Theory of Science, vol. IV

Bernard Bolzano Theory of Science Volume Four

Translated by Paul Rusnock and Rolf George

OXFORD

Theory of Science: Overview of Contents

UNIVERSITY PRESS

Great Clarendon Street, Oxford, OX2 GDP, United Kingdom Oxford University Press is a department of the Universiry of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Paul Rusnock and Rolf George 2014 The moral rights of the authors have been asserted

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(Vol. (Vol. (Vol. (Vol. (Set)

INTRODUCTION BOOK ONE: THEORY OF FUNDAMENTALS PART I: Of the Existence of Truths in Themselves PART II: Of the Recognisability of Truth BOOK TWO: THEORY OF ELEMENTS

first Edition published in 2014

ISBN ISBN ISBN ISBN ISBN

VOLUME ONE

1) 2) 3) 4)

As printed and bound by CPI Group (UK) Ltd, Croydon, CRO 4YY Links to third parry websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibiliry for the materials contained in any third parry website referenced in this work.

Part I: Of Ideas in Themselves Chapter 1: On the Concept of an Idea in Itself Chapter 2: Internal Attributes of Ideas in Themselves Chapter 3: Distinctions between Ideas that Stem from their Relation to each other Chapter 4: Distinctions among Ideas that Result from their Relation to other Objects Appendix: Previous Treatments of the Subject Matter of this Part VOLUME TWO BOOK TWO: THEORY OF ELEMENTS (continued) PART II: Of Propositions in Themselves Chapter 1: General Characteristics of Propositions Chapter 2: Differences between Propositions which Arise from their Internal Constitution Chapter 3: Distinctions among Propositions which are Based upon their Relations to each other Chapter 4: Several Types of Propositions Stating Relations between other Propositions Chapter 5: Some Further Propositions whose Linguistic Expression Warrants Special Comment Appendix: Previous Treatments of the Subject Matter of this Part PART III: Of True Propositions PART IV: Of Inferences Appendix: Previous Treatments of the Subject Matter of this Pait

VOLUME THREE BOOK THREE: THEORY OF KNOWLEDGE PART PART PART PART

I: Of Ideas II: Of Judgements III: Of the Relation between our Judgements and Truth IV: Of Certainty, Probability, and Confidence in Judgements

Contents

BOOK FOUR: THE ART OF DISCOVERY PART I: General Rules PART II: Particular Rules

xxiii

Introduction to Volume Four

VOLUME FOUR BOOK FIVE: THEORY OF SCIENCE PROPER PART PART PART PART

I: General Rules II: On the Determination of the Extensions of the Sciences III: On the Choice of a Class of Readers for a Treatise IV: On the Propositions which Should Appear in a Treatise

Chapter l: Chapter 2: Chapter 3: Chapter 4:

On the Essential Propositions of a Treatise On Supporting Propositions On Occasional Propositions Constituents of a Treatise whose Special Character Derives from Other Factors

PART V: On the Divisions of a Treatise PART VI: On the Order in which the Propositions Belonging to a Treatise Should Appear Chapter l: General Rules of Order Chapter 2: Particular Rules PART VII: Theory of Signs or, On the Signs Used in or Recommended by a Treatise Chapter 1: On the Signs Recommended in a Treatise for the Reader's Own Use Chapter 2: On the Signs Used in a Treatise PART VIII: How the Author of a Treatise Should Behave PART IX: On Scientific Books that are not Genuine Treatises APPENDIX

BOOK FIVE: THEORY OF SCIENCE PROPER

1

§. 392.* Content and divisions of this book

Part I: General Rules

3

§. 393. * Definition and justification of the concepts of science and treatise . . . . . . . . . . . . . . . . . . . . . . . . . §. 394. Other definitions of these concepts . . . . . . . . . . . . . §. 395.* The highest principle of the entire Theory of Science . . . §. 396.* Immediate consequences: l) The science we intend to present in a treatise must merit inclusion in the ranks of the sciences §. 397.* 2) The class of readers for whom our book is intended should be appropriately chosen . . . . . . . . . . . . . . . . . . . §. 398*. 3) A suitable treatise must make what it presents in print as easily and securely understandable to its readers as possible §. 399.* 4) It must attempt to make the most important ideas, judgements, and inferences distinct . . . . . . . . . . . . . §. 400*. 5) It must confer the appropriate degree of confidence upon every thesis, and make its degree of reliability evident . . . §. 401.* 6) A suitable treatise must indicate the objective connection between truths insofar as this is possible . . . . . . . . . §. 402.* 7) A suitable treatise must attempt to counteract any disinclination the reader may have to recognise the truth . §. 403.* 8) A suitable treatise must also make it as easy as possible to locate, retain, and recall the theses it presents . . . . . . Vll

3 8 14 18 18 19 19 20

21 22 23

Contents

Contents

§. 404.* 9) There must be signs for the concepts occurring in the

§. 405.* §. 406.* §. 407.*

§. 408.*

science in question which the reader will find convenient for his own use . . . . . . . . . . . . . . · · · · · · · · · l 0) One must also take care to ensure that the reader obtains appropriate images of the objects that are dealt with . . . . 11) The book must be arranged in a way that promotes as much as possible its correct use by the reader . . . . . . . 12) A suitable treatise must be organised in a way that ensures that any faults it may have cause the reader the least possible harm . . . . . . . . . . . . . . . . . . . . · · · · 13) A suitable treatise must permit its readers to see the reason for most of its features . . . . . . . . . · · · · · ·

Part II: On the Determination of the Extensions of the Sciences §. 409.* §. 410.* §. 411.*

§. 412.* §. 413.*

§. 414.* §. 415.*

§. 416.* §. 417.* §. 418.* §. 419.*

Consequences of various ways of delimiting the extensions of sciences . . . . . . . . . . . . . . . . . . . · · · · · · · 1) No science need be specified for a truth that cannot be expressed in writing . . . . . . . . . . . . . . . . . · · · · 2) Every truth communicable by writing that is noteworthy not merely as a supporting proposition should belong to at least one science . . . . . . . . . . . . . · · · · · · · · · · 3) Having too small an extent is not a sufficient reason for rejecting a science, though having too great an extent may be 4) The fact that all of its theses are already known is not a sufficient reason for rejecting a science . . . . . . . . . . . 5) The fact that truths are quite similar is not a sufficient reason for uniting them . . . . . . . . . · · · · · · · · · · 6) A great difference between truths, in particular, the fact that they come from a completely different source of knowledge, is not a sufficient reason to separate them . . . . . . 7) There may be sciences which have certain theses in common, and even one science wholly contained within another 8) One science may depend upon another either from the subjective or the objective point of view, or both . . . . . . 9) There may even be sciences which are dependent upon each other . . . . . . . . . . . . . . . . · · · · · · · · · · 10) One should not demand that a science which contains a truth also contain its applications . . . . . . . . . . · · · · Vlll

§. 420.*

24 25 25

26 26 29

11) One should not demand that all the truths of a science depend upon the same objective or subjective principle . . §. 421.* 12) It is a very good thing to classify truths according to attributes that can be used to inquire about them . . . . . . §. 422.* 13) If a pure concept, particularly a simple one, occurs exclusively in certain truths, then one may expect that these truths deserve to be united in a single science . . . . . . . §. 423.* 14) For every inquiry there is a place in a science where it may most fruitfully be presented . . . . . . . . . . . . §. 424. Investigating whether a given science meets its purpose . . §. 425. Devising a concept of a suitable science . . . . . . . . . . §. 426. Division of the entire domain of truth into individual sciences §. 427. Other views . . . . . . . . . . . . . . . . . . . . . Part III: On the Choice of a Class of Readers for a Treatise

29 30

32 34 35 35

37 39 41 42 43

44 45

46 47 48 50 51 54

58

§. 428.* Consequences of various ways of determining our class of §. 429. §. 430. §. 431.

readers . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rules for judging the appropriateness of a given class of readers . . . . . . . . . . . . . . . · .. · . · · · · · · · · Several classes of readers that must be distinguished in almost all treatises . . . . . . . . . . . . . . . . . . . . . . . The most common mistakes made in carrying out this task

Part IV: On the Propositions which Should Appear in a Treatise

58 59 60 61

63

§. 432.* Content and chapters of this part . . . . . . . . . . . . . §. 433. The signs we use in a treatise must refer immediately to

63

complete propositions . . . . . . . . . . . . . . . . . . . . §. 434.* Various ways propositions can occur in a treatise . . . . . §. 435.* Three ways in which the reader may use the propositions occurring in a treatise . . . . . . . . . . . . . . . . . . . . §. 436.* Three kinds of relations in which the propositions we intend to present may stand to our science . . . . . .

63 65

Chapter 1: On the Essential Propositions of a Treatise §. 437.

In every treatise some propositions must be advanced as essential . . . . . . . . . IX

67

68 70

70

Contents

Contents §. 438. §. 439. §. 440.* §. 441 .* §. 442.* §. 443.* §. 444. §. 445. §. 446. §. 447.

§. 448. §. 449. §. 450. §. 451. §. 452.

How do we judge whether a given proposition belongs to . ?. . . . . . . . . . . . . . . . . . . . . . . . . . our science What does it mean to say that a proposition is sufficiently noteworthy? . . . . . . . . . . . . . . . . . . . . . . . . . When is a proposition important enough to justify the demand that the reader impress it upon his memory? . . . . . When is a proposition worth presenting even if the reader is meant to consider it only once? . . . . . . . . . . . . . . . When is a proposition worth presenting in a book so that it can at least be consulted upon occasion? . . . . . . . . . . More precise determination of these rules according to the nature of the readers . . . . . . . . . . . . . . . . . . . . . Does the more general truth merit precedence over the particular truth? . . . . . . . . . . . . . . . . . . . . . . . . . Do truths that follow immediately from a truth deserve to be presented along with it? . . . . . . . . . . . . . . . . . Do equivalent propositions deserve to be presented alongside one another? . . . . . . . . . . . . . . . . . . . . . . Whether merely analytic or identical propositions, as well as propositions with redundant and imaginary ideas, may be presented as essential doctrines . . . . . . . . . . . . . Can a purely negative proposition sometimes merit presentation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . May we also present propositions that are merely probable in our treatise? . . . . . . . . . . . . . . . . . . . . . . . . Does the mere possibility of an attribute sometimes deserve to be presented? . . . . . . . . . . . . . . . . . . . . . . . May we present propositions we regard as essential in any way other than advancing them? Warning against several mistakes

Chapter 2: On Supporting Propositions §. 453.* What degree of confidence in the reader's mind should we attempt to confer upon a proposition which we advance as essential? . . . . . . . . . . . . . . . . . . . . . . . . . . §. 454. What influence does the nature of our readers have on the nature of our supporting propositions? §. 455. General rules . . . . .

x

§. 456.

70 §. 457.

71 72

§. 458. §. 459.

73 §. 460.

May we use our readers' opinions as supporting propositions even if we consider them mistaken? . . . . . . . . . May we employ empirical supporting propositions in a science that concerns only purely conceptual truths, and vice versa? . . . . . . . . . . . . . . . . . . . . . . . . . . . . Where should grounds of proof based on authority be used? Which supporting propositions should we merely refer to, and which should we prove? . . . . . . . . . . . . . . . . In what ways may supporting propositions occur in a treatise?

92

93 95 97 97

73 74 76 78

78

79 82 83 85 87 88

89

89 90 91

Chapter 3: On Occasional Propositions

98

§. 46 l. * General rule . . . . . . . . . . . . . . . . . . . . . . . . . §. 462. * I. Determination and justification of the concept of our science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 463.* II. Determination of the relations between our science and other sciences . . . . . . . . . . . . . . . . . . . . . . . . §. 464.* III. Historical information about our science . . . . . . . . §. 465.* IV. Indication and justification of the rules we followed in composing our book . . . . . . . . . . . . . . . . . . . . . §. 466.* V. Determination and justification of our class of readers . §. 467.* VI. Description of the usefulness of our science and our treatise . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 468.* VII. Acknowledgement of the deficiencies of our science and our treatise . . . . . . §. 469. VIII. Requests to the reader . . . . . . . . . . . . . §. 470.* IX. Applications . . . . . . . . . . . . . . . . . . . §. 471.* X. Warnings against misunderstandings and misuse §. 472. XI. Divisions in the book . . . . §. 473. XII. Transitions and questions §. 474. XIII. Repetitions and references §. 475. XIV. Overviews . . . . . . . . . §. 476. XV. Fiction . . . . . . . . . . . §. 477. XVI. Propositions setting out terminological requirements §. 478. XVII. Indication of our name and other information about ourselves . . . . . . . . . . . . . . . . . . . . . . . . . . §. 479. XVIII. Indication of an idea that applies exclusively to our book . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 480. Further information concerning our book as merchandise

98

Xl

98 100 100 102 102 103 104 105 107 109 110 111 113 114 115 118 118 119 120

Contents

Contents §. 48 l.

The various ways in which occasional propositions may occur in a treatise . . . . . . . . . . . . . . . . . . . . . . .

III. On Determinations l 20

Chapter 4: Constituents of a Treatise whose Special Character Derives from Other Factors 121 121

§. 482. * Contents of this chapter

I. On Basic Propositions

121

§. 483.* The concept of a basic proposition, or principle; various kinds of these and their uses . . . . . . . . . . . . . . . . Principles may belong to any of the three previously con§. 484. sidered kinds of proposition . . . . . . . . . . . . . . . . . §. 485. Principles [basic propositions] must always be true propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 486. But they need not be basic truths . . . . . . . . . . . . . . §. 487.* Nor need such principles and their relation to the science be immediately evident . . . . . . . . . . . . . . . . . . . . . §. 488. Must such principles always be purely conceptual truths or provable from concepts alone? . . . . . . . . . . . . . . . §. 489. What degree of certainty should we confer upon a principle? §. 490. En-ors in this business . . . . §. 491. Other views on these matters

II. On Comparisons and Distinctions §. 492.* Concept and use of comparisons and distinctions . . . . . §. 493. These may belong to any of the three kinds of propositions §. 494. Incorrect comparisons are generally more harmful than inc01Tect distinctions . . . . . . . . . . . . . . . . . . . . . Comparisons and distinctions can also be useful if we merely §. 495. indicate but are unable to prove them . . . . . . . . . . . . §. 496.* When making comparisons or distinctions, we will do well to place the point of comparison or distinction under its own concept . . . . . . . . . . . . . . . . §. 497. May similes be included in a treatise? §. 498. Mistakes in carrying out this task §. 499. Other views . . . . . . . . . . . . . .

xii

12 l 129 130 131 131 132 134 134 135

140 140 141 142 143

143 144 145 146

146

§. 500.* Concept and use of determinations . . . . . . . . . . . . . §. 50l. They may also belong to any of the three kinds of propositions §. 502. Determinations concerning the essence of an object are of the highest value, though others, even analytic ones, are not to be scorned . . . . . . . . . . . . . . . . . . . . . . . . §. 503. Do determinations stating relations or the mere possibility of an attribute merit inclusion? . . . . . . . . . . . . . . . §. 504. Do negative determinations merit inclusion? . . . . . . . . §. 505. Are determinations involving classifications worthy of inclusion? . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 506. May determinations in a treatise contain redundancies? . . §. 507. How should determinations that are also supposed to furnish indicators be constituted? §. 508. Mistakes in this business §. 509. Other views

IV. On Descriptions

146 147

148 150 152 153 154 154 155 156

159

§. 510.* Concept and use of descriptions . . . . . . . . . . . . . §. 511. * The circumstances in which descriptions should be added, and how they should be constituted . . . . . . . . . . . .

V. On Proofs

159 161

163

§. 512.* Concept and use of proofs in a treatise . . . . . . . . . . §. 513. To which of the three kinds of proposition may the proofs in a treatise belong? . . . . . . . . . . . . . . . . . . . . . §. 514. Which propositions in a treatise should be proved? §. 515. Which presuppositions and modes of inference may be used in a proof? . . . . . . . . . . . . . . . . . . . . . . . . . . §. 516.* Proofs in a treatise must make it as easy as possible for readers to achieve the stated degree of conviction . . . . . §. 517.* Proofs in a treatise should emphasise the grounds upon which they are based as distinctly as possible . . . . . . . . . . . §. 518.* To which propositions and inferences in a proof should we call particular attention? . . . . . . . . . . . . . . . . . . . §. 519. Proofs in a treatise must prevent the harmful influence of the inclinations . . . .

xiii

163 164 165 166 170 170 172

173

Contents

Contents §. 520. §. 521. §. 522.* §. 523.* §. 524.* §. 525.* §. 526. §. 527. §. 528.

§. §. §. §. §. §. §.

529.* 530.* 531.* 532.* 533. 534. 535.

§. 536.* §. 537.*

If possible, proofs in a treatise should themselves determine the proper degree of confidence . . . . . . . . . . . . . . . It is always fitting for the proofs in a treatise to appear united in a single proposition . . . . . . . . . . . Other virtues of such proofs: a) Ease of retention . . . . . b) Comprehensible steps . . . . . . . . . . . . . . . . . . c) Explaining how the proposition might have been discovered . . . . . . . . . . . . . . . . . . . . . . . d) Explaining the objective ground of the truth . . . . . . . e) Imparting other knowledge . . . . . . . . . . . . . . . . Do the proofs we give in a treatise always have to be the very ones that convinced us? . . . . . . . . . . . . . . . . What should we do when more than one proof is available? Proofs with a mixed, or progressive and regressive, procedure Proofs by reduction to absurdity . . . . . . . . . . . . Proofs by induction and analogy . . . . . . . . . . . . Proofs from mere concepts and proofs from experience Proofs based on authority . . . . . . . . . . . . . . . . Proofs based on the reader's conceptions . . . . . . . . Proofs that are only supposed to show that the probability of a proposition exceeds a given quantity . . . . . . . . . . Survey of the most common flaws that may afflict proofs in a treatise. a) Pertaining to matter b) Pertaining to form . . . . . .

l 75 177 178 178 179 180 181 182 182 184 186 197 198 203 203 205 206 207

§. §. §. §.

546. 547. 548. 549.

§. 550.

How examples may also be used to abridge our presentation How examples can promote attentiveness . . . . . . . . . How examples also facilitate retention and recall . . . . . How examples must be constituted if they are to serve as confirmations or proofs . . . . . . . . . . . . . . . How examples should be used to spread other truths

VIII. On the Consideration of Mere Ideas and Propositions

223 224 225 225 226 227

§. 551.* On the necessity of considering mere ideas and propositions 227 §. 552. Which ideas and propositions should be the object of special consideration in a treatise? . . . . . . . . . . . . . . . 228 §. 553. To which inner and outer attributes should such consideration be extended? . . . . . . . . . . . 229

A. On the Definition of Ideas and Propositions §. 554. * Which ideas and propositions in a treatise merit definition? §. 555.* Which definitions require a special proof of correctness? . §. 556.* How such proofs should be carried out; in particular, a) if the idea is claimed to be simple . . . . . . . . . . . . . . . §. 557. * b) How to prove a definition that indicates how a complex concept is composed . . . . . . . . . . . . . . . . . . . . §. 558. c) How proofs of the correctness of a definition of a given proposition should be carried out §. 559. Other views . . . . . . . . . . . . . . . . . . . . . . . . .

230 230 231 232 233 235 236

208

VI. On Objections and Replies

B. On Comparing and Distinguishing Mere Ideas and Propositions 244 §. 538.* Concept and use thereof . . . . . . . . . . . . . . . . . . . 208 §. 539. Which objections and replies should be included? . . . . . 210 §. 540. How should the objections included in a treatise be com212 posed? . . . . . . . . . . . . . . . . 213 §. 541.* How objections must be constituted 218 §. 542. Mistakes in this business 219 §. 543. Other views

219

VII. On Examples

§. 544.* Concept and use of examples . . . . . . . . . . . . . . . . 219 §. 545. How examples must be composed in order to facilitate understanding . . . . . . . . . . . . . . . . . . . . . . 222 XlV

§. 560.* When and in what manner comparisons and distinctions concerning mere ideas and propositions should be added

C. On Classifications

244

245

§. 561.* Various kinds of classifications; their benefits . . . . . . . §. 562. * Attributes of classifications that are supposed to acquaint us with noteworthy objects . . . . . . . . . . . . . . . . . . . §. 563.* On the constitution of classifications that are to be used in proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 564.* On the constitution of classifications that are supposed to facilitate retention and recall . . . . . . . . . . . . . . . .

xv

245 248 249 250

Contents

Contents §. 565.* Attributes of classifications that are supposed to make it easier to locate truths . . . . . . . . . . . . . . . . . . . . §. 566. Further virtues of classifications . . . . . . . . . . . . . . §. 567. Does the idea of the terms always have to be composed of the idea of the whole to be divided? . . . . . . . . . . . . §. 568. Is it a flaw if a classification contains terms that can also be looked upon as terms of a sub-classification? . . . . . . . . §. 569. It is often necessary to divide the same whole in different ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 570. Whether and in which cases a classification should be accompanied by the basis of the classification . . . . . . . . §. 571. May the difference between the terms of the classification be based upon a mere relation and, in particular, based on quantity? . . . . . . . . . . . . . . . . . . . . . . . . . . . §. 572. * Most of the classifications presented in treatises must be justified . . . . . . . . . . . . . . . . . . . . §. 573. How should such justifications be carried out? §. 574. Mistakes in this business §. 575. Other views . . . . . . . . . .

D. On Indicating Objective Connections §. 576.* When should we indicate objective connections? . §. 577. How should such indications be composed? §. 578. Mistakes in this business . . . . . . . . . .

Part V: On the Divisions of a Treatise §. §. §. §.

579.* 580.* 581.* 582.

§. 583.

§. 584. §. 585. §. 586. §. 587.

Contents of this part . . . . . . . . . . . . The utility of divisions . . . . . . . . . . General rules for the business of division . Particular kinds of divisions: I. Those based upon the special way in which propositions are brought forward . . . . II. Divisions based upon the inner nature of the parts they create . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Divisions based upon the relations between the parts they create . . . . . . . . . . . . . . . . . . . . . . . . . IV. Divisions based upon the objects that are dealt with . V. Divisions based upon the way propositions are known VI. Divisions based upon the use of propositions . XVl

§. 588.

251 251 253

§. §. §. §.

589. 590. 591. 592.

253 §. 593.

255 §. 594.

255 §. 595.

257 258 259 260 265

VII. Divisions based upon the relation of propositions to the reader's sensibilities . VIII. Divisions that facilitate understanding IX. Divisions aimed at making it easier to locate propositions X. Divisions that are supposed to promote retention and recall XI. Divisions based upon the relations of propositions to our science XII. Divisions based on the relations between the parts they create and our treatise . Survey of the most common mistakes in the business of division A glance at other presentations of this topic

284 285 285 286 286 288 293 295

Part VI: On the Order in which the Propositions Belonging to a Treatise Should Appear 297 §. 596.* Contents and chapters of this part . . . . . . . . . . . . . . 297 §. 597. * What should one understand by the order of propositions at issue here? . . . . . . . . . . . . . . . . . . . . . . . . . 297 §. 598. * The importance of adopting one order rather than another 298

269 269 270 272

274 274 274 276 280 280 281 282 282 284

Chapter 1: General Rules of Order §. 599. * On the various ways a proposition that is later to be advanced may be brought forward . . . . . . . . . . . . . . . §. 600.* On the various ways we may present a proposition which has already been advanced . . . . . . . . . . . . . . . . . §. 601. * What other kinds of propositions should always be presented before advancing a proposition? . . . . . . . . . . . §. 602. What influence should the objective connection between our propositions have on their order? . . . . . . . . . . §. 603.* To what extent should the usefulness of propositions be considered when ordering them? . . . . . . . . . . . . . . §. 604. To what extent should the relation of the reader's sensibilities to our propositions be taken into consideration when ordering them? . . . . . . . . . . . . . . . . . . . . . . . §. 605.* To what extent should propositions which are more certain precede others? . . . . . . . . . . . . . . . . . . . . . . . §. 606.* To what extent should easier propositions be placed earlier? XVll

299

299 303 304 306 308

310 311 312

Contents

Contents

§. 607.

To what extent must we always claim more in succeeding propositions? . . . . . . . . . . . . . . . . . . . . . . . . 312 §. 608. To what extent should more general propositions precede those which are more particular? . . . . . . . . . . . . . . 313 §. 609. * To what extent should simpler truths precede those that are more complex? . . . . . . . . . . . . . . . . . . . . . . . 315 §. 610. To what extent should conceptual propositions precede empirical ones? . . . . . . . . . . . . . . . . . . . . . . . . . 316 §. 611. To what extent should propositions we can prove a priori, or from mere concepts, precede others for which this is not the case? . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 §. 612. To what extent should the similarity between certain propositions have an influence on their order? . . . . . . . . . . 318 §. 613. * To what extent should the objects dealt with by certain propositions exercise an influence on their order? . . . . . . . . 318 §. 614. To what extent should propositions be presented in the order in which they were or could have been discovered? . . . . 320 §. 615. How the mere ordering of our theses can facilitate understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 §. 616. How the mere ordering of our theses can make it easier to locate them . . . . . . . . . . . . . . . . . . . . . . . . . 322 §. 617. How the mere ordering of our theses can facilitate retention and recall . . . . . . . . . . . . . . . . . . . . . . . . . . 325 §. 618. Limits to be observed in striving to attain the above-mentioned aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 §. 619. Should the love of the familiar or of novelty have any influence on the ordering of our propositions? . . . . . . . . . . 328 §. 620. What influence do the divisions made in our book have on this order? . . . . . . . . . . . . . . . . . . . . . . . . . . 329 §. 621. Sometimes the nature of the theses themselves provides no basis for their ordering . . . . . . . . . . . . . . . . . . . 330 §. 622. We should almost always indicate the rules of ordering we follow, and often must justify them . . . . . . . . . . . . . 330 Chapter 2: Particular Rules

331

§. 623.* What is special about the ordering of essential theses §. 624. Whether supporting propositions should ever be presented

331

before the reader knows why they are necessary . Special rules for ordering occasional propositions

332 334

§. 625.*

XVlll

§. 626. §. 627. §. §. §. §. §.

628. 629. 630. 631. 632.

§. 633. §. 634. §. 635. §. 636.

The place for principles . . . . . . . . . . . . . . . . . . . The place for comparisons and contrasts, as well as determinations . . . . . . . . . The place for descriptions . . . . . . The place for proofs . . . . . . . . . The place for objections and replies The place for examples . . . . . . . The place for mere consideration of ideas and propositions, in particular: a) Definitions . . . . . . . . . . . . . . . . . b) The place for comparing and contrasting mere ideas and propositions . . . . . . . . . . . . . . . . . . . . c) The place for classifications . . . . . . . . . . d) The place for indicating objective connections A glance at other presentations . . . . . . . . . .

338 339 340 340 341 342 343 344 344 346 346

Part VII: Theory of Signs or, On the Signs Used in or Recommended by a Treatise 352 §. 637. * Contents and chapters of this part . . . . . . . . . . . . . . 352 §. 638.* Survey of the most important benefits that may be had through §. 639.

a suitable choice of signs . . . . . . . . . . . . . . 352 The order in which we should pursue these benefits . . . . 353

Chapter 1: On the Signs we Recommend in a Treatise for the Reader's OwnU~

~4

§. 640.* Various kinds of signs we may recommend in a treatise for

the reader's own use . . . . . . . . . . . . . . . §. 641.* Attributes which all of these signs must possess . . . . . . §. 642.* Special properties of spoken signs . . . . . . . . . . . . . §. 643. On the relations between the various signs we recommend for the reader's own use among themselves as well as with those which we ourselves use in our book . . . . . . . . . §. 644. The factors which must be taken into account when determining the signs we recommend for the reader's own use . §. 645. How these recommendations should be made . . . . . . . §. 646. That and how our recommendations should be provided with a certain justification . . . . . . . . . . . . . . . . . . §. 647. Where should such recommendations and justifications appear? . . . . . . . . . . xix

354 355 356

357 358 360 361 361

Contents §. 648.

Whether and in what manner we should recommend a special name for our book to our readers

Contents §. 670.

361 §. 671.

Chapter 2: On the Signs Used in a Treatise

363

Section 1: General rules

363

§. 649.* §. 650.*

§. 651.* §. 652.* §. 653. §. 654. §. 655. §. 656. §. 657. §. 658. §. 659. §. 660.

§. 661. §. 662.

§. 663. §. 664. §. 665. §. 666.* §. 667. §. 668.* §. 669.

General attributes of the signs used in a treatise; they must be 1) written signs . . . . . . . . . . . . . . . . . . . . 2) They should not be difficult or costly to produce . . 3) In addition, they should promise a certain longevity . 4) They should be easily recognisable . . . . . . . . . 5) A precise connection between the sign in the designated idea should either exist or be easily established . . . . . . 6) They should not give rise to any detrimental secondary ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7) A sign should never have two easily confused meanings 8) One should never connect signs which are too similar with different ideas . . . . . . . . . . . . . . . . . . . . . 9) The signs in the treatise must also inform us about the order in which they should be considered . . . . . . . 10) Further highly commendable attributes . . . . . . . . . Do we sometimes require several signs for one idea? To what extent should we follow previous authors in our written presentation in a treatise? . . . . . . . . . . . . . . To what extent should we avoid using terms of art in a treatise? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . As far as possible, we should express our thoughts using signs that are already known to our readers, taken in senses they already know . . . . . . . . . . . . . . . . . . . . . . How we should use ambiguous signs . . . . . . . . . . . . When it is permissible to depart from customary modes of designation . . . . . . . . . . . . . . . . . . . . . . . . . When an additional meaning may be allotted to a sign our readers already know . . . . . . . . . . . . . . . . . . . . How we should proceed in devising new signs . . . . . . . Among several interchangeable ideas, which one especially deserves to be designated? . . . . . . . . . . . . . . . How to ensure that readers learn the sense of our signs Mistakes in this business xx

363 364 364 365

§. 672. §. 673. §. 674. §. 675. §. 676. §. 677.

366 366 367 368 368 369 371 372 372

373 373 374 374 376 378 380 386

Our choice of designations must often be accompanied by a special justification . . . . . . . . . . . . . . . . . . . . How may we ensure that the connection between a sign and the idea it designates is sufficiently intimate? . . . . . . . . The signs towards which we direct the reader's attention should insofar as possible already be known . Which language should one write in? . . . . . How to choose among several individual signs Spatial relations between signs . . . . . . . . How we must strive to achieve other aims through designation in addition to understanding . . . . . . . . . . . . . . Some features of designations due to the customary form of printed books . . . . . . . . . . . . . . . . . . . . . . . .

389 391 392 392 394 395 396 397

397

Section 2: Special theses §. 678.* Peculiarities of written presentation which stem from the relations of a proposition to our science . . . . . §. 679. Written presentation of principles . . . . . . . . . . §. 680. Written presentation of comparisons and contrasts . §. 681. Written presentation of determinative propositions . §. 682. Written presentation of descriptions . . . . . §. 683.* Written presentation of proofs . . . . . . . . §. 684. Mistakes in the written presentation of proofs §. 685. Written presentation of objections and replies §. 686. Written presentation of examples . . . . . . . §. 687.* Written presentation of considerations of propositions and mere ideas; in particular: a) Definitions . . . . . . . . . . §. 688. b) Comparisons and contrasts between propositions and mere ideas . . . . . . . . . . . . . . . . . . . . . . . . . . §. 689. c) Classifications §. 690. d) Indicating objective grounds . . . . . . . . . §. 691. Written presentation of the divisions of a book . §. 692. Written presentation of questions and answers §. 693. Written presentation of reviews and surveys . . §. 694. Written presentation of fiction in a treatise . . . §. 695. Written presentation of information about the author . §. 696. Written presentation of the title . . . . . . . . . . . . §. 697. The most common mistakes in written presentation in treatises . . . . . . . . . .

xxi

397 398 399 399 400 400 404 405 405 405 407 407 408 408 409 409 410 411 411 412

Contents §. 698.

Other treatments of this subject .

Part VIII: How the Author of a Treatise Should Behave §. 699.* Contents and necessity of this part . . . . . . . . . . . . . §. 700. * Morality also has its place in the composition of a treatise . §. 701. * All the rules prescribed by the art of discovery must also be followed here . . . . . . . . . . . . . . . . . . . . . . §. 702.* What must be done before one begins to write the book .. §. 703.* Note taking . . . . . . . . . . . . . . . . . . . . . . . . . §. 704.* The order in which one should work on the individual parts of the book . . . . . . . . . . . . . . . . . . . . . §. 705.* Examination of the individual features of the book . §. 706.* Parts of the book which are related to one another . §. 707.* Consulting one's predecessors . . . . . . . . . . . §. 708. * The special care that verbal expression in a treatise merits . §. 709. * Making use of the judgements of others . . §. 710.* Finishing the work. Publication . . . . . . . §. 711. * The most common mistakes in this business

Part IX: On Scientific Books that are not Genuine Treatises §. §. §. §. §.

712. 713. 714. 715. 716.

Contents of this part and its connection with the foregoing On essays . . . . . . . . . . . . . On texts for use in oral instruction On handbooks . . . . . . . . On scientific entertainments .

Appendix §. 717. §. 718.

414

INTRODUCTION TO VOLUME FOUR

420 420 420 421 422 423 424 426 427 428 429 430 431 433

436 436 437 442 444 445

447 A glance at previous arrangements of the Theory of Science Proper . . . . . . . . . 447 The dialectical method . . . . . . . . . . . . . . . . . . . 454

Bolzano's Index

462

Bibliography

488

Volume Four of the Theory of Science is devoted to the Theory of Science Proper, the science that teaches us how to divide the entirety of human knowledge into individual sciences and to present these sciences in well-composed treatises. It contains, accordingly, a manual of style, instructing authors in detail on matters ranging from the choice of topic and audience to revision and publication. In addition to this, however, we find sustained reflections on general methodological topics such as definitions, principles, proofs, and classifications. Bolzano's judicious and acute mind, along with his broad learning, are very much in evidence throughout. The imposing bulk of volume four reflects the breadth of Bolzano's inquiries. We note, to begin with, that his conception of a science is considerably broader than ours. Among the sciences he mentions are philosophy, logic (including the theory of elements, the theory of knowledge, the art of discovery, and the theory of science proper), the theory of experience, metaphysics, ethics (including general ethics, special ethics, human ethics, and casuistry), aesthetics, religion, theology, Catholic and Protestant dogmatics, even the catechism for married couples, jurisprudence, pure and applied mathematical sciences (including combinatorics, syntactics, arithmetic, algebra, analysis, number theory, geometry, chronometry, hydrostatics, hydrodynamics), natural sciences (including physics, chemistry, astronomy, mineralogy, natural history, botany, zoology, anthropology, psychology), medicine and various branches thereof, history and allied sciences such as chronology, genealogy, heraldry, numismatics. Given the scope of his treatment, covering empirical and conceptual, theoretical and normative sciences, it is perhaps not surprising that his remarks are often quite general, and that discovering his views on a particular kind of science requires a fair deal of selection. This is in sharp contrast to the more narrowly focussed discussions of mathematical method for which Bolzano is well known. 1 In the Theory of Elements (Vols. 1-2), we have seen Bolzano's interest in the abstract objects of pure logic. One might have expected this interest to show itself in the Theory of Science Proper, and to have led Bolzano to work out an account of sciences in themselves, independent of any reference to 1See,

Index of Names

498

Index of Subjects

501 XX!l

e.g., Contributions to a Better-Grounded Presentation of Mathematics and Purely Analytic Pr~of (English translations of both these works may be found in The Mathematical Works of Bernard Bolzano, S. B. Russ ed. and tr. [Oxford: Oxford University Press, 2004]), "On the Mathematical Method," pp. 40-82 in On the Mathematica/ Method and Correspondence with Exner (Amsterdam: Rodopi, 2004). XXlll

Introduction to Volume Four

Introduction to Volume Four

human beings, their minds, or needs. This is most decidedly not his approach. Rather, in this part of the work we encounter Bolzano's practical side, and find him interested primarily in the question of how useful knowledge might best be organised and presented in order to ensure its efficient spread, for the benefit of mankind. The highest principle of the Theory of Science, which Bolzano states near the beginning of Volume Four, is accordingly a practical proposition:

objects. But when we look at other sciences, the Aristotelian account seems to fail utterly. Consider, for example, a science such as public health. While one may say in a certain sense that such sciences deal with a certain genus of objects, this seems forced at best. A treatise of public health, for example, rightly deals with such heterogeneous matters as law, politics, psychology, cultural beliefs and practices, epidemiology, economics, genetics, microbiology, education, waste treatment, terrorism, and so on. Given this, can anyone say with a straight face that public health deals with a single genus of objects in the same sense that number theory does? Yet it is surely a science, and indeed among the most important. In his own treatment of the question, Bolzano reveals that he has great faith in what has more recently been called the wisdom of crowds. Established practice, he remarks, has created boundaries for most if not all recognised sciences. More often than not, these boundaries turn out to be rather difficult if not impossible to capture by means of tidy necessary and sufficient conditions. Nonetheless, Bolzano thinks, when accepted by all or almost all, they are often drawn in a way that would be difficult to improve upon:

In dividing the entirety of truth into individual sciences and in presenting these sciences in special treatises, everything must be done in the way required by the laws of morality, thus in such a way that the greatest possible sum of good (the greatest possible promotion of the general well-being) is thereby produced. 1 Consider, for example, the methodological problem of establishing the boundaries of the sciences. Where many philosophers of the time sought to deal with this problem by means of pure reason, considering only the intrinsic nature of truths or the objects they treat of, Bolzano's approach is frankly pragmatic. Sciences exist to serve human needs, and their limits are determined by these needs. He explicitly rejects the views that the boundaries of sciences should be fixed by the kind of objects they deal with, 2 by their sources of knowledge, 3 or by dependence on a particular principle. 4 Moreover, the same collection of truths may be divided in different ways, resulting in sciences whose domains overlap, 5 and distinct sciences may enter into all sorts of relations of dependence, even mutual dependence. 6 In short, we find here all the messiness of life and little of the tidiness of the a priori classifications that have given rise to so many wall-charts of the great tree of science, most of them as ambitious as they were short-lived. We also find a theory that holds up much better than its rivals when confronted with data. Following Aristotle, many logicians of Bolzano's time maintained that each science deals with a particular genus of objects. Geometry, for example, would deal with spatial objects, zoology with animals, astronomy with heavenly bodies, and so on. Bolzano would not deny that this is the case for some sciences, adding that he can account for their existence on his own principles, since it is also useful to collect truths about these sorts of 1§395

[IV.26].

2 §393.

3§415. 4 §420. 5 §416.

