Problems and Projects

Citation preview

OTHER BOOKS BY NELSON GOODMAN

The Structure

of Appearance

Ne 1son \Good ma n1 c_

Fact, Fi ction, and Forecast Languages

of Art

~PROBLEMS and

PROJECTS 1 THE BOBBS-MERRILL COMPANY, INC.

Indianapolis and New York

coa.2 9 u .J

9 oo]) o

Copyright © 1972 by The Bobbs-Merrill Company, Inc.

To my mother

Library of Congress Catalog Card Number 73-165221

Sarah Elizabeth (Woodbury) Goodman

Printed in the United States of America

1874-1964

First Printing

1

7

Contents Introduction

xi

I

Philosophy Foreword : 1. The Revision of Philosophy : 5 2. The Way the World Is : 24 3. Some Reflections on the Theory of Systems 4. Review of Urmson's Philosophical Analysis 5. Descartes as Philosopher : 45 6. Definition and Dogma : 49

II Origins Foreword : 57 1. Sense and Certainty : 60 2. The Epistemological Argument 3. The Emperor's New Ideas : 76 4. Review of Armstrong's Berkeley's Theory of Vision

69 .;

80

III Art

•

Foreword : 83 1. Art and Authenticity : 85 2. Art and Inquiry : 103 3. Merit as Means : 120 4. Some Notes on Languages of Art : 122 5. Further Notes : 133 6. Review of Gombrich's Art and Illusion vii

141

33 41

CONTENTS CONTENTS

4. Review of Craig's "Replacement of Auxiliary Expressions" : 322 5. Elimination of Extralogical Postulates (with W. V. Quine) : 325 6. Safety, Strength, Simplicity : 334 7. Science and Simplicity : 337 8. Uniformity and Simplicity : 347

IV Individuals Foreword : 149 1. A World of Individuals : 155 2. Steps Toward a Constructive Nominalism (with W . V. Quine) : 173 3. A Revision in The Structure of Appearance

Foreword : 357

v

1. A Query on Confirmation : 363

Meaning Foreword : 203 1. Talk of Time : 207 2. On Likeness of Meaning : 221 3. On Some Differences about Meaning : 231 4. On a Pseudo-Test of Translation : 239

6. Replies to Comments on Fact, Fiction, and Forecast : 398

Foreword : 421 1. Order from Indifference : 423

VII Simplicity

viii

4. An Improvement in the Theory of Projectibility (with Robert Schwartz and Israel Scheffier) : 389 5. Inductive Translation : 394

IX Likeness

273

Foreword : 275 1. The Test of Simplicity : 279 2. Recent Developments in the Theory of Simplicity : 295 3. Condensation versus Simplification

2. On Infirmities of Confirmation-Theory : 367 3. The New Riddle of Induction : 371

7. Review of Reichenbach's Elements of Symbolic Logic : 413 8. Snowflakes and Wastebaskets : 416

VI Relevance For eword : 241 1. About : 246 2. "About" Mistaken

VIII Induction

199

2. Seven Strictures on Similarity

437

x Puzzle 319

Foreword : 449 The Truth-Tellers and the Liars ix

~:

451

CONTENTS

Sources and Acknowledgments Nam e Index : 459

453

Introduction This book contains most of my previously published papers, excerpts from my three earlier books, and some half-dozen items not before published. These papers have been gathered into chapters according to subject, and a new foreword supplied for each chapter. The forewords have varying functions: to provide historical notes concerning the papers in the chapter or the problem they deal with; to relate these papers to each other, to other papers in the book, and to the current state of investigation of the problem; and sometimes to correct common misunderstandings or to carry investigation of the problem a step further. Thus while these forewords are primarily instruments of organization, they sometimes suggest new ideas. Of my published papers omitted here, "The Calculus of Individuals and Its Uses" (with Henry S. Leonard) and several pieces on simplicity have been superseded by The Structure of Appearance; and "The Problem of Counterfactual Conditionals" has been used as the first chapter of Fact, Fiction, and Forecast. "Sequences" seemed not to fit anywhere, and "Graphs for Linguistics" is unsatisfactory. On the other hand, I have included "The New Riddle of Induction" from Fact, Fiction, and Forecast, "Art and Authenticity" from Languages of Art, and-under the title "Talk of Time"two sections from Structure of Appearance. All these deal with t opics of rather wide interest and are easily understood apart from the books. The papers included here vary in difficulty. Only a few, such as V ,2, VI,1,2, and VII,1,2,3 call for technical training at the graduate level. All the rest should fall well within the competence of the advanced undergraduate student of philosophy; and at least half of these papers, such as most of those in the first three chapters, should give the elementary student or the serious layman no undue trouble.

x

xi

INTRODUCTION

Some supplementary readings are suggested at the end of each chapter except the first and last. These lists are partial and arbitrary but, together with references given in text and footnotes , provide some introduction to the bibliography of the topic in question. On some of these topics, such as induction, the literature is vast; on others, such as aboutness and simplicity, most of the literature is covered by the references and reading lists together. The sources of the several papers making up the book are detailed in a list at the end; and acknowledgement is there made of permissions received. I am grateful to the editors and publishers who have granted such permission, and especially to my collaborators on one or more of these papers: W. V. Quine, Israel Scheffler, and Robert Schwartz. Many of the papers have been corrected or otherwise improved since first published. Graham Roupas did much of the editorial work for the present volume and made many valuable suggestions. The cross-references in the forewords , text, and footnotes have seemed to me to make a subject index unnecessary, but a name index has been included. The following abbreviations are often used in referring to my own books: " SQ" for A Study of Qualities (doctoral dissertation, Harvard University, 1941) ; " SA" for The Structure of Appearance (first edition 1951; second edition, The Bobbs-Merrill Co., Inc., Indianapolis, 1966) ; "FFF" for Fact, Fiction, and Forecast (first edition, 1955; second edition, Bobbs-Merrill, 1965) ; " LA" for Languages of Art (Bobbs-Merrill, 1968) . Where there are two editions, reference is always to the second unless otherwise specified. References to parts of these books are italicized-e.g., " SA IV,3" to distinguish them from references to the present book-e.g., " IV,3" . The more problems, the more projects; the more projects, the more problems. Yet if there are more beginnings than endings in my work, I hope the reader may now and again have an intimation that not all problems of philosophy are immortal.

Harvard University

xii

Problems and Projects

I

Philosophy FOREWORD That most of the items in this chapter are occasional papers is no accident. Except under the stimulus of special circumstance, I have been disinclined to discuss the problem of defining philosophy, its aims, and its methods; for although this is as legitimate a philosophical problem as any other, it has had more than its share of attention in recent times. What I have written on the problem has usually been retrospective rather than prescriptive. To find out what philosophy is I have usually looked back upon how more specific problems have been attacked-by others and by me. Alone among the papers here, "The Way the World Is" points forward as well as backward, and foreshadows Languages of A rt. Statements from two of these papers may appear to contradict one another. In "The Revision of Philosophy", I say "what is w anted is a certain structural correspondence between the world of the system and the world of presystematic language" , while in "The Way the World Is" I say "there is no such thing as the structure of the world for any system to conform or fail to conform to". B ut of course the structure of the world of presystematic language is simply a world-structure under one world-description and not the structure of the world independ~nt of any description; and the correspondence asked for relates two world-descriptions rather than a description to such a world-in-itself. Occasionally the objection is raised that to speak of descriptions of the world implies that there is such a thing as the world. One m ight as well point to pictures of Don Quixote to prove that there is one and only one such person. "Picture of Don Quixote" and

3

I

PHILOSOPHY

" description of the world" are one-place predicates and are better · t ion " " ( se e replaced by "Don-Quixote-picture" an d " wor ld - d escnp further below V,2; also III,4). Rather than there being one and only one Don Quixote, there is none; rather than there being one and only one world , there may be many. This does not mean that all world-descriptions are equally true; but it does raise the question what distinguishes true from false ones (see II ,1). More than once in the present chapter, I urge that isomorphism must replace intensional and even extensional identity as a criterion for constructional definition. In Chapter I of The Structure of Appearance, I argue this point more fully and explain in detail the appropriate variety of extensional isomorphism (see also IV,3 below). This reconception of the nature of explicative definiti_o n transforms or even resolves some philosophical issues; for an mteresting example see the discussion of materialism and the mindbody problem by Lynn Foster.* Other aspects of definition, especially of theoretical terms, are briefly discussed in III ,5b below. In SA (Part Two) , I do not include under "logic" the calculus of classes or the calculus of individuals, as I do in "Some Reflections on the Theory of Systems". The line between what is logic and what is not, as Quine has insisted, i's quite arbitrary. In the book there were advantages in confining the term " logic" to what is co:Umon to all the constructional systems under consideration; in the paper, for a more general audience, I conformed to what seemed to be the more common practice. The two little talks concluding this chapter are just that. They are unlikely to be fully understood by those who did not know the Second World War and its aftermath. And the defense of analysis and blueberry pie will seem quaint in this age of dogma and drugs. * Lynn V. Foster, Constructionalisrn _and the ~ontemporary Min d-Body Id entity Debate, Doctoral Thesis, Brandeis Umvers1ty, 1971.

1

The Revision of Philosophy My title refers not to the reworking but more literally to the revision of philosophy-a new way of looking at philosophy, a new conception of its nature and objectives, and consequently a new appraisal of its methods and results. I am not concerned with the modification of what has been done by able philosophers from Thales on, but rather with a re-vision, in the sense explained, or a new version, of what has been done and is being done . Let me hasten to say that this 'new vision' is not original with me, and that it is not even very new. It has been coming into our consciousness slowly over a long period; and it has, I think, been tacitly adopted for sometime past by those who have contributed most to philosophy. But it is often obscured by what even these same people say when they talk about philosophy ; and it is almost never understood by those whose participation in philosophy consists solely in controversy. Thus I think it may be well to dust off this rather old new vision and hold it up once m ore to public view. What follows below constitutes a paper written for a forthcoming volume on Carnap. 1 Although cast in the form of a discussion of the significance of Carnap's D er Logische Aufbau der We lt,~ it deals almost entirely with the more general questions which must be answered before the significance of such a book can be appraised. Thus I am at the same time implicitly defending the point of view of, for example, my own book 3-which, incidentally, contains an exposition and criticism of a considerable part of the Aufbau. 1. Evil Days for the Aufbau

The Aufbau is a crystallization of much that is widely regarded 1. The Philosophy of Rudolf Carnap, edited by Paul Schilpp, in preparation [LaSalle, Illinois: Open Court, 1963]. These sections are published here with Professor Schilpp's permission. 2. Rudolf Carnap, D er Logische Aufbau der Welt (Berlin, 1928). 3. SA.

4

5

I,l

PHILOSOPHY

as the worst in twentieth-century philosophy. It is an anathema to antiempirical metaphysicians and to alogical empiricists, to analytic Oxonians and to antianalytic Bergsonians, to those who would exalt philosophy above the sciences and to those who would abolish philosophy in favor of the sciences. A good part of current polemical writing in philosophical journals is directed against views found in virulent form in the Aufbau. The Aufbau stands pre-eminent as a horrible example. My purpose here is to survey and appraise the charges against the Aufbau, and to set forth some convictions concerning the significance of the work. This virtually amounts to the unpromising, but welcome, task of defending the Aufbau against almost everybody, including Carnap himself-indeed, including a succession of Carnaps who have belittled this early work for different reasons at different times in the twenty-eight years since it was published. But I am more interested in the current atmosphere of opinion concerning the Aufbau than in what particular people have said at particular times; and my adversary, except where specifically named, is a composite figure encountered as often in conversation as in the journals.

2. Phenomenalism and the Aufbau In place of the 'impressions' or 'simple ideas' of eighteenth-century British philosophy, Carnap based his system on total moments of experience-the elementarerlebnisse-in order to begin as nearly as possible with what he regards as unanalyzed and unprocessed experience. The system is plainly phenomenalistic, and phenomenalism has been under heavy and incessant attack. The most popular objection is that phenomenalism is incompletable. No full and adequate account of the objective and intersubjective world of the sciences can be given, it is contended, upon 4 a purely phenomenalistic basis. Carnap's own first disavowal of the Aufbau expressed the conviction that a phenomenalistic system, unlike a physicalistic one, could not be all-embracing for science; and perhaps nothing else he has ever written has found such widespread agreement. 4. In "Die physikalische Sprache als Universalsprache der Wissenschaft", Er k enntnis, vol. 2 (1931), pp. 432-465.

6

THE REVISION OF PHILOSOPHY

I,l

. The argun:~nts commonly adduced to support the charge of the mcompletab1hty of phenomenalism cannot, in the nature of the c~se, be very cogent by themselves; for the thesis they are designed to prove is not precise enough, and there is available no developed body of theory within which a sound proof might be given. Proof that a complete system cannot be constructed on any phenomenalistic basis prerequires some precise delimitation of the class of phenomenalistic bases, a full statement of admissible methods of construction, and a clear conception of what constitutes completeness of the kind in question; and none of these requirements is easy to meet. Thus, for example, the argument that phenomenalism is incompletable because the infinite world of ~athem~tics and the sciences cannot be accounted for upon a fimte basis has at first sight the simple force of the statement that an infinite number of things cannot be made out of a finite number. But if we understand that the question is rather whether we can interpret in terms of statements about a finite number of entities all indispensable statements that prima facie refer to an infinite number of entities, the matter cannot be settled so easily. On the other hand, the thesis that phenomenalism is incompletable hardly needs proof. Surely no complete system will be offered _within any foreseeable length of time; and no other means of provmg the possibility of completion is in prospect. The task of constr~ction is so formidable , and the tendency to regard it as hopeless is so strong, that the presumption is all against the claim th_at any phenomenalistic system-or for that matter any system wit~ a v~ry narrow basis-is completable. Even without proof or clar1_fic_at10_n of the thesis that phenomenalism is incompletable, one is Justified in accepting this thesis at least until the opposite is rendered more credible. B~t if the thesis-proven or unproven-is accepted, what conclus10n can be drawn from it? Usually phenomenalism is taken to be. u~terly ~iscredited once its incompletability is acknowledged. It is JUSt this step in the argument-a step commonly passed over as obvious-that I want to challenge. I am ready to maintain that the value of efforts to construct a system on a phenomenalistic or any other narrow basis is very little affected by whether or not the system can be completed. Euclid's geometry is not robbed of v alue by the fact that the circle cannot be squared by Euclidean

7

1,1

PHILOSOPHY

means. Indeed, acceptance prior to Euclid of the impossibility of squaring the circle with compass and straightedge would not in the least have diminished the importance of developing Euclidean geometry; and it would not, I think, have been ground for turning attention solely to the discussion of the adequacy of various bases or to the development of geometry on a basis broader than Euclid's. Moreover, propositions affirming the Euclidean insolubility of certain problems could hardly have been precisely formulated or have been capable of proof except against the background of elaborated, even if incompletable, mathematical systems. But my point is not just that it was psychologically n ecessary or helpful to work in this way. What is accomplished in the incompletable system has permanent value when incorporated into a fuller system. Indeed after a system like Euclid's has been developed as far as possible, questions concerning what can be accomplished with even fewer means (e.g. without a straightedge or without a given postulate) often still have interest. The analogy, I take it, is traPsparent. Incompletability by itself is no decisive objection against the attempt to build a system on a phenomenalistic basis. Only b y positive efforts with severely restricted m eans can we make any progress in construction; only so can we discern the exact limitations of a basis and the exact supplementation needed. And what we achieve may be retained in an expanded system, and will h elp solve parallel problems in alternative systems. Carnap's suggestion that his single chosen primitive might be enough for a complete system was indeed rash and untenable. But his mistake here was no worse than that of people who thought Euclid's basis enough for a complete plane geometry. The incompletability of the system of the Au.fbau. or of phenomenalism in general is n ot a very damaging charge. Incompletability is n ot the only count urged against phenomenalism. Sometimes the objection is rather that a phenomenalistic system, whether completable or not, is epistemologically false ; that it misrepresents the cognitive process. Phenomenal events or qualities, it is held, are not the original elements of knowledge but are products of an artificial and highly sophisticated analysis, so that a phenomenalistic system gives a highly distorted picture of actual cognition.

8

THE REVISION OF PHILOSOPHY

1,1

Any such view rests on the premise that the question "What are the original elements in knowledge?" is a clear and answerable one .. And the assumption remains uncontested so long as we ar e dommated ~y the tradition that there is a sharp dichotomy between the given and the interpretation put upon it-so long as we picture the knower as a machine that is fed experience in certain lumps and proceeds to grind these up and r eunite them in various ways. But .I do not think this view of the matter will stand very close scrutmy. For the question " What are the units in which ex~erienc: is actually given?" seems to amount to the question Wha~ is .the real organization of experience before any cognitive org.an~zat10n takes place ?" and this, in turn, seems to ask for a des~rip:10n of cognitively unorganized experience. But any description itself eff.ects, so to speak, a cognitive organization; and apart from a description, it is hard to see what organization can be. The sea~ch .for the original given is sometimes envisaged as an interrogat10n m which I am first asked what I just saw. I reply, " I saw the worst cr.1mmal alive today", but my questioner complains that I am makmg too many judgments about what I saw· he wants me t? tell him exactly what I could see and n othing m;re. As he continues to press me, I reply successively; " I saw a man", " I saw a human looking animal" , " I saw a moving obJ'ect" , " I saw sue h -an d s~ch a configuration of color patches". But if my questioner is consistent. and persistent, none of these replies-or any other I can g~ve h1~-will satisfy him ; for all m y answers describe my experie~ce m words a.nd so impose on it some organization or interpretat10n . What he 1s covertly demanding is that I describe what I s~w without describing it. All my answers may be true descriptions of what I saw, but no description can be a satisfactory answer to the question what I merely saw;" for the question is a bogus one. But obviously I cannot discuss the whole question of epistemological priority very thoroughly here. And there is no need. For the value and validity of a constructional system do not depend . 5.

T~e

snares .in the kene'5 of Meaning" had e ' mg of the condu,ion of nal paragraph of "On Some Diffe een so~ewhat softened by the The papers on meaning f rences aoout Meaning" of te rms. A compound is aocus attention upon certain compounds · l words. For example , " pie . t urearger term ,,obtain " of a ho "d ed by addi ng oth er part of a house" , "small house" " use , escription of a house" seven gables" a re compounds of , thcorner house" ' a n d "h ouse of, . a"dso, incidentally , a compound of so e term th "house" ,· and eac h is l ll" me o er term-of "pi· t ,, escription" "part" " " bl ,, ' , sma "co ,, c ure e '."'even", and "of ,ev;n g::i::.,' in the final exampl; . meaning-comparison are all a . e compounds relevant s10ns ar e not included . h mong those fo 'f m t e extensio f such that the·ir extenr i two terms such as " ns o the term compound d · e · th "'difference in meanincentaur" i an d " unicorn" are coexten,iv; tensive parallel compoun:s s n~,t revealed in such equally coex '.om" but in 'uch 'mall centaur" and "'mall uni: picture of a centaur" and " . y divergent parallel compound compounds of tw o t erms thuspicture s as cliff ·of a unicorn" · Wh en parallel ser ves are said to differ m . seconda er m extension, the terms th em1 a ists point to as making a differen extension. What intensionexphc1tly formulated as d'ff c~ m meaning can.. often m ore I erence in second ary ex.enswn . be 1

~;;-;;ng

~i;;:'r

b

~a

;~d

exten,iona~:

?

_6. Semantics and the Ph'l . Illmo1s: 7 University of Ill'mo1s .t osophy Press, 1952 of Language ). , ed . L. Linsky (Urbana,

· E.g. see Carnap M e 238. . · · of Chicago Press • 1956, ) , p.amng and N ecessity (2 n d ed ., Chicago·· U mvers1ty

205

v d . "On Likeness of Re· ection of pictures' suggeste m . The 'lan guage theory modification with time. ~ . " has undergone some descriptions and pictures ~et~:~~ew that the distincti:~s:~::e:; conventionalizati~n ~o!~:

