concepts, which we grasp ‘in an intuition’ and those which we grasp only ‘in a thought’; but, according to him, even these latter can only lead to a quite deviant, though ‘to an intuitively quite deviant formation’. Here intuition still plays an important role in the concept, and even though Lotze has raised himself far above the traditional theory of the concept, he did not go far enough. He clearly felt the inadequacy of the view that a concept is a sum of characteristics. Above, we have already accepted his proposal to replace the formula S = a+b+c… by the formula S = F (a,b,c,…) which indicates that, in order to yield the value of S, a,b,c must, in the individual case, be combined in a precisely specifiable way and, in general, in very many different ways. But this does not achieve a great deal. The construction of such formulae cannot in principle bring us closer to a true understanding. The apparatus of letters, circles, etc., invented by logical formalism, may have a certain pedagogical value, but one may easily be tempted to think that such formalism has succeeded in grasping the true essence of logical thought, and this may lead to the gravest errors. After all, Fr. A. Lange (1877), in all seriousness and with a great deal of astuteness, tried to demonstrate that the stringency of syllogisms was based on the mathematical intuition which impresses itself upon us when we consider the diagrams used for explaining the different syllogistic figures. To see the whole theory of characteristics and the related theory of the concept as a general and precisely specified representation is schematic and

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superficial. Let us take something other than animals and plants as our examples, say ‘disease’. Of this too one can have an representation and a concept. One can grasp the representation only by imagining a sick person, and, indeed, one who is suffering from a particular illness, while once more deliberately neglecting this or that individual characteristic. But the concept of disease is about something quite different. If, for example, Cohnheim (1877:6) defines disease as the state of the body in which the “deviation from the regular, i.e. healthy process of living” is so strong that “the controlling facilities of one or more conditions are no longer adequate to regulate the sequence of the different bodily functions without disruption”, it is difficult to understand how the conventional teaching about characteristics can cope with such a concept and how one can still speak of some sensible intuition or other as an essential element. If we have understood it, this definition will evoke above all a series of physiological laws, and if, in the process, we still intuit something, the essential element does not reside in these intuitions but in the type of relations which we have introduced between these intuitions. We have to free ourselves from the opinion that concepts are a matter of a clearly represented, intuitive sensible image and become aware of the fact that we shall have only fully grasped something if we can do without sensible intuition. The examples from the descriptive sciences, from where the whole characteristic-theory originates, show how little the traditional theory of the concept has grasped the process which we call scientific interpretation. When we know that the horse belongs to the class of solipeds, have we understood what a horse is? Is such a specification of a concept more than a makeshift which was introduced simply because there was no truly scientific principle which provided more than a purely external classification? Does not the whole division into classes, genus and species, etc. represent, as we have already seen, merely a crude attempt to gain an oversight over the organic world? And yet, logic persists in choosing its examples from these sciences, which form the lowest stage of human knowledge. Lotze rightly pointed out that the subordination under the general ‘animal’ or ‘plant’ does not really subsume an object under a concept, but only delays the task since animal and plant are only general images. But Lotze does not tell us what a concept really is. He spreads a vague light which permits us to see that not everything is the way it is normally presented, and that in itself is quite stimulating; but no bright ray of light falls upon things, which would permit us to understand how things really are. Apart from the

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exceptions mentioned above,2 the theory of the concept does not seem to have progressed, in principle, beyond the theories of the time when science, in fullest accord with logical doctrines, was hoping to produce gold by boiling up together its “characteristics” such as weight, glitter etc. in a crucible. It is not our task to provide here a detailed theory of the concept. We only wanted to show that the concept is wrongly construed if it is replaced by an intuitive schema or a picture-like outline, into which the characteristics are fitted and from which they can then be read off. We wanted to show that one cannot understand the theory of definition as a tool for scientific conceptual interpretation, if one tries to understand it by means of examples like “a man is a featherless biped”. The additional feature, which most people seem to find in concepts as opposed to definitions, is the intuitive-sensible image, which is always introduced into the standard examples of logic; but this is quite inessential and therefore has nothing to do with the concept itself. We must consider the scientifically defined concept as a distinctive form of judgment. Taking examples from the explanatory sciences, it would be easy to show how concepts can be so completely transformed into judgments, that nothing remains but the thought that these judgments should form a unity. Concepts from physics, such as gravitation and other concepts of physical laws clearly illustrate this point. In its scientific content, the concept of gravitation is identical with the law of gravity, and laws are always judgments. But this example, especially, could raise a doubt, by creating the impression, namely, that our theory only embraces a part of scientific concepts. For this reason, we shall add some further considerations to elucidate our views, and refute objections. We have stressed that the essential content of the concept does not consist of intuitive images which may easily occur in the process of understanding wordmeanings, but is to be found in the relations which we think have been established between the intuitions, or that, to use conventional terms, what is at issue in the case of the concept is not the ‘representations’, but the ‘relations of representations’. In this way we seem to have presented an opposition between relation-concepts and thing-concepts. Sigwart (1890:49–55) has linked an objection to our account of this matter. Although the theory, developed here, that the concept is to be construed as the meeting-point of judgments and that the concept unfolds at this point, the theory goes too far. “Was sollen, wenn jeder Begriff nur ein Komplex von Urteilen ist, die Subjekte und Prädikate dieser Urteile sein?” (If every concept is only a complex of judgments, what are the subjects and predicates of