In discovering the various sciences whose concepts have been introduced among us to the present day, it seems to me that the human understanding has shown a no less brilliant side of its nature than it did in discovering the truths that the sciences were devised to present. It would be difficult to show that even a single one of the sciences whose concept has not merely been suggested by some scholar but approved and accepted by all deserves to be either rejected or restricted. People may well never have been distinctly aware of the rules they followed here, but their correctness is vouched for by their results, that is, the fact that people only seldom disputed for long over the boundaries of a science, and still less often found it necessary to abandon a determination of boundaries once it was settled upon. 1

We are reminded here of the case of a certain university that had constructed a new building to house the faculty of pure mathematics, and wanted to determine the best way to lay out the footpaths in front of the building. Instead of calculating the optimal layout as one might have expected, they simply planted grass in front of the building and waited. After a certain time, the most favoured paths became evident, and these were duly paved. 1§426

6 §§417-418.

xxiv

[IV.81-82].

xxv

Introduction to Volume Four

Introduction to Volume Four

In a case close to Bolzano's heart, namely, mathematics, accepted usage had, as far as he could tell, settled on the following characterisation of the mathematical sciences, even if it was never stated in precisely this form:

intend to present is well-delimited, in particular, whether the truths it contains are of sufficient importance to merit its own treatise, where importance is, as usual, measured in terms of human needs. Should the answer turn out to be affirmative, the next question concerns the class of readers for whom the prospective treatise is to be written. Balzano distinguishes various audiences that authors may aim at, among them: scholars, who seek exhaustive treatments of a science, practitioners, who seek knowledge in treatises in order to apply it, and the general public. In each case, a different treatment is required, and not every person is well-equipped to produce it. Thus authors must ask themselves not only what audience they are aiming at but also whether they have the skills and knowledge required to write a good book for precisely those readers. The next question concerns the content of a treatise. Here, Balzano distinguishes three main kinds of propositions, essential, supporting, and occasional.1 The essential propositions of a treatise are those that belong to a science according to its concept, which are known and sufficiently noteworthy. Supporting propositions, though non-essential in this sense, are included for the sake of proving essential propositions. Occasional propositions, finally, are those that are included for the sake of various other aims, and might include applications, requests addressed to the reader, historical remarks about the science, even fiction in some cases. A treatise of physics, for example, might include Newton's laws as essential propositions, make use of various mathematical theorems as supporting propositions, and present various applications (e.g., to astronomy) as the occasion arises. There follows a more detailed consideration of the contents of a treatise, covering, among other things, principles, comparisons and distinctions, determinations, descriptions, proofs, objections and replies, examples, definitions, classifications, and indications of objective connections. 2 Several of his discussions are particularly noteworthy.

. . . a science deserves to be called mathematical as soon as a considerable portion of its theses are quantitative determinations, among which are some whose correctness can only be seen by means of certain considerations (requiring special instruction) concerning the nature of quantities. 1

In 1810, when he was still something of a philosophical greenhorn and more given to a priori speculation, Balzano had objected to a similar definition because it was based upon a distinction of degree that was left somewhat vague, resulting in a blurring of the boundaries of mathematics itself. 2 Later, approaching the question from a pragmatic point of view, he reversed his position, finding the (admittedly fuzzy) boundary of mathematics well-drawn in view of human needs. Its appropriateness is confirmed by near-universal acceptance as revealed by common usage. A final consideration speaking in favour of retaining the accepted definition is a reckoning of the costs and benefits involved in trying to win acceptance for a different one, one which generally, though not invariably, favours conservatism: This addition [sc., "a considerable portion of its theses are quantitative determinations, etc."] will admittedly be just as objectionable for many as it once was for me (as I have no wish to conceal), since the distinction between mathematical and nonmathematical sciences would then boil down to a matter of more or less. I still do not deny that this is a bad thing; only I do not see how it can be avoided without determining the concept of mathematics in a way that departs far too much from accepted usage, in the end creating more confusion than benefit. 3 A second main concern of the Theory of Science Proper is the composition of the treatises in which sciences are presented. Here, Bolzano asks prospective authors to begin by asking themselves whether the science they 1Bolzano,

GrojJenlehre, Einleitung, §3 (Bernard Bolzano-Gesamtausgabe [BBGA], Series 2A, Vol. 7, p. 29). 2 Contributions to a Better-Grounded Presentation of Mathematics, I, §4: "Who can fail to see that this is an extremely vague and unscientific delimitation of the boundaries of mathematics?" 3 GrojJenlehre, Einleitung, §3 (BBGA, Series 2A, Vol. 7, p. 29). XXVl

Principles Bolzano's discussion of principles, or basic propositions [Grundsatze], to begin with, draws upon some of the distinctions he had so carefully drawn in the earlier parts of the work. Recall his insistence upon our distinguishing between the subjective and the objective dependence of one truth upon others. In the subjective sense, a truth may be said to be dependent upon others if recognition of the latter plays a role in bringing about the recognition of the former (clearly, this sort of dependence is relative both to persons 1§436. 2 §§482-578.

xx vii

Introduction to Volume Four

Introduction to Volume Four

and to occasions). In the objective sense, by contrast, we may say that a truth depends on certain others if it is either deducible from them(§ 155) or, in the strictest sense, if the latter are among the supporting truths of the former, i.e., its proximal or remote grounds (§217). The term 'principle'. similarly, can be used in either a subjective or an objective sense. In the subjective sense, a principle is a truth that is used to convince readers of the truth of all or at least a considerable number of other propositions of the science. In the objective sense, a principle is a proposition that is adduced as belonging to the objective grounds of either all or at least a considerable number of other truths of the science. 1 What counts as a principle in either of these senses is obviously relative to a particular science and to a particular presentation of this science. Though Bolzano admits that principles in his sense may be self-evident (in which case they are called axioms, if theoretical or, if practical, postulates), this is by no means a requirement. Objective as well as subjective principles may thus themselves require proof. Bolzano also maintains that the objective notion of a principle or basic proposition [Grundsatz] must be carefully distinguished from that of a basic truth [Grundwahrheit], 2 the latter notion being absolute, the former relative to a particular presentation of a science. Since he believes that the objective grounds of purely conceptual truths can only lie in other conceptual truths, 3 empirical objective principles occur only in empirical sciences. The proofs of purely conceptual principles may nonetheless make use of empirical propositions, as is the case, e.g., in physics and chemistry. 4 Empirical subjective principles, finally, may also occur in purely conceptual sciences, though Bolzano thinks this is only permissible in the rarest of cases. 5

as in cases of incomplete induction. They may even proceed from premises the author deems false, as in the case of arguments based on readers' preconceptions. 1 This being said, it is clearly better for proofs to have true premises and valid inferential steps. Although, generally speaking, Bolzano considers it a virtue if a proof indicates the objective grounds of a given truth, many perfectly good proofs do not do this. This is obviously the case for many of the proofs in empirical sciences, e.g., when a historian aims to establish the date of a certain event by considering the testimony of independent witnesses. But it will also occur in the purely conceptual sciences, for example, in astronomy or rational mechanics, where empirical data are used to argue for a purely conceptual principle such as the inverse-square law of attraction. Indirect, or apagogical proofs are discussed at some length in §530. Balzano had no objection to the inclusion of such proofs in a treatise (his own mathematical works contain many fine examples), noting that they are often among the most succinct and convincing proofs available. On the other hand, he thought that indirect proofs did not indicate the objective grounds of their conclusions. There are a couple of reasons for this: [l]f from a proposition N eg.M which is false the false proposition Neg.A is to be deducible through the mediation of the true propositions B, C, D, ... , then the true proposition M must be deducible from the true propositions B, C, D, ... , and A. We can see immediately from this that it is more straightforward to prove M directly from B, C, D, ... , and A, than first to prove Neg.A from B, C, D, ... and Neg.M, and then to conclude from the obvious falsity of Neg.A that there must be a false proposition among the others, and since propositions B, C, D, ... are all certain to be true, that this false proposition must be Neg .M, and that M must therefore be true. Moreover, if the opinions voiced in §221 about the inner connection between truths is not in eirnr, it follows that the propositions upon which a given proposition rests in an apagogical proof can never be its objective ground in its pure form. It is certain that the objective ground of a truth cannot lie in the larger number of propositions from which it is deduced in this mode of proof, if it is also deducible from a smaller number. 2

Proofs A proof, as Bolzano defines the notion, is a set of propositions that either is or at least appears to be presented with the aim of convincing someone that a given proposition is true. 6 As such, proofs are inherently subjective in nature, and must be tailored for their audiences if they are to be successful. They come in a wide variety of forms: deductive, inductive, direct, indirect, purely conceptual, empirical, by appeal to testimony or authority, by analogy, and so on. Proofs in this broad sense can contain inferences that are not valid, 1§483. 2 §486. 3 § 133. 4

§488, no. 3.

5 §488.

6 §512;

2

cf. §370, in Volume 3.

xx viii

§534. §530 [IV.270-271].

XX!X

lntmduction to Volume Four

Introduction to Volume Four

In the remainder of the section, Balzano describes a procedure whereby indirect proofs can be converted into direct ones. 1 The resulting direct proofs are usually somewhat more cumbersome to express; nonetheless, Balzano claims that the propositions in themselves expressed in these proofs are in fact simpler. Though he says that he is convinced that all indirect proofs can be transformed into direct ones, his procedure relies on some special assumptions about the forms of the premises contained in the proofs to be converted, and is thus not fully general. He seems to be well aware of this, as he elsewhere supports his claim that all indirect proofs can be converted with an inductive argument, remarking that he has succeeded in transforming all of Euclid's indirect proofs into direct ones. 2 One apparent problem with indirect proofs in Bolzano's system is perhaps worth mentioning at this point. Recall that Balzano defined deducibility as a special case of compatibility: 3 in order for a proposition M to be deducible from propositions A, B, C, ... with respect to variables i, j, k, ... , these premises must be compatible with respect to the given variables. It follows from this that for no proposition X can we have both X and •X deducible from A, B, C, ... with respect to the same variables. For at least one substitution for i, j, k, ... would have to make all of A, B, C, ... true and, if both X and •X were deducible from A, B, C, ... , both would have to be true when the given ideas were substituted for i, j, k, ... , which is absurd. Thus, at least on the face of things, the customary kind of reductio proof taught in logic does not seem to be possible in Bolzano's system. 4 This objection, however, neglects the triadic character of Bolzanian deducibility. While it is admittedly true that there cannot be a case where both X and ·X are deducible from A, B, C, ... with respect to the same variables i, j, k, ... , there is no reason why this could not occur with respect to different

ones. 1 Suppose, for example, we wanted to prove •(AV B) indirectly, from the premises •A and •B, where A and Bare any propositions you like. We add the opposite of our sought conclusion, AV B, as an additional premise, obtaining the set {•A, ·B,A VB}. Now consider the following inference, where the boxes indicate the variable parts (here it is understood that variation of uniform parts is itself uniform):

That is, we consider this inference under the form: '(

The premises are clearly compatible with respect to the indicated variables; equally clearly, every substitution for the variable parts that makes the premises true also makes the conclusion true. So B is certainly deducible from the given premises. On the other hand, •B is also deducible from the given premises when the variables are as indicated below:

l·AI [;]]

1Interestingly,

Frege makes quite similar remarks about indirect proofs, and uses the very same example Bolzano did, in an unpublished manuscript entitled "Logic in Mathematics" (Posthumous Writings, ed. H. Hermes, F. Kambartel and F. Kaulbach, tr. P. Long and R. White [Oxford: Blackwell, 1979), p. 245-246). Though this coincidence is suggestive, we know of no conclusive evidence for the claim that Frege read Bolzano's Theory of Science. Cf. G. Sundholm, "When, and Why, did Frege read Bolzano?" The Logica Yearbook [Prague: Filosofia, 1999], pp. 164--174; P. Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford/New York: Oxford University Press, New York, 1996), p. 234, note 69; W. Ktinne, Versuche iiber Bolzano/Essays on Bolzano (St. Augustin: Academia, 2008), pp. 330-346. 2 §530, note 2. 3 §155, in Volume 2. 4 Cf. M. Siebel, Der Beg riff der Ableitbarkeit bei Bolzano (St. Augustin: Academia, 1996), p. 109.

xxx

IAvBI l·BI i.e., when we consider the inference under the schema:

1

Bolzano makes a related point in §530, note 3.

xx xi

Introduction to Volume Four

Introduction to Volume Four

In this case, then, we have both B and -.B deducible from the given premises, as required for a traditional reductio. From this it does not indeed follow that the given premises are incompatible (as we have seen, they are compatible with respect to the variables indicated above), but it does follow that not all of the original propositions are true. However, once we have established that, among three propositions ex.,~' y, at least one is false, we may infer the the falsity of any one of them (and hence the truth of its negation) from the truth of the other two. 1 This completes the proof of •(AV B).

whose meanings are known, we often find ourselves in a position to determine with more or less precision what the former represents based only on the assumption that the author did not intend to express anything obviously absurd. In such cases one says that we have recognised the meaning of the sign from its use or from the context. 1

Explications and Definitions Treatises are composed of written signs, and Balzano spends a fair bit of time discussing how these signs should be chosen and used. Part of this discussion deals with how an author should communicate the meanings of the signs he uses to his readers. Generally speaking, this will be done with the help of other written signs whose meanings are already known. Bolzano's general term for propositions put forward in order to teach readers the meanings of given signs is 'Verstandigung', which we have translated as 'explication'. Definitions (Erkliirungen) are a special kind of explication. Bolzano describes these as follows: If [an idea is complex], and we succeed in discovering [its] constituents ... , then we choose a very secure means of explaining our meaning if we set out the signs which express these constituents in the very sequence in which the ideas they designate must follow one another in our mind in order to produce a complex idea, and then observe that the idea which the reader obtains through the consideration of these signs is the very one we connect with our new sign. 2

Definitions, then, determine ideas by specifying their parts and manner of combination. Clearly, this means is not available for simple ideas. It also cannot be used for complex concepts in cases where we do not have signs for the parts. Bolzano suggests a variety of ways of conveying the meaning of a sign in such cases. One of the most interesting closely resembles the modern notion of a contextual definition, or a definition in use: It is well known that when we encounter a sign which was unknown to us up to that point, in connection with several others 1Cf.

§252, in Volume 2. no. 8 [IV.546].

2 §668,

xx xii

Bolzano himself uses this procedure to characterise the notion of a proposition in itself. 2 He recognises that this is not a fine-grained means of communicating content: the best one may hope to achieve by it is that the reader will connect an idea with the sign which is equivalent to, i.e., co-extensive with, the idea we connect with it. If, however, we add further determinations, e.g., that the idea is simple, or that it contains a known concept as one of its parts, etc., he thinks it quite possible to convey meanings precisely in this way, at least in certain cases.

* Towards the end of the fourth volume, Balzano includes a small section on the ethics of authorship (§§699-711). Among other things, he reminds prospective authors to reflect on their shortcomings and, if unable to remove them, at least to take measures to counteract their bad influence. The choice of terminology, the propositions to include, the order in which these should appear, the division of the treatise into parts, should not only be carefully considered beforehand, but continually re-examined as composition proceeds. Clarity should be a constant concern, ambiguity and obscurity systematically eliminated. Ignorance and uncertainty should be frankly confessed rather than concealed. We should seek the opinion of many others before publication as well as after, seeking always to make our work more comprehensible and beneficial to our readers. Revision should continue as long as necessary, provided that this does not prevent us from undertaking other, more beneficial tasks. In this, we should be our most severe critics. We should not seek to imitate famous predecessors, nor pursue novelty for its own sake. And so on. One is tempted to think that this section, along with much of the fourth volume, consists almost entirely of things that go without saying. While there 1§668,

no. 9 [IV.547]. in Volume I. For a more detailed example, see Bolzano's explication of the concept "object of an idea" in On the Mathematical Method and Correspondence with Exner, p. 179 et seq. 2 § 19,

xx xiii

Introduction to Volume Four

Introduction to Volume Four

may be some truth in this, it would probably be more accurate to say that what is said there goes unheeded with a depressing frequency. In the age of the Internet and the academic culture of publish or perish, when textbooks are altered for commercial or political rather than scientific or pedagogical reasons, we have all become accustomed to sloppy or otherwise deficient work. But the problem was already well advanced in Bolzano's day, and the fourth volume is accordingly liberally sprinkled with polemics aimed at the contemporary German philosophical culture. The vast expansion of the publication industry, for one thing, has led to a flood of half-baked productions:

draw attention to oneself; and only he who knows how to veil his ignorance in a cloud of fashionable scholastic words, so that the most commonplace thoughts appear to be profound truths in the obscurity of his expression, will be celebrated. Germans! When will you turn back from this aberration which only makes you intolerable and ridiculous in the eyes of your neighbours? l

The great ease with which German authors, with very few exceptions, find publishers for their works results in a great many works being rushed to the press. How many, who haven't even half finished their work, give their permission to have it printed; how many send hastily scribbled signatures to the press! Is it any wonder when treatises produced in this way are full of unripe thoughts, contain mistakes which a second look would certainly have revealed, books in which there is no genuine order, whose beginning and end contradict each other, etc. l

Bolzano did not merely denounce the ills of his age, nor did he rest content with giving instructions on how others should proceed. For, even disregarding his other works, it should not be forgotten that the Theory of Science is itself a treatise, and meant to exemplify the very precepts it advances. It is, as Anders Wedberg has observed, a model of philosophical composition: I know of no earlier and few later writings which are composed throughout with such clarity and precision, such dialectical acumen and such attention to earlier and contemporary literature. To turn from, for example, Kant's Critique of Pure Reason to Bolzano's work is like coming from a jungle to an open and well-planned community. Clarity prevails not only in the general organisation of the work but also in every detail. Hardly any concept is introduced without a thorough explication with illuminating examples or, whenever possible, a concise definition. Hardly any assertion is made without an account of his reasons for it. Conceivable objections are answered. Throughout Bolzano gives attention to other writers who have dealt with the same topics, whether or not their points of view agree with his own. His prose, disdaining all literary embellishments, advances with a quiet and somewhat heavy matter-of-factness. To read the Wissenschaftslehre is also to receive a lesson in intellectual morality. 2

Obscurity, moreover, afflicts almost all philosophical writing in German. In Bolzano's opinion, much of the blame for this belonged to Kant, who had by precept and example exempted philosophy from the precision required in mathematics. But Kant had at least acted with good intentions. After him, things had gone rapidly downhill. Among the worst offenders is Hegel: It is with regret that I say this, but the duty to forthrightly criticise what I recognise as pernicious nonsense compels me not to conceal it when I find almost all of the flaws indicated above in the writings of one of the most esteemed modern philosophers, namely HEGEL; indeed, what is even more depressing is that these features are not considered flaws by a large part of the German public, but rather are admired as virtues. Alas, it is so. Authors as well as readers today in Germany seem to take pleasure only in a kind of writing which envelops every thought in a cloud formed of obscure words, which they cannot even see through halfway, so that books in the field of philosophy whose authors will not cater to such a depraved taste almost run the risk of remaining unread. One must speak in riddles if one wishes to

1§697,

note [IV.589-590]. Cf. §718. A. Wedberg, A Hist01y of Philosophy; Volume 3: "From Balzano to Wittgenstein," tr. B. Weclberg (Oxford U.P., 1984), pp. 53-4. 2

1§711

[IV.617].

xxxiv

xx xv

Introduction to Volume Four

Rolf George began work on this translation over a half century ago, resulting in an abridged edition of 1972. 1 He had originally planned to make a complete translation, but, as he then wrote, he "was eventually persuaded that early complaints about the unnecessary bulk of the work had their point." When I (Rusnock) came to study with him in the early nineties, I argued the contrary case, and we discussed the possibility of completing his work, in the end resolving to get around to it someday. Other projects intervened, including translations of shorter works, but finally the time came to see it through. It is with some satisfaction that we write these final words of introduction, knowing that one of the great works of nineteenth-century philosophy will now be fully accessible to English-speaking scholars and students.

Dr. B. Bolzano's

THEORY OF SCIENCE Attempt at a Detailed and in the main Novel Exposition of

LOGIC With Constant Attention to Earlier Authors Edited by

Several of his friends

Volume 4 Sulzbach J. E. v. Seidel Publishing

1837

1Themy

of Science, ed. and tr. R. George (Berkeley/Los Angeles: University of California Press, 1972). XXXVI

Book Five:

THEORY OF SCIENCE PROPER

§.392.*

Content and divisions of this book

0 wi(f3ten Aile doch, die w1s beneiden, Wie schmerzenreich des Dichters Leben ist! Wie 's ihn bekiimmert, dafi sein Werk so fern Von dem geblieben, was es werden sollte Und konnt' und mujJte! Also bessert er Und ewig mocht' er bessern an dem Werk, Um seinem Urbild niiher es zu bringen. Raupach, Tassos Tod, II, 3 1

1If those who envy us only knew how full of woe the poet's life is! How it pains him that his work remains so far from what it should and could and must be! So he emends his work, and might do so forever, to bring it closer to his ideal.

Given all that has been said in the previous parts of this work, I believe I may proceed directly to the presentation of what I take to be the proper subject of logic, namely, the presentation of the general rules governing the determination of the domains of individual sciences as well as the elaboration of these sciences in treatises. 1) I shall first define concepts such as that of a science and of a treatise somewhat more precisely than I did in § l, and shall also investigate whether there is some highest principle from which all the rules that must be observed in the cultivation of individual sciences as well as in the elaboration of treatises of these sciences may be derived, as consequences are derived from their grounds. Should this search prove successful, I shall follow the presentation of this highest principle with that of some of its immediate consequences, those which it shall be beneficial to have continually before our eyes when developing the other rules. 2) Afterwards, I shall present the rules governing the division of the entire domain of truths into individual sciences, as well as those covering 4 how to judge whether a proposed science is appropriate. 3) Having presented these rules, logic shall turn to its second (and more more extensive) task, that of indicating the rules which must be observed in the composition of a treatise of a science. Since in composing a treatise one must have a definite class of readers in mind, and since it is possible to err in choosing these readers, we must first present rules which will allow us to judge whether or not a certain class of readers for whom we intend to write our treatise is well-chosen. 4) If we have convinced ourselves that it is worthwhile to compose a treatise of a given science, the next question that arises is: which propositions merit inclusion in our book, and how should these be organised? One sees immediately that quite lengthy instructions are required on this point.

THEORY OF SCIENCE PROPER. §. 392. 5) It shall turn out, however, that we almost always do well to present the entire collection of propositions that belong in a treatise in different sections. Thus some instruction must be given on the business of division. 6) But since the propositions that should be included in a treatise cannot be presented to the reader all at once, but must instead appear one after the other, we must determine the sequence in which they appear, i.e., we must order them. And we should not omit instruction on this matter either. 7) Furthermore, since all books are a kind of written presentation, and s since it is necessary in a treatise to recommend to the reader certain signs for his own use, either written or oral, in addition to those we ourselves use in the book, we shall have to give some instruction on how this should be done. 8) Since in all the rules I just mentioned in nos. 4-7 we shall only speak of how the book should be constituted, but not of how the author should behave, it shall be fitting to give some instruction on the latter as well. 9) Finally, there are also books which, although not genuine treatises, nevertheless aim at scientific instruction, and hence must be composed according to almost the same rules as treatises. A treatise of logic seems the right place to say a few words about these. The nine parts the reader shall find in this book result from these 6 indications.

2

PART I

General Rules §.393.*

Definition and justification of the concepts of science and treatise l) I already stated in § 1 that by a science in the proper, objective sense I mean nothing other than a collection of all the truths of a certain kind, which are of such a nature that the part of these truths that are noteworthy and known to us is worth writing down in a special book, combined, if necessary, with other truths necessary for the understanding or proof of the former, in a way that makes them as comprehensible and convincing as possible. According to this definition, I presuppose: (a) that sciences can only contain truths. False propositions and opinions can have a place in a science only insofar as something true is said about them, e.g., that they are held by certain people, or that they arose in a certain way, etc. I also count as belonging to a science only (b) truths of a special kind, the various kinds of truths serving to distinguish different sciences. Just how these kinds are to be determined I leave undecided in the concept. It might be the object the truths concern or some other attribute of these truths. But I do always count the entire collection of truths with the stipulated attribute as belonging to the content of the science, regardless of whether they are known to us or noteworthy. I only require of each sci- 7 ence (c) that it be a collection of truths among which there are at least some that are humanly attainable and noteworthy enough to merit presentation in a special book as indicated above. Thus it must not merely be possible but also useful to collect such truths in a book, to order them appropriately, and if need be, to add others to them, so that they appear quite comprehensible and convincing to the reader. 2) I believe I can justify my taking the concept of a science in precisely this way as follows: (a) No one will censure the first determination that only truths belong in a science. For this requirement only forbids us from presenting in a treatise something we hold to be false, and from using deceptive means to make it appear plausible; it by no means prevents us from presenting something that is merely probable to us as being such. And who could find this requirement unfair? Even if someone does not doubt that there are beneficial e1rnrs, must he not object to the presentation of such errors, supported by all possible apparent grounds, in a book

3

THEORY OF SCIENCE PROPER. Part I§. 393.

that claims to expound a science? For according to prevailing usage, this word is connected with the idea of a specific collection of truths. Thus someone who presents something he himself does not believe under the shingle of a science is guilty not of a mere deception but of an outright lie. (b) I do not say precisely how one should determine the kind of truths belonging to a science; in particular, I do not say this should be determined by the object dealt with in these truths. The reason for this is that I have observed that although the object of a given truth decides in some 8 cases whether or not it belongs in a given science, this is not the case for the vast majority of sciences. Who would, for example, want to say that the object a given truth deals with determines whether or not it belongs in the domain of ethics, or law, or religion? It is not the object dealt with in a truth that determines whether it belongs in ethics, but rather a completely different attribute, namely, whether it expresses an obligation. This is not only the way things should happen according to the present concept of a science, but also according to the nature of the case. For what awkward collocations of truths we would obtain if we divided them into sciences based only on the objects they concern (i.e., according to their subject-ideas). We should then have to unite in a single science truths of the most diverse kinds, which cannot help in the slightest to illuminate or contribute anything to one another. Think, for example, of the various truths-physical, mathematical, moral, political, historicalwhose object is man. (c) Whether a truth is known or unknown to us, as well as whether it is noteworthy or not, also has no influence on the question of whether it belongs to a certain science, though these factors are taken into consideration when the science is presented in a treatise. It seems to me that prevailing usage requires that a science be unchanging, while the sum of what is known and noteworthy to us changes every day. (cl) I do not admit every collection of truths of a certain kind into the ranks of the sciences, but only those containing at least some truths that are humanly attainable and merit presentation in a treatise in the way I have described. This too is required by accepted usage, or in any case is not 9 a harmful restriction of the accepted concept. If we were to call any collection of truths of a specific kind a science, regardless of whether these truths were attainable for us or merit presentation in a special book, there would be infinitely many things that common usage would never call sciences which would nevertheless be able to claim this name. There would be a science for every object, even the most negligible; and one could call any multitude of truths, no matter how little they might be connected

4

THEORY OF SCIENCE PROPER. Part I §. 393.

to one another, a science. For there would always be a certain generic concept that subsumes these truths exclusively (§ l 0 I), so that one could quite rightly say that this multitude of propositions constitutes a determinate species. Obviously, we do not do this. Rather, whenever we call a collection of truths a science, we always think that these truths stand in a special connection to one another, by virtue of which no other collection of truths could be better united than these. Admittedly, it is true that we sometimes speak of sciences, some of which are worth presenting in their own treatise and others of which are not. This seems to show that the concept of a science does not stipulate that a science must be a collection of truths that merits presentation in a special book. I nevertheless think that in such cases we use the word "science" in an improper sense, roughly in the same way we occasionally use the words "proof", "principle", and hundreds of others to speak of things that are not really what these words designate, but rather only for things that are looked upon as or claimed to be such. We say that a science is not worth presenting in a special treatise when we at bottom only wish to say that the collection of truths in question does not merit the name of a science at all. In any case, what harm can come of refusing the name of a science to a collection of truths that is either unattainable for us or does not contain any truths 10 are are noteworthy for us or does not merit presentation in a treatise? (e) But many readers will find my concept too wide rather than too narrow, and will ask: why should a mere collection of truths be called a science? Should we not also demand that these truths be appropriately ordered and combined with the others required to understand and prove them? To this I reply that it seems to me that the propositions that are merely required to render comprehensible and to prove the others do indeed belong to the presentation of the science in a treatise, but not to the science itself. The reason for this is that otherwise the science would cease to be something unchangeable. For according to the diverse circumstances of the time at which we write, and the various readers for whom our book is intended, quite different propositions must be used for the sake of understanding or proof. It would admittedly be something else altogether were someone to recommend that in addition to the truths included in a science according to my conception one should also count those which contain the objective ground of the former. In that case, every science would be thought of as a collection of truths containing not only all the truths of a certain kind but also all those from which the former follow as consequences. On such a conception, a science would remain unchangeable, since objective

5

THEORY OF SCIENCE PROPER. Part I §. 393.

11

12

grounds, unlike subjective grounds, do not themselves change. This concept would also make it clear why in written presentations of a science we must strive to point out the objective grounds of the theses we present, as far as these are known to us and accessible to our readers. But since the duty of indicating the objective grounds of the truths can also be recognised from my concept of a science (as I hope to show in the sequel), it is at least not necessary on this account to depart from my definition and increase the content of an already quite complex concept with the suggested addition. 3) The concept of a treatise is intimately connected with this concept of a science. In § 1, I defined a treatise as a book which either is or at least appears to be composed with the definite intention of presenting all the known truths of a science which are noteworthy for its readers in a way that makes them easiest to understand and accept with conviction. From this definition, one sees: (a) that I do not demand that a treatise must contain all the truths of the science in question that are known or noteworthy, nor that they be presented in an order or connection which is at least as comprehensible and convincing as any other that is possible. Rather, I call a book a treatise even if it achieves the above ends only partly, to a certain degree, enough to make us think that it was composed with these aims in mind. But it must (b) at least seem to be the case that the author intended to present all the truths belonging to the science which are known and noteworthy for his readers; for if it were obvious that the book covered only a part of these theses, I would not call it a treatise of the entire science, but only an essay on this part of the science. Furthermore, (c) it must appear to be the case from the entire presentation of these theses, their order, and the way that they are connected with each other, that the principal aim of the author was to make them as distinct and convincing as possible. For if, on the contrary, it is evident that the author had some other principal intention, i.e., the easiest way to find these truths, or the best way to retain them in memory, etc., subordinating the ends of distinctness and conviction to this, then we do not call his book a treatise, but rather a dictionmy or a mnemonic table, and so on. In saying this, I make it clear that (cl) a book may still be called a treatise even if we observe that its author had other ends besides those of making truths comprehensible and convincing. Indeed, it seems to me that a treatise is all the better for providing benefits of the most varied kinds to its readers. (e) Finally, it should be noted that the expressions used in my definition indicate that a book I call a treatise should be able to be understood by and

6

THEORY OF SCIENCE PROPER. Part I §. 393.

to engender conviction in the class of readers for whom it is intended on its own, without the help of oral instruction. A book which is obviously intended merely to help in oral instruction I would instead call a manual, handbook, or outlines of lectures, etc. 4) There should not be too much difficulty in justifying this concept. (a) First, the fact that I do not require a treatise to be perfectly suited to the indicated ends is reflected in accepted usage, which distinguishes between more and less successful treatises. On the whole, it seems to me that we give the name of a treatise to a book on account of an attribute of the book itself, namely, a certain aptness in meeting the aims indicated above, deciding nothing about whether this aptness was the aim prompting the composition of the book. If it were possible for a book such as the Elements of EUCLID to be produced by pure chance, the fact that it was not produced with the intention of achieving these ends would not prevent anyone from calling it a treatise and indeed an excellent one. (b) Prevailing usage also tells us that we look upon the aims of comprehensibility and conviction as the principal ones. For we only say that someone has 13 taught us a truth when he has not merely produced an idea of this truth in us but also produced in us a conviction of its truth. It is also obvious that our ability to find a truth quickly, or to retain it in memory, is of little use where conviction is lacking; precisely for this reason, it is of the highest importance that there be books which are devised primarily for the sake of engendering conviction-and what better name might there be for these than treatises? (c) It is most likely that objections will be raised to my demand that a treatise be suitable for self-instruction. Two considerations were responsible for my decision here. First, it is not only useful but also necessary that we have books that are suitable for self-instruction; for not all people have the opportunity or the inclination to receive oral instruction. In addition, it is certainly more difficult to compose a book in a way that makes it suitable for self-instruction than it is to compose one suitable for supplementing oral instruction. And if we have appropriately set out the rules that must be followed in producing a work of the first kind, those to be observed in writing a manual will be easily gathered from them. Thus if the instructions we give in the theory of science are to be as useful as possible, they must teach us how to compose books from which people may instruct themselves, without the benefit of a teacher. Note I. It is not my intention in framing this definition to forbid the use of the defined words in other senses. It is customary, and shall always remain so, to speak of progress as well as regression in the sciences; we

7

THEORY OF SCIENCE PROPER. Part I §. 394.

say that someone expands a science, that another grounds it more deeply, and that a third makes it more fruitful, etc. Here by science one means 14 roughly the total content of existing treatises. Note 2. Because the word art is closely related to the word science, it behooves me to elucidate the former somewhat more precisely than I did in § 11. With respect to human beings, we give the name of art to any sort of skill in performing activities, developed through practice and effort, which is suitable for attaining a specific end. By contrast, we call activities of animals artful, and say that they result from a special instinct, when we see them accomplish things that humans could only bring about after repeated attempts and practice. In a second sense, we also call a theory which instructs us how to develop such skills an art, e.g., the art of brewing beer. It is only when the word art is taken in this second sense that it is closely related to the word science; for in this sense, many sciences can also be called arts, namely, when they instruct us in such a way that we cannot become truly skilful in performing the activities without practice. Many connect the word art with the associated concept that the instructions concerning the activity in question are not specified in accordance with the strictest requirements of science, and in particular do not indicate the objective grounds of its precepts. Thus, for example, silviculture as it is usually presented would be an art in this sense, while medicine as presented in the most accomplished treatises would be called a science. Clearly, though, there are sciences which are not arts either in the wider or the narrower sense of the word, namely, all those that contain no instructions for action, but only theoretical theses, as well as those which do indeed give such instructions, but only concerning actions that may be performed without previous practice. Finally, there are also arts that cannot be elevated to sciences, either because the activities they deal with cannot be taught through written instruction, or because they aim at something harmful to the human race, and for that very reason 15 should not be taught, and still less made the object of a science, e.g., the art of deception. §.394.

Other definitions of these concepts

THEORY OF SCIENCE PROPER. Part I§. 394.

as one through which knowledge is engendered; he defined knowledge as a cognition that arises from the idea of the actual ground of the truth. Accordingly, the Greeks had attained the concept of the perfection of scientific instruction which appears to me to be the highest there is, since ARISTOTLE would only acknowledge a science in cases where one sees the actual ground of every truth, where one not merely knows that (o·n) but also why (5t6-n) it is so. As much as we admire this, we should not imitate it exactly. For in my opinion it is asking too much to require that we indicate the objective connection between truths in all scientific instruction. This indication, moreover, does not belong to the science as such, but rather to the presentation of the science in a suitable treatise. In recent times, the word system has been used instead of science, and WOLFF, among others, gave the definition: Systema dicitur veritatum inter se et cum principiis suis connexarum congeries. Connexae autem dicuntur veritates, si cognitio unius pendet a cognitione alterius (Phil. rat., §§889, 877). 1 One sees that here a science is confused with its presentation in a treatise. We also lack a determination of how many truths must be assembled in order to obtain a complete system (the presentation of a complete science). According to this definition, for example, a single geometrical theorem together with the premises from which it is 16 proved would constitute a system. Obviously moved to repair this defect, CRUSIUS wrote (W z. G., §21 ft): "By a science, insofar as one gives an objective definition, one understands a collection of scholarly truths, which is of considerable extent, when there is a rational ground for dealing with them together. A cognition is scholarly if it is raised above the common, either on account of its matter-when one knows a number of things that one does not come to know in the course of everyday life-or else on account of its kind, when it is deep and subtle." To this I would object that the determination "of considerable extent" is quite vague, especially since a very small number of truths may sometimes be sufficient to construct a science, while a very large number may not. An example of the former is the pure theory of time, an example of the latter, any part of geometry, e.g., the theory of parallels. Furthermore, while it may be true that every science offers us some knowledge that ordinary life does not, what justifies the claim that every presentation that offers such knowledge is scientific? In STEINBART' s Anl. z. Selbstdenken (3rd

If we ask how the concept of a science was understood by the Greeks, i.e., the oldest people from whom we have scientific works, we find that ARISTOTLE (An. post., I, 2 and elsewhere) defined scientific instruction

1An aggregate of truths which are connected with each other and with their principles is called a system. We say that truths are connected if the knowledge of one depends upon the knowledge of the other.