1

MEANING

is a matter o~lpr_ese~~~ o~ome Differences abofut Me:::~~alizae m . deg,-ee o conv fi rm ., but whi . t be a difference in " (I 2 bove) the distinction is taken o . "The Way the World Is ' a tions tion, seven years later : rather a matter of ~i~e,-e~t c_onv~:scrip~ distinction is heldAt~ ~he specific features distmguishi~! them as In Languag es of 1t, 11 the features common . as we as . from pictures, tions . d . detail t and desymbols, a.re. ~~:d~:e~~or deaiing with picftuMres ~~:;,~ r:~~rs in my The oasic i . "0 Likeness o ea in

~~~~~t~:;k~f e~;!~~:~;): '.i'~~~:'.:'.::n~ :ih~~~~~·:.r~~:~~::~~

"About" (VI,1 bel~w an _ Languages of A,-t, as we . as i lions. and of representation-as m e about hunting non-exis:ent ·efublished correspondenc d b Quine in construing beli unp ·d ·s use Y f aper on Much the same i ea i bert in the course o a p . n h ffier however, m s db Meyer C1nd Lam contexts, ~n f y existential assumptions. _Sc e ather from the freeing logic rom . 10 takes his cue r h . "T lk of Time" ' and as . l . of indirect discourse , . his ana ysis d ephcas m a dr g treatment of inscriptions ~n r rovide a better way of han m . d Q ine 1 1 that this may p convince u ' belief-contexts. 44 espe-

1960) ' § ' . MIT · · · Press, d Object (Cambn·d ge, Mass .. 8. Word an . " Journal cially P· 216 . d ·d Quantification Theory ' Logic and Stan ar 9 "Universally Free S--26. of S.ymbolic Logic, Vol. 33 (1968), PP· I d "r ect Quotation", Analysi~, Volv!~ l A roach to n i t tion" Ana•ysis, . 10 "An lnscriptiona . P? l" m and Indirect Quo a t.' n" Philosophy . 83-90· "lnscnpttona 1s d Indirect Quota 10 ' (1954), PP· ' d "On Synonymy an 19 (1958)' P~· ~2zi8i1~~5)' PP· 39-44. ff fkka (Dorof Science, o. . d D Davidson and J . ml d 5 and Objections, e . . 334 11. See W or . D Reidel , 1969), P· · d recht · ·

Talk of Time Time and Language In ordinary discourse we often indicate the time of events not by explicit description but by such a word as " now", "yesterday", "next week", "past", "later", or by the tense of a verb. As a result, we have quite unequivocal statements that nevertheless seem, paradoxically, to change in truth value. For example, when I say "The Red Sox now lead the American leagu e", I am being quite definite; I am not saying that they lead at some unspecified time but am indicating the time unmistakably. What I utter is thus not an open statement like "x is yellow" but a closed statement that is eith er true or false. And yet although it be true when I first utter it, it may be false when I repeat it later. The point is, of ocurse, that we must be more careful to distinguish between a statement and other statements that resemble it. In the last example, we have two statements, not one. Each of the utterances is a distinct, definite statement; an d the two in fact have differen t t r uth values. These utterances may be exactly alike in sound pattern; but it is each utterance and not anything common to the two that constitutes a statement. Similarly, it is each of the utterances of "now"- not anything common to the twothat constitutes a word and refers to a certain time. In platonistic terms, the distinction between the general pattern or type of a word or sentence and its particular instances or tokens was drawn many years ago by Peirce. Too often, however, those who have noticed the distinction have looked u pon it as a matter of isolated academic interest and assumed that thereafter one need b e concerned only with the types. More recently,i we h ave been for ced to recognize th at often-as in the example above-it is th e tok ens that function as words or sentences; for we find different tokens of the same type naming and affirming different things. 1. See, for example, SQ, pp. 594-623; and Reichenbach's Elements of Symbolic Logic (New York: Macmillan, 1947 ), pp. 284-298.

~I V,l MEANING

do without. Actual discourse, Indeed, it is the types that wehc~ndiffer from and resemble each after all, is made up of tokens t S e " now"s and others · tant ways ome ar other in various impor f f. 't are desks and others articles o urm ure l "very"s just as some predicate to severa h l' cation of a common h chairs; but t e app i . l of furniture-does not imply t at tokens-or to several a~tic es d b that predicate. And we shall there is a universal designate y d to be construed as a rd or statement nee s h find no case w ere a wo l . of types not only does token The exc us10n . type rather t h an as a . 1 lts I think in clariaway with some excess baggage but a so resu ' , fying o.ur immediate pr,~b~:.. is no longer appropriate. It is both Obv10usly the term to sand inscriptions are no fluous · for utterance , d misleading an d super , 1 b t as the actual wor s or b d d as mere samp es u h' h longer to e regar e nd the linguistic universals from w ic statements themselves, a. 1 ger to be countenanced d' t' hed are no on they were to be is mguis . th f t that words and stateat all. Nevertheless, to emphas~ze. e ~c events of shorter or or inscnpt10ns-i.e., d ments are utterances . h terms as "wor . I hall sometimes use sue d longer durat10n- s ' t ,, " 'Paris' -events" an so " ts" " 'here -even s , events", noun-even , . 11 dundant in all these cases. on even though the suffix is r ea y ret . Paris but certain utter' f are not even s in "Paris"-events, o course, h "P ·s"s A word-event sur. t' namely t e an . ances and inscnp ions- . ' d ' t applicable to utterances rounded by quotes-events is a pre ica e and inscriptions; and any ,, " 'Paris' consists of five l etters is short for any

. . ,, . f fi e letter-inscnpt10ns "Every 'Paris' -inscription consists o v d inscri ti on is a separate word (or Although each utterance and.ff p between two words often tc ) the i erence d' statement or 1ett er, e . ' es we need not is. l . t e For most purpos ' has no practica impor anc . " . ,, 11 f which name the same the several Pisa s, a o d . h tinguis among . widel in size, shape, color, sou~ ' thing, even though they differh d y must carefully distinguish lace date, etc. On the other an h' we . n "Pisa" may be more P ' " · ,, It· true t at a give a "Pisa" from a Pans . i~ th "Pisa" just as a given . k . "Paris" than hke some o er ' h ea given

208

V,l

TALK OF TIME

m ushroom may look more like a given toadstool than like some other mushroom; but in both cases we must discern just that overt difference that is correlated with a difference in appropriate use. In the case of "Pisa"s and "Paris"s, and in many other cases, some certain difference of shape or sound-pattern is the clue to a difference in what the words name. Yet by no means every difference of extension is accompanied by a difference in shape or sound-pattern. The nominata of two " Paris"s that are exactly alike in shape may be as different as those of a "Paris" and a "Pisa"; for some "Paris"s name a city in France while others name a town in Maine. To note from its shape that a given word is a "Paris" thus is not enough. In order to determine which of two places the particular "Paris" in question names, we must look to the context-i.e., to the surrounding words and to certain attendant circumstances. Similarly, the various "this continent"s name six or more different individuals; the various "John Smith"s a still greater number; and the various ''I''s name vastly many different individuals. For convenience, let us speak of words (or letters or statements, etc. ) that are catalogued under a single label as replicas of one another , ~ so that any "Paris" (or any "I say") is a replica of itself and of any other "Paris" (or "I say") . Roughly speaking, a word is an indicator if (but, as will be made clear later in this section, not necessarily only if) it names something not named by some r eplica of the word. This is admittedly broad, including ambiguous terms as well as what might be regarded as indicators-proper, such as pronouns; but delimitation of the narrower class of indicators-proper is a ticklish business and is not needed for our present purposes. What has been said above will suggest that almost every name has a replica somewhere that names something different, and that therefore almost every name will be an indicator according to this criterion . But the distinction between indicators and nonindicators becomes effective when applied to a limited discourse. Within such 2. This usage differs from that of C. S. Peirce, who speaks of inscriptions or utterances as replicas of a word type; see Collected Papers of Charles Sanders Peirce, Vol. II (Cambridge, Mass.: Harvard University Press, 1932 ), p. 143 .

209

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t · d'

MEANING

. ll be many names that are no m la discourse there will norma y ll as pronouns will still often roper nouns as we . d' cators-alth ough P . . ally be nonm icators. ns wi11 occasion . . be indicators, an d pronou the personal md1cators, Among the commonest indicators arel indicators. Of the perand the tempora . . d' t the spatial m ica ors, 11 fers to its own utterer, "I" or "me" norma y r e . d sonal indicators, an d ertain others determine "we" or "us" refers to the utterer an c dd sed by the utterer, a " " lies to those a res . . by the context; a you app th gh there is no variation . t' cally even ou . . and so on. Characteris i . . ' names there is a wide variation in what a given personal md1~~:~rindicat~r name. Much more ~e­ in what several replicas of th' g the person in question 'd f course F or one m ' b mains to e sa1 ' ? . 'tin the ostensible rather th an the . i·ng i·n a copy or transcripis sometimes, as m ghostwri g, h . d' tors appear h actual utterer; and t em i~a aker of these inscriptions but tot e tion relate not to the ~ctua. m utterance. Furthermore, some maker of the original inscription lor but serve a prepositional . h "h' " not on y name . di indicators, like t e is s, d f the same shape as m Again some wor s o . bl . a purpose as we 11 . '. . ll but simply varia es, ot md1cators at a h cators are actua11y n . "If anyone disapproves, e may · m · to . is . t h e "he " in an case in pomt t ' s divides temporarily ription some ime d d . leave". Finally, an msc 1 if a given placar r ea · d' ators· for examp e, several different m ic .' d' ff nt persons on th ree suc. car r1ed by 1 ere . "I h te Hit1er" is d · g "l"-inscr1.pmg a three day-parts of the en urm . B ll this is by way of subscript to cessive days , then the tion name different persons . ut a the main point. . . h to be taken into account . f atial md1cator as . d' t The location o a sp d f a personal in ica or. as the pro ucer 0 r . in much the same way . h "h "s name regions they ie m, Some spatial indicators like dt e " ere discrete from the r egions .k the "yon ers are s while others l i e . h t g· on a given indicator name ases JUSt w a re 1 l tary they name. I n mos t c , . 1 ding su ch su pp emen on its context, me u rt of l depends party up h "h "s one may refer to pa t t . E n among t e ere ' aids as pointing. ve 11 t a town a county, as a e, a ' b t h f r sever a Y o a room wh ile ot ers re e 11 th subsidiar y remarks a ou continent, etc. Analogues of a t' ~ indicators. F or example, a ersonal indicators apply to spa ia ally refers to the place P rsonal letter norm f . . . "here" -inscription m a pe " " . delivered telegram re ers ritten· , a here in a where it was W

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r ather to the place where the original was written or spoken; and if a "No Parking Here" sign is moved about, certain different temporal parts of the "Here" name different places. But we are primarily concerned with the temporal indicators. P art of what is to be said concerning them is already evident from our glance at the personal and spatial indicators. The "now"s, for example, behave much like the "here"s; each "now" names a period in which it lies, and the periods named by different " n ow " s range f rom a moment to an era. Other terms , like the "yesterday"s and the "soon"s, name periods earlier or later than th emselves; but in every case the time of utterance of a temporal indicator is relevant to what it names. We need hardly review the oth er points of analogy between temporal indicators and those already discussed, but certain temporal indicators require special att ention. In the first place, the "past"s, "present"s, and "future"s lend themselves to frequent abuse in theoretical discourse. Most "present"s function exactly like most "now"s, naming some period they lie in; and the various "present"s name many different periods of varying lengths, some of them remote from others. A "past", however , most often names all the time preceding-and a "future" all the t ime following-a certain period in which it lies. Thus the period named by a given "past" overlaps, and indeed includes or is included in, the period named by any other "past"; and the same holds for "future"s. This fact , that what is once past is always ther eafter past (and that what is once futur~ was always theretofo re future), creates an illusion of fixity and leads to treating the "past "s (or the "future"s, or even the "present"s) as if, like the "Eiffel Tower"s, all named the same thing. Metaphysicans have capitalized on this confusion for some very purple passages on The Past, The Present, and The Future. We must be careful to remember that nonsimultaneous "past"s (or "present"s, or "future"s) commonly name different even if not discrete periods. V ery often, however, temporal indication is accomplished in a sentence not by any word devoted solely or chiefly to that purpose but rather by the tense of the verb. A "Randy ran" tells us not only who did what but also when, i.e., prior to the period of production of the sentence itself. The "ran", besides specifying

211

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.

.

.

_

emporal mdicator, non d erves a l so as a t f . the action performe ' s . . . d. t different periods o time. simultaneous " ran" s ordinarily in ica e may also serve t h e th"ird Incidentally, verbs i_n s.om~ l~nf~~a;::mple, a creo in Spanish ~n­ purpose of personal mdfiica.t io;, that the pronoun yo is customarily dicates its utterer so de mte y omitted. ally indicates a period within A verb in the present tense n.o rm b . the future tense nor. d ced while a ver m . which the verb is pro u ' . roduction. The mterpreh · d after its own P d b mally indicates t e perio d f binations of tense ver s tation of compound t~nses an o o:oe~mes requires care but is with other temporal mdicators s seldom really difficult. A "Randy had been running" . ment-presum. k lace prior to a mo tells us that the runnmg too p t that is in turn prior to the .fied in the contex ably further speci t itself An isolated time of production of the sen ence . "World War II was present" '

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t ime, we may truly say that it will be past. Of course, a combination such as a "was future " or a " will be past" is usually set within a r estrictive context. In many statements the tense is merely grammatical, the verbs not actually functioning as temporal indicators. This is true more often than not in formal discourse. For one thing, generalizations ar e usually without effective tense; an "All men have spines" refers not only to all men contemporary with the statement but also to all who preceded or will follow it. In many singular statements also the verb , although in the present grammatical tense, is adequately translated by a purely tenseless symbol. For example, where an " a" and a "b" are proper names, a given "a overlaps b" may speak simply of the overlapping of the two individuals, without indicating anything about the date of their common part; that is, the sentence may just say that a o b. On the other hand, another " a overlaps b"

· ply as does a however' te11 s us sim ' "World War II is past" , . The . . . the sentence in question. that World War II is pno:, to " present" or a "will be "' sent or a was 1. h d "present" in an is pre l indication accomp is e present" in no way affects the temhpodra an "is past" or an "is fuby the verb al one . On the other an as 'respectively, a " was" or a ture" functions in the same way f' ch combinations need be "will be" . No exhaustive survey o s~ th t some may result in . but it should be note a attempted here, ts For instance, a virtually vacuous statemen . "World War II was future" . . n context determining what prior mo-if unaccompamed by a y W ld War II-says only what ffi d to precede or t ment is being a rme h t did not begin at the first momen may be said about any event t a th t does not run to the end of of time. Likewise, of any event a

212

may have effective tense, being used to affirm not just that a and b have some common part but that they have some common part that is contemporary with the sentence itself. The context makes the difference. Now one may say that two things overlap [tenseless] if and only if they did or do or will overlap; an "a o b" is implied by an effectively tensed "a overlaps b" or "a overlapped b" or "a will overlap b", while an "a o b" implies none of these but only such a disjunction as an "a overlaps b, or a overlapped b, or a will overlap b". But parallel principles do not hold for all other verbs, indeed , an "a l b" obviously is not implied by an effectively tensed "a is discrete from b" or "a was discrete from b" or "a will be discret e from b", but implies them all. Moreover, even though each verb that is effectively in, say, the past tense indicates a period preceding the verb, the relationship affirmed to obtain between

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·

th erein are also translations of it. Indicators and statements cont aining them are not freely repeatable. Ordinarily, when we want to make continued or renewed use of a given term or sentence that occurs earlier in our discourse, we just repeat it, i.e ., introduce a replica to take its place. If the term or sentence is freely repeatable, then for most purposes we n eed not distinguish between it and its replicas; we proceed as if all were numerically identical. But in the case of an indicator or a sentence containing one, where not all the replicas are translations, this is obviously dangerous. Often, indeed, no available replica of a given term or sentence is a translation of it, so that an inaccessible original cannot, in effect, be brought back into play by repeating it. For this reason, although indicators are of enormous practical utility, they are likely to be awkward for formal discourse. Various remedies may be applied. One lies in supplying a freely repeatable name (or description) of the indicator, or of the sentence containing it, and thereafter, instead of repeating th e term or sentence, referring to it by m eans of a replica of this name. For example, a given "now" might be identified by any

. d and other individuals referred to in a sentence vanes such a peno Wh'l 'd bly with different verbs. i e an consi era " a overlapped b"

. f d b within such a period, an places a common part o a an

"a was earlier t h an b " places a within such a period; a "color c was at place p" places the (color-spot)

. d da sum of c and p within such a peno , an

"color c matched color d" SA VII 2) d (b t not their sum, of course-see ' seems to place c and u . d These examples will perhaps at a moment wi'thi n such. a peno .. kinds of hasty general'izat'ion be sufficient warning agamst certam about tense. licas of other temporal indicators Like some verbs, some rep 1 . a selves indicators. For examp e, m them t are no h t' or . . e onl what is past at t at ime "We can know ~t a give:~; is fu:Ure at that time", present at that time, not . Rather the " d "future" name no times. , the " past" , " present , an " and the "is future at" are tense" is past at" , the "is present at , cti'vely be translated by . t that may respe less two-place pre d ica es . . h " "is at" and "is later " is earlier t an , , . t the tenseless pre d ica es · of words from t h an" . b ' t 1£ prevent a strmg Effective tense does not y l se A t d statement has as con. · tatement. ense t constitutmg a genume s nd a tenseless statemen, stant a truth value as a ten.seless one;t ~ ti'me The difference is d · 1s an even m no less than a tense one, h t t ments with indicators are t d ot er s a e that tensed statemen s an bl , Now of course no term or not so to speak, 'freely re~eatha e . a quale is repeated; for a ted m t e way 1 0 , statement is ever repea . t d not a universa. n · particular even an ed term or statement is a d tatements are much repeat the other hand , nearly all ten:ris anB~t a term or statement is ~aid in that they have many r:plicas.' discourse if all its replicas to be freely repeatable m a given

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"The 937th word uttered by George Washington in 1776''. A later repetition of that "now" is not a translation of it; but any r eplica of this descriptive name is a translation of every other, and names just the particular "now" in question. And using such a nam e, we can readily arrive at a repeatable translation of the indicator; e.g., the "now" in question is translated by any "Th e period referred to by the 937th word uttered by George Washington in 1776"; or alternatively, if the period is a day, by any "Th e day on which George Washington uttered his 937th word in 1776". Or we may seek a translation that contains no name of the indicator itself, but rather another name for what the indicator names. Thus a certain "here" is translated by any "Philadelphia"; and a certain "ran" is translated by any "runs [tenseless] on January 7, 1948 at noon EST''.

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Against such translations , it is sometimes urged that they do not really convey the content of the originals. A spoken "Randy is running now" tells us that the action takes place at the very moment of speaking, while a "Randy runs [ tenseless) on October 17 , 1948, at 10

P .M.,

EST."

does not tell us that the action takes place simultaneously with either utterance unless we know in addition that the time of the utterance is October 17, 1948at10 P.M., EST. Since-the argument runs-we recognize the tenseless sentence as a translation of the tensed one only in the light of outside knowledge , we have here no genuine translation at all. But this seems to me no more cogent than would the parallel argument that "L' Angleterre" is not a genuine translation of " England" because we recognize it as a translation only if we know that L' Angleterre is England. A different question may arise from the auxiliary function of tensed verbs as indicators. If two tensed predicates are coextensive but indicate different times, are they translations of one another? Do we demand that the two a gree in what they apply to , or do we demand that they agree also in what they indicate? It is to be noted that ordinarily predicates that indicate different times differ also in extension; for to say that a tensed predicate indicates a time is a convenient way of saying that the application of a tensed predicate is restricted to individuals at that time. Nevertheless in some cases predicates that-according to the looser locution-'indicate different times' may agree in extension. A clear if unimportant

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and " or d ers a unicorn steak" are translations of each other Both . pr oblem concerning the c .t ·. f questions illustrate a general g. en era 1 problem lies outside n ena m or the . u se of " translation". That m passing that I think (1) th y provmce here. I can only remark siderably with the na t ure and e appropriate may var Y con. purpo f criteria h cn teria much more st . h se o t e discourse and (2) f ormu 1ated within the rmgent t an simpl e coextensiveness can b f ramework of extensionalism a e The Passage of Time Wefl have still to deal with s t a t ements th t r e ect the temporal flow of a seem most patently to of events movmg . from the f tevents. . . One spea k s of time passing p t f . u ure mto th , as. , o thmgs growing steadil old e pre~ent and on into the be mterpreted? y er. How is such language to say that time passes see ms t o amount t of To f ime · that a moment progresses constant! . f o saymg Yet obviously a time d y m a ~ture-toward-past direction other time'; it fa identifi~~,w~~~ i~.!t 't' po,ition with mpect to and if any time moves then all pos1t10n m the temporal series how a move together. Now we ha v e seen, " Time t is future " , a later

"T ime t is present" ,

example is that of a

and a still later

"stood still while walking"

"Time t is past"

and a simultaneous

may all be true; and how the con. . as a translation any Junction of the three might have

"will stand still while walking". Since neither applies to anything, they are coextensive. The question whether they are translations of each other is quite analogous

"Washington's 27th 'future , is . ear her . th t' · is at time t ; and his 49th 'past' . 1 an ime t ; his 13th 'present' is ater than time t"

to the question whether

3. For a furth er discussion, see the other articles in this chapter.

" orders a centaur steak" 216

217

V,l · l Th motion of time t ostensib y exhe f t that t has different rela(where each " is" is. tenseless): · ply mt e ac pressed here consists sim d "fferent verbal events. A gain a tionships of precedence to i " . resent and will be past "Time t was future, is now p ' . . s at time t is later than some says merely that thi~ utterance i later tim:. On the other hand, earlier time, and earlier than some

MEANING

a statement like

t then becomes then becomes presen ' " A time is at first f uture , past"

· e final clause, for example, says ~e~is quite a different matter. Th . t. lar "past" nor that it is . l° er than this par icu h t ther that a time is ear i " h The clause says rather t a a "past or ot er· · st earlier t h an some h d this as we have seen, JU , l .me or ot er an t time is past at some i , t. r other What the c ause . r r than some ime 0 . says that a time is ear ie d d on the time of its own or thus does not epen t al in question says ed the whole sentence contains no ac ~ any oth er utterance. Inde bl Of course we may quite t . freely repeata e. indicators at a 11 b u is 1 . f ·t free of words having many t a trans at10n o i . d"ly understand ably wan . . . nd such a translation is rea i . that are actual mdicators , a rep1icas provided: · 1 ter identical with some time a . " A time is later than some time : ' f " ier than some still later ime . than x , an d earl In the case of a

,, recedes further into the past , " Time t is past and constantly . . t " ast" is an actual indicator while the uttered at time s, the firs p th t time t is earlier than s ; and second is not. The sentence say~ a and q is later than r, then t that if q and r are times later t dan s: Wh t has been said here of th n t prece es 1 • a . precedes q by mor~ a. easily be adapted to the mterprestatements concerning times can . g events nts concern1n · tation of paralle1 stat eme t l"ke So far I have not considered statemen s i "While it endures, a thing constantly grows older" .

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This again is normally a tenseless, freely repeatable statement, saying in effect that if two times r and s are within the period of du ration of a thing, and r is earlier than s, then a larger part of that period precedes s than precedes r. Thus are sentences that express the passage of time or the fl.ow of events translated by sentences that merely describe relationships of precedence in the temporal series . The suggestion of fl.ow or of passing or of ageing disappears ; and just for this reason, it may be felt that we are missing something important about time. Most efforts to formulate just what is missed end in vague poetry or in hopeless confusion over temporal indicators. Yet I think that u nderlying these efforts there is a certain peculiarity of time that deserves attention. Strangely enough it turns out not that time is more fluid than (say) space but rather that time is more static. W e saw that the analogy between space and time is indeed close. Duration is comparable to extent. A thing may vary in color in its different spatial or in its different temporal parts. A thing may occupy different places at one time, or the same place at diffe r ent times, or may vary concomitantly in place and time. The relation between the period of time occupied by a thing during its entire existence and the rest of time is as fixed as the relation between the region the thing covers during its entire existence and the rest of space. And yet there is this difference: two things may approach and then recede from each other in space, may grow mor e and then less alike in color, shape, etc.; but two things never become nearer and then farther apart in time. The location or th e color or the shape of a thing may change, but not its time . This may seem to depend on a mere verbal accident. Why not sim ply generalize the use of " change" a little so that a thing chan ges in a given respect if different parts of the thing have differ ent qualities of the kind in question? Because, it may be fairly answered, this ignores the distinction between a minute mobile thing that travels over a given region, and a spatially large thing that occupies a comparable region at a single instant. Each of the two things has parts that differ from one another in location; but according to ordinary usage, only the former undergoes change. 219

V,1 MEANING

" h ge" in the one case but not the other, l . g the term c an y app ym n im ortant distinction. ordinary usage marks a .p •t t variation in time and h ge 1s concom1 an In other wor d s, c an . . s· e time lS a1ways 0 ne of the variant facsome other respect. me . h tever is the other variant k of change in w a tors in change, we spea h lth gh there is no change that does factor in the given case. T us a ou. . not involve time , there is no change m time.

2

B

On Likeness of Meaning {Every so often someone steers m e into a quiet corner and asks whether I am really in earnest about my paper "On Likeness of Meaning". There is perhaps some ground for the feeling that I am rather less in earnest about it than are some of my opponents. The problems it deals with do not seem to me to have quite the paramount importance that is commonly attached to them these days. And while I am increasingly convinced that any reasonably adequate explication of the misbegotten notion of synonymy is likely to yield the conclusion that no two terms in a natural language are exactly synonymous, I care very little whether that particular conclusion stands or falls. Nevertheless my paper is a serious and I hope not wholly unsuccessful attempt to deal with what I consider an interesting question. If we resolve to confine ourselves to terms and the things they refer to, renouncing concepts, intensions, senses, meanings, criteria in mind, and the like, how are we to do justice to the ostensible difference in meaning between two words, such as "centaur" and "unicorn", that have the same extension? Since my second paper was published, my attention has been called to two further points that may need clarifying. First, if all we seek is some difference between "centaur" and "unicorn", why not point simply to their shape or their spelling? The answer is that degrees of difference in shape or spelling do not correspond even approximately with what we ordinarily regard as degr ees of difference in meaning. Comparison of the extensions of the various parallel compounds into which the two terms enter is much more pertinent. And when simple extensional agreement of two terms is not a strong enough relation for a given purpose, what is usually wanted in addition is extensional agreement of certain parallel compounds. The set of compounds for I am deeply indebted to Morton G. White and W. V. Quine, with whom I have fr equently and profitably discussed the problem dealt w ith in this paper.

220

221

V,2 MEANING

.

. d d what in practice constitutes suffih" d d is ma e-an which t is eman .· f om discourse to discourse. cient synonymy-:-va~ies . r h t recisely by entertaining expres;, The second ob3ection is t a ,,P h eplacement of "centaur . " · f a centaur , w ere r . h sions like picture o . . l"k " unicorn" will c ange by an extensionally identical terhm i el d y transgressed the h h le I ave a rea the extension of t e w o ' d thus forfeited my goal. ional language an fi boundaries of ex t ens k ·ng within the con nes ll t to propose eepi But I did not at a wan d R th I have tried to suggest · l"it Y so construe .. a ter' t that are indee d nonof extens10na T of certain con ex s . how the recogni wn . . evertheless enables us to explam extensional by this cntenon n h words as "centaur" and . ing between sue . the difference in m ean . . of the more distressing " . rn" without involving us in any unico . )1 aspects of intensionalism. (1954 J d two names or predicates in an Under what circumstances o . g? Many and widely h the same meanin · ordinary language ave . t this question but they have . d have been given o ' vane answ:rs . the are all unsatisfactory. one feature in common. y . to the effect that two predicates One of the earliest answers is d f the same real Essence · · f they stan or have the same ineaning i t help very much unless . Id b t this does not seem o or Platonic ea; u h to find out whether two we know ' as I am afraid we do ~ot, ow d f th me Platonic Idea. terms stan or e sa l . th t two terms have the same A more practical proposa is a t l idea or image; or in meaning if they stand for the sadmffe m:nma eaning only if we have predicates i er in h h t other word_s, t at wo . that satisfies one but not the ot er a mental picture of somethin~ in fact all and only pelicans have of the two . Thus even thoug_l . g· e a sparrow or a kangaroo ·11 e can easi y ima in " d gallon-size d b 1 s , w h d" cates "is a pelican an with a gallon-sized bill; and thus the prhe iatisfied by exactly the · d bill" even t ou g s "has a gal l on-size ' h e meaning. There are . d. . d l do not have t e sam . same actual in iv1 ua s, . h I the first place, it is not . . d"ffi lti s with this t eory. n two familiar i cu e d h t cannot imagine. Can we . t hat we can an w a we very clear JUS w . h t? Can we imagine a tone we imagine a man ten miles h1g or no . . only to be confronted have never heard? To decide thehse cases dis and more serious difs But t e secon by new and h ar d er one . l l have no correficulty is that of predicates that pretty c ear y

222

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ON LIKENESS OF MEANING

sponding image, such as "clever" or "supersonic". Of course there is imagery associated with these terms; but that is hardly to the point. There is imagery associated with nonsense syllables. The image theory thus sometimes gives way to the concept theory: the theory that two predicates differ in meaning if and only if we can conceive of something that satisfies one but not the other. This enables us to transcend the narrow boundaries of imagination, but unfortunately it hardly seems to provide us with any criterion at a ll. Presumably we can conceive a fivedimensional body since we can define it although we cannot imagine it. But similarly we can define a square circle very easily (as a rectangle with four equal sides and such that every point of it is equidistant from a center) or a five-sided triangle. If it be objected that because such definitions are not self-consistent they do not represent genuine concepts, I must point out that the claim of inconsistency here can be supported only by appeal to just such meaning-relationships as we are trying to explain. We cannot use them in trying to define them. If the objection is put rather in the form that although we can define a square circle there is no possible thing that can satisfy the definition, then it is clear that we are not judging possibility by conceivability but rather judging conceivability by possibility. Our criterion of sameness of meaning has thus changed: we are saying that two predicates have the same meaning if and only if there is nothing possible that satisfies one but not the other. The possibility theory is somewhat ambiguous. Does it say that two terms differ in meaning only if it is possible that there is something that satisfies one but not the oth er? If that is all, then any two terms we know to have the same extension have the same meaning. If I know that Mr. J ones is in New York, I no longer regard it as possible that he is n ot in New York; and similarly if I know that two predicates are satisfied by exactly the same individuals, the possibility is excluded that they are n ot satisfied by the same individuals. But this formulation seldom satisfies proponents of the possibility theory, who will cite cases of terms that, even though acknowledged to have the same extension, have different meanings. The thesis, they say, is rather that two predicates differ in meaning if there 'might have

223

V,2 MEANING

.

. fied one but not the other; or m other been' something that satis t 1 entity that does satisfy ·ble but non-ac ua .. . words if there is a poss1 . Th t·on of possible entities ' h red1cate e no 1 one but not the ot er p .1 . h rd one for many of us to ot be actua is a a d that are not an cann d .f do accept it how are we to understan d or accep t · An even l h we . ot such' a possib1e t h at . nd when t ere is n decide when t h ere is a ? We have already seen t th ther of two terms . satisfies one but no eo r g to conceivability as a test of that we get nowhere by appea i~ hether two predicates " P" C then determme w d. possibility. an, we ' "bles by asking whether the pre land " Q" apply to the same ~o~~" is self-consistent? This is hardly cate "is a p or a Q but ~~" :nd " Q" are different predicates the helpful; for so l~ng as. lo ically self-consistent, and we have compound predicate is g . . h ther it is otherwise selffor determmmg w e d no rea y means f amounts to the very quesconsistent. Indeed the latter quesh1on meaning And since we "P" d "Q" have t e same · tion whether an . h two predicates have the began by asking how to determme w ent t d back where we s ar e . d"f . same meanmg, we are h · ght try the very l All these difficulties suggest t at we ~~ tes have the same ferent and radical theory that two pre t{cathe same things-or meaning if and only if they apply to ex~c yThis thesis has been the same extension. h in other wor d s, h ave b d anced· but some of t e than it has een a v ' l attacked more oft en . t worthless. An examp e ainst it seem o me familiar argument s a g t . of a term is different d t that the ex ension is the absur argumen th b this thesis two terms may at different times and t~at erde yt t another. The extension t t one time an no a be synonymous a f everything past, presen ' · ts of course o h . of a predicate cons1s ' J.· "ther the making or t e and future to which the term app .1es; nf e1h t "cake". h the extension o t e erm eating of cakes c anges 1 ot against the thesis Certain other similar arguments a~p y .; they have the same h the same meanmg l that two terms ave . th . -that does not concern . b t ainst the different es1s l extension, u ag . ·t meaning For examp e, us here-that the extension of a term is is follo~s : before we hesis one may argue as . against the 1atter t ' . d . t "P" applies to a given . t h ther a given pre ica e . f can investiga e w e " P " means, and if the meanmg o thing a we must .know what t k ow the extension of " P"-and " P" is its extension we mus n 224

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ON LIKENESS OF MEANING

therefore must know whether it applies to a-before we can set about finding out whether " P " applies to a. But this argument does not apply against the weaker thesis that two predicates have the same meaning if they have the same extension; for obviously we may decide by induction, conjecture, or other means that two predicates have the same extension without knowing exactly all the things they apply to. And yet, while many of the appar ent objections seem to me unsound, I think we cannot maintain the unqualified thesis that two predicates have the same m eaning if they have the same extension . There are certain clear cases where two words that have the same extension do not have the same meaning. "Centaur" and " unicorn", for example, since neither applies to anything, have the same (null) extension; yet surely they differ in meaning. I do not mean to suggest that identity of extension with difference of meaning occurs only where the extension is null, but such cases are enough and are the most striking. Now the precise way in which the proposed thesis failed must be particularly noted. Obviously if two terms have the same meaning they have the same extension; the trouble is that two terms may have the same extension and yet not have the same meaning. Extensional identity is a necessary but not a sufficient condition for sameness of meaning. In other words, difference of extension does not draw distinctions as fine as those drawn by difference of meaning. Does this mean, then, that we must return to the dismal search through Never-Never land for some gh ostly entities called "meanings" that are distinct from and lie between words and their extensions? I don't think so. D espite the obvious inadequacy of the thesis we have b een considering, I think that difference in meaning b etween any two terms can be full y accounted for without introducing anything beyond terms and their extensions. F or while it is clear that difference in meaning of two terms " P " and "Q" is not always accompanied b y difference in extension, I think it is always accompanied by difference in the extension of certain terms oth er than " P " and " Q" . L et me explain. Since there are no centaurs or unicorns, all unicorns are centaurs and all centaurs are unicorns. Furthermore, all uncles 225

V,2

MEANING

nicorns· and all feet of unicorns are of centaurs are uncles of u , lize on this? L eaving H w far can we gen era h feet of centaurs. o . I . t' ns we must exclude t e g ammatica varia io , d " aside absur. tor un of r "thought s " or "concepts" or even meananalogu es m erms . ntee that thou ghts of centaurs are ing" itself; for there is n~ g.u arauall attributed to the m ental refthoughts of umcorns. This is uhs y W have in logic the th eoh . ess of sue terms. e er en ce or t e vaguen ll th th'ngs that bear the relation r em that if all a's are f3's , then ah el t. I p to a /3 (see Principia . th t bear t e re a ion . p to an a are thmgs a . . h t rally be supposed that this Mathematica , 37.2) ; and it m1g t nal'k~ those we have been conh t th of sentences i e g . . 'ded the phrases mu ar antees t e ru nd unicorns prov1 sidering about centaurs a_ 1 b . ts if to anything. But actu ally volved apply only to phys_1ca o Jee aintings drawings, pr ints, this is not the case; for pictures-1.et. pll ·ctur,es of centaurs are · 1 b · ts yet no a P1 statues-are physica o 3ec , 11 . t es of unicorns pictur es of f · nor are a pie ur f pictures o unicorns, . . I t the cited theorem o . ht this seems to vio a e fi centaurs. At rst hat s1g it . shows is . th a t "picture of" is not always a 11 logic. Actua y, w " l'k "f t of" or unc 1e of" · If ·xis a foot of a centaur, relation-term i e of" to some y that is a centaur. Thus then x bears the r elation foot 1 of a centaur then there if there is a ny foot of a centaur orf ahny un_c e and indeed there isontrast i t ere isB t · is a centaur. u m c ' cannot infer that there something that is a picture of a cenlta:ir, wte A phrase like "picture there certam Y is no · . . d the fact that it applies to is some centaur-as ,, . · gle predicate an h of a centaur is a sm ' t ble us to conclude t at one or many things plainly does no ena_ tures of To avoid the b . t th t these thmgs a r e pie . h d there a r e o 3ec s a . t'fi d inferences perhaps we a . t ake such un3us i e , f' temptation o m. of our discussion not of 'pictu res o better speak durmg the rest f 'centaur-pictures' and 'unicorncentaurs or umcorns but rather o

o? "

pictures', etc. . f ·corn-picture not by virtue · t differs r om a um · A centaur-pie ure and lack of resemblance to a umnor centaurs. " Centaurof its resemblance to a c.entaur . h . neither umcorns . corn; for t ere are . " 1 apply to different ob3ects " d " unicorn-picture mere y d d picture an. " d " d sk" apply to different objects, an we nee just as " chair an e h . the other. The simple fact k h · the one case t an m d no n1ore as w Y in ,, d " . is that although "centaur an umcorn ,, apply to nothing an so 226

V,2 ON LIKENESS OF MEANING

have the same extension, the term "centaur-picture" applies to many other s. things and th e term "unicorn-picture" applies to many Now the important p oint h er e is this: although two words have the same extension , certain predicates composed by making identical additions to th ese two words m ay h ave different extensions. It is then perhaps the case that fo r ever y two words that differ in m eaning eith er their extensions or the extensions of some corresponding compounds of them are different. If so, difference in meaning among extensionally identical predicates can be explain ed as difference in the extensions of certain oth er predicates. Or, if we call the extension of a predicate by itself its primary extension, and the extension of any of its compounds a secondary extension, the thesis is formulated as follows: Two terms have the same meaning if and only if they have the same primary and secondary extensions . Let us, in order to avoid entanglem ent with such terms as " thought of . . .", "concept of . .. ", "attribute of . . .", and "m eaning of .. .", exclude from consideration all predicates that apply to anything but physical things, classes of these, classes of classes of these, etc. If the thesis is tenable, we have answered our qu estion by stating, without r efer ence to an ything other than terms and the things to which they apply, the circumstances under which two terms have the same meaning. · This explanation takes care of well-known cases discussed in the literature. For instance, Frege has used the terms "(is the) Morningstar" and "(is the) Eveningstar " as examples of two predicates that have the same extension-since they apply to the same one thing-but obviously differ in m eaning. This difference in meaning is readily explained according to our present thesis, since the two terms differ in their secondary extensions. Ther e are, for example, Morningstar-pictures that are n ot Eveningstarpictures-and also, indeed, Eveningstar-pictures that are not Morningstar-pictures. But is our thesis satisfactory in general ? Perhaps the first question that arises is wh ether it takes care of cases where we have two terms "P" and " Q" such that there are n o ?-pictures or Q-pictures-say where "P" and " Q" are predicates applying to odors or electric charges. These present n o difficulty; for 227