2. See: Note 1 of this chapter.

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these supposed to be?). What is overlooked is the fact that “daß in unseren Begriffen bestimmte Synthesen enthalten sind, die allein möglich machen, eine Anzahl von Urteilen wirklich in eine Einheit zusammenzuschließen” (that our concepts contain determinate syntheses which alone make it possible to join a number of judgments together into a unity). Admittedly, the concept of gravitation is identical with the law of gravity, “so ist er es nur darum, weil er ein Relationsbegriff ist, kein Dingbegriff; er setzt gravitierende Massen voraus” (but this is only so because it is a relation-concept and not an thing-concept; it presupposes gravitating masses). In short, Sigwart thinks our presentation is onesided because it only stresses one, in itself correct, point of view. In order to evaluate this objection we have to separate two arguments. Regarding the distinction between thing- and relation-concepts: this must not be confused with the distinction between ‘representations’ and ‘relations between representations’. Not only does the concept of gravitation, as the concept of a relation between masses, consist of judgments, but the scientific content of the concept of ‘mass’ is itself a complex of judgments, as far as it is a defined concept of mass rather than an indeterminate general representation, in the sense we have explained. Or, expressed more generally, not only are concepts of relations concepts, in respect of their logical content, compound out of relations, but a concept, consisting of relations or relations of representations, can also be formed of any other object at all, and thus of a thing as well. This in itself already proves that our theory is not only applicable to relation-concepts, i.e. concepts of relations. One must not confuse the structure of the content of a concept with the structure of objects, which fall under the concept. The definitions of concepts of things also transform the ‘general representations’ of things into judgments, in the same way as the definitions of the concepts of relations do. In this context, the distinction between thing- and relation-concepts is irrelevant, however important it may be in other respects.3 But this does not refute another more general objection raised by Sigwart. If every concept is a complex of judgments, what then are the subjects and predicates of these judgments supposed to be. This is a justified question and may then lead to the view that one would, finally, have to arrive at concepts which could no longer be transformed into judgments. But this objection does not affect the theory of concepts which we have advocated here, for by ‘concepts’ in this connection, we only understand defined

3. See also: Rickert (1929), especially p. 66ff., the section: ‘Dingbegriffe und Relationsbegriffe’, where I discuss Sigwart in some detail.

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concepts. This is unobjectionable, even from Sigwart’s viewpoint, because he too distinguishes the concept from general, as yet unspecified representations. It is therefore certainly also true that the analysis of concepts into judgments cannot be continued indefinitely, and that consequently not all judgments have subjects and predicates consisting of defined concepts, i.e. of judgments. We must, indeed, eventually arrive at judgments, of which the subjects and predicates are indefinable elements of our knowledge. But our theory does not have to fit these ‘concepts’. We distinguish between defined concepts and simple word meanings as indefinable elements of concepts. Then everything fits, because our objective here was to show that the concept, in as far as it is defined, consists of judgments. The truth of these statements is unaffected by Sigwart’s objections. In order to attack our theory it is necessary to abandon the field of Sigwart’s logic. According to Sigwart (1911: 68, 104 & 162) a judgment is a synthesis of representations made in the consciousness of objective validity, and since the valid relation of the representation also constitutes the logical content of the defined concept, the scientifically valid concept must, according to Sigwart himself, have the form of a judgment. The distinction then lies only in the linguistic expression and can then be reduced to the logically inessential distinction between word and sentence. But, if the logical content of the judgment, with which we are concerned here, is meant to be something fundamentally different from the logical content of the concept, it would have to be proved that judgments are more than valid syntheses of representations. Then, it would indeed be the case that we have only followed the tradition which considers the judgement to be a synthesis of representations, and shown concepts to be a form of judgment. If we want to arrive at a comprehensive theory of concepts, we shall, finally, also have to face the question whether judgments can be conceived simply as relations of representations or not, and to do this, it seems to be necessary to abandon tradition also in this respect. It is, in fact, possible to prove that for each true or false sense of an assertion, i.e. for every content of a judgment, there belongs a ‘yes’ or ‘no’ which adds itself to the relation of the representations as a new element.4 This is decisively important for the relation between concept and judgment, for it is precisely this ‘yes’ or ‘no’ which seems to be missing in concepts. In this respect then, the defined concept cannot count as a complex of judgments.

4. See also: Rickert (1828:165ff.) and the references cited there. In this essay I have discussed in some detail Sigwart’s theory of judgments which recognises the ‘no’ as a fourth element beside the subject, predicate and copula, and which denies the corresponding ‘yes’ in positive judgments.