8

9

THEORY OF SCIENCE PROPER. Part I §. 394.

ed., 1793, §279) we read: "A collection of truths of reason which have the same principal subject is called a themy; if it is presented following a scholarly method, it is called a science. If all the main concepts and propositions belonging to the science that have been discovered so far are ordered in a way that permits us to distinctly recognise their connection and grounding, or any remaining gaps, we obtain a system or doctrinal edifice." I need not say how close this definition is to the one I gave. However, the requirement that one point out any gaps belongs not to the science but rather to treatises of the science. It also seems to me that it is never necessary to include this requirement in the definition. Rather, it suffices that we are able to derive it from the definition. In KANT' s Logik 17 (§95) one reads: "A science is a whole of cognition which is a system and not a mere aggregate. It requires a systematic cognition, i.e., one produced following carefully drawn up rules." Accordingly, a science would be nothing other than a whole of cognitions or truths produced according to carefully drawn up rules. But until it is specified more precisely what the nature of these rules should be, and indeed, even if it is tacitly presupposed that the rules must be rational, this definition remains far too wide. For is not, e.g., every dictionary a whole of cognitions produced according to carefully drawn up and very reasonable rules? This fault is not addressed in my opinion when one calls the rule that unites all the parts into a whole an Idea, the Idea of a whole or of the unity of the whole, as some, e.g., KIESEWETTER (W A. d. L., p. 430) and JAKOB (L., §343) have done. I must still ask for a more precise determination of this Idea, or else I shall claim that according to this definition the times table is also a science. Prof. KRUG gives the same definition in his Log. (§ 119); but in the Fundamentallehre he says (p. 6): "A science in the objective sense is a collection of homogeneous and connected cognitions which relate to a certain object." And on p. 267, "A science in the formal sense is a systematic collection of evident cognitions." But if one does not specify more precisely what the required homogeneity is supposed to amount to, this entire clause becomes superfluous, since any things, no matter how diverse, may be called homogeneous in some respect. The clause "which relate to a certain object," if it is taken to mean that all the truths belonging to a science must concern the same object, seems incorrect to me for reasons already indicated (§393, no. 2, b); and even KRUG himself seems to sense (in the note) that the requirement that all cognitions of a science be evident is too strong. In MAAB' s Logik, §447, we read: "A collection 18 of methodically combined truths is called a system; a system of truths of

10

THEORY OF SCIENCE PROPER. Part I§. 394.

the same kind, i.e., concerning one and the same object, is called a science in the broadest sense of the word." And in §433, he tells us that method is order in a collection of ideas, and order is simply conformity to rules. It follows that a science would simply be a rule-governed combination of several truths concerning the same object, a definition which can be seen to be incorrect from what has already been said. Councillor FRIES (Syst. d. L., p. 289) writes: "The forms of thought that contain the completeness of the subordination of the particular under the universal I call the scientific forms or the forms of systematic unity. A whole of cognition ordered in accordance with these forms is called a system with respect to its form, and a science with respect to its content." With his requirement of the "completeness of the subordination of the particular under the universal," it seems to me, F. wanted to specify more precisely wherein consists the conformity to rules or unity that others required. But even if FRIES' requirement is understood in the widest possible sense, it does not include everything that one must observe in a truly scientific presentation. For there are certainly many other things that must be attended to-in a good presentation, for instance, when one truth follows another it should not merely be deducible from it, but evidently so, etc. CALKER' s definition (Denki., § 169) is similar: "Science in general is a cognition of the connection between the manifold in the existence of things with unity, constructed in accordance with the laws of thought, or an ordered whole of cognitions, in which the connection of the manifold with unity is represented." A scientific presentation must certainly indicate the connection mentioned here-but is this the only, or even the most important demand we should make of such a presentation? When SCHULZE (L., § 100) says that "every plurality of cognitions which constitutes a whole according to logical laws" is called a science, this is certainly true and the converse also holds, if one defines logic as I have done above, and 19 as SCHULZE himself actually does, namely, as instruction on scientific presentation. But I do not see how one may use the concept of logic when defining that of a science without falling into a circle. In STIEDENROTH's Theorie des Wissens (Gott., 1819), p. 19, we read: "Science is not a single item of knowledge, nor a mere aggregate of knowledge with no inner connection. In a science, all the parts must mesh together, and this so completely and exactly, that each stands in the place where it follows with necessity from what precedes and from which the next item proceeds with necessity." This definition also only applies to the presentation of a science in a treatise. And if following of necessity is supposed

11

THEORY OF SCIENCE PROPER. Part I§. 394. to indicate a relation of deducibility, then at most pure rational sciences such as mathematics may be presented in an order which can be said in a certain sense to meet STIEDENROTH' s requirements-but what about history, natural history, or other empirical sciences? In RITTER'S L. (p. 155), we read: "A science is a combination of several acts of thought into a unity, through which what is lacking in each individual act is supplemented." It cannot be denied that our individual acts of thought (ideas and judgements) can attain a completeness through our learning a science that individually they lack. But can this be reversed? Can we say that every combination of several acts of thought, through which an incompleteness they had individually is "supplemented", is a science? Yet R. defined the deficiency of individual acts of thought that science is supposed to fulfill more precisely by saying: (a) that no individual act of thought represents an existence which can be thought in and of itself, and (b) that no individual act of thought can be fully convincing. I can admit neither of these things. Does not the concept "God" represent an existence which can be thought in and of itself? And if no individual judgement is completely 20 certain on its own, how could it become any more certain if combined with others? For the conclusion is never more certain than its premises. In the Phanomenologie des Geistes (Bamb. and Wtirzb., 1807), HEGEL made it his goal to provide a deduction of the concept of a science. "Pure science presupposes liberation from the opposition of consciousness; it contains the thought inasmuch as it is identical with the thing in itself, or the thing in itself inasmuch as it is identical with the pure thought; or, the concept of science is that truth is pure self-consciousness, and has the form of the self, that what exists in itself is the concept and the concept is what exists in itself." I must confess that the liberation from the opposition of consciousness, the abolition of the distinction between an idea and its object appears to me to be not only unnecessary for a genuine science, but indeed destructive of all rational thought. What Prof. FISCHER says in the preface of the 2nd edition of his Lehrb. der mech. Naturl. (Berlin, 1819) about the concept of science is well worth taking to heart: "Science," we read on p. viii, "is based upon a very special striving of the human mind, and thus differs from other branches of mental activity in art, religion, and the practical matters of life. The author can only locate this difference in the fact that in science, the mind strives to attain the greatest possible distinctness in its basic concepts and principles, whereas in other kinds of mental pursuits we require only clear or indeed (as is sometimes the case in art) obscure concepts to attain our ends." On p. vii, we read

12

THEORY OF SCIENCE PROPER. Part I §. 394. that "distinctness is the actual aim of scientific presentation." Striving for distinctness, as this worthy scholar urges us, cannot be recommended too strongly, especially in our time. All the same, I believe that it is not the highest goal of scientific presentation, but rather a subordinate one. We strive for distinctness only because and only to the extent that it promotes 21 the goal of certainty, and the even higher goal of furthering the general well-being by spreading secure and useful knowledge. What we originally require of a presentation if it is to be called scientific seems to me always to be that the truths contained in it appear in an order and connection that makes them the easiest to grasp and that engenders the most firm conviction. Now we cannot attain this end if we never raise our concepts to the level of distinctness. Hence striving for distinctness also becomes a duty, but only insofar as it furthers the end we mentioned or else promises some other benefit. This is why this striving for distinctness has different degrees and takes different directions in different sciences. We would not, for example, think it necessary to give a precise definition of the concept of a state in a work of history, no matter how many times that concept might occur there, but we would do so in jurisprudence. Why is this? In order to arrive at the truths we want to learn from history, a merely clear idea of what we call a state suffices; but in jurisprudence, claims are made about states that can only be properly judged if we have acquired a distinct concept of a state. Prof. BACHMANN (Syst. d. L., p. 270), writes: "Knowledge becomes science when the cognitions that exist separately are united, enter into such an intimate connection that they penetrate and animate each other, and are meshed together as the functions of the various parts of a healthy and strong organism.* One calls this whole, with its symmetrical anatomy, a system." It seems to me that such figurative expressions and similes are ill-suited to produce a distinct concept. What is it supposed to mean when one says that cognitions penetrate and animate one another? In what respect can the interaction of the various parts of an 22 organism be imitated in a science (or a treatise)? For while the premises may be a condition for the conclusion, the latter is not a condition for the former. And why should the anatomy of a science be symmetric? As the foregoing shows, only the fact that people have hitherto almost universally confounded the concept of a science with that of a treatise can explain why no logic (at least as far as I am aware) has given a definition of the latter concept, i.e., that of a treatise. Even WOLFF, otherwise so scrupulous in giving definitions, who wrote a special, quite lengthy *One of the best-loved catchwords of our time!

13

THEORY OF SCIENCE PROPER. Part I §. 395. section in his Logic de libris conscribendis, 1 where clearly libris means treatises (books which present sciences), begins this section directly with an inquiry into the various kinds of such books; he says nothing about the question of what a treatise is in general. We must thus tum to the teachers of rhetoric; here it suffices to consider only one of them, MAAB, who writes (Rhet., §256): "A dogmatic discourse in which all the principal truths of a science are presented summarily, i.e., for the most part quite briefly, is a Lehrbuch (Compendium) of this science; one in which the individual truths are developed in detail is called an Abhandlung." 2 In my opinion, neither common usage nor the etymology of the word requires us to think of a treatise [Lehrbuch] as a very brief presentation of only the principal truths of a science. In any case, it will be admitted that it is necessary to have a word to designate any written presentation of all the known truths of a science that are noteworthy for a certain class of readers, if it is composed in a way that makes these truths easiest to understand and recognise as true, regardless of whether this book is espe23 cially concise or detailed. We need such a word, I say, at least in logic; for it is the principal aim of this science to speak of books, to describe their organisation, and to give instructions on how to compose them. Thus even supposing that the word Lehrbuch (treatise] has hitherto been applied to books that merely give a summary presentation (perhaps because such books are far more numerous then detailed ones), it must still be permissible here at least to use the word in a broader sense, namely, that in which the stipulation of brevity is omitted. §.395.* The highest principle of the entire Theory of Science

Whenever the instrnction we impart concerns an art, or even only some activity or other, i.e., whenever we give instructions on how an end achievable through free human activity can and should be attained, we hope in vain to dispose of this task properly and to give complete instructions, if we do not take account of the relation in which all the means we recommend for the attainment of the encl, if not also the end itself, stand to the laws of morality. For merely by indicating the end, i.e., the effect that is supposed to be produced through a certain procedure, the procedure itself 1on

the writing of books of these words might be translated as "treatise" (tr.).

2 Both

14

THEORY OF SCIENCE PROPER. Part I §. 395. is rarely if ever completely determined. For almost always there are several ways of acting which, no matter how different they may be in other respects, are neve1theless equivalent with respect to producing a given effect. Mechanics, for example, teaches us that innumerable ways may be devised to lift a given weight with a given force, and daily experience shows us that men not only strive for but also reach the same end through the most diverse ways and means-one, for example, draws the attention 24 or the sympathy of others in one way, while another does this differently, or they know how to move people's wills in different directions, etc. Thus if in giving instructions on how to attain a given encl we paid heed only to this encl itself, in such a way that any suitable means for attaining it were equally recommendable, we would have to leave many things completely undetermined. But since no one denies that the laws of morality cover all of the acts a person pe1forms wittingly and willingly, it may be conjectured in advance that the various ways of proceeding that might be indicated as suitable for attaining a given encl are not always equivalent in relation to the higher encl prescribed by morality. Now, even if this is not expressly stated, we must look upon it as a requirement that goes without saying that among the various means suitable for attaining a given encl, we should prefer those that best promise to meet the ends prescribed by morality. For even if the person who asks us for instruction does not ask whether or not the means we shall indicate to him are compatible with the laws of virtue, we, in giving such instruction, are in no way permitted to recommend means that directly contravene those laws, at least not if there are other equally suitable means that are compatible with them. From this it follows that even if we are not in a position to inquire whether the aim for whose realisation we suggest means is moral in itself, we should certainly never fail to consider the relation of these means to the moral law when we recommend them. Consequently, every instruction given for any sort of art or kind of conduct must be accompanied by the following proposition: "One must proceed here in such a way that, in addition to the originally specified encl, 25 as many good ends (i.e., ends indicated by the moral law) as possible may also be achieved." And this proposition is, I claim, of such a kind that one may actually deduce from it, as consequences from their ground (at least their partial ground), all the rules contained in our instruction, so that we may consider it the highest principle of the entire instruction. For this single truth already puts us in a position to determine the procedure required to attain the indicated encl as precisely as possible, if these determinations

15

THEORY OF SCIENCE PROPER. Part I §. 395.

are not arbitrary, but based upon rational grounds. For what other respects might there be from which we might deduce requirements for this procedure? The first that might come to mind is that there could be some other benefits for the agent or for others that are attainable through an appropriate application of the available resources, in which case we would also have to give instruction on these, even though no mention whatsoever is made of them in the task we set. This is certainly fitting; yet who can fail to see that all of this can be deduced from the above truth? The laws of morality require that everything that will benefit an individual be done, provided that it harms no one else. Thus if it is possible for an agent to obtain further benefits for himself or others by using the resources he was prepared to apply to the originally determined goal, it follows from the above proposition that we should point these out in our instruction and recommend them. Thus it is clear that this proposition determines everything which may reasonably be stipulated in the procedure for which we are giving instructions. What it leaves indeterminate is by its very nature not determinable on any grounds. For if doing something one way or an26 other neither furthers the goal that the agent himself indicated better than anything else, nor complies more fully with the laws of morality, i.e., if there is no conceivable benefit either to the agent or any other being if it is done one way or another, then this must be looked upon as a matter of indifference, and here we must be guided not by rational grounds but merely by arbitrary choice. If what has just been said holds for any instruction, then it must also hold for that given in the theory of science. Hence we may set out the following proposition as the highest principle of the theory of science: In dividing the entirety of truth into individual sciences and in presenting these sciences in special treatises, everything must be done in the way required by the laws of morality, thus in such a way that the greatest possible sum of good (the greatest possible promotion of the general wellbeing) is thereby produced. Note 1. What I say here is in no way new. Rather, rational people have always proceeded as I have required, or at least held themselves to be obligated to do so, even if they were not distinctly aware of it. In each of our actions we have some goal or another in mind which leads us to exert forces we think to be suitable means for attaining it. If we attempt to determine more precisely the ways in which these forces may be applied, it becomes clear to us that our goal requires many additions; we provide these, and govern, or should govern ourselves by the following principle:

16

THEORY OF SCIENCE PROPER. Part I §. 395.

"It should be done in a way that produces the greatest possible sum of good that we can produce through our efforts." In this consideration, it not only happens that we add to our original goal, but also that that origi nal goal is somewhat altered. Indeed, we may even reject the original goal and replace it with another one entirely. In no case, however, do we regard it as a failing of instruction someone gives us for reaching our goal and the associated expenditure of our energies, if this instruction teaches how to 27 bring about other good things we had not thought of in addition to those we expressly asked about. On the contrary, we shall only be inclined to give the name of a completely well-devised instruction to one which instructs us on a given topic in such a way that none of the benefits that can be produced by a work of this kind has been lost. The whole world will consider someone a fool if he passes up the opportunity to produce a permissible benefit simply because he intended to produce nol it but rather something else. Although I know of no treatise of logic that expressly states the principle I have set out here, I nevertheless believe that my departure from received practice concerns only the presentation, not the matter itself. For have not other logicians believed that in presenting a science one must proceed as required by the laws of morality, and do not the rules they set out amount to the same as mine? Note 2. The laws of morality, i.e. , the rules concerning what we should do, belong to the class of truths that we easily recognise, and find almost universally acknowledged by everyone on the globe. All the same, there have been many disputes among philosophers concerning the highest of these truths, or the practical proposition from which all the others may be deduced as consequences from their grounds, and the view that seems correct to me enjoys very few adherents at the present time. I believe that the highest moral law demands nothing other than the promotion of the general well-being. And if this is correct, the highest principle of the theory of science may be expressed somewhat more precisely, as I have done above in the last sentence preceding the notes. Certainly, even those who do not look upon the promotion of the general well-being as the only duty will not dispute that in cases where promoting of the general well-being does not contravene any other higher law, we are obligated to do so. And someone who admits only this will, I hope, hardly 28 feel tempted to raise objections to the rules I shall set out in the sequel merely because we hold differing views on the highest moral law.

17

THEORY OF SCIENCE PROPER. Part I§. 396, 397. §.396.*

Immediate consequences: 1) The science we intend to present in a treatise must merit inclusion in the ranks of the sciences Now that we know the principle from which all the theses to be presented here may be deduced, it may be helpful to begin by adducing some o~ its immediate consequences that we will do well to keep constantly before our eyes when developing the other rules. The first .o.f these is th~ 01~e contained in the title to this section. For since the wntmg of a treatise is a job that requires a not inconsiderable expenditure of time and effort, it is obvious that we cannot approve of such an undertaking if we have not first convinced ourselves that some benefit will come of it. Now it cannot be denied that even if the science we choose is not at all fruitful, and does not deserve to be included in the ranks of the sciences, many good and useful things may still be contained in our book. But who can fail to see that we could have brought about far more good and useful things if the subject we had selected were itself worthy of our efforts? And is not ~he very concept of a treatise presenting a science that is not worth presentmg an absurdity? No matter how it may otherwise be constituted, no matter how perfectly it attained its immediate end, it could never live up to th~ remoter end that would have justified our writing it. Thus it cannot ment 29 the name of a good and complete book. Thus, e.g., the art of deception is certainly not worthy of being presented as a science; hence a book that attempted to convey this art to readers would on that account alone not merit the name of a good and sound one. §.397.*

2) The class of readers for whom our book is intended should be appropriately chosen

THEORY OF SCIENCE PROPER. Part I §. 398, 399. for whom we intend it must clearly be appropriately chosen. It must be possible to indicate some rational ground why we have decided to write for precisely this class of readers, why we have not written for a larger or smaller number of readers, and why we have not included readers of other kinds among them. For if the opposite were the case, and our choice inept, if, for example, we thought of kinds of readers that do not exist, it is obvious that our book will either not be useful at all, or at least far less useful than it would have been had a suitable choice been made. 30 §.398*.

3) A suitable treatise must make what it presents in print as easily and securely understandable to its readers as possible According to its concept, a treatise is supposed to be a book which, if read, can convince one of certain truths. But in order to be convinced of a truth, the idea of the proposition of which it consists must first be called forth in our minds. For if we do not even represent a proposition, we certainly cannot see that it is true. Thus nothing is more necessary when writing a treatise than expressing ourselves in a way that our readers understand just what we intended to present through our signs. According to the principle of §395, however, we will not be satisfied with expressing ourselves in a way that our readers may well understand, but only after taking much trouble to figure out what we meant. Rather, we must strive to make this understanding as easy and secure as is possible without destroying other, greater benefits. For proceeding in this way increases the usefulness of our book. The more readily and securely our reader understands us, the less frequently he will misunderstand us and be misled, the more pleasant the instruction our book provides will be, and the longer he will stick with it. This will leave him with more energy to devote to reflection about the truth of what we tell him, and thus speed his progress from one proposition to another, and thus enable him to learn the entire science more quickly, etc.

It lies in the very concept of a treatise not only that it must be devoted to a particular science, but also that it must be destined f~r a particular. class of readers. And only for this class of readers must it be appropnately organised so as to make it as comprehensible and convincing as possible. If it does not appeal to other readers for whom it was not intended, the fact that it was not possible to make one and the same book comprehensible and instructive for everyone will not be held against it. But just as the science we intend to present must be appropriately chosen if our book is to merit the name of a good and complete one, so too the class of readers

It goes without saying that every treatise must awaken a great number of ideas, judgements and inferences in the mind of its readers-how else

18

19

§.399*.

4) It must attempt to make the most important ideas, judgements, and inferences distinct

31

THEORY OF SCIENCE PROPER. Part I §. 400.

THEORY OF SCIENCE PROPER. Part I§. 401.

could it be said to be a book which instructs them? But they will not attain a distinct consciousness of every idea, judgement and inference; rather, a significant part will and must remain at the level of obscurity. But insofar as it is possible, we must strive to raise the most important ideas, judgements, and inferences to clarity, and indeed to distinctness. For this will provide the following benefits: (a) first of all, people find it pleasant in and of itself to have a clear awareness of what is going on in their minds, and still more pleasant to have a distinct awareness. (b) But this is not merely pleasant, but also a very useful exercise in the necessary art of self-observation. For in this way we become more skilful in obtaining a clear and distinct awareness of other things that occur within us, something which puts us in a position properly to judge, as well as to reject what is baseless or unjustifiable, etc. (c) When we strive to make the propositions and inferences upon which our claims are based as distinct as possible, it often happens that we discover and are able to correct mistakes. (d) If truth is in our doctrine, then it is to be expected that the distinct presentation of the grounds upon which it is based will create a more solid and lasting conviction. (e) Now they will be in a position to discover other truths in ways similar to those we used to discover the truths we presented to them. (f) If we have made mistakes, and the inferences we made are plain for them to see, they can not only more easily notice the fallacy that misled us, but also be more aware of it in similar cases. And so on.

source of error. We only err when we take something to be true which is merely probable, or ascribe a higher degree of probability to something than it actually has. Thus if someone is to protect us from error, he must teach us to hold what is probable as only probable, and never to overestimate the degree of its probability.

§.400*. 5) It must confer the appropriate degree of confidence upon every thesis, and make its degree of reliability evident It follows from the very concept of a science that in a well-constituted 32 treatise we may only present propositions we hold to be true. But since we can only be completely certain of the truth of a very small portion of our judgements, and since, moreover, opinions that are merely probable can also be useful, it will be permissible to present propositions that in our eyes are only probable. But we must present these in a way which makes it clear to the reader that we are speaking of merely probable things. For only when we proceed in this manner, only when we present what is not completely reliable but only probable as such do we protect our readers from the danger of error as much as lies within our power-for overestimation of the degree of probability of a proposition is the most common, and indeed (if the attempted explanation of §309 is correct) the only

20

§.401.* 6) A suitable treatise must indicate the objective connection between truths insofar as this is possible According to the definition of the concept of a treatise given in §395, it suffices that one present the theses of the science to which the treatise is dedicated in such an order and connection with other truths that anyone with the prerequisite knowledge who thinks through them in this order will accept them with the appropriate degree of confidence. I did not require that one indicate the objective ground upon which each of them rests. This was done intentionally; for according to what we saw in §216 and elsewhere, one cannot demand that an objective ground be indicated for every truth, since there are some truths that have no such ground. Fur- 33 thermore, even when such a ground exists, it may be exceptionally difficult or even completely impossible for us humans to discover it. On the other hand, it is both useful and pleasant to learn the objective connection between truths, and indeed this has so many advantages for research itself, that we are by no means unreasonable if we demand that every author of a treatise should at least strive to discover this connection and make it known to the reader whenever this is feasible and not forbidden for higher reasons (i.e., for the sake of benefits of greater importance). The benefits of these efforts are, roughly speaking, the following: (a) With many propositions we hold to be true and are prepared to include in our treatise, striving to discover their objective ground may lead to the discovery that they are false. Thus, e.g., in metaphysics, one might seek the ground of the claim that an infinite sequence of successive states cannot be completed, and in so doing discover that this claim is false, since there are no tenable grounds for it, while one can discover such grounds for the opposite claim. (b) If, by contrast, in reflecting on the ground of a proposition held to be true we learn not only of proofs which show that it is so but also of grounds that explain why it is so, it is obvious that our confidence must be greatly increased by the latter. Thus inquiring into the ultimate grounds of our duties can have the important benefit of convincing us 21

THEORY OF SCIENCE PROPER. Part I §. 402.

34

all the more firmly of the truth of the unanimous judgements of human understanding on such matters. (c) Indicating the grounds of a proposition can in many cases be the most instructive and convincing, if not the shortest, proof of its truth. Thus the most concise proofs of certain geometrical truths about the similarity of spatial objects are precisely those that one finds by seeking the objective grounds of these truths. (d) Still more importantly, the discovery of the objective grounds of a truth often puts us in a position to discover a number of other useful truths. This is especially the case with empirical truths, where the discovery of the ground is at the same time the discovery of the cause of a phenomenon. Knowledge of causes, however, puts us in a position to realise many of our wishes and aims, to free us from innumerable evils, and to bring about conditions that are beneficial for ourselves and others. For thousands of years people have known that lightning strikes high towers and buildings; since FRANKLIN taught us why this occurs, we possess a means for protecting ourselves from the greater pait of the damage lightning can cause. In the realm of purely conceptual truths, too, the discovery of the ground is rarely completely fruitless. The discovery of the reason why the base angles of an isosceles triangle are equal, for example, led to the discovery of the important truth that any two triangles in which two angles and the included side are equal are themselves equal, and from this truth to thousands of other previously unknown truths. Inquiring into the ground of the truth that the forces on a lever in equilibrium are in inverse proportion to their distances from the fulcrum led to the most important discoveries in mechanics, and so on. (e) In any case, someone who knows not only that something is so but also why it is so knows one more truth. Seeking this truth provides a special exercise in thinking, and finding it provides a special pleasure.

away from the proofs we adduce and keep the genuine or merely apparent grounds against our proposition before their eyes, and they will never be convinced. And how can they be convinced, if they discover through their acumen that our proofa contain gaps we have overlooked, and if not only our proofs but even some of our theses require c01Tection? Can it really be expected that our readers will overlook these failings, and despite our poor advocacy for good things nevertheless not misjudge them? Can we expect them to fill the gaps they discover in our proofs through their own reflection, and undertake to correct whatever errors they find? Indeed, will they even want to do these things if they do not possess pure love of truth? It is obviously our duty to prevent these evils as much as we can. Obviously we should strive to awaken a quite impartial love of truth in our readers, and if this should prove impossible, we should at least strive to diminish and render less hai·mful every kind of resistance their passions put up to the recognition of truth. Were we to fail to do this, we would (a) hope in vain to bring our readers to acknowledge our theses merely through the force of our reasons in cases where one of their passions is opposed. For, as noted above, they need only ignore these reasons in order not to feel their weight. (b) Be that as it may, the love of truth is so noble and excellent that we would be wrong to miss any opportunity to 36 promote it. §.403. *

8) A suitable treatise must also make it as easy as possible to locate, retain, and recall the theses it presents

Be the presentation we give in our book ever so comprehensible, be the grounds we adduce as proof for our claims ever so decisive and evident, should we even take pains to give proofs that show not only that something is so, but why it is, we still should not expect our readers to acknowledge all of the truths we present if they are predisposed not to accept them, if they will not be convinced. They need only direct their attention

In most sciences there are certainly individual propositions which it remains beneficial to learn even if they will later be lost to our memory, and even if we shall have no means of ever recalling them. For the benefit of the exercise our faculty of judgement obtains through learning them remains even when they are forgotten. Even when a proposition is destined to serve as a premise in the deduction of many other truths, among them some that cannot simply be lost sight of, and indeed some that we should never lose confidence in, even then it may not be inevitably necessary for us to impress this proposition as firmly upon our memory as we do all the truths proved from it in our treatise. Even if, when it is evoked, we can only recall it as a previously demonstrated proposition, this can be enough to make us accept with conviction the truth that is now proved, and to permit us to rest assured in its truth in later years when we can no

22

23

§.402. * 7) A suitable treatise must attempt to counteract any disinclination

the reader may have to recognise the truth 35

THEORY OF SCIENCE PROPER. Part I §. 403.

THEORY OF SCIENCE PROPER. Part I §. 404.

longer recall the premises from which it was proved. It is undeniably an advantage, however, if we are clever enough to order our book in such a way that an attentive reader who is not possessed of an exceptional memory can nevertheless in a short time retain in his memory a great many theses, including those that promise to be of use. With some sciences the principal benefit of learning them is that the truths proved in them 37 shall never disappear from our memory, but rather shall occur to us on every suitable occasion, along with the grounds they rest upon, or at least that we shall be in a position easily to find them. The truths of religion or ethics, for example, can only render the service they are intended to if they were not only known once in the past, but rather are constantly borne in mind. Certainly, then, we shall greatly increase the benefits produced by composing a treatise if its very arrangement is such that it ensures that at least the theses which can be advantageous to know in the future are lastingly impressed upon the mind, and shall at the appropriate times either occur immediately to the reader or else be easily refreshed. §.404. *

9) There must be signs for the concepts occurring in the science in question which the reader will find convenient for his own use

THEORY OF SCIENCE PROPER. Part I §. 405, 406.

doubt that for each idea presented to the reader in a well-constituted treatise we must recommend a sign which is of a nature to be conveniently used. §.405.*

10) One must also take care to ensure that the reader obtains appropriate images of the objects that are dealt with The images we mentioned in §284, which we involuntarily connect with most of our ideas insofar as we believe them to be objectual, and which have such a great influence on our judgements that we must always be attentive to them in our own reflection (§345), also deserve to be considered in the instruction we give to others in a book. We must, namely, bring it about that all the images the reader forms of the various objects dealt with in our treatise-either through what we say about these objects or for other reasons, among them often what we do not say-are as correct and adequate as possible. If we fail to do this, even if everything else we have expressly presented is correct, we may not boast that we have done everything in our power to prevent the formation of incotTect ideas. 39 §.406.*

If the theses and truths we present in a book are to be retained in the reader's memory so as to be recalled at suitable times, if he is to be in a position to reflect upon them, to communicate them to others, or at least to speak to others about them, and so on, then, for each of the ideas occurring in the book, he must procure signs that are useful not merely for us and in a book, but rather also useful for him for the purposes of reflection or ordinary conversation. The reader does not come by this asset merely by being acquainted with the signs we use in our book, even if these should be highly suitable. For signs that are suitable for presentation in a book are for that reason not suited for use in one's own reflection or in the oral communication of thoughts, and may not even be suitable 38 for the reader's aim of presenting his own thoughts in writing. Thus if we wish him to procure signs that are truly useful to him, we must ourselves take the trouble to acquaint him with them. For it would certainly be unfair to leave it to the reader to discover such signs, not only because he could hardly be expected to chance upon useful ones, but also because the greatest confusion must ensue if each of our readers introduced different signs for lack of any uniform recommendation. There is therefore no 24

11) The book must be arranged in a way that promotes as much as possible its correct use by the reader Plainly, the most important benefit a well-organised book may produce can only be brought about if it reaches the readers it is intended for and if they make appropriate use of it. For how can a book produce benefits if the persons it is intended for either do not read it at all or do not read it in the appropriate order and with the appropriate attentiveness, if the truths one presents to these readers, even if understood and accepted with the appropriate degree of confidence, are not retained afterwards, but rather confused with others, or used to derive faulty conclusions, etc.? Now it is true that the organisation of a book cannot always be blamed for the fact that it does not find appropriate readers-for our book cannot compel people to read it, and this in the correct way. Yet we can do something to contribute to these goals. We can, e.g., attract or repel readers through the inner constitution we impart to our book; we may bring it about that they either attend to our presentation with heart and soul or turn from it 25

THEORY OF SCIENCE PROPER. Part I§. 407, 408.

with disgust and weariness. Thus no proof is required that a book which may lay claim to excellence must be organised throughout in a way that is necessary to allow its readers to make appropriate use of it. §.407.*

12) A suitable treatise must be organised in a way that ensures that any faults it may have cause the reader the least possible harm As much care as we may take in composing a book to ensure that we 40 do not make even a single false claim, and that we cannot be blamed for making any false inferences, we certainly cannot completely avoid errors of these and many other kinds. And it will happen only too often that the reader will become infected with our errors, taking the propositions we claim to be true for truths and adopting our modes of inference as his own. Indeed, if we have not taken suitable precautions, the very advantages of our book may be the reason why its errors are so unthinkingly repeated and persisted in so firmly. The history of every science provides us with examples of how the errors of those who made outstanding contributions have been the most dangerous, since they were generally accepted and difficult to root out. A little reflection will allow one to see that it does not lie completely within the power of an author to prevent such evils, though he can do much to diminish them. To do this, then, to take care to ensure that his errors produce the least possible harm, is a duty binding every author of a treatise. §.408.*

13) A suitable treatise must permit its readers to see the reason for most of its features

41

It is well known that an object can be called well-constituted or pe1fect only insofar as each of its features has a rational ground, i.e., insofar a~ it may be shown that this feature had to be exactly as it is and not otherw1se if the object was to attain a given end. A treatise, too, insofar as it is to be well-constituted, must be put together in such a way that a rational ground may be indicated for each of its features. Now I claim that a good feature in such a book is that it permits its readers to recognise these rational grounds in most, if not all, cases. Accordingly, I re~uire the a~tho: of a treatise to take care to ensure that the reasons for h1s proceedmg m the

26

THEORY OF SCIENCE PROPER. Part I§. 408. way he does is evident as far as this is possible. Proceeding in this way will have the following benefits: (a) Many features of the book which might have appeared completely arbitrary to the reader had his attention not been drawn to their appropriateness will now appear to him to be the result of a rational rule, and therefore be found agreeable to his sense of order and regularity, and thus encourage him. (b) If the ground that led us to introduce this or that feature, i.e., the rule we followed in introducing it, is expressly indicated, then on the one hand we give the reader the opportunity to consider the correctness of the rule, and on the other he may check to see whether we have fully complied with it. Both of these provide him with exercise in thinking. (c) If the reader encounters an error in our book, then, because we have indicated the rules we intend to follow, he will find it easier to decide where the cause of our mistake lies, that is, whether it lies in the incorrectness of the rule or in our failure to follow it properly. (d) If our rules are rational and worthy of imitation, and new at least in part to our readers, we may, by making them known, create the benefit of leading our readers to adopt and follow some of these rules in similar cases. (e) If we undertake to draw the reader's attention to the grounds for all or almost all of the features of our book, we must ourselves become distinctly conscious of these grounds. In many cases, this permits us to detect any deficiencies in these grounds, or to see that we have not faithfully followed them, etc. The reader will gather for himself why I do not require that the ground of everything that appears in our book be indicated. This would have to proceed ad infinitum-and how 42 many features are based upon grounds we cannot render comprehensible! Finally, can it not be a good thing in a few cases if the reader is unable to guess why we do what we do, at least beforehand? Examples of this shall be given in the sequel. Note. One often hears it said in our day that great masters usually cleverly conceal the rules they follow in producing their works. Indeed, many say that such masters are not even aware of any rules, but instead always find the right way by instinct. If one does not take the latter to mean that a distinct knowledge of the rules one should follow is incompatible with a true mastery, or would detract from this mastery rather than promoting it, if instead one only takes it to mean that it is possible to accomplish great things in the sciences without being distinctly aware of the rules one follows, one will see from §9 that I by no means dispute this. But I completely reject the first claim and say that the great masters-with a very few exceptions, where the opposite was necessary due to peculiar

27

THEORY OF SCIENCE PROPER. Part I §. 408.

PART II circumstances-rather than concealing the rules of their methods, instead make them as evident in their works as the nature of the rules permits. Why should they think to conceal? And why should a work be called more perfect if the laws according to which it was produced are not as distinctly perceptible as their nature permits? The truth contained in the above statement is that truly accomplished works are produced not merely according to a single rule, but many, with the result that it is not an easy matter to say of each feature of the work which pmticular consideration led the master to introduce it in just the way he did. By contrast, works which are prepared following only a few simple rules must for this very reason visibly bear their imprint. And if the craftsman had conceived his rules in a limited way, applying them in places where they do not belong, 43 the flaws of his work will be obvious (from the violation of other rules), no matter by what self-made rules he may have governed himself. Thus one is in fact not wrong in saying that a work in which one may too easily perceive the rules of its construction is only mediocre. But it does not follow from this in the least that we must intentionally conceal the rules 44 we follow in order to impart greater perfection to our work.

28

On the Determination of the Extensions of the Sciences §.409.' Consequences o.f various ways of delimiting the extensions of sciences The first thing that must be clone in order to put oneself in a position to rationally resolve to write a treatise of a science is to assure oneself that the science we intend to present in writing is called for, i.e., that it truly merits a place in the ranks of the sciences, and this precisely as it is delimited. This part shall give brief instruction on how to make such judgements, how to tackle the associated problem of discovering a concept of a science that is worth developing, and finally how to approach the still more general problem of dividing the entire domain of truths into a sufficient number of appropriate sciences. This last task shall not be undertaken here; rather, I shall only indicate the rules that are to be observed in this business, and even then only those that must be generally observed. But since the correctness of rules for some task or other may be better judged if we are more distinctly aware of the various beneficial and harmful effects that may ensue if the task is done one way or another, we shal! begin by addressing this point, and consider the various advantages or disadvantages that arise if the extensions of the sciences are determined in one way or another. 1) First, concerning the advantages that an appropriate determination of the extensions of the sciences may bring, (a) the most important is 45 doubtless that everyone who desires to learn a certain species of truths shall have the opportunity of finding all of them that are currently known and noteworthy in a specific place, and he shall find them all the more reliably if they are separated from everything that does not belong with them. (b) A second benefit is that an appropriate separation of what is foreign and the assembly of truths of the same kind makes it much easier to understand as well as prove the latter, since knowledge of one of these truths prepares the mind for the others. (c) The relation of ground to consequence between truths is also easier to notice when circumstances permit ground and consequence to be presented one after the other as belonging to the same science. (cl) When we see before us a whole of truths that bears its own name and is of moderate extent, there arises in us the desire to become fully acquainted with it, and (e) when we hear, in

29

THEORY OF SCIENCE PROPER. Part II§. 410.

addition, that this small number of truths is the entirety of what is known about the given topic, this prompts us to inquire whether we can discover others. These inquiries can succeed all the better because (f) the truths are classified in such a way that the discovery of those belonging to a given science require the same talent, skill, previous knowledge and resources. (g) If we go about the business of dividing the entire domain of truths into individual sciences in a skilful way, and in particular in such a way that every humanly attainable truth belongs in one or the other of them, we may be led to many hitherto unnoticed species of truths that are quite noteworthy. 2) Most of the disadvantages stemming from an inappropriate delimitation of the boundaries of the sciences follow from what has just been 46 said by mere opposition. In addition to these, however, there may also be the following: (a) If the number of the sciences is too great, it can cost a great deal of of trouble merely to become acquainted with the concepts of all of them, and to know enough of each in order to be in a position to judge where we should look for a truth we would like to know. (b) An overly narrow delimitation of the sciences makes one lazy and leads to the flaw of making one satisfied with learning one or a few truths, even though one in fact needs to know far more and is quite capable of doing so. (c) The most damaging thing occurs when instructions that are necessary in the same circumstances are separated, e.g., the theory of healing internal and external afflictions.

§.410.*

1) No science need be specified for a truth that cannot be expressed in writing If one admits what I have said so far, one should also agree with the following propositions, which I consider to belong to the principles that must be followed in the business for which I shall give instructions. The first proposition is: No science need be specified for a truth that cannot be expressed in writing. No one will deny that there are truths, among them some very useful ones, that cannot be put in writing, because the ideas of which they consist cannot be expressed through written signs. Who, for example, would want to describe the smell of a given plant merely in words, unless we already know of some similar smell? I do not claim here, however, that it is a mistake for the concept of a science 47 to be defined in such a way that truths that cannot be expressed in writing 30

THEORY OF SCIENCE PROPER. Part II§. 410.

can be counted as part of the content of a science; rather, I only claim that in dividing the entire domain of human knowledge into individual sciences it should not be considered a mistake if such truths are overlooked, no matter how useful or even necessary they might be otherwise. I claim this because if the nature of a truth is such that it cannot be expressed in writing, the fact that we have not indicated any particular science to which it belongs cannot have any harmful consequences. For since this truth cannot appear in a treatise even if it does belong to a given science, the only apparent harmful effect its absence from the science could have would be that the truth in question could not bear the name of a scientific truth. But it does not follow from this that we cannot recognise its worth, or that we cannot use means more effective than writing to spread it should it merit being more widely known. Indeed, even if some benefit can come of giving instructions on how one can come to know this truth, it may be permissible to give these the name of a science. Thus, e.g., we shall not introduce a science that describes the various colours, odours, and other sensible but difficult to describe qualities of medicines; yet we shall not only ask doctors to become acquainted with these characteristics through their own observation, but even give particular instruction on how they may do so in some special science. Note. It might well occur to whoever compares the claim of this paragraph with the remark made in §75 that intuitions, as such, are incommunicable, that according to my view empirical truths may never lay claim 48 to the distinction of forming the object of a particular science. For since I only call truths empirical if they contain some intuition or other as a component (§133) and, as is almost too obvious to mention, a truth is not communicable if the individual ideas of which it is composed are not communicable, one might well believe that all empirical truths belong to the class of those that cannot be communicated in writing, and that as a consequence one would not be justified in claiming that there is a science to which they belong. Nevertheless, this is not my opinion. Rather, I claim that a great many empirical truths are important enough to merit presentation in individual sciences (which are accordingly called empirical sciences). Things stand thus: it is certainly well-founded to say that a single empirical truth grasped by one person can never be completely grasped by a second person, nor indeed by the same person at another time. Strictly speaking, it is no longer the same proposition that I put forward when I now say that Alexander was born roughly 2190 years ago (before this instant) as when I said this an hour ago. For the intuition

31

THEORY OF SCIENCE PROPER. Part II §. 411.

of that which I presently feel or think that lies in the words "before this instant" is now different from that of an hour ago. Similarly for the propositions "Sirius is a fixed star," inasmuch as the intuitions found in the idea "Sirius" are different for every man. But these differences will be seen not to matter here, and so can be disregarded. To say that we have communicated the truths contained in these propositions to the reader, it suffices that he form ideas that, if not composed of the same intuitions as ours, are nevertheless equivalent to them, i.e., that he thinks of the same object (the same subject), and attributes the same property (the same predicate) to it that we do. Thus in the first example it will be sufficient if our reader 49 connects an idea with the word "Alexander" that in fact refers only to this Macedonian king, and learns that this man was born 2190 years before the year this was written. Thus understood, there can be no doubt that empirical truths can also be communicated through writing, and merit presentation in their own sciences. §.411.*

2) Every truth communicable by writing that is noteworthy not merely as a supporting proposition should belong to at least one science When a truth is of the sort that can be communicated by writing and in addition deserves to be spread in this way, when, more precisely, we may expect that a written presentation of the truth in a suitable place will be of use not only to one, but to many people, then there must be at least one science in which this truth is presented. Now there are three ways in which this can occur: we may present the proposition as one that is indigenous to the science, i.e., one that already belongs to it on account of the concept we form of this science; or we may include it merely as a supporting proposition, that is, only because it is used to prove other truths; or, finally, we may mention it as the occasion arises, e.g., as a useful consequence of a truth that is indigenous to the science. I claim that if the reason why a truth is noteworthy does not consist in the fact that it is used as a premise in a proof of a truth that belongs to a given science, then it is never enough merely to present it merely as a supporting or an occasional proposition. Rather, there must always be a science in whose treatises it 50 is presented and proved. Were we to present it merely as a supporting proposition or an occasional proposition, people would at most become acquainted with it by reading the treatises in which it is used in this way. But if we are striving (as we should in this case) to permit anyone who 32

THEORY OF SCIENCE PROPER. Part II §. 411.

wanted to become acquainted with the truth in question to find it easily, such a provision obviously will not suffice. If the proposition were only noteworthy insofar as it is used as a premise in a certain proof, then there would be no harm in our not knowing where to find it. For it is not the subject of our interest, except perhaps when we happen upon it. In the opposite case, however, where the knowledge of the truth in question is of considerably broader use, it is to mankind's essential advantage to make it as easy as possible to locate, and thus it is fitting that it be presented in a place where anyone who needs it can find it easily and reliably. But this is only the case if there is a science to which the truth is indigenous. In this case one only need know the concept of the science in order to know immediately that the truth in question is to be sought in treatises of that science. Thus, for example, there is no need of a special science in which the approximate values of certain integrals that aid astronomers in obtaining highly valuable results are determined, if these integrals have no other applications. For in that case, we would not inquire about these integrals except when we turn to those astronomical problems. If it should subsequently turn out, however, that these integrals can be applied elsewhere, it will be fitting to consider them in pure analysis (namely, in the part called the integral calculus). The truth that things whose determining elements are equal (or similar) must themselves be equal (or similar) provides an example of a truth which is so noteworthy, both on account of its simplic- 51 ity and of the many consequences belonging to different sciences that can be deduced from it, that we could not escape censure were we to present it merely as a supporting proposition, and had no special science (such as metaphysics) in which it could be presented and proved as an indigenous truth. Note. If I am not mistaken, it follows from this principle that, as numerous and varied as are the sciences that have already been introduced, still more sciences should be introduced, which will with time find their discoverers and contributors. If you have discovered even a single purely conceptual truth, the knowledge of which will be welcome or beneficial to others, or if you have made even a single observation in your life which promises to cast light on some obscure part of our know ledge, and there is no existing science which might incorporate your discovery in the expectation that anyone to whom it might be useful would seek and find it therein, then it is immediately apparent that there is a gap in our present system of sciences, and that it is worth some effort to think about how to fill it.