V,2

MEANING

. t " Q" consist not merely nsions of a pre d ica e f the secon d ary ex t e . ,, b t also of the extensions o · f " Q picture u b of other such comof the extension o ,, "Q bol" and any num er . "Q-diagram ' -sym ' l ·d inscriptions are as genume pound terms. Indeed act~a wlo7 . - d o if there is such an b. t anythmg e se, an s physical o ~ec s. as . . n that is a P-description and is .not a actual physical mscriptio "P" d " Q" differ in their secQ-description, or vice-versa, then . an . d thus in meanmg. . ondary extensions an d as if every difference m k .t ear more an more d This ma es i app d·ff ce in primary or secon ary meaning will be reflected by a i eren how this to be true. extension. Indeed, I think we canh ntsoo:v:r say " P " and " Q", . t predicates w a ' . . ion of the phrase "a p that is not F or, given any .wo . do we not have m an mscript . . nd not a Q-description? a Q" something that is a P-descridptio~ at.on" applies while the d. t "centaur- escrip i . . Clearly the pre ica e . . ,, does not apply to an inscription predicate " unicorn-description . ,, Likewise the predicate of "a centaur that is not a unl~corn h. ·1e the p;edicate "acrid. t. " app ies w i " pungent-odor-descnp ion . . t. of "a pungent ly to an mscnp ion t odor-description" does no app r". and thus the two predicates odor that is not an acr~d oddo ,, ' hatever may be the relation,, nd "acrid-o or -w · " pungent-o d or a . d"ff ·n secondary extension · extensions- i er i · ship of their primary . 1 ,, d "trilateral" differ m · Again " triang e an and thus in meanmg. h . t trilateral" is a triangle"triangle t at is no meaning b ecause d . t" We do not however, . . b t t a trilateral- escrip ion. ' " . description u no " . l ,, differs in meaning from tnget the absurd result ~h~t tria~g e that " triangle that is not a gle"· for of course it is not t e case . ... an ' . t t . ngle-description:·· . triangle" is and is no a ria If difference of meanmg far we have come. fl But now see h ow d then no two di erent . d i· n the way I have propose ' is exp1ame " - - that is .. · " is a . . · . iy phi-ase such as ,, · b th a " One basic prmcip1e is. a'. . Th " -- that is not a · · · is 0 • -descnption. us descnp- desci-iption an d a . . . -description. Being a not-a- ... d -description and a not-a- · · · · -description. By a secon ~is no-t a sufficient condition for not ?etmg ~s ·~;t a ... -description unless a -descnp ion · t· Formuprinciple, however, a no t - - , , . makes it also a . .. -descnp 10n. , , the first principle (or some other) . 1 deciding whether any phrase is or 15 1 tion of complete and exact pnnc1p els d . either possible nor necessary a t - description would be d1fficu t an is n no a . . · here.

228

ON LIKENESS OF MEANING

V,2

words have the sanie meaning. We have assuredly answered the complaint that in terms of extensions alone we cannot draw fine enough distinctions. Here we get distinctions that are as fine as anyone could ask. But now we risk the opposite complaint: for can we accept the conclusion that a word has the same meaning as no word other than itself ? Before we decide that we cannot tolerate this conclusion, let me note that in the course of developing our criterion we have incidentally shown that there are no two predicates such that each can be replaced by the other in every sentence without changing the truth-value, even if we exclude the so-caLLed intensional contexts in which such words as "necessary", "possible", "attribute of", or "thought of" occur. Thus if we maintain that two different words h ave the same meaning, their lack of interreplaceability in some context other than these can immediately be offered as evidence that the words do not have the same meaning. It seems apparent, therefore, that the demands we commonly make upon a criterion of sameness of meaning can be satisfied only if we recognize that no two different predicates ever have the same meaning. Theoretically, then, we shall do better n ever to say that two predicates have the same meaning but rather that they have a greater or lesser degree , or one or another kind , of likeness of meaning. In ordinary speech when we say that two terms h ave the same meaning, we usually indicate only that their kind and degree of likeness of meaning is sufficient for the purposes of the immediate discourse. This is quite harmless. But we must remember that the requirements vary greatly from discourse to discourse; often it is enough if two terms have the same primary extension; in other cases, identity in certain secondary extensions or others is also required. If we overlook this variation and seek a fixed criterion of sameness of meaning that will at once conform to these differing usages and satisfy our theoretical demands, we are doomed to perpetual confusion. To repeat, it is commonly supposed that a satisfactory definition of synonymy must meet two requirements: that some predicates be synonymous with others, and that either of a pair of synonyms be r eplaceable by the other in at least all non-intensional contexts

229

V,2

3

MEANING

B t we have seen that these two without change of. truth-va~~;· T~e sound course seems to be t o requirements are incornpati e. t peak degr ee of intern ymy as so o s ' construe degree of syno ' t d-and to recognize th at .. 1 g lines above sugges e . 11 replaceab1hty-a on m between diverse predicates is nu . the relation of exact synony y t bearing this paper has words to sugges a l.k Just a few fu r th er . ·d that a sentence i e . It is sometimes sai . on another questwn . . . h · g of B is contained m ' " . Lytic if t e meanm "All A's are B s is ana 1 that two different . . f h as shown n ot on y . that of A. Our mv estiga ion h quite the same m eamng; . "A" nd " B" n ever ave . h predicates like a k "th r is meaning-included mt e but further that , so to spea : n~i teh t i· s not a B -description, and · A descnpt10n a d other; for there is an d . t. Thus at least accor . t an Aescnp wn. ' t . f" lytic" no non-repetitive a B -description t h a is no ing to the su~gested inte1:"Pr~~:i:o~t ::~an sa; is that it is mor e, statement will be an~ly~~". will b e enough to convince many of u s or less , nearly analytic. . i_s t t ment is never absolutely necesthat likewise a non-repetitive s a e sary, but only more or less nearly so.

On Some Differences about Meaning In the light of many discussions of my paper " On Likeness of Meaning'', I want to clarify and amplify some of its main points, th en (in 2 below) answer briefly certain specific comments, and finally (in 3 below) suggest a minor but perhaps welcome amendment. 1

1 The hopeless confusion of attempts to define synonymy in terms of images, concepts, possibilities, etc. leads us to seek a definition solely in terms of actual, even of physical, objects. Yet we must face the fact that some clearly non-synonymous names or predicates apply to exactly th e same objects; the most striking but not the only examples are those where, as in the case of " centaur" and " unicorn", neither term applies to anything. One main point of my earlier paper is that difference in meaning even between such terms can be explained without reference to anything but physical objects. Pictures, for example, are physical objects and yet some (indeed most) pictures of centaurs are not pictures of unicorns. In other words, while "centaur" and "unicorn" apply to exactly the same objects, " picture-of-a-centaur" and "picture-of-a-unicorn" do not by any means apply to exactly the same objects. This suggests that we should take into account not only what is denoted by a given term itself but also w hat is denoted by compou nds containing that term (otherwise 1. Concerning two articles published since the first appearance of the present paper, one by Lester Meckler in Analysis, Vol. 14 (1954 ), pp. 68-78, and one by David Shwayder in Philosophical Studies, Vol. 5 (1954 ), pp. 1-5. I can say only that these w riters have, in different ways, seriously misunderstood m e.

230

231

V,3

MEANING

than in quotation marks). My proposal is that two terms are synonymous if and only if (a) they apply to exactly the same objects, and (b) each compound term constructed by combining certain words with either of the terms in question applies to exactly the same objects as the compound term constructed by combining the same words in the same way or with the other of the terms in question. This criterion recommends itself by accounting, without reference to anything but physical objects, for differences in meaning between coextensive terms. But if we can picture these differences that are not exemplified in actuality, just where does the power of pictorial differentiation end? The limits of realistic or representational depiction may seem rather narrow; but as a matter of fact there is no purely representational depiction. Conventionalization to some degree is always present, and increases rather gradually from the realistic painting through the sketch, the semi-abstract picture and the ideographic sign to the word in ordinary language. The string of inscriptions that we call a description is in effect merely a highly conventionalized picture. But description, or word-picturing, is so delicate and potent an instrument that there is virtually no limit on the distinctions it can make. The difference between a man twenty feet tall and a man twenty and one one-hundredth feet tall is hard to paint but easy to state. Indeed, we can even find triangle-descriptions that are not trilateral-descriptions. A couple of rather clear examples are : (i) plane figure or three angles and four sides (ii) triangle that is not a trilateral. That these apply to nothing doesn't matter; centaur-descriptions likewise apply to nothing. And, even if we allow ourselves to speak of possibility for the moment, it doesn't matter that these descriptions apply to nothing possible . All that matters is that despite the nonexistence and even impossibility of triangles that are not trilaterals we have actually before us in (i) and (ii) descriptions of such triangles. "Triangle-description", then, applies to some strings of inscriptions that "trilateral-description" does not. Thus

232

V,3 " .

"

ON SOME DIFFERENC ES ABOUT MEANING

. at 1east on f th . . triangle and "trilateral" cliff er m d . eo eir correspondm g secondary extensions . . ' an accordmgly d'ff . our criterion. By similar ar um i er m meaning by m eaning. g ent, every two terms~ will differ in Now of course I cannot define . . than I can define pictures . ldescript10ns precisely any more f . precise y E t d . o pictures of centaurs wo ld b . xac an mclusive definition elusive definition of d ~ . e no less difficult than exact and in. escript10ns of triangl B r equired here is that th b es. ut the most that is ere e an app . bl cases, and that anomalou d recia e number of clear b y reasonable rules Int: an paradoxical cases can be dealt with tions that have be~n r . en dext secti~n, I shall discuss some quesaise concerning th 1. ang e- escription"· but th e app ications of "tri1 d . ' ere are other c d m ore easily specifiable ranges of l' . ompoun terms, having b e used in carrying through th app icat10n, that may equally well graph. For example, "literal Ene trgun:ent of the preceding paraas applying just to those inscri ; ish triangle-word" may be taken and "literal English t ·1 t 1 p wns that are tokens of "triangle" :i 1 . " . n a era -word" ' tn ateral". Since thes as app ymg just to tokens of 1 . e corresponding extens10ns, the terms "t . 1" compounds have different riang e and "t ·1 t l" ar gument, every two terms-cl· ff . n a _era -and, by similar Now I am well a h i er in meaning. . ware t at various 1 'bl m g out these examples need to b ~ ausi e grounds for rulpr oving that every two te d'ffe c~nsidered (see 2 below)· but . rms i er in me . ' primary goal. The anmg is no part of . f paramount problem i t d 1 my isons o meaning without f _s o ea with comparmodalities. The proposed re _etrence to mtensions, attitudes or s d en enon in te f ' . rms o primary and econ ary extensions meets th' ex · evi d ent difference . is requirement a n d yet successfully · P1ams .

terms like "centaur" and ~' m ~eanmg even between coextensive . . to accept th unicorn" . In view · am willmg of these virtues I e apparent conse h ' are synonymous. Anyone h h quence t at no two terms not follow at all, or that it w obs ows that this conclusion does can e precluded by suitable pro . 2. That is every tw v1sos, ' o names . a natural language l'k E g]'ish . R estricted artificial Jang or pr ed.ica tes m terms will have the same be_ so constructed 3. Whereas "l't ers Y this cntenon. angle'". I era! English 'triangle'- term" applies to tokens of " 'tri-

mea~i~~e:sc~~h easi~ 233

th~t eso~~

V,3

lly acceptable. But . . more genera 1 will simply render my c.ntenon how disqualified by the resu t I h ld that the criterion is not, any ' mous· for this result

MEANING

tha~

no two terms. ~re a~~~~u~~~n ~~~:i:~able. The extreme difto me unfamiliar ra that surely have ex.seems t' e any two terms h . th way to acceptance of t e view ficulty of finding in prac ic actly the same meaning opens e ms but only ter~s that have that there are no absolute s;:::yor another kind, of likeness of a greater or lesser degree , o meaning. 2

. g~ that I should want any Richard Rudner is correct ~: ~=~:.mulated in terms of a strict final statement of my views d as actual inscriptions or events, nominalism that reg~rds ~or . s 1icas5 of one another rather tha~ which are said to e 7 ep h' k that Mr. Rudner is some Of But I now t m h 'tokens' of a common ty~e. G • ht about the consequences of sue everly Robbms ng two word-events wrong an d B · t that every a restatement. What foll~w~~st ::ery two word-events that are differ in meaning but on ~ ~n meaning. not replicas of each o;~er diff~r D Rollins cites' are theref~~e notf . . . that a defimt1on o The "wild results . that b' fon however, is ts forthcoming. His chief o lec 1w; terms differ in meaning depar . synonymy that makes every t.. acie this is reasonable enough, too far from ordinary usage. Pnma f ge is less drastic and better but the departure from ordmary ose we have a pile of logs, motivated than at first appears.t' f~urposes of the same length em being for all prac ica of measurement some of th . · t a process as others. Will Mr. Rollms re)eCf these logs are of exactly the that gives the result that no /wo ·~ to ordinary usage is indeed h re the usage is much e length? A certain con orm1 y ~::anded of any definition; but even w e .

;:a

. f Meaning'', Analysis, Vol. 4. "A Note on Likeness o 5. See V,1 above.

10 (1950), PP· 115-llB.

d ts" Analysis, Vol. 12 (1952), PP· 98-100. 111 · g" Analysis, Vo · 6 "On Synonymy 0 fWm-"m' . . l f Sameness of Meanm ' 7. "The Philosophical Denia o (1950), PP· 38-45·

234

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ON SOME DIFFERENCES ABOUT MEANING

clearer and more constant than in the case of meaning, what is commonly spoken of as sameness may turn out, according to a perfectly good definition, to be only approximate sameness. 8 And resistance to the conclusion that no two terms are exactly alike in meaning ought to be softened considerably by the recognition that some terms like " triangle" and "trilateral"are, through being interreplaceable in most compounds, very much alike in meaning. Thus I think the second of Kingsley Price's!J objections can be answered by saying that dictionary definitions are useful because they join expressions that are much alike in meaning-although the degree of likeness varies considerably. Mr. Price's first objection I cannot follow. He says that "glub" and "gloob" differ in meaning by my criterion because "glub that is not a gloob" will be a glub-description but not a gloob-description. But I am dealing with names or predicates in a language. When nonsense syllables are incorporated like words in a phrase, the phrase itself is nonsense. Or in other words, if "glub" is not in the language then neither is "glub-description''. Mr. Thomson says 10 that two words are synonymous because they have the senses they have. This is much like saying that a city is north of another because of the locations they have; and it seems to me misleading and irrelevant. It obscures the fact that cities in quite other places are such that one is north of the oth er ; and it appears to deny that we can define the predicate "is north of" in an appropriate and useful way without reference to the location of any particular city. Mr. Thomson seems to be objecting to all definition of general terms rather than pointing to any special difficulty about synonymy. Nor can I accept Mr. Thomson's argument that if I am willing to use "centaur-picture" without being able to define it precisely, I should be equally willing to use "Platonic Idea of a centaur''. One difference is that 8. Mr. Wienpahl, I take it, in "More about the Denial of Sameness of Meaning"-Analysis, Vol. 12 (1951), pp. 19-23-is making this same general point that clarification as well as conformity is required of a definition. 9. "A Note on Likeness of Meaning", Analysis, Vol. 11 (1950 ), pp. 18-19. 10. "Some Remarks on Synonymy", Analysis, Vol. 12 (1952) , pp. 73-76.

235

V,3 MEANING

. . h the term "centaur-picture" clearly I do know some things to whic . h. h the term "Platonic d 't know anything to w ic . applies an d I on . h ld b glad to have a full exphca" plies I s ou e d Idea of a centaur ap . 1 b d line cases· but I nee an . f " icture" in order to sett e or er . ' tion o .P " . c Idea" before I can apply it all. explication of Platoni . d bout what constitutes a de. h e been raise a Many questions av . . . d d If the animal befor e t definition is nee e . scription.11 No comp1e e h . t. whether there are polar b t e ques ion us is clearly a po1ar ear, d th gh we neither know how . 1 d ·s settle even ou . bears on our is an l h ther it applies to certam " 1 b " nor are sure w e . to define po ar ear dary extensions differ we . T h that two secon other animals. o s ow "f mber that we can peroint Now i we reme . need only a case m p . . t t al or even possible, then fectly well describe w~at ils n~thaacnugles totalling 110 degrees", · 1 " "tnang e wi . . "isosceles tnang,e .' . l" are all triangle-descriptions and "triangle that is not a tnlaterah s like th at of "tr iangle . sage In ot er case ' . ld d according to or. mar~, u . . a eal to ordinary usage may y1e that is not a triangle ' direct ppf l t rules that fit ordinary Th we must ormu a e . . no firm decision. en b ·ected to decide th ese . . l d that can e pro) usage where it is c ear an t way of doing this, but a · one correc doubtful cases. Th er e is n 0 t stion runs as follows: Any 1 overing the presen que d bl reasona e rue c . ,, . b th a_ description an a··· " that is is 0 phrase of the f orm ·d· · d . t. on is not a soandso-d e. . d t-a-soan so- escnp i h description; an a n~ b b the first clause of this rule. T us scription unless reqmred to 1 ~' .y b th a triangle-description and a "triangle that is not .a t~ian:,~il: "t~ilateral that is not a triangle" not-a-triangle-description, . t t . gle-description. As exand is no a nan . f is a trilateral- d escnp ion . 11 h plications abou t descripy avoid a t ese com plained above, we ma . h nds as our examples. tions by choosing certain ot er compo~ h as "triangle-descripMr. Clarke argues12 that a compoun sue h . the Journal of Symbolic Logic, Vol. 11 For example, by Mr. Churc m estions that Mr. Smullyan desThe vanous s~gg V l 11 (1951) PP· 69-72-are 15 ( i950 ) ' pp. . 150-151. . " S bols"-Ana,ys1s, o. ' patches so easily m

130. Putnam here takes a different approach from mine, his basic concepts being those of a state-description and of amount of information. Where he touches on the same problems I consider here, his conclusions are in general also quite different from min e .

246

2. A Dilemma The statement . . Maine has many lakes IS obv10usly about Maine s· statement . mce Aroostook County is in Maine, the Aroostook Count seems also to b b Y grows potatoes e a out Mame So 1 . England ' do the two sta t ements . a so, smce Maine is in Ne'"·v

~=w England is north of Pennsylvania

A

w England States are small pparently we speak about M . . any thing contained in (wh th ame whenever we spea k about member, etc.)' Maine and eh er as part, member, member of that contains Maine B' ut t w enever we speak about anything · o accept th· · · an obvious H empel synd " IS prmciple is to overlook clusion that any statem rtombe- and to be saddled with the conen a out anyth· · M · a me. Consider the statement mg Is a statement about . Florida is Democratic. Accordmg to the principle stated h ' . and therefore about Maine Th , t is Is about the United States · e statement . Satellites are planets smce it is about th . ' . e umverse and thus b t h um verse, would also be ab t M . a ou w atever is in the J ou ame ust where did we go wrong? p perhaps only some not II . er aps we generalized too fast· . ' a , statements ab t h ' ou w at contains or is contamed in Maine are about M . N E ame. For example d ew ngland borders on New yo k ' oes not say that Maine borders on N r Portland is with out lakes ew York; nor does

h

2. The paradox arising here . Hempel in a quite differen JS analogous to that pointed out b Confirmation" M' d V t connection. See his "St d' . y C. G. ' in ' ol. 54 (1945) , pp. 102_ 104 . u Jes m the Logic of

247

Vl,1 RELEVANCE

say that Maine is without lakes. Yet to say of the biggest city in Maine that it has no lakes, or t o say of the section of the United States containing Maine that it b orders on New York , is pretty clearly to say something about Maine. Our dilemma is rather deep-seated. Given any statement, we can argue plausibly that it is about Maine . On the other hand, to admit that every statement is about Maine is to make utterly pointless any assertion that a given statement is or is not about anything in particular. The w ay out, I think , is to distinguish between two cases: the one , of a statement that is independently or absolutely about a given thing; the other, of a statement that is, Telativ e to certain other statements, about a given thing. Let us begin, then, by try \ng to formulate accurately enough what it is for a statement t o be absolutely about an object. 3. Absolutely About Certain words or combinations of words in a sentence designate. Just which expressions are taken as designating, and what they are taken as designating depends upon how the sentence is construed and upon what view is held concerning predicates. For example, in Maine is smaller than Texas the first and last words may be taken as designating States, the middle three as designating a dyadic relation, the first four as designating a class, and the last four as designating another class. All but two of these are excluded, however , if we insist upon treating the last four words as an indissoluble one-place predicate of which no component word designates. And others are excluded or supplanted if we regard predicates as not designating anything or as designating attributes rather than classes. But it is no part of our task here to decide such questions, either in a particular case or in general. Our sole problem (and it will prove troublesome enough) is to determine what a sentence is about, giv en what its terms designate. Purely for purposes of illustration in most of what follows, any normal analysis of a sentence into predicates and arguments will be considered admissible, and a predicate will be considered to designate its

248

ABOUT

Vl ,1

extension : th e c1ass of those elements it applies to or denotes.a A sentence may be said to mention whatever any expression in it designates. Thus Maine prospers Utah is west of the Pine Tree State !he Atlantic State farthest from Florida is agricultural aH mention Maine, by name or descri ti tions the class of things th t p on. The first also menth e class of things west ~ Mpr?sper; the second mentions Utah o~ ame the clas f th' , tah is west, and the relation , . s o mgs of which U west of'" and the th ' d (class of pairs) designated b y "is , Ir sentence mentions a th S eral other classes and relations. no er tate and sevTh~t a statement mentions a class does . not by any means imply that it mentions ever y , or even any partic 1 b t u ar, mem er of that c ass. For example .1 d . ' a sen ence mentions wh t . it es1gnates (the whol t . a a predicate in e ex ens10n of th d' necessarily what that d' e pre icate) but not pre icate denotes (th l e severa things the pr edicate applies to) M t' . en wn of the New E gl d S not constitute mention of M . N n an tates does of Maine constitute ment' am;- h or, conversely, does mention thermore a sentence It~ n o t e New England States. Fur' men wns only what a . . used to designate in that t n express10n m it is sen ence not what th t e sew ere or normall d . ' . a expression may 1 h " . ,, Y esignate; for instance, . Mame has five letters ment10ns a word but not a Sta t e.

As . a first attempt to explain absol . ment10n, we might say th t ute aboutness m terms of M . . a a statement S · b 1 1 ame if some statement T th . is a so ute y about ically from S But this ·n dat ment10ns Maine follows logstatement . w1 nee a good deal of amendment. The

3. But in 8 below shall d' ment of "about" if ,,.;eI counten1scuss the .modifications called for in the treatp~;iti'd0.n'. which I favor, is ex;~a~~;;t~~~g :~t i~ddivi~uals. This nominalistic o n iv1duals" [IV,1 above]. Int ~. en .e m my paper "A World I3 follow Quine's in "De s1·g na t ion ' he ate" and "denote" a b ove andusage E · tof design ,, . .(1.939 ), pp. 701-709. Like him I /1s :nee.' Journal of Philosophy, Vol'. 6 md1v1dual with its m ember so th t a so ~re identify the unit-class of an its exten sion designates wh;t it de:ot:s~red1cate having such a unit-cl ass as

249

ABOUT

VI,l

VI,1

RELEVANCE

(1) It either is or is not the case that Maine prospers mentions Maine without saying anything about it. Moreover, since (1) follows logically from any statement whatever, even

from

(2) Florida is Democratic, every statement would qualify by this criterion as being absolutely about Maine. The remedy that immediately suggests itself is to require that the T in question must not follow from logic alone , must not be logically true. But this will not suffice; for from (2) follows the conjunction of (1) and (2) , which mentions Maine and is not logically true. Thus (2)-or any other sentence that is not logically true-will still qualify as absolutely about Maine. To exclude as a permissible T every statement that contains a logically true clause, or a 'non-essential' occurrence of an expression designating Maine, will accomplish very little. F or this requirement and all others so far imposed will be met by another consequence of (2) , namely (3) Maine or Florida is Democratic . Clearly some such statement mentioning Maine follows from any S that is not logically true ; and every statement that is not logically true will thus satisfy all the stated conditions fo r b eing absolutely about Maine. Why, now, even though (3)-unlike (1) , and unlike the conjunction of (1) and (2)-is sur ely about Maine, are we unwilling to accept the fact that (3) follows from (2) as showing that (2) is genuinely about Maine? Notice that we can put for "Maine" in (3) a name or description of anything (or even a univer sally quantified variable) and still have a statement that follows from (2). In other words, whatever (2) says about Maine it says about everything else as well. Now we must seriou sly raise the question whether a statement can properly be regarded as saying about an y par ticular thing what it says about everything else. Or is a statement genuinely about Maine only if it says something about Maine that it does not say about someth in g else? Offh and, the answer might seem to be dictated by an elementary pr inciple of logic. From "Every x is P" we can infer "111, is P" by the rule of instantiation upon universal statements. What

250

holds for everything holds for ea . fo llows that if a statement S is a~~ particular thing. Surely, it Mame. But this does n ot .t ut every x , then S is about qui e settle th aves somewhat as " ch " d e matter; for " about" beh oose oes If I k J as ohnny to choose some presents and he r eplies "I chosen anything Ch . c oose everything", he has not . · oosmg something i I somethmg else. That J h nvo ves not choosing 0 nny chooses eve L i"k ew1se, . · saying so and so a b out an ob· ry t · x is always false · so and so about some oth N h. Je_c mvolves not saying . .d er. ot m g said b is sa1 about Maine Yet . I . a out every object . no v10 at10n of th I f . occurs here . What hold f e rue o mstantiation s or every x h Id · d d does it hold for every x that Johnn o s m ee form ; but never ment says so and so about y chooses x , or that a state-

h

However , although nothi:·g can b . can be said about the class of all thine s~d about every x , much The universal class is infinit!s. uch a statement as says something about th . ab out all other classes · theu u_nt1versall ~lass that it does not say b ' s 1 is se ectivel d h a out the universal class B t y an ence genuinely th e universal class or so~e utha statement about a class, whether her of that class Many o t. er, is not thereby about any mem. · ques ions remain 0 f JUSt what statements ar b , ' course, concerning e a out everythin ' · h a out the universal class d h g m t e sense of being b ' an t ese I shall t e moment we need only t th come to presently. At h . no e at what a t t everythmg (about th . s a ement says about e universal class V) .t d any particlular object m. th t h ' i oes not say about ticular object m it doe ' ta w at a statement says about a par' s no say abo t h. nothing can be said about b ~ everyt mg (V) ; and that objects, or about every class eo~e:fas~e;e;ft, o~ about every class of In sum the troubl .h objects, etc . . . ' e wit our definition f b 1 ies m the absence of . o a so ute aboutness I· any reqmrement f I · yield logically a state t . . o se ectivity. That s men rn.entionmg M · · condition for S to be b I I ame is not a sufficient . a so ute y about M . . S speakmg, yield such a statement witho must, roughly m ent for everything else T b bl ut y1eldm g a parallel state· 0 e a e to p t th. we s all need to mak f h u is more precisely h e use o t e not" f h ' statement with respect t . wn o t e generalization of a S h . o a given express· 10n of a sentence Q with wn. u c a generalizarespect to an expression E i· s arrive . d at t

_am~,

251

VI,l

RELEVANCE by putting an appropriate variable for E ever ywhe.re in Q an~ r efixing to the result a universal quantifier governing that fable. Now a statement S is absolutely about Mame ?nly if. S yields logically some statement T in which some. ex~ress10n des~gh. 'thout so yielding the generalizat10n of T wit nates M aine, wi respect to that expression. In the cases considered above, although (3'~ fo~o':':s fr.om }2), . t'on of (3) with respect to Mame -viz . (x) h t h e genera1iza i (x or Florida is Democratic) "-also follows from (2) · H ence t e fact that (3) follows from (2) cannot now be adduced to show that (2) is absolutely about Maine. Again, although (1) and the · t' of (1) and (2) follow from (2), so also does the genercon]unc ion " . ,, h ed alization of each of these with respect to Mame ; t us we ne no further provision to discount a T that is logically tr~e or t~at contains only non-essential occurrences of an express10n desig-

:ar

nating Maine. . . Nevertheless, our formula still needs strengthening m one way. From the statement ( 4) Aroostook County grows potatoes follows the statement k (5) Everything that is a State and contains Aroostoo County contains a county that grows po~atoes. Our present formula rightly precludes taking (5) as evidence t~at ( 4) is about the class of States, for the generalization of ( 5) with respect to "State"-namely, . (6) (a) (Everything that is an a and contams Aroostook County contains a county that grows potato~s) -also follows from (4). However, (5) also has in it the exp~ess~on " . and the generalization "State and contains Aroostoo k C ount Y , of (5) with respect to this-namely, (7) (a) Everything that is an a contains a county that grows potatoes) . -certainly does not follow from ( 4). Hence ( 5) may still be cited as showing that ( 4) is absolutely about whate~er is a State and contains Aroostook County-that is, about Mame. ~urthermore, by use of this same device, we could show that (4) is about anythink that contains or is contained in Aroostook County. "W_e must rule out statements like (5) as evidence that statements like (4) 252

ABOUT

VI,l

are about Maine (or about oth er things containing or contained in Aroostook County) on the ground that the generalization of (5) with respect to a part of the expression designating Maine in ( 5) also follows from ( 4) . Let us say that a statement T follows from S differentially with respect to k if T contains an expression designating k and follows logically from S, while no generalization of T with respect to any part of that expression also follows logically from S. Then our final definition of absolute aboutness runs: S is absolutely about k if and only if some statement T follows from S differentially with respect to k. Furthermore, the consequences that a statement yields differentially with respect to k embody what that statement says about k. What S says about k and what Q says about j are the same if and only if the consequences that S yields differentially with respect to k and the consequences that Q yields differentially with respect to j are the same except for the expressions in them designating k and j. What Sand Q say about k is the same if the consequences they yield differentially with respect to k are the same except (at most) for the expressions in them designating k . By our definition of absolute aboutness, the statement (8) Crows are black is about black things as well as about crows;~ and this seems to me quite as it should be. R yle and Putnam treat (8) as about crows but not about black things; and no doubt it is often thought of as 'telling us something about' the one but not the other. But whether we consider it on a given occasion as telling us about crows or black things may depend upon whether the statement occurs as an answer to a question like "What is the color of crows?" or to a question like "What are some black things?" By our definition above, absolute aboutness is purposely made independent of such shifting psychological emphasis and of the grammatical position of the designating expressions in a sentence. Since absolute aboutness has been defined in terms of certain logical consequences, what a statement is absolutely about will 4. That is, about the class of black things and the class of crows. I shall usually omit "class of", as understood, between "about" and a plural noun.

253

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RELEVANCE

depend in part upon what logic is presupposed; and this is to some extent aribtrary. No satisfactory criterion for distinguishing just what is logic from what is not has been discovered. Rather, logic is specified by listing the signs and principles that are to be called logical; and the lists given by different logicians are not all the same. A major point of difference concerns the theory of membership. I here regard logic as including the usual theory of statements and all of quantification theory (with identity), but as stopping short of the full theory of classes. Thus although logical constants are here treated as non-designatory, the signs "V" and "A", for example, are treated as non-logical terms designating the universal and the null class. However a different specification of logic may perfectly well be used in conjunction with the above definition of absolute aboutness. Even with the boundaries of logic explicitly drawn, what a statement in ordinary language is absolutely about can be determined only after its interpretation has been settled with regard both to its logical form and to the particular designation of its terms. The remarks on this point earlier in this section will bear some further illustration. The statement Men are earthbound, if construed as of the form "(x) (Mx ::i Ex) '', is about men and earthbound things; but if construed as of the form "(x) (Mx ::i Bxe) ", is about the earth, men, and the relation of being bound. We are safe in saying that (9) Paris is growing is absolutely about Paris; but we cannot tell whether it is about the capital of France or about a certain town in Maine unless we know which place is named by the particular "Paris"inscription5 occurring in (9). That is , baldly: what ambiguous statements are about is ambiguous. But while (9) would rarely be used unless the context resolved the ambiguity, many statements in ordinary discourse remain ambiguous in a way that gives little practical difficulty. In Maine prospers 5. Concerning the difference in reference and truth-value that may obtain among the several utterances of a statement, see VI,1 above.

254

VI,1

ABOUT the word "Maine" ma ll b an individual or anyyo~:rmofa Y e tbaken equally well as naming ' a num er of classes ( · class of counties) or an f e.g. a certam Seldom does the ' conte:t one o a n~mber o.f classes of classes, etc. we may usually decide at remove t ~ ambiguity completely, and our convemence which of 1 1 natives the sentence ment· d . severa a ter10ns an is about A · ·1 . of statements like · simi ar case is that Everything is material w here we are normally free to ;e ard " . " as naming the universal cl g l . everythmg equally well q uantified variable B t ass or fas hp. ay1~g the role of a universally · u none o t is raises d.ffi 1 definition of absolute aboutness. any I cu ty for our All this discussion of h t scure the fact that ment:n aof ~ s~nt;~ce ~entions should not obsufficient condition for S to b by 1 islne1ther a ~ecessary nor a . e a so u te Y about k " Th t . is not a necessary c d T . · a ment10n out that a statement~~: wn is most easily illustrated by pointing Cows are animals is absolutely about no . . 7 entially w·th n-cows, smce this statement yields differ1 respect to non-cows the consequence Non-animals are non-cows . That mention is not ffi · . . logical truths as 1 a su c1ent cond1t10n is evident from such of (1) whi h ( ) ' and from such contradictions as the denial

a lso fr~m

su~h :s~:~~:e:~:sare

not absolutely about Maine; and

h. Maine and everything else prospers, .th l . w ich ment10ns Maine and . tory and yet, since it iel;: ne1 er og1cally true nor contradicrespect to Maine is not ~b 1 nt ol cobnsequen:e differentially with ' sou e ya out Mame M ere1y b ff f · . . y pu mg or one expression a suitabl h s10n with the same desig f Y c osen expresna wn, we can transform any statement 6. Contra Carnap, who (in the . . tion of Babylon by S a b th passage cited m footnote 1) regards men to be about Babylon ~ar~ap'sa ;~es:_ary ;nd a sufficient condition for that logically equivale.nt statem t e m 10~ oes not meet the requirement is designed to meet this requir:~:n~re .a out t~e same things. My definition that every statement about an th. :-v1thbout y1eldmg the anomalous result Y mg is a out every thing.

S

7. See 4 below for further discussion of this result and its accep tability.

255

RELEVANCE

VI,1 VI,l

that is absolutely about Maine into another that says the same thing about Maine and yet is also about, say, Florida . For example, although both the statement (10) Maine prospers and the statement (11) The Atlantic State farthest from Florida prospers are absolutely about Maine, and although what they say about Maine is the same, (11) is also absolutely about Florida. However, (11) does not follow logically from (10) ; the additional premiss Maine is the Atlantic State farthest from Florida would be n eeded. Accordingly, (11) cannot be cited to show that (10) is absolutely about Florida ; and no paradox arises. Furthermore, absolute aboutness as defined above is purely extensional: if S is absolutely about k , and k is identical with j, then S is absolutely about j. The question of intensional contexts of "about" will have to be dealt with later, in 7, but much else needs to be done first . 4. Some Consequences A self-contradictory or a logically true statement is not absolutely about anything. No statement follows from it differentially with respect to k ; for a self-contradictory statement yields all statements as consequences, while for every consequence a logically true statement yields, it also yields a generalization of that consequence with respect to an expression designating k. Furthermore, the conjunction of any statement with a self-contradictory one is not absolutely about anything. But the conjunction of a statement S with a logically true statement is absolutely about just what S alone is absolutely about. Since a compound of two statements, neither of them self-contradictory or logically true, may be self-contradictory or logically true, the compound may not be absolutely about k even though one or each of the component statements is. Even if a conjunction of statements is not self-contradictory or logicaly true, and one or each is absolutely about k, still the conjunction may not be absolutely about k. Consider the statements: 256

ABOUT (l 2) Maine is material (13) Everything is material (14) Everything b t M · . Although (12 ) and (l ) u ame is material. 4 are absolutely ab t M . · . . Junction is not; nor is the co . . ~u ame, their confor that matter is the con . nt~unct10n of either with (13)-nor h ' June wn of all th Th ' ree. us from the fact t at a conjunction is not absol t 1 b u e y a out k, we cannot conclude that its conjuncts are not. Conversely, from the fact that a co . . . k we cannot conclude th t . n3unct10n is absolutely about conjunction of the two staat any con3unct of it is. For example the em en ts ' (15) Northern states are cool . ld (16) Atlantic states are cool ' y1e s the statement States that are either h differentially with r nort ern or Atantic are cool espect to the I f northern or Atlantic Th . ~ ass o states that are either f b . e con3unct10n of (15) d ore a solutely about th· 1 an (16) is therestatements that are not ab:~l ~rfer :lass. Thus a conjunction of u e Ya out k may itself be absolutely about k. An atomic as well as a molecular may of course follow from and . statement absolutely about k absolutely about k F ' yield, statements that are not Someth." ~r exam~le, neither of the statements mg is material . Everything is material is absolutely about Maine th and yield (12) But i· f t ' t ough they respectively follow from · · a s a ement T f ll f w1th respect to k, then T (as well a o. ows rom S differentially moreover T then foll f . s S) is absolutely about k. and k ' ows rom itself differ . 11 . ' ' . Every statement that f 11 entia y w1th respect to ·h o ows from a t t s a ement differentially w1t respect to k, mentions k bu h Accordingly every stat ' t t e converse do-es not hold ' ement absolute! b · q uence T such that T f 11 y a out k has a conse. o ows from T d"ff to (and hence mentions) k I erentially with respect We have seen that no st~tement i that a statement like s absolutely about every x but (17) Everything ages

257

VI,1 RELEVANCE

may be construed as saying that the universal cla~s is identical with the class of things that age, and therefore as bemg absolutely about the universal class. Again, from the statement All minnows are fish th e statement . Everything is either a fish or not a minnow, . . which may be construed as saying that the universal class is identical with the class of things that are either fis~ or non-mmnows, follows differentially with respect to the umversal class. viously singular and existential statements can be shown m similar ' fashion- to be about the universal class. Indeed, e:ery statement that is absolutely about any object or class of obiects is absolutely about the universal class; and furthermore, every such statement is also about the null class. L ogically equivalent statements a;,e a~solutely ab ~ut exactly the same things . Hence since (10) ( Mame prospers ) is equiv-

O?-

1

alent to

. Everything that does not prosper is not Mame, . (10) is absolutely about the class of things that are not Mame, as

well as about Maine. And (18) All crows are black, in view of the equivalent Everything is either a black crow or a black non-crow or a non-black non-crow, is absolutely about the class of non-crows, the class of non-black things the class of black non-crows, the class of things that are 1 eith er black crows or non-black non-crows, and so on. In general, a statement absolutely about any class or classes is absolutely 8

about each Boolean function of them. • That logically equivalent statements should ~h.us be about iust the same things would seem a minimal cond1t10n of _a dequacy that any acceptable definition of aboutness must satisfy. And 8. A further question I left open_ wh en this pape~ ;as s~~{~~d ~':ob~~~ lication has since been settled by Hilary Putnam an osep. 's also ab0 proved thabt thekne gatetohf _a solutely a out . 5 ee eir Vol. 62 (1965), pp. 305-310.