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This raises a completely new problem. A well-founded answer to this question would, however, take us far beyond the limits of an inquiry into definitions. Here we only wanted to show that the defined concept already contains the relation of representations, which is widely held to be decisive for the logical essence of judgment, and that, to this extent, there is no difference between judgment and defined concept in respect of its logical content. We have only implicitly referred to the ‘yes’ or ‘no’ in the sense of each true judgment, and to the corresponding acts of affirmation and negation, when we called the concept a complex of judgments “thought as resting”, and therefore separated it from the “living” judgment; the explicitly performed acts of affirmation and negation must obviously be absent from the concept. Nevertheless, this circumstance does not exclude the logical content of concepts having the same validity as the judgment in which the affirmation or the negation have been performed and thus are to that extent living.5 Indeed, this validity will have to belong to it if it is to have scientific value, i.e. truth, and our theory has been developed only for scientifically valuable concepts, not for arbitrary ‘complexes of characteristics’. We can therefore continue to maintain that the logical content of the defined, scientifically valuable concept consists in the logical content of judgments.

4.4 Concept and word But if the defined concept is, in respect of its logical content, nothing apart from the judgments which form it, and if the task of combining these judgments into a unity cannot be performed, what is the relevance of the concept for our knowledge? What does it mean to think conceptually? Conceptual thinking would, indeed, be impossible, if there were not another element, which has not so far, in this context, been explicitly considered, additional to the judgments thought of as resting. This element is language, which up to now has been deliberately kept in the background because we had to separate definition as specification of a concept from definition as the explanation of a word. We can now easily understand the special significance which the word has for the conceptual thought-process, quite apart from its

5. In Rickert (1929), I have presented the proof that scientific concepts also have the validity of judgments. See, in particular, the section ‘The validity of concepts’, p. 52ff. In this essay I have also tried to show that the theory of concepts presented here can be proved and represented independently of the theory of definitions.

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function in the communication of thought. The unity of the thought, which we have recognised as an insoluble problem, is replaced by the unity of the word. We would, indeed, never be able to arrive at such complex conceptual thoughtprocesses if we did not have language at our disposal for designating the complexes of judgments, which we would never be able to grasp as unities, by means in each case of a word, which replaces this concept, which is eventually inseparably associated with the judgements thought of as resting, and which can now be used as a building block in the process of thinking. With the assistance of words, we can readily use the results of scientific investigations which have been combined into a concept, for, from the meanings attaching to them, we can, by means of their synthesis, form new judgments and eventually establish a whole system of judgments, the subjects and predicates of which are complexes of judgments, and the necessity of which arises of its own accord, as soon as we dissolve the complexes into their separate judgments, i.e. as soon as we define the concepts, and thereby make manifest their relations to other concepts or complexes of judgments. On the other hand, the fact that each concept must necessarily be designated by a word, is another explanation for the fact that even with a defined concept, the essential logical content of which is free from sensible intuition, we believe that we are still dealing with a unity; the word alone constitutes the unity and obscures the fact that, apart from the language, one only has judgments before one which, when explicitly expressed, have to take the form of propositions. It is probably unnecessary to point out that our theory is not ‘nominalistic’. An insight into the true logical nature of the concept may, however, help us understand why nominalism is still being defended with tenacity and a certain plausibility. If the question is asked what really corresponds to the general concept, one does not find anything general there because everything real is individual. For this reason people tried to find the ‘essence’ of a thing, which a concept is supposed to express, in the word and thought that the general was merely a complex of sounds. We recognise, however, that the word is merely a means enabling us to use a complex of judgments as something unitary and permanent in the process of thinking and that the general consists in the content of judgments. Let us return to the definition which we have called analytical and the nature of which we can now easily understand. As we already know, it is the judgment, which extracts from the concept the results of thinking deposited in it, and we now understand that in the sentences, by means of which it is linguistically formulated, the grammatical subject must always be the word which forms the substitute unity for the judgments thought of as at rest. In this sense, it is

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correct to say that a definition is a definition of the word. But only in this sense. For this reason we shall continue to consider the expression ‘word-definition’ to be unsuitable, for here the word has only appeared as external means and could be replaced by anything else. It is not the word-explanation which is logically essential, but the analysis of its meaning which consists of judgments, i.e. the specification of the concept. We also see that in respect of its linguistic formulation, the analytical definition has rightly been called an ‘identical judgment’, for it explicitly conveys, in a series of judgments, the same logical content that is implicitly understood by the word, and presents itself therefore as an analytical judgment in the Kantian sense. But from this fact it does not follow, as Sigwart (1890:54) believes, that the definition is not an explanation of a concept, but only the explanation of a word, because, on the one side of the equation, we do not have only the word, and, on the other, its explanation. The logical sense of the defining sentence is rather that, on the one side we have to imagine the meaning of the word as a unity of the concept, and, on the other, by contrast, the same meaning analysed into its elements. On this assumption, and only on this assumption, is it then correct to say that every linguistically formulated definition must be reversible; its subject designates the same thought-content as its predicate, only in a different form. It is indifferent to whether the thoughtcontent appears as subject or predicate in one of the two forms. In each case they are judgments, in the one case, thought of as at rest in the concept; and in the other case, as explicitly realised and enumerated.