33

THEORY OF SCIENCE PROPER. Part II§. 412.

THEORY OF SCIENCE PROPER. Part II§. 413, 414.

§.412.*

§.413.*

3) Having too small an extent is not a sufficient reason for rejecting a science, though having too great an extent may be

The mere fact that the extent of a contemplated science would turn out to be quite small, because there are only a few truths of the sort the science would unite according to the concept we form of it, is not a sufficient reason to reject it. For the worst harm that can come of this is that by increasing the number of sciences we will also increase the number of fields of which people must at least know the concepts in order to decide where 52 to seek a truth they wish to consider. This minor inconvenience can be outweighed, however, by much more important benefits, e.g., people will be more attentive to truths for which we have established a special science, or will perceive more distinctly the inner connection that obtains between them, or they will be more likely to resolve to learn such a science if they think this can be done quickly, etc. If, on the contrary, the extent of a science is too large, so large in fact that it contains more truths than a single person is capable of grasping, this leads to the extremely harmful consequence that everyone will be deterred from learning it. The only benefit that uniting many truths in a single science can have is that of making it easier to find the individual truths belonging to the field that people desire to know. When this benefit is no longer present, perhaps because the same goal can be reached by using a special order such as that followed in dictionaries, then uniting truths in such a way must certainly be censured. According to this principle, for example, the theory of time deserves to be developed in a special science (the pure theory of time), although it is true that this science can only consist of a very small number of propositions.* By contrast, it would be misguided to want to unite all the purely conceptual truths, or all the truths drawn from experience, into a single science. For who would not shrink before a science 53 of such vast scope? And what can be said about those who would herd together all the truths there are in a single science, namely the science of eve1ything or (as they themselves call it) philosophy?

4) The fact that all of its theses are already known is not a sufficient reason for rejecting a science

When I required in §411 that for every thesis that is at least somewhat noteworthy there be a science to which it is indigenous, I did not wish anyone to draw the conclusion that less noteworthy truths, and in particular those that everyone already knows, deserve to be excluded from the domain of a science or at least from written presentations of sciences. This is so far removed from my opinion that, on the contrary, I maintain that there can be sciences in which the greater part or indeed all of the theses they contain are already known to everyone. For even treatises of such sciences can be of use to mankind. It can be worthwhile to present truths that are known to us if it turns out that other, unknown truths can be deduced from them, or if we are thereby made aware of the objective grounds of these truths, and the connection between them. These objective grounds and this connection are often unknown, even for the best known truths, and it is worth the trouble to become acquainted with them, in part because this provides an excellent exercise in thinking, in part because insight into the connection between known truths is often a means for discovering new truths, and deciding disputes of the highest importance. Thus though one might object that all of the theses of the pure theory of time (§412) are already known to everyone, one nevertheless learns something new and worthwhile when, in elaborating this science, 54 one becomes acquainted with the objective grounds of why time has just the nature it does, or indeed just what concept it is that we designate with the word time. Similar remarks hold for the science I have already mentioned several times under the name of the theoty of experience. Logic, the very science I am now presenting, may be cited as an example of a science where many, though not all, of the theses may be supposed to be already known, even though no one considers it superfluous to present logic in detail. But metaphysics too, as well as ethics, aesthetics, arithmetic, and geometry, among others, are also sciences that contain truths that are known at least in part. §.414.* 5) The fact that truths are quite similar is not a sufficient reason for uniting them

*It is odd that KANT, who saw that the propositions about time are just as special as those about space. to which a science has exclusively been devoted for centuries, refused to grant the same right to the theory of time. He gave no other reason for this in his Abhandlwzg Uber Philosophie (see his Kleine Schrijien, eel. Starke, Vol. 2, p. 250) than that the science of time would have too few propositions.

1) It might seem that the more similar truths are, the more suitable it would be to combine them in a single science, but closer consideration

34

35

THEORY OF SCIENCE PROPER. Part II§. 414.

shows this thought to be baseless. Even when certain truths are quite similar, it can nevertheless be the case that they are not all useful for one and the same person, or that it is necessary to divide them into two or more parts in order to avoid confusing them, or that such a division is at least beneficial in that it permits us to grasp more distinctly the difference between the truths, or the differences between their grounds or the consequences that flow from them. In such cases it will certainly be praiseworthy to separate rather than unite the truths. Thus the truths assign~d exclusively to ethics and to jurisprudence are closely related and qmte similar, yet it is a good thing to deal with them in separate sciences, in 55 part because this helps prevent us from confusing them and taking something that is merely legal to be morally good. In this book, for similar reasons, I separated the theory of ideas and propositions in themselves from that of thought ideas and propositions. So too, one should, in my opinion, treat the ascetical and the historical interpretation of the Scriptures (where the former sets itself the problem of indicating the edifying ways a given passage may be used, while the latter investigates the question of which ideas the author probably intended to stimulate in his readers) as a pair of separate sciences. 2) In particular, the fact that certain truths deal with the same object, or with several intimately connected parts of one and the same whole, is not a sufficient ground for uniting them in a science, nor is the fact that they all have the same predicate-idea, or that they are all deducible from the same premises. Despite all the similarity such truths might have, one might still require previous knowledge of widely varied kinds in order to have full insight into them; the circumstances in which they might be of use might be so opposed that it would be far more suitable to deal with them in separate sciences than to unite them in a whole. Thus it would certainly be quite misguided to unite all the truths of natural history, history, medicine, ethics, politics, and theology dealing with human beings merely because they treat of one and the same object. For even if some of these truths are noteworthy for everyone (and these could certainly be united in a single science, say, anthropology), this certainly does not hold for all of them. A prince, his ministers, and his people form an intimately connected whole-but does this mean we must always deal with the nature, rights and duties of all of them in the same context? The duties that 56 attach to gender, age, and station all flow from the same supreme moral law; nevertheless, it can be quite advisable to deal with them separately in individual treatises.

36

THEORY OF SCIENCE PROPER. Part II§. 415.

Note. From this one may judge how vague and incorrect it is to say that truths r~f the same kind belong to the same science. The most diverse truths can, from a certain point of view, be reckoned to be of the same kind. But even if this saying had the sense that the more similar truths are, the more they deserve to be united in a science, it would (as we just saw) still not be justifiable. §.415.*

6) A great difference between truths, in particular; the fact that they come from a completely different source of knowledge, is not a sufficient reason to separate them 1) Just as great similarity does not immediately justify our uniting truths, so too a great difference between truths does not immediately justify our separating them. For no matter how different theses may be in certain respects, it can still be useful and indeed necessary to present them as intimately connected, either because each illuminates the other, and might be misunderstood or misapplied if presented alone, or else because a man who lives in circumstances which require him to know one of these truths also finds himself in circumstances where he needs to know the other. What diversity we find, for example, in the truths of the various branches of medicine! Yet how necessary it is to unite all these varied branches in one and the same whole, since only one who knows all rather than merely some of these is in a position to make fruitful use of them as a physician. 2) In particular, the circumstance that given truths have a completely distinct source of knowledge, e.g., that some are known based only on purely conceptual truths (a priori) and others only through experience, is in my opinion not a sufficient reason for always placing them in dif- 57 ferent sciences. This distinction is certainly important enough that one should never overlook it. But to this end it suffices simply to point it out; it is by no means necessary to teach the different kinds of truths in different sciences. That they are to be presented together can often be required most decidedly simply because the exercise of the profession which requires knowledge of the one also requires knowledge of the other. Furthermore, we cannot always tell from which source of knowledge we draw our knowledge of a given truth, as this often depends upon quite accidental circumstances, and changes with time. For the very same truth that we drew from experience today we may perhaps learn how to derive [entwickeln] tomorrow from the nature of the concepts of which it 37

THEORY OF SCIENCE PROPER. Part II§. 415.

is composed (that is, a priori). Thus it would be permissible, even for example in pure number theory, to accept based on experience the theorem that any number can be expressed as a sum of no more than four square numbers. so long as no one yet knows of a proof for it based on pure concepts. This will occur far more often in metaphysics, where there are many purely conceptual propositions that can be shown with great probability from experience, although we are in no position to produce a proof of the sort we seek for all purely conceptual truths, and which in some cases we have already found. Note. What KANT says in his Critique of Pure Reason (A 842/B 870 seq.) seems to stand in direct contradiction with this claim: "It is of the utmost importance to isolate cognitions that are distinct according to their genus and origin, and to prevent them from getting mixed up with others that they are commonly associated with in use. One must admit that the difference between the two elements of our knowledge, of which one is completely in our power a priori, the other of which can only be gathered a posteriori from experience, has remained very indistinct even 58 for professional thinkers, and consequently no one was ever able to draw the boundary of a particular kind of cognition, and with it to arrive at the correct idea of a science (namely, metaphysics) that has occupied human reason so much for so long. The mere degree of subordination (the particular under the general) cannot determine the boundaries of a science, rather only the utter heterogeneity and distinctness of origin can do so." One will not find these remarks of KANT to be so utterly opposed to the views I expressed above when one considers that what he meant by a priori cognitions was nothing other than what I have called purely conceptual propositions. It was only because he did not become distinctly aware of how these propositions are objectively distinguished from empirical propositions that he was unable to give any other definition than one that appealed to the subjective manner of their formation in our mind. Accordingly, he rightly designated empirical cognitions as those "that can only ever be known from experience; a priori cognitions, however, are designated simply as the opposite of these." This by no means says that a priori cognitions cannot also sometimes be supplied to us by experience, only that they do not necessarily have to be. Thus when KANT stressed so highly that metaphysics should incorporate only a priori cognitions, he really only claimed that one should present in metaphysics only propositions that are according to their nature purely conceptual. This I too require, only I wish to add that, where necessary, we should not hesitate

38

THEORY OF SCIENCE PROPER. Part II§. 416.

to admit into our presentation of this science purely conceptual truths that we have come to know only from experience, and for which we do not yet possess a rigorous a priori proof. Obviously, though, it must be added that one should still seek the objective ground for these propositions. §.416.*

7) There may be sciences which have certain theses in common and even one science wholly contained within another ' As we saw in §§410 and 411, the problem of dividing the entirety of human knowledge into individual sciences should by no means be under- 59 stood to mean that not a single humanly attainable truth, no matter how insignificant, should be left out, as if every one had to be incorporated into some science or other. We must also remark that one can by no means require that this division be carried out in a way that if a truth appears in one field, it must not appear in any other. If one were of a mind to enforce this, and require that every truth be indigenous to at most one science, one would have to approach the division of the entirety of human knowledae in s~ch a w~y that the domains of individual sciences were mutually e:clus1ve. At first glance, this seems to be not only achievable, but also advantageous, since it would give rise to no more sciences than is absolutely necessary to exhaust the noteworthy portion of human knowledge. If we do. not proceed thus, but rather permit the same truth to appear in several sc1en~es, we seem to do something superfluous, needlessly expanding the domams of at least some sciences. Yet closer consideration reveals that ~he expansion of the domains of individual sciences caused by incorporat1~g the same truth into several of them is never a significant disadvantage, smce those who already know a truth from one science will not be Iona detained when finding it dealt with in another; indeed, they may even find the repetition welcome. With the opposite procedure, by contrast, there is the worry that many people would never learn a truth that is of the greatest importance to them (a far greater disadvantage), simply because they do not get around to the one science where it is presented. The most weighty consideration, however, is that there are certain situations in life in which it is clearly necessary to know truths A,B,C, ... , Mand others 60 in which one needs to know other truths M,N, 0, ... , z. Given this, i~ would be unforgivable were we not to incorporate both the former and the latter into their own sciences, the study of which we can recommend to all who find themselves in such circumstances, or can foresee that they 39

THEORY OF SCIENCE PROPER. Part II§. 416.

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soon shall. Now what should we do if it turns out that these sets of truths have one or more truths, such as M, in common? If we wanted to prevent any two sciences from having a truth in common, we would have only two options: either we separate out the truth or truths that the sciences have in common, and present this common part as a separate science, or else make a single science out of the two. I do not deny that the former course is feasible in some cases. If the collections A, B, C, ... , M and M.N, 0, ... , Z have a significant number of theses in common, we can usefully construct a separate science containing them, the study of which can be recommended to people who find themselves in the former as well as those who find themselves in the latter set of circumstances; and if we can be sufficiently sure that both groups will heed our recommendation, we may omit the common part from the two collections A,B,C, ... , M and M,N,O, ... , Z. But this means is not always applicable, because the number of common theses may be so small that not much is gained by separating them out; because one may not always assume that every reader we urge to obtain knowledge from another book will actually do so; and, finally, because the truths that belong exclusively to these collections often have such an intimate connection to the truths common to both sciences that it would be impossible to present the former without adverting to the latter. The second means is still less applicable. For if we wanted to combine all the knowledge that might be necessary in the most varied situations in life merely because each of these situations requires knowledge that is also required in some other, how great a totality would we not have to produce! And would not every reader find far more of what he does not need than of what he does? Would not readers indeed stumble across knowledge that is harmful rather than beneficial to them? Thus, I conclude, among the sciences we introduce, there must inevitably be some which have theses in common. I claim, furthermore, that it is not futile to have sciences such that one is completely contained within the other. It can be the case, namely, that we rightly combine the truths A, B, C, ... , Z into a single science, because there are circumstances in which it is necessary to know all of them; but we may with equal right present only a part of these, A, E, J, ... as a distinct science, because there are circumstances in which it is better to consider these truths alone, or because we may expect that emphasising the truths A, E, J, ... in this way will draw more attention to them, etc. If what has been said so far is correct, then there are three relations sciences may bear to one another with respect to their extensions. First,

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THEORY OF SCIENCE PROPER. Part II§. 417. there are sciences that have not a single truth in common. These may be called separate or completely distinct. But, according to what has been shown, there may also be sciences which, for all their diversity in some parts, nevertheless have some truths in common. These may be called linked or overlapping. Finally, there may and should be sciences where the extension of one is entirely contained within that of the other. The former may be said to be subordinate to the latter or one of its branches. A science that is not subordinate to any higher one may be called a principal 62 science [Hauptwissenschaft]. Geometry and Ethics are a pair of sciences whose extensions are completely separate, while astronomy and geography are overlapping, since both contain theses about the Earth. The catechism for married couples is a science subordinate to special ethics, and exists for good reason alongside the latter. For although we cannot be censured for uniting all the duties of mankind, in whatever varied circumstances, into a single science (namely, special ethics), because there are people, e.g., priests, who must know all of these duties, it is nevertheless appropriate to present certain duties, e.g., those of married couples, in separate treatises, and thus to form the concept of a science dealing with these duties. Pure general ethics may serve as an example of a principal science, for there is to my knowledge no science which contains it as a part. §.417.* 8) One science may depend upon another either/ram the subjective or the objective point of view, or both Even when the truths that belong in a science by virtue of the concept we form of it (i.e., those that are indigenous to the science) do not appear in any other science, they may nonetheless be dependent upon theses from other sciences either in the subjective or in the objective sense, or in both. I say the first when we need the theses of the other science in order to see that the theses of our science are true, i.e., in order to prove them (make certain of them); the second, when the truths of the other science contain the objective grounds of the truths of our science, thus when we need them to objectively ground the latter; the third, finally, when both of these things occur. Now I claim that it should not always be consid- 63 ered a mistake when the boundaries of sciences are fixed in a way that such relations result. It is indeed true that such an arrangement obliges us frequently to refer to theses that do not occur and are not proved in 41

THEORY OF SCIENCE PROPER. Part II §. 418. our book but rather in some other, and that this is more or less unpleasant to most readers. Nor can it be denied that, because our readers do not yet know these other truths, and may, either from lack of opportunity or from lethargy, not come to know them, they will either fail to be sufficiently convinced of the truths we present to them or else fail to gain insight into their objective grounds. But it is impossible in any case to avoid this problem. For if we wanted to determine the content of every science in such a way that every truth required for the proof or objective grounding of its theses also belonged to its content, how immense the presentation of every science would have to become, and how frequently the same theses and proofs would have to be repeated in the most varied sciences! Think of how large a treatise of astronomy would have to grow if we were to adopt all of the theorems it draws upon from mechanics, geometry, and analysis as indigenous to astronomy and requiring proof within that science. But the necessity of proceeding in the way I have described has always been recognised, and most sciences are delimited in a way that makes them dependent upon not only one, but two, three, or more others, either in the subjective or the objective sense, or both. For this reason, I call them dependent or derivative sciences, and those upon which they depend supporting sciences. I call a science that is not dependent upon any other completely independent. I say, for example, that 64 history is a supp01ting science of anthropology, though only subjectively, since the peculiarities of human nature, although not grounded in history, can and frequently must be shown through history. A science which in my opinion depends only objectively on another is the science of space with respect to the science of time. For although it is true that we do not require the properties of time in the least in order to prove the properties of space, if by prove we mean making them certain, the nature of space is nevertheless (in my opinion) objectively grounded in the nature of time. Note. In a narrower sense, a science is called a supporting science only if we usually learn it merely for the sake of the aid it affords us in the pursuit of some other science. In this sense, e.g., chronology, genealogy, heraldry, numismatics, etc., are called supporting sciences of history. §.418.* 9) There may even be sciences which are dependent upon each other

I venture to claim that even if two sciences were specified in a way which resulted in their being mutually dependent, they would not for that reason 42

THEORY OF SCIENCE PROPER. Part II§. 419. merit rejection. Admittedly, the truths of one science which we use to prove or objectively ground the theses of another cannot themselves be proved or grounded by the latter's theses, since this would be contradictory; but nothing prevents certain theses of one science being deduced from certain theses of another, and certain other theses of the latter science deduced from other theses of the former. The only disadvantage of referring back and forth in this way is that the reader might fear that we are leading him in a circle (§371). But this worry can be dealt with by pointing out, whenever necessary, how the theses one calls upon have not 65 been demonstrated from the theses one is in the process of proving, but rather from others. Even among sciences that have attained the highest degree of perfection, some cannot avoid the relation of mutual dependence, as is shown by the example of pure number theory or analysis and the so-called combinatorics or theory of order. No one can say that the domains of these two sciences have been incoITectly determined, yet we must inevitably use theorems of combinatorics when proving certain theorems of analysis (e.g., the theorem about the permutation of factors, or the binomial and polynomial theorems, etc.); and just as inevitably must we call upon analysis to ground various theorems of combinatorics. §.419.* 10) One should not demand that a science which contains a truth also contain its applications

The applications of a truth, i.e., the consequences that follow from it (either objectively or merely subjectively) which have a particular practical usefulness are the principal reason why we become acquainted with the truth. Nothing is more desirable, therefore, than that everyone who learns a truth also come to learn its applications, at least those that may become useful to him given his particular circumstances. Yet it should not be required that the applications of a truth always be included in the science to which the truth itself belongs. For even if the concept of a science is understood in a way such that many of the applications of the truths indigenous to the science do not belong to its content, this need not prevent us from mentioning these applications in our treatise, should we deem these appropriate for the class of readers for whom our book is intended. 66 If, on the contrary, such applications all belonged to the science according to its concept, we could not pass over a single one of them, and would thus infinitely expand the extent of every science. To this it may be added

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THEORY OF SCIENCE PROPER. Part II §. 420. that there are many applications that are well worth knowing but which require a considerable amount of other knowledge in order to be understood or proved. This preliminary knowledge might be presupposed in some, but not in all, of our readers. Or it might happen that an application which is important for one reader is not so for another. Furthermore, people who have an excellent ability for working with truths of a certain kind may not have the gift of devising applications for them. Finally, there are truths that we only accept with complete impartiality because we do not know what applications they might have. Here are reasons enough why it can often be advantageous to separate a truth from its applications, at least far enough to found a particular science for the former but not the latter. Thus it is certainly a good thing that we devote a science exclusively to the truths dealing with the nature of space, namely, geometry, while consigning the applications of these truths, which help to explain the most varied phenomena in nature, to improve our arts and industries, etc., to other sciences. This does not prevent us from incorporating some of these applications which can be easily understood as occasional theses in a presentation of the science of space. §.420.*

11) One should not demand that all the truths of a science depend upon the same objective or subjective principle It was already remarked in §414 that the fact that certain truths can all be derived either subjectively or objectively (i.e., as consequences from 67 their (partial) grounds) from one and the same premise is not a sufficient reason for uniting them in a single science. I now venture to claim that, conversely, it is not a sufficient reason to place truths in different sciences merely because there is no common premise from which they either objectively follow or else are subjectively deducible. In other words, one should not demand that every science have a single (objective or merely subjective) highest principle from which all its truths are derived. For how could one prove the opposite? Could it be that truths that do not have a common objective ground or indeed even truths that do not spring from a common source of knowledge could not in any respect form an intimately connected whole, so that it would be appropriate to form a science to contain them? Could they not merit being united for other reasons, e.g., because it is useful for a certain class of people to know all of them, since knowing some but not the others would either be pointless or dangerous and harmful? History and many other empirical sciences furnish

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THEORY OF SCIENCE PROPER. Part II§. 421. examples of truths that do not spring from the same source of knowledge (unless one wants to play with words and say that they all come from experience), and where there is no single proposition from which all these truths may be derived. Note. It may appear here that I contradict a claim that is made quite generally, since one almost always hears it said that all the truths of a science which is worthy of the name must be comprehended in a single principle-sometimes one also says in a single concept-since otherwise the collection of truths would lack genuine unity. Closer inspection reveals, however, that the word principle is used here in a much wider sense, and at bottom people only require that it be possible to indicate a proposition which precisely determines which truths belong in a particular science and which do not. I too claim this, for certainly the kind of truths be- 68 longing to the content of any science must be determined with the utmost precision by virtue of its concept. §.421.*

12) It is a vety good thing to classify truths according to attributes that can be used to inquire about them If the division of the entire domain of truth into individual sciences is to deliver the benefit of allowing us easily and reliably to discover which of these varied fields contain the truth we wish to know in a particular instance, then the attributes used to determine whether a given truth belongs in this field or that must be of the sort that we are usually acquainted with before we know the truths themselves. If we desire to learn not some truth or other but a particular one, it lies with the nature of the case that we already know an attribute of the truth that makes it precisely this one and no other. If, for example, we want to know the memorable feats of Alexander the Great, we stipulate through this very desire that the truths we seek must concern Alexander. Thus this attribute of the truths is known to us before we know the truths themselves. Now, if among the sciences that we know at least by their concepts there is one which promises to deal with all of the truths with the given attribute or a broader one, we may expect that we shall find the instruction we seek therein. If this is to happen always, or at least frequently, the attributes used to determine whether a truth is dealt with in one science or another must be of a kind that enable us to locate these truths even when they are unknown to us. In the 69 contrary case, the concept of a science or, what amounts to the same, the

45

THEORY OF SCIENCE PROPER. Part II§. 422.

kind of truths that are supposed to be dealt with in the science, depends upon an attribute of these truths that we never or only seldom recognise unless we know the truths themselves, and it is obvious that such a science could never be used by us in order to locate knowledge that we seek. We usually say that an attribute of certain truths which can be recognised without knowing the truths themselves and which is useful for distinguishing these truths from others is an attribute by means of which we may inquire about these truths(§ 145) or, more briefly, though less properly, that it is an attribute according to which we may inquire. Thus one says, e.g., when someone desires to know a truth concerning expressions of quantities, that he asks about quantitative-expressions, and so on. In this way of speaking we may also express the proposition set out above more briefly as follows: "If the division of the entire domain of truths into individual sciences is to help us to seek any individual truth you like, then our division must be based upon our customwy ways of inquiring." Now, since the benefit we attain in this way is one of the most important that the division of the entire domain of truth in individual sciences can deliver (§409), we will do well to use this basis of division whenever other aims do not require us to do otherwise. Thus the historical sciences have been divided according to peoples and periods, since these are features we usually know in advance when seeking historical truths. In making such inquiries, we for the most part wish to learn the course of events in a situation where at least the people concerned or the period is known to us. If, by contrast, we were to put forth a concept of a science that con70 tained all the truths that are not consequences of any others, but rather are genuine basic truths, this would be in many respects a most noteworthy science; yet it would rarely be suitable for helping us to locate truths that we seek, since only in the rarest of cases do we know that a truth belongs to the class of basic truths before we know the truth itself.

THEORY OF SCIENCE PROPER. Part II §. 423.

ness of another thinking being the intuitions are different. The same cannot be said about the concepts that occur as constituents in propositions. When we become aware that a certain class of truths may lay claim to a certain concept as their exclusive property and when, in addition, this concept is not composed of parts which might well occur frequently elsewhere in different combinations, but rather is a simple concept, this circumstance must always appear to us to be highly significant, and as a reason speaking in favour of making this species of truths into the object of a special science, provided that other circumstances do not forbid this. For when truths contain a particular concept which does not occur elsewhere, there is already a certain close connection between them. Furthermore, it must be the case that in the entire collection of such truths there will be at least one which stands in the relation of ground to consequence to the others. Thus simply to indicate the objective connection between these truths appropriately, it will be necessary to consider them as a whole of a special kind. On account of this very connection, we may also surmise that the objects these truths deal with will belong together to a greater or lesser extent, that the discovery and elaboration of these truths will require similar talents, and that they will be applicable in similar circumstances. Thus we will do well to raise the collection of these truths to a special science, provided that they are not too numerous and no other circumstances forbid uniting them in this way. §.423.* 14) For eve1y inqui1y there is a place in a science where it may most fruitjit!ly be presented

From the remarks of §410 the reader will gather for himself that the intuitions contained in a proposition do not deserve to be seriously considered when dividing the entire domain of human knowledge into individual sciences, since every time such propositions are grasped in the conscious-

When we formulate a problem aimed at the discovery of new truths, the science in which it is stated is not a matter of indifference. The mere circumstance that we think that the inquiry in question belongs essentially in a particular science already contains the assumption that all the truths to which we might be led to in our inquiry are constituted as they must be to belong to the science according to its concept. Thus even if the problem is itself capable of leading us to many other useful truths, it is to be expected that in attempting to solve it in this place we would only look at truths that belong to the science we are currently studying. Observations of other kinds, no matter how near we may be to making them, will be missed, since from the point of view we occupy when considering the matter they are hidden from us. Accordingly, if we wish every

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§.422.* 13) If a pure concept, particularly a simple one, occurs exclusively in certain truths, then one may expect that these truths deserve to be united in a single science

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THEORY OF SCIENCE PROPER. Part II§. 424.

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humanly useful truth to come to light sooner rather than later, we must, when undertaking the division of all truths into individual sciences, indicate a place for each inquiry in some science, the concept of which is wide enough that we may be assured that none of the truths to which the inquiry may lead shall be excluded. Rather than giving a number of examples, I shall provide only one, drawn from a science that may boast of the greatest perfection. The investigation of the various ways in which a given multitude of things (elements) a, b, c, ... may be combined with one another was, as is well known, first considered a problem belonging to analysis (the theory of numbers), and the immediate consequence of this limited point of view was that no one thought of ways of combining the given elements other than those by means of which they may be made to form a sequence, i.e., one assumed that among the things combined one always had to be the first, another the second, yet another the third, and so on. All the same, the reader will concede that there are many other ways of combining things that certainly merit consideration on account of their frequent occurrence. Thus, e.g., with two or more things a, b, c, ... a new one l may certainly be combined in such a way that the relations between I and each of the individual things a, b, c, ... are equal. We encounter examples of such combinations in geometry, for instance, when we are given two points a, b and asked to find a third c which bears the same relation (lies at the same distance) to each. I believe I can show that many important theorems of geometry, if rigorously presented, that is, derived from their objective grounds, require certain propositions belonging to Syntactics, which have been completely overlooked in previous treatises of the science. In Syllogistics, too, if it is approached from a more general point of view, theorems of Syntactics are required, as may be seen even in some of the inferences included in my presentation. §.424. Investigating whether a given science meets its pwpose

After these preliminaries, we shall now attempt to solve the three problems of this part (indicated in §409) as well as we can. If we must judge whether a science, the concept of which we have before us, should be accepted in the ranks of the sciences, developed, and presented in special treatises, we must investigate: (1) whether the given concept of a special kind of truths that are supposed to appear in the science is emp(v, i.e., whether there are in fact some truths of this sort; but 48

THEORY OF SCIENCE PROPER. Part II§. 424. also (2) whether these truths, or at least some of them, belong to the class of those that are humanly attainable, and (3) at the same time noteworthy; finally (4) whether they are constituted in such a way that they can be presented in writing and thereby communicated to those who do not yet know them. It is obvious that if any one of these four conditions is not met, the proposed science must be rejected (§410). If, however, we find that all these conditions are met, we may inquire, further: (5) whether the collection of these truths would be too large to constitute a single science (§412), i.e., whether there are too many of them for the cognitive faculties of any man to comprehend. If this is the case, and should it appear that the aims of making these truths easier to locate, rightly understand, or apply (§419, 421) do not require that they be united in such a whole, it would be decided that the proposed science is misguided. (6) Even if its extent is not so great as to surpass human cognitive abilities, it still does not follow that the proposed science is appropriate. This would only be the case if some benefit comes of uniting these truths, and indeed a greater benefit than would have been produced had some of them been omitted, 74 or others added. This, then, is the question we must look into. To this end, we must consider whether establishing this science, i.e., uniting all and only the truths of this kind in a treatise, would bring the benefit of facilitating our search for such truths whenever we have need of them. We know from §421 that this is only the case if we have only taken account of attributes of truths that are usually known to us even when we do not know the truths themselves. Thus we must see if this is the case here and indeed to such an extent that any alteration of the extent of the science, either expanding or contracting it, would reduce the benefit. We must also consider whether there is a set of circumstances that occurs frequently in life where it would be necessary for people to know all and only these truths, in which case it is to be hoped that the introduction of our science will bring about a more complete acquaintance with these truths for precisely those people who need them. Furthermore, we must investigate whether uniting these truths would bring the advantage of making them more easily comprehensible or convincing, or whether one would do well to unite these and only these truths, either on account of a certain important peculiarity of precisely these truths, because it would illuminate the manner of their connection, because it might be expected that the gaps in our knowledge of this field would be more obvious, thus facilitating inquiries to fill them, or because the investigations we combine in this way require similar mental abilities, preliminary knowledge, and external

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THEORY OF SCIENCE PROPER. Part II§. 425. circumstances, and so on. (7) If it should turn out that the given delimitation of the domain of our science is advantageous in every respect, then obviously the science merits approval. If, however, there are both advantages and disadvantages (and this is the most common case), nothing can 75 be done except to weigh these in order to determine which predominate. §.425. Devising a concept of a suitable science

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When we are confronted with the second problem (§409), that of thinking up a concept of a science that would be worth pursuing and presenting, it will most easily be solved by directing our attention to various truths that are humanly attainable, noteworthy and communicable through writing, and asking the following questions about them: are there several other truths it would be important for us to know in situations where it is important to know the truths in question, or which, if presented along with them, would make all of them more comprehensible and convincing, or which it would be pleasant to learn alongside the truths in question, or which would better allow us to perceive the connection among them? Would presenting these truths alongside the given ones stimulate and provide opportunities for new discoveries? Would the same equipment, previous knowledge, and opportunities, etc., be required to discover them? If we answer in the affirmative to one or more of these questions, we then proceed to inquire what characteristics these truths have in common, and seek especially to discover a characteristic that can be used to inquire after them, i.e., one which can be recognised without knowing the truths themselves. If we succeed in discovering such a characteristic, we form the concept of a science in which all the truths possessing this characteristic are united, and then investigate, following the rules of §424, whether this concept of a science is appropriate, or whether it needs to be broadened or narrowed. To give but a single example, anyone who reflects on the origin of his own judgements will hit upon some noteworthy truths of the sort I considered in §303. But he will immediately become aware that the concept of a science which would explain the origin of all our judgements would be too broad. Seeking a suitable restriction of this domain, he will perhaps hit upon the concept of a science that explains the origin of all the empirical judgements usually thought to be immediate, since we are not aware of their being mediated, and closer examination may well confirm the appropriateness of such a science, which one might call the theory of experience.