~~~~~e~b~~~ ~:!.t~~t~~~~~:r~:arl:;a~hilosophy, 258

ABOUT

VI,l

although a statement about k will therefore also be about the class of all other things, we have already seen why this does not imply, under our definition, that the statement is about every particular thing. Yet the result that a statement absolutely about a class is absolutely about the complementary class may not seem altogether welcome. Is there no reasonable sense in w hich a statement may be about a given class and yet not about the complementary class? Such a sense may indeed be readily defined . A statement S may be called immediately about k if S follows from itself differentially with respect to k-and therefo re both mentions and is absolutely about k . By this definition, statement (18) will be immediately about the class of crows but not immediately about the class of non-crows . Statement (18) does not follow from itself differentially with respect to the class of non-crows; it does not mention though it is absolutely about that class. This increased narrowness had to be achieved, however, by sacrificing the equivalence principle; logically equivalent statements are not always immediately about the same things. As a result, the usefulness of the notion of immediate aboutness is severely limited. F or example, while a statement may in this sense be about a class without being about the complementary class, no statement can have consequences that are immediately about a class without having equivalent consequences that are immediately about the complementary class. On the w hole, then, immediate aboutness has only occasional utility and moderate theoretical interest, and need not detain us longer h ere. Obviously a statement may be absolutely about another statement as in the case of the four statements: John said, "Maine prospers" "Maine prospers" is true Statement S is short Statement Sis absolutely about k. Other examples are: (19) Statement S is not absolutely about k (20) Statement S is false. But if S itself happens to be (19), it is false; while if S happens to 259

Vl,l

t bsolutely about any. t and so no a d 20) it is self-contra ic ory be ( ' · b o thing .° l about a statement that is a s A statement that is absolutMe ~ the second power, or square, . bears to ame M lutely about ame lute aboutness. The statement of the relation of abso ". b t Maine . "Maine prospers is a ou d wers of the relat10n . h h first and secon po H bears to Maine bot t e f the relations may occur. owObviously, any higher power o . . tatement like l . material" is doctrinaire S eve r , a th·ng e se is . " Maine and every i b t ess but is simp1y a f absolute a ou n ' l o t that is not absolute y b e ars to Maine no lpower about a statemen statement absolute y about Maine .