Chapter Five: Genus proximum and differentia specifica 5.1 Genus and essence in the empirical sciences After having seen what definition has to be in respect of its logical content, if it is to be more than a mere word-explanation, we shall now investigate the form in which it customarily appears. As is well known, in the act of defining we do not separately list all the judgments contained in the concept, but name another concept with the aid of a word and then add one or several judgments. This may happen simply for practical reasons. As we observed above in the discussion of the linguistic formulation of a thought, methodology would then only be able to demand that one should proceed as expediently as possible, i.e. to choose the concept one gives first in such a way that it contains as many judgements as possible about

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the concept to be defined, so that as few as possible have to be listed separately. But logic is not content with this demand. Aristotle (Topics I, 8, 103b 15) says O}ismoÌ| eÎ~ geÈnou| ~aiÌ diajo}w Í n eÎsti (the definition consists of the genus and the difference), and modern logic too requires the specification of the concept of the genus and the specific difference. We must try to understand this demand and shall first limit our examination to the so-called analytical sciences. We see first that there are cases in which the specification of the genus proximum may only be demanded for reasons of external convenience, provided that the content of the respective science has a form similar to Plato’s pyramid of concepts. In Linné’s system, for example, it is convenient to specify plants or animals by means of genus and species because this is the shortest way of indicating the position of the respective organism in the system. The requirement that one specify the genus is explained by the fact that this form of definition is also the easiest. But, as already mentioned, this complete articulation of the descriptive sciences is only the result of a more or less arbitrary classification, and even though it is very useful, because it permits a clear overview and an easy allocation of most newly emerging phenomena, it contributes almost nothing towards our knowledge of the essence of things. We must therefore leave aside the systems of the descriptive natural sciences and ask what value the specification of the genus proximum has in sciences which do not attempt to classify their objects. Aristotle demands that one specify the genus, precisely because he considers it the expression of the ‘essence’, and by subordinating an object to the genus, the definition should also therewith grasp the object. Now the question arises: apart from external reasons of convenience, does the specification of the concept by the genus have, in addition, any independent theoretical value for those who have abandoned Aristotle’s metaphysics and who can no longer demand that definition should specify a metaphysical essence in the traditional sense? The word ‘essence’ has several meanings. Methodology has no use for it if it means the archetypal ground of things, as it does for Aristotle, or absolute being in contrast to the merely given existence of things, as it does for Hegel. But the empirical sciences also say that they want to explore the essence of a thing. By this they apparently mean the highest degree for which they can possibly strive, namely, insight into the coherence of the laws of nature. Does the specification of the generic concept, as the specification of the essence of an object, make sense for the empirical sciences, so that we would be entitled to retain the Aristotelian formula for modern logic? We shall answer this question by means of two examples. When zoologists define a dog as a mammal with such and such characteristics, we cannot possibly

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say that this specifies the ‘essence’ of a dog; but we have already remarked that the descriptive natural sciences do not provide the examples necessary to understand the process of true scientific knowledge. Let us rather clarify what physicists achieve when they subordinate a phenomenon under a concept and let us again choose the concept of gravitation. Everybody knows, even without physics, that bodies have the tendency to move towards the earth, and that they fall unless they are prevented from doing so. So we say that, everybody has a ‘general representation’ of this fall of bodies because he has often seen stones, apples, feathers or other bodies drop and knows among other things that some fall faster and others more slowly. From this general representation of ‘falling’ physicists shape a scientific concept, i.e. they eliminate the faster or slower speed as inessential and retain a uniformly accelerated movement of bodies towards the centre of the earth. They understand each separate fall of a body as a particular expression of the general movement of falling, and determine the specific differences by reference to the nature of the medium through which it falls and to the magnitude of its specific weight. Physicists then discover that all bodies in space seem to behave as if they were attracting each other and in such a manner that their attraction is in proportion to the product of their masses and in inverse proportion to the square of their distance. With this discovery, the general fall towards the earth is understood as a special case of a still more general gravitation. What does this mean logically? When physicists define the fall of an apple by specifying the general concept of falling, and the concept of falling, in its turn, by general gravitation, they have obviously achieved much more than when zoologists, in the definition of ‘dog’, mention the concept ‘mammal’ and define this, in its turn, by reference to ‘vertebrate’. Zoologists subordinate their object under a general representation, which does not need to be more than an indeterminate image, and with the definition they indicate its place in the system. Physicists, however, have understood the individual case as a manifestation of general laws governing the entire physical world. We can, therefore, easily understand the sense, even today, of the demand made by logic that we should define by specifying the genus. The definition which specifies the genus, achieves, in the last example, the same as the Aristotelian oÏ}ismoÈ| (horismos). For Aristotle the geÈno| (genus) was the expression of the ever existing pure form which manifested itself in transient individual things; for us the concept of the genus is the expression of the timelessly valid law which we are constantly finding in changing phenomena. If, therefore, the concept consists of judgements containing a law, then it, indeed, furnishes the highest knowledge of the essence of things, to which the empirical