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THEORY OF SCIENCE PROPER. Part II§. 426. §.426. Division of the entire domain of truth into individual sciences The last problem of §409 is no doubt the most difficult. It requires that we indicate not just this or that science, but rather all sciences that deserve to be pursued, and at the same time show that together their domains exhaust the sum of all humanly attainable truths. It has already been said that I shall not attempt to solve this problem here, but rather shall only present the rules which must be followed in solving it. According to §411, we may only demand that every humanly attainable and noteworthy truth that is communicable by writing appear as indigenous to at least one science, but not that there be only one such science for every such truth. From this one may gather the sense in which it is said that the domains of the individual sciences must together embrace the sum of all humanly attainable truths, namely, only in such a way that every noteworthy truth lies within the domain of at least one of these sciences, not that it may not also appear within the domain of another science. Thus these domains need not be mutually exclusive, but rather may have parts in common, indeed the domain of one science may lie wholly within that of another. Nor do we require that all the subordinate sciences be enumerated, but 77 rather only that as many principal sciences be indicated as are necessary to exhaust the domain of humanly attainable truths. Now in order to meet these requirements, we must: 1) strive to discover a difference between truths which, though it may not yet be ready for use in determining the domain of an individual science because the number of truths that would have to be united would be much too large, at least does not separate truths that belong together in a science. We may believe that we have found such a difference when none of the grounds for uniting truths in a science speaks against our division. It seems to me that the division between conceptual and intuitive truths is an example of this. For not only do these truths differ essentially with respect to their objective connection, in that the former can never be grounded in the latter, but also the discovery and elaboration of one of these kinds of truth requires completely different materials, previous knowledge and aids than the other. For this reason, as soon as people became better acquainted with this distinction, they made it the basis for the division of human knowledge into individual sciences. Sciences dealing with purely conceptual truths have for this reason been called conceptual

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THEORY OF SCIENCE PROPER. Part II §. 426. or a priori sciences, while those that deal with empirical propositions are called empirical or experiential sciences. 2) If such a division has been found, we seek to subdivide each of the divisions in the same way, and continue until we arrive at collections of truths that are not too great to become the object of an individual science. 78 Here we must proceed according to the precepts of §424. 3) In order to discover the sort of differences between truths that are required for this end, we must fix our eye upon the following factors: (a) the constituents of which a given truth is composed, and above all the simple concepts it contains. If we notice a concept in one truth that does not occur in all other truths, it is worth inquiring further (§422) whether all the truths that have this concept in common are so intimately connected that uniting them all in a science would not be misguided. (b) Equal attention should be paid to the object the given truths deals with. For even though we are not justified, according to §414, in immediately uniting all the truths dealing with the same object in their own science, this may indeed happen in many cases, especially when, in addition to the subject, some further attribute is indicated as a common characteristic of these objects. (c) Very special attention should be paid to the applications that flow from a given truth, and the particular circumstances in life in which these applications may be made. For even though, according to §419, is is not always necessary to include the applications of a given truth in the science to which the latter belongs, nevertheless one of the most important benefits at which we aim in dividing the entirety of human know ledge into individual sciences is that of uniting all the truths that anyone needs in his particular situation in a special science (§409). Thus whether a certain truth may be applied in this or that set of circumstances deserves serious consideration in this business. (d) The same holds for the question of the aptitudes, previous knmvledge, and aids which are necessary for the discovery and development of a certain class of truths. For, as was pointed out in §409, it is of great benefit if truths that are equal 79 in these respects are united. (e) Finally, with truths that we may not look upon as already known, which the reader will first learn from us, we will do well (§421) to inquire which attribute of these truths can best be used as a characteristic by someone who does not know these truths in order to seek them when necessary. If we have several such characteristics in mind, we form the concept of a science that contains all the truths that have these characteristics in common, and proceed to investigate whether

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THEORY OF SCIENCE PROPER. Part II§. 426. such a science would be appropriate in all other respects, following the instructions of §423. 4) If we discover several ways that the domain of truth or some part of it can be divided, and if each of these divisions has its own advantages, nothing prevents us from declaring all of them to be valid, even if this leads to the appearance of sciences whose domains are composed uniquely of parts of the domains of other sciences (§416). Thus, e.g., the division of truths into conceptual and empirical mentioned in no. 1 should not hinder us from effecting a second division between theoreticctl or speculative sciences on the one hand and practical or technical sciences on the other, even though the latter division would cut across the former in many places. 5) It should not be forgotten that in pursuing the business of dividing the entire domain of humanly attainable truth into individual sciences we should not only fix our eye on the part of these truths that is already known to us; rather, we should also pay heed to those that are so far unknown, but which we may hope to discover sooner or later. Encouraging and providing occasion for investigations aimed at discovering such truths by introducing sciences to which they belong is not the least benefit that may come of instituting various sciences (§414). Here, however, we must be especially careful (§423) to ensure that every search for useful truths is 80 assigned a place in a science that in no way narrows the point of view from which the search is to be undertaken, so that one may rely upon it that it is here or nowhere that such an inquiry can be most fruitfully pursued. According to this rule, in particular, we must always judge when instructions for a certain procedure should be considered part of special ethics and dealt with accordingly, or when they merit the institution of a new science existing in its own right. The latter will be the case whenever the instructions in question cannot be communicated without first having set out a good deal of purely theoretical knowledge. Thus, it would be quite inappropriate to deal (as some would have it) with the entirety of logic, pedagogy, agronomy, the theories of the arts and sciences, etc., as mere branches of ethics. 6) Finally, the question arises of how we may be assured that we have succeeded in our task at least to the extent that no more principal sciences are wanting. I know of no other means of acquiring this assurance than when repeated experiments show that every useful truth we can think of belongs to one or another of the sciences we have set out. Some may

53

THEORY OF SCIENCE PROPER. Part II§. 427.

think that it would be possible to have a science which would, according to its concept, contain all the truths that do not belong to the content of any other science. But this cannot happen, since the appropriateness of such a science cannot be justified according to the rules set out above. Note. In discovering the various sciences whose concepts have been introduced among us to the present day, it seems to me that the human understanding has shown a no less brilliant side of its nature than it did in discovering the truths that the sciences were devised to present. It would 81 be difficult to show that even a single one of the sciences whose concept has not merely been suggested by some scholar but approved and accepted by all deserves to be either rejected or restricted. People may well never have been distinctly aware of the rules they followed here, but their cotTectness is vouched for by their results, that is, the fact that people only seldom disputed for long over the boundaries of a science, and still less often found it necessary to abandon a determination of boundaries once it was settled upon. The reader must decide whether I have succeeded in indicating these rules in accordance with truth. Most will, I fear, have expected much more detailed instructions concerning such a large and difficult problem as that of dividing the entire domain of truth into individual sciences. Some will also demand rules of a completely different kind, rules that are drawn not from the merely subjective needs of men but from the objective constitution of the truths themselves. I am therefore resigned to the fact that what I have said here will be looked upon with scorn by some, and mocked by others. I wish from my heart, however, that they would rather find something that not only appears to be learned, but is also c01Tect and useful. §.427. Other views In previous treatises of logic, even in those whose authors recognised clearly enough that logic is simply a theory of science, the topics I speak of here, namely, the way in which the entire domain of truth should be divided into various sciences, as well as the question of how to judge whether or not a science is appropriate, have been almost universally passed over in silence. ARISTOTLE touches upon this subject with the brevity he so cherished in An. post., I, ch. 28: M(cx OE smcn~µrl so-clv ~

THEORY OF SCIENCE PROPER. Part II§. 427. µ~-cc: EX -ci.0v CX\Jc0)v, µ·~·W 2:-cc:pm EX -c6)v E-cspc.)v. 1 Thus he merely re- 82

quires that the truths combined in a science be homogeneous. But how indeterminate this concept of homogeneity is! In modern treatises the only claim that may be regarded as pertinent here is the proposition that in every science there must be only a single principle from which all these truths are deducible. This claim would make it possible to determine not only the number of the sciences but also the extent of each, since there would be just as many sciences as there are such highest principles, and each science would contain only the truths deducible from their paiticular highest principle. This was the view of RUDIGER (De S. V. et F, IV, 1), MAAB (§447), METZ (§23), GERLACH (§281), KLEIN (p. 203) and others. The reason usually adduced to support this claim is that without the presence of such a highest principle from which all the individual truths may be deduced, there would be no connection between these truths and hence no unity. This reason has often been quite correctly refuted, e.g., already by SAVONAROLA (Comp. L., VIII, 47), by remarking that such unity might be attained in other ways, for instance, through the homogeneity of the truths (the unity of the object they concern). ARISTOTLE seems not to have sensed the necessity of such a principle, for in every science he distinguishes (An. Post., I, 10) proper and common principles and says (An. Priot:, I, 30) that there are many of the former in every science. Among the moderns CHRISTIAN WEIB (Log., §318), STIEDENROTH (Theorie des Wissens, p. 47), ESSER (L., p. 243), TWESTEN (L., §257), and others have declared the requirement of a single principle for each science unjust, and rightly pointed out that on this view mathematics, logic, and philosophy would not deserve to be called sciences. Incidentally, it was already noted in §420 that many take the above claim in a sense according to which I too could accept it. In CRUSIUS' Weg zur 83 Gewifiheit, §§23-25, there is a brief "Theory of the delimitation of sciences," which I shall discuss only briefly here. According to CR., "there can be only four reasons for uniting several truths in one science, namely: (a) because these truths stand under a common concept, whose species and individua one now intends to consider. This is the case in physics and geometry. (b) Because all of these truths are parts or determinations or results of a real whole, as in physiology and thelematology. (c) Be1

hos ysvous, oocx sx -cwv i1pc0-c0w o6yxc:t-cm, xcxt µspri fo-clv ~ i1Cf0YJ wi'.mw xcxu' w'rccx. 'E-cspcx OE bto-c~µY) so-clv hspcxs, OCTL1)V ext ctpxcxl,

A single science is of a single genus-however many items result from the first principles and are parts or essential attributes of these. One science differs from another when their principles neither come from the same items nor those of the one from those of the other.

54

55

.

THEORY OF SCIENCE PROPER. Part II§. 427.

cause these truths are derived from a universally determined principle, as in jurisprudence. ( d) because they are all means to a given end, as in ethics and algebra. Whether these grounds are sufficient and reasonable is determined by the following rules: (ex) The extent of the science must be large enough, i.e., sufficiently rich in truths that surpass the common understanding. (~) The segregation of certain truths in a science must in every case be of real use for the knowledge of learned truths, namely, that such truths are aggregated which must be thought together for the sake of some useful end. Among these the most general end consists in obtaining a comprehensive understanding of a subject from a certain point of view, where every truth is found in a place where it is most easily proved or where its ground can most readily be seen. (y) One should not depart from accepted usage without weighty reasons. (0) But if a noteworthy class of truths would not be sufficiently noticed, or there would be confusion concerning certain truths when this is harmful, it is rational to favour the aims of thoroughness over custom. (s) Since the application of these rules is based upon certain postulatis, namely, on factors that honest and knowledgeable people are indeed aware of, but cannot prove to others, every scholar must have so much love of truth that he decides matters im84 partially. But even when all of this is observed, there is still much freedom of choice in fixing the number and boundaries of the sciences." Everyone will see that these claims come quite close to what was said above. Nevertheless, I must point out that the reason for uniting truths in a science adduced in (a) in fact encompasses all the following ones, so that the latter should have been considered subordinated to rather than coordinated with the former. For if truths may be considered parts of a real whole, or consequences flowing from a single basic truth, or means for a certain end, do they not stand under a common concept? I also doubt whether (as is claimed in ex) the extent of a science is only large enough when it is rich in truths that surpass the common understanding. No one would say that the pure theory of time was rich in such truths, yet it pe1fectly merits the status of a special science. Only what is said in ~ and o seems to me to reveal the true factors that lead us to determine the extensions of sciences in a certain way. And when it is said at the end that much remains open to choice, I can at most concede this with respect to the subordinate sciences. For how far one proceeds in subdividing, and whether one looks upon a number of truths merely as a branch of a science or as a science in its own right, can in fact sometimes be immaterial, and thus arbitrary. But the way that the principal sciences are delimited does not seem to

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THEORY OF SCIENCE PROPER. Part II§. 427.

leave much room for choice, as the widespread agreement on this point indicates. AMPERE, in his Essai sur la Philosophie des Sciences (Paris, 1834, Vol. 1) attempts a classification of all the sciences, in which he defines concepts for many new sciences; yet I did not find anything there on the rules to be followed in this business. 85

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THEORY OF SCIENCE PROPER. Part III §. 429.

PART III

On the Choice of a Class of Readers for a Treatise §.428.*

Consequences

c~f various

ways of determining our class of readers

If it turns out that a science is appropriate, we then consider whether the special class of readers for which we intend to compose our treatise is appropriately chosen or needs to be altered. For reasons similar to those mentioned in §409, I shall begin with a brief survey of the advantages and disadvantages of determining the class of readers for a book in one way or another. 1) If the class is suitably determined, and there are a good number of readers of the kind our book requires, then this brings the benefit of helping many people at once. The exact opposite will occur if we think of readers of a sort that do not exist at all, or else may not be expected to read our book. 2) If it is not possible to write in one and the same book in a way that is suitable for the needs and mental capacity of all who want to learn the science, an appropriate choice can at least permit us to do so for those who need this science the most. With an unsuitable choice, we might well 86 completely miss out on this opportunity. 3) A well-considered choice can unite readers who are quite similar in terms of their previous knowledge and needs, and it is easy to see that we can satisfy these readers better and at less expense than would have been the case had we called together readers who are far too different. Explanations that are salutary, indeed necessary for one person may well be quite dispensable or even vexatious for another; this is all but inevitable in dealing with religious subjects, if readers of quite different levels of education are invited. Concepts that must be presupposed as known to some people lest we offend them must be explained in detail to others; proofs that one person finds quite illuminating are incomprehensible to the other. One person welcomes it if we attempt to indicate the ground of each truth we present, indeed he requires this of us, while others become bored or indeed confused when we do so. One person expects us to take account of this or that doubt or objection, or to point out applications to this or that situation, while all of this is dispensable for another. With this

58

in mind, we might want to write some sections of our book for one sort of reader and others for other sorts, hoping in this way to meet the needs of all; but this would have at least the disadvantage that every reader would find that our book contained a great deal that was dispensable for him, something which would make it unpleasant to use and more costly to produce. 4) Finally, we might also choose a class of readers that is best suited to our own abilities, a class of readers we are best suited to satisfy; or, by contrast, we might have the misfortune to write for a class of readers that is not suited to our abilities and circumstances-for example, readers we do not know well enough, or whose concepts and ways of thought we 87 cannot adjust ourselves to, or who require a more comprehensive, more deeply grounded instruction than we are able to provide, etc. We cannot satisfy such readers, while we might well have been able to write a quite agreeable book for another circle of readers. §.429.*

Rules for judging the appropriateness of a given class of readers 1) From what has been said, it is clear that there are four points which must be considered when we want to judge whether the class of readers we have chosen for our book is appropriate. We must consider, namely: (a) how numerous this class is, i.e., how many people have the assumed attributes, and may be expected to get a hold of our book and read it? (b) How useful does our book promise to be for those of them who desire instruction in our science? (c) What further benefits may come from the circumstance that we bring together readers with such similar previous knowledge and requirements? And how much more useful could our work become for each of them due to this circumstance? (d) What is the relation between the task we set ourselves in writing for readers of this class to our own energy and abilities? Are we better suited to this task than others? 2) If the class of readers for whom we intend to write is so constituted that we would decrease the benefits or at least not increase them were we to alter it, then it is doubtless appropriate. But this happens only seldom; most often, a change would be beneficial in some respects, harmful in 88 others. Here we must compare and weigh the opposing advantages, and opt for the ones that seem most important. 3) When it may be expected that if we ourselves do not undertake to compose a suitable treatise for a certain class of readers, someone else 59

THEORY OF SCIENCE PROPER. Part III §. 430. would take it upon himself to do so, factor lb may be set aside completely, and we need only consider a, c, and d. If the number of readers remaining in our class when we only include those who are sufficiently alike in their previous knowledge and requirements is large enough to justify the trouble of composing a treatise and the expense of printing a special book for it, we need not consider factor a, but need only take c and d into account. If, finally, the question is merely whether a given class of readers is appropriate, regardless of who will undertake to write for them, only a, b and c must be considered; and under this last condition, the given class will be all the more appropriate, the more its members are assumed to be alike with respect to their previous knowledge and needs. For the common good it is all the more beneficial the more different classes we make, supposing that we form them according to the precepts set out above and take care that books that are as suitable as possible are devised for each. §.430. Several classes of readers that must be distinguished in almost all treatises When the small number of readers or some other circumstance prevents us from introducing as many classes as the diversity in previous knowledge and needs makes desirable, we should nevertheless distinguish at 89 least two or three classes for any science of considerable extent, especially if it is generally useful, and compose just as many kinds of treatises for them. l) First, there is for every science a number, perhaps small, of people who want to receive complete instruction, who desire a book which presents all the truths pursued in the science that are known so far. All the people, namely, who intend to further develop the science as specialists in the field rightly demand that nothing of what others have discovered before them be held back, regardless of whether it has found even the slightest application, provided only that it is possible that it may someday lead to something beneficial. Treatises written for such readers may be called comprehensive or scholarly. 2) If a science is of considerable extent, and contains a large number of truths which must be presented in scholarly treatises even though they have found no application so far, there may be a class of people who would like to find a treatise presenting only a part of these truths, namely,

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THEORY OF SCIENCE PROPER. Part III§. 431. those that have known applications. Among these are all those who intend to learn and apply such a science for the ends of life. Thus, e.g., a conscientious doctor who does not pursue his science as a scholar, but intends to apply it in life, wishes us to communicate to him everything in his field that is known and useful. We can, however, spare him the truths that do not yet have any known applications, since learning the former truths already demands a great deal of time and energy. Thus one should compose treatises for sciences of considerable extent which omit subjects 90 that are of purely scholarly importance, and contain only what is known to be applicable. Such treatises are said to be composed for practitioners. 3) Finally, with sciences that are not only of considerable extent but also contain some truths that everyone needs to know, there is a third, quite numerous class of readers who desire a book that presents neither everything known in this science, nor everything known to be applicable, but rather only as much as could be learned through an appropriate use of their time without preventing them from doing something even more useful. One may say that books of this kind contain only what is generally useful or applicable. As long as such books have not yet been composed for every science and distributed in sufficient numbers, one will wish in vain for the fortunate day when everyone shall know what is most beneficial for him, and when for this very reason millions of people will no longer make themselves unhappy simply because they did not know what they should and easily could have learned. The hefty works required for readers of the second and especially the first class could never find their way into everyone's hands, nor is it convenient or feasible for many other reasons to ask everyone to extract from such books precisely what is useful for him. §.431. The most common mistakes made in carrying out this task I remarked at the end of the previous part that people had followed the rules according to which one should proceed in dividing the entirety of human knowledge into individual sciences quite well, though they were hardly aware of them. The same can scarcely be said for the subject of which I presently speak. People have frequently come up short in determining the class of readers that a book they are composing is intended for, and continue to do so. Countless books are published every year which must be failures simply because the author either did not determine the 61

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THEORY OF SCIENCE PROPER. Part III §. 431.

PART IV circle of readers for his book at all, or else did so in an inappropriate way. It will be useful to draw attention to the most common mistakes made in this business. (1) It happens all too often that the author of a book never becomes distinctly aware of the question of which readers he intends his book for. Thus he sometimes writes things that would have been suitable had he been thinking of one class of readers, and sometimes things that would be suitable for another class, so that in the end his book is not fully satisfactory for anyone. (2) Other authors do think of a specific class of readers, but imagine them to be of a sort that is either to be found nowhere, or at least not among those who will obtain the book. What is more common than an author who imagines his readers to be far more eager to learn his subject than they actually are, or takes them to have more patience and perseverance, as well as more previous knowledge, than they actually do? How many authors suppose that all the premises that they take to be indisputable are so in the eyes of their readers as well? etc. (3) Still other writers, either from vanity or the good intention to be of greater use, aim at a number of readers that is too large. They want to compose their book in a way that will make it equally useful to people of the most diverse previous knowledge and requirements, with one or the other or both of the following results: either readers find the book too vast, since each of them finds a good deal there that could have been left out, or else they find it incomplete, since each finds that something that 92 would have been useful to him is lacking.

On the Propositions which Should Appear in a Treatise §.432.*

Content and chapters of this part Once we know that the science we intend to deal with as well as the class of readers have been suitably chosen, we may proceed to determine the content our book should have. But since every book is according to its concept a collection of signs, namely, written signs, which have been chosen by us as a means of awakening certain ideas, it is obvious that we cannot answer the question of which signs our book is to be filled with until we have decided which ideas we wish to awaken in the minds of our readers. The present part, which deals with this question, will have to be much more lengthy than the two previous parts. I shall begin by presenting some of the general considerations intended to prove that all of the signs appropriately included in a treatise refer uniquely to complete propositions or, strictly speaking, to ideas of propositions. Then I shall attempt to determine more precisely the various ways such propositions may appear in a treatise, how readers may use them, and the varied relations in which they may stand to the science in question. Since it shall turn out that, from this last point of view, there are three kinds of propositions which occur in almost all treatises, namely, essential, supporting, and 93 merely occasional propositions, I shall set out in three separate chapters the rules according to which one should judge which essential, supporting or occasional propositions belong in a treatise. Finally, there are three other kinds of propositions and collections thereof. These can occur so frequently in treatises for such varied reasons that they may sometimes be counted among the essential, sometimes among the supporting, and sometimes among the occasional propositions. Their particularity is thus not determined from the point of view considered above, but rather from another. I shall therefore deal with these in a fourth chapter, adding the rules governing how these parts should be composed. §.433.

The signs we use in a treatise must refer immediately to complete propositions A treatise is supposed to be a book that is suitable for instructing us in a certain science. Now to instruct means to bring it about that we acquire

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THEORY OF SCIENCE PROPER. Part IV §. 433. knowledge we otherwise would not have. And knowledge can only arise in us if judgements do. Judgements, finally, can indeed be brought about in countless ways; but if one speaks only of the changes that must occur in one's mind, judgements can only arise in two ways: either judgements we have previously formed, and from which the given judgement may be derived, are reawakened in us, or else various kinds of ideas that our power of judgement uses to form certain judgements are awakened in us. A book, since it is nothing other than a collection of signs, can only give rise to knowledge in us in the latter way, namely, in that our looking upon the signs it contains awakens many ideas in us. If this is not to occur merely by chance, and thus imperfectly, the ideas one relies on here, i.e., 94 the ideas that are produced in us by looking at the signs in the book, must be precisely those that the signs are especially suited to awaken, which are called the meanings of the signs. And if these meanings are to prove as suitable as possible for the goal in view here, namely, instruction, I claim that they can only be ideas of complete propositions. Ideas of complete propositions that someone awakens in our mind have the peculiarity of moving us involuntarily to determine whether this proposition is true or false. If this may be judged without much trouble, if the propositions represented are such that their truth is immediately apparent to us, or can easily be recognised from the context in which they appear, these ideas lead us to make these judgements our own. Other ideas, on the contrary, which do not represent propositions, may indeed lead us to form judgements, even quite instructive ones; but this occurs haphazardly, and it is not in the power of the person who brought about these ideas in us through those signs to determine what sort of judgements they may be. Thus, e.g., someone who merely leads us to form the unconnected ideas "soul", "body" and "mortal" may lead us to form the judgement "The soul is not mortal like the body is" or to form the opposite judgement "The soul is just as mortal as the body", or some other judgement altogether. This is why even in everyday discourse we demand that people speak in complete sentences, and pardon the use of single, detached words at most in cases where the circumstances make it sufficiently clear which complete proposition the speaker had in mind. How much more strictly 95 must this demand press on the author of a treatise! Thus everything that is represented through signs in a treatise must be a proposition, though I shall leave it undetermined whether they must all be truths. They need not represent a complete proposition on their own, though they must at least do so in connection with neighbouring signs.

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THEORY OF SCIENCE PROPER. Part IV§. 434.

Note. Perhaps some of my readers will put forward an objection against the claim made here. For there do seem to be cases where the author of a treatise does not aim at setting out entire propositions, but only concepts, e.g., when he defines certain concepts or makes known the signs he will use to represent them. Closer consideration reveals, however, that even in passages where the author of a book seems to deal with mere concepts, propositions, complete propositions, may be recognised. Thus, e.g., every definition contains an entire proposition, namely, a proposition in which we state which concepts we take a certain concept to be composed of; so too we express a complete proposition when we let the reader know that we have chosen this sign for this concept, etc. §.434.*

Various ways propositions can occur in a treatise Now that we know that propositions are the only thing that appear in a treatise, namely, by being represented through the signs occurring there, it is necessary to get to know a little better the various ways that such propositions can appear. 1) In general, we may say that a proposition occurs in a book, that it is contained therein, or that it is raised there if there is a certain specific sign in the book which is suitable for producing an idea of this proposition in the minds of our readers. 2) The mere occurrence of a proposition does not amount to an actual acknowledgment of this proposition on the part of the author; rather, this 96 only happens when we awaken the idea of this proposition in the reader's mind in a way that allows him to gather that we hold it to be true. Understandably, though, we may allow propositions to occur without doing the latter, indeed, we may do the exact opposite, namely, give the reader to understand that we hold it to be false, e.g., if we expressly refute it. 3) Advancing a proposition, in the strict sense of the word, is more than simply raising, and even more than merely acknowledging it. In the broader sense, the mere raising of a proposition is indeed called advancing it. But in the narrower sense in which I intend to take the word here ' I shall say that a proposition is advanced at a certain place in our treatise if at that place we for the first time acknowledge it in such a way that we allow our readers to perceive our expectation that they too will come to accept it with at least a certain degree of confidence, provided they have not done so already. It is obvious that we do not do all of this every time

65

THEORY OF SCIENCE PROPER. Part IV §. 434. we acknowledge a proposition, still less when we merely raise it. Do we not quite often remark that we ourselves hold a certain proposition to be true, even though we do not (at least not yet) expect the reader to do the same? In such cases, we acknowledge the proposition without advancing it at the same time. 4) If a proposition is simply raised at a certain place, without being advanced, I say that we merely mention it. Now this can happen both if 97 we acknowledge it and if we do not. 5) A special kind of mention which is at the same time acknowledgment occurs when we invoke or refer to a proposition. I say that we invoke or refer to a proposition when we awaken an idea of it in a way that conveys that we not only hold it to be true, but also expect that our readers will themselves have been convinced of its truth elsewhere, or shall be convinced of it now. The proposition we refer to at a certain place may or may not be found in some other place in the book. In the latter case, i.e., when we refer to a proposition that we do not advance anywhere, I say that our book presupposes it. 6) If a proposition we have not mentioned must be added in thought to the premises we do mention at some point in order for the conclusion we draw from the latter to be valid, we usually say that we tacitly refer to the added proposition. If the additional proposition is never advanced in our book, we may be said to tacitly presuppose it. By contrast to such tacit reference or presupposition, the kind of reference described in no. 5 might be called explicit reference or presupposition. Whether and when we make use of these varied ways of mentioning a proposition in a treatise must be determined more precisely in this part. I must again point out that it was only for the sake of clearly distinguishing the concepts defined here that I used the above terminology; I by no means demand that these terms always be used in just these senses. Indeed, I will not even do so myself, but rather, for the sake of variety, sometimes use the expressions "mention", "raise", "advance", and so on 98 with the same meaning, provided that I think that the reader may gather with sufficient precision from the context what I mean. Note. The concept of advancing a proposition which I introduce here may give some readers pause. They may ask where in previous treatises, even the best, may one find such a laborious way of presenting propositions? I reply that in all treatises which are composed with some precision, one may find not only propositions advanced as I demand here but also all the other ways of presenting propositions, which can gener-

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THEORY OF SCIENCE PROPER. Part IV §. 435. ally be distinguished clearly enough, though there may occasionally be some obscurity. This is the case, notably, with mathematical treatises, especially those composed according to the old method. Propositions that appear there under the headings: principle, theorem, corollary, etc., are essentially presented in the way I require above in order to say that they are advanced. Or do not the above-mentioned headings, and still more the proofs that accompany the propositions, already indicate that the author not only himself holds these propositions to be true, but also hopes that the reader may accept them with conviction? It is true that there is never an explicit indication of the degree of confidence with which the author accepts such propositions and hopes that the reader will. But this is omitted only because the high degree of confidence with which such propositions may be accepted is quite obvious. In treatises of empirical sciences, e.g., of history, where the degree of reliability of various propositions is quite different and cannot be judged by the reader unless he is informed about it, one usually finds an estimation of this probability (admittedly, only an approximate one, since this is the only kind possible). Finally, no further proof is needed of the fact that in all treatises there appear propositions that the author merely mentions, or which he acknowledges without intending to prove them to the reader, i.e., without advancing them. 99 §.435.* Three ways in which the reader may use the propositions occurring in a treatise It is obvious that the theses we present in a book in one of the ways I have just described will only be useful to readers if they can make appropriate use of them. There are essentially three ways that this can be done, when we are speaking of direct use that is to be of some benefit. (a) First, we may demand that the reader not merely represent a thesis we present to him, but also consider the grounds we adduce for it and, should he find these satisfactory, to repeat the judgement that he is compelled to make as often as necessary in order to impress this truth upon his memory so that it may be recalled at appropriate times. (b) But we may also content ourselves with asking the reader merely to consider the proposition in question several times, not with the intention of retaining it, but only in order to see whether he could accept it, and in particular whether it follows from the grounds adduced for it. (c) Finally, we can include 67

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JOO

something in a book merely in order to provide someone who has not read the book with a place to look in the future should he need to find it. Every other kind of use is either indirect, and involves one of these three kinds, or else is not of the sort we could ask of our readers. Thus, e.g., the actual application of our theses in life is a use that we certainly may demand, but it only occurs if one of the above kinds of use has preceded it. We may and should also desire that our theses will serve for exercise in thinking, that people will, perhaps by discovering something incorrect in them, contribute to the progress of the sciences, but this can only happen if they become acquainted with these theses and examine them. Finally, to give an example of a kind of use that we may not even secretly intend, because it is not morally good, I will mention misunderstanding and misinterpretation. §.436.* Three kinds of relations in which the propositions we intend to present may stand to our science

A third and most important difference among the propositions that may appear in a treatise stems from the relations in which they stand to the science to be presented. It is already quite clear from what has been said on many occasions how mistaken it would be to believe that a wellconstituted treatise of a science can contain no other propositions than those which belong to the science or whose presence is required by the goal that is explicitly indicated in the concept of a treatise. If we were of a mind to include no other propositions in our treatise than those belonging to the science to which it is devoted, we should only very seldom be able to convince the reader of their truth, yet this goal is explicitly indicated in the concept of a treatise. For in order to prove the truth of the propositions of a given science, propositions belonging to other sciences are quite often required. But a treatise that may lay claim to perfection will not merely fulfil the goal indicated in its concept, but will also foster as much good as it can without working against that goal (§395). It may and should accordingly contain a number of other remarks, provided that these are constituted in a way that increases the usefulness of the book in all respects. If we analyse this more carefully, we soon find that there are three kinds of proposition that may claim a place in a well com101 posed treatise: (a) First, there is for every treatise a completely special kind of truths which we promise to impart simply by virtue of the science

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THEORY OF SCIENCE PROPER. Part IV§. 436. to be presented, since we assume the obligation of not overlooking any truth of this species which is known and sufficiently noteworthy. I call such propositions the essential or indigenous propositions of the book, and also, when no misunderstanding is to be feared, the object of the science, though the object of a science in the proper sense of the word is something completely different (§ 12). One might also say that these propositions are proper to the science, if this designation were not better reserved for the theses the science does not share with any other. (b) In order to prove these propositions to the reader with the same degree of probability they have in our eyes, we are almost always obliged to add a greater or lesser number of other propositions that do not belong to the science at all. I permit myself to call these supporting propositions. (c) Finally, there can also be quite a few propositions which, though they belong neither to the first or second class, nevertheless bring some benefit or other through their presence, and may thus not unjustly be included, e.g., propositions which make the reader desire to attend to our instruction, or which show useful applications of what has been learned, etc. I shall call this kind of propositions occasional. In a treatise of the theory of space, e.g., we must present all of the known truths expressing attributes of space that are sufficiently noteworthy for our readers simply by virtue of the concept we have of this science. Thus I say that these truths are indigenous or essential to such a treatise. Alongside these we must also present many other truths which, though they do not express any attributes of space, are nevertheless necessary to produce conviction in the former, e.g., the proposition that equals added to equals produce equal sums, and the like. These are the supporting truths. Finally, we shall also add re- 102 marks which concern neither the attributes of space nor serve to prove propositions about space, but rather deliver some other benefit, e.g., noteworthy applications, reports on the discoverer of a proposition, and so on. I say of such truths that they appear merely occasionally.

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Chapter I

On the Essential Propositions of a Treatise §.437 In every treatise some propositions must be advanced as essential Since every treatise must be a written composition from which the most noteworthy truths of a science may be learned, it is obvious that in such a book we must put forward at least some propositions that belong to this science in such a way that the reader will become convinced of their truth, provided he is not so already. To this end it is, if not necessary, at least beneficial that we leave the reader in no doubt concerning what we think of these propositions, that we quite clearly give him to understand the degree to which we are convinced of their truth, and that we are presenting them to him for this very reason. If we proceed in this way, however, one is fully justified, according to the definition given in §434, in saying that these propositions are advanced by us. Thus it is clear that in every treatise some theses must be advanced. This by no means says that we must advance all of the truths belonging to our science that we mention in our book. There can be circumstances (as we shall see more clearly in the 103 sequel) which lead us merely to mention some theses, without expecting the reader to accept them as true based on our say-so. §.438. How do we judge whether a given proposition belongs to our science? Before we may decide whether it is worth advancing a given proposition in our book as essential, we must first investigate whether it may be considered a truth belonging to our science. Understandably, this depends on two circumstances: (a) whether the proposition is true and (b) whether it belongs to the species of truths encompassed by our science according to its concept. The investigation of the first question occurs following the rules set out in §369. To look into the second, however, it is first necessary to bring the concept of our science to the level of distinct awareness. This concept already contains the species of truths that belong to the science, and in most cases it will then be easy to recognise whether

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or not a given proposition is of this species. Thus, e.g., nothing is easier than to judge whether a given proposition belongs to the science of space once we know that according to its concept this science is supposed to contain all the truths that express attributes of space. For then we need only consider whether or not the given proposition expresses an attribute of space. In particular cases, though, it may be difficult to decide this question properly, namely, when the domain of the science is determined according to the objects certain truths deal with, and when it is doubtful whether the object the given proposition deals with really belongs to this species of objects-either because we do not yet know the object well enough, or because the species itself is not sharply delimited. Thus, e.g., it is doubtful whether the description of a given body belongs in natural history when, due to lack of knowledge, we are not certain whether it is a 104 product of nature or of human art, etc. Zoophytes, e.g., cause some embarrassment for Botany, as do Phytozoons for Zoology, since the division we assume to exist between plant and animal is itself somewhat shaky. It goes without saying that in such cases of uncertainty we err on the side of excess, i.e., we incorporate rather than omit a noteworthy truth when we are in some doubt concerning whether it belongs to our science. For it is obviously a smaller error to have learned something notable in the wrong place than not to have learned it at all. §.439. What does it mean to say that a proposition is sufficiently noteworthy? According to the definition of §393, we need not advance all the truths belonging to a science in a treatise, though we must advance all that are known and worth presenting. Now from the principle set out in §395 it follows that a truth is only worth putting forward if we can be assured that the benefits of presenting it outweigh the disadvantages this would cause. If we call this attribute of propositions their SL{fficient noteworthiness, we may certainly say that among all the truths in the domain of our science that are known to us, we should single out those that are sufficiently noteworthy. When using this manner of speaking, one should not apply "note" to the reader of the book, i.e., take it to mean that only if a reader should take note of a truth, i.e., impress it upon his memory, does it deserve to be presented. For we are not justified in demanding that the reader impress upon his memory all the truths we include in our book; indeed, we are not even justified in demanding this for the essential I OS 71

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 441, 442.

propositions. Rather, as we have seen in §433, theses may be advanced in order that the reader may consider them once or twice, or indeed skim over them, only using them when the time comes to apply them. Thus we must understand note to mean simply that the author takes note of them, i.e., writes them clown, so that noteworthy simply means worthy of being written clown in our book.

the means they used to progress. But since the number of useful things that human industry daily requires increases continually while the number we can retain in memory never surpasses a certain limit, one sees that many things that were once important enough for us to recommend that the reader impress them upon his memory do not remain so for all time. Rather, we find that gradually room must be made for other, even more important truths-not to mention that there are truths whose usefulness is based upon certain circumstances, and thus begins and ends with the latter. Thus cumbersome methods of calculation are replaced when better ones are discovered, and medicines for diseases that no longer occur lose their interest for practising physicians.

§.440.*

When is a proposition important enough to just(fy the demand that the reader impress it upon his memmy? From what was just said, it is clear that in order to c01Tectly answer the question whether a given proposition is sLdficiently noteworthy, one must first decide which of the three kinds of use described in §435 one intends the reader to make of it. For even if a proposition is not important enough to justify the demand that the reader impress it upon his memory, it may still be worth presenting for the sake of one of the other two aims. Now a proposition is worth advancing with the definite aim of the reader storing it in his memory when the following two conditions are met: (a) it is certain that impressing the proposition upon the memory brings advantages that would not be present were it merely considered a few times or looked up only at the moment when it is to be applied. We have examples in almost all moral and religious truths, since these are only of use to us if they are borne in mind, and indeed in such a way that they occur to us on every occasion when they should have an influence on our feelings or actions. Similar remarks hold for many truths of medicine and 106 other sciences, since we would miss many occasions to apply them if we did not have them in our memory, but instead had to look them up in a book every time. (b) In addition, we must have good reason to suppose that the time and effort that the reader would have to expend on learning these truths could or would not be applied elsewhere to better use. For this second condition to be met, it is necessary, among other things, that there not be any truths in the same science that are more useful than those we have selected, and which might justifiably have been chosen instead. Thus one could never justify a treatise of history which recounted events that are less instructive than other events from the same time, concerning the same people, which are omitted, if, e.g., it limited itself to names and elates, only reported battles and wars, and said not a single word about the actual condition in which the people found themselves, and of

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§.441. *

When is a proposition worth presenting even if the reader is meant to consider it only once? For us to be justified in encouraging the reader, not to impress a proposition contained in our book upon his memory, but rather simply to consider 107 it once, the following must be the case: (a) Some benefit must arise from merely considering the proposition in this way, which is large enough to outweigh the time and energy the reader expends in doing so. Merely grasping the proposition, e.g., might provide a certain amount of exercise in thinking, or it might serve as a premise to a conclusion that is wo1th knowing, or as an example which illuminates some general truth that has just been advanced, and so on, and these benefits must persist even when the proposition has later been forgotten. In addition, (b) there must not be anything else that could deliver the same or greater benefits. Thus if we have a choice between several truths which are equally well suited for the immediate ends such occasional consideration should meet, but one provides a special advantage if retained in memory, then we must certainly privilege the latter. §.442.*

When is a proposition worth presenting in a book so that it can at least be consulted upon occasion? When it is a matter of judging whether a truth should be presented in our book merely so that readers may find it if they need it, the following factors must be considered: (a) the effort it would cost us ourselves if we

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 443. had to look for a truth we had not yet located; (b) the effort required to present the proposition in writing, both by ourselves and by those who reproduce it; (c) the increase in the price of our book, which weighs even upon those who will not need this addition; (d) how great the benefit is for those who find something here that they put to use; (e) how great the 108 number of such people is compared to the number of those for whom the proposition has no use. If the only reason keeping us from presenting a truth in our book is the greater bulk and cost of the latter, we say that we must omit the truth/or want of space. In most sciences, there are in fact so many truths that would be noteworthy enough to merit inclusion merely for the sake of occasional reference, that it is in the end only lack of space that forbids us to incorporate them all. But since "lack of space" actually means something quite different than what these words say, one sees that deciding how many of these truths should or should not be included depends upon the most varied circumstances, e.g., on progress in the art of printing and other related arts and handicrafts, on the financial means of the citizens, etc. In our day, when the arts of paper-making and printing have reached a high degree of perfection, incomparably more merits inclusion in a treatise than would have been the case in an age where books were reproduced by hand-copying, using costly materials. And how will things stand when society has devised and instituted arrangements which make the use of books all but free for individuals? Actually, some of the factors mentioned in this section should also have been considered in the previous two sections. But so long as it is a question of theses that are considered important enough to lead us to urge the reader either to impress them upon his memory or at least consider them a few times, the difficulties occasioned by the author's presenting them in writing or the increased cost of the book fall so far into the back109 ground that they can in most cases simply be ignored. §.443.*

More precise determination of these rules according to the nature of the readers Clearly, the investigations we have just discussed must take into account the class of readers for whom our book is intended, or those whom we can foresee will use it. For depending upon the diversity in the readership, the same truth will sometimes be worth presenting in our book and sometimes not. 74

THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 443. 1) If we intend the book for scholars who are experts in the field, then we undertake (§430) to present every known truth of the science in question, unless the truth be of the sort that communicating it would not only now be of no benefit, but also can be expected to be of no benefit in the future. But in order to be able to say that one may expect some benefit, more is required than that we cannot see its impossibility. For a complete impossibility is nowhere to be found; and if the mere (problematical, § 182) possibility of a future application were enough to oblige us to include a truth in a scholarly treatise, we would have to include all, which would cause such works to swell to monstrous proportions. The degree of probability with which we expect that our truth will find some application must be at least large enough that if we wished to include all the truths that had the same right to be included there would not be such a large number of them that the disadvantages of including them outweighed the benefits. It is certainly true that this is often hard to judge, and we are for the most part obliged to follow a mere obscure feeling. Nonetheless I believe that the reader will concede me the following two rules: (a) Every purely conceptual truth which is not of the sort that can be discovered anew through one's own reflection whenever necessary de- l LO serves to be recorded in a scholarly treatise. (b) Every empirical truth that cannot be explained on the basis of the concepts we already have is for that very reason worth setting out in such a book. 2) If our book is not intended for scholars, but rather for the second or third classes of readers distinguished in §430, i.e., when we only write for those who intend to learn the science in order to use it in life, we must, for each truth we present, consider whether the benefit which leads us to include it might be more fully delivered by presenting some other truth, in which case we must understandably choose the latter rather than the former. Here it should be noted that in almost every science discoveries are made from time to time which have the effect of depriving many of the things that formerly were necessary to know of their usefulness in life. In medicine, for example, remedies are frequently discovered that prove to be more efficacious than the previous ones, at which point knowledge of the latter ceases to be of use to the practising physician. Such older remedies may remain important to the scholarly physician, since it is not impossible that continued research will discover that they are useful in special cases. But they can be passed over in silence in a treatise intended not for scholars but for practising physicians.

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 444.