RELEVANCE

5 . Relatively About absolutely about Maine are neverMany statements that are not b t Maine . Such a statement as ortant sense a ou . . theless in some imp rows potatoes, (21) Aroostook Coui:ty g. bout Maine relative to the while not absolutely about Mame, is a statement

kC ty is in Maine. . lly with respect to ( 22) Aroostoo tounent di·ff erent ia no sta emh . ld differentially with respect T hat is ' (21) yields d ( 22) toget er yie Maine; but (21) an h ·ther yields by itself: t t at nei to Maine a statemen . M . e grows potatoes. 23) Some county m am . ( . ew Hampshire relative to . Similarly (21) is abouCt N t borders on New Hampshire; . d Aroostook oun y come readily to mm . . h f l t've aboutness and other examples o . re a i k relative to Q if and only if t ere Tentatively, then, S is a~o~t ws differentially with respect toh~ is some statement T that o o from either alone. But by t is m S and Q together but not t about anything would f ro . d almost any statemen criterion, as it stan s, ,, . M . rospers is absol'ke " John asserted that ame p one chooses 9. Whether a statement i s" or not w ill depend uphon hhowScheffier ) for "M ·ne prosper C C urc , lutely about . a1 ro osed (e.g. by arnap , P P am on g th e various ways . interpreting indirect discourse.

260

ABOUT

VI,1

be about Maine relative to almost any other statement that is absolutely about Maine. For example, (24) Ghana is tropical would have to be counted as about Maine relative to (25) Maine prospers, since the conjunction of (24) and (25) follows differentially with r espect to Maine from the two together but not from either alone , and so qualifies as T. The remedy for this trouble is not simple. Merely to rule out S · Q as an admissible T will not suffice; for S Q-or, in our example , Ghana is tropical if and only if Maine prospers -likewise follows from S and Q together but not from either alone. The more stringent requirement called for is, I think, that T be a unitary consequence of S · Q in a sense that must now be explained. For convenience, I shall consider only statements in which no scope is governed by more than one quantifier, although completeness would demand treatment of cases of multiple qu antification as well. In the first place, a statement U is an explicitly unitary consequence of W only if U is expanded to eliminate all descriptions and class-abstracts, all statement-connectives other than conjunction and disjunction, and all negations applied to expressions containing another negation , a quantifier, a conjunction or a disjunction. Every statement has a logical equivalent satisfying this requirement. In the second place, every disjunctionsign in U must be outside the scope of every existential quantifier. Here again, every statement may readily be transformed to meet this condition, which is imposed for the sole purpose of simplifying the formulation of the third and crucial requirement. This third r equirement is that every conjunction-sign in Ube, with reference to W, irrevocably within the scope of an existential quantifier. Conjunction-signs are thus captive only if they can be neither freed from the scope of existential quantifiers nor eliminated. Now in the present context there are four circumstances under which conjunction-signs can be so removed from the scope of existential quantifiers: (i) if any of the conjoined clauses dupli-

=

261

VI,l RELEVANCE

cates another; for example, " ( 3x) (Ax· Bx· Ax)" can be supplanted by " ( 3 x) (Ax · Bx) "; (ii) if any of the conjoined clauses lacks the variable governed by the quantifier; for example, " ( 3 x) (Ax· Be) " can be supplanted by "( 3 x) (Ax)· Be"; (iii) if any of the conjoined clauses identifies the governed variable with a constant; for example, "(3x) (Ax · x = c)" can be supplanted by "Ac"; (iv) if merely exchanging the existential quantifier for a universal one results in a statement that still follows logically from W ; for example, "( 3 x) (Ax · Bx)" can be supplanted in the present context by "(x) (Ax · Bx) "-although the are of course not equivalent-provided the latter is likewise a consequence of W. If none of these four circumstances obtains, then all conjunction-signs within the scope of the existential quantifier

~wo

111

areIncaptive. short, an explicitly unitary consequence of a statement is a consequence expanded in the way described and such that all its disjunction-signs are free of the scopes of its existential quantifiers and all its conjunction-signs ar e captive. Since any consequence can be transformed to meet the first two conditions, the third is the really effective requirement. Its motivation is clear. We want to exclude any consequence that is merely a loose composite of statements that derive separately from two others. The requirement that all conjunction-signs be captive guarantees that, so to speak, the consequence cannot be broken in two, that it does not consist of two statements lightly stuck together. A unitary consequence may now be defined as one that is logically equivalent to some explicitly unitary consequence. Thus T is a unitary consequence of S · Q if and only if T is logically equivalent to some explicitly unitary consequence of S · Q. This definition will not determine whether T is a unitary consequence of S · Q until the logical structure of the statements concerned is settled. As observed earlier, there is usually some 10. One might expect that a conjunction-sign could also be freed in the relevant sense if a statement that likewise follows from W is obtained by dropping the existential qu antifier and replacing the variable by a constant throughout the scope. But this gives unwanted results. The pertinent difference between this and the fourth case above seems to be that introduction of a non-logical constant would be involved here but is not involved in the mere exchange of an existential for a universal quantifier.

262

VI,l

leeway in deciding th 1 . ABOUT language. The statem:n;gical structure of a statement in ordinary . (26) Maine prospers and Gh . might be construed t . ana is tropical no as a conJ t. ment with a single one-piece orunc wn but as an atomic statewill not at all affect th two-place predicate But th. ab t e operatwn of . · is ou ness; for (26)' if thus tak ou.r criterion of relative ;all~ from (24) and (25) and ,:n ;:1t:o~>c, will not follow logi. or here anyway. Thus (26 ai to meet the conditions it·fis deemed to h ave, cannot )be ' no matter add d which logical s t ructure m wn to support the unwelcome uce ~nder our present defiMame conclus10n that (24) is . a b out W hrelative to (25) . . e. ave now arrived at a defin· is entirely symmetrical with iens for relative aboutness that put ·. S a n d Q are about k rel respect to S and Q ' an d that may be f some unitary consequence T o~ i;~ to each other if and only if r espect to k from S. Q but not fr ~ follows differentially with Incidentally wh·l om either S or Q alone h ' 1 e we have . w ere a statement not absolutelconcentrated so far upon cases anothe~ statement, the defi.nitio: ~bout k is about k relative to determine whether a statemen giv~n operates equally well to abo.ut k relative to another statte:;:at is absolutely about k is also Mame relative to eith er of the st tent. F or example, (25) is about Wh t a ements a prospers grows If Maine prospers M . but not relative to (24) ' . ame votes Republican Maine h or to either of the following· ' as many lakes . At th b Maine . . pro spers and gr ows. e egmmng of our stud On the one hand , the qu f y, we found ourselves in a dilemm •:out a given object k w:ether a given 'tatement S :; t e other hand, we seemed foroc e a perfectly genuine one. On statement is about anything Th ed to th.e conclusion that any now clear. Only statements. e resol.ut10n of this dilemma is absolutely about k B t of a k 1 . u any state certain t b well-defined class are . r e ative to some statement or men a out anything is about given statement S is absolutel bother. !he qu estion whether a and unambiguous. The quest:'a: ;~t ~given. object k is complete et er S is relatively about k

,e:~~~nt

263

RELEVANCE

VI,l

VI,1

is incomplete; rather we must ask whether S is about k relative to some other given statement Q. 6. Some Further Consequences Suppose S and Q are about k relative to each other. Then if h is identical with k , S and Q are about h relative to each other. And if W is logically equivalent to Q, then S and W are about k relative to each other. As already noted, relative aboutness is symmetrical with respect to S and Q. It is irreflexive, since S·S yields nothing that S does not. The relation is non-transitive, as will be evident from the following example: of the three statements, Houlton is in Aroostook County Aroostook County is in Maine Caribou is in Aroostook County, the first and second are about Maine relative to each other, and so are the second and third; but the first and third are n ot. If one or each of S and Q follows from the other, then they are not about anything relative to each other; for their conjunction will then yield no statement not yielded by at least one of them alone. Since every statement follows from a contradiction, and since a logically true statement follows from every statement, no statement is about anything relative to any contradiction or logical truth. And of course, two statements that contradict each other are not about anything r elative to each other; for their conjunction is not absolutely about anything. All relative aboutness traces back to absolute and even to immediate aboutness; for two statements are about k relative to each other only if their conjunction is absolutely about k, and in that case the conjunction yields a statement that is immediately about k. Obviously, however, from the fact that a conjunction of the two statements is absolutely about k, we cannot infer that they are about k relative to each other. Furthermore, that two statements are about k relative to each other does not imply that either is absolutely about k. For example, neither of the following two statements: Northern States are cool

264

Atlantic States d ABOUT are amp and St is a b 1 land damp are foggy , ates that are both cool so ute Y about th 1 Atlantic. y et th e c ass of States th at are b other, since th .ese st~tements are about this cl oth No:thern and this class the con1unction yields differentia~:ys re~tahtive to each S ry consequence w1 respect to tates that are bo h zom the fact that and Alantic are foggy. other. ' we cannot infer that they are n~t s:~:ements is absolutely . ut k relative to each Ob v10usly t wo statements a statement. Also th may, relative to each o as well as of b 1 ere are second and high ther, be about r elative abo ~ so u:e aboutness. And Jamin ;~ powers of relative absolutely ub ness mtermixed occur as . a IOns of absolute and a out a s d ' m a state fo urth that is ab 1 econ that is, relative to hment that is to leave it at thatso utely about k. The reader wha ~ ird, about a may construct his ow ·11 . o oes not want n 1 ustratwns.

u::;a

:b:~st

nei~h~o::~:,n

So far

7. Rh etorically About , we have been d 1" sta tement and somethin ea mg .so.lely with the relation construed "a bout" g that It is about. All our d . ~etween a sion has been mad ;s a semantic two-place pred· efimt10ns have Pickwick, or centa:r~rHstatements about fictions li~=~~ No provi. . ow can we sa h gasus, Mr. (27) Pickwick fell y t at the statement . b is a out Pickwick, when there i . . Ryle concludes that s h s no Pickwick for it to b b Pickwick- but th uc a statement only e a out?

~lace,

if there is : 0 ::~~~hcan hardly be left ~~:~e~ ~: ~~ a~out e about, neither is m g as Pickwick for the e rst seem t 0 b there any such th· statement to e about I h mg for th statement . n t e second place e statement to between

c:see:~i~:ot~:s :bodut Pickwick gl~st:ess~~e::~=l~· tt~at ~he

Maine and eve~ v:ry different ones like is mct10n which may seem t b ythmg else is material p . . o e about Maine or l"k , that . b aris IS growing , I e a certain utterance of is a out a town in M . ame but may seem to be abo t . . u a city m

265

VI,1

RELEVANCE

VI,l

ount must be given of the France. Some more satisfactor~ acc. k d ot about Pegasus, sense in which (27) is about P1ckw1c an n centaurs, or Maine. nalo ous situation may help.11 A .picture A glance at a closely a ~ h b a certain relat10n to a f bJect t at ears 1 . Abraham Lincoln is an o . f p · kwick? We can comp a0P b t icture o ic · k erson. But what a ou a p f thing since there is no Pie Cent · is · a p icture o any ' of the sense m · w h.ic h ly deny that it nt ·th ut any accou . wick- but this leaves us w1. o f M Pickwick and not a picture the object in question is a o we must r ecognize is that . b er 01. Abraham Lmco n. of M1caw

p1~turel W~~t

while

. . . icture of Lincoln ( 28) The front1sp1ece lS a p d" t "-is a ·n the two-place pre ica e may be construed as. apply1. g (the frontispiece and Lincoln)' . picture of · · ·" to a pair of obJects

the statement . . . icture of Pickwick ( 29) The frontispiece is a P. h e-place predicate d a plymg t e on .is ra ther to be construe as p . . b' t (the frontispiece). . . k" to a smgle o Jee . "-is a picture of P1ckw1c an le itimately infer that there is From (28) but not from (29) w.e c. g picture. For (29) says something of which th~ fronti:~1:c~t:k:ick-picture; the "Pickl ·n effect that the frontispiece bl t of a longer predicate and h . epara e par wick" occurs here as an ms . d tly designating term t an can no more be treated as an mdepen en can one of its syllables or le~ters.t 'cture of anything and yet is · · e is no a .P1 p · kwick-picture-so (27) Just as the frontisp1ec · k better is a ic · a picture of Pickw1c -or . ' b t Pickwick-or better, is . is not about anyt h mg. an d yet lS a OU Pickwick-about. In saymg " .

Maine

"Maine prospers is about d "cate to relate a statewe are app lying a semantic .two-place pre i ment to an object; but in ~~ymg ickwick "Pickwick fell is about p d. te to a statement. . ntactic one-place pre ica . th t we are applymg a sy . infer that there is somethmg . a And since we cannot validly h " . mind hereafter by saymg (27) is about, we may w~ll keep t is m rather that (27) is Pickwick-about. · " [V2 11. Cf. my paper "On Likeness of Meamng ' above].

266

ABOUT

If this suggests an approach, the characteristic of being Pickwick-about still has to be defined . As a first try, we might 1 say that S is absolutely ~ Pickwick-about if and only if some statement T that contains the expression "Pickwick" follows from S while the generalization of T with respect to "Pickwick" does not. This is too loose, however; for "Pickwick" must not merely be somehow contained in T but must occur as a genuine term of T rather than in some such oblique way as in quotation marks or in "Pickwick-picture"-or, for that matter, in "Pickwick-about". Obviously we cannot stipulate that "Pickwick" d esignate in S ; the needed restriction must be effected in some oth er way. Let us say that an expression E occurs as a term of S if and only if E either is a predicate in S or occupies one of the ar gument places of a predicate in S. Occurrence of an expression in quotation marks, or as part of a word, etc. does not constitute occurrence as a term. A statement T follows term-differentially from S with respect to a term E of T if and only if T, but no gener alization of T with respect to any term of T that is part of E, 1 follows logically from S. :i Our definition of "Pickwick-about" may now run: S is Pickwick-about if and only if it yields some statement T term-differentially with respect to "Pickwick". "Pickwick-about" is one of a large family of predicates that end in "-about"; and the definition given is a sample application of a general schema for defining any such predicate.Sis --about if and only if some statement T follows from S term-differentially with respect to "- ", where both blanks are filled in any one case by the same expression. The expression in question must occur as a term of T, but may or may not designate. Hence the schema may be used to define "Maine-about" as well as "Pickwick12. The distinction between absolute and relative Pickwick-aboutness parallels that between absolute and relative aboutness. But I shall often omit, as understood, the modifiers "absolute" a nd "absolutely". 13. Note that for T to follow from S term-differentially with respect to E, E must occur as a term of S ; but for T to follow different ially from S with r espect to k, even if k is an expression, T must rather contain a term designa ting k. Occttrr ence as a t erm and d esign ation are related in the following way : if E occurs as a term of S, and if the result of putting an appropriate variable for E in S and prefixing an existential quantifier governing that va ria ble is a statement that follows from S, then E d esignates in S.

267

VI,1

RELEVANCE

about"; and even though "Maine" design ates, saying that S is about Maine is n ot equivalent to saying that S is Maine-about. For example The Pine Tree State prospers is about Maine but not Maine-about. (Incidentally, The usual name of the most northeasterly State has five letters is about " Maine" but n ot " Maine"-about; while "Maine" has fiv e letters is about " Maine" and is " Maine"-about.) We saw that absolute aboutness as defined earlier is extensional; i .e. if S is absolutely about k, and k is identical with h, and S is identical with Q, then Q is absolutely about h. Now clearly, a statement that is Maine-about may not b e Pine-TreeState-about even though Maine is identical with the Pine Tree State. But this no more shows that " Maine-about" is nonextensional than the fact that an object that is a cane may not be a tine, even though tin and can are identical, shows that "cane" is non-extensional. " Maine" is an inseparable p art of the oneplace predicate " Maine-about" as "can" is of the one-place predicate "cane". And just as "cane" is extensional in that if a is identical with b then a is a cane if bis, so " Maine-about" is extensional in that if S is identical with Q then S is Maine-about if Q is. The extensionality of the rhetorical " -about" predicates is gained, of course, at the price of having a multiplicity of them. A statement that is H omer-about is absolutely about Homer only if " Homer" designates Homer; and even if " H omer" designates Homer, a statement absolutely about Homer is not necessarily Homer-about. Thus, whether or not "Homer" designates Homer, being H omer-about is n ot equivalent to being absolutely about Homer. The positive relationship between absolute aboutness and the "-about" predicates can be summarized in two schematic principles. If Sis - - -about and "- - " designates k, then S is absolutely about k. If S is absolutely about k , then for some expression"--" that designates k, Sis - - -about. The question whether or not a statement is - - -about and the question whether or not that statement is about what "--" designates assume different relative importance with different terms . Since " Pickwick" does not designate, the answer to the

268

VI,1

question wh eth er a statemen . ABOUT negative, and thus only the fir t t is about Pickwick is always hand, once the qu estion wheth:r ~u~st10n counts. On the other answered, the qu estion whether i s _ateme~t is about Maine is . t is Mame-about-which . effect merely asks furt h 1 . er concern1ng h m w at words happen to be en:p oyed-1s of minor interest sa1d fo r all_ terms that designate . " U o':'eve~; th~ same cannot be class of umcorns; and the state~ent mcorn designates the (null) . b Ever y unicorn h as a h orn is a ou t the class of unicor a_b out the class of centaur~s,B:~n~~. ab_out the null class, h ence smce every statement abo u~ an I~ is n ot very informative abou t the null class and he b y ob3ect or class of objects . ' clas f nee a out the cl f is . s o centaurs. What matt h . ass o unicorns and the um~orn-about rather than ne~~ ere is that the statement is Agam, if "material" d . u -class-about or centaur b E es1gnates the universal 1 -a out. . b verything material is ta "bl c ass, the statement is a out the universal class· but ng1 e so are all statements that are about an y object or class of ~b . is that this statement is mat ~ e~tsb and the important point h ere of ab~olute aboutn ess takes ena -a ou t. !n gen eral, the question quhest10n of the application otrtheceedendc_e m our inter est over the w ere " - " . pre icate "_ ,, d esign ates the null 1 about except does not designate at all. c ass or the universal class or Each "ab t" S - ou predicate satisfies h . th:nd Q ~re logically equivalent th!ne :qmv~le~ce condition: if oth er is. Predicat es b . e ther is -about" .f earing to th ese " -a b out" P d. l S ame re1ation th t . . r e icates the a immediate about ;:;s.s may, of course, be defin d· f ness bears to absolute about.ame-about if and only if S fo~l ' or exa_mple, S is immediately with r espect to "Maine" H ows from itself term-differ ent1·a1J t· f h · owever m y ~a is_ Y t e equivalence condition ~nd a~y. such predicates fail to em1c. But it is worthwhile t' t e1r value is largely aca0 statements: compare the follow· f m g our 3 ( 0) Sis absolutel b . . . ya out non-crows (31) S is immed1ate1 b (32) Ya out n on-crows is non-crows-about 33 < ) Sis immediat 1 Statement (30) . 1" e y non-crows-about imp ies that S yields a statement containing (a)

H

s·

269

Vl,l

s

. (31) implies that it1 of non-crows, (32) word(s) designating tdh(e )c da::ignating the class of non-crowds;"non. (a) wor s . · g the wor self contains . statement contamm d " non. lies that S yields a. S itself contains the wor . imp ( ) '" and (33) implies that 33 . lies (32). But only with crow (s) ,,' (31) implies (30) ' and ( ) i,~P crows" designates the crow s · ·ss that n on) d the help of the additional pre~~ ly (30)' and (33) imply (31 an on-crows does (32) p c1ass of n h . h theorems or sue hence (30 ) . k ace to consider furt er b noted in It may e I h 11 not ta e sp mat:er: as relative rhetorical ~~~;t~::ds to consider rhetorical assing that the practical manse be filed, for example, he may Where documents are to t . these documents p aboutness. th at all terms of statem.en s m -that what they simply assume . he can consistently 1 l4 It is d-wherever · rsal c ass. designate , an . ull class n or the umve . fact of designate is neither the n h has to deal w ith the gnm d 't h ·1 oph er w o t he unworldly p i osthat may se em to designate on . life that some terms

RELEVANCE

S Nominalization

. . f redicates and some ism 1:. speaking 0 p to this The language of platon 1 ' has b een used freely up d" of c asses , t be mo iother terms as names of aboutness h ave o . . t H ow would our treatment . 1. ? The nominahst can pom. d f omina ism. b t" t the deman s o n ,, d "relatively a ou " b 1 t ly about an . . fi.ed to mee "fy the definitions of a so u e f utterances or inscriptions pun h · terms o 1 · ndi. for him there are on y i . . mply by casting t em m

:i~the; t~anste:i~::~~~~~ !u~;~:~,ethese t~o-p:.~:~:~;~~ra:~~c:ii;~

:id~~ ~e:: often ; and the rhet?ric:l ro:i:-1ir::ever , the way o~ de-

:i~l play a correspondingl.y bigge ·n ha~e to be somewh at revised. . h " -about" predicates wi fining t ese . . f the present pap er upon concerning the bearing o . The Transformat10ns etrieval is planned _for nns lvania, spon14. A memorandum . problems of fin~or~~~o~e:rieval, at t~e UnSivers~~E:fo:Ue to this chapter.] Proj ect on In o1ma 1 . e Foundation. ( ee F d by the National Sc1enc . d discussion here see sore . d nominallsm un er . the platonism an 15. Concernmg. d als" (cited in note 3) . "A World of Ind1v1 u

270

VI,1

ABOUT

For the nominalist, there is no appropriate variable that may be put for a predicate or other ostensible class-name in a statement, and therefore no generalization of any statement with respect to any such term. Thus for no statement T that follows from S will any generalization of T with respect to any such term also follow from S. H en ce if S yields a statement T in which "--" occurs as a predicate (or as filling a place of a predicate of classes), and no term of T that is part of"--" designates an individual, then S will yield T term-differentially with respect to "- -". As a result, every statement whatsoever will be prosper-about, for example, since Maine either does or does not prosper will follow from every st at ement term-differ entially with r espect to " prosper". The central feature of the r emedy to be administered h ere is that rather than a generalization of T with r espect to "- -", we speak of statements obtained from T by putting for "- -" any other expressions of the same syntactical category. Rou ghly, two terms are of the same syntactical category if either may r eplace the oth er with out disrupting th e syntactical structure of any statement. A fuller and more precise definition of the appropriate notion of being of the same syntactical category is wanted, of course, and it will have to b e in terms of inscriptions or expr essionoccurrences rather than of expressions. This cannot be attempted here; but suggestions toward such a definition are to be found in the work of several recent writers on logic and structural linguistics. 1 G The next and final step is to revise the schema for defining the 16. See Tarski, Logic, S emantics, Metamathematics (Oxford: Clarendon Press, 1956) , pp. 215 ff. and the discussions referred to in his footnote on p. 215. See also Zellig H arris, Methods in Str1lCt1lral LingtListics (Chicago: University of Chicago P ress, 1951 ), Chaps. 15 and 16; and Noam Chomsky, "Systems of Syntactic Analysis", JotLrnal of Symbolic Logic, Vol. 18 (1953), pp. 242-256. The structural linguists have undertaken the difficult task of providing a method for determining, given only samples of discourse in any language, wheth er two expressions in that language are of the same grammatical category. Our problem above is much simpler since we understand the language we are dealing with, know its general grammatical structure, and suppose it to consist of statements determinate in logica l form .

271

Vl,l RELEVANCE . about if and only . t ead: S is . . l" b ut" predicates or h ' h "- " is a term, rhetonca -a o tatement T of w ic . l ically . . ld logically some s f" " does S yield og if S yie s E of T that is part o ~ f E an expression while for no terr; btained from T by putting orh" about" predi-

:~et~e sstd::fie;'~:~~:~.~~~~~~~~ ~~d'~:, ::~hE~'a:::J~~:~!: cate is e ne h me statements as w virtually 17 t e sa same or . d platonistic scheme. to the unrevise

9 . Conclusion

·m

est some of the di I have tried to sugg h seemingly In the foregoing pages, th t arise in setting fort " b t" to adoxes a ge of a ou , culties an d par derlying our ordinary usa . . t' ely well. · ciples un bl d mtui iv obvious prm . t t and servicea e a~ theoretical fine some consis en and to outline some d' de · g the term, h ve not is1 f founded ways o usm of these definitions. a b t etc. . l consequences t hrases a ou , ., and practica . bout questions abou ' p lt' The probl ations a , d'fficu ies. cussed exc am resent any grave new i c is another but these should not p beliefs about, feelings about, et . lem of thoughts about, . ff d for what is o ere matter. than completeness . . in too I claim finality no more d licate and its investigation 1 d , of the devices emp oye , b . the problem is too e a ove , On the other hand, some f 'tary consequence, early a stag~ . a differential and o a uni is such as the ideas of l' with other problems. ove useful in dea mg . may pr . for "same syntactic particular definition given 17 Depending upon the . " d by Noam Chomcategory . tes several suggestions ma e who were kind 18. The above text incodor~empel, and Israel Scheffler, 1 am particularly Burton Dreben, C. . . t 3'ust before it went to prd~ssg. to an important ky s ' d the manuscnp ents lea in enough to rea d Chomsky for comm (5 above) · indebted to ~rebend a;nition of a unitary consequence modification in the e

2

"About" Mistaken Nicholas Rescher contends 1 that my definition 2 of "absolutely about" yields the anomalous result that the sentence "Pa" is absolutely about all individuals. His argument is entirely specious. He begins by proposing that we adopt the postulate that there are at least two individuals in the world. Then, he says, "Pa" is by my definition absolutely about any other individual in the world, since for any such individual, say b, the statement" ( 3 x) (x -=!=- b · Px)" follows logically from "Pa". Now the assumption that there are at least two individuals is not very risky or restrictive. But "follows logically from" is so used in my definition of "about" that one sentence follows logically from another only if the inference holds in every non-null universe of discourse . The fact or the assumption that the world contains two or millions of individuals does not affect this usage ; and infe rences depending on any assumption of more than two individuals cannot be counted as logical in determining what a sentence is absolutely about under my definition. In the second place, Professor Rescher has concealed a premiss in the words "for any other individual". Either he must adopt some such general assumption as that no two individuals have the same name-a drastic assumption that is false for English-or he must admit "a -=!=- b" as an explicit premiss. From "Pa · a -=f=- b" the statement " ( 3 x) (x -=!=- b • Px)" does follow logically; but this, far from showing that "Pa" will be absolutely about every individual, yields only the unobjectionable result that "Pa" is about b relative to the statement "a -=!=- b", or that "Pa· a -=f=- b" is absolutely about b. 1. In "A Note on 'About'", M i nd, Vol. 72 (1963), pp . 268-270. 2. In "About", Mind, Vol. 70 (1961), pp. 1-24.