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sciences can aspire. On this assumption methodology rightly demands of a definition that it specify the genus. In this case, defining means grasping the essence of an object, as it did for Aristotle. At the same time this also confirms again our view of the concept as a distinctive form of judgment. The claim that the concept expresses a law is nonsensical if the concept is seen as a ‘general representation’ with precisely fixed characteristics, and it is hard to see what such concepts are supposed to achieve for our knowledge of essence. But if we have recognised the concept as the judgments which have been used as a unity by means of a word, it is immediately clear that science must aim to create concepts consisting of necessary judgments, and that such concepts are in respect of their content, no longer different from laws, and therefore provide definitive knowledge. The concept of gravitation and the law of gravity are, as we have explained, in respect of their theoretical content, completely identical as knowledge, and a definition which subordinates a phenomenon under this concept has thereby expressed its ‘essence’, in as far as this word can have any sense in the empirical sciences.

5.2. Genus in mathematics In the so-called synthetic sciences the situation is naturally different. Logic, it is true, frequently chooses examples from mathematics in order to explain the relation of subordination and superordination of concepts. So, the square is defined as an equilateral rectangle, the equilateral rectangle as a right-angled parallelogram, the parallelogram as a quadrilateral with equal diagonals, etc. It is hardly necessary to point out that such subordinations under the genus proximum only have a value in logic as examples, and that they cannot express the methodological peculiarity of concept-formation in mathematics. Just as it makes no sense in mathematics to speak of the essential and inessential characteristics of objects or, correspondingly, of the essential or inessential elements of concepts, we may suppose that in mathematics, definitions by genus proximum and differentia specifica are only used for reasons of external convenience. This fact is related to the peculiar nature of the objects of mathematics. In mathematics there are no real ‘accidental’ phenomena, which one tries to grasp as special occurrences of a general law, and in which, for this purpose, one separates what is essential from what is inessential; on the contrary, each individual object, if it is a mathematical object, constitutes, as an ‘ideal’ being, the complete expression of the mathematical concept. The inessential does not

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feature in a mathematical object. In this respect, mathematical being differs in principle from sensible-empirical being. This does not mean that in mathematics concept and object coincide. The mathematical concept of a triangle is something other than a mathematical triangle itself. This is already apparent from the fact that the mathematical concept of the triangle is always the same and identical, whereas there are any number of mathematical triangles, which may, at best, be equal.1 Here, we only want to state that there can be no doubt about the way in which a mathematical concept, once arrived at by synthesis, should be analytically defined. One only has to specify what was done when it was formed; the superordinate genus has no logically distinctive significance; it becomes important at best when it is linguistically formulated for reasons of convenience. This observation may encounter opposition because, in the discussion of mathematical definition, that which is the externally useful linguistic formulation, tends to be separated even less than is usually the case from what properly belongs to the logically necessary form of concept-determination. If both are kept apart, it will be obvious that in mathematics the definition by genus proximum and differentia specifica can only be required if one wishes by its means, to attain one’s goal as quickly as possible, i.e. where it is the most convenient form of linguistic expression. This is unlikely to be the case in all instances, and, besides, in mathematics the general generic concept does not have the logical significance it can have in the natural sciences. There, as we have shown, the old Aristotelian requirement was given a new meaning by substituting the comprehension of conformity to natural laws for the metaphysical knowledge of essence. On this assumption the generic concept as concept of law is the highest form of knowledge which grasps essence, and in this way subordination is logically justified. What has been seen as a special case of the general law, has, in truth, been grasped in respect of its ‘essential’ constituents. In this case, the requirement that one specify the genus proximum and differentia specifica does not in any way arise from the need for the most convenient linguistic expression. In mathematics, on the contrary, there can be no question of such a ‘hierarchy’ of concepts, such as exists in the natural sciences. Without it, however, the requirement that one define by genus and differentia cannot be logically justified. In the case of mathematical concepts, all their constituent judgements have to be performed explicitly if they are to be completely defined, i.e. if we are to become fully conscious of them in respect of their scientific content. This observation justifies

1. Where one thinks differently and, for example, does not differentiate the concept of singular from the number one, one confuses equality with identity. On this topic see Rickert (1924).

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the statement that mathematical concept-analyses have to mirror precisely mathematical concept-formations, and that, for the rest, there are no special methodological rules for mathematical definitions by means of concept analysis.