3) Finally, if we intend to write a treatise which contains only what is generally usejitl, i.e., only those things that it is appropriate for everyone to know (§430), then in order to justify including a thesis in our book it is far from sufficient that it be beneficial, or even more beneficial than any other thesis of the science that might supplant it. Rather, if we are to decide this question, we must fix our eye on the entire multitude of truths that are known at this point, and which are of such a nature as to promise some benefit for anyone who learns them. We must then compare this multitude with the cognitive capacities we humans enjoy, and with the length of time that may be devoted to mere learning without detriment to action. If we imagine all truths, regardless of which science they belong to, arrayed before us, according to how worthy they are of being known by all people, and if we start with the most necessary and continue through this sequence until we have a collection that cannot be made any larger if people of average powers are to be in a position to learn them all without unduly interfering with action, it becomes apparent which truths have every right to be included in our treatise, namely, those which appear in this collection.

be discovered from the former, but the former cannot be discovered from the latter. And not only these particular truths, but many others, often infinitely many, are contained in the more general truth, and can be recognised from it through a very simple consideration. To this one may acid that the recognition of the more general truth usually provides more exercise in thinking than would instruction concerning the particular truth. On the other hand, experience teaches us that overly general truths are applied very infrequently, that when a case arises that is subordinate to this general truth we often fail to expend the meagre effort required to deduce the appropriate particular truth from it, even though this would involve very little reflection, especially when the truth in question is one that threatens to curtail our pleasures. It follows from this, in my view, that we may not always privilege the more general truth over the particular in instruction; rather, we may only do this with full right when it is more a matter of exercise in thinking than applying the truth that is to be learned, and where, further, when no disadvantage comes of the fact that we do not immediately recall the appropriate particular truth that follows from it when a case arises, when no passion speaks against the recognition of the particular truth, and when one may suppose the reader to possess enough skill in thinking that we may expect that he will himself deduce the particular truths that flow from the more general one. Thus, e.g., in the science of space one may present the more general proposition mentioned above rather than the particular ones, especially in a treatise that presents this science primarily for the sake of sharpening the intellect. If, however, one or more of the above conditions is not met, it will be advisable to present the particular truths instead of the general truth in cases where limitations of space or some other factor make it impossible to present both. In presentations of ethics, for example, it would certainly not be a good thing to spend so much time on very general moral truths that no time remained to mention the particular duties people have in various circumstances; for it is precisely the latter that we most need to know, and if they are not expressly presented to us, it can scarcely be expected that we will ourselves deduce them from certain general truths, no matter how easy this may be. For this reason, and because in certain complicated cases we often fail to determine the conect behaviour, I am of the opinion that casuistry (i.e., the branch of ethics charged exclusively with discovering and resolving such complicated cases) is wrongly neglected in our days.

§.444.

Does the more general truth merit precedence over the particular truth?

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The harder it turns out to be to apply the previously stated rules, the more necessary it is for us to attempt briefly to answer a number of particular questions on this subject. Of this sort, first of all, is the question of what we should do when confronted with a choice between two truths belonging to our science, one of which is more general than the other, which is subordinate to it. Thus, e.g., the geometrical truth that all lines, surfaces and bodies are related to each other as the lines, surfaces, and bodies similarly constructed from them is far more general than the well-known propositions that one finds in previous treatises of the science of space that the perimeters of two similar polygons are related as a pair of their similarly situated sides, that their areas are as the squares on these sides, and similar prisms are as the cubes on these sides, etc. Supposing that this more general truth is as easy to prove (or even easier) than the particular ones, the question arises whether one would do well to present the former instead of the latter. At first glance, one might think that the answer should be an unconditional yes. The more general truth, one might think, is without exception more useful than the particular. The latter can 76

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THEORY OF SCIENCE PROPER. Part IV, Ch. l §. 445. 446.

THEORY OF SCIENCE PROPER. Part IV, Ch. l §. 447.

§.445.

indeed must sometimes present equivalent propositions in one and the same treatise.

Do truths that.follow immediately from a truth deserve to be presented along with it?

The following question should be answered in much the same way as the previous one: when we have presented a truth, in which cases (if any) should we expressly present others that follow immediately from it, either in the strongest sense of the word or else in the sense that we may l 14 certainly suppose the reader to know the intermediate propositions that mediate the inference? We shall omit such consequences when either of the following conditions is met: if we may suppose that the reader will be mentally active enough to hit upon them himself, or when no harm will come if he fails to do so. Thus, e.g., it would be superfluous in a treatise of geometry if, after having presented the theorem about the sum of the angles in a triangle, one expressly added that a figure whose angles sum to more than two right angles is not a triangle. For what reader would be incapable of deducing this consequence if he had to? In other cases, by contrast, we will do well to explicitly adduce the consequence. Thus, e.g., once we have demonstrated the simplicity of our soul, it easily follows that it cannot perish through dissolution. Since it is so important that no one overlook this consequence, it befits a treatise of psychology to mention it expressly. §.446. Do equivalent propositions deserve to be presented alongside one another?

Among the propositions that are immediately deducible from a given proposition are those which are equivalent to it(§ 156). This is the name I give to propositions that are deducible one from the other, as, for example, the proposition "A is true" is deducible from A itself, and conversely. Now if the proposition that is equivalent to a given one differs from it no more than in the above example, it is obvious that it would be quite superfluous to present it along with the other. Yet it would be quite mistaken to believe that all propositions that are equivalent to each other may be so easily deduced. The propositions "This figure is a triangle" and "The angles of this figure sum to two right angles" are equivalent, when 115 only the idea "this" is considered variable in both. Yet will everyone who knows the first also know the second? Thus it can certainly be worthwhile to present both of them. Thus there is no doubt that one may and 78

§.447. Whether merely analytic or identical propositions, as well as propositions vvitlz redundant and imaginary ideas, may be presented as essential doctrines I) In the broader sense defined in § 148, a proposition is called analytic if there is even a single one of its constituents that may be replaced ad libitum by any other without destroying the truth or falsity of the proposition, provided that one only chooses ideas that do not make the proposition become objectless. In this sense, I must even count propositions such as the following among the analytic truths: The soul of Socrates is a simple substance; the angles of an equilateral triangle together sum to two right angles; if ~ = b, then a = ± ./2b, etc. For in each of these propositions there is an idea (Socrates, equilateral, 2) that may be replaced by any other without affecting the truth of the proposition. And these examples already suffice to show that not every analytic proposition expresses an obvious truth, so that it would be superfluous to communicate it to anyone. Rather, one sees that merely analytic propositions sometimes are not only sufficiently noteworthy to merit presentation in a treatise, but that we may indeed be obliged to provide proofs for them. It cannot be denied that an analytic proposition whose truth is not immediately obvious can sometimes be recognised as true if one has first learned a certain synthetic truth from which it follows. Thus the above propositions fol- 116 low quite easily from the following synthetic ones: Every soul is a simple substance; the angles of any triangle together sum to two right angles; whenever~ = b, we also have a= ±VJ;. But it would be hasty to conclude from this that one may rightly dispense with presenting analytic propositions in a treatise. For, in the first place, circumstances do not always permit us to present and prove the synthetic proposition before the analytic one that follows easily from it. We also know from §444 that it is not always permissible or even advisable to pass over a truth in silence merely because it is an easy consequence of others one has already presented. It may be a matter of great importance that a truth, though it be analytic, not be overlooked by anyone. It may be that we require this truth as a link in a chain of inferences, and cannot count on our readers' power of independent thought to derive this premise from what we have

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already said elsewhere without being prompted by us. In all such cases, we are not to be censured for expressly presenting this analytic truth. In sciences, such as analysis, which contain concepts that are so complex that not even the most practised thinkers may hold all their components before their mind, the presentation of analytic truths is quite frequently necessary. And often one needs not merely to present them, but to provide quite detailed proofs of them as well. Only analytic claims of the sort that anyone who is rational can form for himself, which are not noteworthy, and which are not required as premises in a deduction may in fairness be passed over in silence. But in cases where we deem it fitting to present a purely analytic claim, it also behooves us expressly to remind our readers 117 that the truth we express belongs to the class of the merely analytic truths, provided that we may suppose our readers to possess the knowledge required to understand such an observation, and that we aim to impart the greatest distinctness to all the concepts occurring in our presentation. For if the reader is able to overlook this, and take the truth before him to be synthetic, this could only happen because he has formed false ideas either of the constituents of the concepts occurring in the proposition or of their objective connection to each other. 2) What I have just said about all analytic propositions also holds for the special kind usually called identical propositions (§ 148, no. 2). We are ashamed when in daily life we find ourselves unwittingly making a merely identical judgement; this would be even more inexcusable for an author of a treatise, especially one who is setting out the essential theses of his science. This is by no means intended to say that in cases where the identity of concepts is concealed, as in many mathematical equations, we would not be justified in putting forth such a proposition (while expressly mentioning that it is identical). But in that case it is not really the proposition itself but rather the statement that it is an identical proposition that we teach on this occasion. 3) Propositions with redundant ideas (§69) deserve even less to be mentioned in any scientific presentation, though I do believe that this can be justified in exceptional cases. If, e.g., we wanted to show that an idea was redundant, then one certainly cannot avoid mention of a proposition containing the idea. But there may be other cases where we cannot effectively prevent the reader from connecting a redundant idea with a given 118 sign. We might not be able, e.g., to assume that he has the knowledge required to show how the idea would have to be constituted were it not to be redundant; or perhaps this is not the place to enter into such in-

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quiries, since there are more important things to think about. In every other case, however, it must be counted as an error when a proposition containing redundant ideas is used instead of one that is free from redundancy, but equivalent to the former in all other respects. For not only is the proposition that we obtain by omitting the redundancies simpler and hence preferable on that account alone, but also it must mislead the reader when he hears it said that something is determined through many characteristics m, n, o, p, ... when it is already determined by the fewer characteristics m, n, .... The reader will not suppose us to do something superfluous, and precisely for this reason our way of proceeding will lead him to the mistaken idea that the characteristics m, n, ... do not suffice by themselves to determine the object in question, but rather that others 0 , p, ... must be added to them. Suppose, e.g., we put forward the proposition: "An action that promotes the common good and at the same time does not contravene any moral law, deserves to be performed." Would not the reader suspect in this case that we think that there are actions that promote the common good yet contravene some moral law? 4) Some have also objected to all propositions which contain either objectless or indeed imaginary ideas (§§67, 70). On this view, many beautiful theorems about imaginary quantities would have to be expelled from the domain of mathematics. I at least would not want to have to answer for this. On the contrary, I am of the opinion that such propositions can and often do express truths, indeed quite noteworthy truths. Thus, for example, the equation: (cos x ± sin xv=T )"

cos nx ± sin nxH

in my opinion states that the two quantity-ideas standing on the left and right hand sides of the equals sign, if transformed (in a way that would not 119 destroy the equality if real quantities were involved, i.e., in a lawful way) so that they designate two actual quantities, are always equal. And every mathematician knows that this is a very useful truth, from which a number of most useful formulas may easily be derived. Of course when using such objectless or imaginary concepts it is always necessary to inform the reader of this fact. For it would certainly be a gross error if our silence on this point led them to imagine, e.g., that 0, Vi were actual numbers or that 0, J=T, etc., were actual quantities. 1

1,

1 For Bolzano, "actual number" covers what we would call the positive natural numbers, and "real quantity" quantities expressible by positive real numbers. (ed.)

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 448.

Note. Thus the presentation of one and the same proposition can merit either praise or blame, depending upon the varied natures of our readers and the remarks that we make along with it. If in a treatise of ethics intended for readers who are practised in thinking we express the highest moral law as follows: "Promote virtue and happiness", and we fail to point out that this expression is redundant, because the addition "virtue" could actually be left out without diminishing in the slightest that which the proposition requires of us, then we deserve reproof, since our silence gives the reader occasion to imagine the contrary, and thus to confuse his ideas. But if we note that the addition is made merely in order to make the formula all the more fruitful in its actual application, namely, by reminding the reader of the important truth that when we can make others disposed to act ethically through our own action we have a duty to do so, then our way of proceeding merits approval. In giving instruction in medicine, it is perhaps impossible to repeat too often the warning that a doctor should never seek to disturb natural operations, that he is only their servant, and can only assist nature, etc. Only one should never forget that in saying this he at bottom merely announces the identical proposition that a doctor must always take care to avoid applying the means of his art in the wrong place, i.e., where they are really not called for. But it 120 would be a truly ridiculous error to present such rules as principles from which one can objectively derive the proper behaviour of a doctor at the sick bed.

THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 449.

and prove this negative or corrective proposition(§ 141) in our book as an indigenous thesis. Beginners in geometry, for example, easily fall into the mistake of thinking that surfaces and solids with equal boundaries must have the same content. Thus it is appropriate to warn expressly against this error in a treatise of the science of space. But since the proposition that denies this error, namely, that figures with equal boundaries need not have the same content, may itself be considered a geometrical truth, since it too expresses a certain attribute of space, we may advance it in our book 121 as an essential theorem. §.449.

May we also present propositions that are merely probable in our treatise?

Even those who have worked on a science for a long time may happen to accept as theses propositions that aren't even true. It is thus understandable that the readers who will learn the science in the first place from our book, which we tailor to their needs, will go astray even more often. Given, then, that we may foresee to some extent that many of our readers will commit certain errors, and when we sense that they may form an opinion which has the appearance of a truth belonging to our science and yet is erroneous, it becomes our duty to warn against this opinion, and to show its baselessness, at least insofar as we fear that its acceptance would bring certain disadvantages. Now if the proposition that expresses the negation of this error also has the form of a proposition that should belong to our science (and when we have correctly determined the concept of our science, this will often be the case), then we must advance

According to its concept (§393), a science may only contain true propositions; it does not follow from this, however, that in attempting to present a science in writing, i.e., by composing a treatise, we may only include propositions concerning whose truth we entertain not the slightest doubt. For when we write a treatise for any science, we do not actually claim that the truths belonging to this science are constituted exactly as we present them. Rather, we say only that we believe that they are so constituted; and here, on account of the fallibility of our intellect, we may be certain in advance and admit to our readers that we may well be mistaken on many points. For this reason, we do not lie when we present as truths propositions that are merely probable for us, provided that we really do hold them to be true with a greater or lesser degree of probability. Also, if we did not do this, and only advanced as theses of our science propositions that we could state with complete confidence, the content of every science would tum out to be very small indeed. For, as I have pointed out several times, most of our important judgements, namely, those that we must deduce by means of a long sequence of inferences, do not enjoy such complete confidence. Now if the views put forth in §319 are correct, we only hold propositions to be true when the degree of probability they have for us is greater than one-half. Only such propositions, i.e., propositions whose probability is greater than that of their negations, may be advanced in a treatise as theses of the science it deals with. Other propositions, whose probability is smaller, may not be advanced, though this does not exclude the possibility of mentioning them in some other way, as shall be shown in the sequel. Now, when we advance a proposition,

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§.448.

Can a purely negative proposition sometimes merit presentation?

THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 449.

we convey that we not only ourselves hold it to be true, but also hope that the reader will place his trust in it. Hence, in order to advance a proposition it is not enough, strictly speaking, that it be probable merely for us; rather, we must expect that we will succeed in rendering it highly probable in the eyes of the reader as well. Thus if it is not feasible to present the grounds for the probability of a given proposition, at least not to the extent required to make it rational to accept it, we shall not advance it but rather content ourselves with simply mentioning it (§434), adding at most the assurance that we ourselves possess convincing grounds for it, and perhaps with the request that the reader examine it carefully. The case I describe here can even occur with propositions we have derived from pure concepts, even in a science that is reckoned the most certain, indeed, the science whose proofs compel assent, i.e., mathematics. When, e.g., someone takes up one of those mathematical problems for which there are innumerable failed attempts at a solution, e.g., the theory of parallels or devising a purely algebraic expression for all the roots of every algebraic equation, then his purported solution should not be announced with the tone of a man who is certain that others must agree with him; in other words, he should not advance his solution, but rather convey in 123 some other way that he is not yet fully certain that he will succeed in convincing the reader of the correctness of his claims. This will even be the case when he is permitted to present the grounds supporting his conviction in full; for if he is modest, he will himself not look upon these grounds without some misgivings until they have been examined by many others and recognised as correct. We find ourselves hindered far more often, and for a different reason, from advancing propositions we have merely drawn from experience. Because concepts are communicable (§75), what we have deduced from mere concepts may be supported by grounds which we can communicate to others in exactly the form they appear to us, at least when we are able to express ourselves perfectly on the subject. This is not the case with empirical propositions. The same immediate perceptions that we made, and from which we have deduced a certain conclusion, the grounds that lead us to make a certain judgement, cannot be awakened in others' minds; rather, only one of the following two things can occur: either the others must take our word for it that we had perceptions that would justify such a conclusion, or else they must have the opportunity and the will to place themselves in external circumstances where intuitions will arise in their soul that are either similar to ours or in any case of the sort that lead to the same conclusion. Now if

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 450.

the proposition that we have deduced from our own intuitions has a high degree of internal improbability, it is not only futile but unwise, indeed wrong, to demand that others accept it based on our testimony. If, in addition, the circumstances in which one must find oneself in order to decide the matter based on one's own observation are not of the sort that our reader can bring about when he pleases, we should most certainly not advance our proposition. For in such cases we must not only entertain justified doubts about the correctness of our conclusions, given that these 124 are by their very nature based upon merely probable inferences, but even when the degree of this probability is high enough that we ourselves do not doubt, we cannot expect the reader to agree with us, since we cannot communicate our reasons. As examples I would point to the phenomena of the so-called animal magnetism, as well as the efficacy of infinitely small doses of medicines as prescribed by homeopathy. Both of these have such a great inner improbability that the first observers had no right to claim that these were genuine, i.e., to appear before the public with the expectation that others would and should accept them on their word alone, even if they were themselves fully convinced (a matter I shall not make any attempt to decide). Homeopathy indeed presents us with a most embarrassing proof of the obstacles such an impassioned and uncritical procedure (followed by certain writers, and imitated by the public either in the same or the opposite sense) raises to the secure determination of the truth. Is it not incredible that such an easily answered question as whether the principle Similia similibus curantur 1 is correct, nay even the far simpler questions of the efficacy of medicines in such extremely small doses and whether the efficacy increases as the dose decreases, still await an answer after more than 30 years, when so many demands have been made to investigate them and so many opportunities to do so hourly present themselves? §.450.*

Does the mere possibility of an attribute sometimes deserve to be presented? Even though there are many attributes which we cannot definitely say belong to certain objects, we may still be able to say definitely that it is not 125 impossible for them to belong to these objects, i.e., that the assumption 1

Like cures like.

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 450.

that one or more of these objects has the attribute in question does not contradict any purely conceptual truth, or at least not any such truth that is known to us (§ 182). The possibility that an attribute belongs to a thing is often itself regarded as an attribute of the thing, and we accordingly say that the former property can belong to it. Thus we say, e.g., that human beings have the attributes of possibly erring, possibly sinning, etc. At bottom, this is not stated quite correctly. For the possibility of having a certain attribute b is not an attribute of the things comprehended under the idea A, but rather a relation obtaining between the ideas A and b or, more precisely, an attribute belonging to the proposition "The idea [A ]blacks objectuality", namely, that it is not a purely conceptual truth; or even, finally, a mere relation between this proposition and our knowledge, namely, that no purely conceptual truth which contradicts this proposition is known to us. We may still be permitted to use this manner of speaking, however, provided that we interpret it as just indicated. Now no one will deny that propositions merely stating possibilities in this sense can be noteworthy enough. How important it is to know, e.g., the possibility of erring or sinning that we have in certain circumstances! The further question arises, however, of whether we may present the possibility of certain things possessing a given attribute in a treatise of a science in which, according to its concept, we are only supposed to deal with attributes that actually belong to these objects. I believe that I may answer in the affirmative. For though it may be true that in such propositions 126 we do not state attributes of the things supposed to be dealt with in the science themselves, we may at least advance such propositions insofar as they may lead us to the recognition of such attributes. Once the possibility of an attribute b belonging to the things A is demonstrated, we have reason to investigate more carefully whether this attribute can be shown actually to belong to one or more of these objects. In certain cases, when there are a great many things standing under the idea A, and when these things occur in the most varied circumstances, one may infer the actual presence of an attribute b in certain individuals from its mere possibility with a greater or lesser degree of probability. If, e.g., it is merely possible that a certain failing will be found in some people, we may suspect with considerable probability that it will actually be met with in one of us, etc. Such propositions will occur all the more frequently in sciences which deal only with the nature of a definite class of objects. Note. When I claimed above that the possibility of a thing possessing a given attribute was not really an attribute of the thing in the strictest

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THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 451.

sense of the word, I depart from the opinion of others. In MAAB' s Log., §163, one reads: 'The possibility of certain attribute can be a necessary characteristic of the thing and hence may be used in defining it." Since I do not have the first edition of this book at hand, where M. appends some examples, I cannot look into the examples he may have cited there. Instead, I shall deal with the example of the fallibility of men which has already been used. 1 I freely admit that we frequently speak as if we regarded this fallibility as an attribute of men themselves. But what do we actually mean? That men have often erred and continue to do so? That is not what is meant here, for otherwise we would have spoken not of the mere possibility of error, but rather of its actuality. We can only speak of 127 a mere possibility of error if we turn our attention to specific men and specific cases. But to say that a specific man can err in a specific case simply means that we know of no truth which stands in contradiction with the assumption that he will err. But this is obviously not an attribute (property) of this man, but rather an attribute of our concepts and our knowledge of him. §.451.

May we present propositions we regard as essential in any way other than advancing them? Since there are almost always so many truths that would merit inclusion in a treatise for occasional reference if not for the sake of being impressed upon the reader's memory that only considerations of space prevent us from incorporating them all (§422), one might think that every truth belonging to our science for which we find space in our book would deserve for that very reason to be advanced. For in order to advance a proposition, especially when we do not require the highest degree of confidence, but can content ourselves instead with the confidence engendered by our testimony in favour of the proposition, it does not require many more words than merely mentioning and acknowledging it. Closer consideration reveals, however, that this is not the case. It is indeed true that few words are required to advance a proposition if the only reason we give the reader to accept it is our own authority. But it is equally true that in many cases advancing a proposition in this way is not only useless but also unseemly and offensive. When the reader's passions resist the acceptance of a truth, we will accomplish little or nothing by urging him to take our word for 128 In §182 [II.230] (ed.).

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THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 453.

THEORY OF SCIENCE PROPER. Part IV, Ch. 1 §. 452.

it. If, in addition, the proposition is of the sort for which everyone can determine, using his own reason, whether it is true or false, at least if he is sufficiently practised in thinking, then many will indeed be ashamed to believe something based on our testimony that they should have been able to discover just as well as us through their own reflection. If they are even propositions it would be dangerous to accept on the testimony of a single person, or if this would set a bad example, then it would in fact be wrong for us, and offensive to the reader, were we to demand that he adopt our opinion simply because it is ours. Doubtless it is often better and wiser merely to mention a proposition we hold to be true, or to acknowledge it ourselves without even beginning to express any expectation that the reader will adopt it as well. For it may well be that we are prevented from adding another kind of grounds for our proposition, grounds which, if presented, would entitle us to expect the reader to adopt our opinionfor instance, by constraints of space, because the readers do not know the premises required for a proof, or because they are not very practised thinkers, etc. One may well ask, though, what might reasonably lead us merely to mention a proposition we consider essential to our science if we do not intend to advance it? To this I reply that many benefits may lead us to do so. Sometimes, the mere mention of a proposition serves to draw the reader's attention to it, and to awaken his spirit of inquiry; sometimes we can lead him to reflect on how little he still knows, and thus stimulate his desire to learn; sometimes the proposition may be used as an example intended to illustrate a general truth, etc. Finally, it is obvious that propositions we do not advance need not possess a degree of 129 probability which is > ~'insofar as we do not even claim them to be our opinion, but rather mention them simply in order to recommend that they be examined further. For a proposition can still be worthy of examination if its probability is considerably smaller than one half.

areater claim on the reader's attention than is merited ' and thus prevent the reader from learning other, more useful truths. (2) Self-love often leads us to deem experiences we have had, changes we have helped to bring about, and discoveries we have made worthy of mention in places where this is not merited, e.g., in a treatise which is intended not for scholars but instead for practitioners or the general public. (3) In treatises intended for the general public (§§430, 433) nothing is more common than to include too much, even in cases where we do not actually overestimate the value of our science. This happens sometimes because we overestimate the mental capacity of people in general, hastily concluding that because something is possible for some it is possible for all, sometimes because we do not make a thorough survey of all that is worth knowing in the domain of human knowledge, and sometimes because we do not give enough consideration to the fact that man is born to act rather than 130 to learn. (4) Sometimes, clinging to tradition, due perhaps to ignorance of new advances and discoveries or laziness, we stick with theses that were previously dealt with in our science, even though these should now rightly be set aside because the concept of the science has been essentially changed or better theses are known. (5) But people are also guilty of the opposite error, abandoning established theses merely for the sake of novelty in favour of others that have neither been examined properly nor are as useful, etc. b

Chapter II

On Supporting Propositions §.453.*

§.452.

Warning against several mistakes

What degree of confidence in the reader's mind should we attempt to confer upon a proposition which we advance as essential?

These are the most important rules that we must bear in mind when we want to judge whether or not a given proposition merits to be included in our book as a thesis which is proper to our science. It will not be superfluous to end by listing the most common mistakes made in such judgements. (1) All too often, an exaggerated estimate of the importance of our science or some of its individual theses leads us to make a far

Since every supporting proposition in a book only appears there as a means through which we may confer the appropriate degree of probability upon the theses we have advanced as essential, it is obvious that we can only decide which supporting propositions we should include once we have decided the degree of confidence with which we wish each of the essential theses to be accepted by the reader. Thus I must give some

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THEORY OF SCIENCE PROPER. Part JV, Ch. 2 §. 454.

THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 455.

brief instruction on how to decide this matter. Now the degree of confidence we should seek to confer upon a given proposition in the minds of our readers may vary considerably depending upon which of the two following cases obtains. The first is when we ourselves are sufficiently convinced that the proposition is either true or certainly so constituted that, even if it were false, no harm could come to our readers from accepting it. The second case occurs when we are not even sure that our claim is true, nor that erring on this point would always be harmless for our readers. 1) If we are sure enough that the proposition is true, or that accepting it even if it were false would produce no bad consequences, then even though it is not our duty to do so, we are at liberty to infuse the reader with as high a degree of confidence in our proposition as the nature of the grounds we adduce for it and the time allotted to prove it permit. But if the proposition is of the sort that is more beneficial the more confidently it is held, or perhaps of the sort that is only genuinely fruitful if it is not doubted in the least, then it is not merely permissible but rather a duty to support it with all the true grounds that promise to engender conviction in the reader. If, finally, the proposition is of the sort that many passions in the human heart speak against, then it is even necessary to apply every permissible means we can to oppose these passions and bring about the necessary recognition of the truth. 2) If, by contrast, we are not even ourselves sure whether our proposition is true, nor whether belief in it would be harmless even if erroneous, we cannot look upon it as an unconditional benefit if our presentation succeeds in convincing our readers to trust in it as much as we do; rather, we must take care to make known the reasons that speak against it as well as those that speak in favour of it, so that in the end the reader will only find it as probable as we do-provided, that is, that he does not know grounds for or against it of which we ourselves are ignorant.

when others are unable to, some may find them fully reliable while others think them uncertain, and so on. Thus if we intended to proceed in the same manner in all cases, not taking any account of those for whom we write when choosing our supporting propositions, we would often go astray, and waste a good deal of time and effort without attaining our aoal. Jn particular, when we write a book for scholars, and indeed for b scholars who intend to fully master the science, we may not fail to mention anything that speaks either for or against the propositions we advance as essential, no matter how insignificant it may be. In this case, then, the number of supporting propositions which we expressly mention or refer to must be very large indeed; we must not only include proofs that are easy to follow, but also proofs based upon quite intricate inferences and the most diverse premises, be they ever so remote. When, by contrast, we write a treatise that is supposed to be useful for the general public, we must always choose those supporting propositions that are the simplest and the best known (if necessary, merely the testimony of other people), through which we can engender a sufficient degree of confidence. §.455. General rules

The nature of the readers for whom our book is intended must also be taken into consideration when we seek to determine which supporting propositions would be appropriate. For the same premises can be familiar to some but unknown to others, some can grasp them with little effort

From what has been said, one may immediately derive the general rules according to which we must judge which propositions merit inclusion in a treatise as genuine supporting propositions. Namely, we have every reason to include as many and as varied supporting propositions as are necessary to engender in our readers' minds an appropriate degree of confidence in the propositions we advance as essential, and no reason whatsoever to include any more. Suppose that the proposition M deserves to be accepted by our readers with a degree of probability = m; suppose, further, that we have two independent arguments that can be used to prove it, one of which is based upon supporting propositions A, B, C, ... , with probabilities a, b, c, ... , while the other is based upon D, E, F, ... , which have probabilities d, e, j; .... In these circumstances, the probability of the proposition M based on the grounds A, B, C, ... will be = a · b · c · · · , and its probability based on the grounds D, E, F, ... will be d · e · f · · · . If these grounds are independent of one another, the probability of M relative to both sets will be = abc +def - abcdef. Thus if the value of this quantity is equal or very close to M, we may content ourselves with these supporting propositions. It goes without saying, however, that these probabilities can only very rarely be calculated exactly; usually, we must make

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§.454. What influence does the nature of our readers have on the nature of our supporting propositions?

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THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 456.

THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 457.

do with a rough estimate. If we can attain the same degree of certainty through different kinds of supporting propositions, then we must understandably choose those which are preferable in some other respect, e.g., because they may be more succinctly presented, are easier for our readers to recall, may be used to deduce many other truths, etc. But since these rules are still so general that they do not tell us enough about how to proceed in individual cases, we must move on to address several particular questions.

at least for the present, and for not producing any harm in the future. As long as the reader adheres to that opinion, he will place more trust in our proposition, and when his error is revealed to be such, the other grounds will still remain. On this view, it would not be a bad thing, e.g., in a book on religion for the uneducated, to deduce the existence of God from the almost universally held opinion that the universe must have a beginning in time, provided only that we do not claim that this opinion appears fully correct to us. For in this case we do not misrepresent the truth, and those who are not yet practised in thinking find a comfort in this proof that only disappears once they have advanced to a point where they can better appreciate the force of the other proofs we adduce.

§.456. May we use our readers' opinions as supporting propositions even if we consider them mistaken? It often happens that the truth we want to demonstrate to our readers can easily be deduced from a premise that seems false to us, but is accepted as true by them. Thus the question arises whether and in what cases it is permissible to use an opinion we ourselves believe to be erroneous to convince our readers, and hence to include it as a supporting proposition in our book. I reply that this may never take place in a way that has us profess the opinion ourselves, i.e., claim it for our own. For in that case we would be lying, which is always impermissible. But we cannot be said to advance (§437) a proposition that we do not profess; thus it is obvious that we may never advance such supporting propositions, but rather may at most refer to them, without claiming that they seem true to us. But even this would not be acceptable if there were other proofs that could confer the appropriate degree of confidence upon the proposition in the minds of our readers. For even though we do not lie when we say directly to the reader that we ourselves do not hold the premise we now permit ourselves to draw conclusions from to be true, but rather only adduce it because he looks upon it as a truth, this way of proceeding still has the disadvantage of securing the reader's conviction only for the present, not for all future time. When he sooner or later finds out that the opinion he now holds is mistaken, what will become of the confidence that our 135 proposition in fact deserves? In my view, it is only fully appropriate to refer to a ground that we ourselves think incorrect in a single case, namely, when the other grounds that the reader is capable of grasping and which we actually adduce, are by themselves incapable of conferring upon our proposition the degree of confidence it merits. Only such a way of proceeding can be praised for producing some benefit, if not forever

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§.457. May we employ empirical supporting propositions in a science that concerns only purely conceptual truths, and vice versa? It is well known that purely conceptual truths can often be deduced from empirical premises with a greater or lesser degree of probability. Indeed, it is not seldom the case that experience is either the only or at least the best means to become convinced of the truth of a given proposition of this kind. Thus the question arises whether we may proceed in this way in a treatise, and apply empirical supporting propositions in a treatise of a science which concerns only purely conceptual truths. I have no hesitation in answering yes, provided that one of the following cases occurs: (a) no way is yet known of deducing the proposition in question from purely conceptual truths, of (b) if such a proof is known, but cannot be used, perhaps because it requires too much previous knowledge, or finally (c) if a proof from mere concepts would not suffice to confer the degree of confidence upon our proposition for our readers that it merits and must have in order to be genuinely useful. Thus we would certainly be blameworthy were we to deduce the existence and properties of God in a treatise of religion from mere concepts, and not from the purposeful arrangements in the structure of the universe, and from so many phenomena that announce the existence of this being. Just as little should one have any qualms about advancing the thesis of the mutual attraction of all matter in a treatise of metaphysics, and using a probable proof for it based on experience if we do not know how to prove this through mere concepts. Even mathematicians are not ashamed when, finding themselves unable to solve a problem a priori, i.e., from mere concepts, they 93

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THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 457. have recourse to experience, and hold a formula to be true or likely true if experience provides confirmation for it. We have many examples of this in dynamics and hydrodynamics. It remains true, nevertheless, that the empirical supporting propositions we use in the presentation of a purely conceptual science are an imperfection, at least when they are used not merely as confirmations of other proofs but instead as the only grounds for our claim. Things stand otherwise in the opposite case, when we apply purely conceptual truths as supporting propositions in an empirical science, i.e., one that deals with empirical truths. In order to know most 137 empirical truths (namely, all that are not immediate judgements of perception), certain conceptual truths are not only indispensable; rather, we must regard it as an advantage when we are able to prove an empirical proposition without calling upon a large number of immediate perceptions, deducing it instead from several purely conceptual truths along with a few immediate perceptions. Thus, for example, it was an undeniable perfection, when the great NEWTON taught us how to prove (admittedly with less than complete certainty) the truth that diamonds are flammable from mere concepts along with the experience that this body refracts light very strongly. This is only a mistake when the pleasure we take in a proof from mere concepts or some other reason leads us to take conceptual propositions to be true when they are not sufficiently certain, and hence to scorn the possible confirmation or disconfirmation of our conclusion through experience. Thus in medicine people have often wanted to decide the appropriateness or inappropriateness of a suggested therapy from mere concepts, without waiting to see what experience tells us about it; and in the natural sciences up to the time of BACON, people found it more convenient to explain everything with conceptual propositions than to make appropriate observations and experiments. Note. The mistake I just mentioned has given rise to so much mischief in these and other sciences, that we cannot thank enough the men who saved us from it, and insisted upon a steady progress with the help of experience. But when they said that no a priori inferences were valid in empirical sciences, this expression was not quite correct, and led to contradictions. In empirical sciences, a priori inferences are not only permitted, but meritorious, and indeed sometimes indispensable. Everything depends upon not overestimating the reliability of such inferences, and 138 using observations and experiments to confirm or disconfirm them whenever possible. But since we easily slide from one extreme to the other, we find that here and there, especially in France and England, people have

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THEORY OF SCIENCE PROPER. Part IV, Ch. 2 §. 458. gone so far as to want to ban all reference to purely conceptual truths in the empirical sciences. The only exception is mathematical truths; but some of them do not even reckon these to be conceptual truths, and for this reason oppose them to the so-called philosophical or metaphysical theses. These scholars have failed to notice that even the simplest propositions b, it is false that cb =ca.

If we consider it in relation to form ( 1) introduced in no. 2, we see that A in (i) is here replaced by the idea of a D.acb, in which a > b, while B is replaced by the idea of a D.acb in which cb = ca. The false assumption from which we start in an apagogical proof, according to no. 2, is the 275 proposition: (2) The idea of a triangle in which a

> b and cb

= ca has

(8) {A D.acb, in which ca = cb, has a = b A D.acb, in which a =I- b does not have ca = cb The proof of these two propositions should not be difficult. The first is El., I, 5, the second is generated from the first by transposition, from which we can see that in this particular case we do not even need both of them to arrive at the proposition

objectuality. The idea of a D.acb in which a =I- b, but ca= cb, is objectless. In Euclid, this proposition is tacitly contained in the hypothetical proposition: "If in D.acb it were the case that cb = ca, then it would have to be the case that a = b." Proposition (3), whose truth is presupposed in this proof, is the proposition

From this we obtain the desired conclusion The idea of a D.acb, in which a> b, but ca= cb, is objectless, or

(3) In every D.acb in which cb =ca, a= b. Hence the idea R is the idea of a D.acb, in which cb = ca, and Sis the idea of a D.acb in which a= b. Hence (4) takes on the following form: (4) The idea of a D.acb, in which cb objectuality.

ca, and a =I- b, has

We arrive at this conclusion from the assumption (2) without any further premises simply by replacing the idea "A D.acb, in which a > b, and cb = ca" by the wider idea "A D.acb in which a =I- b, and cb =ca". This is now the X, or the idea (7) of an A' which is a B', so that in this special case B' and B (and R) are the same, and only A' was generated by extending A, where a > b was replaced by the more general a =I- b. Hence (5) and (6) are as follows: (5) The idea of a D.acb, in which cb objectuality. 190

= ca,

but a =I- b has

A D.acb in which a

> b does not have ca = cb.