273

272

Reading for Chapter VI Suppldemeton::f:;ences within text and footnotes seco n ary . ' 'About' " ' Mind, Goo mans d " A Reply t o R esc h er on Patton , T ., Vol. 74 (1965), pp. 592-593.

VII Si.mplicity FOREWORD Simplicity, like "about", is a rather new topic for philosophy. With virtually no relevant literature available, no inherited respect for or even recognition of the problem, and no established guidelines, the problem itself had to be d efined , its importance explained, and workable concepts and strategies developed, almost from scratch. In the course of the struggle, a number of interesting aspects of philosophical method were illustrated; for example, the search for a root principle in the criteria implicitly used in rej ecting the first attempts at definition, the need for making some arbitrary, along with many grounded , d ecisions on the way from an intuitive notion to a technical concept, and the progressive building of a calculus by looking to the consequences of the initial axioms for cues to appropriate additional axioms. Since, however, each paper on the structural simplicity of sets of extralogical primitive predicates tends to be amended or superseded b y the next, and since all m y papers on the subject are superseded b y the treatment in the second edition of SA, I have included only four of them here: two that are in effect surveys, one note that clears up a fairly common misunderstanding, and a review . To amplify the review a little, William Craig's device for eliminating extralogical primitive predicates consists not in defining them but in providing a separate postulate for proving each theorem that amounts in effect to the expansion, under such definition, of a theorem containing them. While the number of primitive terms is thus decreased, the number of primitive propositions is increased. This might be remedied if these postulates could in turn be disposed of by the device of "Elimination of Extralogical

274

275

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SIMPLICITY

. . k 1 for finite sets of postulates, Postulates" but that device wodr slton ~th here will normally be in1t ts to be ea wi · while the postu a e se h" . licity and systematiza. ' cedure ac ieves s1mp f finite. Thus Craig s pro . . . l" .t and systematization o tion of terms by sacnficmg s1mp ic1 y statements., d . 1 recent l y d.1scuss ed by Bohnert~ and by Ramsey s ev1ce, h 1. . at1·on of extralogical bases "ble t e e imm · d Scheffler,-~·. ma k es t poss1 . . . d. tes in favor of quantifie t pnm1tlve pre ica . f by droppmg cons an . of the complexity o . bl S h reduct10n to zero P redicate vana es. uc f · h logic For example, . d d upon use o a nc . . extralog1cal bases epen s f . d . . d als will require quanhf d "cates o m iv1 u removal of a set o pre i h d. cates and-depending on · bl s for sue pre 1 ' . ti"fication over variables fication over vana e 1t d ay require quan how these are re a e -m . y is thus paid for by E tralog1ca 1 econom of much higher type. x C lexity of the latter sort . . h 1 g· cal apparatus. omp . complexity m tble o 1 1 1 f extralogical simplicity; but by our ca cu us o h is not measura e to suggest does t e t what Bohnert seems ' neither, contrary o t ominalistic prejudice. By 1 ·t here res on n . judgment of comp ex1 y . . h more complex, all else bemg quite neutral standards, logic is t e equal, the higher its order. . t t bearing on the whole ' 1 does have impor an f Ramsey s proposa . . d . r "t . (1) it remm s us that our calculus o question of Slillp lCl y. b l° d only for comparing sysextralogical simplicity sho~ld e aplp ie 1 tive to a minimal logic . .h e logic-or on y re a terns havmg t e sam . h t· how simplicity of extrab th . (2) it raises t e ques 10n · adequate to o , t be measured agamst . d f 1 · al apparatus are o .. logical basis an o ogic h t finite set of primitives I th . k the fact t a any one another. Here, m ' ed b a single one of higher type sugof a given type can be re_plac ~ ses complexity much more gests that raising the logical type ilnc~eal imitives at lowel levels. drastically than does adding extra og1ca pr " . " . The Foundations of Mathematics, ed. R. B. 1 F. P . Ramsey, Theories m l p l 1931 ) pp. 212-236. ' . . d . Routledge and Kega au ' Braithwaite (Lon on. . . . M th d" Journal R 's Ehmmat10n e 0 • 2. H. G. Bohnert, "In Defense ~~5-;;;r;_~e~nd "Communication by Ramseyof Philosophy, Vol. 65 (1968) , pp. . V l 34 (1967) , pp. 341-347. Cl " Philosophy of Science, o . Sentence ause • k· K f ) pp. 203-222; and 1967 3. Th e Anatomy of Inquiry ~~~':; Jo~~r~al ~o~l~ilosophy, Vol. 65 (1968), "Reflections on the Ramsey Met ' pp. 269-274.

VII SIMPLICITY

(3) It underlines the extravagance of unlimited logical apparatus by showing that with such apparatus available, extralogical simp licity hardly matters. With unlimited funds, who need worry about the cost of living? While work on the simplicity of sets of primitive terms has resulted in some progress, "The Elimination of Extralogical Postulates" has up to now eliminated hope of progress toward measuring the simplicity of sets of primitive statements. Occasional attempts to dismiss the result of that paper have not, I think, touched the central point; and we are still very much where we were thirty years ago. Quine has recently 4 reaffirmed and expanded upon our conclusion and our despair. My insistence that simplicity is of the essence of science has sometimes been misinterpreted as the claim that simplicity is the only or the always-overriding factor in the choice of basis. Obviously, considerations such as brevity, clarity, convenience, familiarity, and utility for a special purpose usually enter also, and may sometimes exact a considerable sacrifice of simplicity; but these other factors, unlike simplicity and truth, are minor aids rather than major aims. A scientific system may be cumbersome, difficult, strange; but with no simplicity we have no system and no science at all. The remaining papers in this chapter are concerned with simplicity as an integral factor in the inductive rather than in the deductive aspects of science. Whenever we make any inference beyond the evidence, we must choose among countless alternative hypotheses; and the choice must favor simplicity. For just as every admission of an undefined term or unproved statement detracts from definitional or deductive systematization, so every admission of an exception in a hypothesis or of an irregularity in a curve tends to detract from the inductive subsumption of the particular under the general. Science proceeds by definition, demonstration, and generalization; and abhors the undefined, the unproved, and the ad hoc. But just as a measure of structural simplicity must disallow the 4. "Implicit Definition Sustained", Journal of Philosophy, Vol. 61 (1964), pp. 71-74; reprinted in Th e Ways of Paradox (New York: Random House, pp. 195-198. 1966)'

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spurious gain that results from compressing a number of primitives into a single one-place predicate of higher type, so a measure of inductive simplicity must disallow the spurious gain that results from smoothing out a rough curve or hypothesis (such as "All emeralds are examined before t and green, or not so examined and blue") by use of specially contrived coordinates or terms (such as "grue"). What we must do is take into account the familiarity of the predicates along with the grammatical composition of the hypotheses. Of course, sheer familiarity cannot be taken as the measure. What we must do, as discussed further in VIII below and in FFF, is to develop-on the basis of familiarity in relevant contexts and other considerations-a criterion that does not exclude new terms wholesale and that allows for trading some elementary simplicity for higher-level simplicity and consequ ent overall simplicity of theory. Curiously enough, familiarity of another sort entered into the measurement of the structural simplicity of bases in a different way, by distinguishing between the routine replacement that does not and the non-routine replacement that may result in genuine economy. Criteria for the simplicity of hypotheses will obviously h ave a direct bearing upon the distinction between significant and superficial similarity (see IX,2 below) , between natural and artificial kinds, and between regular and random sequences. A ll these questions are part of the general scientific problem of classifying and ordering nature and making it lawful. Long neglected, as a matter for effete and gratuitous speculation, simplicity has recently been taking on pressing importance. The response to the earlier papers was a deafening silence, and simplicity is still far from a popular topic among philosophers; but scientists and technologists are becoming acu tely aware of the need to measure simplicity in the course of their own work. Publication of "The Test for Simplicity" brought inquiries from all fields of science from all over the world. "Uniformity and Simplicity" resulted from a request from geologists. Engin eer s and educators have asked for help. Unfortunately, while we know somewhat more about simplicity now than we did thirty years ago, we still know far too little to meet all su ch needs.

1

The Test of Simplicity Simplicity is a test of the ff t " . t h e test of simplicity? e ec iveness of s cien · t"fi h w·h at is i c t eories; but

All scientific activity amounts to th . . among systems of hypotheses 0 efrnhventr_on of and the choice g .d. h . ne o t e primary .d m mg t is process is that of sim 1" . . cons1 erations m ore mistaken than the t d 't· pl ~city. Nothmg could be much ra I wna idea that fi system and then for the sak f 1 we rst seek a true W e are inevitably concerne~ ow~t~g:ince ~lone, seek a simple one. concerned with syste t 11 f mphcrty as soon as we are m a a · or syste · h. extent that the basic b 1, m rs ac reved just to the . d . voca u ary and set of fi t . . m ealmg with the given b" rs prmcrples used su 3ect matter are · l"fi d . . P 1rcrty of basis vanishes t h . srmp 1 e . When sim. d . o zero-t at rs whe t rs enved from any of th th ' n no erm or principle Systematization is the same ~h. ers-system also vanishes to zero. Furthermore in th e h . e mg as simplification of basis. . 1 ' c orce among alternat· srmp icity are not alwa 1 1 rve systems, truth and ys c ear y di t . . h b often, simplicity is one test of truth ;hmgms a le factors. More over what bodies in th . . e martyr-bestrewn debate . e umverse are fixed h qu est10n : What ch oice of . t f as turned into the 1 d porn s o referenc ·11 · P est escription of th · e wr give us the sim. D h err re 1atrve motion ? A d b u em and Poincare l it is 1 t s· n Y now, after . f ' a mos a commonpl th h t IOn o a mode of scientifi 1 . ace at t e refutaNewtonian consists not ~ exhp an.at10n such as the Ptolemaic or the rn s ow mg th t ·t · · sh owing that its application would b . a I rs mapplicable but in The case can be put e mtolerably complex. even mor e strongly W system or hypothesis that not onl ." e want to select a dence but also predicts 1 yhagrees with the established evit' correct y t e outco ff wns and experiments Th 1 . me o urther observa. · us se ect10n of a th ma d e m advance of the d t . . eory must always be d e er m m at10n of some 0 f th f an , accordingly, some crite . h e acts it covers; non ot er than conformity with such M ~i Pd. Duhem, Systeme du Monde e io e (Pans, 1908). (Paris, 1913) ; H. Poincare, Science et

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oints akin the selection. After as many p . . facts must be applied mm d ex eriment concerning the c~r­ as we like have been plotte y of time and deterioration relation of two factors (for hexamp e.'ning points by choosing one predict t e remai d . t . ) of radioactivity , we that cover the plotte porn s. among all the infinitely many curves dinal factor in making d' f . .t f some sort is a car Obviously, simp1ici y o " e) The very vali ity o . k th " smoothest curv . this choice (we pie e h th the choice is properly made ac. d epen d s upon w e er l' .t here is not a consi'd era t'ion the choice cording to such criteri~ . Thus simp ~c~~t is one of the standards of ft truth is determine h applicable a er . . h effort to discover trut . validity that are applied m t e

SIMPLICITY

t i

Nature of the Problem . th and systematization, what is the But if simplicity is a test of.tru f th standards of simplicity contest of simplicity? Explication o e t problems in the philosok h t pressing curren h ay be inclined to as w y S titutes one of t e mos · t' t however m phy of science. The scien is 'G. tw~ alternative systems coverthere is any 1JrobLeni here. ive~ . 1 pretty clear which, if . tter isn t it a ways ing the same sub3ect ma ' le arise? There are two .h . the simpler? How does any puzz eit er, is good answers. d'ffi ult and significance of formulating In the first place, the i . c 1· Y. more depend upon trouble . · f simp icity no h precise general cntena o . 1 . dgments of simplicity t an k g particu ar JU . d d . encountere d m ma m f codifying deductive logic epen. the difficulty and sigmfican~e o d' reasoning. The systematic upon trouble encountered m or i;a~h'tehead and Russell? and logic developed by Aristotle, Bolofe, dr~wing or correcting infer. . d tally a too or h others is only mci en . . th laboratory. Event e enerences needed in ordinary hfo or iln . e . ce MilP aims much less ... . f . ductive ogic sm getic investigation o . m akin redictions than at ehcitmg at providing instructions for ::U g pk The utility that the re. . f their own sa e. h the laws of induction or . . t ral scientist is as muc a ve for the practicing na u su1ts may ha hought (London , 1854); · · to the L aws 0 f T ·d E G Boole An Investigation in . . . Mathematica (Cambn ge, n2. . . ' d d B Russell Principia A. N . Wh1tehea an · ' gland, 1910-1913 ) · · (London 1843) · 3. J . S. Mill , A System of L og1c '

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b y-product as the utility that the scientist's results may have for the technologist. Investigation of the canons of deduction or induction or simplicity no more derives its main interest from the h elp it may give to physicists or biologists than investigation of th e laws relating mass and energy derives its main interest from the help it may give to munitions-makers or surgeons. But in the second place, comparative simplicity is often not very readily and surely judged. Of course if we succeed in deriving one of a set of hypotheses or concepts from the others, the saving is obvious. But comparison of theories incorporating different hypotheses or concepts can offer great difficulty. For example, just w hen does introduction of the concept of a new fundamental particle simplify and when does it complicate physics? Sometimes d ifferent aspects of over-all simplicity may set up competing claims. Is a genuine simplification achieved by deriving mathem atics from a few logical concepts at the cost of three big volumes of complicated formulae? We may smugly reply "yes" , on the ground that what counts is the simplicity of the basic n otions and postulates, and that the derivation is bound to be complicated just to the extent that the basis is simplified. But we seem to take the opposite view when we rate description of the motions of astronomical bodies in terms of ellipses as simpler than that in terms of circles; for the advocates of epicycles might argue that the more elaborate constructions and computations required by his system ar e the symptomatic r esult of the greater simplicity of his elementar y concepts. If we are tempted to dismiss the idea that circles are simpler than ellipses as a mere superstition, we shall be embarrassed by the fact that, as remarked above, we employ some such notion of the relative simplicity of different curves when we choose one to fit to plotted points in order to extrapolate from determined data to untested cases. Plainly, simplicity is not a single easily estim ated characteristic of systems but several different interrelated characteristics, few of them easy to estimate. Thus simplicity is a problem for the scientist as well as for the philosopher. Canons of simplicity need to be formulated not only for their intrinsic interest but also as means for making needed judgments in actual scientific investigation. A more plausible excuse for neglecting the study of simplicity has been that the problem, far from being too easy, is h opeless.

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Simplicity, the argument runs, is vague, ambiguous, variable, and subjective, and therefore too elusive for measurement. But exactly the same argument might have been urged in primitive times against the possibility of measuring temperature or size. Ordinary judgments of size vary with distance, perspective, atmosphere, color, eyesight, and e~en with interest. Size may mean total bulk, or it may mean maximum diameter, or it may mean height and length and breadth or any of many other quantities. And size changes with temperature, pressure, growth, and wear. The arguments against the measurability of simplicity would, indeed, have been equally strong against the measurability of almost anything. Precision, fixity of meaning, verifiability, and objectivity are the results of measurement, not preconditions of it. What the problem of simplicity needs is a lot of hard work . So far, just a little has been accomplished; the entire bibliography of contributions to the subject hardly lists more than a dozen items,4 most of them published during the past fifteen years. Thu s the problem is not only one of the most important in the philosophy of science but also one of the newest to be tackled seriously. We are still seeking proper formulation of some aspects of the problem, still exploring avenues of approach to others. Yet this gives the whole matter added interest; for here we can observe philosophy, and therefore science, in the early, formative stages of a typical development from a nebulous cluster of difficulties into articulated questions and on towards an organized discipline.

Simplicity of Basic Terms We must begin by staking out a very small part of the pr oblem for concentrated attention. 5 A theory is a system of statements. I shall be concerned here solely with the simplicity of the set of 4. The articles mentioned in reference 5 together contain a fairly complete listing of relevant publications, from an article by Lindenbaum in 1935 to the present: Karl Popper's discussion in Logik der Forschung (Vienna, 1935) should be added to the list. 5. The treatment of simplicity to be outlined here has been developed in the course of several of my publications, from "On the Simplicity of Ideas", Journal of Symbolic Logic, Vol. 8 (1947 ), pp. 107-121, through "Recent Developments in the Theory of Simplicity" [VII,2 below]. [Further revisions were made in the second edition of SA , which appeared eight years after the present paper.]

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concepts, or the vocabulary of term . ments. Furthermore since s s, employed m these state"or" "not" ,.,. f ' h " ,~me words and symbols-like "and" ' ' i .... t en all" "some" " - " ' these-are logical appa t' ' ' - , or translations of . ra us common to all th t . tion, we need consider I h e sys ems m quesI · on Y t e remaini Such of these as are not defi cl . h ng, extra og1cal terms . ne in t e system ar .It.ive " and constitute th t l . e ca II e cl " prime ex ra og1cal bas· f th the system. It is the simplicit of is o .e vocabulary of want to examine here. y such extralog1cal bases that I Among the extralogical terms of terms like " .... is acid" cl l . a system may be propertylike " .... is larger than ~~; aa~~o~-term~ of various degrees, and ------------" . The exam les i .... hes halfway between place, a two-place and a th pl g ven are respectively a one~ee-p ace predicate and may be abbreviated in stand "B ( ' ) " ar symbolic notation, as " A(x)" " L( )" d x,y,z . In general, an n-place a· ' x ,y ' an iables and stands for an a· 1 ~re !Cate has n blanks or Varn-a ic re at10n Many of the most familiar terms occ~rri . . . . are not predicates but th ng m scientific theories "the father of " "thrat er nonassertive function-terms like · · ', , e Hemperature of · · · · " , "th e a·istance ' between . . . . and· ·_ ments there is a corres~on~:ever, fo_r e~ery fu.nction of n argufo rmity, let us suppose th t g. n +hl ad1c relat10n; and for uniconsideration all funct1'on t a inh t e extralogical bases under - erms ave be I' · corresponding predicates Th th f en e immated in favor of . us e unct10n-term "the father of . .. . " or "f n . ' .r gives way to the predicate " . · · · . ", or "F(x y) '" and th f . is the father of ' "d ,,e .unct10n - te rm "th e cl·istance between .... a n d -'" or th a· t b , '"11 gives way to the predicate " . e is ance etween a d ,, ------------ is ·· ·· n or"D( )" Some predicates like " . ' " x ,y,z . . ' · · · . Is wooden and " · c1ent fossil than _ ,, . · · · · is a more anOthers, like " are r , ~:e pdr~~1cates of things or individuals. d' ···· are an · · · . is a subset of ,, pre icates of classes of things. Still other . ' are of classes, numbers and S s ~re predicates of classes ' so on. ome pred1cat l'k " temperature of _ , , and " . es, 1 e .... is the h t · · · · Is a member of " e erogeneous, relating numbers to h' . , are and so forth. To avoid h . cl t m gs, or thmgs to classes . . avmg to eal at the st t 'th d'ff , h d'ff . ar w1 I erences m simplicity arising from . sue i erences in ty 1 t . th at m every case und . cl . pe, e us require er cons1 erat10n all the predicates be of

cl

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Vll,l

th' gs that is that all apply solely to m ' and so on. Our results can a single homogeneous type- f h' ' or all apply solely to classes o t mgs, where predicates of differafterwards be extended to cover cases

THE TEST OF SIMPLICITY

VII,l

SIMPLICITY

ing types are involved. . . that all predicates in a basis assumption is ,, One more temporar Y l'k " is a centaur or h t . that none are i e . . . . h h' are applicable-t a . is , " . l ing to nothing. Althoug t is is a square circle m ap~ ~ ay sometimes want to " ···· restriction, we m may seem a common-sense d. t when we are not sure determine the simplicity of a Hpre ica e the restriction can later . t ything owever, wheth er it app1ies o an . be easily removed. . . ediate problem is needed. .fi ti on of our imm t' Still further speci ca l matical construe ion with the pure y gram .h h . d We are not concerne . N we concerned wit t eir . th their length or are h of predicates or wi prehending them. T ese asrelative familiarity or the ease ohcom ay be philosophically and nd many ot ers m h 'th . 't pects of simp1ici y a nt concern is rat er wi . ·fi ant but our prese d' tly scientifically sigm c ' . licit of bases which pertains irec that logical or structural si~p . y f theories founded upon them. to the degree of systematization to l simplicity is indeed part of Just what constitutes such struc urad learer as we proceed. But d ·n have to be ma e c . our problem an wi d' t ·n different languages, or lil enwe may note here that pre ica es i l ge or quite unlike in d of the same angua ' d tirely different wor. ~ a be equally simple in this sense, an grammatical composition, m y . tly the same instancesthat for two predicates to apply ~n exac tension-is a sufficient h denotation or ex d that is , to have t e same . . for their having the same egree though not necessary condition of structural simplicity. fi d way of measuring the want to n a f h In summary, t en, we f d fined extralogical terms o . . t f the set o un e structural simp1ici y o . t to be able to assign to any That is we wan l ·t of t a theory or sys em. , hat will indicate the comp exi y. such set of terms a number t . 'ficant aspect of the complexity that set and, accordingly, one sig~i h t ll the terms under conb · by assummg t a a of the theory. We egm d' t s belonging to some one homosideration are applicable pre hica e number of places, and of t they may ave any h th B geneous type. u . . iven basis need not ave e everal predicates m a g course the S same number of places. 284

f

A Clu e to an Answ er The first step toward measuring the size of objects must have b een to fix upon a single elementary clue: the application of a yardstick of some sort directly to the object in a certain way. This at once abstracted from apparent differences resulting from variations in distance and perspective, picked out as standard certain among the innumerable dimensions of familiar objects, and provided a unit for the numerical expression of size. We seek some comparable clue to the problem of measuring the simpliicty of extralogical bases . From the primitive predicates constituting the extralogical basis of a system, with the help of the specified logical apparatus, all other extralogical terms of the system are generated by definition. One might first think of measuring complexity by definitional yield, or defining-power, on the principle that if one set of predicates is definable from a second but the second is not definable fro m the first, then the second is more complex, and that interdefinable sets of predicates are equally complex. For example, we can define " .... differs by one from - - - " , among the natural n umbers, from " .... is the immediate successor of - - - " (as follows: "x differs by one from y if and only if x is the immediate successor of y , or y is the immediate successor of x") but not vice versa; hence a basis consisting solely of the former predicate would be rated simpler, according to this proposal, than a basis consisting solely of the latter predicate. Yet, plausible as this idea may be in a few such cases, it is quite mistaken as a general principle, for it would have the consequence that no simpler basis could be found fo r a system than that arrived at by merely taking all the extralogical terms of the system as primitive! For, since any adequate basis for a system must yield, through definition, all the terms of the system, the set of all these terms would, by the proposed criterion, be as simple as any adequate basis of fewer of these predicates. What is actually maximal complexity would be accounted m aximal simplicity. Counting the predicates in a basis seems, offhand, a better test of simplicity, but this likewise fails. For if the number of predicates 285

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in a basis were the sole measure of complexity, ultimate simplicity would always be achieved in a purely trivial way. Any number of predicates can be readily combined, so to speak, into a single predicate. A basis consisting of a one-place predicate (say, " . . .. is copp er " ) and a two-place predicate (say, " - - - is more durable than -------·----" ) can always be replaced by a single three-place predicate (say, " .... is copper, and - - - is more durable than ____________ "). And , taking all the predicates in a system, we can immediately construct a single many-place predicate from which all are definable by a routine procedure. If the total number of places in those original predicates is ni, the single primitive thus arrived at will have m places. Obviously, no genuine increase in simplicity is effected in this way; many simpler predicates are replaced by a single correspondingly more complex one. The proposal to measure the complexity of bases by a mere counting of the predicates they contain fatally ignores all differences of complexity among predicates. The idea that inevitably suggests itself next is to measure complexity by the total number of places in all the predicates in a basis. The spurious simplicity effected by artificially combining several predicates into one will thus be properly discounted. But the new proposal is again too hasty. While we have clear grounds for not regarding an m-place predicate as simpler than a set of several predicates with a total of m places, we have no such grounds for not regarding the set of predicates as simpler than the single predicate. Replacement of set by single predicate is always possible, but not replacement of single predicate by set. For example, the two one-place predicates ". . . . is a parent" and ". . . . has a parent" will not serve instead of the two-place predicate " ... . is a parent o f - - -"; to say that x is a parent and y has a parent is not to say that xis a parent of y. The complexity of bases seems to vary not only with the number of places but also in some manner with the way this number is distributed among predicates in the basis. Several plausible formulas for this variation can easily be devised, but there is no immediately obvious method for choosing among them . Not only have our efforts been unsuccessful so far, but trying out one rule of thumb after another begins to look like an unprom-

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ismg method of attacking our robl "':"'e want has lain ready in our ~and:~~ ye:, a~l the time, the clue m g of primitives as a meas f . re3ectmg the mere counture o complexity se t of predicates can alw b ' we argued that any ·1 ays e replaced by · t acit Y appealed to th · . a smg1e predicate We . e prmc1p1e that repl . l"fi . acement of a basis by anot h er effects no genu · me s1mp I cat10n wh h ment can always be m d b ere sue a rep1acef a e Y a purely applications of the sam . . 1 rou me procedure. Other e prmc1p e come qui kl t ca t es can always be add d t b . c y o mind. Predie o a as1s without d t . quacy, and a predicate can al b es roymg its ad emore places; obviously in n ~~ys e r~pla~ed by another having other words an elemen,ta e1. e~ clase is simplicity increased. In ) ry prmc1p e ap r d . . measures of complexity is this· If ever pie. Judging proposed always be replaced by some basis likey basis like a given one can not more complex than th d . a second, then the first is . e secon . This ma n egative a principle to Y seem too meagre and carry us very far but .t . h k pro bl em. More carefully f l d ' I is t e ey to our 1.t w1·11 constitute the fu d ormu ate .' clarified ' an d supplemented n amenta 1 ax10m of a calculus of simplicity'.