Chapter Six: Nominal and real definitions 6.1. Name, thing and concept All that remains to do is to closely examine the widely discussed theory of nominal and real definitions in the light of our results. We have shown that the term ‘nominal definition’ should not be used in the sense given it by Sigwart. Otherwise ‘definition’ would mean something which no longer has anything to do with Aristotle’s oÏ}ismoÈ| (horismos). It is, of course, also possible to say, as Sigwart (1890:50) does, that it is linguistically desirable to differentiate between concept-formation and definition, and to understand by definition only the sentence which establishes the meaning of a word. But one may not seek to justify this, as Sigwart does, by saying that what one finds in textbooks under the title of ‘Definitions’ is not thought-processes but linguistically formulated propositions, for what one finds in books under all titles are linguistically formulated propositions; nevertheless, logic is never concerned with sentences themselves but with the thought attaching to them, or their logical content. But, as Sigwart himself admits, the question for what the word ‘definition’ may best be used is a terminological question. If one wishes to call only explanations of words ‘definitions’ one is free to do so. But then, the expression ‘nominal definition’ becomes completely misleading, because it gives the impression that there is another type of definition; besides, in this case, the theory of definition is no longer an essential part of methodology. It no longer deals with a logical problem at all, but merely with the most expedient form of linguistic expression. That which under the label of oÏ}ismoÈ| (horismos) belongs to logic, is always a concept-specification, or more accurately, as we have seen, both concept-formation and concept-analysis. As long as people disputed whether the concepts that were being defined were only general names or general things, the designations ‘nomimal definition’ and ‘real definition’ made good sense. Without this metaphysical problem of general realities, these designations have lost all meaning and should therefore be abandoned. Nor can they be justified by saying that a definition is ‘real’, if a real object

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corresponds to the defined concept, and ‘nominal’, if it there is nothing in reality corresponding to it, only a name. For we have seen that the word or some designation or other is necessary in every definition, and that to this word there always corresponds something, namely its logical meaning. When the concept has been defined, this logical meaning consists of the content of the judgments considered to be at rest, which means that it is neither a thing nor a name. The distinction one has in mind would rather be tantamount to saying that the judgements which form the concept could be wrong in one case and correct in another. For this alternative, which cannot be decided by methodological reasoning, the names ‘real definition’ and ‘nominal definition’ would certainly be unsuitable. One might, with the same justification, speak of nominal and real judgments and nominal and real inferences. In short: Neither the name, nor the thing, but only the concept is defined.

6.2 Provisional and definitive definitions Nevertheless, the expressions nominal and real definitions are meant to denote something that, in Lotze’s words, is of some value (Lotze 1880:202), but it has not always been fully clarified what this may be; otherwise these names would not have been preserved. We have pointed out the relative nature of concepts in the analytical sciences in contrast to the absolute concepts of mathematics. Lotze wanted to retain the expressions nominal and real definition only for the analytical sciences. There they would serve as a useful warning, whereas in mathematics real definition could no longer differ from nominal definition. It is obvious what Lotze means. With ‘real definition’ he wants to designate the definition which has an absolute validity, similar to that of the definitions in mathematics. Real definition would, accordingly, represent a kind of higher knowledge. Sigwart (1890:330) too, holds a similar view, though he expressly stresses that the demand for real definition involves a mixture of logical and metaphysical thoughts. He considers three senses of the word ‘concept’. First, the word designates a ”natürliches psychologisches Erzeugnis” (natural psychological product), a general representation. This “empirische Bedeutung” (empirical meaning) is contrasted with an “ideal meaning”, according to which the concept designates “den Zielpunkt unseres Erkenntnisstrebens” (the goal of our striving for knowledge), inasmuch as in that striving we are searching for an “adequates Abbild des Wesens der Dinge” (adequate image of the essence of things). Between the ‘empirical’ and the ‘metaphysical’ concepts, Sigwart places the “logical” concept which is determined by the sole requirement “daß unsere

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Urteile gewiß und allgemeingültig seien” (that our judgments be certain and universally valid). As we know, he requires of this concept that it be firm and determinate, so that agreement would be assured among all thinking individuals, specifically excluding the question of the relationship between thought and actual reality. He wants to distinguish “die formale Brauchbarkeit der Begriffe zum Zweck des Urteilens” (the formal utility of concepts for the purpose of judgments) from their metaphysical adequacy. This would appear to be unobjectionable if this distinction were meant to serve only to deflect epistemological and metaphysical questions. But this is not the case. The distinction is based on a specific epistemological or metaphysical presupposition and this presupposition has contributed to Sigwart’s theory of the concept, and especially to his theory of definition, not being as convincing and clear, as other parts of his work. The distinction between empirical and the strictly logical concept has already been examined and we know that Sigwart does not want to call the ‘natural psychological product’ a concept, but prefers to call it ‘general representation’, reserving the term ‘concept’ for the representation the characteristics of which are precisely fixed and which concludes the work which natural thought has already started everywhere. We have felt obliged to alter this view to take account of the fact that we speak of concept where a relationship between representations or elements of concepts has been realised, and that we consider the true essence of the concept to lie in this relation, i.e. in the logical content of the judgment. But what is the relationship between the logical and the metaphysical concept? According to Sigwart, every metaphysical concept must also be a logical concept, though it differs from it in that it is also an adequate image of the essence of things. We shall leave aside the question whether the phrase “adequate image of the essence of things” is meaningful; this question does not belong to methodology but to epistemology or transcendental philosophy. Instead we shall examine whether, like Sigwart, one may oppose to the logical concept within logic, and logic which, to boot, pretends to be methodology, a concept which is in principle different from it and which is the object of our knowledge. Such an opposition always gives the impression that the knowledge furnished by a logically formed concept were at most a second-order knowledge, as if the definition of such a concept were something incidental, and as if true understanding were furnished by this metaphysical concept, which then attempted to grasp the true essence of things in this ‘real definition’. This distinction seems to be questionable not only because it does not make sense without a specific metaphysical presupposition, but also because it hides another distinction which is