Hence the entire proof can be briefly stated in this way: A D.acb, in which ca = cb, has a = b, hence A D.acb in which a =I- b does not have ca= cb, consequently A D.acb in which a > b does not have ca = cb. 4) We can see that the only thing required to transform an apagogical proof in such a way that no false propositions have to be considered is to express some of its propositions in a simpler way, such that we delete from their subject-ideas certain characteristics which make them objectless. In the usual exposition of this subject such propositions are put in hypothetical form, such as the following: "If something which has attributes o., ~, y, ... had the attribute µ, then it would also have non-y, where non-y designates an attribute which is incompatible with y. But 191

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 530.

in such a proposition the attribute y of the subject-idea of the antecedent can obviously be deleted, and this simplification will not make the proof of this hypothetical proposition nor the deduction of other propositions 277 from it any more difficult. From this it follows that the apagogical mode of proof should never be used where a really clear understanding of the grounds (if only the subjective grounds) of a given truth is intended. For it obviously does not meet this purpose, and through a few alterations in the propositions, which are actually simplifications, and which reduce the number of inferences, it is always possible to generate a proof which does not include a reduction to absurdity. But I do not wish to say that it is not permissible to proceed in the apagogical manner where the point is not so much to generate a clear insight into the ground, but to produce conviction in a familiar and succinct manner. For, although in this case the thoughts are more complex, they can still be more familiar to the reader, and their linguistic expressions can often be shorter, since frequently a more complex idea can be more briefly expressed than a simpler one. If it seems advisable to retain the apagogical proof procedure, then I should like to see at least the following rules obeyed: (a) Our mode of presentation alone should make it obvious, even to the inattentive reader, that the propositions which we deduce from the assumed negation of proposition M, up to and including the proposition Neg.A, are not asserted by us, but are merely introduced for the sake of argument. (b) The proposition Neg.A, whose absurdity is taken for granted, must be chosen very carefully. Since it assumes a rather prominent position in the proof, it always attracts the attention of the reader above all others; hence, if they are not sufficiently convinced of its falsity the whole proof would be without effect. Note 1. Some, such as CALKER (Denki., § 190) call any proof purgative or apagogical as long as it contains an inference in modo tollente, 278 i.e., one of the following form: Among the propositions M, N, P, ... , at least one is true. The propositions N, P, ... are all false. Hence M is true. In this wider sense not every apagoge in our proofs is avoidable. Indeed, I claim that there are truths that are objectively grounded in such considerations. This may well be the case with Proposition 19 of Book I of the Elements. For the objective ground of the truth that we must have cb > ca if a > b can hardly lie in anything other than the truth that on this assumption, we can have neither cb = ca nor cb < ca. But for this very reason it would not be correct to count such proofs as reductions to absurdity. ARISTOTLE (Anal. post., I, 26) distinguishes the two, calling the

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 530. former cm65t:l~lc; CT'Lt:prrnx~ and the latter :::tc; cXOUVCXWV. Latin logicians use the terms privativa and per exhaustionem. The confusion between the two may well have been caused by that fact that in proofs in modo tollente one quite often uses the deductio ab absurdwn, namely, when the falsity of the propositions N, P, ... is shown. Note 2. It has always been recognised that apagogical proofs in my sense do not indicate the objective ground of the proposition proved. E. REINHOLD (L., p. 409) would not even give them the name of logical proofs. ARISTOTLE (Anal. prim:, II, 14) attempts to demonstrate that every apagogical proof can be converted into an ostensive one. LEIBNIZ, on the contrary (Nouv. Ess., IV, viii) is of the opinion that this is at least difficult in many cases. Here I would like to side with ARISTOTLE. For even though I think his proof is inadequate (being based upon the presupposition that all inferences are in only one of the three syllogistic figures), I have succeeded in every case I tried (namely, with all the proofs per absurdum in EUCLID'S Elements) in converting apagogical into ostensive proofs. Here I suppose that a proof is to be called ostensive provided only that it is not apagogical, and that a proof is called apagogical only if it uses a hypothetical premise with the denial of the proposition to be proved as its antecedent and an absurdity as its consequent. On this conception a proof is still ostensive even if it contains a so-called inference of conversion, be it the usual one, where from a proposition of the form "Ev- 279 ery A has B" one deduces the conclusion "Whatever is not Bis not A" or the following, which is not spoken of much in previous treatises of logic: "Every A which is B is also a C. Hence an A which is not a C, is not a B either." By using such conversions and the means detailed in no. 2, I have so far always succeeded in rendering superfluous the assumption of the negation of the proposition to be proved and the reduction of this assumption to other false propositions. But even if an altered proof of this kind no longer deserves to be called apagogical, can one boast that it indicates the objective ground of the truth? I believe, for example, that the proof of Elements, I, 19 given in no. 3 above can be called a genuine grounding; at least I know of no other that deserves the name more. - Although recent logicians agree in holding ostensive proofs in higher esteem than apagogical ones, some have shown a preference for some features of the latter, for reasons that seem incorrect to me. Thus in KIESEWETTER' s W A. d. L. (p. 493), we read that the necessity of the proposition to be proved is always made evident in apagogical proofs; while TWESTEN (p. 156) writes: "Indirect proofs produce a stronger consciousness of neces-

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 530. sity." Why should this be? Is the inference "M is true, since the denial of M leads to the absurdity Neg.A" in fact any more convincing than when I simply deduce the truth of M from the evident truth A? It also seems false to me to assume that apagogical proofs always show the necessity of the proposition proved. This will only be the case if the proofs involve only pure concepts, and in this case, the ostensive proof would do the same thing. But we also use the apagogical method to prove propositions that can only be proved from experience, as when we show that someone has lied because otherwise, had he spoken the truth, he would not have been so disconcerted when it looked as if we would investigate the truth of what he said. Incidentally, the reason why such proofs are most commonly used in mathematics, namely in geometry, might well 280 be explained by the observation made in no. 4 that such proofs can often be expressed more concisely than direct proofs, which are actually more simple. The symbols commonly used in mathematics (letters in algebra, figures in geometry) frequently provide the opportunity for abridgement. How concise the geometer can be! He draws some scalene triangle or other, calls one of the larger angles a, a smaller one b, and the third c, and says: If a > b, then we must have be > ac. For if be were not > ac, we would have either be = ac or be < ac. be = ac would yield a = b, while be < ac would yield a < c. Both of these conflict with the assumption. Hence we have neither be ac nor be < ac; hence be > ac. Who can fail to see how many words are dispensable because of the letters? But if one wanted to proceed ostensively, one would have to explain various things verbally instead of using letters. For one could only proceed roughly as follows: Because a > b, we have neither a = b nor a < b. Since a i= b, we cannot have cb = ca; for if-not merely in the present triangle, but in any other as well-two sides are equal, then so are the opposite angles. Because a is not < b, it is also not the case that cb < ca; for if-in any triangle at all-one side is smaller than another, then the angle opposite the former is also smaller than the angle opposite the latter. Therefore, etc. Note 3. Since there are many propositions, both true and false, that may be deduced from a single false assumption Neg.M, depending upon whether one adds these or those additional propositions B, C, D, ... , and employs this or that mode of inference, people have always thought that there may be cases where from a false assumption Neg.M, in combination with some true premises B, C, D, ... , one might be able to deduce a conclusion that is the contradictory opposite of Neg.M itself, namely 194

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 530. M. It is clear that in such a case, where even the assumption of the opposite of M leads to the proposition M, that the truth of the latter must be recognised all the more clearly. Because of this, they considered this a special kind of proof, which was only counted as apagogical due to the similarity between the two methods. WOLFF, e.g., puts things this way (Phil. rat., §558.9). Others, however, have raised serious doubts about 281 such a procedure. LAMBERT, notably (N.O., Dian., §383 ff.) tried to prove at length that what is here assumed is in fact impossible, and all examples that had been adduced were merely apparent ones. Concerning the example derived from EUCLID'S Elements (IX, 12) and other similar ones, I would claim that they are incorrectly cited as examples. For if one considers these proofs more closely, one finds that the proposition Neg .M, though it is indeed listed among the premises from which the opposite, M, is deduced, does not really belong among them, at least not insofar as only exact inferences (§ 155, no. 27) are supposed to occur here. It appears, namely, that this proposition can simply be set aside, since the required conclusion M either already follows from the remaining premises B, C, ... , or that in order to deduce M from B, C, ... , one does not require the additional premise Neg.M itself, but rather only a certain proposition m that follows from Neg.Mand says less than it, and on account of that is compatible with M. In order to show that the number a must be divisible by a prime number e if the power an is, the assumption made in the Euclidean proof that a is not divisible by e is not required at all. Rather, it suffices to note that an = a 11 - 1 ·a can only be divisible by a prime number e if either an- I or a is; one then shows that each of these two assumptions either is the proposition to be proved, or else leads to it. - But LAMBERT did not prove that the same applies to every case where one begins with a false proposition along with certain other true ones and deduces the opposite of the former, and thus has not shown that what so many logicians and mathematicians assume to be possible is in fact not so. The proof he gives in §384 runs, in brief, as follows: "If from the false proposition A is B, in combination with some true propositions, the opposite of the former may be deduced, i.e., the proposition A is not B, then the additional propositions must be of the forms B is C, C is D, D is not B. From this it would obviously follow that A is not B. But one sees that not all of these intermediate propositions could be true, since they already contradict each other, as the conclusion B is not B follows from them." I 282 detect several errors in this proof. First of all, since the proposition "A is B" is supposed to mean nothing other than "Every A is B", it is not 195

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 530. necessary that the conclusion must be of the form "A is not B", i.e., "No A is B", if it is to contradict the former. Rather, it would suffice to deduce a conclusion of the form "Some A are not B." In the second place, not all modes of inference can be reduced to a sorites; here it is also simply assumed that there must be several intermediate propositions. Why should it be impossible to obtain from the false major premise A is B, along with a single minor premise, a conclusion that contradicts the former? But the most important mistake, in my opinion, is that L. tacitly assumes that the assumed (false) major premise is supposed to contradict the conclusion in the same respect, i.e., relative to the same variable ideas i, j, ... with respect to which the latter is deducible from the former. It is obvious that this is impossible; it is another question altogether whether it be possible to come up with a false proposition which, in combination with certain true propositions and with respect to given variable ideas i, j, ... , yields a conclusion which stands in the relation of contradiction to the former with respect to certain other (fewer) variable ideas. Only this is required for the possibility of the mode of proof spoken of here. That such a case arises, and this indeed with the simplest mode of inference Barbara, is shown, it seems to me, by the example that L. himself (§387) allowed to be the most plausible of all, and against which his only objection was that the two premises were incompatible and hence could not be united. But is this reason sound? The example runs as follows: "Every proposition is false. That every proposition is false is itself a proposition. Therefore it is false that every proposition is false." I cannot see how it can be denied that in this case we deduce from a false major premise through mediation of a true minor premise, and in conformity with the most con-ect mode of inference, a conclusion which negates the major premise. It must also be 283 admitted that this major premise shows its falsity simply because it produces a conclusion which is its denial. Thus the possibility of this kind of apagogical proof is demonstrated by a single example. However, since I have stated that all apagogical proofs can be turned into ostensive proofs, the reader will wish to know whether this proof, too, can be stated ostensively; this question is all the more reasonable as I have used this proof in §31 for a very important purpose, namely, in order to show that there are any truths at all. In order to produce the conclusion, the major premise which we assumed there is not really required; all we need is the following undeniably true proposition: "If the claim that no thing of a certain kind A has attribute b is itself a thing of this kind, then this claim itself lacks attribute b." If we add to this major premise the minor premise:

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 531. "The claim that no proposition has truth is itself a proposition", then the desired conclusion follows, namely, "The claim that no proposition has truth itself lacks truth." Note 4. The fact that in an apagogic proof we learn the truth after starting out from a false hypothesis led many, for instance DESCARTES (Prine., III, 47), TACQUET (Elem. Geom., Amsterdam 1683, Appendix), to say "that one here deduces the true from the false"; and the example I spoke of in the previous note was the occasion for the claim that "one can often directly deduce the true from the false," claims to which others strongly objected. In my opinion, one should not say that the truth is deduced from false propositions (neither from the false proposition N eg.M one begins with, nor from the false proposition Neg.A where one ends), but rather only from the observation that were we to take the first proposition to be true, this would oblige us to take the second, obviously false one to be true as well. The consideration of a proposition provides the occasion for a certain cognition, but this is not the same thing as deducing the latter from the former. Yet some seem to think they have an example of discovering the true from the false in mathematical procedures which begin with mere approximations of or even completely false assumptions 284 about the true value of a quantity, and produce ever more exact calculations (as in the regula falsi or the approximation methods of algebra, astronomy, etc.). I would observe that in these procedures one does not actually make false assumptions, since the inferences employed are valid when the value used in the calculation is merely close to the true value; in some cases, they are even valid for all values. §.531.* Proofs- by induction and analogy

The procedure of induction (§328) renders the most valuable service when we attempt to convince ourselves that a given proposition is true (or false), and it is not to be scorned when we want to convince others. Proofs that follow this procedure may be called inductive. We may present such proofs: (a) in all cases where we know of no other proof, and thus (Cl.) for all truths of experience, which by their very nature we can only be convinced of by means of induction, and indeed for the most part by means of an incomplete induction which actually only confers probability on a proposition. (~) Also for certain purely conceptual truths that we do not yet know how to prove from mere concepts, and instead seek 197

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 532.

to confirm them through experience. Even in pure mathematics, namely, in the theory of prime numbers, there are propositions which highly regarded mathematicians are not ashamed to accept even though they have no generally valid proof, accepting them only because every example they have tried has confirmed the rule. (b) Even when another proof is already known, it is permissible to have recourse to a proof by induction, either complete or incomplete, if the other proof is beyond our readers' capaci285 ties, or requires previous knowledge that they lack. With all such proofs by induction, we have to assure that the reader can appropriately judge the completeness of our enumeration of individual cases. If the induction is complete, i.e., if we prove the truth of the proposition that every 5 is a P by dividing all the objects standing under the idea 5 into certain groups 51 , 5", 5111 , ••• , and showing of each group in turn that the attribute P belongs to each individual they contain, we must point this out to the reader. . purpose, It . IS . not necessary to s110w th at the I'deas 5 1, 5" , 5 111 , ... , For thIS are mutually exclusive, nor that each of these ideas is subordinate to 5. But we must show that there is no 5 which does not stand under any of the ideas 5', 511 , 5111 , ••• , or (what amounts to the same) that the idea of an 5 which is neither 5' nor 5" nor 5111 ••• is objectless. But if our induction is incomplete, we must at least show that it is able to confer the degree of probability on our proposition with which we have advanced it. Similar remarks apply to proofs by analogy. §.532.* Proof~ from

mere concepts and proofs fmm experience

The difference between proofs based upon whether or not the various premises from which one deduces a conclusion by means of perfect inferences are all purely conceptual propositions has an influence on how they should be composed. l) Proofs of the first sort, from pure concepts, doubtless have the advantage of delivering complete certainty (provided only that the premises and the modes of inference used are not incorrect), since the opposite of what they prove is impossible (under the stipulated condition and in the sense of § 182, no. 4 ). When the proof consists of a long sequence of 286 inferences, however, the risk of error in the premises and the inferences can often become so great that we must not only ourselves be extremely distrustful of the proof, but also cannot warn our readers enough not to accept it unless they and others have repeatedly examined the sequence 198

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 532. of inferences we present, and found it to be correct. lf this has not yet happened, and the proposition to be proved can be confirmed in another way, namely, through experience, then we will do well to do this. We shall have to have recourse to such confirmations through other grounds when the inferences from pure concepts lack sufficient certainty for our readers (even if they are certain enough in our eyes). Incidentally, regardless of whether we present a proof from pure concepts alone or along with others, if we wish it to be appropriately understood and to engender a suitable degree of certitude in our readers, it is necessary to present it as distinctly as possible. Thus we should not be content merely to adduce as many premises as are required to deduce our proposition mediately; rather, we must also mention all the intermediate propositions, i.e., all those that can mediate and facilitate the deductions, and make the reader distinctly aware of them. 2) Proofs from experience, or empirical proofs, i.e., proofs in which we deduce something from certain perceptions which follows only by means of an inference of probability, can for this very reason never produce complete certainty, though they may attain a degree of probability that is great enough to permit us to disregard the possibility of the contrary, if not to deny it. Although the confidence that we may demand of our readers in such cases cannot actually be distinguished from complete certainty, and though we may on that account permit ourselves to 287 call the certainty complete, we must, whenever this usage might lead to understanding, never forget to observe that this confidence is only valid under the present circumstances, and might have to be altered if these circumstances were to change. But if the degree of probability that our proposition obtains by virtue of the experiences we adduce is not so great as to permit us to neglect the possibility of the contrary, it becomes our duty to warn our readers not to place too much trust in our proof. We owe them such a warning principally when we suspect that they may be inclined towards a certain kind of credulousness. If, on top of this, the proposition in question is of the sort whose truth can be definitively settled by further experience (observation or experiment), either our own or that of our readers, we will do well to ask them to petform these experiments or observations and, if need be, to give special instruction on how to do so. We may consider such instructions and demands to belong to our proof insofar as they are propositions which, if not immediately, at least mediately (through the actions they prompt) contribute to increasing the degree of certainty the reader attaches to our proposition. Special care

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 532. is required, however, in cases where we deduce the truth of a proposition from our own perceptions and when others, namely, our readers, cannot repeat them. It is understandable that there can be a great difference in such cases between the degree of confidence with which we are entitled to claim the truth of our proposition and the degree of trust we may demand of the reader. Here, in order to become rationally convinced, the reader must enter into considerations of a completely different sort than 288 those in which we had to engage ourselves; they must discover grounds which prove that our testimony is credible, that is, that we had the requisite skill correctly to deduce what actually happened from our immediate perceptions, and that we have the will to communicate the truth we recognised. Thus we must make a special effort to convince them of these two things, and only to the extent that we succeed in this are we entitled to demand their assent. If, as is often the case, we are not in a position to provide a sufficient guarantee of our knowledge of the matter or of our trustworthiness, a guarantee that would be secure enough to outweigh the (perhaps very great) inner improbability of the proposition we advance, we fail miserably if we simply expect them to believe us. And can this not also be most harmful? Readers who conform to our expectation and thus believe without having sufficient grounds for doing so, will only find their credulousness (a most grievous fault) strengthened. Readers who quite correctly notice that we have no right to demand their assent in the present circumstances will only come to distrust us all the more, the more they gather from our behaviour that we do not know what is required here. How much better, then, to freely admit in such cases that we know that the present circumstances are not of the sort that permit us to demand that others trust us and that our intention is merely to draw the reader's attention to this matter, and communicate something which might, in combination with other observations that have yet to be made, eventually contribute to a decision of the matter. 3) If all the premises used in our proof are purely conceptual and all the inferences used are of the sort I called perfect (§253, no. 2), then if we make these two features known to the reader, he will recognise that 289 our proof establishes the necessity of the proposition that is proved or the impossibility of the contraty in the sense of § 182, and this with as much certainty as we have that the premises are true and the inferences correct. If only some of the premises A, B, ... are empirical, while the rest, along with the modes of inference, are as before, the reader will, when we point this out to him, recognise that our proof establishes the

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 532. necessity of our thesis or the impossibility of the contrary conditionally (conditional, namely, on the propositions A, B, ... ), again with as much certainty as we have that the remaining premises are all purely conceptual truths and the modes of inference perfect and correct. Understandably, it is sometimes useful to make our readers aware of this, since in this way they learn more distinctly what truly matters in an examination of our proof. But we would be mistaken if we intended this observation to make them believe that the certainty our proof engenders is already greater on this account than could be attained with proofs of another kind, which are based upon empirical assumptions and probabilistic inferences. The special thing about proofs from pure concepts using perfect inferences is not the degree of certainty, but rather the way that this degree can be increased. Note I. Some logicians, such as CRUSIUS (§525), RbSLING (§217), and BACHMANN (§350) speak of mixed in addition to a priori and a posteriori proofs; these are supposed to be based upon both a priori and a posteriori principles. In my opinion, every proof based on experience is supported not only by perceptual propositions (a posteriori principles), but also upon certain purely conceptual truths (a priori principles), even though the latter may not be expressly mentioned. For from mere perceptual propositions, if these are not combined with purely conceptual propositions such as "every perception must have a cause'', there can be 290 no experience which imparts knowledge of the nature of things outside us. Thus the difference can consist at most in the circumstance that the so-called mixed proofs contain several purely conceptual truths, among them some that we cannot suppose to be generally known. - The claim that a priori proofs leave no room for the possibility of the contrary, made so unconditionally in many textbooks of logic, can also give rise to misunderstandings, especially when (as in KANT' s Logic, p. 92) one infers from this that "in no science that contains a priori cognitions-thus neither in mathematics, nor in metaphysics or morals-is an opinion to be found. For it is in and of itself absurd to opine a priori. Nor could anything be more ridiculous than, for example, to merely have an opinion in mathematics. Here, as in metaphysics and morals, one either knows or does not know!" Emboldened by such a predecessor, HEGEL allowed himself to speak even more forcefully; and in Volume 13 of his Works, p. 24, one reads: "One ceases to accord a man-even if he be a historian of philosophy-even the rudiments of education the moment he speaks of philosophical opinions. Philosophy is the objective science of truth,

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 532. science is necessary, comprehending knowledge, no opinion, no elaboration of opinions." If one does not wish to accord a meaning to the word "opinion" that departs completely from current usage (and one cannot assume that everyone who claims to be educated will immediately adopt such an arbitrary meaning), then by "opining" one means simply holding something to be true without being completely certain. If so, then I cannot begin to fathom why mathematicians and philosophers, and especially metaphysicians, should not be allowed to opine. Indeed it seems to me that it behooves them to do so. Is one somehow raised above all danger of error in one's opining about mathematical and philosophical subjects? As experience teaches us, hardly. And if we inquire into the circumstances where this danger arises, and how it can to some extent be diminished, we 291 find that the high degree of confidence that the philosopher or the mathematician can claim for some of his theses rests merely on certain factors that are mediated by experience, thus historically-e.g., on the circumstance that others agree with us, and the like. If things stand thus, then I ask: where will the ban on the use of the word opinion and the command to substitute the word knowledge lead, except to give an illusory justification to the adoption of an improper tone of complete certainty in a science that has every reason to speak with the greatest modesty. Note 2. It is well known that KANT (Critique of Pure Reason, A734/ B762 and elsewhere) wished us to recognise an important difference between the proofs used by mathematicians and and those used by philosophers, namely, that the former "proceed through the intuition of the object" while the latter "can only be conducted through mere words (concepts)." For this reason, he thought only the former should be called demonstrative, while the latter might be called acroamatic or discursive. At A727/B755 he even says: "the geometer, applying his method in philosophy, can only build houses of cards, while the philosopher who applies his method to mathematics is merely babbling." The harmful influence such claims have had on philosophy has already been touched upon; here I would like to add that they are no less harmful for a genuinely scientific development of mathematics. For if mathematicians were formerly inclined to appeal to the mere appearance of a figure to prove their theorems, they now believe (not only in Germany, but also in France, England, the Netherlands and wherever else people have accepted something of this Kantian teaching) that they are fully justified in so doing. Now I will not dispute that such a right exists when it is a question of books intended for readers who do not have the required practice in thinking and 202

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 533, 534. previous knowledge to recognise the objective grounds of mathematical truths. But when a presentation is not constrained by such considerations, where the highest degree of scientific rigour is supposed to be attained, I hold it to be a duty to deduce nothing from the mere appearance of a figure, from a so-called intuition, be it pure or of some other kind. In brief, mathematical theorems should be proved in the same way purely philosophical ones are. That people have often failed in this, and that 292 most who attempted to prove mathematical truths with the philosophical method were merely babbling does not prove that it is impossible; but it does follow from what was said in §§79, 315 and elsewhere about the nullity of the Kantian theory of space and time as pure intuitions and the mediation of synthetic judgements through intuition that we should never cease trying to discover such proofs. §.533. Proofs based on authority I have already said in §458 that and under what circumstances proofs based on authority may be used in a treatise. From this is follows directly that we should not ask our readers simply to weigh the number of voices for and against, but rather, if we expect them to accept the view of a given patty, we must point out that the persons belonging to this party have the attributes of credible witnesses on this matter, and indeed are more credible than the opposing party. §.534. Proofs based on the reader's conceptions It has already been noted in §456 that we may sometimes employ supporting propositions in a treatise that we ourselves do not regard as entirely true. In particular, we may give proofs which contain a premise or inference that our readers regard as true and correct but we ourselves do not. I permit myself to call such proofs, which are adapted to the reader's conceptions but not to our own, proofs based on the reader's conceptions. They are also called proofs based upon concessions (ex concessis, ad hominem, xcrr' &v&pwrrov) while, by contrast, all others are called proofs xccc' c:X;A.tj'l'h:tcxv. Understandably, many rules must be carefully observed in formulating such proofs. First, there is no doubt that we may never 293 present propositions and inferences that appear incorrect to us in a way

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that people may take to be an endorsement of them, for this is simply lying. It will often be a duty to tell the reader that we think these grounds and inferences to be false and incorrect. Such an admission will be a duty, I say, when the very propositions and inferences we now wish to use to derive a truth pose some risk, i.e., when there is reason to worry that our readers might afterwards misuse them to support pernicious errors. For example, in a treatise of Christian ethics we might prove that there is a righteous and permissible anger, or that there are various duties, by merely appealing to the scriptures, and thus from the incorrect assumption that the scriptures can on their own teach us our duties, regardless of whether or not what they say accords with reason. But we should expressly remark that we take this assumption to be incorrect. For how easily it might lead to the most pernicious maxims, since the scriptures, if not first assessed by reason to determine whether this or that should be looked upon as a God-given precept or authorisation, could be used to justify the most perverse ways of acting! - If, by contrast, we can foresee no evil consequences that would result from our readers persisting in the error of taking certain inferences and propositions we think incorrect to be true, and when we can use these to prove an important truth that they would be unable to grasp based on other, more correct grounds, or would like to provide them with easier access to that truth, then it would be unwise and unjust for us needlessly to allow our readers to clearly see 294 that we do not accept the opinion we mention. For such remarks would only weaken the reader's trust in the proposition we want to prove. Thus here we must, to the extent that this is possible, speak of such propositions and inferences in language that does not show us to be either for or against them. The usual way of doing this is to discuss such propositions and inferences as something that others have said, without adding that we ourselves also find them convincing. One thing that should be looked upon as an almost unshakable duty, however, is that we should never confine ourselves to such proofs alone, but also add another proof that at least we ourselves take to consist of fully correct propositions and inferences. If it should prove impossible for us to present in detail a proof we find satisfying, perhaps because understanding it would require previous knowledge we cannot expect our readers to possess, it is at least our duty to mention it, so that in any case we have taken care to ensure that no reader who sooner or later becomes aware of the incorrectness of the proof we did present will take offence and reject the proposition itself as unproven.

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Note. Since according to these instructions we do not advance the propositions and inferences that we take to be incorrect, the actual form of our proof is as follows: "You too, who claim such and such, must for the sake of consistency, also admit that ... etc." - Now since ther~ is nothing false in this inference, one may say, strictly speaking, that in this proof we do not use any false premises; at least we do not ;onclude anything from them. This remark should help to make what was said in §§456 and 515 more fully comprehensible as well. §.535.

Pmofs that are only supposed to show that the probability of a proposition exceeds a given quantity . It quite commonly occurs that in giving a proof we do not intend to confer a precise degree of confidence on the proposition we prove, but rather 295 merely want to show that its probability is at least greater than this or that given quantity. We do not intend to show how probable our proposition is, but rather only that it is certainly more probable than a given one. This is the case whenever we claim that a proposition we present is certain eno~gh for us. t.o rely on it in the sense of §317, no. 3, or even simply that its probab1hty is great enough that it cannot be disregarded. The first case occurs, e.g., when we claim that a human body which bears traces of decomposition should not be expected to revive, and may be considered ~ead, with a degree of confidence that is high enough to allow us to bury it: For here what matters is that the harm that would be caused by continumg to store the body obviously outweighs the good that might come if it were to revive. The second occurs when we say that a man who has shown no signs of life for many hours may still be alive. For here we only want to say t.hat the p.ossibility of his revival is not so improbable as to permit us t~ disregard 1t. Understandably, proofs of this special kind require a special procedure. Here, in particular, we may: (a) certainly not overlook any of the grounds that speak against our proposition (which lower its probability); it is not our duty, however, to mention all the grounds that speak for it, as long as we present enough of these to establish that the degree of probability conferred upon our proposition by their preponderance over the grounds speaking against it is decidedly greater than the given quantity. (b) In a proof of this sort it is not necessary to calculate the degrees of probability of the grounds for and against our proposition; rather, if we assign a degree of probability to those speaking in favour of 296 205

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it that is so low that every reader must concede it is not too high, while assigning a degree of probability to the grounds speaking against it that readers will find too high rather than too low, and if, finally, we show that even with this method of calculation the probability of our proposition turns out greater than the given quantity, then it is clear that our proof accomplishes what it is supposed to. If, for example, we wish to show that an institution that provides for widows will not be disadvantaged by the obligations it has taken on it provided that it is sufficiently large, we need only show that we have not assumed the life expectancy of men to be longer, and that of women to be shorter, than experience shows it to be.

un&~cxmc; de; oXAo ysvoc;). This flaw occurs, e.g., when propositions of

§.536.*

Survey of the most common flaws that may afflict proof1· in a treatise. a) Pertaining to matter

In §370 ff., we considered proofs only as means for our own instruction. Because of this, I only spoke in §§371 and 372 of the flaws of proofs that present obstacles to such instruction. Here, I must warn against flaws in proofs that make them unworthy of inclusion in a treatise. These flaws, too, may be classified as pertaining either to matter, i.e., the nature of the propositions used as premises or form, i.e., in other features. For their part, flaws pertaining to matter may be divided into those that concern premises that are not rules of inference and those that concern rules of inference. l) Flaws of the first sort may be placed under the common name of untenable assumptions (cf. §371). We must reject as untenable assump297 tions not only (as in §371) assumptions that are false in themselves or which are backwards or turn in a circle, but also: (a) all assumptions that are not sufficiently evident to our readers; as well as (b) all of those which, though they do confer the appropriate degree of confidence on the proposition to be proved, nevertheless do not permit them to see this as easily as would have been the case with a different selection of premises. This might be called the flaw of ponderousness. All of the proofs that have been given to prove theorems of similarity in previous treatises of geometry deserve this reproof. It is also a flaw here (c) if we prove the given proposition from propositions that are not based upon its objective ground, when it was possible to deduce it from such propositions. This might be called a proof based on foreign grounds (per aliena et remota,

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~he pure theory of quantities are proved from geometrical or mechanical grounds (as is sometimes done). It is also a flaw (d) when we have recourse to propositions that our readers take to be true though we do not, when we are not compelled to do so. 2) The rules of inference that we follow in our proof are not only to be criticised when they are completely incorrect, i.e., when the proof is illogical, which happens in all of the cases enumerated in §371, no. 2, but also when our readers do not take them to be evident or reliable enough. Note. The ancients recognised very clearly that it is a flaw to give a proof based on foreign grounds (E:~ &nou ysvouc;;) when a proof from indigenous grounds (E:~ apxD)v OL){sl(0V) could have been given. But when ARISTOTLE cites as an example of this flaw the case where a geometrical truth is proved through arithmetical considerations, it seems to me that he was mistaken, even if many of the most respected geometers of our time agree with him. The arithmetical theorems are actually more general that the geometrical, and the attributes that belong to spatial quantities are ob- 298 jectively grounded in attributes that belong to quantities in general. Many geometrical truths must, accordingly, be proved through purely arithmetical considerations (i.e., belonging to the general theory of quantities), and indeed precisely in those cases where one intends to indicate their objective grounds. This being said, it is true that such proofs may lack the intuitiveness people like in proofs intended for beginners. It is also true that whenever one is concerned only with conviction, the geometrical proofs are to be preferred, and indeed that for many other reasons these should not be overlooked. §.537.*

b) Pertaining to form Even when the propositions and inferences we use in a proof have been appropriately chosen, we may still fall short either in the way we combine them or in many other respects. First of all, all the flaws that a proof can have in relation to ourselves (§372) can also be urged against it in relation to our readers. It may deserve, e.g., to be criticised for containing gaps if we have omitted various intermediate propositions which would have made it easier to recognise that the proposition to be proved can be deduced from the given premises, even though the extent of previous knowledge we may assume our readers to possess called for us to include

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 538. them. In addition to such flaws, however, there are others. ( 1) Sometimes it is a flaw if we do not remark expressly enough that the reader may take the propositions we present in their full generality, without tacitly adding any limiting conditions. One might say in this case that we create the appearance of unnecessary limitations. Even mathematicians are sometimes guilty of this mistake when, e.g., they fail to remind us that what they claim also holds if the lines that are drawn do not lie in the same plane, etc. (2) It is another flaw in our proof if we do not bring forward 299 everything that we could have in order to stymie the inappropriate influence of our readers' inclinations, if, for instance, we are too forward with our opinions, or tax the reader's vanity too heavily. How many theological writers, especially controversialists, sin in this way! (3) The offence is even more grievous when we use propositions or inferences we think incorrect using the arts of dissimulation and lying. Such proofs are to be taxed with dishonesty. And may theologians search in their hearts to see whether they truly believe everything they ask their readers to accept, especially in the historical parts of their works! (4) It is another flaw when we either fail completely to specify the degree of confidence that our grounds deserve, or do so incorrectly, e.g., estimating it to be too high. No flaw is more common that this one. Note. Since the theory of proofs is of such great importance, it is to be expected that previous logicians will have dealt with it with such thoroughness that their followers would have little to change or add. No wonder, then, that my presentation differs little from previous ones on this subject. And since I have elsewhere indicated the reasons that moved me to make the few changes I did, there is no need to add anything here. It goes without saying that for a full comparison of my views with those of previous authors, one must consider both what was said here and what appears in §§370 ff.

VI. On Objections and Replies §.538.* Concept and use thereof

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The theory of objections and replies is closely connected to the theory of proofs we have just presented. I say that an individual proposition or collection of propositions (which may always be looked upon as a single one) is an objection against the proposition M if these propositions 208

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 538. are put forward with the specific intention of diminishing the degree of confidence of M for anyone who considers them, or if they are at least constituted so as to do this. If the confidence in M is to be diminished but not utterly destroyed (and thus transformed into its opposite, namely, confidence in the proposition Neg .M), one calls the objection a doubt; otherwise one speaks of an objection in the stricter sense. The latter may also be defined as an attempted proof of the proposition Neg.M (§512). By contrast, a proposition or collection of propositions is called an answer; response, removal or reply if it is put forward with the specific intention of counteracting the effect brought about by considering the objection, or is at least constituted so as to do this (cf. §371, no. 2k). That such objections and replies must appear quite often in a well-constituted treatise is evident form the various benefits they produce: (a) When we expect that an objection is already known to our readers or may become known to them in the future, and cannot assume that they will have the insight and good will to answer it themselves, then it is obvious that we must include the considerations that serve to refute it in our book (along with the objection itself, if our refutation would not otherwise be comprehensible) if we wish our readers to look upon our theses with the degree of confidence they merit, not just for now, but into the future as well. (b) Another benefit of no less relevance which we gain by unashamedly 301 presenting all the objections against our claim is that our readers, seeing this proof of our frankness in our behaviour, will be inspired to trust us. (c) By providing suitable replies to objections, we provide our readers with valuable exercise in thinking, we find the occasion to bring many other useful insights to their attention, and gradually put them in a position to find correct responses to objections we have not raised, and which may not even have occurred to us. (cl) Finally, we should not forget about the possibility that we are mistaken in our claim, and that the en-or we have overlooked might be most easily detected by our readers if we present the various objections to our claim and add what we have to say by way of response. Note. That the word doubt is used here to refer to a proposition, while in §306 it is used to refer to a state of mind, should not cause any misunderstanding. The reason why I do not define an objection in the broader sense as a more or less successful proof of the falsity of a proposition (as others do) is that it seems to me that not every objection is made (or could only be made) with the intention of proving the falsity of the proposition against which it is directed, i.e., to bring the reader to form the judge-

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ment that the proposition is false; for the grounds adduced are often far too weak to accomplish this. What is or might be intended in every case, however, is that the degree of certainty attaching to the proposition be decreased. That the concept of an objection does not require that this intention actually be present, but rather only that it might be, is clear from the fact that we ourselves raise objections to our claims before refuting them. But do we bring these objections forward in order to decrease the degree of certainty of our proposition? But just as an objection can be less than an attempted proof of the falsity of a proposition, the refutation of an objection can also be more than a proof of its falsity. For we often 302 know quite well that a given objection is false, yet look for a reply all the same. This reply will accomplish more than merely showing the falsity of the objection; it is supposed to undo the effect produced by the objection (with respect to our confidence in the proposition M) as completely as possible. §.539.

Which objections and replies should be included? 1) First, concerning objections, it is understandable that there may be some we answer and some we are unable to answer. The latter, i.e., objections for which we know no considerations through which the damage they cause to the probability of our proposition can be entirely repaired may only be passed over in silence in the rarest of cases (namely, for the propositions mentioned in §453, no. 1). In all other cases, honesty requires us to mention them. Indeed, it is precisely when we are occupied with a proof of a proposition that we are obliged to bring forward the grounds that lower its probability.* If constraints of space or the reader's inability to understand prevent us from setting out these contrary grounds in detail, we should at least make a general remark that there are grounds which count against our claim, and never claim the degree of reliability of our proposition to be higher than it deserves to be in light of these. But we should also include objections for which we are able to provide completely suitable replies, whenever so doing would bring a consider303 able benefit. This may occur when any of the following factors is present: *I am pleased to find that the late HERMES (in his Einleitung in die christkathol. Theo/., Vol. 2, §7) takes the same view with regard to the necessity of distinguishing grounds against a thesis from (other) objections. ESSER (L., p. 257) takes a different view of this distinction.