I?

First Axioms of Simplicity Let us, then, adopt our first postulate:

Pl. If every basis of a relevant kind K . som e basis of a relevant kind L th .is alway s replaceable by L (that is, K does not have ah/ h en K is no.t more complex than briefly, when "v" stands for "th g er com~lex1ty value than L-or, e complexity-value of", vK ~ vL) Now a good many points here 11 . p lace, to say that every b . f kc.a for explanation. In the first as1s o md K · l some basis of kind L . t is a ways replaceable by . is o say not only th t th eqmvalent basis of kind L b t l a ere always is some · u a so that we l th at is, that given any b . B f k. can a ways find one · . as1s o md K "th f ' tion than that B is of k. d K ' w1 no urther informam we can defi · stated logical apparatus l ' ne m terms of B and the a one some bas· B' f k . we can redefine B from B ' d th l . is o md L such that I th an e og1c alone n e second place, our postulate . A k. speaks of the complexity of k inds rather than of b md has th h. h ases. 1 va ue possessed by any b . f h . e ig est complexityas1s o t at kmd, and a basis has the

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complexity-value of the narrowest relevant kind to which the basis belongs. In the third place, what are the r elev ant kinds ? Structural kinds, certainly, since we are concerned with structural complexity. But not every structural difference constitutes a difference in relevant kind; for if that were the case, then, since definition from a basis always depends upon structural features, our postulate would in effect reduce to the test in terms of defining power that we have already rejected. What will constitute a relevant kind depends upon the fact that our postulate is intended to express the principle that purely routine replacement effects no genuine simplification. Now, replacement of a basis B by another, B', is purely routine if every basis like B , or of the same broad sort or general kind as B , is always replaceable by some basis of the same general kind as B '. The ordinary notion of a " broad sort" or "general kind" is vague and has to be supplanted by something much more clearcut. As a first approximation, we may define such a relevant kind as any class of bases delimited by specifying the number of predicates in a basis and the number of places in each predicate, together with any or no information concerning the three most commonplace properties of predicates: reflexivity, transitivity, and symmetry. (At the moment, the reader need not understand what these properties are; they will be explained presently, and our tentative definition of relevant kinds will be somewhat revised.) Thus, for example, the class of all bases consisting of a two-place and a one-place predicate is a relevant kind; so also is the narrower class of bases consisting of a symmetric two-place predicate and a one-place predicate. Our tentative definition of relevant kinds will presently be further explained and somewhat revised. One may ask what justifies this particular interpretation of "broad sort" or "general kind'', this particular decision in spelling out the imprecise notion of purely routine replacement. Quite plainly, no precise interpretation can claim to be uniquely indicated. Developing any method of measurement is a process of forging a sharp and effective tool from rough practice. We must look to the practice where it offers us guidance and, at the same time, remove obscurities, resolve conflicts, and fill in gaps, by

VII,l THE TEST OF SIMPLICITY

rulings designed to yield the most si .fi measuring anything must meet taxi~m cant results. A me~hod of m ands of faithfulness and . b'lg but somewhat el.us1ve desively correct Th hserv1cea I ity, but no method is exclu. us our c osen defi ·t· f find its justification in the comb ' t' m i;~ o relevant kinds must lation of the rough not · f i~a wn o its plausibility as a trans10n o general kinds" 'th th acceptability of the calc 1 f wi e overall u us o measurement to wh · h th' . pretation is a contributing factor. ic is mterFrom postulate 1, we can show that tw k' plex if every basis of each k' d . o mds are equally comd m is always replaceable by some basis of the other k d in ' an we can derive com l 't l fo r many kinds of bas B t . p exr y-va ue equations es. u m order to der· · l sh ow that certain kind rve mequa ities-to s are more complex tha t . h we need something mor Th . . . n cer am ot ersto the effect that a bas':· rs .rs .provided by a prosaic postulate complex than a b . rs ~o~s1stmg of some predicates is more asrs consrstmg of none a d th h a basis can be computed b dd' ) n at t e value of predicates in it. y a mg the complexity-values of the

P2. Every predicate in an extralo ical b . h .. plex ity-value, and the value of th gb . ~sis as a positive comof the predicates in it. e asis is the sum of the v alues The second clause operates to exclude f . . determining complexit rom consrderatron, in y-va 1ues any intercon t' predicates in a basis But h .' t nee wns among the · sue m erconnections ·11 b k . wr account later, after a primar s e ta en into established. y ca 1e of complexity-values has been

Development of a Calculus Easily proved from thes l rems: such as that wheree p~stu ates are certain elementary theo-

predi~:::sl~~~~~~:i ~:;. c~m:~xi.ty-value of

the class of m-place SISts of a single m-place

d' )G •' m o asrs that conpre icate is less than the complexity-

!

6. shall often abbreviate such a lo . " the kmd of basis that consists of on t c~t1on as . the complexity-value of val u~ of two-pl ace predicates" or et0 \~~hp ace pred1.cate" to "the complexitypred1cate". e complexity-value of a two-place

288 289

VII,1

SIMPLICITY

VII,1

THE TEST OF SIMPLICITY

value of the class of n-place predicates. But the further development of the calculus requires treatment of more specific properties of predicates and becomes highly complicated. I shall describe it here very sketchily, merely to suggest its general character and the results obtained. 1) Two-place predicates are irreflexive if they never relate anything to itself. Thus, " . ... is a parent of---" is irreflexive. On the other hand, the predicate " .... has the same blood type as - - - " is reflexive, since everyone has the same blood type as himself. Some predicates, like " .... has a brother in common with - - - " , are n either reflexive nor irreflexive; every person with a brother, but no person without, has a brother in common with himself. One-place predicates are, degenerately, both reflexive and irreflexive. Reflexive predicates are interreplaceable with irreflexive predicates and hence they are equal in complexity. A two-place predicate " P " that is neither reflexive nor irreflexive has the same complexity-value as a set of two irreflexive predicates : a twoplace predicate that relates every two distinct elements, x and y, related by "P", and a one-place predicate applying to every x that "P" relates to x itself. With predicates of more than two places, the varieties of reflexivity-properties multiply rapidly, for a many-place predicate may be reflexive or irreflexive or nonreflexive with respect to all or to any given selection of its places. But fortunately, the coniplexity of any basis can be proved to be equal to that of a certain basis consisting solely of thoroughly irreflexive predicates (that is, irreflexive with respect to all their places). Every other predicate of n places in the basis gives way to a set of one or more thoroughly irreflexive predicates, each having not more than n places. Thus we can confine our attention to thoroughly irreflexive predicates. 2) A two-place predicate, like " .. . . is greater than---", is said to be transitive, since if x is greater than y and y is greater than z, then xis greater than z. However, transitivity proves to be less pertinent to complexity measurement than (and is now to be supplanted as a defining property of relevant kinds by) a stronger property of predicates that may be called self-completeness. If two

We saw that a self-complete two-place predicate has the same complexity-value as two one-place predicates; but a symmetric self-complete two-place predicate has the same value as a single one-place predicate. Such a predicate merely pairs in both directions every two elements of a set. Likewise, an n-place thoroughly symmetric and thoroughly self-complete predicate merely combines in all directions (or applies to all permutations of) every n elements of a set, and such an n -place predicate has the same comp lexity-value as a single one-place predicate. 4) Complications pile up fast when we consider predicates that are only partially, rather than thoroughly, self-complete or symmetric. For instance, many-place predicates may be symmetric

290

291

one-place predicates-say, " is red" and " - - - is white"are compounded into one two-place predicate-say, " . .. . is red, and - - - is white"-the latter is self-complete. In general, a selfcomplete two-place predicate is such that if it joins x to y and also z t? ~ (and if all these except possibly y and z are different), then it JOms x to w. Predicates of more places may likewise be selfcomplete with respect to all their places: for example the predicate " . d , .... is re , and - - - is white, and ___________ is square" . Such predicates (whatever their number of places) are, figuratively , rather unstable; they break down easily into one-piece predicates. For that reason, thoroughly self-complete predicates do not often occur in actual systems; but for the same reason, their consideration is important for our present purposes. Their resolubility enables us to show that the complexity-value of an n-place thoroughly self-complete predicate is equal to the complexity-value of n one-place predicates . This is a crucial step towards determination of the general relationship between the complexity-values of predicates differing in number of places. 3) A two-place predicate is symmetric if it pairs elements in both directions whenever it pairs them in either; for example " . · . . is a sibling of - - - " is symmetric, since everyone is ~ sibling of anyone he or she has as a sibling. The three-place predicate "· · ·. and - - - and ---·-------- are triplets" is also symmetric with respect to all its places, and a predicate of more places that is similarly entirely order-indifferent is likewise thoroughly symmetric.

SIMPLICITY

VII,1

with respect to some rather than all of their places. If x lies on a straight line between y and z, it follows that x lies on a straight line between z and y , but not that y lies on a straight line between x and z . Thus the three-place predicate ". . . . lies on a straight line between - - - and ------------" is symmetric with respect to the last two, but not with respect to all, of its places. Again, a predicate may be symmetric with respect to sequences of its places rather than with r espect to its places severally. For example, if x is exactly as much greater than y as w is greater than z, then w is also exactly as much greater than z as xis greater than y . The four-place predicate in question here is not symmetric with respect to any two or more of its places severally but is symmetric with respect to pairs of its places: the pair of its first two and the pair of its last two places. Similarly, self-completeness may occur with respect to sequences of places rather than with respect to places severally. And a single predicate may exhibit many varieties of symmetry and self-completeness at once . All this makes the full treatment of our problem very intricate. Briefly, what we do is define the symrn,etry index of a predicate as a certain function of all the symmetries the predicate has, and also define the seLf-conipleteness index of a predicate in a comparable way. We then examine how complexity varies in relation to these indices. 5) This examination, carried out with the help of two supplementary postulat es, yields the means for determining the complexity-value of any relevant kind of basis as a function of two constants (either but not both of which may occur vacuously) : the complexity-value of one-place predicates and the complexityvalue of two-place irreflexive predicates. In order to achieve a fully quantitative measure, a final postulate is needed to fix the numerical value of these constants. This postulate stipulates that all, and only, those kinds of bases that can be shown by preceding postulates to have the same value as one-place predicates shall have the value 1, and that all other kinds shall have the lowest integral value consistent with this requirement and with preceding postulates. Assignment of the value 1 is a mere convenience; 292

VII,1

we might ha

d THE TEST OF SIMPLICITY ve use some arbitrar tegral values is always indicat d . y constant c. The choice of inw here, as h ere, use of noninteg;al m tny scheme of measurement va ues can be avoided.

Resultant Simplicity Formulae S ome of the resulting com lexi m g formulae: p ty-va1ues are given by the fo11ow1) The class of n-place thor . ough1y irreflexive predicates has value 2n - 1.

2) The class of n-place thoro . self-complete pred· t h ughly irreflexive and thoroughly 3) Th ica es as the value n e class of n-place thorou h .. . symmetric predicates has th I g ly irreflexive and thoroughly 4) The e va ue n . class of n-place d. fle xive, self-complete and s pre ica~es that are thoroughly irreMore generally th' l Ymmetr1c has the value l. . ' e va ue of an k. d predicate is 2n _ 1 · Y in of n-place irrefl · d minus an amount d d · exiv e an self-completeness and th . . . epe~ ent upon its symmetry When th I' . . ei1 interrelation. e imitation to irreflexive . values of relevant k' d predicates is dropped th m s are obtained ' e computing the value of . ' as suggested earlier by f . corresponding k' d ' 1 m s of bases consisting o irreflexive predicates Th n b · ese va ues ris um er of places; the clas f 11 e very rapidly with the complexity-value 4 and ths o two-place predicates has the value 15. ' at o all three-place predicates, the

t

Some temporary exclusions

d

.

I~ t~e first place, the restrictio:~o e earlier can now be removed. ehmmated without aff t' applicable predicates is easily . ec mg any of o 1 app 1icable predicates h ur resu ts; for since inave zero compl 't r e1evant kind cannot . . ex1 Y, adding them to 1 increase its compl 't 1 a p ace, where the predicates in . exr y-va ue. In the second geneous type (but meet t . quest10n are not all of one homo cer am cond 't' f fi . met h ods of correlating each P d' r wns o nrtude), available the 1 t re rcate with a set f owes type may be applied d h o predicates of then. be readily calculated. Finali an t e complexity-values may consideration of interconne t' Yb our temporary exclusion from c wns etween predicates is compen-

293

VII,1

2

SIMPLICITY

sated for b y adoption of a secondary rule for choosing in certain cases between bases of different structure but of equal computed

Recent Developments in the Theory of Simplicity

complexity-value. Present Status of Simplicity Study The calculus I have outlined is virtually complete; proofs for all theorems are currently b ein g checked. However, some possibilities for improvement still n eed consideration , and other writers h ave entered some objections and have proposed modifications and alternatives. 7 Investigation of the measurement of the structural simplicity of th e extralogical bases of theories is unlikely to reach its stage of ultimate stagnation for some time. But that can also be said of many an older scientific inquiry. I hope enough has been said to show that our problem has at least been carried some steps away from its stage of initial confusion. We have been dealing, it must be remembered , with a very small corner of the big problem of simplicity. We have considered the simplicity of terms but not of p ostulates framed in these terms. We still have no measure of the overall simplicity of theories. Nor does our calculus answer the crucial questions involved in the 8 fitting of curves and in induction generally. But perhaps the progress made on one aspect of the problem will somewhat alleviate despair concerning the r est. If some of the r emaining questions seem too vague to be amenable t o precise formulation and treatment, we may well r eflect that most scientific problems seemed that way once. The obscurities of problems are due less to the subject matter than to shortcomings in its investigation. 7. [F or a discussion of these see the following paper in this volume.] 8. On this topic, see I. Scheffler , "Inductive Infer ence: A New Approach",

We in_crease the degree of systematization . . of a given subject matter wh achieved m our account enever we mak set o terms left undefined B t . e any reduction in the f the extent of the economy .effu /;ce terms differ in complexity, by counting the undefined t ec e I cannot be determined merely f i I h erms. n a number f i· w ns, ave investigated the problem o_ ear ier publicacomplexity of primitive extr 1 . 1 of measurmg the structural structional systems Re tla ogica vocabularies, or bases of con· cen Y several oth ' ave appeared and some 1 er papers on the subJ"ect new resu ts obt . d I h ment on papers by Patrick S ame . want here to comSvenonius-in this order h' hu~pesh, John Kemeny, and Lars . , w ic is t e re f h pu ication. I shall answ th . . . verse o t eir order of ·bl er e criticism d d pomt out some serious def ec t s m . t h e alt s a vance by Suppes , . comp exity proposed b K ernative measures of y emeny and e 1 . h 1 xp am w y the treatment o efinability and replace b·1·t , ff f d · a i I Y o ered by S · me an important contrib u t 10n. . I n the cou venonms seems to f d. papers, I hope also to mak 1 rse o iscussing these . e c earer some a t 0f ma ly, I want to outli · spec s my own work F 1 ne an improvement in the system proposed 1. These, in order of publication ar . " . n ai of Symbolic Logic Vol 8 (194S) e. On the Simplicity of Ideas" J Vol.' 14 ( 1949), pp.' "The Logical P 1c1 , ibid Vol 14 (l ' n mprovement in th Th 9 edition of SA ; Notes on st9)' the third chapter the eory pp. 189-191; and "Axio:Fhc1ty 'Journal of Symbolic Logic vot1r1si Vol. 52 (1955)' pp. of Simplicity", of

::~~~atl~s'.ty',~?i~. '. "Ne\~ (1~52)' Ph~~:ophy,

0

j:f..4~?~~2 lI;

?~· ~~8-229;

70~~~2~easurement

Simplici~yu~f

o:

Jo~irnai

followm g errata in th" 1 . • dl, }or "(1,3) 4)a,;ti(cble) should ?e noted : rea symmetr " ( ) ' ' · P 717 lme 14 f Y,18, line 2 of T15, after ,;(vKinsert "and (syL-syK) / c"· y .d fc after " ( K . • an or 1s r ead " " (d for "=""" v I~" msert "or (sy L-syK ) 1c" ~re) · ) P· '.18, line 2v of Pc3, . r ea . < ; and for "k" read " " . e p. 719, Imes 2-4 of T16 tlons fr?m this article that the reader · [Postulates, theorems, and appendix to the present paper.] w1 need to refer to can be found in the

~:i~Ppji~;t6y',,line

Science, Vol.127 (1958) , pp. 177-181.

-;fS)

294

(2,~s) " ~s:a:~?i1~)ne(f

~-.

~l

295

L') /o~

defini~

VII,2

RECENT DEVELOPMENTS IN THE THEORY OF SIMPLICITY

VII,2

SIMPLICITY

in my "Axiomatic Measurement

of Simplicity" (hereinafter re-

ferred to as AMS)·

. kS 's "Nelson Goodman on the uppes . . . ..2 Concept of Logical S1mphc1ty

1. On Patric

. l basis is the ratio of its defining The economy of a~ ~xtralohgica t bases are of equal defining · · hcity W en wo l" "t power to its simp .. . l ro ortionate to their simp ici y . ower their economy is direct y p Pt. here is not the mere inP ' . r "ty in ques ion Clearly, then, the simp ici d" t. t factor that must be measured . g ower but a is me h t h verse of d e fi nin p bl f measurement t a as This is the pro em 0 h in some ot er way. N theless Suppes writes: always concerned me . ever ' . in that the basic intuitive i~ea to In my discussion I am assur g t al simplicity. An entirely difcaptured formally _is t~a~ o str~~ ur cern itself with the relative ferent notion of simplicity couf tchon me structure for instance, d" t bases or e sa ' complexity of pre ~ca e t of primitive notions for groups or an evaluation of different ~e :wo measures assign the same comBoolean algebras. Kemeny s d" t bases yielding the same strucplexity value to different prGe ic~ e , paper suffers from a failure ture My own feeling is that oo man s to distinguish clearly these t wo problems. " . . h " ntirel different notion is JUSt t e What Suppes here calls an e YTh" . made very explicit in nsidering is is one I have always b een co d . . . my book a but Suppes . r ity an agam m ' my first article on simp ic . . h" h I had to presuppose some k ly AMS m w ic apparently nows_ ~n 'ined in my earlier publications. of the basic ex~ositi?ns cont~ffer drastically . Consider a set S of The two notions mdeed d1 l A ording to the Suppes . umber of p aces. cc predicates of varymg n . t o simpler basis for S than . r "t we can arrive a n d" t notion of simp ici y . es of S as primitive. Accor mg o we get by taking all the predicat h . pler basis-and struc. arrive at a muc sim d fi my notion, we may "f discover how to e ne . my usage-1 we t turally simpl er, oo , m · terms of others. . t es in some pre d ica t ders many of h"is en·t·iN ot surprisingly, Suppes's false star ren

?e

2. Philosophy of Science, Vol. 23 (1956 ) , pp . 153-159. 68-69 of SA . f the first article cited in footnote 1; and PP· 3. S ee p. 110 O

296

cisms of my work utterly irrelevant. For example, the cases he cites in his Sections 1 and 2 to illustrate the counterintuitive results of my measurement of simplicity are quite beside the point, proving only that complexity in my sense does not vary uniformly with defining power. Likewise, the particular argument he brings against my Postulate 2 (which fixes the complexity-value of a basis as the sum of the complexity-values of the predicates in it ) depends entirely on his misconception of what I am talking about. 4 Incidentally, some things he says convey the false impression that my method of measurement does not apply to bases consisting of several interconnected predicates. Actually the method applies to bases consisting of any number of predicates connected in any way. Suppes's initial misunderstanding also infects his discussion of the notion of relevant kinds. He thinks the notion can well be dropped. But if the notion is dropped from my treatment, so that Pl for example reads "If a basis B 1 is replaceable by a basis B", t hen B 1 is not more complex than B/', then obviously no basis can ever be replaced by a simpler one, and we have a measure not of complexity in my sense but of defining power. The clue to my approach is that replacement of a basis B 1 by a basis B" is a replacement of a more complex by a less complex basis only if bases like B 1 cannot always automatically be replaced by bases like B". Where the replacement is purely routine, as is the replacement of six one-place predicates by one six-place predicate, no genuine simplification is effected. The notion of relevant kinds is a means of spelling out more explicitly such phrases as "a basis like B" ; and t hus it is only with the help of the notion of relevant kinds that a complexity-measure of the sort we want is defined. The question remains, of course, whether the needed notion of relevant kinds is adequately defined. Suppes complains that it is not ; and indeed no full and formal definition is given. What is given , rather, is a rough description, later filled in and modified by a study of certain properties of many-place predicates. In sum, the explanation given is as follows: A relevant kind of basis is de4. For further discussion of this postulate and the matter of connections am ong primitives, see 2 below.

297

. VII,2

SIMPLICITY

. b r of redicates in such a basis and limited by statmg the r:dicate along with any or no the number of places m eac pfl . 't ' symmetry and self. g the re ex1v1 y , ' information concern1n . . d y sum non-null product, " f these predicates, an an ' k' d completeness· o 1 t k 'nds is also a relevant m . or non-null difference of two reh evanl i t k1'nds quite clearly. No 'fy t e re evan . This seems to me to spec1 T t the full definition m new primitive is required, but w~1 ml g l:~orious ould be excessive Y · t. ordinary nota wn w . b t elevant kinds and other . ht k why the lme e ween r One m1g as d . st where I draw it. A proper classes of basis should be rawdn JU . of the whole philosophy ld ·re a long iscuss10n . answer wou requ1 . that developing a precise t All I can say h ere is of measuremen . he basis of rough practice is a method of measurementffon t. t 1 from ordinary intuitive Ju dg. h and e ective oo . of forgmg .a s arp. do them the least violence. This ments while stnvmg to d 't always is a very deli. d 'udicious at once; an 1 . being arbitrary an J d d in being judic10us Y . hether one has succee e 1 cate quest10n w . I have selected relevant kinds reparbitrary. The particular way . d et not unfaithful formulaff t t 0 give a precise an Y resents an e or . make between a mere Y h d ' t' ton we common 1Y 1 tion of the roug is me 1 b . and a replacement that effects mechanical replacement of a as1s

i:um \

pr~cess

mea~s

a genuine saving. . . . th points. His passing reff f ther criticism on o er d Suppes o hers ur . ll t'on of num erica Y detennined is vaguely emark that t e no i f t d b the definition he quotes fined seems to me adequately r~ u ~ Ymerically determined by from my text: briefly, a quantity .1s nu ber for it But h e goes 1 h they fix a umque num . £ given postu ates w en ' f one postulate to re er 1 . th t it is 'not customary or ' on to comp am a . . irable 'for a variety of reasons to to others, and that it is un.de~ t' uch as nurnerical.ly deterintroduce a metamathemahca no wn s

mined into my postulates.cl P2 . T15 does indeed complicate the The reference to Pl an m . akm . to, b u t stron ger) than, 5 Self-completeness is ( P xy transitivity. P zw . x =F yA . 2x place =F w d. · t "P" is self-complete if (x) (y) (z) (w . d fi. ition by the addition pre 1ca e . d'ff from my ear11er e n d . z =F w . :J Pxw) . This . I ers to the antecedent. This change is ma e of the final two non-identity clauses . has shown to result from use of to preclude an a.n?maly that Lars Svenomus the earlier defimt10n.