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indispensable in methodology. We cannot, in fact, totally avoid the question under what circumstances our concepts are actually true, i.e. consist of valid judgments, because obviously there are formally complete concepts which are not true. We do not mean concepts which can be produced as cheap entertainment by some sort of logical game and which are created in the full awareness that they are valueless for our knowledge. We are speaking here only of concepts which have arisen in the course of scientific work, and in the case of which we simply cannot escape the conviction that they will one day be found to be erroneous, however logically complete they may appear in respect of their form. In the case of all empirical concepts we entertain this suspicion; in other words, we consider all empirical concepts as provisional. But is this true of all concepts in general? Not in the sense that other possibilities are excluded. Therefore, methodology will oppose provisional concepts with ‘ideal’ concepts which represent the goal of our knowledge, but not in the sense that we search in them for an adequate image of the essence of things, but in the sense that they are definitive concepts, constituted in such a way that we recognise that they could not have been formed any other way and will never be changed. This assumption does not contain any metaphysical presuppositions. We are firmly convinced that, no matter how our empirical experience may be extended, it is now and for ever in the nature of the concept of the plane triangle that the sum of its angles equals two right angles. But this does not mean that we are dealing with a concept which represents a metaphysically adequate image of what is. The certainty with which we hold such a concept, or, to be more precise, its constituent judgments, to be true, depends exclusively on the necessity of thought, with which they force themselves on us. We shall not here investigate the criterion of the truth of our knowledge, for this is not a methodological problem. But we must ask: may we speak in methodology of a criterion other than the necessity of thought, without introducing confusion? In the introduction to his logic, Sigwart (1911: 8) himself explained how logic should be positioned regarding the relationship between thinking and being. He summarised his opinion as follows: “Wenn wir nichts als notwendiges und allgemeingültiges Denken produzieren, so ist die Erkenntnis des Seienden mit darunter begriffen; und wenn wir mit dem Zwecke der Erkenntnis denken, so wollen wir unmittelbar nur notwendiges und allgemeingültiges Denken vollziehen. Dieser Begriff ist auch derjenige, der das Wesen der ‘Wahrheit’ erschöpft.” (If we produce only necessary and universally valid thoughts, then knowledge of what is is embraced by such thought; and if we think with the

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purpose of knowledge our immediate wish is to perform necessary and universally valid thinking. This concept is also the concept which exhausts the essence of truth.) This is true as long as one limits logic to methodology, i.e. as long as one excludes metaphysical or transcendental philosophical questions, and from this it follows of necessity that no distinction in principle may be made between concepts formed in the individual sciences according to logical rules, as we have been made aware, and concepts which we regard as the goal of our knowledge. The same necessity of thought leads us to both. A logic which puts emphasis on methodology, will only be able to make a distinction of degree between these two types of concepts. The theory of definition does not go beyond methodological considerations and for this reason it can only state the following. When someone establishes a definition, he will always try to know or understand an object by means of his concept, i.e. he will try to form it correctly. Methodology can only specify the rules which have to be followed. It will take note of the fact that definitions in the empirical sciences are, in the nature of things, probably always only provisional, because any empirically new material may overthrow them; and it can contrast these provisional definitions with examples of a type which can be seen to be always valid like mathematical definitions and, in a certain sense, also juridical ones. But methodology may never differentiate between true and false definitions or concept-specifications and then call the one nominal and the other real definitions. If names are needed for the different types of definitions the best names are probably provisional and definitive or conclusive. Finally, the word nominal definition is also often used for the definitions which one places at the head of a science, and which are meant only to indicate the area to be investigated. They usually include a classification as their essential element. The judgments contained in such definitions are then claimed to be only ‘hypothetical’, i.e. provisionally valid. Their correctness can only be recognised at the end of the investigation. Here, too, the term nominal definition could not be more inappropriate. The terms ‘hypothetical’ or ‘problematic’ definition might be more suitable, if one wanted to use the expressions of Kant’s table of judgements at all. Accordingly, the definitions of the natural sciences could be called ‘assertoric’ and those of mathematics ‘apodictic’. One would thereby characterise the judgements which constitute the definition and the concept, and by reference to which alone the definitions are to be distinguished. Beyond this, there are no different types of definitions at all; apart from the mere specification of the meaning of a word there is only the concept-specifications which, as concept-synthesis and concept-analysis we have subjected to close examination.