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(a) if we are not completely sure that the claim against which the objection is aimed is true, and when a faithful presentation of the objection might make it easier for our readers to discover any errors there might be; (b) if we must be concerned that remaining silent about the objection would lead our readers to suspect us of dishonesty, or weaken their trust in our knowledge and judgement to the extent that it would prevent them from attaching the degree of confidence to the theses we present which would be beneficial for them; (c) if the observations we might make with the aim of enabling our readers to answer the objection should they come across it in the future would not only not be appropriately understood by them at present, but even misinterpreted, if we did not simply say what we were talking about; (d) when raising and answering the objection serves as a means to provide the reader with practise in correct thinking or with useful knowledge, etc. If none of these benefits is to be had, if there reason to be concerned that some will only be irritated if we raise an objection that everyone can answer for himself, if the presentation can be composed in a way which permits anyone who knows the objection to see that we have borne it in mind and have in fact refuted it without expressly mentioning it, then it will be permissible, and sometimes even a duty, to omit the objection. 2) From this we may easily gather when a reply to an objection merits inclusion. If we considered it necessary to include the objection, then it is obvious that we must also include its refutation, provided that we know of one and it is not of the sort we may confidently expect any reader to think of himself. What has been said thus far reminds us, however, that sometimes we must present a reply even when we do not present an objection, i.e., include something that serves to calm worries that do or 304 might in the future occur to our readers. Note 1. I must confess that I find the presentations of many sciences to be defective in that they take too little notice of the objections, doubts, and questions that any reflective reader might raise against the theses that are advanced. It is not only in presentations of sciences dogged by controversy such as metaphysics that we must pay constant attention to the more or less plausible objections that oppose our claims, but also in a science in which, as one says, one can prove everything evidently, namely, mathematics. Treatises that are supposed to be suitable for self-instruction, at the very least, must be constituted in a way that deals with the doubts and questions that are likely to occur even to beginners at every step (something which will occur all the more certainly, the more alert they are). If

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 540. this were done, perhaps we would not see so many people, among them some who are quite intelligent, shrinking from the study of mathematics! In asking this, however, I do not wish to be misunderstood, and so point out that what I recommend here is by no means a return to the scholastic form of a continual pro et contra (endless objections with responses, etc.). Note 2. The question whether and in what cases objections and replies belong among the essential, supporting or occasional propositions of a science is easily answered from what has already been said in §513 about proofs, so I will not discuss it here. §.540. How should the objections included in a treatise be composed? If, following the instructions of the previous section, we are agreed that a certain objection deserves to be included in our book, we move on to 305 consider how it should be composed. With all the objections we raise, we must remember that, whenever possible, the readers should notice merely from the way that we present them that: (a) we ourselves have devoted all due attention to them and examined them without prejudice; also that (b) we present them to our readers with complete impartiality, and that our presentation does not make them out to be weaker than they really are; that, on the other hand (c) we do not in our hearts accept them, and only affect to refute them because we do not dare allow our own views to come to light. 2) Bearing these requirements in mind when framing objections leads to a considerable variety in their presentation. Sometimes it is appropriate not to present an objection in detail, but only to allude to it. This will be the case whenever we may assume that the reader will be in a position to supply the rest for himself, and can be in no doubt that our brevity was motivated by considerations of space or some other innocent purpose, rather than dishonesty. By contrast, we we must present an objection in complete detail, even in the very words someone else used to present it previously, and with as much strength as we are able to give it, when any of the following circumstances obtains: (a) when we can only convince our readers in this way that we have given our full attention to the objection and presented it faithfully; (b) when their minds can only be put at rest in this way, and otherwise, e.g., if we did not employ the very words of the opponent, they would worry that we had, either intentionally or by chance, sapped the objection of some of its strength; (c) when

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 541. such a presentation increases the reader's trust in the truth in any case; or (cl) when going into such detail can provide exercise in thinking, etc. 306 §.541.* How objections must be constituted 1) I have already observed that not all objections presented in a treatise need to be accompanied by a refutation. But where it is necessary to add one, it must (a) be so constituted that considering it will completely remove the semblance of a doubt against our proposition that was created by mentioning the objection. For if this were not accomplished, if appearances still counted against the truth of our proposition, the degree of certainty this claim would enjoy in our readers' eyes would have been diminished even more through their becoming acquainted with that objection than it would have been had we simply presented the objection along with our proof. They would then have to think that we had deceived them. For this very reason, our refutation must (b) never consist simply of a repetition of the grounds adduced in our proof of the proposition against which the objection is directed. For although it is quite possible that these grounds were not fully appreciated by our readers at first, and that only now when they are repeated will they appear in their true light, and although this can cause the degree of confidence our readers now bestow on the proposition to be no less than it was before they became acquainted with the objection, since the weight of the contrary grounds contained in the latter is overbalanced by the increased weight our original grounds now have in their eyes, it is still not fitting for us to count on our readers making such mistakes. Instead, our book should be composed so that if the reader places his trust in a proposition because he evaluated its grounds correctly in the first place, he never finds it necessary to withdraw this trust. But this would be the case if we afterwards brought forward contrary grounds and had nothing to reply except what we have already said. (c) In answering objections we ourselves 307 have not thought up but which others have actually urged against us, it is vital to show the reader that we are neither too stubborn to admit mistakes nor motivated by any other ignoble passion (e.g., taking pleasure of discovering other people's mistakes), that we have proceeded with impartiality and are only interested in discovering the truth. (cl) If we suspect that even some readers adhere to the opinion we declare to be erroneous, and have even spoken in favour of it elsewhere, we must also do everything we can to make it easy for them to abandon their previously held 213

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 541. view, or at least not needlessly to make this more difficult. We must, e.g., not stir up their love or controversy or make them feel too ashamed, etc. (e) Finally, a refutation can only be called well-constituted if it gives the reader as much practice in thinking and communicates as much other useful knowledge as the nature of the objection and other circumstances permit. 2) After these general observations, we may specify the constitution of particular kinds of replies more precisely. (a) If the objection was not devised by us but was actually raised by others, and if for this reason we deem it appropriate to present it not in our own words, but instead in the words of those who formulated it, it will often be necessary to present the meaning of the propositions that are advanced, as well as the grounds upon which these are based, more distinctly than was the case in the given presentation. (b) If we have done this, and if we may hope that the reader will now recognise clearly enough what is said in the objection, and the grounds upon which it is based, we proceed to ask whether we ourselves hold any of these to be true. If this is the case, it is usually a good thing to mention it. For in so doing we prove our forthrightness and can of308 ten enrich our readers' knowledge in truly useful ways. (c) If it should turn out that none of the individual claims that comprise the objection is false, then it is obvious that the proposition we advanced cannot conflict with them, but must instead stand in a relation of compatibility with the objection (§154). This can also be the case if those propositions are in fact false, since false propositions are sometimes compatible with true ones. It is easy to see from this that when such cases occur, we will do well to point this out. For when there is no genuine, but only an apparent incompatibility between the propositions that form the objection and the proposition we advanced, and when the nullity of this appearance is made manifest, it is obvious that the objection must lose its force even in the eyes of those who take all of the propositions constituting it to be true. They can now mean at most that these propositions, which they take to be true, reduce the probability of the proposition we advanced, but they can no longer believe that they have shown it to be false. How one should proceed in order to remove this appearance of conflict, and thus to show that the propositions advanced by our opponent are compatible with ours, has been discussed to a certain extent in §368. It should be observed, however, that here, where it is a question of making this compatibility evident to our readers, we must take account of any truths they know which might seem to be such that, if added to the claims contained

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 541. in the objection, would permit one to deduce a conclusion that overturns our proposition. We must show that no such truth indeed leads to such a contradiction. (d) If we have accomplished this and our readers now see that the claims in the objection do not really conflict with the proposition we advanced, they might still (as said above) reduce its probability in their eyes, and this indeed to a much greater extent than should be the case according to the correct rules of probability, supposing the objection 309 to be true. Here, for reasons similar to those mentioned in part (c), we will do well (if possible) to show through a separate argument that even if these propositions were true, they not only would not refute our claim, but also fail to render it improbable, or at least that the reduction in probability clue to these propositions would not be as great as they imagine. (e) If we find ourselves compelled to declare all the propositions and modes of inference employed in the objection to be correct, then the only possible way of completely refuting it is to show that these propositions not only do not conflict with ours, but do not even reduce its probability; or, in case we cannot accomplish the latter, by adducing new grounds which increase the probability of our proposition by at least as much as the objection reduced it. If, by contrast, we are able to show that some of the propositions or inferences in the objection are incorrect, or at least improbable, then we may weaken the objection in a quite different manner. Namely, we must show the falsity or the improbability of these propositions or the untenability of the inferences. The way that this can be clone has been sufficiently discussed elsewhere. (f) It not seldom occurs that a proposition claimed by our opponent is only false when understood in the generality he gives it, but can be made true by adding various restrictions. For reasons that are easy to gather, it is appropriate to point this out, and thus to show which determinations and restrictions would make our opponent's claim true, and acceptable to us as well. This procedure is usually called limitation or restriction. (g) If the propositions in an objection that we declare to be false or improbable are furnished with proofs, it will be necessary to expose the insufficiency of these as well, i.e., to point out their flaws to the reader. For only when we do this will 310 they fully place their trust in the proofs we offer for the falsity of those propositions; we also take advantage in this way of another opportunity to give them practice in correct thinking. (h) Yet even with the propositions contained in the objection that we must accept as true, it may be that the proofs given for them contain flaws. In such cases it is almost always appropriate at least to mention these flaws briefly, though not to

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discuss them in detail-the latter so as not to tax our readers unnecessarily, the former to prevent them from being misled, as well as to bring their estimation of the trustworthiness of our opponent's opinions in line with truth, in case it should have been set too high to begin with. (i) Sometimes is turns out on closer consideration that the propositions used by our opponent not only fail to conflict with our claim, but instead, regardless of whether they are true or false, they can be used to deduce it. We should not fail to point this out to our readers, not in order to show them the opponent's ignorance, but rather to increase their trust in the correctness of our claim. One might call this turning the opponent's weapons on himself (Retorsio). (k) Sometimes the propositions brought forward by our opponent are not even mutually compatible. We will do well to bring their inconsistency to light, for this already shows that not all of them are true. (1) If all of this has been done, and we have thus shown that our opponent's arguments against our claim are incorrect, we still have to explain how it may have come about that he raised this objection. If we are able to explain this in a truthful and convincing manner, this must not only increase our readers' confidence in our proposition, but also be very instructive for them in other respects, e.g., by learning the source of one error, they may be put on their guard against making similar ones, etc. I have already spoken about how to look for such explanations in Book 4 on Heuristic. Sometimes our opponent makes one error that leads to all the others. It is easy to see that it is always helpful to point out this first mistake (rrpw-wv tj;i::u8oc;). It should be observed, however, that in communicating what seemed the most probable to us after inquiry, we must be careful not to wrong our opponents or to goad them more than necessary, or make others suspect us to suffer from certain ethical shortcomings. Thus we must take care not to speak in a tone that expresses too much confidence in the correctness of our opinion; we may never know for certain that the error arose in precisely this way and not otherwise; and if we believe that our opponent's error stems from a condemnable passion, or that he claims something that he does not believe in his heart to be true, we should never say this unless circumstances absolutely compel us to, etc. Note. The case spoken of in no. 2c, where someone puts forward as truths propositions that are supposed to be incompatible with our claim but turn out not to be occurs quite often. Usually we then explain that the burden of showing this alleged incompatibility falls on the opponent, while we believe ourselves justified in believing the propositions to be

compatible without proof. This, it seems to me, is the the most justifiable sense that is connected with the well known logical canon: Neganti incumbit probatio. For it is obvious that when one speaks of a denier (Negrms) this cannot simply mean anyone who advances a negative proposition, since as a rule positive propositions require proof no less than negative ones. Thus here a denier means someone who declares another's claim to be false. This, we mean, should not be done without proof; and since is is not supposed to be enough simply to state that the proposition proved by the other person is false because it contradicts this or that truth, he must prove that this contradiction exists. This request seems reasonable to me in most if not all cases, and especially in written presentations in treatises. When an opponent objects that there are certain truths A, B, C, D, ... , which are incompatible with our claim M, it goes without saying that he assumes that our proof of Mis not decisive (which would mean that M was true), since otherwise the propositions A, B, C, D, ... as well as M would be true and hence compatible, contrary to his assertion. He must thus claim one of two things: either that from the propositions A, B, C, D, ... , which he takes to be objective truths, a truth Opp.M, which conflicts with M, follows objectively, without requiring any universally valid rule of deduction (§221); or else he must admit that there is a universally valid rule of deduction according to which we may infer a proposition of the form Opp.M from propositions of the forms A, B, C, D, .... If the former, he may dispense with the trouble of a proof by adding that this ground-consequence relation between the truth Opp.M and the truths A, B, C, D, ... can and must be immediately apprehended. But the ability to form immediate cognitions, at least those involving purely conceptual propositions, may be assumed to be essentially the same in all men. Hence, should the consideration of the propositions A, B, C, D, ... fail to compel us immediately to form the judgement Opp.M, we may certainly distrust the opponent's assurance, and have no fear that our readers will agree with him. If, by contrast, he claims that a proposition of the form Opp.M can be derived from propositions of the forms A, B, C, D, ... , he must indicate which ideas contained in these propositions he considers arbitrarily variable. And once we know this, we may attempt to refute him in two ways. One is to examine the entire sequence of inferences which make it seem that the proposition Opp.M can be deduced from A, B, C, D, ... , and to show its untenability. Understandably, this procedure can never engender complete certainty, since we can never be sure that we have thought of all possible ways that a conclusion such as

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 542.

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 543, 544.

Opp.M might be deduced from the propositions A, B, C, D, ... , especially if these are combined with other truths E, F; G, .... Our confidence will be assuaged at best if we are sure that our opponent knows of no modes of inference other than those we have examined. Since in conversation this would be most easily accomplished by asking our opponent to demonstrate how Opp.M is supposed to be deducible from A, B, C, D, ... , it is also natural to have recourse to this here. In other words, we should oblige the opponent to provide a proof of the existence of the claimed incompatibility of M with A, B, C, D, .... In written treatments, however, and especially in a treatise, we cannot wait for the opponent's reply, but rather wish the reader to form a judgement immediately upon reading our book. Thus here we must adopt the opponent's position, and attempt proofs of the existence of this incompatibility until we may expect that the reader will consider the subject exhausted, and no longer worry that the opponent, even if he were present, would be able to adduce something beyond what we brought forward on his behalf. The second method of refutation is far more brief and decisive, namely, when we are able to replace the ideas i, j, ... by certain others which make the propositions A, B, C, D, ... as well as M indisputably true even in the eyes of our opponent. For when we do this, we have directly proved the compatibility of A, B, C, D, ... with M.

follow (sophistry). (d) We make a man look ridiculous or depict him as dangerous, when he in no way deserves this, etc. (e) We at least give the impression that we have not examined the opponent's reasons with due attentiveness and impartiality, that we will not back down because we will not even entertain the thought that we might be mistaken, or that we take a condemnable pleasure in discovering his en-ors, etc.

§.542. Mistakes in this business

1) The most commonly encountered mistakes in presenting objections in 314 a treatise are as follows: (a) we raise objections that are easily dealt with while ignoring those that are difficult to refute; (b) we do not present them faithfully, but in a weaker form; (c) we devise objections that no one has raised or ever will (fencing with the wind). 2) When refuting objections, by contrast, it happens all too often that we (a) do not correctly grasp the sense of the objection (ignoratio, mutatio elenchi). In particular, we often hastily assume that someone else holds a view that contradicts our own merely because he uses different words. Such disputes are said to be merely verbal (Aoyoµa.x(cx). (b) We often believe that we have sufficiently refuted an objection when we have merely shown that not all of the claims contained therein have been properly proven, or that some of them are false; or we attempt (c) to show the falsity of the objection by drawing consequences from it that do not really 218

§.543. Other views

The circumstance that objections and replies are most often necessary in oral instruction caused logicians to misplace this theory in the part of logic, which they rightly or wrongly believed had to be devotee! to oral instruction. But there are some who deal with the topic in a more appropriate place. The presentation of CRUSIUS (W. z. G., §§539-565) is especially thorough and has many singular features. One of these is when (§540 ff.) he attempts to show that one may deduce contrary conclusions 315 from propositions that must be assumed to be true using perfectly correct inferences. The offensiveness of this claim disappears when we remark that C. does not speak of true propositions, but instead of propositions that must be taken to be true, and recall what was said in§§ 167, 301, 309, and 319, namely, that a proposition which is in fact false may nonetheless possess a great probability relative to certain assumptions and hence must be taken to be true by all those that know only the assumptions upon which this probability is based.

VII. On Examples §.544.* Concept and use of examples

Another kind of proposition that deserves mention here are those customarily called examples. If I am not mistaken, we call a particular proposition an example with respect to a general one if consideration of the former makes the latter easier to judge, even if it is not necessary for this purpose. Thus we call the particular example that four times three is the same as three times four an example of the general proposition that changing the order of factors does not alter the product, since consideration of 219

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 544. the former helps us to understand the sense of the latter, and indeed to recognise its truth, even though it is not necessary to consider the former in order to recognise the truth of the latter. In a broader sense, I give the name of example not only to propositions of the sort just described, but also to objects dealt with in these propositions. Thus we say of a reprobate who has come to a bad end that he is an example that confirms the truth that no vice remains unpunished. That such examples should be included in well-constituted treatises becomes clear from a consideration 316 of the benefits they can bring. To begin with, (a) appropriate examples can often considerably facilitate the comprehension of our theses. For if we add to the more general proposition a particular one containing the same ideas, combined with some new ones, many expressions that were perhaps obscure in the expression of the former will become definite and distinct thanks to those used to express the latter. On top of this, the particular proposition may already be familiar to our readers, or at least composed of ideas that are more familiar to them, and this has the advantage of making it easier to grasp. Thus when it appears before them first, they can more readily extract the ideas required to form the general proposition, and hence more readily form this proposition itself. (b) From many examples intended to illustrate a general proposition, the truth of the latter becomes so clear that clever readers will hit upon it themselves, even if we do not expressly add it. Thus we can sometimes completely omit the more general truth. Many convoluted rules, e.g., for extracting square roots, are easiest to understand through a single example. (c) Another benefit of examples is that they hold the reader's attention more firmly, and are more important and pleasant in his eyes. Hence they make the entire instruction, and in particular the general truths that prompted us to give the examples, more significant and memorable. (d) An example added to a general truth forces the reader to bear in mind the ideas of which the latter is composed for a longer time, and to repeat them frequently; and we may hope that in this way the general thesis will make a more lasting impression on his memory. (e) The more diverse the secondary ideas occurring in the examples, and the more these turn up in other contexts, the more they will facilitate recall of the general proposition. (f) If we deduce the truth of the particular proposition used as an example not from the more general one, but instead from independent 317 grounds, then the confidence our readers place in the general proposition will always increase somewhat. (g) Frequently, such examples are the only thing we can adduce in order to prove our claim, either due to

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 544. constraints of space or our readers' lack of requisite knowledge; sometimes, indeed, we know of no other grounds than these particular cases, which may be so numerous as to fully prove the general proposition, or at least to confer a high degree of probability on it. (h) The falsity of a general proposition can sometimes be proved by means of an example as well, namely, when this example is an obviously false proposition which is subordinate to the general one. Examples used in this way are given the special name of counterexamples [Jnstanzen]. (i) Well chosen examples can be used to acquaint our readers with truths that are very important for them, or at least to draw their attention to them, etc. Note. ARISTOTLE (An. P1:, II, 24) as well as older logicians (see, e.g., REUSCH, §577) understand by an example (m::xp&5i::tyµa.) a mode of inference according to which one infers from a given case others that are similar to it. WOLFF (L., §790), HOLLMANN (L., §45), and FRIES (p. 118) mention examples as a means for making ideas clear and vivid, without giving any definition of the concept. That examples in the broader sense include not only propositions but all s01ts of other objects is beyond any doubt; in a treatise, however, where nothing but propositions may occur, examples will consist of propositions alone. EB ERHARD' s definition (Synan. W.B.) of an example as an individual wherein that which belongs to the genus or species can be intuited, seems to me to be too narrow, since examples can be general as well as individual propositions, provided only 318 that their extension is smaller than that of the proposition for which it is to provide an example. Thus the formula (a+ b ) 2 = a 2 + 2ab + b2 is a general proposition, and yet with respect to the more general binomial formula it is only a single example. In my opinion, it also belongs to the sense of an example that consideration of it is apt to facilitate recognition of the more general truth, though it is not necessary for this. Against this last clause, some might point to the case discussed under (g), where our entire proof consists of adducing examples. I think, however, that we call these individual propositions examples only insofar as we believe that it is possible to know the proposition that is proved without knowing these examples. Another objection against this definition that might be raised is that we also look upon propositions that show the actuality of a certain object as examples which prove propositions stating that such objects are possible. Thus we say that the proposition that a scholar can also be depraved by citing the example of the scholar N., who is actually depraved. Yet the proposition that the scholar N. is depraved does not stand to the proposition that a scholar may be depraved in the relation I

221

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 545.

described in § 152 as that of a particular proposition to a corresponding more general one. I reply that we use the term "example" in an improper sense in such cases, prompted perhaps by the circumstance that we look upon the proposition that scholars can also be depraved as equivalent to the proposition that some scholars are depraved, and look upon "scholar" as the subject-idea of the latter proposition. Admittedly, this does create the appearance that the relation of the two propositions is that of a particular to a corresponding more general one. I have intentionally described examples as propositions rather than as true propositions, for in my view false propositions may also serve as examples, as when we show the falsity of some general rule through the obvious falsity of one of its individual cases. 319 Note 2. In my opinion, the figures customarily used in geometry and related sciences should also be looked upon as examples, though of a particular kind. For when we refer our readers to a figure, we do not, or at least should not, do so with the intention that the single truth that they learn from looking at this figure, namely, that the spatial objects drawn there undoubtedly have such and such relations (e.g., that this angle is only a part of that one, etc.), should be considered a completely valid proof of the general truth that these relations obtain in every spatial object of that kind. Rather, we merely intend for this single truth to serve as a corroboration of a general truth deduced on completely different grounds, or that it serve as a means for making the latter easier to recall, etc. §.545.

How examples must be composed in order to facilitate understanding

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 546.

those that make it easier for the reader to form our proposition in his 320 mind, either because they are customarily present or because the idea of them is connected for whatever reason with the ideas that compose our proposition. Features of this sort should instead be sought, and used in our example, as a serviceable means for promoting not only understanding of our proposition but also its retention in memory. Thus it facilitates not only understanding but also retention of the truth that the period of a pendulum is proportional to the square root of its length when we adduce the example that a lamp which hangs from a chain four times longer than another swings twice as slowly. It is good, however, to emphasise the determinations of our example that are essential, i.e., which do and must occur in the general proposition the example is supposed to help us to understand, and to seek to distinguish the remaining, merely contingent features (those which might be this way or another). Among other things, this means that we should not simply tacitly assume the essential determinations if they will not be obvious to everyone, but should instead expressly mention them. Thus if the example mentioned above were destined for novices, we should expressly mention that 2 is the square root of 4. In addition, determinations that are not included in the general proposition and which are therefore arbitrary should not occur according to an easily detectable rule, since this might give the impression that they must always, according to this rule, be present, or because this rule might distract them from the matter at hand. Thus, when giving an example of the addition of fractions, it would be a mistake to make the denominators equal and the numerators unequal, etc. Finally, it scarcely needs to be said that even when examples are chosen only for the sake of facilitating understanding, we will sometimes do well to adduce several, since an obscurity left behind by one might well be dispelled by a second or a third. 321

The above indication of the various benefits that examples can bring should permit us to judge when we should use them. The question of how they should be composed, however, will be easiest to answer if we go through the benefits in order, and ask what attributes these require in an example. Here it is clear that if some of these attributes conflict with one another, we should prefer those that promise a greater benefit, all other things being equal. Concerning, first, the benefit of facilitating understanding of what we present, it is obvious that this goal can only be reached if the chosen example is itself easy to understand. Ease of comprehension is thus one of the first virtues an example should never lack. Now in order to ensure ease of understanding, our example should not be overly complex, and hence not weighed down by superfluous features. We should not, however, reckon among the superfluous features

The second use of examples that I mentioned is that sometimes they can replace the general proposition we wish to bring to mind, and be expressed more concisely than the latter. It is easy to see that such use will only be possible when the ideas that occur in the example, though more complex than those occmTing in the general proposition, nonetheless have simpler designations, or when we may expect that the concepts

222

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§.546.

How examples may also be used to abridge our presentation

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 547.

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 548, 549.

we tacitly pass over in our example will naturally occur to the reader. This is the case, for example, when we present the rule for extracting square roots in a single example rather than in terms of general concepts. Here we can be much more concise, without fear of misunderstanding.

§.548.

§.547.

How examples can promote attentiveness 1) If examples are to draw more attention to the general propositions we advance, two things are necessary: (a) the propositions of which they consist must draw the reader's attention more than the corresponding general propositions; (b) in addition, one of two things must be the case: either the more general proposition must come to the reader's mind when he considers the example, or he must understand that knowing the more general proposition will put him in a position easily to judge the truth of the example and other similar ones. 2) If our examples are to meet the first requirement, they must: (a) consist of propositions which the reader finds pleasant to learn or recall. This will be the case when they deal with objects that are pleasant 322 in and of themselves or which give free rein to the reader's cognitive powers, principally the imagination, allowing them to feel these powers. (b) If there are several examples, they must be sufficiently diverse; for too great uniformity is tiring even with the most pleasant of objects. 3) If the reader is supposed to understand that acquaintance with our more general proposition is supposed to put him in a position to judge the truth of our particular example and other similar ones, then he must see that our example really stands to the general proposition in the relation of a subordinate to a superordinate proposition, and that it is not the only one that does so. Hence we must choose our examples in a way that makes this relation obvious or at least easy to point out, and in the latter case, we should not fail to point it out. Thus a beginner in arithmetic will not feel greatly impelled to attend to our theses unless we show him with all sorts of examples how many important questions he might answer with their help. But we will accomplish this only if our examples contain important questions and if the beginner sees that innumerable other problems can be solved in the same way.

224

How examples also facilitate retention and recall 1) The fourth and fifth benefits of examples that I mentioned above were that they can make it easier to retain the more general truth in memory and to recall it at appropriate times. I shall consider these two aims together, since the features that promote them are for the most part the same. I mentioned in §545 that it can facilitate understanding if the example concerns 323 a thing, and the idea of this thing awakens in our minds the concepts that are to be combined to produce the general truth. Clearly, this will also facilitate its retention in and retrieval from memory. Thus we must give the preference to such examples when they are to be had. (2) Examples that are not so constituted must, if they are to be impressed upon the memory and recalled, give the reader occasion to dwell longer upon the general truth, and indeed to form the idea of this truth repeatedly. Admittedly, not every example is well suited for this. Thus we must take care to choose examples that the reader cannot consider without thinking of the general truth they are supposed to impress upon them, and we must also present them in a way that makes this almost impossible to avoid. (3) In order for truths to be recalled at appropriate times, it is necessary to draw examples from objects that appear frequently to our readers, and which they will encounter at precisely those times when they want to bear our truth in mind. It is obvious, by the way, that for the most part this can only be accomplished by adducing many examples.

§.549.

How examples must be constituted if they are to serve as confirmations or proofs 1) The sixth, seventh, and eighth uses of examples were: to increase the reader's trust somewhat in the general truth we have proved or, in default of another proof, to confer some degree of credibility on it or, finally, to show the falsity of a general proposition through the obvious falsity of a particular proposition that is subordinate to it. These can be dealt with together. If an example is to serve any of these purposes, two things are necessary: (a) our readers must above all see that the proposition that con- 324 stitutes the example is in fact subordinate to the general proposition; and (b) they must be convinced of its truth (in the first two cases) or its falsity 225

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 550. (in the third) based upon grounds which do not include the proposition to be proved (in the first two cases) or its opposite (in the third). (2) In order to meet the first requirement, we must prove that our example is related to the general proposition we wish to corroborate or refute as a particular proposition subordinate to a general, unless we can foresee that this will be obvious to our readers. (3) Concerning the second requirement: even in the case where the grounds we use to prove the particular proposition are a mere repetition of those used to prove the general proposition, they can still increase the reader's confidence in the latter, insofar as repeated consideration of these grounds makes him more convinced of their correctness. The benefit will be far greater, however, when the grounds that are used to prove the example are of another kind and, even better, if they are based on a very short sequence of inferences or in general attain a high degree of certainty. The same holds for the denial of the proposition if our example is supposed to be used to refute the general proposition. (4) If these two requirements are met equally well by two examples, the reader will be best served by the one that gives the least ground for suspicion that it was taken into consideration when framing the general proposition. Thus a reader will not gain much confidence in an algebraic formula which we claim to be valid for all values of its variable quantity x if we show that it holds for x 0 or x 1; for one may assume we thought of these values when arriving at the formula. We should therefore choose others. (5) If examples are supposed to provide the only proof for our 325 claim, we must provide several of them. For the purpose of refutation, however, a single counterexample obviously suffices to show the falsity of the general proposition. Still, we must choose one whose falsity can be made quite obvious. §.550.

How examples should be used to spread other truths The benefit that we may produce by using examples not only to make a single truth evident to our readers, but also to bring other useful truths to their attention, is far from unimportant. Since every example is, as such, a proposition that is subordinate to the corresponding general proposition, and since it is well known that such subordinate propositions can be produced from the general ones by adding the most diverse concepts and propositions to the latter, one can easily see how in our examples many things can be spoken of, among them some that do not have the slightest 226

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 551. connection to our science. If we are of a mind to use this opportunity, and we have the necessary knowledge, we will find opportunities when presenting almost all sciences to remind our readers of salutary truths they may already know, or to bring new ones to their attention. But as useful as this can be, the following rules should not be overlooked: (a) our examples or, more precisely, the ideas they contain, should not distract the reader too much from the truth that constitutes the actual object of our presentation; (b) when communicating new, previously unknown, truths to our readers, we should not become incomprehensible. In order to avoid doing so, it must at least be clear that the new things we say, no matter how obscure they may appear to our readers, are genuine examples cor- 326 responding to our proposition. Thus in a treatise of arithmetic one may certainly give as an example of division the problem of determining how many times greater the speed of light is than the speed of sound, even if we foresee that may beginners will not understand how we came up with this problem. (c) Finally, we must also take care to ensure that people will not mistake the good intention we may have in striving for such variety, namely, to provide a benefit for the reader, and think we are merely showing off our varied knowledge.

VIII. On the Consideration of Mere Ideas and Propositions §.551.*

On the necessity of considering mere ideas and propositions Among the many objects the reader must consider attentively in receiving instruction, mere propositions and their possible parts, i.e., ideas occur quite frequently. For even if there are no other propositions and ideas upon which we need to fix our eye in order to meet the goals of our science, there are always at least the propositions occurring in our book which we want the reader to grasp with a truly clear consciousness, to impress upon his memory, or to determine their degree of probability, etc., which we must not merely express, but also examine from various perspectives. But if we are supposed to consider certain propositions more closely, and bring it about that they are grasped with truly clear awareness by our readers, then we cannot neglect their proximal parts, namely, ideas. There can be no doubt, then, that in almost every treatise there will have to be propositions and entire collections of propositions that deal with nothing other than other propositions and ideas, considering 327 227

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 552.

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 553.

whether these propositions and ideas are simple or complex, the connections between them and so on. I permit myself to call such propositions and collections of propositions considerations concerning mere ideas and propositions. The species of theses to which they belong will depend upon other features as well as their relation to our science. One may expect, however, that each of the three kinds of propositions enumerated in §436 can contain some propositions of this sort-that is, they may be essential, supporting or occasional propositions.

other truths in addition to those we have presented, etc. Thus I take it to be .a mis~ake when in the usual treatises of geometry the concepts of direct~on, distance, length, content (of a surface or solid), among many other important concepts that occur so frequently in this science, are not given sp.ecial consideration; (cl) when we have some other noteworthy observation to make about these ideas.

§.552.

To which inner and outer attributes should such consideration be extended?

Which ideas and propositions should be the object of special consideration in a treatise? Since whenever a proposition needs to be considered more closely we also need to do the same for the ideas of which it consists, the question of which ideas in a treatise deserve special consideration cannot be answered until we have determined which propositions are worthy of such consideration. 1) Now, among the propositions worthy of special consideration, there are: (a) all of those we advance (§434), if we also wish the reader to become clearly aware of them or to consider their degree of probability. For obviously this can only happen if these propositions become objects of which we say something, i.e., if we make them objects of consideration; (b) all the propositions we do well to compare or contrast with others, including (c) all truths whose objective connections with others we should 328 show. For when we state the objective connection in which they stand, we say something about them; (d) any propositions we must first analyse in order to show their truth; (e) all erroneous propositions we need to warn our readers about; (f) all propositions, finally, with respect to which we wish to point out or commend something to our readers. 2) A mere idea, too, merits special consideration in our book when it (a) occurs as a constituent in a proposition which itself deserves special consideration of the kind that requires consideration of its constituents; (b) when the consideration of this idea is the best means to prepare the reader to see the truth of a proposition we are to prove or to see its objective connection with others; (c) when it is desirable for this idea to become clear and familiar to our readers. This is desirable if the idea occurs in more than one of the propositions belonging to our science, or when a fluent knowledge of it puts the reader in a position to discover 228

§.553.

What exactly we should fix our eye on when merely considering ideas and propositions can be decided by taking into account our aims in presenting them. Under different circumstances, the most diverse remarks may be necessary, concerning either their inner attributes or their relations to other things. It will suffice to mention here only the most commonly 32 9 occurring sorts of inquiries. 1) First of all, we often have simply to consider the inner constitution of given ideas or propositions. Here, the most common problem is to specify whether the ideas or propositions are simple or complex, and in the latter case, what their parts are and how they are combined. Considerations in which we teach this elevate the ideas and propositions in question to distinctness (§281 ), and might be called distinct(fications or determinations of content; to use a more common term, I shall generally call them definitions. In the special case where the definition indicates parts in the concept or proposition in question, they may also be called analyses. The questions whether a given idea is objectual or objectless, whether it is imaginary, etc., also occur quite often. 2) It often becomes necessary to direct the reader's attention to certain outer attributes, or relations, of given ideas or propositions. Thus we must (a) often speak of their similarity or dissimilarity to others, i.e., we must make comparisons and distinctions. (b) For reasons similar to those that apply in the case of our own reflection (§346), it is often quite useful when instructing others through our treatise to make them aware of the logical relations obtaining between ideas and propositions. Thus we will (ex) almost always deserve our readers' gratitude when we point out to them that these or those ideas are equivalent or stand in a relation of subordination, overlap or exclusion. We can especially often help 330 by pointing out the relation between the extension of a single idea to 229

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 554.

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 555.

the sum of the extensions of several others; earlier (§ 140), I gave the name of classifications to such propositions. (~) Among complete propositions, the relations of deducibility, equivalence, exclusion, complementation, along with other similar ones (§§ 154-160) are of such importance that we should almost never neglect to mention it when propositions we have advanced stand in one of these relations. (c) With respect to true propositions, according to §401, the consideration of their objective connection (i.e., the decision of the question of whether they belong to the class of grounding or grounded truths, and in the latter case, the grounds from which they follow) is of great importance in any rigorously scientific presentation. Some of the considerations mentioned here, namely, (a) definitions, (b) comparisons and distinctions concerning ideas and propositions, (c) classifications, and (cl) the indication of objective connections, are so important and at the same time so difficult that we must now give special instruction concerning their composition.

readers who have the greatest facility in thinking would be fatigued by such a beginning. Such analyses would be even more out of place in a science where the principal reason for expounding and learning it is to be found in its applications to life. How absurd it would be, for example, if in a treatise of medicine or practical wisdom we were not content with the clear concept the reader connects with the word "means", and sought to analyse it! It is well known that WOLFF made such ridiculous errors. - Thus we should give definitions of concepts and propositions, attempting to analyse them into their proximate if not their ultimate constituents, only when this is required either to prove a truth or to indicate its objective ground, or where we may expect that such inquiries will not be unwelcome to the reader. 332

A. On the Definition of Ideas and Propositions

When, as not seldom happens, we point out to the reader that we wish a certain sign (or word) to be used to express precisely the idea that arises from the combination of such and such ideas (which are designated by other, already known signs), then we certainly know the parts of which the new idea is composed. If we call the proposition in which we indicate this a definition, then there is no doubt that there are also definitions that require no proof of correctness. For an idea which was not composed of the indicated parts would for that very reason not be the idea we are speaking of. Such definitions are called synthetic. It is another matter when the idea we wish to be connected with a given sign or, more generally, of which we now speak, is not determined in this way (i.e., through the indication of its constituents) but in some other, e.g., through the use we make of the sign or through our stating that we mean the idea that common usage actually connects with this word, and so on. Here, according to §281, even though we may be certain that our readers know the idea, we cannot draw the conclusion that they also know whether it is simple or composed of such and such parts, nor that they will find this obvious if we tell them. By contrast, when we consider how varied the judgements of scholars have been concerning the simplicity or the particular kind of compositeness of a concept, we must conclude that few things are more difficult than discovering a correct definition of a given concept and-in case one has actually found such a definition-convincing others of its 333 correctness. Thus if we present a definition that does not first give rise

§.554.*

Which ideas and propositions in a treatise merit definition? The first sort of consideration of mere ideas and propositions that I shall say more about are the definitions. By this I mean, as I said above, nothing other than propositions which specify whether a given idea or proposition 331 is simple or composed of parts and, in the latter case, what these parts are and how they are combined. To begin with, I must indicate which ideas and propositions deserve to be furnished with their own definitions. As noted in §333, we can go too far in striving for distinctness in our own reflection; this is all the more certainly the case when communicating our thoughts to others, where we may lose ourselves in definitions and analyses, a blameworthy excess. Obviously, definitions should be given for ideas and propositions only if this creates a benefit that outweighs the time and energy the reader must expend to grasp them. Thus in no science, even one mainly concerned with providing exercise in thinking, shall we undertake to give complete definitions, extending right clown to the ultimate, simple parts, of every concept and proposition, nor even of all the essential theses. For, at least given the present condition of human knowledge, we are in no position to do this; and certainly even those 230

§.555.*

Which definitions require a special proof of correctness?

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 556. to a concept but rather of a concept that is already given (such definitions are usually called analytic), without saying something by way of proving its correctness, we should not really expect the reader to accept our definition with conviction. Such a definition can at most serve to make our view known, and to invite others to examine and reflect on it. If, however, we wish our readers to agree with us, and base inferences on our definition, we should never fail to furnish a proof of its correctness. Clearly, what has just been said also applies to entire propositions. §.556.* Hovv such proof~ should be carried out; in particular; a) if the idea is claimed to be simple One easily gathers that the grounds we must present in order to convince the reader of the c01Tectness of a given definition are roughly the same ones that we had to use to convince ourselves when engaged in an impartial examination of our own opinion. In the first place, if we have to justify the definition of a mere idea which we have declared to be simple, then there is no other way to show the coITectness of this claim than to convey to the reader the impossibility of producing this idea through a combination of several others. This will occur if we first ask our readers to attempt as often as they like to produce the idea in question by combining several others. But in order for them to understand the rational basis for our expectation that they will fail, we must tell them that we and many others before us have already attempted to do this many times without success. 334 In order to ensure that our readers find this credible, to show them how such attempts should be caITied out and the resulting complex concepts examined, and to counteract the indolence that might lead them not to make any such attempts, we must put several such attempts before their eyes, i.e., we must ourselves examine a complex concept that we or perhaps someone else formed in the belief that it was identical with the given one (or at least a concept the reader might think identical with the given one), and show the difference between it an the concept to be defined. Clearly, these examples will be all the more noteworthy for our readers, and do more to convince them, the more it seems at first glance that the complex concept we set out is actually identical to the given one. But we can show the two to differ by pointing to a difference in their extensions or content. A difference in extension can be pointed out by adducing an object that stands under one of the concepts but not under the other. If we 232

THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 557. are unable to do this, because the concepts have the same extension, then if they are different, they must differ in their content. There is no other way to establish this than by calling upon the readers' own awareness. We must, namely, ask them to pay close attention to what occurs in their rninds when they think each of the concepts. If they are compelled to admit that different things take place within them when they think the two concepts; if they must admit that the difference between these thoughts does not merely consist in their having different designations, nor in the secondary ideas associated with them, then it is proven that the concepts are not identical. If, for example, we wanted to convince our readers that 335 the concept of something in general is completely simple, we would do well to look carefully at several attempts to form this concept through composition. One might attempt, for example, to define something as the negation of nothing, so that "something" would be the same as "not nothing". Here we must in fact admit that the latter concept has the same extension as the former, which was to be defined. Yet we could observe that there is a difference in the content of the two concepts, notably, that the given concept appears as a part of the other concept we formed. For when we attempt to become distinctly aware of what concept we think by the word "nothing", we notice that by "nothing" we think of the mere negation of something. The concept "nothing" is nothing other than the concept "not something", and hence it is absurd to say that the concept of something is composed of the concept of negation, "not" and the concept "nothing". §.557.* b) How to prove a definition that indicates how a complex concept is composed If we are still supposed to define a mere idea, but claim in our definition that it is complex, composed of such and such parts combined in a certain way, then we must prove that the concept which is produced by the indicated combination is in fact identical to the one we are supposed to define. We will accomplish this if we show our readers that no difference can be indicated between the concepts, which would have to be the case if they were in fact two. To this end, we ask the reader to look for such differences himself, reassuring him that we ourselves and many others 336 before us have already tried without finding any. On top of this, we deal with several of these attempts in our book, thereby showing that there is

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THEORY OF SCIENCE PROPER. Part IV, Ch. 4 §. 557. no difference even in the features one would be most likely to think might contain one. We accomplish the former by pointing out how every object that the reader is inclined to place under one of the concepts may with equal right be placed under the other. The latter can only be accomplished by asking our readers to engage in introspection in order to become aware of whether anything different occurs inside them when they think the two ideas, setting aside differences that derive from the different designations or associated secondary ideas that do not belong to the ideas themselves. If, after the most rigorous examination, they detect no differences apart from the last-mentioned kinds, they must conclude that the concept they form according to our instructions and the concept to be defined are one and the same, and hence that our definition is correct. They will be even more assured of this if we bring forward a number of other definitions that have been or might be proposed and show that they are incorrect, namely, by showing, in a way similar to that discussed in the previous section, that these complex concepts are different from the concept to be defined. If, e.g., we have proposed the definition that the concept of space is composed of the concepts "possibility" and "location", combined so that a space is thought of as the possibility of a location, and if we further claim that the concept of a location is composed of the concepts "relation", "time", "force", "effect", among others, so that by a location of things one means the relations of things which are the reason [Grund] why, en337