VII,2

RECENT DEVELOPMENTS IN THE THEORY OF SIMPLICITY

theorem and make the route of proof long and arduous. Later in this paper I shall explain how, without essentially changing the structure of the system, both T15 and P3 can be simplified to eliminate the reference to earlier postulates. However, when Suppes proposes the odd device of strengthening P3 to incorporate w hat has already been established, 6 he shows that he has quite overlooked the positive purpose served by the atypical features of my set of postulates. If my aim had been to axiomatize an already accepted set of complexity-values, this could have been accomplished most effectively by taking as axioms some of the principles that come out as theorems (e.g ., T21) in my system. I was concerned rather to axiomatize the foundations from which a set of complexity-values could be derived. The problem was to develop a systematic method of measurement out of unsystematic comparative judgments; and a primary function of the axiomatization is full disclosure both of the elementary clues elicited from such judgments and also of the evolution of a system of measurement out of such clues. The fundamental clue I have adopted is the replaceability principle formulated in Pl.7 P2 has the status of an auxiliary instrument. Just how far can we go without further assumptions? T15 shows that we can go a good way: that for all the many varieties of predicate that can, so to speak, be taken apart and examined, complexity diminishes directly as symmetry increases. Our next step is to add P3, which is suggested by Tl5 and projects the same principle over all other predicates as well. 6. He points to the lack of proof for TIS as partial justification of his proposal. I stated in AMS that I had no proof; but soon afterward I found a proof for an even stronger theorem, so that P3 could be weakened. This I explained during the discussion at the meeting of the American Philosophical Association on December 28, 1955, when AMS and the papers by Suppes and Kemeny formed the basis of a symposium. However, see the fu rther proposal outlined in 4 below. 7. Note especially that the replaceability principle provides a sufficient but not a necessary condition for comparing complexity-values. Indeed, much of the subsequent development of the system is devoted to determining comparative values where no replaceability relationship obtains. This point is missed by people who suppose, for example, that to give bases consisting of a single 2-place predicate the same value as bases consisting of four I-place pr edicates must be wrong because bases of neither kind are in general re placeable by bases of the other kind.

298 299

.! i

VII,2

SIMPLICITY

Thus P3 is in effect a 'smooth extension' of T15 and, more remotely, of Pl. Such a dynamic development, with successive additions to the foundations being guided by the r esults of prior assumptions, may not be characteristic of axiomatizations of mathematical theories, but it seems peculiarly well-suited to the purpose in hand. Finally, Suppes balks at adopting anything like my fifth postulate. What I should do, he thinks, is to state within what b oundaries the first four postulates determine complexity values, and leave it at that. He holds that to add a postulate that turns the comparative theory into a quantitative one is to confuse " the axiomatic problem of formulating the intuitively essential conditions an adequate complexity measure should satisfy and the different problem of choosing one among the possible large number of measures satisfying the essential conditions". I cannot see that there is any such sharp distinction. Each successive postulate narrows further the range of admissible measures by bringing to bear an additional intuitive principle. Choice of some among the many measures that satisfy Pl-P3 is so accomplished by P4; and choice of some-in this case one-among the many measures that satisfy Pl-P4 is so accomplished by P5. Thus the role of P5 is entirely parallel to that of the other postulates. The only question, then, is whether P5 itself is intuitively well-founded; and this perhaps deserves to be explored here a little more fully than in my article. The first four postulates provide for the complexity-value of each relevant kind an expression of the form "xc + ys", where "x" and "y" stand for positive integers (except that either but not both may be 0), and where "c" stands for the complexity-value of 1-place predicates (i.e., for v{l pl.}) and "s" for the complexityvalue of 2-place irreflexive symmetrical predicates (i.e., for v{2 pl. irref. sym.}). No non-numerical sign other than "c" and "s" occurs, regardless of how many places there are in predicates of the kind in question. Thus the first four postulates fix all complexity-values relative to c and s. Is there, now, any sound basis for selection among all the possible values for c ands? First, consider c. No simpler extralogical basis can be found than a single one-place predicate; and the complexity-value of one-place predicates is the natural unit of complexity-measure300

VII,2

RECENT DEVELOPMENTS IN THE THEORY OF SIMPLICITY

ment. We might assign c any numerical v 1 be no good reason for ch . a ue, but there seems to h oosmg a value other th 1 H t ere is no call to labor the oint· f an . owever, leave c without any numer· p l ' . or we can almost equally well to give c the value 1 or leav~c~ :~~~:m;nt. The decision whether m terms of c makes no m d'ff e and express other values ore i erence tha th d · · when measuring in miles t . . n e ecis10n whether, ' o assign a mile the val 1 d our m easurements e.g. "13" "27" "584" u e an write · l ' ' ' or leave 'th out nu71 menca value and write "13 " "27 " " i wi Th . m ' m , 584m" en the smgle crucial question l f . postulates is the fixing of l . e t unresolved by the first four s re ative to c Of c . postulates yield the equation: · ourse, smce earlier s + c = v{2 pl. irref.} , a determination of the value of 2. . tive to c would serve equall lt~ce irreflexive predicates relation of the value relative / we f. s a matter of fact, determinastricted to self-complete p od.c o any one relevant kind not rewould d B · . focus our discussion upon re icates S f o. ut it is easiest to . . s. o ar we kno f P2 positive. Let us see whether th . ' . w rom that s is us to fix s relative to c. e1e are considerations that may help

. In the first place, the very character f . . vites integral measurement Th b o om subJect matter inand the number of pla . . e n.um er of predicates in a basis ces m a predicate ar · t l every predicate has th l e m egra . Furthermore e same va ue as . t oughly irrefl exive predicates· th ;n m egral number of thora predicate is integral. and th' e se -completeness measure of · ' e symmetry m f is a sum of multiples of the values of ..easure o a predicate [This no longer holds in th . d partit10ns of the predicate. . e revise treatme t f SA h t e discussion of the combi . n 2nd ed. See book.] Finally we have alr ndatiohn factor on pp. 94-103 of that ' ea Y s own that th l · of each relevant kind i's 'bl e comp exity-value expressi e as th f and a multiple of s. In the face of . e sum o a multiple of c tegers are indicated anywa h all this and of the fact that inclearly required very st y w erever other numbers are not . ' rong reasons would b d d . non-mtegers into this h A . e nee e to mtroduce sc eme. ccordmgl · fi · ourselves to the positive It' l Y, m xmg s we restrict taken as 1. mu ip es of c, or to integers if c be

°

301

,

SIMPLICITY

.

VII,2

Thus to give 2-place irreflexive symShall we now, set s at 1. 1 s 1-place predicates seems the same va ue a h d. t metrical pre ica es . . . b I h ld not like to rest t e · t itive · ut s ou pretty plainly counterm u . ' 1 . dgment of this sort. I should h ·1 one parhcu ar JU ld 1 'd ti' on· To sets at 1 wou case very eavi yon genera consi era · d add another an . 1 pre di'cates the same value . flmore · symmetnca give 2-place irre ~xive t . 1 self-coniplete predicates. In 1 1 . . fl ive symme nca as 2-p ace irre ex ld t reduce the comp exity d ]f mpleteness wou no . other wor . s, se . B u t from this together with Pl-I fl -co · predicates. of 2-place irre exive . fl . redicates in genera h t place irre exive P P4 ' it would follow t a nplace . fl . elf-complete predicates irre exive s 1 have the same va ue as n1 redicates-in other words, and hence the same value asl n 1-p acehpatever has any simplifying f that n o degree o se lf-c omp . .eteness . fw t of many-place pred'iTh th distmctive ea ure f effect ath all. h us e 11 the possible sequences o lect some among a A d cates-t at ft ey h . se 1 esS-wou ld b e completely discounted. n occupants o t eir p ac d the proposal to set s at 1. this result is enough to con emn l' gai·nst setting s at 2. This b . t' however ies a ·m No such o Jee wn,. f b' . the lowest that avoids the virtue o emg f di . t h assignmen h . as t and there seems t o be no good reason or any . culties of t is sor ' b id From our ear ier 1. . B t till more can e sa · higher assignment. u s h . l'f . g effect of symmetry is . f 11 that t e simp i ym . . postulates, it o ows ,,, If set s at 2 the simphfymg ' exactly the symme t ry measure."ll· likwewise be exactly the selfeffect of self-completeness wi h be 1 ced system of valuation, * That sue .a a ant is no inconsiderable completeness measure. . T21 Its from setting s a 2 reflected m of' an resu . that recommends itself anyway at point in favor assignment

VII,2

RECENT DEVELOPMENTS IN THE THEORY OF SIMPLICITY

?

the lowest feasible one. d f r settings at 2-or at 2c. Here, then, is a sketch of the ghroun 1s ond perhaps not the best t · ly not t e on Y a P5 is one way-cer am d d fixing this value. I have way~-of formulating these groun s an . pom · t see pp . 99-100 of SA. . 8 For further explanation of this .

stulate that restricts our choice of values that v{2 pl. irref. self-com.} . xPt z ).

Now instead of Pt, the relation 0 of spatial overlapping might be taken as primitive; Pt could then be defined in terms of 0 as follows: (2)

Pt = dr ;;y(z) (x 0 z . ::> • y 0 z).

The transitivity of Pt then follows by purely logical principles from the definition. The statement (1) with its " Pt" clauses expanded according to (2) is a purely logical theorem, demonstrable independently of any stipulations or assumptions concerning the properties of 0. Thus, through a change involving neither increase nor decrease of primitive ideas, the need of adopting (1) as a postulate is removed. Or again, consider a system comprising ordinary logic and the sole extralogical primitive S , the relation of simultaneity, governed by a sole postulate to the effect that Sis symmetrical: (3)

(x)(y)(xSy. ::> .ySx) .

Here instead of S the relation N, " is no later than" , might be taken as primitive; S could then be defined in terms of N as follows:

(4) 324

S = rtr;;y(xNy . yNx). 325

SIMPLICITY

VII,5

Postulation of (3) then becomes unnecessary, for (3) is an abbreviation, according to ( 4), of a theorem which is demonstrable within pure logic. A system having a single extralogical postulate of transitivity, or of symmetry, is of course very trivial. In practice one would be interested in a good many properties of Pt besides transitivity and its consequences, and hence would have adopted further postulates. Here, contrary to what we observed in the case of a single postulate, the shift to 0 will in general net no real saving; for, the desired properties of Pt other than transitivity remain to be supplied in an indirect and probably more complex way by appropriate postulation regarding 0. But suppose we write together in a single conjunction all the desired postulates regarding Pt, thus getting a single postulate adequate to all the properties of Pt which we may ever care to prove. What if we were able to eliminate this entire conjunctive postulate by shifting our primitive and appropriately defining P t, just as we eliminated the transitivity postulate above? This would be surprising enough to challenge the whole notion of postulational economy. We shall show that a wholesale elimination of this sort is in fact possible in a wide range of cases. Indeed, it will turn out that in order to be wholly eliminable in the described fashion a postulate set need fulfill little more than the requirement of consistency. 2. Conditions on the Definiens

What sort of definition, in general, will serve thus to eliminate a set of postulates? We may limit our attention to extralogical postulates, in interpreted systems presupposing logic. Only extralogical postulates are amenable to the kind of elimination suggested; for this consists in reducing the postulates via definitions to statements that are validated automatically by the logic presupposed. For the present let us limit our attention further to the case of a system involving no extralogical primitives beyond a single constant term, say "d". This term is to be thought of as given some specific extralogical interpretation. Now if a definition of "d" in terms

326

VII,5

ELIMINATION OF EXTRALOGICAL POSTULATES

of a ~ew primitive is to eliminate the postulates of the system the defimens must be such that its substitution for " d" t th , t 1t . t 1 . urns e posu a es m o og1cal truths. Let us use "P (d)" bb : . th · . as an a revrat10n for . e con3unct10n of the postulates; and let us represent the definiens mquest10nas"f(a)" h ",, . h . , w ere a is t e new primitive and no other extra 1og1cal express· · · d "P a " . wn rs mvo1ve . The requirement is, then, that ~f( )) b: logically true; in other words, that "(x) P(f(x)) "wh1ch contams only logical signs-be true . But th~ definiens must also meet a second requirement. No matter what mterpretation may have been chosen for " d" d h · , un er t e t · t' res nc wns imposed by the postulates, we must be able so to r·n terpret " " th t h ~ a t e proposed definiens will provide the chosen r·nterpretat10n f " d" F . or example, in the case of the system that . or ~ad .(1) as ~ts only postulate, it would have been unsatisfactory to e ilmt1.nateKt e postulate by defining Pt in terms of a new primitive re a 10n as /\/\

xy ( .3 z) ( .3 w) (x K z . y Kw),

because we could not find any interpretation for " K" d' h' h h' . accor mg t "o w ic,, t rs defimens would have the intended interpretation f pi:rt o~·. Indee~'. we can ~e sure that there is no relation K satisy g t rs cond1t10n; for if there were the part-whole relat· would be t · 10 ' 10n . symme n~a . ur second requirement, then, is that there be some mterpretat10n of "a" such that d = f (a). in h t . . quired that (.3x) (d = f(x)). ' s or, itis re-

cai!~~iefinitio~ of '_'d" as "f.(a)" will reduce the postulates to logi-

. s, and st~ll yield the mtended interpretation of "d" (under some mterpretahon of "a"), if and only if (5)

(x)P(f(x)). (.3x) (d = f(x)). 3. Existence of Logical Models

. The contemplated elimination of "P (d)" · f f m avor o a 1og1cal truth cannot of course be effected unless "P (d)" . . f . " P(d)" . . ism act true; mdeed rs a l?g1cal consequence of (5). But granted that P(d) , ~nder what circumstances will there be a definiens "f ( ) " , mg th · · t a meete con3om requirement (5)? We shall show that there will be 327

VII,5 SIMPLICITY

. . l . f the postulate set has a logical model; such a defimens if and ~n y i . l t t fulfilling the conditions . .f and only if there is a logica cons .an i.e., i th rimitive that the postulat~s impose on e p bb evi.ated say as "c"; so that a r . S uppose there is suchd aficonstant, . "f(a)" conforming to (5) is : " P(c) " is true. N ow a e mens ( 1y )[ p (a) . y

= a . v . ,_, p (a)

.y

= c J'

(1y) [P(x ) . y

= x. v. ,_,p(x)

.y

= c] ;

and clearly the statement: (7)

(x) [P(x) . f(x) = x. v. ,_,p(x). f(x) = c]

"f(x)" is construed as (6) . But (7) implies that . h is true w en (x) [P(f(x)) v. f(x) = c] , which in turn implies that (x)[P(f(x)) v. P(c) ~ P(f(x))] and hence that (x)P(f(x))

4. The Case of Many Primitives Suppose n ow that our postulate set "P(d 1 , • • · , d,, )" involves many primitive terms " di'', · · · , "d 11 ". Definitions of these in terms of new primitives "ai'', · · · , "a.11 " will b e appropriate for the purpose of eliminating the postulates if and only if the respective definientia " f1 (a1, · · · , a ,, )", · · · , "f,, (a1 , · · · , a,, )" are such that (8)

( 3x)(d = f(x)) "f(x)" 1·s construed as before. · "f (a) ,' conf ormw h ere f that there will be a d e fi mens . . We see t h ere ore · l odel Now it is 1 ing to (5) whenever the postulatets:~ti~a:eo;~~~i:l model wheneasy to see ~onverselyd tfiha_t th~ sfe r the first part of ( 5) implies e.g. there is such a e mens , o d l f th ever h t "f (A)" is a logical mo e o e that p (f(A)) and hence t a postulates. 328

(x1) · · · (x,, )P(f1(x1 , · · ·, x ,, ) , · · ·,

f,, (xi, · ·· , x ,, )). Ox1 )

· · · ( 3 x ,, ) (d1 = f1(x1 , · · · , x ,, ). · · · . d"

= f,, (x1, · · · , x ,, )) ,

as may be seen by reasoning analogous to that of §2. Paralleling the case of one primitive, it is easy to show that there will be such definientia if and only if the postulate set has a logical model; i.e., if and only if there are logical constants, say "c 1", · · · , "c,,", such that P(c1, · · · , c,, ). Suppose, first , that there are such constants. Then definientia "f1(a 1, · · · , a,, ) " , "fo(a 1, · · · , a,, ) ", etc. conforming to (8) are:

. "P( c ) " · in view of our h ypothesis · · · that . . " P (d)" is b y h ypothesis true, it is obviou s Agam, smce d = ( 1y )[ p ( d) . y = d . v • ,..._, p ( d) . y = c J and hence that

ELIMINATION OF EXTRALOGIC AL POSTULATES

The choice of "A" here is of course arbitrary; any other logical constant, say "V" or "32", would have served as well. Our choice is indeed subject to certain r estrictions so lon g as logic is thou ght of as requiring a theory of types; but these restrictions vanish when the theory of types is abandoned in favor of one or another of the alternative theories. 1

.

. t f the following considerations. The expression as is apparen r om corresponding to " f(x)" is n ow: (6)

VII,5

(1y) [P(a1, · · · , a 11 ) . y

=

a 1 . v. ,....., P(a 1, · · · , a11 ) • y = ci] ,

(1 y ) [P(a1, ·· ·,a,, ) . y = a~. v. ,.....,P(a 1, ···,a,, ) .y = cJ ,

etc ., as is apparent from the following considerations. The expressions corresponding to "f1 (x1, · · · , x ,, ) ", "f 2(x 1, · · · , x ,, )", etc. are now: (9)

(1y) [P(x1, . ..

(10)

(1y) [P(x1, · · · , x ,, ) . y

,X11 ). y

=

X1.

v. ,..._,p(X1, .. . , x ,, ). y = C1] ,

= x~. v. ,.....,P(x1, · · · , x 11 )

•

y

=

cJ,

etc.; and the statement : 1. For one such theory and citations of others see Quine, Mathematical Logic (New York : W. W. Norton, 1940), pp. 128-132, 155-160, 162-166.

329

VII,5 SIMPLICITY

(x 1) · · · (x,, ) [P(x 1, · · · , x ,, ) . f1 (x1, · · · , x ,, ) = x ~ L(x1, · · ·' X·_• • . ..• v. -P(x1 , ... 'x ,, ) . f1 (X1, ... ) x ,, ) - Ci • t~(Xi, = x ,, ) · · · , x 11 ) = c~. · · · J (11}

" "f :.! (x 1, · · · ' x 11 ) " ' etc. are construed , X 11 ) , is true when "f1(x1, (11), by reasoning analogous to that in §3, as (9), (10) , etc. From we conclude that (x 1) ··· (x,, )P(f1(X1, ··· , x ,, ), ··· ,f,, (x1, ·· · , x ,,)). The other half of (8) can likewise be established by fol~owi.ng the analogy of §3. We see therefore that there will be ~efimenha conforming to (8) whenever the postulate set has a logical model; a~d it is easily shown conversely, again following the analogy of ~3, that the p ostulate set will have a logical model whenever there are such definientia . . . We have confined our consideration to postulates m which the sole extralogical primitives are terms ; i.e., signs capable of occurring alongside " = " or " E" or indeed wherever a free variable can occur. Our findings would not apply directly to a syste~ · 1ves e .g . a primitive functor "EB" , admissible. only m . h invo w h ic . contexts of the form "x EB y" and not separable as a term m its o.wn right. Actually this is no restriction, however, for extralogical · 'tives other than terms are always readily avoidable. Instead pnm1 h h . .f t rm of "x EB y" , e.g., we can write "d'(x ;y )" w ere t e pr~mi ive e " d" designates the relation of x EB y to the ordered pair x ;y. 5. Cons equences we have seen that we can reduce a set of extralogical postulat_es to logical truths, by defining ideas identical wit~ the erstwhile primitives in terms of new primitives, if and oi:ly if the postulates have a logical model. The variety of extralog1cal syst~ms w~ose postulates can be eliminated in this way is thus exceedingly wide. This is especially apparent when we reflect that logical constants of any degree of complexity may be used as mod~ls, and th~t a constant is a model for any set of postulates that ascribes to a primitive a subclass of the properties of that constant. . Consider e.g. the calculus of individuals, which has the smgle

330

VII,5

ELIMINATION OF EXTRALOGICAL POSTULATES

primitive discreteness ( (12) (13) (14)

l ) and

the following three postulates: 2

(w)(3x)(y)[y l x . = (z)(z Ew. ::J . y l z)] , (x)(y)[(w)(x l w. = . y l w). ---(x l x). ::J . x = y ], (y )(z){(3x)[(w)(y l w.v. z l w: ::J . x l w). ---(w) (x l w)] = ---(y l z)} .

;;.y

The relation of class exclusion, (x C y), is a logical model of these postulates. It must be borne in mind that the elimination of the three postulates does not proceed by defining l as (x C y) ; such a definition would give " l " a meaning different from that originally intended. Discovery of a logical model assures us rather (by §3) that some definition can be found that will r eproduce the original interpretation of " l " and still reduce the three postulates to logical truths. A definition to which the reasoning of §3 directly leads is:

;;,y

l = dt

( 1Z) [ P

(Sep) . z

= Sep . v . ,_, P (Sep)

.z

= ~y (x

C y)]

w here "Sep" ("separateness") is a n ew primitive synonymous w ith the original" l ",and "P(Sep)" is short for the conjunction of (12)-(14) with "Sep" put for" l "·Since "P(Sep)" is in fact true, this definition preserves the intended interpretation of" l "; but the definition makes (12)-(14) logically true independently of t he truth of "P(Sep)". What postulate sets, now, cannot be eliminated b y this procedure? Inconsistent sets, obviously; for they have no logical models. But there are also other postulate sets, presumably consistent, for which no logical model is known. One such set consists of the postulates " d Ed" and "d E 2" ; models are readily found for t hese respective postulates taken singly, viz. V and iA U iV, but no model is known for the pair. A still simpler set of the same kind 2. Cf. Leonard and Goodman, "The Calculus of Individuals and its Uses", Journal of Symbolic Logic, Vol. 5 (1940), pp. 48-49. The postulates of that paper are here rendered in unabbreviated form , and so recast as not to depend on the theory of types. On abandonment of the theory of types the range of "x", " y", etc. ceases to be limited to individuals ; hence we find ourselves called upon to decide whether to construe "x l y" as vacuously true or as false when x and y are not both individuals. We h ere arbitrarily adopt the former alternative; thus the clause "~ (x 1 x)" in (13) and (14) stipulates that x is an individual.

331

VII,5

SIMPLICITY

consists of the single postulate "d = id". In general, of course, the question whether a postulate set has a logical model will de~end on details of the underlying logic. Logic might be constructed m so strong a fashion as to endow these bizarre examples with l~gical models or in so weak a fashion as to deprive even more ordmary postul;te sets of their models. Particularly weak logics aside, however we have yet to find a postulate set that has a plausible extralogic,al interpretation but still lacks a logical model and thus resists elimination. In any case, a postulate set that can be proved consistent by the usual device of citing a logical (e.g., arithmetical) model is ipso facto eliminable in the described fashion. Such elimination cannot be dismissed as trivial on the ground that postulates and ciefinitions are somehow essentially the sa~e. They differ formally in the obvious circumstance that we can dispense with the need for definitions, but not the need for postul~te~, simply by writing all statements of the system in terms of pnn:1tives. A general proof of the possibility of eliminating postulates m favor of definitions, far from being rendered unnecessary by the claim that the two are essentially alike, is precisely the kind of evidence that would be needed to substantiate such a claim. The discovery that extralogical postulates are ordinarily eliminable challenges conventional notions of postulational economy as applied to extralogical systems. We are thus moved to seek a standard according to which the economy achieved by the present method can be distinguished as in some sense trivial-like the economy achieved by simply conjoining many postulates as one. We first observe that the present method of eliminating postulates leads to a system that is incomplete, in the sense that many statements involving the new primitive are neither demonstrable nor refutable. Hence we may be tempted to rule that postulational economy is significant only in complete systems; but this is unsatisfactory, because any extralogical system contains logic and is therefore necessarily incomplete in view of Godel's theorem. A better solution is suggested by the concept of categoricity, or the related concept of synthetic completeness. 3 An extralogical system is synthetically complete if and only if it is as complete as the

VII,5

ELIMINATION OF EXTRALOGICAL POSTULATES

underlying logic; i.e., if and only if every statement of the system is demonstrable or refutable or demonstrably equivalent to a statement containing only logical signs. It would seem that economy of extralogical postulates is significant only in systems that are synthetically complete. This constitutes a considerable modification of old standards of postulational economy. Discrimination between real and apparent economy comes to depend upon proof of the synthetic completeness of the system in question. Not only is such a proof normally very difficult, but most of the useful extralogical calculi on record are in fact synthetically incomplete. Unless some quite different criterion in discovered, the extensive economies here shown to be possible will in practice seldom be distinguishable from those effected in any of the usual ways. and is relate~ to the .latter as follows: systems that are categorical (with respect to a given logic) are synthetically complete, and synthetically complete systems possessed of logical models are categorical. These matters were set fort? by Tarski at the Harvard Logic Club in January 1940, and discussed m a paper On Compl et eness and Categoricity of Deductive Theories. See also Lindenbaum and Tarski, Ober die Beschriinktheit der Ausdrucksmittel d eduktiver Theorien, Ergebnisse eines mathematischen Kolloquiums Heft 7 (1936) , pp. 15-22, wherein "Nichtgabelbarkeit" answers to "syntheti~ completeness".

3. The latter notion, under the name "completeness relative to logi