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Translator’s note 1. I am deeply indebted to Dr. David Walford for his meticulous revision of my translation. Without his generous assistance I would not have been able to undertake this task. 2. The author’s original idiosyncratic indications of emphasis by means of bold lettering and quotation marks have been preserved in order to maintain the distinctive style of his writing. The syntax and punctuation of the original have been altered where sentences would have been uncomfortably long for an English reader. The German text of the authors quoted has been preserved and translations have been provided in brackets after the original. Bibliographical references have been converted to a separate list of references.

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Name Index

A Alembert, Jean le Rond d’ 7, 9–10 Aristides 133, 157 Aristotle 3, 7, 11, 25–90, 100, 156, 161, 181, 196–7, 199, 201, 205–6, 240–4 Arnauld, Antoine 6, 147 Avenarius, Richard 212 B Beckett, Samuel 14 Berkeley, George 159–162 Borel, Peter 121–2 C Carroll, Lewis 4–5 Cassiodorus, Flavius M. A. 11, 91 Chambers, William & Robert 7 Cicero, Tulio 7, 11, 93, 129 Clavius, Christopher 121–2 Cohnheim, Julius 224–5, 233 Condillac, Étienne Bonnot de 174 Conring, Hermann 145 Cournot, Antoine 9 Crusius, Christian A. 171 Cuvier, Georges de 178 D Darwin, Charles 224, 223–5 Descartes, René 114–5, 130, 154 Diderot, Denis 7, 10, 12 Dionysius 66

E Eleatics 202 Euclid 9, 106, 150, 182 F Fries, Jakob Friedrich 193 Furetière, Antoine 7 G Goethe, Johann Wolfgang 13, 193 Gorgias 203 Guyton de Morveau 3 H Hegel, Georg Wilhelm Friedrich 13, 240 Heraclitus of Ephesus 202 Hobbes, Thomas 4, 181 Jhering, R. 200, 218–221 I Isidore of Seville 11, 91–5 J Jacquet, Andrew 121 K Kant, Immanuel 3, 8, 13, 163–172, 193, 212, 231,239, 248 L Lalande, André 10

256

NAME INDEX

Lange, Friedrich, A. 232 Lask 231 Lavoisier, Antoine Laurent 3 Leibniz, Gottfried Wilhelm 7, 145–158, Linné, Carl von 240 Locke, John 4, 7, 125–144, 152–9 Lotze, Rudolf Hermann 199–200, 206, 210, 212, 233, 245 Lycophron 89 M Mackintosh, Sir James 187 Malebranche, le Père Nicolas 147 Mallarmé, Stéphane 3 Mill, John Stuart 9, 14, 173–190 Montaigne, Michel de 114 N Nicole, Pierre 6 Nietzsche, Friedrich 13 Nizolius, Marius 148 Nominalists 181 P Pascal, Blaise 5, 9, 95–118 Plato 4, 14, 15–24, 66, 98, 134, 157, 186, 204–5, 240 Port Royal 6,7, 8 Protagoras 203–4 Pythagoreans 37 Q Quintilian, Marco Fabio 11 R Rey, Alain 1–15 Rey-Debove, Josette 10 Rickert, Heinrich 13–14, 191–250 Robinson, Richard 2, 4, 8

Riehl, Alois 231 Rümelin 219 S Sacchieri, Giovanni G. 8–9 Schuppes, Ernst, J. W. 231 Searle, John 13 Sigwart, Christian 191–2, 199–201, 208–10, 229, 231, 234–6, 244–7 Socrates 15–23, 203–4 Spinoza, Benedict de 119–124 Sophists 203–4 St. Augustin 114, 166 Stewart, Dugald 187 Stoics 3, 8 T Theaetetus 15–23 Tschirnhaus, Walter von 146 U Ueberweg, Friedrich 197–8, 213, 227 V Varro, Marcus Terentius 7 Victorinus, Caius Marius 11, 91–5 Volder, Burcher de 147 Vries, Simon de 124 W Whateley, Richard 181 Whewell, William 186 Windelband, Wilhelm 191, 202, 231 Wittgenstein, Ludwig 3 Wundt, Wilhelm 231 X Xenocrates of Chalcedon 52, 74 Xenophanes of Colophon 204

In the series TERMINOLOGY AND LEXICOGRAPHY RESEARCH AND PRACTICE (TLRP) the following titles have been published thus far or are scheduled for publication: 1. CABRÉ, M. Teresa: Terminology. Theory, methods and applications. 1999. 2. ANTIA, Bassey Edem: Terminology and Language Planning. An alternative framework of practice and discourse. 2000. 3. TEMMERMAN, Rita: Towards New Ways of Terminology Description. The sociocognitive approach. 2000. 4. SAGER, Juan C.: Essays on Definition. 2000.